Cosmology with CMB and Large Scale
Structure
Yin-Zhe Ma
Institute of Astronomy and Kavli Institute for Cosmology,
Trinity College
University of Cambridge
A thesis submitted for the degree of
Doctor of Philosophy
May 2011
2
I hereby declare that my thesis entitled ‘Cosmology with CMB and Large Scale Struc-
ture’ is not substantially the same as any that I have submitted for a degree or diploma
or other qualification at any other University. I further state that no part of my thesis
has already been, or is concurrently being, submitted for any such degree, diploma or
other qualification. Most of the original material in this thesis is based on papers that
have either been published or submitted for publication as follows
1. Large-Angle Correlations in the Cosmic Microwave Background,
G. Efstathiou, Y.Z. Ma, and D. Hanson, MNRAS 407 (2010) 2530;
2. Testing a direction-dependent power spectrum with Planck,
Y.Z. Ma, G. Efstathiou, and A. Challinor, Phys. Rev. D 83 (2011) 083005;
3. Constraints on the standard and non-standard early Universe models
from CMB B-mode polarization,
Y.Z. Ma, W. Zhao and M. Brown, JCAP 10 (2010) 007;
4. Peculiar velocity field: constraining the tilt of the Universe,
Y.Z. Ma, C. Gordon and H. Feldman, Phys. Rev. D 80 (2011) 103002;
5. Cosmic Mach Number as A Sensitive Test of Growth of Structure,
Y.Z. Ma, J. P. Ostriker and G.B. Zhao, 1106.3327 [arXiv:astro-ph.CO], submitted to
Phys. Rev. Lett;
Although the original work listed here has been done in collaboration as described
above, most of it has been done by the author. Various figures throughout the text are
reproduced from the work of other authors, for illustration or discussion. Such figures
are always credited in the associated caption. This thesis contains fewer than 60,000
words.
Acknowledgements
This thesis is about the stories that happened in the early and late time of the Universe, and it
should be started by acknowledging the help, support and encouragement from all of my supervi-
sors, family and friends which were invaluable to me.
Beyond all of the acknowledgement, I would like to make my greatest thanks to my thesis
supervisor, George Efstathiou. As a distinguished cosmologist and accomplished supervisor, he
always mentors me to choose projects with the highest level of impact, and teaches me to tackle
the problems in a distinctive way. His inspiring speeches and broad knowledge of astronomy also
influence me a lot. I am looking forward to learning more from George in the years to come.
I would also sincerely thank my second supervisor, Anthony Challinor for his help on many
difficult problems that I could not solve by myself without his hints and inspiration. At the start
of my PhD, I also enjoyed very much his lectures on large scale structure formation which got me
quickly involved in real precision cosmology. I would also like to thank the cosmologists at IoA
Cambridge with whom I had invaluable discussions: Michael Brown, Donald Lynden-Bell, Simeon
Bird, Michael Bridges, Martin Haehnelt, Steven Gratton, Duncan Hanson, Lindsay King, Antony
Lewis, Andrew Pontzen, Martin Rees, Richard Shaw. Among these people I would make special
thanks to Duncan Hanson, who really helped me a lot with CMB programming at the start of
my project; and Prof. Donald Lynden-Bell, for telling me of many interesting topics of astronomy
and for inviting me to family gatherings.
The Institute of Astronomy at Cambridge University is such an encouraging and thoughtful
place where many outstanding scientists have been working. During discussions with Andrew
Fabian, Gerry Gilmore, Paul Hewett, Robert Kennicutt and Carolin Crawford, I felt my knowledge
of astronomy, was greatly broadened. I also enjoy the friendships with my peer graduate student,
Becky Canning, Alex Calverley, Ryan Cooke, Stephanie Hunt, James Mead and James Owen.
In addition, I thank the computing team for solving so many technical problems during my 3-
year PhD (Hardip, Andrew and Sue). Furthermore, sincere thanks to the administrators of the
Institute, Mrs. Margaret Harding, Susan Leatherbarrow, Sian Owen, Paula Younger, and Mr.
Mark Hurn for their patient help and great strength of support for my travelling and for my
scientific career in general. Margaret is always helpful in making travel arrangements for me to
conferences and visits. Paula worked very hard for me to send references and arrange supervisor
meetings even when she was several months pregnant. Mark always brings useful books to me
when I need them. Before I came to Cambridge, I suffered a lot from bureaucracy imposed by
various administrators, and I never encountered supporting staff who were as efficient and helpful
as those at Institute of Astronomy. I can say that they are the best secretaries that I have ever
met.
College life and activities are also very much an important part of three year PhD at Cambridge.
Trinity College Cambridge, is not only a prestigious and wealthy college, but also a wonderful
place to meet many smart and eminent people in every different subject, and develop personal
and professional networks. It is a place to learn about the world and I always feel that I was so
lucky to be admitted. I would sincerely give thanks for the financial support from the College and
the Cambridge Overseas Trust, and I could not have come to Cambridge for my degree without
these funding schemes. In the college life, I am grateful to my English supervisor, Ms. Margaret
de Vaux, for her efforts in improving my English reading, speaking and writing. I especially
thank Lord Martin Rees, the Master of Trinity College and Astronomer Royal, for discussions on
many exciting and important topics in astronomy as well as general science. As a world-famous
cosmologist, Martin delivers articulate speeches on various occasions which have taught me how
to make the science understandable and appreciated by the general public. At the end of my PhD,
I got a “best-speaker” award at Trinity College Annual Science Symposium which I attribute to
Martin’s inspiration and influence. His well-known formula, “ability×impact” on how to search
for suitable projects, also impressed me a lot. I would like to thank my friends from Trinity with
whom I have shared my three-year time with. In particular, I would like to thank my fantastic
friends Johanna Tudeau, Juggs Ravalia, Lindsay Chura, Peng Zhao, Ena Hodzik, Sina Bonyadi,
Sara Ahmadi, Rebekah Clements, Yang Xia, Xuxiao Zhang, Fangyuan Cheng, Hao Wu, Mao Zeng,
Bo Zheng and Mingjie Dai et al.. I know they are less likely to read this thesis but they really
deserve thanks.
Collaborations are essential to a cosmologist and I have been lucky to work/discuss with many
excellent researchers throughout the world during the PhD course. First I must thank Christopher
Gordon, Hume Feldman and Douglas Scott for mentoring me closely on a number of occasions,
offering new ideas, revising paper drafts, and supporting me in navigating postdoc jobs. I especially
thank Douglas Scott and CITA office to support me as the CITA national fellow subsequent my
degree. I must also thank Jeremiah Ostriker, Gongbo Zhao, Wen Zhao, Laurie Shaw, Houjun
Mo, Xingang Chen, Baojiu Li, Xuelei Chen and Pengjie Zhang for the help and discussion they
have given me on various occasions. I really gained much valuable knowledge of cosmology and
astronomy from these collaborators.
Last but not least, completing a PhD is not possible without the unwavering support from my
family and closest people. I would like to thank my parents, Jian-Ming Ma and Yu-Qin Lu, as
well as other family members, for their constant love and support.
Abstract
Cosmology has become a precision science due to a wealth of new precise data from various
astronomical observations. It is therefore important, from a methodological point of view, to
develop new statistical and numerical tools to study the Cosmic Microwave Background (CMB)
radiation and Large Scale Structure (LSS), in order to test different models of the Universe. This
is the main aim of this thesis.
The standard inflationary Λ-dominated Cold Dark Matter (ΛCDM) model is based on the
premise that the Universe is statistically isotropic and homogeneous. This premise needs to be
rigorously tested observationally. We study the angular correlation function C(θ) of the CMB
sky using the WMAP 5-year data, and find that the low-multipoles can be reconstructed from
the data outside the sky cut. We apply a Bayesian analysis and find that S1/2 statistic (S1/2 =∫
[C(θ)]2d cos θ, used by various investigators as a measure of correlations at large angular scales)
cannot exclude the predictions of the ΛCDM model. We clarify some issues concerning estimation
of correlations on large angular scales and their interpretation.
To test for deviation from statistical isotropy, we develop a quadratic maximum likelihood
estimator which we apply to simulated Planck maps. We show that the temperature maps from
Planck mission should be able to constrain the amplitude of any spherical multipole of a scale-
invariant quadrupole asymmetry at the 1% level (2σ). In addition, polarization maps are also
precise enough to provide complimentary constraints. We also develop a method to search for the
direction of asymmetry, if any, in Planck maps.
B-mode polarisation of the CMB provides another important test of models of the early Uni-
verse. Different classes of models, such as single-field inflation, loop quantum cosmology and
cosmic strings give speculative but testable predictions. We find that the current ground-based
experiments such as BICEP, already provided fairly tight constraints on these models. We inves-
tigate how these constraints might be improved with future observations (e.g. Planck, Spider).
In addition to the CMB related research, this thesis investigates how peculiar velocity fields can
be used to constrain theoretical models of LSS. It has been argued that there are large bulk flows
on scales of & 50 Mpc/h. If true, these results are in tension with the predictions of the ΛCDM
model. We investigate a possible explanation for this result: the unsubtracted intrinsic dipole on
the CMB sky may source this apparent flow, leading to the illusion of the tilted Universe. Under
the assumption of a superhorizon isocurvature fluctuation, the constraints on the tilted velocity
require that inflation lasts at least 6 e-folds longer (at the 95% confidence interval) than that
required to solve the horizon problem.
Finally, we investigate Cosmic Mach Number (CMN), which quantifies the ratio between the
mean velocity and the velocity dispersion of galaxies. We find that CMN is highly sensitive to
the growth of structure on scales (10, 150) Mpc/h, and can therefore be used to test modified
gravity models and neutrino masses. With future CMN data, it should be possible to constrain
the growth factor of linear perturbation, as well as the sum of the neutrino mass to high accuracy.
Contents
Contents v
List of Figures ix
1 Introduction 1
1.1 Friedmann-Robertson-Walker models . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Problems of the Big Bang model . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1.1 Horizon Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1.2 Flatness Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1.3 Homogeneity Problem . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Inflation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2.1 Inflationary Scenario . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2.2 Single field description of inflation . . . . . . . . . . . . . . . . . 8
1.2.2.3 Inflation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.3 The properties of the observable Fluctuations . . . . . . . . . . . . . . . . 16
1.2.3.1 Gaussian Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.3.2 Linear perturbation . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.3.3 Adiabatic Initial conditions . . . . . . . . . . . . . . . . . . . . . 17
1.2.3.4 Close to Scale-invariant . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.3.5 Small but non-zero tensor modes . . . . . . . . . . . . . . . . . . 18
1.3 The Cosmic Microwave Background Radiation (CMB) . . . . . . . . . . . . . . . 19
1.3.1 CMB temperature power spectrum . . . . . . . . . . . . . . . . . . . . . . 19
1.3.1.1 Scenario of recombination . . . . . . . . . . . . . . . . . . . . . . 19
1.3.1.2 Mathematical Treatment . . . . . . . . . . . . . . . . . . . . . . . 22
1.3.2 CMB polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.3 Prospects of constraining inflation . . . . . . . . . . . . . . . . . . . . . . . 29
1.4 The Growth of Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.4.1 Mass Fluctuations and Matter Power Spectrum . . . . . . . . . . . . . . . 31
1.4.2 Linear growth of structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
v
CONTENTS
1.4.2.1 Jeans equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.4.2.2 Sub-Hubble-length modes . . . . . . . . . . . . . . . . . . . . . . 35
1.4.3 Non-linear growth of the structure . . . . . . . . . . . . . . . . . . . . . . 38
1.4.3.1 Spherical Collapse model . . . . . . . . . . . . . . . . . . . . . . 38
1.4.3.2 Dark Matter halo abundance . . . . . . . . . . . . . . . . . . . . 43
1.4.3.3 Gas Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.5 Dark matter and Galaxy formation . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.5.1 Dark Matter Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.5.2 Dark Matter on Small Scales . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.5.2.1 Population of sub-halos and satellites: how cold is cold dark matter? 47
1.5.2.2 Cusp and Core problem . . . . . . . . . . . . . . . . . . . . . . . 48
1.6 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.6.1 Classification of problems in Cosmology . . . . . . . . . . . . . . . . . . . 49
1.6.1.1 Known Physics but unknown details . . . . . . . . . . . . . . . . 49
1.6.1.2 Unknown Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.6.2 Aims and Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 51
2 Large-Angle Correlations in the CMB 55
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2 Estimators of the Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . 56
2.3 Reconstructing low-order multipoles on a cut sky . . . . . . . . . . . . . . . . . . 62
2.4 Analysis of the S1/2 statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.4.1 Approximate Bayesian analysis . . . . . . . . . . . . . . . . . . . . . . . . 70
2.4.2 Frequentist analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3 Testing a direction-dependent primordial power spectrum with CMB observa-
tions 75
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2 The anisotropic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2.1 Covariance matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2.2 Quadratic estimators and the Fisher matrix . . . . . . . . . . . . . . . . . 78
3.3 Forecasts for Planck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3.1 Constraints on the anisotropy amplitude . . . . . . . . . . . . . . . . . . . 79
3.3.2 Constraints on scale-dependence . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3.3 Axisymmetric models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
vi
CONTENTS
4 Constraints on the standard and non-standard models of the early Universe
from CMB B-mode polarisation 87
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 B-mode polarization and its observations . . . . . . . . . . . . . . . . . . . . . . . 89
4.2.1 B-mode polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2.2 Constraints on the B-mode signal . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.3 Data Analysis Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.2.3.1 χ2 analysis and hyper-parameters . . . . . . . . . . . . . . . . . . 92
4.2.3.2 Fisher information matrix . . . . . . . . . . . . . . . . . . . . . . 94
4.3 Constraining the SFI model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3.1 Scalar and tensor primordial power spectra in the SFI model . . . . . . . . 95
4.3.2 Constraints on SFI from current data . . . . . . . . . . . . . . . . . . . . . 97
4.3.3 Prospects for future observations . . . . . . . . . . . . . . . . . . . . . . . 98
4.3.4 Single-field slow-roll inflationary models . . . . . . . . . . . . . . . . . . . 102
4.4 Loop Quantum Gravity and its observational probes . . . . . . . . . . . . . . . . . 105
4.4.1 Primordial tensor perturbations in LQG models . . . . . . . . . . . . . . . 105
4.4.2 Constraints from current data . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.4.3 Prospects for future experiments . . . . . . . . . . . . . . . . . . . . . . . 109
4.5 Cosmic strings and their detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.5.1 B-mode polarization from cosmic strings . . . . . . . . . . . . . . . . . . . 111
4.5.2 Constraints from current data . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.5.3 Prospects for future experiments . . . . . . . . . . . . . . . . . . . . . . . 114
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5 Peculiar Velocity Field: Constraining the tilt of the Universe 117
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2 Likelihood and mock catalogues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6 Cosmic Mach Number as A Sensitive Test of Growth of Structure 127
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2 Statistics of Cosmic Mach Number . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.3 A sensitive test of growth of structure . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7 Conclusions and Outlook 135
vii
CONTENTS
A Instrumental Characteristics of CMB Experiments 141
B Statistics of the conventional χ2 and the hyper-parameter χ2 145
B.1 χ2 statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
B.2 Hyper-parameter χ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
C Analytic formulas of the correlated Window Function f12(k) 149
References 153
viii
List of Figures
1.1 WMAP 7-year Internal Linear Combination (ILC) Map. The colour scale is from
-200 K to 200 K. Figure taken from [196]. . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Solution of the horizon problem. The scales of interest were greater than the Hubble
radius before a ∼ 10−5, but fairly early on, all of the scales were inside the Hubble
radius and were sensitive with microscopic physics. . . . . . . . . . . . . . . . . . 7
1.3 Generic shapes of potentials for different inflation models. Left: large Field (Chaotic)
Inflation; Middle: small field inflation; Right: hybrid inflation. Figure taken from
[42]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Left: Expected 95% uncertainty on the inflationary parameters ns and r from
WMAP and Planck in 1990s. Each model has a unique prediction for ns and r.
Figure taken from [66]. Right: Current constraints on r−ns parameter space from
WMAP 7-year results. Figure taken from [72]. . . . . . . . . . . . . . . . . . . . . 15
1.5 The CMB temperature power spectrum (red line), which displays the amplitude
of the CMB temperature fluctuation ∆T/T as a function of angular scale on the
sky. The acoustic peaks suggest the interplay of the gravity and radiation pressure
and they are very sensitive to cosmological parameters. The black dots are WMAP
7-year data, and blue band is the cosmic-variance. Figure taken from [52]. . . . . 19
1.6 Density contrast in conformal Newtonian gauge. The photon, baryon, and CDM
density perturbations are plotted in red, green and blue respectively. The upper
figure is long wave length case (k=0.01 Mpc−1), and lower figure is the short wave
length case (k=1.0 Mpc−1). Figure taken from [62]. . . . . . . . . . . . . . . . . . 20
1.7 Contribution of different terms in Eq. (1.81) to the temperature anisotropies from
adiabatic initial conditions: δγ/4+ψ (Sachs-Wolfe effect, magenta line); vb (Doppler
effect, blue line); Integration term (Integrated Sachs-Wolfe effect, green line); total
effect (black line). Figure taken from [62]. . . . . . . . . . . . . . . . . . . . . . . 21
1.8 Thomson scattering of radiation with a quadrupole anisotropy generates linear po-
larization. Red colors (thick lines) represent hot radiation, and blue colors (thin
lines) cold radiation. Figure taken from [76]. . . . . . . . . . . . . . . . . . . . . . 26
ix
LIST OF FIGURES
1.9 The power spectrum of the cross-correlation between temperature and E-mode
polarization anisotropies. The anti-correlation at 100 < l < 200 and the positive
correlation at 200 < l < 400 are due to the phase coherence of fluctuation generated
during inflation [53]. The data is WMAP 7-year result [52]. . . . . . . . . . . . . . 27
1.10 WMAP 5-year measurements of the temperature and polarization power spectrum
to constrain the tensor-to scalar ratio. Left: Contours show 68% and 95% C.L of
the parameter space. The gray region is the constraint from the low-l multipole
data (TE, EE, BB at l ≤ 23) only, and the red region is from the low-l data plus
the high-l TE data at 24 ≤ l ≤ 450. Finally, the blue region is the constraint
from the low-l polarization data, the high-l TE correlation data, and the low-l
temperature data at l ≤ 23. Right: the gray curves show (r, τ)=(10,0.050), the
red curves (r, τ)=(1.2,0.075) and the blue curves (r, τ)=(0.2,0.080), where τ is the
optical depth τ =
∫ η0
0
neσTadη. Figure taken from [3]. . . . . . . . . . . . . . . . . 28
1.11 The marginalised likelihood of tensor-to-scalar ratio r in WMAP 7-year data and
5-year data, from the polarization data (BB, EE and TE) alone. All the other
cosmological parameters, including the optical depth, are fixed at the 5-year best-
fit ΛCDM model [3]. The vertical axis −2 ln(L/Lmax) is the standard χ2 under the
Gaussian approximation. The dashed line −2 ln(L/Lmax) = 4 corresponds to the
2σ CL. The solid, dashed and dot-dashed lines show the likelihood as a function of r
from the BB-only, BB+EE, and BB+EE+TE data. (Left) The 7-year polarization
data. WMAP 7-year data shows r < 2.1, 1.6, and 0.93 (95.4% CL) from the BB-
only, BB+EE, and BB+EE+TE data, respectively. (Right) The 5-year polarization
data. r < 4.7, 2.7, and 1.6 (95.4% CL). Figure taken from [72]. . . . . . . . . . . . 28
1.12 Comparison of matter power spectrum P (k), inferred from the CMB temperature
anisotropy to constraints from other observations, including the observed galaxy
distributions. The consistency of independent results shows that the density per-
turbations are the seeds for structure growth. Figure taken from [5]. . . . . . . . . 30
1.13 Power spectrum (upper-left), transfer function (upper-right), and root-mean-square
of mass fluctuation in linear perturbation theory at different redshift. Here we adopt
ΛCDM model with WMAP 5-year parameters Ωb = 0.0449, Ωc = 0.222,ΩΛ =
0.734,h = 0.71,σ8 = 0.801 and ns = 0.963 [3]. . . . . . . . . . . . . . . . . . . . . . 33
1.14 Inhomogeneous 1-D density fluctuation. The dark (light) curve is the field smoothed
(not smoothed) on large scales. Small scale fluctuations that have density contrast
over δcrit collapse first, while the large scale fluctuation cannot collapse since their
overdensity does not exist this threshold. Figure taken from [44]. . . . . . . . . . . 43
x
LIST OF FIGURES
1.15 Left: Two-dimensional slice of the baryon distribution at z = 2.5 in a ΛCDM hy-
drodynamical simulation of structure formation. The low and high density regions
are shown as dark and bright colours. The voids and filaments are typically as-
sociated with material in the Inter-Galactic Medium (IGM) and are observable in
quasar absorption spectra. The luminous baryonic material (stars and galaxies)
forms in the highest density regions, which is shown as the knots and sheets where
many filaments converge. Right: The underlying dark matter distribution. The
baryon distribution is slightly more diffuse (less small scale structure) compared
to the underlying dark matter due to gas pressure support. In the simulation, the
cosmological parameters are: Ωm = 0.26, ΩΛ = 0.74, Ωb = 0.0463, h = 0.72. The
box size is 60 Mpc/h. The number of gas and dark matter particles are 2 × 4003.
The softening length is 2.5 kpc/h. Figure taken from [87]. . . . . . . . . . . . . . 44
1.16 The cumulative number of Milky way satellite galaxies as a function of halo cir-
cular velocity. Left panel: Missing satellite problem; Right panel: the effect of
reionization. Figure taken from [104]. . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.1 The figure to the left shows the correlation functions computed using the pixel
estimator (2.3) applied to the 5 year WMAP ILC map (degraded as described in
the text) over the full sky and with the WMAP KQ85 and KQ75 masks imposed.
The figure to the right shows the correlation functions computed from the pseudo-
power spectra (2.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.2 Covariance matrix for the pixel correlation estimator (2.3) computed for full sky
maps. The scale on the right is in units of (µK)4. . . . . . . . . . . . . . . . . . . 60
2.3 The figure to the left in the top row shows the WMAP 5 year ILC temperature map
smoothed by a Gaussian of FWHM 10◦ and repixelized to a Healpix resolution of
NSIDE=16. The figure to the right in the top panel shows the degraded resolution
WMAP KQ75 mask used in this Chapter. The remaining figures to the left show
the reconstructed all-sky maps computed from data outside the KQ85 sky cut,
using the harmonic coefficients computed from equation (2.19) with the coupling
matrices truncated to (from top to bottom) ℓmax = 5, 10, 15 and 20. The figures to
the right show equivalent plots for the reconstructed all-sky maps computed from
data outside the KQ75 sky cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4 Reconstructions of the correlation function from equation (2.22) for various choices
of ℓmax and sky cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.5 As Figure 2.1 but using the QML estimator of equation (2.14). . . . . . . . . . . . 68
xi
LIST OF FIGURES
2.6 The temperature power spectrum at low multipoles computed from the low resolu-
tion WMAP 5-year ILC map. The points (corrected for the 10◦ FWHM smoothing
and slightly displaced in ℓ for clarity) show QML power spectrum estimates for
three sky cuts: no mask; KQ85 mask; KQ75 mask. Error bars show the diagonal
components of the inverse of the Fisher matrix (2.29). The solid red line shows
the power spectrum computed from the all-sky ILC map (which is identical to the
QML all-sky estimates). The solid green and blue lines show the power spectra
computed for the ℓmax = 10 reconstructions of Fig. 2.3. . . . . . . . . . . . . . . . 69
2.7 (a) Posterior distributions of ST1/2 computed as discussed in the text. The red (solid)
histogram shows the distribution of ST1/2 from an analysis of the whole sky. The
blue (dotted) histogram shows the distribution computed with the KQ75 sky cut
applied. The vertical dashed lines in the figures shows the value ST1/2 ∼ 49000 (µK)4
for the best fitting ΛCDMmodel as determined from the 5-year WMAP analysis [3].
(b) Frequency distributions of S1/2 for statistically isotropic, Gaussian, realizations
of the ΛCDM model [3]. The red (solid) histogram shows the frequency distribution
for the pixel ACF estimator applied to the whole sky. The blue (dotted) histogram
shows the distribution computed with the pixel ACF estimator with the KQ75 sky
cut applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.8 Pixel based estimates ACF estimates for the WMAP ICL map corrected for the
local ISW contribution for redshift z < 0.3 as described by [132]. . . . . . . . . . . 72
3.1 Noise-free simulation of a model with scale-invariant quadrupole asymmetry in
the primordial power with gLM = 0.1δL2δM0. The isotropic component of the
temperature map is shown on the left, and the anisotropic component on the right.
The colour scales are in µK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.2 Anisotropic components of the Q (left) and U (right) polarization maps in a noise-
free simulation of the model in Fig. 3.1. The maps have been smoothed here with
a Gaussian beam of FWHM 3◦ to enhance the imprint of the preferred axis in the
Q map. The colour scales are in µK. . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.3 Estimates of the G20 (left) and G2−2 (right) anisotropy parameters (shown with
points) and their (one-sigma) Fisher errors ([Red] solid lines) as a function of lmax
from five simulations of the model in Fig. 3.1 for one year of Planck data. The
input parameters G20 = 0.1 and G2−2 = 0.0 are shown with horizontal [Blue] solid
lines. From top to bottom we analyse temperature only, E-mode polarization only
and temperature plus polarization. . . . . . . . . . . . . . . . . . . . . . . . . . . 81
xii
LIST OF FIGURES
3.4 Fisher errors for G20 from temperature, E-mode polarization, and temperature plus
polarization in models with power-asymmetry spectral indices q = 0 (left), q = 1
(middle) and q = 2 (right). For q = 0 and q = 1 we show results for one and two
years of observations; for q = 2 we show only the one-year errors since they improve
very little with further observing time. Note that the one- and two-year errors from
temperature alone are indistinguishable when q ≥ 0. . . . . . . . . . . . . . . . . 82
3.5 Marginal distributions for the direction (left) and amplitude (g∗2, right) from a
simulation of the nominal (one-year) Planck survey for a model with an axisym-
metric quadrupole asymmetry aligned with the polar axis (gLM = 0.1δL2δM0). We
parameterise the direction with the equal-area projection x = 2 sin(β/2) cos(α) and
y = 2 sin(β/2) sin(α) and show in solid lines the 68% [Red], 95% middle [Blue]
and 99% outer [Green] contours from the temperature alone; dashed-line contours
are from the E-mode polarization alone. For the amplitude (right), we plot the
marginal distributions from T alone (solid line) and E alone (dashed line). . . . . 85
4.1 Comparison of different theoretical predictions, and the currently available data for
the B-mode power spectrum. The unit for m is Mpl and k∗ is Mpc
−1. Note that
the error-bars are plotted in log-log scale. . . . . . . . . . . . . . . . . . . . . . . . 90
4.2 Polarization noise power spectra for forthcoming experiments. Note that these
curves include uncertainties associated with the instrumental beam. The blue
curves show the B-mode power spectrum for the standard inflationary model with
r = 0.03. In the left panel, we plot the instrumental noise for the Planck and
CMBPol satellites, as well as the lensing B-mode signal and the noise level for
the ideal experiment. In the right panel, we plot the instrumental noise for the
ground-based PolarBear and QUIET experiments, and the balloon-borne Spider
experiment. We fix the other parameters at WMAP 7-year best-fit values (Ωbh
2 =
0.02258, Ωch
2 = 0.1109, ns = 0.963, As(k0) = 2.43 × 10−9 (pivot scale k0 =
0.002,Mpc−1), h = 0.71, and τ = 0.088). . . . . . . . . . . . . . . . . . . . . . . . 91
4.3 The current constraints on r from BICEP and QUaD data. . . . . . . . . . . . . . 97
4.4 The signal-to-noise ratio r/∆r for different experiments, calculated using the Fisher
matrix of Eq. (4.13). Left: The parameter r is treated as a free parameter but nt is
kept fixed at its fiducial value; Right: Both r and nt are treated as free parameters.
The atmospheric noise in PolarBear and QUIET is not included in the plot. . . . 100
4.5 The signal-to-noise ratio r/∆r for different experiments, and the combination of
Planck and ground-based experiments, calculated using the Fisher matrix of Eq.
(4.13). Left: The parameter r is treated as a free parameter but nt is kept fixed at
its fiducial value; Right: Both r and nt are treated as free parameters. . . . . . . . 101
xiii
LIST OF FIGURES
4.6 Forecasted joint constraints on the parameters of the SFI model. The contours
indicate the 68% (1σ) confidence levels. The input models are indicated by the
black points (Left: r = 0.01 and nt = 0, Right: r = 0.001 and nt = 0). . . . . . . . 102
4.7 Forecasted uncertainty on the tensor spectral index, ∆nt, for different experiments,
calculated using the Fisher matrix of Eq. (4.13). Here, both r and nt are treated
as free parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.8 Left: the primordial tensor power spectrum for different models of inflation. Right:
the corresponding B-mode angular power spectrum generated by different infla-
tionary models. Note the lensing B-mode signal which acts as an effective noise for
detecting the primordial CBBl signal. The units for m and k∗ are Mpl and Mpc
−1
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.9 Probability distribution function (PDF) for the parameters in the LQGmodel. Left:
constraints on the mass parameter m˜ = m/Mpl. Right: constraints on k∗. The red,
green, and brown curves overlap with each other, indicating that the vast majority
of the constraining power for the combined data sets comes from the BICEP data. 108
4.10 2D constraints on the parameters m˜ = m/Mpl and k∗[Mpc
−1] of the LQG model.
The red line (BICEP 1σ CL) overlaps with the brown line (BICEP+QUaD, 1σ CL).109
4.11 Predicted signal-to-noise ratios for the parameters in LQG for forthcoming and
future experiments. Here m˜ is the mass parameter, and k∗ is the position of the
bump in the BB power spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.12 Forecasted 2D constraints for the LQG model. The input models are indicated with
the black points. (Left: m˜ = 10−8 and k∗ = 0.002Mpc
−1. Right: m˜ = 10−8 and
k∗ = 0.0002Mpc
−1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.13 Current constraints from B-mode polarization on the cosmic string tension. To
obtain the constraints, we have fixed the wiggling parameter at α = 1.9. . . . . . . 113
4.14 Forecasts of the signal-to-noise ratio for the cosmic string tension Gµ potentially
achievable with future CMB polarization experiments. . . . . . . . . . . . . . . . . 114
5.1 The SN data plotted in Galactic coordinates.. The red points are moving away from
us and the blue ones are moving towards us. The size of the points is proportional
to the magnitude of the line-of-sight peculiar velocity. ”X” is our estimate of the
direction of the tilted velocity estimated from the SN data. . . . . . . . . . . . . . 119
5.2 The simulation of the bulk flow velocity and test of likelihood (5.4). We input
a particular set of parameters (σ∗, u) and do 300 simulations of the underlying
density field with the mean value of this set of parameters. One can see that the
peaks of the average likelihood are very close to the input values ([Blue] dashed lines).120
5.3 The 1-D marginalized posterior probability distribution functions of the parameters
: (a) σ∗, (b) magnitude of u, (c) Cos(θ), (d) φ. . . . . . . . . . . . . . . . . . . . . 121
xiv
LIST OF FIGURES
5.4 (a) The 68%, 95% and 99.7% Bayesian confidence interval contours for parameters
σ∗-u. (b) The 68% contours for Cos(θ)-φ. The black dot is the direction of the bulk
flow found by [244]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.1 (a): The 1-D posterior distribution of the CMN M(r) from 5 different catalogues.
The unit of r is [Mpc/h]. (b): The comparison between the CMN data (3σ CL.) and
theoretic prediction: blue data points are the CMN data from posterior distributions
shown in panel (a); red data points are the data used in Ostriker and Suto 1990
[270]; the black line is the CMN prediction from WAMP 7-yr best-fit values [72]; the
green dashed line is calculated by using ‘popular’ CDM cosmological parameters
ns = Ωm = 1 and h = 0.5 in 1990s [270]; the brown dashed and purple dot-dashed
lines are the CMN from f(R) model with B0 = 10
−4 and from ΛCDM model with
neutrino mass
∑
ν mν = 0.2eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.2 (a): Forecast 1σ errors of the CMN from 6dF survey, in which we assume the
data is around our local region of the universe. Beside ΛCDM model, we plot
B0 = 10
−4 ([Brown] dashed line) and 10−3 ([Red] dashed line) for the f(R) model,
and
∑
ν mν = 0.2eV ([Purple] dot-dashed line) and 0.3eV ([Green] dashed line).
(b): Pθθ [Black], and f(k, a) [Red] in ΛCDM (real line), f(R) (dashed line) and
massive neutrino (dot-dashed line) models. (c): fractional difference between the
f(R) and massive neutrino models with ΛCDM. . . . . . . . . . . . . . . . . . . . 131
7.1 Demonstration of potential effects associated with Malmquist bias. Panel (a): bulk
flow magnitude as a function of depth R. By placing the galaxies at their redshifts
([Blue] triangle points) rather than distance ([Red] square points), the bulk flow
magnitude is consistent with ΛCDM prediction. However, if we place the galaxies at
their TF distances, we find large flows on scales & 50Mpc/h. Panel (b): If we place
the galaxies at their redshifts, the directions of the observed bulk flow at different
depths is consistent with CMB dipole direction. Panel (c): Parameter estimation
of σ8 by using the COMPOSITE catalogue, SFI++ group galaxies (SFI++G), and
SFI++ field galaxies (SFI++F) (see description of the catalogues in [244] for de-
tails). By using COMPOSITE catalogue, placing the galaxies at TF distance, and
generating mock catalogues from Gaussian window function, Ref.[244] obtains the
distribution ([Black] dashed line) which excludes WMAP 5-yr best-fit σ8 = 0.796 at
3σ CL. However, by using the redshift as the true distances and adopting a top-hat
window function, one finds a consistency with ΛCDM ([Black] solid line). Here we
fix the cosmological parameters to be WMAP 5-yr results (Ωm = 0.258, h = 0.719,
ns = 0.963, Ωb = 0.0441) in order to compare with the results in [244]. . . . . . . . 137
xv
LIST OF FIGURES
xvi
Chapter 1
Introduction
In the last two decades, cosmologists have been making great efforts that have established a
standard cosmological model (ΛCDM) which describes the contents of the Universe. A wide
variety of observations have tested the model to a very high degree of precision. These include
measurements of the cosmic microwave background temperature anisotropy (hereafter CMB) [1;
2; 3]; observations of galaxy clustering, for example, from the 2dF Galaxy Redshift Survey [4]
and the Sloan Digital Sky Survey [5; 6]; evidence for the accelerating expansion of the Universe
from the distant Type Ia supernovae [7; 8; 9]. In addition, complementary CMB experiments [10],
weak lensing measurements [11; 12; 13], X-ray measurements of rich galaxy clusters [14; 15; 16;
17] and Lyα forest data [18; 19; 20; 21] have given further precise constraints on cosmological
parameters. The observational data provide strong support for a Universe which is spatially flat
and accelerating at the present day, consisting of approximately 73% dark energy, 23% cold dark
matter (hereafter CDM), and 4% baryons. In addition, the data favour the cosmological constant
Λ as the candidate for dark energy, which leads to the generic concordance model ΛCDM.
According to the concordance model, the inhomogeneous large scale structure of the current
Universe was seeded by tiny quantum fluctuations in the early Universe, which were amplified
in scales during a special period of time when the Universe was expanding almost exponentially.
This period of time is known as inflation [22; 23]. The primordial perturbations generated during
inflation leave two important signatures on the sky that we can observe today. One is the imprint
on the CMB, where acoustic oscillation peaks are generated in the angular power spectrum of
the CMB. The other is the cosmic structure that we observe today, which was amplified by
gravitational instability to produce non-linear structures, such as galaxies and clusters [26]. The
simplest inflationary paradigm predicts an adiabatic, Gaussian and nearly scale-invariant power
spectrum, which is consistent with the CMB observations.
Information from the CMB sky can be used to probe the physics of the early Universe. Many
observational projects have been done to measure the power spectrum, Cl, of the CMB temper-
ature fluctuations, such as the satellite observations (WMAP [1; 2; 3] and Planck [27]), ground-
based telescopes (ACBAR [10], BICEP-II [28], QUIJOTE [29], PolarBear [30], QUIET [31]), and
1
1. INTRODUCTION
balloon-borne experiments (EBEX [32] and Spider [33]). The theory for the prediction of the
power spectrum is now well developed and so it is possible to compute the linear evolution of the
CMB anisotropy to a high precision. Various authors have developed Boltzmann codes, such as
CMBFAST [24] and CAMB [25], to compute the linear part of the power spectrum. Therefore,
it is feasible to estimate cosmological parameters accurately by using the current CMB data from
WMAP and other experiments.
In addition to the CMB observations, the cosmic structure that we see today is also a mani-
festation of primordial perturbations. According to structure formation theory, after the photons
decouple from baryonic matter, gravitational instability causes the dark matter to concentrate,
and the curvature perturbation provides the potential wells for baryons to collapse and form galax-
ies and stars. The matter power spectrum is one of the simplest and most important measures of
large-scale structure. Many observations have been made to infer the matter power spectrum of
the cosmic structure [4; 5; 6] to a fairly high precision, which can be used to constrain cosmological
parameters quite precisely. In addition, it is possible to go beyond the matter power spectrum
and measure the peculiar velocities of galaxies which become another important probe of cosmic
structure.
In this chapter, we shall first introduce the standard cosmology, and then present the infla-
tionary paradigm and its predictions. Then we discuss measurements of the CMB angular power
spectrum, and how these can be used to constrain cosmological parameters. We review the linear
and non-linear theory of structure formation, and discuss how the ΛCDM model predicts cosmic
structure from large (Gpc) to small (Mpc) scales. Then we discuss how observations of galaxies on
kpc scale or smaller can be used to constrain the properties of dark matter. Finally, we summarize
the aims of this thesis.
1.1 Friedmann-Robertson-Walker models
In the standard hot big bang theory, the Universe is expanding from the initial hot and dense
state to the current cool and clumpy state. Mathematically, the standard big bang cosmology is
formulated under the assumption of the cosmological principle, which assumes that the Universe
is homogeneous and isotropic on large scales. Under this assumption, the space-time interval
between two events is expressed as
ds2 = −dt2 +R2(t)
[
dr2
1− kr2 + r
2(dθ2 + sin2 θdφ2)
]
, (1.1)
where (r, θ, φ) are comoving coordinates and R(t) is the scale factor which describes the dynam-
ical expansion of the Universe, and we use units in which c = 1. The dimensionless parameter k
describes the curvature of the Universe: k > 0 describes a finite, positively curved space which cor-
responds to a closed Universe; k = 0 describes an infinite, zero-curvature space which corresponds
2
Content EoS a ρ (density) H(t)
Matter w = 0 a ∼ t2/3 ρ ∼ a−3 H(t) = 2/3t
Radiation w = 1/3 a ∼ t1/2 ρ ∼ a−4 H(t) = 1/2t
Λ w = −1 a = exp(H(t− t0)) ρ ∼ const H(t) =const
Table 1.1: The evolution of each type of matter.
to a flat Universe; k < 0 describes an infinite, negatively curved space which corresponds to an
open Universe. The scale factor R(t) is related to the cosmological redshift z, and the normalized
scale factor a(t) as R(t)/R0 = a(t) = (1+ z)
−1. If we substitute the metric (1.1) into the Einstein
field equations, we can derive the following Friedmann Equation
H2 =
(
R˙
R
)2
=
8πG
3
ρ− k
R2
, (1.2)
where H = H(t) is the Hubble parameter which describes the expansion rate of the Universe,
k/R2 is the spatial curvature for any time slice, and Λ is the cosmological constant. We can
interpret the cosmological constant Λ as another component of the Universe with energy density
ρΛ = Λ/8πG. By assuming the content of the Universe is an ideal fluid, the energy conservation
equation T µν;µ = 0 gives the relationship between the energy density ρ and the pressure p
ρ˙+ 3H(ρ+ p) = 0. (1.3)
Therefore, by differentiating Eq. (1.2) with respect to time and using Eq. (1.3), we determine the
second derivative of the scale factor with respect to time
R¨
R
= −4πG
3
(ρ+ 3p), (1.4)
which characterizes the acceleration of the expanding Universe. From Eq. (1.4), we can see that
the sign of (ρ + 3p) determines whether the Universe is decelerating (R¨ < 0) or accelerating
(R¨ > 0). If p > −1
3
ρ, the expansion of the Universe will gradually slow down under the action of
gravity; on the other hand, if p < −1
3
ρ, the pressure can act as an ‘anti-gravity’ term leading to the
accelerated expansion. For instance, the pressure from cosmological constant (pΛ = −ρΛ) always
acts as an anti-gravity which drives the Universe accelerated expansion. For each component of
the Universe, cold dark matter, radiation and dark energy, we can define their equation of state
(hereafter EoS) parameter w = p/ρ. Since the Friedmann equations (1.2) and (1.4) are related to
the energy density and pressure, it is easy to relate the EoS parameter to the density and scale
factor evolution, as shown in Table 1.1.
Here we assume the dark energy EoS w = −1, i.e. the cosmological constant. However,
constraining the dark energy EoS and providing an explanation for dark energy are active current
areas in astrophysics. If we substitute the above relations between energy density and scale factor,
3
1. INTRODUCTION
we can get the following evolution equation for Hubble parameter
H(z) = H0[Ωm0(1 + z)
3 + ΩΛ0 + Ωr0(1 + z)
4]
1
2 , (1.5)
where H0 is the present day Hubble parameter. Ωm0 is the present day fractional matter density,
Ωr0 is the present day fractional radiation density, and ΩΛ0 is the present day fractional vacuum
energy density. They are defined as the ratio between the present-day matter density and critical
density: Ωi0 = ρi0/ρcrit (ρcrit = 3H
2
0/8πG, i = m,r,Λ). Here we assume the spatially flat Universe
(k = 0, Ωm0 +ΩΛ0 +Ωr0 = 1) which is motivated by the inflation model and also consistent with
the current observational data [3].
Since the density of different types of matter varies with the scale factor in different ways
(see Table 1.1), the Universe does not have a constant EoS. In reality, the Universe went through
a radiation dominated phase at early times, then a matter dominated phase and finally a dark
energy dominated era. There are three classic pieces of evidence to support this standard model,
the expansion of the Universe [34; 35], the abundance of the light elements [36; 37; 38], and the
CMB radiation [2; 39; 40; 41].
Despite the observational support for the standard hot big bang scenario, explanations of the
flatness, isotropy and homogeneity of the Universe requires a deeper understanding of cosmic evo-
lution. Therefore, theoretical physicists proposed a new model, termed the inflationary Universe,
which provides solutions to these problems and also provides a viable mechanism for structure
formation.
1.2 Inflation
Let us first examine the problems with the standard hot big bang model and then introduce the
inflationary scenario. We will discuss how inflation can provide solutions for these difficulties.
1.2.1 Problems of the Big Bang model
1.2.1.1 Horizon Problem
To analyze the horizon problem, let us consider the distance from the last scattering surface to us
in the comoving coordinates (see [42] for more discussion)
∫ t0
tLS
dt
a
=
∫ τ0
τLS
dτ = τ0 − τLS, (1.6)
4
Figure 1.1: WMAP 7-year Internal Linear Combination (ILC) Map. The colour scale is from -200
K to 200 K. Figure taken from [196].
where τ0 is the current conformal time and τLS is the conformal time at last scattering surface. A
given comoving scale projected on the last scattering surface corresponds to an angular scale
θ ≃ λ
τ0 − τLS , (1.7)
where the curvature effect has been neglected. Since before recombination baryons are tightly
coupled to photons, the sound speed in the plasma is very close to that of a relativistic fluid
cs = c/
√
3. Take λ to be the comoving sound horizon which has the magnitude csτLS ∼ τLS/
√
3,
the angular acoustic scale becomes
θ ≃ csτLS
τ0 − τLS ≃ cs
τLS
τ0
, (1.8)
where the last equality holds since τ0 ≫ τLS . At the time of last scattering, the Universe has
already entered into the matter dominated phase, and so the scale factor a and the temperature
T scale with the conformal time as a ∼ τ 2 ∼ T−1. The angle corresponding to the sound horizon
at the last scattering surface is
θhor ≃ cs τLS
τ0
=
1√
3
(
T0
TLS
) 1
2
≃ 10, (1.9)
where we use T0 ≃ 3 K and TLS ≃ 3000 K. This corresponds to a multipole lhor of
lhor =
2π
θhor
≃ 300. (1.10)
This means that the two photons at last scattering surface separated by angle θ > 10 (or
equivalently lhor ≤300) were not in causal contact. However, from measurements of the CMB
temperature field (Fig. 1.1), we know that temperature anisotropies on these scales are not bigger
5
1. INTRODUCTION
than ∆T/T ∼ 10−5. This means that photons at the last scattering surface that were apparently
not in causal contact have very nearly the same temperature. This is the horizon problem.
1.2.1.2 Flatness Problem
The current CMB experiments and SNe Ia constraints strongly favour a spatially flat Universe,
which imposes strong constraints on the initial conditions. To understand the nature of this
fine-tuning problem, we transform Eq. (1.2) into the following form
Ω(a)− 1 = k
(aH)2
. (1.11)
In the standard FRW model, the Universe contains only matter and radiation, so the comoving
Hubble radius (aH)−1 grows with time, and the quantity |Ω(a)− 1| must diverge with time. Thus,
Ω(a) = 1 is an unstable point. If the current spatial curvature is very close to zero, the density
parameter needs to be close to unity to incredibly high precision. For instance, if we consider
the time of Big Bang nucleosynthesis (BBN, T ∼ 0.1 MeV), the Grand Unification era (GUT,
T ∼ 1015 GeV) and the Planck scale (T ∼ 1019 GeV), the fractional density Ω(a) should satisfy
the following conditions [43; 44]
|Ω(aBBN )− 1| ≤ O(10−16), (1.12)
|Ω(aGUT )− 1| ≤ O(10−55), (1.13)
|Ω(aP l)− 1| ≤ O(10−61). (1.14)
Thus, if Ω0 ∼ 1, the energy density at early times needs to be close to unity to an extraordinary
fine precision. This needs an explanation.
1.2.1.3 Homogeneity Problem
The entropy of the current observable Universe is about 1080 in Planck units (estimated by con-
sidering the entropy contributed by radiation) [45]. If we put all of the matter of the observable
Universe into a black hole, the entropy is related to the surface area of the black hole as
S =
A
4l2p
∼ 10120, (1.15)
where A is the surface area and lp is the Planck length. This is very much larger than the
observable value 1080. This tells us that our Universe had an an extraordinary low entropy state
at early times. Our Universe could have been more disordered and inhomogeneous than observed.
This is the homogeneity problem.
6
1.2.2 Inflation models
1.2.2.1 Inflationary Scenario
Based on the hypothesis that in the early Universe there was a short period of time (perhaps
about 10−35 second) when the Universe was exponentially expanding, it is possible that a patch
as small as the Planck scale would have grown to be many orders of magnitude larger than our
observable Universe. As the exponential expansion ended, matter and radiation were created
during a period of “reheating”, and the expansion rate slowed down to the moderate rate of the
radiation dominated era [46]. The flatness, horizon and homogeneity problems can be solved if
the Universe underwent an inflationary phase of quasi-exponential expansion.
According to Eq. (1.11), during the inflation period, the Hubble parameter H is nearly a
constant, and the scale factor evolves as a ∼ eHt, so it will drive the Universe towards flatness
since
|1− Ω(a)| ∼ 1
a2
= e−2Ht → 0 as t→∞. (1.16)
Thus, if inflation occurred, Ω(a) = 1 is an attractor, solving the flatness problem.
Figure 1.2: Solution of the horizon problem. The scales of interest were greater than the Hubble
radius before a ∼ 10−5, but fairly early on, all of the scales were inside the Hubble radius and
were sensitive with microscopic physics.
During the inflationary period, physical scales grow faster than the Hubble radius (in comoving
coordinate, the Hubble radius is (aH)−1), so quantum fluctuations are inflated on scales beyond
the Hubble radius, when they become the classical fluctuations. After inflation ended and the
Universe entered into the radiation dominated era, the frozen fluctuations reentered the Hubble
radius and evolved under the action of gravity and pressure (see Fig. 1.2). Thus, scales that
were apparently outside the Hubble radius at the decoupling time were in fact well within the
Hubble radius during inflation. Inflation therefore provides an explanation of the origin of classical
fluctuations and the horizon problem.
In addition, inflation partially solves the homogeneity problem. As we have shown above, the
7
1. INTRODUCTION
current Universe entropy 1080 is many order of magnitude smaller than the entropy 10120 if all of
the cosmic structure is put into a black hole, which is a puzzle for standard ΛCDM cosmology. If
the curvature of the Universe before inflation is not very fractal like, the Universe can be inflated
to a homogeneous low-entropy state. However, this requires a low-entropy initial state, which is
not explained by inflation so needs to be explained in the new physics.
1.2.2.2 Single field description of inflation
To understand how inflation might work, one should bear in mind that in General Relativity,
gravity couples to both energy and pressure. Inflation might therefore arise if a scalar field
dominates the dynamics of the early Universe. Therefore, we shall consider simple models in
which inflation is driven by a single scalar field. First, let us write down the action for a single
scalar field
S =
∫
d4x
√−gL =
∫
d4x
√−g
[
1
2
∂µφ∂
µφ+ V (φ)
]
, (1.17)
where
√−g = a3 for the spatially flat FRW metric. From the Euler-Lagrange equation, we can
derive the equation of motion for the scalar field φ
φ¨+ 3Hφ˙− ∇
2φ
a2
+ V ′(φ) = 0, (1.18)
where V ′(φ) = dV (φ)/dφ. In Eq. (1.18), the 3Hφ˙ term describes the ‘friction’ caused by the
expansion of the Universe as the scalar field rolls down the potential V (φ). Since the energy-
momentum tensor is
Tµν = ∂µφ∂νφ− gµνL, (1.19)
we can write down the energy density ρφ and the pressure pφ as follows
1
ρφ = T00 =
φ˙2
2
+ V (φ) +
(∇φ)2
2a2
, (1.20)
pφ = −Tii
a2
=
φ˙2
2
− V (φ)− (∇φ)
2
6a2
. (1.21)
Now we can split the inflaton φ into the following two parts
φ = φ0(t) + δφ(x, t), (1.22)
where φ0 is the the classical scalar field and δφ(x, t) is the quantum fluctuation. The classical
scalar field φ0 is the expectation value of the field at the initial homogeneous and isotropic state
[42], whereas the δφ(x, t) is the much smaller quantum fluctuation around the classical field φ0.
1In the space-time with metric (1.1), the isotropic energy-momentum tensor can be written as T µν =
diag(−ρ, p, p, p).
8
φ0 plays the main role in determining the evolution of the ‘background’ (unperturbed) Universe,
whereas the quantum fluctuation δφ is relevant to the formation of structure. Since at the moment
we are interested in the dynamical evolution of the Universe, we shall neglect the δφ term and
drop the subscript to keep the notation simple. Thus, the energy density and pressure can be
expressed as
ρφ =
φ˙2
2
+ V (φ), pφ =
φ˙2
2
− V (φ). (1.23)
If
V (φ)≫ 1
2
φ˙2, (1.24)
the EoS of the scalar field is
p ≈ −V (φ) ≈ −ρ, (1.25)
leading to exponential expansion (Eq.(1.4)). Thus, inflation happens when the potential energy
dominates over the kinetic energy, so that the Universe is dominated by the vacuum energy.
Since the spatial derivatives of the classical field are assumed to be negligible, the equation of
motion for inflaton is simply (Eq. (1.18))
φ¨+ 3Hφ˙+ V ′(φ) = 0. (1.26)
Under “slow-roll” approximation, we require φ¨≪ V ′(φ), so that Eq. (1.26) becomes
3Hφ˙ ≃ −V ′(φ). (1.27)
In simple single field models, inflation ends when the potential energy of the inflaton becomes
smaller than the kinetic energy. Couplings between inflaton and matter provide a friction for
the motion of inflaton, which make the inflaton decay into relativistic particles. Generally, the
dynamic friction effect is much greater than the Hubble expansion, so at the end of inflation the
Hubble expansion can be neglected, and the energy of the Universe is dominated by the coherent
oscillation of the inflaton around the minimum of its potential. This process is called reheating.
To distinguish different inflation models and to constrain these models, one needs to compute
the power spectrum of the primordial fluctuation and other observables that can be tested by
observational data. To do this, we define the following slow-roll parameters
ǫ = − H˙
H2
≃ M
2
pl
2
(
V ′
V
)2
, (1.28)
η =M2pl
(
V ′′
V
)
, (1.29)
δ = η − ǫ = − φ¨
Hφ˙
, (1.30)
9
1. INTRODUCTION
where M2pl = 1/8πG. The number of e-folds which characterize the duration of accelerated expan-
sion can be calculated as
Ne = ln
(
af
ai
)
=
∫ tf
ti
Hdt
≃ H
∫ φf
φi
dφ
φ˙
≃ −8πG
∫ φf
φi
V
V ′
dφ, (1.31)
where in the last step we use the slow-roll approximation 1 and Friedmann equation H2 ≃
(8πG/3)V .
As we have seen, the classical field is the driver of inflation, whereas the quantum fluctuation
(δφ in Eq. (1.22)) induces the metric perturbation around the homogenous and isotropic FRW
background as follows
ds2 = gµνdx
µdxν
= −(1 + 2ψ)dt2 + 2aBidtdxi + a2[(1− 2φ)δij + Eij ]dxidxj , (1.32)
where φ and ψ are scalar perturbations, Bi is the vector perturbation, and Eij is the symmetric
and trace-free tensor perturbation. It is also proved that since the FRW metric possesses a great
deal of symmetry, perturbations can always be decomposed into scalar, vector and tensor parts
[57]. We shall describe each of the scalar, vector, and tensor perturbation and see how they affect
the growth of structure.
For the scalar perturbation, the spatial part of the perturbed metric can be written as [56; 62]
gij = a
2(t)[1 + 2ζ ]δij, (1.33)
where ζ is the spatial curvature of the constant hypersurface density, and ζ = −φ. The comoving
curvature perturbation is defined as the spatial derivative of this spatial curvature as
(3)R =
4
a2
∇2ζ. (1.34)
The key property is that ζ is constant outside the Hubble radius [57; 62]. The power spectrum of
1Usually, we say inflation at least should last about 60 number of e-folds, which is the number of e-folds between
the comoving scale of our observable Universe crosses the Hubble radius and the time that inflation ends. The
total number of e-folds of inflation can be much longer. Also see discussion in Chapter 5.
10
the scalar fluctuation is
〈ζkζk′〉 = (2π)3δ3(k− k′)2π
2
k3
Ps(k), (1.35)
and the scale dependence of the power spectrum can be described by the scalar spectral index ns
ns − 1 = d lnPs(k)
d ln k
, (1.36)
and its running
αs =
dns
d ln k
. (1.37)
A full calculation of the primordial power spectrum for single scalar field models of inflation [42]
leads to the following relation between the spectral index and the slow-roll parameters
ns = 1− 4ǫ+ 2δ. (1.38)
Then, the power spectrum is approximated as a power law form
Ps(k) = As(k∗)
(
k
k∗
)ns(k∗)−1
, (1.39)
where k∗ is an arbitrary pivot scale and As is the normalization. The WMAP team chose
k∗=0.002Mpc
−1, and determined the amplitude of scalar fluctuation at this scale (WMAP 7-year
data, see [72])
As(k∗) = (2.43± 0.11)× 10−9. (1.40)
If the primordial fluctuations are Gaussian, 〈ζζ∗〉 (power spectrum) contains all of the statis-
tical information [56]. However, if primordial fluctuation is not purely Gaussian, the higher order
correlations of ζ will encode information on this non-Gaussianity.
To analyze vector perturbations, well within the Hubble radius, we can describe the non-
relativistic matter by using Newtonian perturbation theory. In this theory, the linearized Euler
equation reduces to [62]
∂tv +Hv = − 1
aρ¯
∇δP − 1
a
∇φ, (1.41)
where φ is the gravitational potential. We can decompose the velocity perturbation into the scalar
part and the vector part as
v = ∇v + v⊥, (1.42)
where ∇ · v⊥ = 0. The vector part of v does not lead to the clumping of the matter since the
vorticity decays as the Universe expands. Taking the curl of the perturbation equation (1.41), we
shall get
∇× ∂tv = ∂t(∇× v) = −H∇× v⊥. (1.43)
11
1. INTRODUCTION
It follows that the vorticity decays by the Hubble flow since ∇ × v⊥ scales as 1/a. For general
types of inflation models, the vector modes are not excited by inflation initially, but they can be
excited by the perturbations introduced by topological defects [56; 62].
Tensor perturbations are described by a trace-free metric perturbation hij (∂ihij = h
i
i = 0) as
[42; 56]
gij = a
2[δij + hij ], (1.44)
where hij corresponds to the gravitational wave perturbation. The power spectrum for the two
polarization modes of hij (hij = h
+e+ij + h
×e×ij, h = h
+ , h×) is
〈hkhk′〉 = (2π)3δ3(k− k′)2π
2
k3
Pt(k), (1.45)
and its scale-dependence is defined as
nt =
d lnPt(k)
d ln k
, (1.46)
i.e., the tensor power spectrum is the following power law form
Pt(k) = At(k∗)
(
k
k∗
)nt(k∗)
. (1.47)
CMB experiments can, in principle, measure the ratio of the tensor to scalar fluctuation amplitude
r =
Pt
Ps
. (1.48)
The value of r depends on the energy scale of inflation, and is important for distinguishing different
inflationary models (see Section 1.2.2.3 and Chapter 4). We shall describe how CMB B-mode
polarization observations can be used to constrain or measure the tensor-to-scalar ratio r (see
Section 1.3.2).
Suppose that inflation is driven by a single inflaton field, then a calculation gives the following
result for the primordial power spectrum of density fluctuations for slow-roll inflation [62]
Ps(k) ≃ H
4
(2πφ˙)2
∣∣∣∣
k=aH
≃ 9
(2π)2
1
(3M2pl)
3
V 3
V ′2
∣∣∣∣∣
k=aH
≃ 8
3ǫ
(
V
1
4√
8πMpl
)4
, (1.49)
12
and the tensor fluctuation is
Pt(k) ≃ 8
M2pl
(
H
2π
)2∣∣∣∣∣
k=aH
≃ 2
3
V
π2M4pl
∣∣∣∣∣
k=aH
. (1.50)
Then we can calculate the tensor-to-scalar ratio
r =
Pt
Ps
≃ 8M2pl
(
V ′
V
)2
. (1.51)
The tensor-to-scalar ratio r is related to the slow-roll parameter ǫ as
r ≃ 16ǫ. (1.52)
On the other hand, since the power spectrum for scalar perturbation is normalized at the current
Hubble radius to be Ps ∼ 2 × 10−9, we can use this normalization with the Eq. (1.49) to derive
the relationship between energy scale and r as
V
1
4 = 1.06× 1016GeV
( r
0.01
) 1
4
. (1.53)
Thus, a detectable large tensor-to-scalar ratio would show that inflation happened at very high
energy scale, comparable to the Grand Unification energy scale ∼ 1016GeV.
In addition, we can use the slow-roll approximation to obtain the following relation which
characterizes the distance in field space from the end of inflation to the time when scales of CMB
fluctuation were created [63; 64]
∆φ
Mpl
&
( r
0.01
) 1
2
. (1.54)
This is called Lyth bound [63; 64]. Thus, if the tensor-to-scalar ratio is greater than ∼ 0.01,
it directly indicates a super-Planckian field evolving from φCMB to φend, which may give some
observational clues about the nature of quantum gravity. Thus, the benchmark value r ∼ 0.01 is
so important that it may rule out a wide class of high field inflation models (see discussions in
Section 1.2.2.3).
13
1. INTRODUCTION
Figure 1.3: Generic shapes of potentials for different inflation models. Left: large Field (Chaotic)
Inflation; Middle: small field inflation; Right: hybrid inflation. Figure taken from [42].
1.2.2.3 Inflation Models
For single-field inflation, a large number of models can be constructed, we can summarize their
features with the following general potential
V (φ) = Λ4f
(
φ
µ
)
, (1.55)
where Λ4 is the vacuum energy density (characterizing the energy scale of inflation), and µ is a
‘width’ which characterizes the change in magnitude of the field value ∆φ during inflation. Many
variants of single field inflation can be encoded in the function f . Figure 1.3 summarizes three
general types of simple inflation potential (see also [42]):
Large Field Inflation (Chaotic Inflation) Large field inflation corresponds to potentials
(such as the left panel of Fig. (1.3)), where the inflaton is displaced initially from the minimum
of the potentials by an amount of order the Planck mass. The potential in the left panel of Fig.
(1.3) satisfies V ′′(φ) > 0 and −ǫ < δ ≤ ǫ, and specific potentials that satisfy the conditions
include V (φ) = Λ4 exp(φ/µ) and V (φ) = Λ4(φ/µ)p(p > 1). In chaotic inflation, the Universe is
assumed to be generated at the quantum gravitational state with energy density comparable to
Planck density [42], i.e. V (φ) ∼ M4pl and ∆φ ∼ O(1)Mpl. The simplest type of chaotic inflation
is a free scalar field with quadratic potential V (φ) = m2φ2/2, where m is the mass of the scalar
field. However, from the constraints on the amplitude of the CMB power spectrum, the mass of
this scalar field should be of order m ≃ 10−6Mpl. If the Universe was driven by a chaotic inflation
potential at an early time, the present observable Universe is only a portion of the entire Universe.
In addition, since the change in magnitude of the field is order of Planck mass Mpl, it may be
possible to probe Planckian physics in this model.
Small Field Inflation The small field inflation models correspond to the type of potentials that
arise from the spontaneous symmetry breaking [67; 68] and from the pseudo Nambu-Goldstone
14
modes [69]. The field starts from an unstable equilibrium state and rolls, initially very slowly,
down the potential to a stable minimum [42; 67; 68; 69]. Thus, the potential is characterized
as V ′′(φ) < 0 and δ < −ǫ as shown in the middle panel of Fig. 1.3. The generic potential is
V (φ) = Λ4[1− (φ/µ)p](p > 1) which can be viewed as the lowest order expansion of any potential
around the origin.
Hybrid Inflation The hybrid inflation model has been proposed in the light of supersymmetry
[70] and supergravity [71]. In the hybrid inflation model, the scalar field responsible for inflation
evolves toward a minimum with nonzero vacuum energy, and ends as the result of the instability
of a second field (Right panel of Fig. 1.3). The typical features for the hybrid inflation models
are V ′′(φ) > 0 and 0 < ǫ < δ.
All of the above models are examples of slow-roll inflation, and can be distinguished by their
predictions of the spectral index ns and the tensor-to-scalar ratio r.
Figure 1.4: Left: Expected 95% uncertainty on the inflationary parameters ns and r from WMAP
and Planck in 1990s. Each model has a unique prediction for ns and r. Figure taken from [66].
Right: Current constraints on r − ns parameter space from WMAP 7-year results. Figure taken
from [72].
Figure 1.4 shows the predictions of different inflation models and compares them with the
observational constraints. In the left panel of figure, the whole parameter space r-ns is divided
by three lines. The dark blue line corresponds to the model with δ = ǫ, i.e. the inflaton potential
is exponential form V (φ) ∼ eφ, so the region above the blue line corresponds to the models with
the potentials V ′′/V > (V ′/V )2, i.e. δ > ǫ, which corresponds to the hybrid type of potentials.
The purple line corresponds to the model with δ = −ǫ, i.e. V ′′ = 0, which leads to the potential
V (φ) ∼ φ. So the region within the two boundary lines corresponds to the model with 0 < V ′′/V <
(V ′/V )2, i.e. −ǫ < δ < ǫ, which corresponds to the large field inflation potentials. Finally, the
light blue line corresponds to the model with r = ǫ = 0, and the region above the light blue
15
1. INTRODUCTION
line and below the purple line corresponds to the models with η < δ < −ǫ, i.e. V ′′ < 0, which
corresponds to the small field inflation potentials. Therefore, for each single field inflation model,
we can compute the prediction of the model on the r-ns plane and compare it with observational
constraints. The left panel shows the forecasted error of Planck and WMAP data in 1990s, while
the right panel shows the WAMP 7-year constraints, which nearly rules out λφ4 inflation model.
However, these slow-roll inflation models are purely schematic. Many important theoretical
questions remain unsolved. For example, what is the nature of the inflaton? How is the shape of
the potential related to the fundamental physics, etc? At present, the vast majority of inflationary
models are phenomenological with tenuous links to fundamental physics.
Other inflationary models In addition to the scalar field models described above, theorists
have attempted to embed the inflationary scenario within string theory. For instance, brane-
inflation models have been developed in the context of Type IIB superstring theory. The D3-
branes in a warped geometry break supersymmetry, then lift the AdS vacuum to a metastable de
Sitter vacuum with a sufficiently long lifetime. The anti-brane is fixed at the bottom of a warped
throat, while the brane is mobile and experiences a small attractive force towards the anti-brane.
Inflation ends when the brane and the anti-brane collide and annihilate, initiating the hot Big
Bang epoch. The annihilation of the brane and anti-brane causes the Universe settle down to the
string vacuum state that describes our universe. During the process of the brane collision, cosmic
strings may be copiously produced [47; 48].
The key difference between the scalar field description of inflation and models of brane inflation
is that the potential of the latter can, in principle, be derived from speculative but nevertheless
microphysical principles. In addition, in this type of model, the inflaton has a fundamental
geometric interpretation. For reviews of recent progress on brane inflation, see [49; 50].
1.2.3 The properties of the observable Fluctuations
At present, observations suggest that the fluctuation have the following properties: Gaussian
statistics, linear on large scales, adiabatic, nearly scale-invariant power spectrum with small tensor
modes. We shall illustrate how these properties are related to the predictions of inflation.
1.2.3.1 Gaussian Fluctuations
In the linear perturbation theory [44; 62], the perturbation is considered to be small so that the
different Fourier modes of the curvature perturbation ζk evolve independently. Statistically inde-
pendent Fourier modes of ζk imply that the curvature perturbation in real space ζ(x) satisfy Gaus-
sian statistics. However, higher order perturbations will induce a small degree of non-Gaussianity.
16
The bispectruum of curvature perturbation ζ is
〈ζk1ζk2ζk3〉 = (2π)3δD(k1 + k2 + k3)Bζ(k1 + k2 + k3). (1.56)
There are various shapes of the primordial bispectrum. Physically motivated models for producing
non-Gaussian perturbations often produce signals that peak at particular configurations of triangle
defined by the three wavenumbers in (1.56). The three often-discussed types of non-Gaussianity
shapes are
Local form: This type of non-Gaussianity can arise from a curvature perturbation of the form
ζ = ζL + f
local
NL ζ
2
L (ζL is the linear Gaussian curvature perturbation), which is expanded in a local
region of space-time. This type of bispectrum peaks at the squeeze limit k3 ≪ k2 ≈ k1. WMAP
7-year best estimate for the local form is f localNL = 32± 21 (68% CL).
Equilateral form: The amplitude of this form of bispectrum peaks at equilateral triangle shapes
(k1 = k2 = k3). The fNL is constrained by WMAP 7-year data as f
equil
NL = 26± 140 (68% CL).
Folded shape: The signal of this form of bispectrum peaks at folded a shape k1 ≈ 2k2 ≈ 2k3.
In the future, the Planck satellite is expected to reduce the uncertainty of the non-Gaussianity
parameter by a factor of four [27; 73]. Note that although the standard single field slow-roll
inflation model produces very small non-Gaussianity, there are various other physical mechanisms
which can generate large non-Gaussianity, such as double-inflation models [74] and topological
defeats [75]. Proposing and testing various types of non-Gaussianity mechanisms is one of the
important frontiers of cosmology.
1.2.3.2 Linear perturbation
Since inflation causes the Universe to expand exponentially (Fig. 1.2), the comoving Hubble
radius decreases during inflation while the comoving perturbation wavelength remains constant,
therefore perturbations are stretched beyond the Hubble radius, remain frozen until inflation ends
and then enters the Hubble radius at later time. Therefore, the observed anisotropy on the very
large scale of the CMB should be fairly small (O(10−5)).
In ΛCDM cosmology, the temperature anisotropy on the CMB last scattering surface is dom-
inated by the Sachs-Wolfe effect on very large scales δT/T ≈ −ζ(xls)/5, where ζ(xls) is the
comoving curvature perturbation and xls is the position vector of any point on the last scattering
surface [62]. In Section 1.3, we will review the imprints on the CMB sky produced by the linear
perturbations.
1.2.3.3 Adiabatic Initial conditions
Imagine at the starting point of structure formation, the Universe is full of a uniform distribution of
matter and radiation; the simplest way to perturb the density would be to compress (or expand)
17
1. INTRODUCTION
a region within the volume adiabatically. This would perturb the curvature of the Universe
while keeping the entropy (also the equation of state) constant. The complimentary mode is to
perturb the entropy of the Universe while keep the curvature unchanged, which is the isocurvature
perturbation.
For adiabatic mode, since the entropy per-baryon
S =
nr
nb
∼ ρ
3
4
r
ρb
(1.57)
is fixed, one requires δm = 3δr/4.
For the isocurvature mode, since the total energy density ρm + ρr is constant, δρm = −δρr at
the time that the perturbation is generated. Single-field inflation model predicts that the initial
condition of fluctuation should follow adiabatic initial conditions, while the double-field inflation
and topological defects can generate isocurvature modes.
1.2.3.4 Close to Scale-invariant
As we have seen in Eq. (1.49), the power spectrum of scalar perturbation Ps is proportional to the
Hubble parameter H and field “speed” φ˙, while these two quantities are nearly constant during
inflation. Therefore, slow-roll inflation predicts an almost scale-invariant spectrum of curvature
perturbations. The current tightest constraints on the spectral index (Eqs. (1.36) and (1.38))
from WAMP 7-year data [72] (at pivot scale k0 = 0.002 Mpc
−1) is
ns ≃ 0.967± 0.014. (1.58)
A scale-invariance spectrum ns = 1 is excluded at 2.4σ. This slight violation of scale-invariance
suggests that during inflation, there is some variation of the Hubble parameter H and inflaton
speed φ˙. The logarithmic scale-dependence of ns(k), known as the “running” of spectral index αs,
is constrained to be [72]
αs =
dns
d ln k
≃ −0.034± 0.026, (1.59)
which suggests that there is marginal evidence of the running of spectral index. However, these
constraints only apply on relatively large scales and actually say nothing about the spectrum on
small scales. Therefore, to test for scale-invariance, one needs more small scale observations for a
larger lever-arm.
1.2.3.5 Small but non-zero tensor modes
As we have already seen in Section 1.2.2.2, the tensor perturbation in the space-time metric can
also source primordial fluctuations. According to Eqs. (1.48) and (1.52), the amplitude of tensor
modes is suppressed by a factor related to slow-roll parameters; it is therefore expected that the
18
Figure 1.5: The CMB temperature power spectrum (red line), which displays the amplitude of
the CMB temperature fluctuation ∆T/T as a function of angular scale on the sky. The acoustic
peaks suggest the interplay of the gravity and radiation pressure and they are very sensitive to
cosmological parameters. The black dots are WMAP 7-year data, and blue band is the cosmic-
variance. Figure taken from [52].
tensor modes predicted by inflation are small but non-zero. Physically, the tensor modes are the
tensor perturbations of space-time in the primitive Universe, where the stretching and squeezing
of space-time produces gravitational waves, propagating at the speed of light. In Chapter 4, we
shall see how the current and future CMB polarization data can be used to constrain different
models of the early Universe.
1.3 The Cosmic Microwave Background Radiation (CMB)
1.3.1 CMB temperature power spectrum
1.3.1.1 Scenario of recombination
From the measurement of the Far-InfraRed Absolute Spectrophotometer (FIRAS) on board the
Cosmic Background Explorer (COBE), the CMB was found to be accurately described as the
blackbody radiation, which is characterized by the Planck function
Iω =
~ω3
2π2c2
1
e
~ω
KBT − 1
. (1.60)
From [58; 59; 60], we know that the data points match the blackbody shape with T = 2.725K
incredibly well over the frequencies range 1 to 600 GHz. This tells us that before last scattering,
the photons were in thermal equilibrium so the Universe was in a hot and dense status. This
provides strong empirical support for the hot Big Bang model.
To understand the physics of CMB, we describe the following statistical aspects. (see [44; 46]
19
1. INTRODUCTION
for more discussions).
First, let us expand the temperature fluctuation on the sky in terms of spherical harmonics as
∆T (n) =
∑
l,m
almYlm(n), (1.61)
where (θ, φ) are the spherical coordinates, and Ylm(θ, φ) are the spherical harmonics. If the
coefficients alm describe a pure Gaussian random field which preserves the rotational invariance
and translational invariance, all of the statistical information will be contained in the angular
power spectrum
Cl =
1
2l + 1
∑
m
〈|alm|2〉. (1.62)
From the analyzes of the WMAP data, we know empirically that the temperature anisotropies
are close to Gaussian [3; 72].
Figure 1.6: Density contrast in conformal Newtonian gauge. The photon, baryon, and CDM
density perturbations are plotted in red, green and blue respectively. The upper figure is long
wave length case (k=0.01 Mpc−1), and lower figure is the short wave length case (k=1.0 Mpc−1).
Figure taken from [62].
In Fig. 1.5, we see that the power spectrum of temperature anisotropies have peaks and
20
troughs between the multipole moments l from 100 to 1500. The explanation of these peaks
is as follows: On super-Hubble-length scales, the comoving curvature perturbation is conserved
for adiabatic scalar fluctuations, and on the sub-Hubble-length scales, the CDM perturbation
grew very slowly. (This is known as the Meszaros effect, see Section 1.4.2.2). However, the
gravitationally-driven collapse of the perturbations in the tightly coupled photon-baryon fluid is
resisted by the pressure of the radiation, so the gravitational compression and collapse of the fluid
is halted by the pressure of the photons. The pressure of the radiation causes the fluid to rebound
till recombination when photons become decoupled from the baryons (Fig. 1.6). Therefore, before
decoupling, since the perturbations were larger than the Hubble radius at earlier times and formed
coherent phase oscillation, the density perturbation of baryons and radiation underwent a period
of ‘baryon acoustic oscillation’, which leave an imprint in the CMB power spectrum in the form
of acoustic peaks (Fig. 1.5).
After the photons decouple with baryons and the Universe is deeply into the matter dominated
era, the photon perturbations decay rapidly as a result of free-streaming. In contrast, as the Uni-
verse entered the matter dominated era, the CDM perturbations grew in amplitude proportional
to the scale factor a. The baryons fell into the potential well of CDM perturbations, so the baryon
perturbation quickly tracks the perturbation of CDM. The baryons and CDM then grew together
at late times (Fig. 1.6).
Figure 1.7: Contribution of different terms in Eq. (1.81) to the temperature anisotropies from
adiabatic initial conditions: δγ/4 + ψ (Sachs-Wolfe effect, magenta line); vb (Doppler effect, blue
line); Integration term (Integrated Sachs-Wolfe effect, green line); total effect (black line). Figure
taken from [62].
21
1. INTRODUCTION
1.3.1.2 Mathematical Treatment
In this section we calculate the CMB temperature fluctuation caused by scalar perturbation (For
more detail descriptions, see [44; 62]). In Eq. (1.32), we retain the scalar potential φ and ψ,
and set the vector perturbation Bi and tensor perturbation Eij , to zero. We first construct an
orthogonal frame of 4−vector (E0)µ and (Ei)µ in the perturbed metric (Eq. (1.32)). We take the
timelike vector (E0)
µ to be the 4−velocity uµ of an observer at rest with respect to the coordinate
system, so it should be parallel to the δµ0 . To first order, it is
1
(E0)
µ = a−1(1− ψ)δµ0 , (1.63)
since to first order gµν(E
µ
0 )(E
ν
0 ) = 1, the spacelike orthogonal basis is
(Ei)
µ = a−1(1 + φ)δµi . (1.64)
[We can easily confirm that gµν(Ei)
µ(Ej)
ν = −δij , and gµν(E0)µ(Ej)ν = 0.]
Based on the above orthogonal basis, we parameterize the photon 4−momentum in terms of
the energy E seen by an observer in the observer’s rest frame, and the propagation direction ei
seen by the same observer on the (Ei)
µ basis of spacial coordinate [62]. Note that δije
iej = 1. We
have the photon 4−momentum
pµ = E
[
(E0)
µ + ei(Ei)
µ
]
= Ea−1
[
(1− ψ)δµ0 + ei(1 + φ)δµi
]
. (1.65)
Now we introduce the comoving energy ǫ ≡ Ea which is constant in the background evolution
[62].
Photon moves along geodesics of the perturbed metric so
dpµ
dλ
+ Γµνρp
νpρ = 0, (1.66)
where λ is the affine parameter of the geodesic so that pµ = dxµ/dλ. Using the parameterization
of Eq. (1.65), we have the following equations in the first order
dη
dλ
=
ǫ
a2
(1− ψ),
dxi
dλ
=
ǫ
a2
(1 + φ)ei,
⇒ dx
i
dη
= (1 + φ+ ψ)ei. (1.67)
1The perturbed space-time metric is written as ds2 = a2(η)[(1 + 2ψ)dη2 − (1− 2φ)δijdxidxj ].
22
Therefore the geodesic equation to first order is
(1− ψ) ǫ
a2
dpµ
dη
+ Γµνρp
νpρ = 0. (1.68)
Now we can figure out the 0 and i component of the geodesic equation (1.68). By using an equation
of the time-derivative of the gravitational potential
dψ
dη
= ψ˙ + ei∂iψ, (1.69)
one can reach the simplified geodesic equations
1
ǫ
dǫ
dη
= −dψ
dη
+ (ψ˙ + φ˙), (1.70)
dei
dη
= −(δij − eiej)∂j(φ+ ψ), (1.71)
where the overdot means ∂η.
Equation (1.70) describes how the comoving energy ǫ evolves along the photon path in the
presence of metric perturbations. In absence of perturbation, ǫ is constant, but this is modified if
there is either (a) metric perturbations along the photon path, or (b) time-evolving gravitational
potentials. The latter case can be important at the later dark energy dominant era and at the
matter-radiation equality epoch. This leads to the Integrated Sach-Wolfe effect that we have seen
in Section 1.3.1.1 and Fig. 1.7. Equation (1.71) describes how the photon direction is deflected if
the gradient of gravitational potential is perpendicular to the line of sight.
Integrating Eq. (1.70) from emission point E at the last scattering surface to the observed
point 0, we get
ǫ0
ǫE
= 1 + (ψE − ψ0) +
∫ 0
E
(φ˙+ ψ˙)dη. (1.72)
Therefore, the comoving photon energy changes due to either (a) gravitational redshifting, or (b)
time-varying gravitational potentials. For the former case, if the gravitational potential at the
emission point is deeper than observing point ψE < ψ0, there is a net loss of energy, and vice versa.
For the latter, for a decaying gravitational potential (ψ˙+ φ˙ < 0), a photon will be blueshifted since
it gains more energy when falling in a potential and loses less energy when climbing out [44; 62].
Therefore, in the CMB rest frame, the ratio of the temperature of the CMB received by an
observer with zero redshift T (e) to the mean temperature of the CMB at last scattering surface
T ∗ is just the ratio of the proper energy of photon received by an observer at z = 0 to that emitted
at the last scattering surface (in the CMB rest frame):
T (e)
T ∗
∣∣∣∣
CMBframe
=
ǫ0
ǫE
aE
a0
23
1. INTRODUCTION
=
aE
a0
(
1 + (ψE − ψ0) +
∫ 0
E
(φ˙+ ψ˙)dη
)
. (1.73)
Now let us take into account two different effects that could change the observed temperature
anisotropy [62]. The first thing we should consider is the Doppler effect. The key quantity here
is the Lorentz-invariant quantity pµuµ (p
µ is the photon’s 4-momentum and uµ is observer’s 4-
velocity). Suppose in the CMB rest frame a photon is moving in direction n with energy E,
pµ = E(1,n), and observer is moving at velocity v, so uµ = γ(1,v) and uµ = γ(1,−v) where
γ = (1− v2)−1/2 is the Lorentz-factor. Therefore,
ECMB = p
µuµ = Eγ(1− v · n), (1.74)
and photon distribution (also Lorentz-invariant) is
f(pµ) ∝ 1
eECMB/TCMB − 1 =
1
eEγ(1+v·e)/TCMB − 1 , (1.75)
which makes the photon distribution like a blackbody but the observed temperature varies over
the sky as T (e) = TCMB/γ(1 + v · e) ≃ TCMB(1 − v · e). Since v = −vb, in the observer’s rest
frame, the CMB temperature on direction e is
T (e)
T ∗
=
aE
a0
(
1 + (ψE − ψ0) + eivib +
∫ 0
E
(φ˙+ ψ˙)dη
)
. (1.76)
The second thing is to determine how the scale factor a varies on the perturbed last scattering
surface, i.e. we need to find the relation between aE and a∗, where a∗ is the scale factor of the
unperturbed space-time at the last scattering surface [62]. Due to the perturbed space-time, the
photon energy density at the last scattering surface E is
ργ(η∗ +∆η)(1 + δγ) = ργ(η∗), (1.77)
where we have used the fact that energy density of photons is uniform over the last scattering
surface and equal to the background value. Taylor expanding the above equation one can get the
first order perturbation
∆η = − δγ
ρ˙γ/ργ
=
δργ
4H
, (1.78)
therefore we have
aE = a(η +∆η) = a(η)(1 + H∆η) = a∗(1 +
δγ
4
). (1.79)
Substituting this equation into Eq. (1.76), we have
T (e)
T ∗
=
a∗
a0
(
1 +
δγ
4
+ (ψE − ψ0) + eivib +
∫ 0
E
(φ˙+ ψ˙)dη
)
(1.80)
24
=⇒ T (e) = T 0
(
1 +
δγ
4
+ (ψE − ψ0) + eivib +
∫ 0
E
(φ˙+ ψ˙)dη
)
,
where T 0 = T ∗a∗/a0 is the background CMB temperature at the reception point. Thus, the
observed temperature fluctuation is [62]
∆T (n)
T¯0
=
T (n)− T¯0
T¯0
=
1
4
δr + (ψE − ψ0) + eivib +
∫ 0
E
(φ˙+ ψ˙)dη, (1.81)
where T (n) is the observed temperature in direction n, T¯0 is the CMB background temperature
at the point of reception. The four contributions in Eq. (1.81) can be summarized as follows:
δr/4 is the intrinsic temperature fluctuation on the last scattering surface, ψE is the gravitational
redshift when the photons climb out of a potential well. The two terms together δr/4 + ψE are
called the ‘Sachs-Wolfe’ effect. eivib is the Doppler effect due to the relative motion of the observer
with respect to CMB rest frame. Finally, the integration term is the ‘Integrated Sachs-Wolfe’
effect which describes the gravitational redshift arising from the time-varying potential.
It is important to recognize that the perfect fluid approximation of photons and matter is
idealized and that in reality, the two components are not perfectly coupled. This imperfect
coupling leads to the damping of the temperature fluctuations on small scales (often called Silk
damping [65]). In fact, on scales smaller than the damping scale (λdamp = c
√
trects, trec and ts are
recombination time and photon scattering time scale), the photons diffuse and erase small scale
perturbations so that the temperature power spectrum at high multipoles (l > 2500) decline as
l−4 (see Figs. 1.5 and 1.7).
The CMB power spectrum therefore encodes much information on early Universe physics, the
matter content of the Universe, as well as geometrical information that relates physical scales to
angular scales on the sky.
1.3.2 CMB polarization
CMB polarization has become a very powerful tool to probe the physics of the early Universe. Since
the CMB temperature anisotropies are sourced by the primordial fluctuations, the anisotropy of
Thomson scattering can generate quadrupole anisotropy of the CMB photons [56; 76]. Therefore,
the polarization signal and the cross-correlation of the polarization with temperature provide
an effective check of the standard cosmology paradigm, and supply a powerful tool to constrain
cosmological parameters. Furthermore, since the polarization pattern on the sky encodes much
information on the primordial perturbations, it may strongly constrain models of inflation.
The relationship between the temperature anisotropies and polarization pattern is determined
by the properties of Thomson scattering [56; 76]. If the incoming radiation field is isotropic, then
the outgoing radiation remains unpolarized since the orthogonal polarization pattern cancels out.
However, if the incoming radiation field has a quadrupole component, such as shown in Fig. 1.8,
25
1. INTRODUCTION
Quadrupole
Anisotropy
Thomson 
Scattering
e–
Linear 
Polarization
ε'
ε'
ε
Figure 1.8: Thomson scattering of radiation with a quadrupole anisotropy generates linear polar-
ization. Red colors (thick lines) represent hot radiation, and blue colors (thin lines) cold radiation.
Figure taken from [76].
the outgoing radiation field will gain a net polarization pattern due to the anisotropic Thomson
scattering. Thus, a linear polarization pattern of the radiation field is generated when the photons
decouple from the baryons just before the recombination. The polarization anisotropies correlate
with the temperature anisotropies since both of them are related to the density fluctuation.
The polarized radiation field can be described by a 2 × 2 intensity matrix Iij(n), where n
denotes the direction on the sky, and Iij(n) is based on the two orthogonal basis e1 and e2 that
are perpendicular to n. Linear polarization is related to two Stokes parameter Q = 1
4
(I11 − I22)
and U = 1
2
I12, whereas the temperature anisotropy is T =
1
4
(I11 + I22). Thus, the polarization
magnitude and angle are P =
√
Q2 + U2 and α = 1
2
tan−1(U/Q). The temperature anisotropy is a
scalar field which is invariant under the rotation perpendicular to n, so we expand the temperature
in terms of spin-0 spherical harmonics as Eq. (1.61).
However, the Stokes parameters Q and U transform under a rotation ψ as (Q ± iU)(n) →
e∓2iψ(Q± iU)(n). Thus, (Q± iU)(n) requires an expansion with spin-2 spherical harmonics
(Q+ iU)(n) =
∑
lm
a
(±2)
lm ±2Ylm(n). (1.82)
We can introduce the linear combinations of the two coefficients a
(2)
lm and a
(−2)
lm as
aElm = −
1
2
(a
(2)
lm + a
(−2)
lm ), a
B
lm = −
1
2i
(a
(2)
lm − a(−2)lm ), (1.83)
then the polarization field is expressed in terms of two scalar fields E and B instead of spin-2
26
quantity Q and U
E(n) =
∑
lm
[
(l + 2)!
(l − 2)!
] 1
2
aElmYlm(n), B(n) =
∑
lm
[
(l + 2)!
(l − 2)!
] 1
2
aBlmYlm(n). (1.84)
Figure 1.9: The power spectrum of the cross-correlation between temperature and E-mode po-
larization anisotropies. The anti-correlation at 100 < l < 200 and the positive correlation at
200 < l < 400 are due to the phase coherence of fluctuation generated during inflation [53]. The
data is WMAP 7-year result [52].
The E and B-modes completely describe the polarization field. The E mode is the curl-free
mode with polarization vector radial for a cold spot and tangential for a hot spot. The B-mode
is the divergence-free mode with polarization vector vortical around a given point on the sky.
Since both the temperature fluctuation and the polarization pattern are sourced by the pri-
mordial fluctuations, it is possible to cross-correlate these different modes. Due to the reason of
symmetry, there are only four non-zero cross-correlation: TT , TE, EE, BB.1
The angular power spectrum is defined for temperature and polarization as
CX,Yl =
1
2l + 1
∑
lm
〈
aXlma
Y
lm
〉
, X, Y = T,E,B. (1.85)
Figure 1.9 shows the WMAP 7-year measurement of the TE correlation [52]. We can see that
the theoretical prediction for the phase coherence generated during inflation is consistent with the
observational data to high accuracy, providing strong support for the inflationary model. As we
have seen in Fig. 1.5, the standard ΛCDM prediction of the temperature power spectrum fits the
data at all angular scales very well. However, the other polarization power spectra have not been
measured as accurately as the TT power spectrum. So far, the EE spectrum has been measured,
but only upper limits have been placed on the BB spectrum (Fig. 1.10 and 1.11) [3; 51; 52; 56].
1This assumes the parity-conservative processes in the early Universe. Since both T and E has parity factor
(−1)l under rotation, while B has parity (−1)l+1 under rotation, the TB and EB should vanish for symmetry
reasons. See more discussion in [3].
27
1. INTRODUCTION
Figure 1.10: WMAP 5-year measurements of the temperature and polarization power spectrum
to constrain the tensor-to scalar ratio. Left: Contours show 68% and 95% C.L of the parameter
space. The gray region is the constraint from the low-l multipole data (TE, EE, BB at l ≤ 23)
only, and the red region is from the low-l data plus the high-l TE data at 24 ≤ l ≤ 450. Finally,
the blue region is the constraint from the low-l polarization data, the high-l TE correlation data,
and the low-l temperature data at l ≤ 23. Right: the gray curves show (r, τ)=(10,0.050), the
red curves (r, τ)=(1.2,0.075) and the blue curves (r, τ)=(0.2,0.080), where τ is the optical depth
τ =
∫ η0
0
neσTadη. Figure taken from [3].
The dependence on cosmological parameters of these various power spectrum differs. The
current measurement of TT and TE power spectrums provides the most powerful constraints on
cosmological parameters. In the near future, Planck will measure the EE and BB polarization
spectrum more precisely than WMAP, which will help to break parameter degeneracies [27].
Figure 1.11: The marginalised likelihood of tensor-to-scalar ratio r in WMAP 7-year data and
5-year data, from the polarization data (BB, EE and TE) alone. All the other cosmological
parameters, including the optical depth, are fixed at the 5-year best-fit ΛCDM model [3]. The
vertical axis −2 ln(L/Lmax) is the standard χ2 under the Gaussian approximation. The dashed
line −2 ln(L/Lmax) = 4 corresponds to the 2σ CL. The solid, dashed and dot-dashed lines show
the likelihood as a function of r from the BB-only, BB+EE, and BB+EE+TE data. (Left) The
7-year polarization data. WMAP 7-year data shows r < 2.1, 1.6, and 0.93 (95.4% CL) from the
BB-only, BB+EE, and BB+EE+TE data, respectively. (Right) The 5-year polarization data.
r < 4.7, 2.7, and 1.6 (95.4% CL). Figure taken from [72].
28
Parameters WMAP7 Only WMAP7+BAO+H0
ns 0.967± 0.014 0.968± 0.012
ns 0.982
+0.020
−0.019 0.973± 0.014
r < 0.36 (95% CL) < 0.24 (95% CL)
ns 1.027
+0.050
−0.051 1.008± 0.042
αs −0.034± 0.026 −0.022± 0.020
ns 1.076± 0.065 1.070± 0.060
r < 0.49 (95% CL) < 0.49 (95% CL)
αs −0.048± 0.029 −0.042± 0.024
Table 1.2: WMAP 7-year constraints on ns, r and αs [72]. All of the other cosmological parameters
in ΛCDM model are marginalized.
In addition to the temperature and E-mode power spectrum, the B-mode power spectrum
can also provide a very powerful tool to constrain the inflation model and probe the physics
of the early Universe. Scalar perturbation produces only E-mode polarization pattern, while
tensor perturbation produces both E and B-modes. Therefore, probing B-mode polarization
is an important test of primordial gravitational waves. The current measurement of B-mode
polarization can provide only upper limits (Figs. 1.10 and 1.11), but the next generation of CMB
polarization measurements should improve these limits substantially [27; 56].
1.3.3 Prospects of constraining inflation
Current cosmological observations of the temperature power spectrum, combined with the other
astrophysical measurements, such as the distant Type Ia supernovae and the angular diameter dis-
tance data from baryon acoustic oscillations (BAO) can place precise constraints on cosmological
parameters. Here, we give a brief review of the current tightest constraints.
Komatsu et al. [72] used the WMAP 7-year data, combined with the angular diameter distance
data of BAO at z = 0.2 and 0.35 [78] and a Gaussian prior on the present-day Hubble constant
H0 = 74.2±3.6 km/s/Mpc (68% CL, [77]) to constrain cosmological parameters. Here, we present
their 1σ constraints on spectral index ns, its running αs (Eq. (1.37)) and tensor-to-scalar ratio r
in Table 1.2, since these parameters are related to the inflation models.
Several points should be noticed concerning their results [72]:
• Assuming zero running of the spectral index (αs = 0), the best-fits of scalar spectral index
are around 0.967 (i.e. a red power spectrum). If no tensors (r=0) are assumed, the scale-
invariant Harrison-Zel’dovich power spectrum, ns = 1, is excluded from the peak likelihood
at the 99.5% confidence level.
• There is no strong evidence for the running index αs, since for any combination of data sets,
αs is always very close to zero.
29
1. INTRODUCTION
Figure 1.12: Comparison of matter power spectrum P (k), inferred from the CMB temperature
anisotropy to constraints from other observations, including the observed galaxy distributions.
The consistency of independent results shows that the density perturbations are the seeds for
structure growth. Figure taken from [5].
• The current tightest constraint on r (r < 0.24) comes almost exclusively from the combined
measurements of TT+TE+EE+BB power spectrum (Fig. 1.11 and Table 1.2) and BAO
measurement and H0 prior. The direct B-mode polarization limits make a negligible con-
tribution to the constraints on r. WMAP data is able to rule out λφ4 inflation model [72],
while a large class of models is still consistent with WMAP 7-year data.
• Since r is degenerated with ns and τ(optical depth, defined as τ =
∫ η0
0
neσTadη), a better
constraint on these two quantities will improve the constraints on r.
By using the WMAP parameter constraints, we can compute the matter power spectrum
according to the WMAP type parameters. This is found to be in good agreement with power
spectrum measurements from galaxy surveys (Fig. 1.12). This agreement suggests that the
primordial density perturbations were the seeds of structure formation in the late Universe.
The future for CMB observations is promising, since many high precision observations are
planned over the next few years. The Planck satellite of European Space Agency [27] will provide
very much improved measurements on the E- and B-modes polarization, and it may be possible
to detect a tensor-to-scalar ratio of about 0.05 magnitude [54; 55]. The Planck satellite will be
supplemented by many ground-based measurements of the small scale fluctuations and polarization
data (such as EBEX [32] and Spider [33]). In addition, the CMBPol satellite proposes to improve
the sensitivity of B-modes by almost two orders of magnitude [46; 56].
30
1.4 The Growth of Structure
1.4.1 Mass Fluctuations and Matter Power Spectrum
We define the density contrast of the fluctuations which source the structure formation
δ(x) =
ρ(x)− ρ
ρ
, (1.86)
where ρ is the mean density and ρ(x) is the density at position x. Inflation predicts Gaussian
fluctuations whose power spectrum is statistically homogeneous and isotropic. This means that
each of the Fourier modes of the fluctuation δk has no correlation with other modes. Thus, we
can further understand why it is convenient to work in Fourier space
δk =
∫
δ(x)e−ik·xd3x, (1.87)
rather than in real space. Statistical isotropy implies
〈δkδk′〉 = (2π)3P (k)δ3(k− k′), (1.88)
i.e. the power spectrum is described by a scalar function P (k).
Now we can calculate the mass fluctuation within some volume to quantify the amplitude of
the fluctuations1 [79]
〈(
δρ
ρ
)2〉
= 〈δ2(x)〉 =
∫
d3x
V
δ2(x)
=
∫
d3x
V
∫
d3k
(2π)3
d3q
(2π)3
δkδ
∗
qe
i(k−q)·x
≃ V −1
∫
d3k
(2π)3
|δk|2 = V −1
∫
k3|δk|2
(2π2)
dk
k
,
where the second equality is valid for sufficiently large volume 2 V . The integrand is the dimen-
sionless power spectrum
∆2(k) =
V −1
2π2
k3|δk|2, (1.89)
which is the power spectrum popular in the early Universe community. In the large scale structure
community, the k−3 factor is often added so that it has the dimension k−3 [44], i.e. P (k) =
1In the above calculation, we have used the Fourier transformation δ(x) =
∫
d3kδke
ik·x/(2pi)3, and delta
function δ3(x) =
∫
eik·xd3x/(2pi)3.
2Here the second equality actually uses the “Ergodic Hypothesis”: the ensemble average of many different
realization can be expressed as an average over a sufficient large volume of a single realization.
31
1. INTRODUCTION
V −1|δk|2. In the following, we shall use the dimensional power spectrum P (k).
To quantify the magnitude of the fluctuation, we calculate the fluctuations on some particular
scale R, i.e. we filter the density fluctuation with some window function. In real space, the
simplest window function, a top-hat window function is defined by
W (r) = 1 if r ≤ R,
= 0 if r > R. (1.90)
The Fourier transform of this function is
Wk(R) = 4πR
3
[
sin kR
(kR)3
− cos kR
(kR)2
]
. (1.91)
Now the total mass in a volume R with window function W (|y|) is
MR =
∫
d3yρbW (|y|) = Vwρb, (1.92)
and the mass fluctuation within a volume centered at some point x is
δMR(x) = ρb
∫
d3yδ(x+ y)W (|y|). (1.93)
Therefore, the Fourier transformation of mass contrast is
(
δM
M
)
R
=
∫
d3k
(2π)3
(
δkWk
Vw
)
eik·x, (1.94)
from which one can reach the mean square fluctuation as
σ2(M) ≡ σ2(R) =
〈(
δM
M
)2
R
〉
=
∫
dk
2π2
k2P (k)
(
3j1(kR)
kR
)2
, (1.95)
which characterize the strength of fluctuations on different scales R.
We will now turn to the matter power spectrum. In Sections 1.2 and 1.3, we have shown
that the primordial power spectrum from inflation is nearly scale-invariant (Eq. (1.39)), i.e.
the strength of primordial perturbation is approximately equal on different scales. However, the
cosmic evolution acts like a filter which can suppress and enhance the strength of the perturbation
spectrum on different scales by a variety of processes: growth under self-gravitation, Silk damping,
free-streaming etc. The “filtering” causes the spectrum in the late times to have more power on
some scales, and less power on other scales (We shall summarize the main effect in the next
32
10-4 0.001 0.01 0.1 1 10
0.001
0.1
10
1000
105
k @hMpcD
PH
kL
@M
pc
h
D3
z=50
z=10
z=1
z=0
10-4 0.001 0.01 0.1 1 10
10
1000
105
107
k @hMpcD
TH
kL
z=50
z=10
z=1
z=0
1 2 5 10 20 50 100 200
1.
0.5
0.1
0.05
0.01
0.005
0.001
R @MpchD
Σ
HR
L
z=50
z=10
z=1
z=0
Figure 1.13: Power spectrum (upper-left), transfer function (upper-right), and root-mean-square
of mass fluctuation in linear perturbation theory at different redshift. Here we adopt ΛCDM
model with WMAP 5-year parameters Ωb = 0.0449, Ωc = 0.222,ΩΛ = 0.734,h = 0.71,σ8 = 0.801
and ns = 0.963 [3].
subsection). The overall filter can be encapsulated in the transfer function T (k, z) 1 , so that the
power spectrum at redshift z can be written as
P (k, z) = Ps(k)T
2(k, z), (1.96)
where Ps(k) is the power spectrum of primordial fluctuations. To accurately calculate the transfer
function and thus matter power spectrum, one needs to solve a set of Boltzmann hierarchy equa-
tions to follow the evolution of various components of the Universe. People have developed several
Boltzmann codes to solve this problem, such as CMBFAST [24] and CAMB [25]. In practice, it
has been figured out semi-analytic formulae that fit the numerical results for the transfer function
quite precisely [81].
In Fig.1.13, we plot the matter power spectrum (upper-left), transfer function (upper-right)
and variance of mass fluctuation (lower panel) from linear perturbation theory at different redshift
z. The cosmic evolution as a filtering effect can be seen from the upper-right panel of Fig. 1.13.
Since the growth function D(t) ∝ a for matter-dominated Universe, the difference between power
spectra at different redshifts is just a time-evolution factor which enhances or suppresses power
on all scales. In the lower panel of Fig. 1.13, we plot the root-mean-square of mass fluctuation
1Only when all of the physical process responsible for filtering has finished, the transfer function can be separate
as T (k, t) = T (k)D(t), while in general this is not true.
33
1. INTRODUCTION
as a function of scale R [Mpc/h] through Eq. (1.95). The density perturbations detach from the
background when they reach the overdensity of unity, the perturbations become non-linear and
begin to form dark matter halos.
1.4.2 Linear growth of structure
Now let us briefly summarize the evolution of adiabatic perturbations of a perfect fluid in the
expanding Universe, i.e. Jeans theory. We will then apply this theory to the spatially flat, CDM
dominant Universe. We shall first ignore the effect of cosmological constant Λ, since it makes a
negligible contribution to the energy density before z ∼ 2, then describe the growth of structure
for models with nonzero Λ in Section 1.4.2.2.
1.4.2.1 Jeans equation
Linear perturbation theory gives the following Jeans equation from the combination of continuity,
Euler and Poisson equations for a single perfect fluid in an expanding Universe [62; 79]
δ¨ + 2
a˙
a
δ˙ +
(
c2sk
2
a2
− 4πGρ
)
δ = 0, (1.97)
where cs is the sound speed of the perfect fluid, ρ is the background density, the a is the cosmic
scale factor, and the dot refers to the derivative with respect to cosmic time. This is the equation
for a damped oscillator in an expanding Universe provided that c2sk
2/a2 > 4πGρ, since in this
situation, the pressure support leads to acoustic oscillation in the fluid system. In contrast, if
c2sk
2/a2 < 4πGρ the system is unstable and gravitational collapse will occur. Therefore, the
critical wavelength, the Jeans wavelength is defined as
λJ ≡ cs
√
π
Gρ
, (1.98)
so that if the proper wavelength 2πa/k exceeds λJ , gravitational collapse will take place. It is also
common to express the Jeans criterion in terms of Jeans mass as
MJ =
4π
3
ρ
(
λJ
2
)3
. (1.99)
In an expanding Universe, the Jeans wavelength changes with time. The evolution of a given
perturbation with wavelength λ can be studied by solving the full relativistic perturbation equa-
tions [44; 79]. In the following discussions, we will discuss the evolution of the sub-Hubble radius
modes (λ < H−1) in various cases. For super-Hubble-length modes, the curvature perturbation
remains constant until it crosses the Hubble radius.
34
1.4.2.2 Sub-Hubble-length modes
CDM perturbation
Radiation dominant era— Meszaros effect The Meszaros effect describes how the CDM
grows only logarithmically during the radiation dominant era on scales smaller than the Hubble
radius [62]. Since the radiation is assumed to dominate over CDM, and the pressure of CDM can
be neglected, the Jeans equation (1.97) becomes
δ¨c + 2
a˙
a
δ˙c − 4πG
∑
j=r,c
ρjδj = 0, (1.100)
in which the summation includes both the radiation and matter. However, the Jeans length for
radiation in the baryon-radiation fluid (cs = 1/
√
3) during the radiation dominant era is the order
of the Hubble radius (Eq. (1.98)), so perturbations with scales much smaller than this will oscillate
as sound waves so that the average contribution on δc is negligible. Therefore, in the summation
of Eq. (1.100), we can neglect the radiation term so the equation becomes
δ¨c +
1
t
δ˙c − 4πGρcδc = 0, (1.101)
where we use a ∝ t1/2 and H = 1/(2t). Since this is radiation domination (ργ ≫ ρc),
δ¨c ∼ 1
t
δ˙c ∼ H2δc ≫ 4πGρcδc. (1.102)
Therefore, we can neglect the last term in Eq. (1.101), and the final equation has the solution
δc ∝ const, and ln(t), (1.103)
i.e. the rapid expansion due to the unclustering radiation leads to logarithmic growth rate of δc.
Matter dominant era Perturbations within the Hubble-length can grow more rapidly once
the Universe becomes matter dominated. This can also be easily derived from Jeans equation
(1.97) by using the fact a ∝ t2/3, H = 2/(3t) and 4πGρ = 2/(3t2)
δ¨c +
4
3t
δ˙c − 2
3t2
δc = 0. (1.104)
By substituting a power law form δc ∝ tp, one can get two independent solutions δc ∝ t−1(decaying
mode) and δc ∝ t2/3 ∝ a. The growing mode of CDM perturbation grows as a and t2/3 in an
expanding Universe, which is fundamentally different from the non-expanding Universe, since in
35
1. INTRODUCTION
Sub-Hubble-length Radiation Matter Λ
t < teq teq < t < tdec tdec < t < tΛ t > tΛ
δb oscillation oscillation a constant
MJb a
3 a3/2 a−3/2 constant
δc const a constant
Table 1.3: The summary of the evolution of sub-Hubble-length density contrast and Jeans mass.
the latter case, Eq. (1.97) has exponential solution (a˙ = 0)
δ(t) ∝ e±t/τ , τ = 1/
√
4πGρ0. (1.105)
Baryonic perturbation
Radiation dominant era The baryons and photons remain tightly coupled through Thom-
son scattering until the epoch of decoupling zdec ≃ 1100. The sound speed for the mixed baryon-
photon fluid is [44]
cs ≃ c√
3
(
1 +
3
4
ρb
ργ
)− 1
2
. (1.106)
Therefore, during the radiation domination, ργ ≫ ρb so cs = c/
√
3, i.e. the tightly coupled
photon-baryon plasma supports the propagation of sound waves.
Now let us compute the evolution of Jeans mass. Since Jeans wavelength is λJ = (c/
√
3)
√
π/Gργ ∝
a2, Jeans mass becomes [79]
MJb = 3.2× 1014MJ
(
Ωb
Ωm
)(
Ωmh
2
)−2( a
aeq
)3
. (1.107)
Such a high mass threshold prohibits the growth of structure.
Matter dominant era Let us calculate how the Jeans wavelength and Jeans mass changed
during this period. Since the decoupling of photon-baryon plasma can substantially change the
sound speed and Jeans mass, we separate this period into two stages: (A) teq < a < tdec (before
decoupling); (B) tdec < t < tΛ (after decoupling).
• teq < a < tdec. Although the dark matter has already started to collapse, the pressure
from the photons prevents the collapse of the baryons, and causes the perturbation in the
photon-baryon fluid to oscillate as sound waves. Therefore, the sound speed is close to that
36
in radiation cs ≃ c/
√
3, so the Jeans wavelength becomes λJ = cs
√
π/Gρrad ∝ a3/2 and
the Jeans mass MJ ∝ a3/2 is the order of 1014MJ. Therefore, the solution of the density
contrast δb from non-relativistic perturbation theory shows oscillatory behavior over this
period of time, which can be seen in the lower panel of Fig. 1.6.
• tdec < t < tΛ. This is the period when the photons decouple from the baryon, so the
sound speed in the baryons is no longer the sound speed of relativistic fluid, but rather
the velocity dispersion in the gaseous mixture of hydrogen of helium, which drops as the
Universe expands. Thus, the baryonic Jeans mass drops dramatically after recombination
and the baryonic perturbation grows and catches up the CDM perturbation.
Since the perturbations in dark matter during this stage grow as the scale factor a, the
baryons decouple from photon-baryon plasma and fall into the dark matter potential wells
quickly, so the growth of the baryon perturbation also scales as factor a. To see this, one
can apply Jeans equation (1.97) to both baryon and dark matter as [62]
δ¨b +
4
3t
δ˙b = 4πG(ρbδb + ρcδc), (1.108)
δ¨c +
4
3t
δ˙c = 4πG(ρbδb + ρcδc). (1.109)
We can decouple the equation by introducing the “total density contrast” δm = (ρbδb +
ρcδc)/(ρb + ρc) and density contrast difference ∆ = δc − δb, then by subtracting Eq. (1.109)
from (1.108) one can get
Ƭ +
4
3t
∆˙ = 0 =⇒ ∆ ∝ const, or t− 13 . (1.110)
δm can be solved from Eq. (1.104) therefore has the solution δm ∝ t−1 and t2/3. Therefore,
the ratio between δc and δb becomes
δc
δb
=
ρmδm + ρb∆
ρmδm − ρc∆
−→ δm
δm
= 1, (1.111)
so we see that δb is catching up δc and therefore falling into the dark matter potential well
quickly. This can be seen from the lower panel of Fig. (1.6).
We summarize these two solutions in the 3rd and 4th column in Table 1.3.
Λ dominant era At a later time when the cosmological constant Λ becomes the dominant
component in the Universe, the baryon perturbations have caught up with the perturbations in
the dark matter, so we can consider them as a single matter perturbation δm. In the Λ dominant
37
1. INTRODUCTION
era, H2 ≫ 4πGρm, the Jeans equation is simplified as
δ¨m + 2Hδ˙m = 0, (1.112)
for which the solutions are δm ∝ const and δm ∝ e−2t
√
Λ/3 ∝ a−2. Therefore, Λ suppress the
growth of structure which implies that the gravitational potential decays as a2ρm ∝ a−1. This
leaves an imprint on the CMB sky as the Integrated Sachs-Wolfe effect, which can be seen in Fig.
1.7.
1.4.3 Non-linear growth of the structure
The linear perturbation theory developed in Section 1.4.2 fails when the density contrast reaches
order unity. The cosmic structures that we see today — galaxies, clusters etc — have density
contrasts much greater than unity. They can be analyzed by performing numerical simulations.
However, non-linear evolution can be studied analytically if some simplifying assumptions are
made. We shall first review the spherical collapse model (Section 1.4.3.1), and then discuss dark
matter halo abundances (Section 1.4.3.2). Finally, we will review briefly some of the gas physics
associated with the collapse of baryonic systems (Section 1.4.3.3).
1.4.3.1 Spherical Collapse model
In Section 1.4.2.2, we have seen that during the matter dominant era, the density perturbation
δ grows as the scale factor a, therefore at some point, the density contrast will exceed unity and
the linear perturbation description will fail.
Consider the density contrast δ(x, ti) of an overdense region at some time ti. In the overdense
region, the self-gravity of the local dense region make the matter within this region expand more
slowly than the background expansion [79], i.e. the self-gravity works against the cosmic expansion,
which makes the overdense region “decouple” from the background. The properties of collapsed
object should be related to the initial condition of the overdense region. We therefore have to
specify this condition before we analyze the time evolution.
Initial condition Let us suppose that the overdense region we are interested in is spherically
symmetrical and has the initial density distribution at time ti ρ(r, ti) = ρm(ti) + δρ(r, ti) =
ρm[1 + δi(r)], where δi(r) is the initial density contrast at radius r. Therefore, the total mass M
interior to radius r and total density contrast δi of the region (radius ri) become
M =
(
4π
3
ρmr
3
i
)
(1 + δi), δi =
(
3
4πr3i
)∫ ri
0
δi(r)4πr
2dr. (1.113)
38
In the spherical collapse model, we assume no shell-crossing, so that M is a constant during the
evolution of the shell. The specific energy of the shell 1 is
1
2
(
dr
dt
)2
− GM
r
= E, (1.114)
where the sign of E determines whether the mass shell will expand forever (E > 0) or decouple
from expansion and collapse (E < 0). Suppose the peculiar velocity of the shell vi is negligible at
initial time t = ti, then r˙i = (a˙/a)ri = H(ti)ri = Hiri, so the initial kinetic energy per unit mass
is
Ki =
(
r˙2
2
)
=
H2i r
2
i
2
, (1.115)
and the potential energy at t = ti is (U = −|U |)
|U | =
(
GM
r
)
ti
= G
4π
3
ρm(ti)r
2
i (1 + δi)
=
1
2
H2i r
2
iΩi(1 + δi) = KiΩi(1 + δi), (1.116)
where Ωi = ρm(ti)/ρc(ti) is the initial matter density parameter of the background Universe.
Therefore, the specific energy of the mass shell becomes
E = Ki − |U | = KiΩi
[
Ω−1i − (1 + δi)
]
, (1.117)
from which the condition of collapsing object (E < 0) can be expressed as δi > [Ω
−1
i −1]. Obviously,
in a closed or a flat Universe (Ωi ≥ 1), this condition is always satisfied for an overdense region
δi > 0, but for an open Universe (Ωi < 1), the above condition sets a threshold for the overdense
region which can collapse eventually. Let us consider a collapsing mass shell with E < 0, which
will expand till some maximal radius and collapse. One can get the “turn-around” radius by
setting r˙ = 0 in Eq. (1.114) and using Eq. (1.117)
rturn = ri × (1 + δi)
δi − (Ω−1i − 1)
. (1.118)
Since we work in a matter dominated Universe, we shall set Ωi = 1 in the following calculations.
The time evolution of the mass shell can be found by solving Eq. (1.114). The solution can
be given in a parametric form as [79; 80]2
r = A(1− cos(θ)), t = B(θ − sin(θ)); A3 = GMB2, (1.119)
1energy per unit mass of the shell
2In principle, an integration constant can be added to either r or T expressions, but the constant can be proved
to be very small if the initial condition of the mass shell is set up to (ti,ri) in a matter dominant Universe, see [79].
39
1. INTRODUCTION
where A and B are constants. The time t increases if the parameter θ increases, and r increases to
rturn before decreasing to zero. Therefore, from the condition r(π) = rm = 2A, and A
3 = GMB2,
one can find out the coefficients from the initial condition
A =
ri
2
(
1 + δi
δi
)
≃ ri
2δi
, B =
1
2Hi
1 + δi
δ
3
2
i
≃ 3ti
4δ
2
3
i
, (1.120)
where the “≃” is under the approximation of a small initial fluctuation.
Time evolution With the above initial conditions, one can work out the evolution of the mean
density contrast within the shell. The mean density within a shell is
ρ(r, t) =
3M
4πr3
=
3M
4πA3(1− cos(θ))3 , (1.121)
and the background density is ρm(t) = 1/(6πGt
2) (a ∝ t2/3). Therefore, the mean density contrast
within the mass shell is
1 + δ(r, t) =
ρ(r, t)
ρm(t)
=
3M
4πA3
6πGt2
(1− cos(θ))3 , (1.122)
and by substituting Eq.(1.119) into Eq.(1.122), one finds that the density contrast evolves as
δ(r, t) =
9
2
(θ − sin(θ))2
(1− cos(θ))3 − 1, t = B(θ − sin(θ)). (1.123)
The linear evolution of the density contrast can be recovered in the small limit of θ, and by
expanding Eqs. (1.123) and using Eq. (1.120), one can find that in the linear approximation,
δ =
3
5
δi
(
t
ti
) 2
3
∝ a(t), (1.124)
which recovers the linear growth law for the linear perturbation theory (Section 1.4.2.2).
Now let us change the notation of initial conditions to match the current observables. We
use the current comoving radius of the mass shell to replace ri: x = ri[a(t0)/a(ti)], and current
magnitude of the initial density perturbation to replace the δi: δ0 = (a(t0)/ati)(3δi/5) = (1 +
zi)(3δi/5). From these relations, one can find the time evolution equations (1.119) and (1.120)
become
r(t) =
3x
10δ0
(1− cos(θ)), t =
(
3
5
) 3
2 3t0
4δ
3
2
0
(θ − cos(θ)). (1.125)
From Eq.(1.124) and the relation (t/ti)
2/3 = (1 + zi)(1 + z)
−1, one can find the relationship
40
between parameter θ and redshift z as
(1 + z) =
(
5
3
)(
4
3
) 2
3 δ0
(θ − sin(θ)) 23 . (1.126)
Therefore, Eqs. (1.123) and (1.126) give a complete solution for the density contrast evolution,
by assuming an initial value of perturbation δ0, one can get the time evolution δ(z). Equations
(1.125) give the solution for how the radius of the mass shell evolves under self-gravity.
The mass shell of the overdense region expands due to cosmic expansion, but at some point it
reaches its maximum radius rm (θ = π) and begins to collapse due to self-gravity. By substituting
θ = π into Eqs. (1.123) and (1.126), one finds the “turn-around” redshift, radius and density
contrast as
(1 + zturn) ≃ δ0
1.062
, rturn =
3x
5δ0
.
1 + δ =
(
ρ
ρm
)
turn
=
9π2
16
≃ 5.6. (1.127)
After the mass shell reaches its “turn-around” radius, it begins to contract and Eq. (1.125)
suggests that at θ = 2π all of the mass collapses to a singular point. However, in reality, the
spherical collapse scenario will break down well before the system reaches a singularity because of
density and velocity irregularities. The dark matter will experience a collisionless process called
“violent relaxation”[85], which broadens the energy distribution of the dark matter particles and
finally realizes the virialised state. The final virialised state can be modelled to describe the dark
matter structure that we see today. At the “turn-around” point, the total energy is equal to the
potential energy Eturn = Uturn ≈ −(3GM2)/(5rturn). From energy conservation equation and the
virial theorem, one has
Eturn = −K ≃ −Mv
2
2
≃ −3GM
2
5rturn
, U = −2K ≃ −Mv2 ≃ 3GM
2
5rvir
. (1.128)
Therefore, the random velocity of the particles within the dark matter halo and the virial radius
become
v ≃
(
6GM
5rturn
) 1
2
, rvir ≃ rturn
2
. (1.129)
Important quantities There are several important quantities that are predicted from this
spherical collapse model [79; 83]. The first one is the time taken for the fluctuation to reach virial
41
1. INTRODUCTION
equilibrium θ = 2π. From Eq. (1.126) we find
(1 + zcol) =
δi(1 + zi)
(2π)
2
3 (3
4
)
2
3
= 0.63(1 + zturn) =
δ0
1.686
. (1.130)
Therefore, δcrit = 1.686/D(z) (D(a = 1) = 1 is the current growth factor) is the density contrast
extrapolated from linear theory, at which the perturbation reaches turnaround 1. The second
important quantity is the mean density of the collapsed object, which is ρcol = 8ρ(tturn) = 8 ×
5.6ρm(tturn) = 8 × 5.6 × [(1 + zturn)/(1 + zcol)]3ρm(tcol), where the ρ(tturn) is the density of the
collapsing object at turn-around time, ρm(tturn) is the cosmic background density at the turn-
around time, and the ρm(tcol) is the cosmic background density at collapsed time. From these
relations, we can find the density of the collapsed object
ρcol ≃ 170ρm(tcol) = 170ρm0(1 + zcol)3, (1.131)
where the ρ0 is the present matter density. Therefore, at the present epoch, the density of a
collapsed and virialised object should be about 170 times the background matter density.
We can summarize the above results as follows: the comoving radius of the initial mass shell
with mass M is
x = ri
(
at0
ati
)
=
(
M
4π
3
ρm0
) 1
3
= 0.086
(
M
108MJ
) 1
3
Mpc. (1.132)
The virial radius of the collapsed object rms of the velocity dispersion of the dark matter particles
are
r =
3x
10δ0
= 1.5
(
M
108MJ
) 1
2
(
1 + zcol
10
)−1
kpc, (1.133)
v = 13.0×
(
M
108MJ
) 1
3
(
1 + zcol
10
) 1
2
km/s. (1.134)
And the binding energy of the halo is
E = 3× 1053
(
M
108MJ
) 5
3
(
1 + zcol
10
)
erg. (1.135)
Finally, we consider the gas properties during the collapse. The gaseous mixture of hydrogen
and helium develops shocks and provides pressure to prevent further collapse 2. Therefore, the
thermal energy can be comparable to the gravitational potential energy. Let µ be the mean
1The real density contrast is greater than 1.686 since at turnaround point it already reaches 9pi2/16 (Eq.
(1.127)).
2Under the assumption that the cooling can be neglected
42
Figure 1.14: Inhomogeneous 1-D density fluctuation. The dark (light) curve is the field smoothed
(not smoothed) on large scales. Small scale fluctuations that have density contrast over δcrit
collapse first, while the large scale fluctuation cannot collapse since their overdensity does not
exist this threshold. Figure taken from [44].
molecular weight in units of hydrogen mass,
µ =
mHnH +mHenHe
(2nH + 3nHe)mH
=
1
2
(
1 + Y
1 + 0.375Y
)
≃ 0.57, (1.136)
where we use helium faction Y = 0.25. Therefore, the pressure equilibrium suggests 3ρgaskBTvir/(2µmH) =
ρgasv
2/2, from which we can obtain the virial temperature
Tvir = 3.9× 103
( µ
0.57
)( M
108MJ
)2(
1 + zcol
10
)
K. (1.137)
The spherical collapse model captures some of the physics responsible for the formation of
halos. In reality, cosmic structures form hierarchically: at early time, low mass halos formed, and
they have continuously accreted and merged to form higher mass halos. In Section 1.5, we will
discuss hierarchial halo formation and universal dark matter density profile, and review galaxy
formation in a bit more detail.
1.4.3.2 Dark Matter halo abundance
The above spherical collapse model provides a rough description of the evolution of individual
halos, but a theory of structure formation should also predict the statistical properties of halos,
for example, their number density as a function of mass at different redshifts. One can, of course,
compute such statistics from numerical simulations, but to gain a physical understanding, an
analytic model is needed [83].
A key advance in computing halo abundances was made in the seminal paper by Press and
Schechter in 1974 [86], which is based on the linear perturbation theory, spherical collapse model
and Gaussian statistics. Once an object with mass reaches a threshold amplitude, it will collapse
and form a virialised object. Therefore, one can expect to find various density peaks in the final
volume of interest. Small scale density fluctuations would typically have a higher amplitude than
43
1. INTRODUCTION
Figure 1.15: Left: Two-dimensional slice of the baryon distribution at z = 2.5 in a ΛCDM hydro-
dynamical simulation of structure formation. The low and high density regions are shown as dark
and bright colours. The voids and filaments are typically associated with material in the Inter-
Galactic Medium (IGM) and are observable in quasar absorption spectra. The luminous baryonic
material (stars and galaxies) forms in the highest density regions, which is shown as the knots
and sheets where many filaments converge. Right: The underlying dark matter distribution. The
baryon distribution is slightly more diffuse (less small scale structure) compared to the underlying
dark matter due to gas pressure support. In the simulation, the cosmological parameters are:
Ωm = 0.26, ΩΛ = 0.74, Ωb = 0.0463, h = 0.72. The box size is 60 Mpc/h. The number of gas and
dark matter particles are 2× 4003. The softening length is 2.5 kpc/h. Figure taken from [87].
the large scale smoothed fluctuations (Fig. 1.14). Therefore small scale fluctuations would collapse
first and merge to form denser and larger halos. To understand this scenario, we consider a 1-D
density fluctuation as shown in Fig. 1.14. The average of the inhomogeneity is zero, but there are
some regions with large fluctuations whose density contrast is comparable or greater than δcrit,
and also some underdense regions with density contrast negative (Note that δ = (ρ− ρ)/ρ cannot
be smaller than −1). It is just these rare regions with large density contrast that will “decouple”
from the Hubble expansion and collapse first. Since the density contrast in the matter dominated
regime will grow as the scale factor a, the large scale inhomogeneity also grows and eventually
collapses.
The Press-Schechter theory predicts the fraction of the volume that collapses at redshift z, i.e.
the mass fraction above M is
fcol(> M, z) =
2√
2πσ(R, z)
∫ ∞
δcrit
dδ exp
(
− δ
2
2σ2(R, z)
)
,
= erfc
(
δcrit√
2σ(R, z)
)
, (1.138)
where “erfc” is the error function. There are several assumptions involved in computing this
distribution. The most important assumption is relating a filter scale σ2(R, z) to a collapsed non-
linear massM(R, z). Often people use the top-hat filter and the spherical collapse model to derive
44
a relation between σ2 andM , but to do it more accurately numerical simulations are needed. The
second assumption is an “ad-hoc” factor of two in Eq. (1.138) in order to recover the mean
density of the Universe. The excursion set formulism developed by [88] gives a more satisfactory
explanation (see also [83]). If a halo with given mass M has density contrast δM < δcrit, it may
still collapse if it has an inner region with density contrast δL ≥ δcrit and mass ML ≥ M . In
this case, the region with mass M would be counted as belonging to the halo with mass ML.
Therefore, the fraction of the volume that has collapsed is greater than the Gaussian distribution
without the factor of two [83; 88]. However, in practice, Press-Schetcher formulism works pretty
well in matching the numerical simulations [89]. We can get the number density of the collapsed
objects with mass in between M and M + dM as
dn
dM
dM = −ρm
M
dfcol(M(R), z)
dM
dM
= −
√
2
π
ρm
M
d ln σ(R, z)
dM
νe−ν
2/2dM, (1.139)
where M = (4π/3)ρmR
3 and ν = δcrit/σ(R, z). The rms σ(R, z) depends on cosmological param-
eters, so by measuring the galaxy cluster abundance, one can quantify cosmological parameters
[44], in particular the matter content Ωm and amplitude of fluctuation σ8.
The Press-Schetcher formulism can fit observations quite well, but it underestimates the rare
halos that host galaxies at high reshift. Therefore, people have proposed various other mass
functions in order to match simulation results better, one of which was proposed by Sheth and
Tormen in 1999 [82]
fcol(> M, z) = A
ν
σ2(R, z)
√
a′
2π
[
1 +
1
(a′ν2)q′
]
e−a
′ν2/2, (1.140)
where the best-fit parameters are a′ = 0.75, q′ = 0.3 and the normalization requires A = 0.322.
1.4.3.3 Gas Physics
When the dark matter halo collapses, the gas falls into the dark matter potential well with a
streaming velocity of order v (Eq. (1.134)). In order to form stars, the collapsed gas has to cool
until its Jeans mass drops to the mass of a typical star [83]. This is important because if the
cooling is ignored, we would expect the gas to shock and produce a pressure support atmosphere
in hydrostatic equilibrium.
Before the first stars were formed, no element existed to cool the gas (Big Bang Nucleosynthesis
cannot produced enough heavy elements to cool the first stars), so the gas can be only cooled by
atomic transitions in hydrogen and helium and molecular hydrogen transition. However, below
∼ 104 K (Eq. (1.137)), atomic cooling is inefficient and therefore cannot provide an effective way
to cool and fragment the gas into stars. The only way of cooling the gas with T < 104 K is via
45
1. INTRODUCTION
molecular hydrogen cooling. After recombination, there are residual free electrons e− which can
act as a catalyst to combine with the neutral hydrogen and release photons H + e− −→ H− + γ
and H− +H −→ H2 + e−. Through this mechanism, free electrons can catalyze H2 and cool the
gas to the temperature or about a few hundred K [83].
For a halo with virial temperature greater than 104 K, atomic transitions and free-free emission
are effective ways of cooling the gas, but when the temperature drops below 104 K, molecular
hydrogen becomes important [83].
In the ΛCDM model, structures on small scales build up hierarchically and form filamentary
structure (Fig. 1.15). Gas cooling defines a characteristic mass of order 1012MJ for the baryonic
content of galaxies. Below this mass scale, the gas can cool so efficiently that it nearly collapses at
the free-fall rate. For larger masses, the gas cannot cool efficiently so galaxies may still be forming
at very low redshift [84].
In summary, structure formation theory of ΛCDM predicts the hierarchial formation of dark
matter halos, and within these halos stars form in the dense, cold knots of gas. By mapping
the density peaks of the distribution from observations, one can test the theory of structure
formation and measure the cosmological parameters. In addition, mapping the cosmic diffuse gas
by observing the emission and absorption lines in either quasar spectrum or in the 21cm line of
molecular hydrogen is also an important way of probing the first structure formation.
1.5 Dark matter and Galaxy formation
In the above introduction, we have reviewed large scale structure formation and sketched some of
the basic physical processes involved in the formation of hierarchial structure. An essential step
in confirming this scenario is to check whether the theory is consistent with observations in the
nearby Universe, since experimentally nearby objects are usually easier to observe in detail. This
task is closely related to the understanding of dark matter properties on Galactic scales since the
dark matter may reveal more microscopic physics on this scale via its effect on small scale structure
formation. In this Section, we shall first look at dark matter from a particle physics perspective,
and then discuss some potential problems posed by observations of small scale structures. For
more discussion in detail, we refer to [90; 92].
1.5.1 Dark Matter Particles
Dark matter is not new to cosmologists. It was as early as the 1930s that Fritz Zwicky discovered
the existence of the dark matter (“missing mass”) in clusters of galaxies by using viral theorem
[94]. In 1970s, astronomers discovered flat galaxy rotation curves and proposed that they might
be explained by an invisible dark matter halo [95]. Currently, there are two popular dark matter
candidates from the considerations of elementary particle physics perspective. One is the lightest
46
Figure 1.16: The cumulative number of Milky way satellite galaxies as a function of halo circular
velocity. Left panel: Missing satellite problem; Right panel: the effect of reionization. Figure
taken from [104].
supersymmetric partner particle, which is called WIMP (Weakly Interacting Massive Particle)
[93; 101]; and the other is the cosmological axion [102; 103].
The WIMP model is proposed under the supersymmetry (SUSY) principle of particle physics.
SUSY can allow the unification of the electroweak and strong interactions and naturally explains
why the electroweak scales is much smaller than the Planck scale. The predictions of the WIMP
mass is typically between 100 to 1000 GeV.
Axion were first proposed to solve the strong interaction CP problem in SU(3) gauge theory.
Axion like fields appear to be common in string theory and so remain a possible candidate for the
dark matter.
1.5.2 Dark Matter on Small Scales
There are two major issues that have challenged the ΛCDM scenario, namely the abundance of
the dark matter halos and satellite galaxies and the existence of the density cusps at the centres
of galaxies [90]. In this subsection, we shall review these problems and discuss how they might
relate to the properties of dark matter particles.
1.5.2.1 Population of sub-halos and satellites: how cold is cold dark matter?
Many fewer satellite galaxies have been detected in the Local Group than the number of sub-halos
predicted from numerical simulations (Fig. 1.16), and this is considered by some to be a serious
problem for ΛCDM cosmology. However, the suppression of star formation in small dwarf galaxies
after reionization can account for the observed abundance of satellites for ΛCDM [90; 104]. This
is possibly related to the “common mass” scale discovered by [105] which may further reveal the
properties of the baryonic physics.
The properties of dark matter halo depend on the mass of the dark matter particles. For
example, warm dark matter with high thermal velocities, can suppress the formation of sub-
47
1. INTRODUCTION
halos. However, it is not clear whether we need to suppress the formation of small dark halos.
What we can observe is the stars, and not all of the dark matter halos host stars. In particular,
feedback from star formation may drive out most of the gas from a low velocity dispersion dark
halo. Therefore, the ‘missing satellite’ problem may be either due to the thermal velocity of dark
matter, or baryonic feedback resulting from the star formation process.
Sterile neutrinos, provide an example of warm dark matter. Sterile neutrinos may have been
produced in the very early stage of the Universe, and have been proposed as a candidate for dark
matter [96]. The main effect of the thermal velocities of such warm dark matter particles would be
to suppress structures below Mpc scale. The Lyman-alpha absorption produced by the intervening
neutral hydrogen in the spectrum of quasars (also known as Lyman-alpha forest), can probe the
matter power spectrum in the mildly nonlinear regime down to small scales (∼ 1Mpc/h) over the
redshift z = 2−6 [91]. Therefore, the statistics of the Lyman-alpha forest at redshift 2.0 < z < 6.4
can be used to set a lower limit on the sterile neutrino mass ms ≥ 28 keV [91]. On the other
hand, the sterile neutrino can decay into X-rays and light neutrinos, so the null results of X-ray
observations (such as XMM − Newton observation of Ursa Minor and Milky Way [98; 99]) can
give an upper limit on the sterile neutrino mass ms ≤ 3.5 keV [97]. Therefore, the sterile neutrino
is nearly ruled out as a dark matter candidate, although some windows of parameter space remain
open [90].
The epoch of reionsation, can provide another constraint on the dark matter particle mass.
Since the thermal velocities of the warm dark matter strongly suppress the formation of low-mass
halos, the absence of the mini-halos can delay the formation of the first stars, and therefore delay
the reionisation epoch [90].
Therefore, the populations of sub-halos and satellite galaxies can set important constraints on
the clustering properties of dark matter.
1.5.2.2 Cusp and Core problem
In 1996, Navarro et al. investigated the structure of dark matter halos from N-body simulations
of the ΛCDM model. They found a universal profile of dark matter density
ρ(r) = ρcrit
δc
(r/rs)(1 + r/rs)2
, (1.141)
which has a slope r−1 as small radii. However, observations of nearby galaxies [106] suggest a
core-dominated halo with a central profile ρ(r) ∝ (r20 + r2)−1.
One possible reason for this problem may be the neglect of baryonic physics in pure dark matter
simulations. There are three mechanisms that have been proposed to solve the cusp problem [90]:
• Dynamical friction: By comparing the pure dark matter and baryon+ dark matter simula-
tions [107; 108], Ramano-Diaz et al. claim that dynamical friction causes infalling baryon
48
and dark matter clumps to transfer energy and angular momentum to the dark matter. As
a result, the dark matter radial profile at low redshift is transformed into an isothermal with
a flat core at low redshift.
• Starburst removal: The rapid blowout of a large amount of gas in the centre of the density
profile due to a starburst can cause the dark matter distribution in the inner region of the
halo to expand [109].
• Bulk gas motion: Supernovae-driven gas bulk motions can smooth out dark matter cusps
in the forming galaxy regions as a consequence of resonant heating of dark matter in the
resulting fluctuating potential. This may provide an explanation of the dark matter cores
in the dwarf spheroidal (dSph) galaxies such as the Ursa Minor satellites of the Milky Way
[90; 110].
All of these mechanisms are very speculative, however, in order to determine the distribution
of dark matter in low surface brightness disks and gas-rich dwarf galaxies, one needs to compare
numerical simulations that model baryonic process with observations.
1.6 Overview
1.6.1 Classification of problems in Cosmology
In the above we have presented an extensive review of our current understanding of cosmic evo-
lution: from quantum fluctuation in the early Universe to the subsequent formation of non-linear
structure. However, the many unresolved questions in cosmology provide great challenges and
opportunities for further research. Below, I give examples of two types of problems in cosmology,
namely as “unknown physics” and “known physics but unknown details”. Both classes of problem
provide fertile ground for novel research.
1.6.1.1 Known Physics but unknown details
Cosmic reionization As we have seen in Section 1.3, at a redshift about 1100, electrons began
to recombine with protons to form neutral hydrogen, so the free electron density began to drop
and radiation decoupled from the baryons. The dark ages began at this point, since there was no
light source in the Universe apart from the gradually dimming CMB radiation.
When the redshift decreased to about 20, gravitational instability caused the formation of
the first non-linear objects. It is believed that the light from the first stars reionised the neutral
hydrogen at a redshift z & 10. Understanding the process of reionization would provide important
information on how the first galaxies and stars formed [83]. The spectrum of quasars can be an
important way of probing the reionization epoch. Quasars release an extraordinary amount of
49
1. INTRODUCTION
energy and may be the brightest objects in the very early Universe. They emit the light in which
the blueward of the spectrum can be absorbed by the neutral hydrogen in the line-of-sight. This
is known as the Gunn-Peterson effect and the rapid increase in optical depth in the spectrum of
quasar suggests that the end of reionization occurs at a redshift of ∼ 6 [111; 112].
The 21-cm line of neutral hydrogen offers a promising way of studying the dark ages and the
early stages of reionization. The 21-cm transition of hydrogen arises from two separate hyperfine
states of the ground-state. This transition is highly temperature dependent, due to the fact that
as objects form in the “dark ages” and emit photons that heat the surrounding neutral hydrogen,
it causes 21-cm line emissions in the surrounding area [83]. The 21-cm line can be detected in
emission or absorption, depending on the spin temperature of the neutral gas [100]. On-going
experiments, such as LOFAR [113], are constructed to detect such signals.
Gravitational Waves Gravitational waves are fluctuations of the curvature of the space-time
predicted by Einstein’s theory of general relativity. They propagate as the speed of light, while
their frequency is determined by the mass of the object(s) and the power is determined by the ge-
ometry and distance of the emitting object(s). On-going projects such as the Laser Interferometer
Gravitational-Wave Observatory (LIGO), Advanced LIGO, and the future Laser Interferometer
Space Antenna (LISA) will constrain the signals over a wide range of frequencies. The direct
detection of gravitational waves will open up a new window to study strong gravity and compact
astrophysical objects.
1.6.1.2 Unknown Physics
Inflation models and the early Universe process To describe the fundamental field that
drove inflation in the early Universe, theoretical physicists have proposed hundreds of models.
However, these phenomenological models are still far from fundamental microscopic physics. In
addition, some theorists try to avoid inflation and propose a non-trivial early universe mechanism
(such as Cyclic Model [114]) to generate primordial fluctuations. The difference between this
model and inflation is the amplitude of the tensor modes. Single-field inflation predicts a tensor
mode r ∼ O(0.1), while the cyclic model predicts undetectable amplitude r ∼ O(10−5). Therefore,
perhaps the most realistic way to distinguish the models is to measure the B-mode polarization
signals in the CMB map to infer the energy level and field configuration of the inflaton. This is a
frontier of cosmology for the coming decades since space-mission (Planck [27]), and ground-based
experiments (QUIET [31], PolarBear [30]) are dedicated to measuring this signal. In Chapter 4,
we are going to discuss the current and prospective constraints on the CMB B-mode signals and
its constraints on the models of the early Universe.
Dark Matter As we have shown above, dark matter is known to exist around galaxies and in
galaxy clusters. However, although astrophysical observations may set some limit on the micro-
50
scopic properties of dark matter, the nature of dark matter is still very speculative. Perhaps the
most promising way of discovering its nature lies in particle physics experiments such as the Large
Hadron Collider (LHC) [115], and the Cryogenic Dark Matter Search (CDMS) [116].
Dark Energy The dark energy problem is one of the most serious problems in modern physics.
If it is caused by the vacuum fluctuations (cosmological constant Λ), it should take a value of
about M4pl, which is an order of 120 times higher than what we observe today. People have
proposed dynamic field (such as quintessence) to explain dark energy, but these models have to
be contrived and fine-tuned to match the observable fluctuations [118]. In addition, these models
are all phenomenological and lack a basis in fundamental physics. The combination of on-going
CMB observations Planck [27] and future galaxy redshift surveys can constrain the dark energy
equation of state to a high precision [117].
1.6.2 Aims and Structure of the Thesis
In the above sections, we have reviewed the standard ΛCDM cosmology in detail, including the
inflation, CMB cosmology and large scale structure formation. We have also given a brief review
of the aspects of the ΛCDM model that requires new physics.
However, although the ΛCDM model can successfully pass most of the observational tests
of cosmology, a number of inconsistencies between the ΛCDM model and different astronomical
observations have been discussed in the literatures. Some authors argue that there is a lack of
large angular correlations in the CMB, and some authors argue that there is a broken symmetry
pattern in the CMB. In addition, on & 50 Mpc scales, it has been argued that there is a very large
amplitude of the bulk flow, which seems to be inconsistent with the linear perturbation theory of
structure formation.
Discovering inconsistencies with an established model is of course extremely important. It is
only by establishing such inconsistencies that the subject can develop and a new paradigm can
emerge. In this thesis, we will compare the ΛCDM model with a number of cosmological tests,
and investigate the constraints on the plausible new physics. In the remaining Chapters of the
thesis, I will investigate the following topics:
• In Chapter 2, we discuss the issue of whether there is really a lack of angular correlations
in the CMB sky. We compare various estimators of the temperature correlation function
showing how they depend on assumptions of statistical isotropy and how they perform on
the WMAP 5-yr Internal Linear Combination (ILC) maps with and without a sky cut. We
further reconstruct the low-multipole harmonics that determine the large scale features of
the temperature correlation accurately from the data that lie outside the sky cuts. The
reconstruction results are in good agreement with those computed from the ILC map over
the whole sky. A Bayesian analysis of the large-scale correlations is presented, which shows
51
1. INTRODUCTION
that the data cannot exclude the standard ΛCDM model. Therefore, we conclude that
irrespective of the assumption of statistical isotropy, there is marginal evidence for the lack
of large angular correlation which is inconsistent with ΛCDM model.
• In Chapter 3, we discuss the test of the statistical isotropy of CMB and develop a method to
access the statistical significance of any broken isotropy. We construct simple quadratic esti-
mators to reconstruct asymmetry in the primordial power spectrum from CMB temperature
and polarization data and verify their accuracy by using simulations with quadrupole power
asymmetry. We show that the Planck mission, with its millions of signal-dominated modes
of the temperature anisotropy, should be able to constrain the amplitude of any spherical
multipole of a scale-invariant quadrupole asymmetry at the 1% level (2σ). Almost indepen-
dent constraints can be obtained from polarization at the 3% level after four complete sky
surveys, providing an important consistency test. If the amplitude of the asymmetry is large
enough, constraining its scale-dependence should become possible. In scale-free quadrupole
models with 1% asymmetry, consistent with the current limits from WMAP temperature
data (after correction for beam asymmetries), Planck should constrain the spectral index q
of power-law departures from asymmetry to ∆q = 0.3. Finally, we show how to constrain
models with axisymmetry in the same framework. For scale-free quadrupole models, Planck
should constrain the direction of the asymmetry to a 1σ accuracy of about 2 degrees using
one year of temperature data.
• Chapter 4 investigates at the issue of how well the current and future CMB B-mode polariza-
tion experiments are able to detect the features of inflation and early Universe models. We
investigate the observational signatures of three models of the early Universe in the B-mode
polarization of the CMB radiation: standard single field inflation, loop quantum cosmol-
ogy (from loop quantum gravity) and cosmic strings as predicted from brane inflation. We
further use current B-mode polarization data from the BICEP and QUaD experiments to
constrain the parameters of these models. We also examine the detectability of the primor-
dial B-mode signal predicted by these models with future CMB polarization experiments.
We find that although the current CMB experiments cannot distinguish between the models,
they can already set strong constraints. Future space-satellites (e.g. Planck and CMBPol),
ground-based experiments (PolarBear and QUIET), and balloon-borne experiments can set
important limits on the fundamental parameters of inflation and other alternatives and
perhaps even distinguish them.
• A large bulk flow, which is in tension with the ΛCDM cosmology, has been reported by
some authors. In Chapter 5, we provide a physically plausible explanation of this bulk flow,
based on the assumption that some fraction of the observed dipole in the cosmic microwave
background is due to an intrinsic fluctuation, so that the subtraction of the observed dipole
52
leads to a mismatch between the CMB defined rest frame and the matter rest frame. We
investigate a model that takes into account the relative velocity (hereafter the tilted velocity)
between the two frames, and develop a Bayesian statistic to explore the likelihood of this
tilted velocity.
By studying various independent peculiar velocity catalogs, we find that (1) the magnitude
of the tilted velocity u is around 400 km/s, and its direction is close to what is found from
previous bulk flow analyzes; for most catalogs analyzed, u = 0 is excluded at about the
2.5σ level; (2) constraints on the magnitude of the tilted velocity can result in constraints
on the duration of inflation, due to the fact that inflation can neither be too long (no dipole
effect) nor too short (very large dipole effect); (3) under the assumption of a superhorizon
isocurvature fluctuation, the constraints on the tilted velocity require that inflation lasts at
least 6 e-folds longer (at the 95% confidence interval) than that required to solve the horizon
problem. This opens a new window for testing inflation and models of the early Universe
from observations of large scale structure.
• In Chapter 6, we investigate the potential power of the Cosmic Mach Number (CMN), which
is the ratio between the mean velocity and the velocity dispersion of galaxies as a function
of cosmic scales, to constrain cosmologies. We first measure the CMN from 5 catalogues of
galaxy peculiar velocity surveys at low redshift (z ∈ (0.002, 0.03)), and use them to contrast
cosmological models. Overall, current data is consistent with the WMAP7 ΛCDM model.
We find that the CMN is highly sensitive to the growth of structure on scales k ∈ (0.01, 0.1)
h/Mpc in Fourier space. Therefore, modified gravity models, and models with massive
neutrinos, in which the structure growth generally deviate from that in the ΛCDM model in
a scale-dependent way, can be well differentiated from the ΛCDM model using future CMN
data.
• In the conclusion Chapter 7, we will give an outlook of the further tests of standard ΛCDM
cosmology and other possible variations.
53
1. INTRODUCTION
54
Chapter 2
Large-Angle Correlations in the CMB
2.1 Introduction
Following the discovery by the COBE team of temperature anisotropies in the cosmic microwave
background (CMB) radiation [41; 119], Ref. [120] noticed that the temperature angular correlation
function, C(θ), measured from the COBE maps was close to zero on large angular scales. This
result attracted little attention until the publication of the first year results from WMAP [1; 61].
The results from WMAP confirmed the lack of large-scale angular correlations in the temperature
maps and led [1] to introduce the statistic
S1/2 =
∫ 1/2
−1
[C(θ)]2d cos θ. (2.1)
The form of the statistic and the upper cut-off, µ = cos θ = 1/2, were chosen a posteriori by
[1] ‘in response’ to the observed shape of the temperature correlation function computed using
a particular estimator and sky cut (as described in further detail in Section 2.2). To assess the
statistical significance of the lack of large-scale power, [1] computed a ‘p-value’, i.e. the fraction
of models in their Monte Carlo Markov chains which had a value of Smodel1/2 < S
data
1/2 , using the same
estimator and sky cut that they applied to the data. For their standard six-parameter inflationary
ΛCDM cosmology, they found a p-value of 0.15%, suggesting a significant discrepancy between
the model and the data.
This problem was revisited by [121]. The main focus of the [121] was to improve on the pseudo-
harmonic power spectrum analysis used by the WMAP team [122] by using quadratic maximum
likelihood (QML) estimates of the power spectrum, with particular emphasis on the statistical
significance of the low amplitude of the quadrupole anisotropy. As an aside, Ref. [121] computed
angular correlation functions from the QML power spectrum estimates and showed that they
were insensitive to the presence of a sky cut. Similarly the S1/2 statistic computed from these
correlation functions was found to be insensitive to the size of the sky cut giving p-values of a few
55
2. LARGE-ANGLE CORRELATIONS IN THE CMB
percent. Ref. [121] concluded that the correlation function and S1/2 statistic offered no compelling
evidence against the concordance inflationary ΛCDM model. Ref. [121] did not explore in any
detail the low p-value for the S1/2 statistic reported by [1], but commented that it was probably
simply an ‘unfortunate’ consequence of the particular choice of statistic, estimator and sky cut
chosen by these authors (in other words, a result of various a posteriori choices).
The CMB temperature correlation function and S statistic have been reanalyzed in two recent
papers [124; 125]. The arguments in the two papers are quite similar, and so for the most part we
will refer to the later paper [125] (since, as in this work, it analyzes the 5-year WMAP temperature
data, [123]). The [124; 125] papers are largely motivated by evidence for a violation of statistical
isotropy in the WMAP temperature maps, in particular evidence of alignments amongst the low
order CMB multipoles (e.g. [126; 127; 128; 129]), although the statistical significance of these
alignments has been questioned ([130; 132; 133]). Refs. [124; 125] make the valid point that
statistical isotropy is often implicitly assumed in defining what is meant by the term ‘correlation
function’ and in defining estimators. They argue further that different estimators contain different
information. They then focus on pixel-based estimates of the correlation function applied to the
WMAP data, including a sky cut, and find p-values for the S1/2 statistic of ∼ 0.025–0.04%,
depending on the choice of CMB map and sky cut. If no sky cut is applied, they find p-values of
∼ 5% (similar to the p-values reported in [121]). Ref. [125] comments that the full-sky results are
apparently inconsistent with the cut-sky analysis suggesting a violation of statistical isotropy.
Any analysis which claims to strongly rule out the simple inflationary ΛCDM model deserves
careful scrutiny, since a confirmed discordance would have profound consequences for our under-
standing of the early Universe. The purpose of this Chapter is to investigate carefully the analysis
presented in [125]. In Section 2.2 we discuss estimators of the correlation function and relate
the pixel-based estimator used by [125] to the pseudo-power spectrum computed on a cut sky.
In Section 2.3, we explicitly reconstruct the individual low order multipole coefficients aℓm from
cut-sky maps using a technique first applied by [131]. This allows us to test the sensitivity of the
large-angle correlation function to the presence of a sky cut, largely independent of assumptions
concerning statistical isotropy or Gaussianity. The results of this analysis are compared with the
QML estimates of the correlation function used in [121]. Section 2.4 describes a Bayesian analysis
of the S1/2 statistic and contrasts it with the frequentist analysis applied by [1] and [125]. Our
conclusions are summarized in Section 2.5. A recent paper by Pontzen and Peiris [134] extends the
analysis presented here to general anisotropic Gaussian theories with largely similar conclusions.
2.2 Estimators of the Correlation Function
If we assume statistical isotropy, the ensemble average of the temperature angular correlation
function (ACF) measured over the whole sky 〈C(θ)〉 is related to the ensemble average of the
56
angular power spectrum 〈Cℓ〉 by the well known relation
〈C(θ)〉 = 1
4π
∑
ℓ
(2ℓ+ 1)〈Cℓ〉Pℓ(cos θ). (2.2)
However, we have only one realization of the sky and we may further choose to impose a sky cut to
reduce possible contamination from regions of high Galactic emission. If we relax the assumptions
of statistical isotropy and complete sky coverage, there is no unique definition or estimator of the
ACF. One can write down a number of estimators that, given certain assumptions concerning the
underlying statistics of the fluctuations, may average to the ensemble mean when applied to data
on an incomplete sky.
Ref.[125] uses a direct pixel based correlation function 1 on the cut sky
Cpix(θ) = 〈xixj〉, (2.3)
where xi denotes the temperature value in pixel i and the angular brackets denote an average over
all pixel pairs outside the sky cut with an angular separation that lies within a small interval of θ.
If the underlying temperature field is statistically isotropic, equation (2.3) provides an unbi-
ased estimate of the correlation function, i.e. the average over a large number of independent
realizations is unbiased, irrespective of the sky cut. However, if the fluctuations are statistically
isotropic and Gaussian, Eq. (2.3) is not an optimal estimator of 〈C(θ)〉. To see this, expand the
temperature field in spherical harmonics
xi =
∑
ℓm
aℓmYℓm(θi), 〈|aℓm|2〉 = 〈Cℓ〉. (2.4)
Then, from the rotation properties of the spherical harmonics, it is straightforward to prove
Cpix(θij) = 〈xixj〉 =
∑
ℓ(2ℓ+ 1)C˜
P
ℓ Pℓ(cos θij)∑
ℓ(2ℓ+ 1)W˜ℓPℓ(cos θij)
, (2.5)
where C˜Pℓ is the pseudo-power spectrum (PCL) estimate on the cut sky:
C˜Pℓ =
1
(2ℓ+ 1)
∑
m
|a˜ℓm|2, a˜ℓm =
∑
i
xiwiY
∗
ℓm(θi)Ωi, (2.6)
where wi is a window function that is zero or unity depending on whether a pixel (of area Ωi)
lies inside or outside the sky cut. The function W˜ℓ in (2.5) is the pseudo-power spectrum of the
1More generally, one can define a bi-polar correlation function C(θi,θj), which can be used as a test of statistical
isotropy (see [135])
57
2. LARGE-ANGLE CORRELATIONS IN THE CMB
window function wi:
W˜ℓ =
1
(2ℓ+ 1)
∑
m
|w˜ℓm|2, w˜ℓm =
∑
i
wiΩiY
∗
ℓm(θi). (2.7)
The relation (2.5) is an identity and does not depend on the assumption of statistical isotropy.
In fact, the pixel estimator (2.5) is mathematically identical1 to the PCL estimator used by
[1] and [121]
CP (θ) ≡ Cpix ≡ 1
4π
∑
ℓ
(2ℓ+ 1)CˆPℓ Pℓ(cos θ), Cˆ
P
ℓ =M
−1
ℓℓ′ C˜
P
ℓ′ , (2.8)
where the matrix M is
Mℓℓ′ =
1
(2ℓ+ 1)
∑
mm′
|Kℓmℓ′m′ |2. (2.9)
The equivalence between the estimators (2.5) and (2.9) demonstrates that for isotropic Gaus-
sian random fields, the pixel estimator (2.3) is suboptimal since PCL power-spectrum estimates
are suboptimal when applied to the cut sky (for extensive discussions see [121; 136] and references
therein).
The coefficients a˜ℓm are related to the coefficients aℓm on the uncut sky by the coupling matrix
K,
a˜ℓm =
∑
ℓ′m′
aℓ′m′Kℓmℓ′m′ , (2.10)
where
Kℓ1m1ℓ2m2 =
∑
i
wiΩiY
∗
ℓ1m1
(θi)Yℓ2m2(θi). (2.11)
Under the assumption of statistical isotropy and Gaussianity, it is straightforward to calculate the
covariance matrix of the estimator (2.3)
〈∆C˜(θi)∆C˜(θj)〉 =
(∑
ℓ1ℓ2
(2ℓ1 + 1)(2ℓ2 + 1)〈∆˜CPℓ1∆˜CPℓ2〉Pℓ1(cos θij)Pℓ2(cos θij)
)
/
(∑
ℓ
(2ℓ+ 1)W˜ℓPℓ(cos θij)
)2
(2.12)
where
〈∆C˜Pℓ ∆C˜Pℓ′ 〉 =
2
(2ℓ+ 1)(2ℓ′ + 1)
∑
mm′
∑
ℓ1m1
∑
ℓ2m2
Cℓ1Cℓ2Kℓmℓ1m1K
∗
ℓ′m′ℓ1m1
K∗ℓmℓ2m2Kℓ′m′ℓ2m2 (2.13)
Figure 2.1(a) shows the direct pixel based estimator (2.3) applied to the 5-year WMAP ILC
map after smoothing with a Gaussian filter of 10◦ FWHM and repixelising at a Healpix [137]
1We are grateful to Anthony Challinor for pointing this out. See Appendix B of [134] for a proof.
58
Figure 2.1: The figure to the left shows the correlation functions computed using the pixel esti-
mator (2.3) applied to the 5 year WMAP ILC map (degraded as described in the text) over the
full sky and with the WMAP KQ85 and KQ75 masks imposed. The figure to the right shows the
correlation functions computed from the pseudo-power spectra (2.5).
resolution NSIDE=16. The results of Fig. 2.1 are consistent with those of [125]. With the
WMAP KQ85 and KQ75 masks1 applied (retaining about 82% and 71% of the sky respectively
[138]), there is little power over the angular range 60◦-160◦. However, there is some non-zero
correlation if the pixel estimator is evaluated over the full sky. Fig. 2.1(b) shows the correlation
functions determined from the pseudo-spectra (2.5). This simply confirms the equivalence of the
two estimators (2.3) and (2.5) (apart from minor differences arising from the finite angular bin
widths). The covariance matrix for these estimators for the full sky is shown in Fig. 2.2, using
the Cℓ for the six parameter ΛCDM model that provides the best fit to the WMAP data ([72]).
The large angle ACF for a nearly scale invariant temperature spectrum is dominated by a small
number of modes leading to large correlations between different angular scales. The main effect
of a KQ75-type sky cut on the covariance matrix is to increase its overall amplitude. The angular
structure of the covariance matrix is insensitive to the precise size and shape of the sky cut.
The main differences between the various ACF estimates plotted in Figs. 2.1 come from
application of the sky cuts. With the sky cuts applied, the ACF’s are close to zero on angular
scales ≥ 60◦. This lack of power leads to particulary low values for the S1/2 statistic of about
1000-2000 (µK)4, as listed in Table 2.12. If no sky cut is applied, the value of the S1/2 statistic is
substantially higher at around 8000 (µK)4. It is worth noting that the values listed in Table 2.1
are very similar to the values obtained from the WMAP first year data (see Table 5 in [121]). The
low multipole anisotropies that contribute to the ACF at large angular scales have remained stable
as the data have improved. The low multipoles are signal dominated and stable to improved gain
1We used degraded resolution (NSIDE=16) versions of these masks. The degraded resolution KQ75 mask is
plotted in Fig. 2.3.
2For maps degraded to Healpix resolution of NSIDE= 16 and smoothed with a Gaussian of FWHM 10◦.
59
2. LARGE-ANGLE CORRELATIONS IN THE CMB
0 20 40 60 80 100 120 140 160 180
θ
1
 (degrees)
0
20
40
60
80
100
120
140
160
180
θ2
 
(
d
e
g
r
e
e
s
)
−50000
25000
0
25000
50000
75000
100000
125000
150000
Figure 2.2: Covariance matrix for the pixel correlation estimator (2.3) computed for full sky maps.
The scale on the right is in units of (µK)4.
corrections, foreground separation and small perturbations to the Galactic mask.
S1/2 statistic in (µK)
4
sky cut Pixel ACF Pixel ACF Harmonic Reconstruction QML ACF
(Eq. (2.3)) (Eq. (2.5)) ℓmax = 5 ℓmax = 10 ℓmax = 15 ℓmax = 20 (Eq. (2.14))
full sky 7373 8532 8170 7777 7649 7606 8532
KQ85 1401 1781 8250 6953 7612 6383 7234
KQ75 647 963 7913 6914 8233 5139 5764
Table 2.1: Values of the S1/2 statistic for WMAP 5-year ILC maps.
As discussed in the Introduction, Refs. [124; 125] argue that the pixel estimates of the ACF
computed from the masked regions of the sky lead to p-values with respect to the standard ΛCDM
cosmology of ∼ 0.1% or less, suggesting a significant discrepancy between the model and the data.
However, the p-values computed for the unmasked sky are much less significant (∼ 5%). Various
interpretations of this result have been proposed:
(i) The interpretation put forward by [124; 125] is that either correlations have been introduced
in reconstructing the full sky maps from the observations, or that there are highly significant
departures from statistical isotropy that are correlated with the Galactic sky cut leading to an
ACF that is very close to zero for regions outside the sky cut.
(ii) The interpretation put forward by [121] is that the ACF computed over the whole sky is
accurate and unaffected by Galactic contamination. The low p-values arise from are a consequence
of using a sub-optimal estimator of the ACF on a cut sky, combined with a posteriori choices of
the form of the S1/2 statistic.
In support of point (ii), Ref. [121] used a quadratic maximum likelihood (QML) estimator of
60
the power spectrum, CˆQℓ , and computed the ACF
CQ(θ) =
1
4π
∑
ℓ
(2ℓ+ 1)CˆQℓ Pℓ(cos θ). (2.14)
For Gaussian, statistically isotropic, temperature maps, the QML estimates CˆQℓ have a significantly
smaller variance than the PCL estimates CˆPℓ if a sky cut is applied to the data
1. Hence the
estimator CQ(θ) will generally be closer to the truth, i.e. closer to the ensemble mean 〈C(θ)〉,
than the estimator (2.8). (Ref. [134] argues more generally that the QML estimator will be
close to optimal for any theory with a power spectrum close to that of the concordance ΛCDM
cosmology). Applied to the first year WMAP ILC map, [121] found that the ACF estimates
derived from (2.14) are insensitive to a sky cut and lead to p-values for the S-statistic of ∼ 5%,
i.e. no strong evidence against the concordance ΛCDM model.
The reason that the QML estimator has significantly smaller ‘estimator induced’ variance than
the PCL estimator is easy to understand (see [136]). For noise-free band limited data, it is possible
to reconstruct the low multipole coefficients aℓm exactly from data over an incomplete sky. This
is, in effect, what the QML estimator does, though it implicitly assumes statistical isotropy in
weighting the aℓm coefficients to form the power-spectrum (see equation (2.26) below) . For low
multipoles, the assumption of statistical isotropy is unimportant, and for the noise-free data and
sky cuts relevant to WMAP, the low order multipole coefficients and the power spectrum Cℓ can
be reconstructed almost exactly from data on the incomplete sky. In this Chapter, we will extend
the analysis of [121] by explicitly reconstructing the low order coefficients aℓm over the entire sky.
This analysis will confirm that the ACF at large angular scales is insensitive to a sky cut and
leads to p-values of marginal significance.
The ‘estimator induced’ variance of the pixel estimator of the ACF (2.3) is also easy to under-
stand intuitively. (This problem has been discussed extensively in the literature in the context of
angular clustering analysis of galaxy surveys [139; 140; 141; 142]). The ACF is a pair-weighted
statistic. Consider the analysis of data on an incomplete sky. An overdensity, or underdensity,
close to the boundary of the sky cut will almost certainly continue as an overdensity, or under-
density, across the cut. If the pair count is merely corrected by the missing area that lies within
the cut region of sky (as in the estimator (2.3)) overdense and underdense regions close to the
boundary will be underweighted. This causes no bias to the estimator, but increases the sample
variance. The analysis presented in the next Section shows that it is possible reduce this sampling
variance by reconstructing the low multipoles across a sky cut in a way that is numerically stable
and free of assumptions concerning statistical isotropy.
1For noise-free data over the full sky, the QML and PCL estimators are identical.
61
2. LARGE-ANGLE CORRELATIONS IN THE CMB
2.3 Reconstructing low-order multipoles on a cut sky
The aim of this Section is to reconstruct the large-scale features of the temperature anisotropies
over the whole sky using only the incomplete data that lies outside a chosen sky cut. This can be
done in a number of ways, for example, by Weiner or ‘power equalization’ filtering [143], Gibbs
sampling [144; 145] or by ‘harmonic inpainting’ [146]. Here we apply a direct inversion method,
which is insensitive to assumptions concerning the statistical properties of the temperature field.
Let the vector x denote the temperature field on the sky and let the vector a denote the
spherical harmonic coefficients aℓm. The vectors x and a are related by the spherical transform
Y,
x = Ya+ n, (2.15)
where n represents ‘noise’ in the data.
Now consider the reconstruction ae
ae = (YTAY)−1YTAx, (2.16)
for any arbitrary square matrix A. The reconstruction is related to the true coefficients a by
ae = a+ (YTAY)−1YTAn. (2.17)
If the data is noise free, (2.16) recovers the true vector a exactly. If, further, we choose A to be
the identity matrix, then
ae = K−1a˜, (2.18)
where K is the coupling matrix (2.11).
The reconstruction of (2.18) is closely related to the problem of defining an orthogonal basis
set of functions on the cut sky, which has been studied extensively in the literature (see e.g.
[147; 148; 149]). If the sky cut is relatively small, and the data are noise-free and band-limited,
the coupling matrix K will be non-singular and can be inverted to yield the full-sky harmonics
a exactly. If the data are noise-free but not band-limited, the matrix K will become numerically
singular on the incomplete sky as ℓmax →∞. (As a rule-of-thumb the matrix will become singular
if ℓmax exceeds the inverse of the width of the sky cut in radians.) This simply tells us that there
are ‘ambiguous’ harmonic coefficients that are unconstrained by the data outside the sky cut. For
noise-free data that are not strictly band-limited, the solution (2.16) truncated to a finite value of
ℓmax will amplify some of the high frequency signal which will appear as ‘noise’ within the sky cut
in the reconstruction xe = Yae. The amplitude of this ‘noise’ can be reduced by an appropriate
choice of the matrix A. If we assume that the signal and noise are Gaussian, the optimal solution
62
of (2.15) is the familiar ‘map-making’ solution [131]
ae = (YTC−1Y)−1YTC−1x, (2.19)
C = 〈xxT 〉 = S+N. (2.20)
If Gaussianity and statistical isotropy holds, the variance of the (2.19) is
〈aeaeT 〉 = Ca = (YTC−1Y)−1. (2.21)
The statement that the estimator (2.19) is ‘optimal’ and the expression for the variance (2.21),
do of course depend on the assumptions of Gaussianity and statistical isotropy. However, as long
as the noise term in (2.17) is negligible, the reconstructed harmonic coefficients will be identical
to the true harmonic coefficients independent of any assumptions concerning statistical isotropy.
Figure 2.3 illustrates the application of this machinery. The upper row shows the smoothed
WMAP 5-year ILC map (to the left) and the degraded resolution KQ75 mask (to the right).
The remaining figures show the reconstructed maps from the harmonic coefficients (2.19) for the
KQ85 mask (figures to the left) and for the KQ75 mask (figures to the right). The figures show
the reconstructions with ℓmax truncated at 5, 10, 15 and 20. The maps for the two sky cuts at
ℓmax = 5 and 10 are virtually identical, and by ℓ = 10 the reconstructions look visually similar to
the ILC map over the entire sky. For ℓmax = 15 and 20, the reconstructions for the KQ85 mask
are stable and, again, look very similar to the WMAP ILC map over the whole sky. For the KQ75
mask, one can see ‘noise’ (i.e. reconstruction errors) beginning to appear inside the sky cut when
ℓmax is increased to ℓmax = 15 and 20.
However, the high harmonics that contribute to the ‘noise’ in Fig. 2.3 make very little con-
tribution to the correlation function at large angular scales. This is illustrated in Fig. 2.4, which
shows the dependence of the correlation functions
Ce(θ) =
1
4π
ℓmax∑
ℓ=2
(2ℓ+ 1)CeℓPℓ(cos θ), C
e
ℓ =
1
(2ℓ+ 1)
∑
m
|aeℓm|2, (2.22)
on ℓmax for each of the each of the sky cuts. In the case of zero sky cut, the correlation function
stabilises to its final shape by ℓmax = 10; higher multipoles make a negligible contribution to the
correlation function at large angular scales. The reconstructed correlation functions for the KQ85
and KQ75 masks are almost identical to the all-sky correlation function for ℓmax = 5, 10 and 15.
For the KQ75 mask, one can begin to see the effects of reconstruction noise in Ce(θ) for ℓmax = 20,
but the correlation function for the KQ85 mask remains stable.
63
2. LARGE-ANGLE CORRELATIONS IN THE CMB
Figure 2.3: The figure to the left in the top row shows the WMAP 5 year ILC temperature map
smoothed by a Gaussian of FWHM 10◦ and repixelized to a Healpix resolution of NSIDE=16.
The figure to the right in the top panel shows the degraded resolution WMAP KQ75 mask used
in this Chapter. The remaining figures to the left show the reconstructed all-sky maps computed
from data outside the KQ85 sky cut, using the harmonic coefficients computed from equation
(2.19) with the coupling matrices truncated to (from top to bottom) ℓmax = 5, 10, 15 and 20. The
figures to the right show equivalent plots for the reconstructed all-sky maps computed from data
outside the KQ75 sky cut.
64
ℓ (∆T 2)ℓ δ(∆T
2)ℓ Harmonic Reconstruction KQ85 δ(∆T
2)ℓ Harmonic Reconstruction KQ75
ℓ ILC ℓmax = 5 ℓmax = 10 ℓmax = 15 ℓmax = 20 ℓmax = 5 ℓmax = 10 ℓmax = 15 ℓmax = 20
2 98.8 0.28 0.04 0.29 0.22 6.5 2.0 5.0 5.4
3 292.2 0.14 0.26 0.31 0.58 2.4 4.0 15.0 14.1
4 150.3 0.57 0.51 0.99 1.09 13.8 3.9 36.2 38.1
5 254.3 0.36 1.18 1.69 1.41 3.4 13.2 23.6 35.8
6 79.6 − 0.88 1.89 1.85 − 4.8 63.1 39.2
7 132.4 − 1.96 2.80 2.28 − 13.9 37.7 42.8
8 55.3 − 1.72 3.65 2.25 − 9.3 67.8 65.6
9 44.0 − 1.77 2.84 2.17 − 10.6 30.0 44.7
10 45.6 − 1.23 2.89 1.69 − 4.3 43.5 33.3
Table 2.2: Residuals of harmonic reconstructions (all in (µK)2).
65
2. LARGE-ANGLE CORRELATIONS IN THE CMB
Figure 2.4: Reconstructions of the correlation function from equation (2.22) for various choices of
ℓmax and sky cut.
In Table 2.2, we list the mean square temperature at each multipole
(∆T )2ℓ =
1
4π
∑
m
|aILCℓm |2, (2.23)
for the first few multipoles (ℓ ≤ 10) computed from the ILC map, and the residuals
δ(∆T )2ℓ =
1
4π
∑
m
|aILCℓm − aeℓm|2, (2.24)
for our reconstructed maps. For the KQ85 mask, all multipoles with ℓ ≤ 10 are reconstructed
to high accuracy, with residuals δ(∆T )2ℓ ∼ 2µK2 or less. For the KQ75 mask, the reconstruction
begins to break down for multipoles ℓ ≥ 6 if ℓmax ≥ 15 because of the coupling with ‘ambiguous’
modes that cannot accurately be reconstructed within the sky cut. Nevertheless, since the shape of
the correlation function is dominated by the low multipoles, the shape remains reasonably stable
even for ℓmax = 20 (Fig. 2.4d).
This analysis shows that it is possible to reconstruct the low order harmonic coefficients that
contribute to the large angle correlation functions accurately from data on the cut sky. The sky
66
cut is basically irrelevant and so the all-sky form of the correlation function can be reconstructed
from the cut sky irrespective of any assumptions concerning Gaussianity or statistical isotropy
and with only a very weak dependence on the assumed shape of the covariance matrix Cij. Values
for the S1/2 statistic for each of the cases shown in Fig. 2.4 are listed in Table 2.1.
Notice that if we define weighted harmonic coefficients,
β = Ca
−1ae = Y ∗ℓm(θi)C
−1
ij xj , (2.25)
then the power spectrum computed from these weighted coefficients is
yℓ =
1
2
∑
m
|βℓm|2 = xpxqEℓpq, (2.26)
where
Eℓ =
1
2
C−1
∂C
∂Cℓ
C−1. (2.27)
In other words, the power spectrum of the weighted coefficients is identically equivalent to the
QML power spectrum estimator [131; 150]. If statistical isotropy holds, and the data is noise-free,
the quantity
CˆQ = F−1y, (2.28)
provides an unbiased estimate of the power spectrum, where F is the Fisher matrix,
Fℓℓ′ =
1
2
Tr
[
C−1
∂C
∂Cℓ
C−1
∂C
∂Cℓ′
]
. (2.29)
Notice that for the complete sky, and for noise-free data
Ca = Cℓδℓℓ′δmm′ , (2.30)
in the limit ℓmax →∞, i.e. the variance on the aℓm is just the cosmic variance. The Fisher matrix
is
Fℓℓ′ =
(2ℓ+ 1)
2C2ℓ
δℓℓ′, (2.31)
and so the QML estimates CˆQℓ are identical to the PCL estimates. For relatively small sky
cuts such as the KQ85 and KQ75 masks, the Fisher matrix at low multipoles will be almost
diagonal [136] and the recovered power spectrum from the cut sky will be almost identical to the
true power spectrum computed from the whole sky. The QML estimator effectively performs the
reconstruction ae of equation (2.19), but uses the assumption of statistical isotropy to downweight
‘ambiguous’ modes that are poorly constrained by the sky cut.
For small sky cuts, we would therefore expect the QML correlation function estimate (2.14)
to be almost identical at large-angular scales to the correlation functions computed from the
67
2. LARGE-ANGLE CORRELATIONS IN THE CMB
Figure 2.5: As Figure 2.1 but using the QML estimator of equation (2.14).
reconstructed coefficients ae. (They are, of course, mathematically identical for zero sky cut.)
CQ(θ) is expected to behave more stably than Ce(θ) as the sky cut is increased, since the QML
correlation function downweights ambiguous modes. This is exactly what we see when we apply
the QML estimate to the WMAP ILC 5-year ILC maps (see Fig. 2.5). The ACF is almost
independent of the sky cut, confirming the results of [121]. Values of the S1/2 statistic for the
QML ACF estimates are listed in the final column of Table 2.1.
The QML power spectrum estimates are plotted in Fig. 2.6. The power spectrum coefficients
CˆQℓ are extremely stable to the sky cut, varying by only a few tens of (µK)
2 for ℓ ≤ 10. The
figure compares these estimates to the power spectrum estimates for the reconstructed all-sky
maps using equation (2.19). We plot the results for ℓmax = 10, since this value is large enough to
determine the shape of the ACF at large angular scales, but small enough to limit the noise in
the reconstructed maps at high multipoles. The power spectra of the reconstructed maps are very
close to the QML estimates at ℓ ≤ 8, though one can begin to see the effects of reconstruction
noise in the KQ75 case at ℓ > 8. (However, as Fig. 2.4b shows, this reconstruction noise has very
little effect on the shape of the ACF at large angular scales.)
The results of this section show that the low-order multipole coefficients that determine the
behaviour of the correlation function at large angular scales can be reconstructed to high ac-
curacy from data on the incomplete sky, independent of any assumptions concerning statistical
isotropy. The usual motivation for applying a sky cut is to remove regions of the sky that may
be contaminated by residual Galactic emission. However, for the KQ85 and KQ75 sky cuts, the
missing area of sky leads to little loss of information at low multipoles. The low multipoles can
therefore be reconstructed from the data on the incomplete sky. The imposition of the sky cuts
does not remove foreground contamination at these low multipoles: any residual Galactic contri-
bution to the low multipoles in the ILC map is, like the CMB signal, faithfully reproduced by the
reconstructions shown in Fig. 2.3. What a Galactic cut can do is to mask out localized Galactic
68
Figure 2.6: The temperature power spectrum at low multipoles computed from the low resolution
WMAP 5-year ILC map. The points (corrected for the 10◦ FWHM smoothing and slightly dis-
placed in ℓ for clarity) show QML power spectrum estimates for three sky cuts: no mask; KQ85
mask; KQ75 mask. Error bars show the diagonal components of the inverse of the Fisher matrix
(2.29). The solid red line shows the power spectrum computed from the all-sky ILC map (which
is identical to the QML all-sky estimates). The solid green and blue lines show the power spectra
computed for the ℓmax = 10 reconstructions of Fig. 2.3.
emission (‘ambiguous’ modes) that could, in principle, couple to the low multipoles in a way that
depends on the estimator (e.g. via the coupling matrix K in the simple inversion of equation
(2.18)). The similarities between the reconstructions of Fig. 2.3 and the full-sky ILC map (and
the correlation functions and power spectra plotted in Figs. 2.5 and 2.6) show that the ILC map
has removed Galactic emission successfully at low Galactic latitudes, since there is no evidence of
high amplitude ‘ambiguous’ modes in the ILC map within the region of the sky cut.
If suitable estimators are applied to noise-free data, a sky cut of the size of the KQ85 or KQ75
masks has little impact on the reconstruction of the low order multipoles or the all-sky ACF. The
imposition of a sky cut does, however, lead to a loss of information if a poor estimator is used to
estimate the ACF. This is what happens when the pixel estimator (2.3) is used to estimate the
ACF on the cut sky [124; 125; 151]. The analysis presented in this Section provides compelling
evidence that the true value of the S1/2 statistic for our realization of the sky is in the region of
6000-8000(µK)4, independent of the sky cut. The remaining question is whether the anomalously
low p-values implied by the cut-sky pixel ACF are the result of a statistical fluke, or indicative of
new physics.
2.4 Analysis of the S1/2 statistic
In this Section we analyze the S1/2 statistic, first from a Bayesian point of view, and then from a
frequentist point of view. We then discuss the interpretation of the low frequentist p-values found
by [125].
69
2. LARGE-ANGLE CORRELATIONS IN THE CMB
2.4.1 Approximate Bayesian analysis
We begin by performing an approximate Bayesian analysis to compute the posterior distribution
of the S1/2 given the data on the assumption that the fluctuations are Gaussian and statistically
isotropic. If the data were noise-free and covered the entire sky then, under the assumptions of
statistical isotropy and Gaussianity, the data power spectrum Cdℓ provides a loss-free description
of the data. Assuming uniform priors on each of the CTℓ , the posterior distribution of the theory
power spectrum coefficients CTℓ is given by the inverse Gamma distribution
dP (CTℓ |Cdℓ ) ∝
(
Cdℓ
CTℓ
) 2ℓ−1
2
exp
[
−(2ℓ + 1)
2
(
Cdℓ
CTℓ
)]
1
CTℓ
. (2.32)
Each of the CTℓ is statistically independent and the mean value is
〈CTℓ 〉 =
(
2ℓ+ 1
2ℓ− 3
)
Cdℓ . (2.33)
The distribution (2.32) will therefore favour theory values that are larger than the observed values
Cdℓ .
The results of the previous Section (and Fig. 2.6 in particular) show that the low multipoles
are well determined and insensitive to the application of a sky cut. We can therefore use the
measurements Cdℓ computed over the whole sky to represent the data
1. The multipole expansion
is truncated at ℓmax = 20 (although as discussed in the previous Section, multipoles greater
than ℓ ≈ 10 make very little contribution to the ACF at large angular scales) and statistically
independent CTℓ values are generated from the inverse Gamma distribution (2.32). These values
are then used to generate Gaussian aTℓm from which we synthesize real-space maps xi at a Healpix
resolution of NSIDE=16 smoothed with a Gaussian of FWHM 10◦. We then compute S1/2 from
the pixel correlation function (2.3). This methodology provides a test of statistically isotropic,
Gaussian models, with no additional constraints imposed on the theory CTℓ apart from uniform
priors.
The posterior distributions of ST1/2 is shown in Fig. 2.7a for the analysis of all-sky maps (red
histogram) and for maps with the KQ75 mask applied (blue histogram). The distribution for the
trials with the sky cut applied is slightly broader than the distribution for the all-sky trials, as
expected since the pixel ACF estimator is sub-optimal on a cut sky. The peaks of the distributions
occur at ST1/2 ≈ 6000 (µK)4 and so low values of ST1/2 are clearly preferred by the data. However,
the posterior distributions have a very long tail to high values (as expected from the inverse
Gamma distribution 2.32). The best fitting six parameter ΛCDM model as determined from the
5-year WMAP analysis [3] has a value of ST1/2 ∼ 49000 (µK)4. At this value (indicated by the
1It is in this sense that the analysis presented here is described as an ‘approximate’, i.e. any residual errors on
the Cdℓ are ignored.
70
Figure 2.7: (a) Posterior distributions of ST1/2 computed as discussed in the text. The red (solid)
histogram shows the distribution of ST1/2 from an analysis of the whole sky. The blue (dotted)
histogram shows the distribution computed with the KQ75 sky cut applied. The vertical dashed
lines in the figures shows the value ST1/2 ∼ 49000 (µK)4 for the best fitting ΛCDM model as
determined from the 5-year WMAP analysis [3]. (b) Frequency distributions of S1/2 for statistically
isotropic, Gaussian, realizations of the ΛCDM model [3]. The red (solid) histogram shows the
frequency distribution for the pixel ACF estimator applied to the whole sky. The blue (dotted)
histogram shows the distribution computed with the pixel ACF estimator with the KQ75 sky cut
applied.
vertical dashed line in Fig. 2.7a), the posterior distribution has fallen to a value of about 0.4. Such
high values of ST1/2 are evidently not favoured by the data, but they are not strongly disfavoured.
Very low values of ST1/2, of ∼ 1000 (µK)4 are also not strongly disfavoured.
From the Bayesian point of view, the quantity S1/2 is a poor discriminator of theoretical models
and so is relatively uninformative. The posterior distributions of Fig. 2.7a are extremely broad
with a long tail to high values. The data, irrespective of estimator or sky cut, clearly prefer low
values of S1/2 but cannot exclude the value of S
T
1/2 ∼ 49000 (µK)4 expected for the concordance
inflationary ΛCDM model.
2.4.2 Frequentist analysis
We now generate statistically isotropic Gaussian realizations with the CTℓ constrained to those
of the best fitting ΛCDM model. The frequency distributions of S1/2 computed from the pixel
estimator are plotted in Fig. 2.7b. The distributions of Figs. 2.7a and 2.7b look fairly similar,
but the frequentist interpretation is very different. For the all-sky analysis, the p-value of finding
S1/2 < 7373 (µK)
4 is 8% and hence is not statistically significant. However, if we apply the KQ75
mask, the p-value for S1/2 < 647 (µK)
4 is only 0.065%. This result appears strongly significant
and, at face value, inconsistent with the p-value for the all-sky analysis.
71
2. LARGE-ANGLE CORRELATIONS IN THE CMB
Figure 2.8: Pixel based estimates ACF estimates for the WMAP ICL map corrected for the local
ISW contribution for redshift z < 0.3 as described by [132].
The low p-value found here and by [1] and [125] come exclusively from analyzing cut sky maps
with ‘sub-optimal’ (in the sense of not reproducing the ACF for the whole sky) estimators. The
sky cuts, ostensibly imposed to reduce any effects of Galactic emission at low Galactic latitudes,
lead to a loss of information and to poorer estimates of the ACF for our realization of the sky.
But as we have demonstrated, the information on the ACF at large angles for our realization of
the sky is contained in the data outside the sky cuts. The imposition of a sky cut therefore has
little to do with reducing the effects of Galactic emission on the ACF at large angular scales. If
there is any cosmological significance to the low p-values, then one must accept that the Galactic
cut aligns with the signal, purely by coincidence, in just such a way as to remove the large-scale
angular correlations for particular choices of estimator of the ACF. This alignment may indicate
a violation of statistical isotropy, as argued by [125], but if this is true the alignment with the
Galactic plane must be purely coincidental.
It seems to us that a more plausible interpretation of the low p-values is that they are a
consequence of the a posteriori selection of the S1/2 statistic by [1] for a particular choice of
estimator and sky cut. It is difficult to quantify the effects of a posteriori choices. However,
numerical tests with the more general statistic
Spµ =
[
2
3
∫ µ
−1
[C(θ)]p d cos θ
]2/p
, (2.34)
(which reduces to S1/2 for the choices µ = 1/2 and p = 2) suggest that it is possible to alter
p-values by an order of magnitude or more by selecting the parameters in response to the data.
It would be possible to raise the p-values even more by varying the size and orientation of a sky
cut.
Is there any way of testing this hypothesis further? In the ΛCDM model, the integrated Sachs-
72
Wolfe (ISW) effect [152] makes a significant contribution to the total temperature anisotropy signal
at low multipoles. The ISW contribution from the time of last scattering (tLS) and the present
day (t0) is given by
∆T
T
ISW
= 2
∫ t0
tLS
dΦ
dt
dt (2.35)
where Φ is the Newtonian gravitational potential (see e.g. [153]). Recently, Ref. [132] has used
the 2MASS near infrared all-sky survey [154], together with photometric redshift estimates to
compute the ISW contribution from local structure at redshifts z < 0.3. If a posteriori choices are
responsible for the low p-values, we should find large changes to the pixel ACF estimates for the
masked sky when the WMAP ILC maps are corrected for the local ISW contribution. As shown
in Fig. 2.8, this is indeed what we find when we subtract the local ISW contribution computed
by [132] from the 5-year WMAP ILC map. The S1/2 statistic computed from the ACFs shown in
Fig. 2.8 are 10360 (µK)4 (all-sky), 6463 (µK)4 (KQ85 mask) and 5257 (µK)4 (KQ75 mask), all
consistent with the concordance ΛCDM model at the few percent level. This is consistent with
our hypothesis that the [125] low p-values are a fluke, unless one is prepared to argue that there is
a physical alignment of local structure with the potential fluctuations at the last scattering surface
that conspires to remove large-angle temperature correlations in the regions outside the Galactic
mask (which seems implausible to us). Francis and Peacock [132] discuss how the local ISW
correction affects a number of other low multipole statistics, in particular reducing the statistical
significance of the alignment between the quadrupole and octopole.
2.5 Discussion and Conclusions
The low amplitude of the temperature autocorrelation function at large angular scales has led to
some controversy since the publication of the first year results from WMAP. This Chapter has
sought to clarify the following points:
[1] We have compared different estimators of the ACF showing: (a) how they depend on assump-
tions of statistical isotropy; (b) how they are interrelated; (c) how they perform on the WMAP
5-year ILC maps with and without a sky cut.
[2] The imposition of the KQ85 and KQ75 sky masks leads to little loss of information on the
low multipoles that contribute to the large-scale angular correlation function. As demonstrated
in Section 2.3, the low multipole harmonics can be reconstructed accurately from the data that
lie outside the sky cuts, independent of any assumptions concerning statistical isotropy and with
only a weak dependence of the shape of the pixel covariance matrix Cij . The ACFs computed
from these reconstructions are in good agreement with the ACF computed from the whole sky
and in good agreement with the maximum likelihood estimator (2.14). There can be little doubt
that the large-scale ACF for our realization of the sky is very close to the all-sky results shown in
Figs. 2.1 and 2.5.
73
2. LARGE-ANGLE CORRELATIONS IN THE CMB
[3] The lack of large-scale structure seen in the cut-sky pixel ACF arises from a particular alignment
of the low order multipoles (see also [134]) which, as we have demonstrated, can be reconstructed
accurately from the data outside the sky cut. The ACF at large angular scales is insensitive to
high order modes localised within the sky cut.
[4] The Bayesian analysis presented in Section 2.4 shows that the posterior distribution of the
ST1/2 is broad and cannot exclude the value S
T
1/2 ∼ 49000 (µK)4 appropriate for the [3] best fitting
inflationary ΛCDM model. The breadth of the posterior distribution ST1/2 distribution shows that
it is fairly uninformative statistic and so is not a particularly good discriminator of theoretical
models.
[5] Unusually low values of the S1/2 statistic are found only if ‘sub-optimal’ ACF estimators are
applied to maps that include a Galactic mask. We have argued that the low p-values associated
with these low values of S1/2 are most plausibly a result of a posteriori choices of statistic. This
seems plausible to us because: (a) S-type statistics are relatively uninformative and hence sensitive
to a posteriori choices; (b) the all-sky ACF (which is compatible with the concordance ΛCDM
model) can be recovered from the data outside the mask and so any physical model for the low p-
values requires a fortuitous alignment of the temperature field with the Galaxy; (b) the analysis of
the local ISW corrected maps presented in Section 2.4 suggests that any physical model of the low
p-values requires a precise alignment of local structure with the large-scale potential fluctuations
at the last scattering surface.
[6] If one does not accept that the low-p values are associated with a posteriori choices, then one
must accept that they may be indicative of new physics as suggested by [125].
In summary, the results of this Chapter suggest to us that, irrespective of the imposition of
Galactic sky cuts or assumptions of statistical isotropy, the large-scale correlations of the CMB
temperature field provide unconvincing evidence against the concordance inflationary ΛCDM cos-
mology.
74
Chapter 3
Testing a direction-dependent
primordial power spectrum with CMB
observations
3.1 Introduction
In recent years, observations of the cosmic microwave background (CMB) radiation fluctuations
by the Wilkinson Microwave Anisotropy Probe (WMAP) and a large number of ground-based and
suborbital experiments have led to a precise measurement of the temperature anisotropy power
spectrum up to multipoles of a few thousand ([10; 61; 155; 156; 157; 158]) . Apart from some
claimed “anomalies” (see below) the observations are consistent with a dark-energy-dominated
cosmology with statistically isotropic, Gaussian, adiabatic perturbations, as expected from simple
models of inflation. (We will refer to this as the “concordance” ΛCDM model.) If statistical
isotropy applies, then the harmonic coefficients of the temperature field,
aTlm =
∫
dΩY ∗lm(Ω)∆T (Ω), (3.1)
must satisfy
CTTlm,l′m′ =
〈
aTlma
T∗
l′m′
〉
= CTTl δll′δmm′ . (3.2)
If the fluctuations are Gaussian and statistically isotropic, their statistical properties are com-
pletely described by the power spectrum CTTl .
There have been some hints of “anomalies” in the WMAP data, perhaps suggesting a violation
of statistical isotropy. These include alignments of low-l multipoles [126; 128; 159; 160], evidence
for power asymmetry [161; 162] and for a deep cold spot in the southern Galactic hemisphere
[163; 164]. For the most part, these anomalies have been found by examining the data without
reference to specific theoretical models. There is, therefore, an a posteriori aspect in computing
75
3. TESTING A DIRECTION-DEPENDENT PRIMORDIAL POWER
SPECTRUM WITH CMB OBSERVATIONS
their statistical significance which is difficult to assess [133]. Some authors have, however, claimed
highly significant discrepancies between the CMB data and the concordance ΛCDM model [165].
Interest in the CMB anomalies has motivated theorists to build inflationary models that violate
rotational invariance, either via the addition of a vector field [166; 167; 168] or via an isocurvature
perturbation [169; 170]. In addition, a number of phenomenological models that violate statistical
isotropy have been proposed, which can be tested against observations (e.g. [166; 171; 172]).
Reference [166] considers the phenomenological model in which the primordial power spectrum
depends on a preferred direction,
P(k) = Ps(k)[1 + g(k)(k · n)2], (3.3)
where g(k) is some arbitrary function of wavenumber, and Ps(k) is the isotropic primordial power
spectrum (1.35). There has been considerable interest in this model recently. The authors of [173]
applied Gibbs sampling to the WMAP five-year maps to test models with g(k) = g∗ = constant,
finding strong evidence for a preferred direction with g∗ = 0.15 ± 0.039 using multipoles l ≤
400. Hanson and Lewis [174] corrected some algebraic errors in the analysis of [173] and applied
a simpler quadratic estimator to the WMAP 5-year data. These authors also found evidence
for a highly significant (9σ) departure from statistical isotropy, but with a preferred direction
suspiciously close to the ecliptic poles, suggestive of some type of systematic effect in the WMAP
data. This analysis was confirmed by [175]. A subsequent analysis [176] showed that asymmetries
in the WMAP beams fully account for the observed violation of statistical isotropy.
Despite this negative conclusion, it is still important to assess the prospects of constraining
violations of statistical isotropy in the CMB with more precise experiments. In this Chapter, we
focus on constraints from the Planck satellite, which was launched successfully in May 2009 and
has recently completed its third full scan of the sky. The Planck satellite has much higher signal-
to-noise ratio in both temperature and polarization than WMAP [27]. It also has higher angular
resolution — 5 arcmin full-width at half-maximum (FWHM) at frequencies ≥ 217 GHz — so
asymmetries on scales of the beam width should have little effect at multipoles l ≤ 1000. It is also
expected that the Planck beams will be calibrated to high precision from scans of bright planets
[177]. Following the successful launch of Planck, the European Space Agency (ESA) has approved
a mission extension until the on-board cryogens are depleted. Planck is therefore expected to
produce almost five sky surveys, compared to the two sky surveys approved for the nominal
mission. This combination of high sensitivity, high resolution and extended lifetime allows greater
scope for testing for systematic effects than was possible with WMAP. For example, it becomes
possible to use polarization maps independently of temperature maps to test for violations of
statistical isotropy.
In this Chapter, we extend the analysis of [178] to assess how accurately an extended Planck
mission can be used to test models with an anisotropic primordial power spectrum. This Chapter
76
is organized as follows: In Section 3.2, we summarize some basic properties of the anisotropic
model. Section 3.3 then applies the quadratic estimator of [174], extended to include polarization,
to compute forecasts for Planck. Our conclusions are presented in Section 3.4.
3.2 The anisotropic model
3.2.1 Covariance matrix
We write the anisotropic primordial power spectrum as
P(k) = Ps(k)
(
1 +
∑
LM
gLM(k)YLM(kˆ)
)
. (3.4)
We assume that parity-invariance continues to hold in the mean so that L is restricted to even
values such that P(−k) = P(k). In this Chapter, we consider a quadrupole modulation, i.e. L = 2,
|M | ≤ 2, with a power-law scale dependence on the wave number gLM(k) = gLM(k0/k)q where the
pivot scale k0 = 0.002Mpc
−1. Scale-invariant modulation corresponds to q = 0.
The harmonic coefficients of the CMB anisotropy can be expressed as
aXlm = 4πi
l
∫
d3k
(2π)3
∆Xl (k)ζ(k)Y
∗
lm(kˆ), (3.5)
where ∆Xl (k) are the adiabatic transfer functions, either for temperature (X = T ) or E-mode po-
larization (X = E). The primordial curvature perturbation is ζ(k) with statistically-homogeneous
but anisotropic correlations
〈ζ(k)ζ∗(k′)〉 = (2π)3δ3(k− k′)2π
2
k3
P(k), (3.6)
with P(k) given by Eq. (3.4). Thus, the covariance matrix of the harmonic coefficients is
CXX
′
l1m1,l2m2 =
〈
aXl1m1a
X′∗
l2m2
〉
= CXX
′
l1 δl1l2δm1m2 + δC
XX′
l1m1,l2m2 , (3.7)
where
CXX
′
l1
= 4π
∫
d ln kPs(k)∆
X
l1
(k)∆X
′
l1
(k) (3.8)
is the usual isotropic power spectrum. The additional term in Eq. (3.7) due to the power asym-
metry is
δCXX
′
l1m1,l2m2
= il1−l2C˜XX
′
l1l2
(q)
∑
LM
gLM
∫
dΩkYLM(kˆ)Y
∗
l1m1
(kˆ)Yl2m2(kˆ)
77
3. TESTING A DIRECTION-DEPENDENT PRIMORDIAL POWER
SPECTRUM WITH CMB OBSERVATIONS
= il1−l2C˜XX
′
l1l2 (q)
∑
LM
gLM(−1)m1
×
[
(2L+ 1)(2l1 + 1)(2l2 + 1)
2π
] 1
2
(
L l1 l2
0 0 0
)(
L l1 l2
M −m1 m2
)
, (3.9)
where
C˜XX
′
l1l2
(q) = 4π
∫
d ln kPs(k)∆
X
l1
(k)∆X
′
l2
(k)
(
k0
k
)q
. (3.10)
Since L is even, the anisotropic covariance is nonzero only for l1 − l2 even as required by parity
invariance. In the following it will also be convenient to introduce 2× 2 matrices Cl1m1,l2m2 and
C˜l1l2 with elements C
X1X2
l1m1,l2m2
and C˜X1X2l1l2 (q) respectively.
3.2.2 Quadratic estimators and the Fisher matrix
Here we assume that the scale-dependence of the power asymmetry (i.e. q) is known. We can then
use the quadratic estimator of Ref. [174], extended to polarization, to form estimates gˆLM of the
anisotropy parameters. For an isotropic survey and in the limit of small primordial anisotropy,
these take the form
gˆLM =
1
2
∑
L′M ′
F−1LM,L′M ′
∑
X1l1m1
∑
X2l2m2
a¯X1∗l1m1
∂CX1X2l1m1,l2m2
∂g∗L′M ′
a¯X2l2m2 . (3.11)
Here, a¯Xlm ≡
∑
X′ [(C
tot
l )
−1]XX
′
aX
′
lm are the temperature and polarization multipoles after weighting
with the inverse of their isotropic total (signal-plus-noise) covariance matrix. The Fisher matrix,
evaluated at gLM = 0, is given by
FLM,L′M ′ =
1
2
∑
l1m1
∑
l2m2
Tr
[
(Ctotl1 )
−1∂Cl1m1,l2m2
∂g∗LM
(Ctotl2 )
−1∂Cl2m2,l1m1
∂gL′M ′
]
. (3.12)
In the limit of vanishing primordial anisotropy, the inverse of this Fisher matrix equals the covari-
ance of the errors on gˆLM , i.e. F
−1
LM,L′M ′ = 〈gˆLM gˆ∗L′M ′〉.
The assumed isotropy of the survey ensures that the Fisher matrix at gLM = 0 is diagonal.
Using Eq. (3.9), the diagonal elements evaluate to
FLM,LM =
∑
l1l2

Tr(C˜l1l2(Ctotl2 )−1C˜l2l1(Ctotl1 )−1) (2l1 + 1)(2l2 + 1)8π
(
l1 l2 L
0 0 0
)2 . (3.13)
Since gLM is complex for M 6= 0, we present our simulation results in the next section in
terms of GˆLM =
√
2Re(gˆLM) and GˆL−M =
√
2Im(gˆLM) for M > 0, and GˆL0 = gˆL0. The GˆLM are
uncorrelated and have the same variance as the gˆLM .
78
3.3 Forecasts for Planck
3.3.1 Constraints on the anisotropy amplitude
In this section, we consider forecasts for the Planck mission. We use the parameters for the
143 GHz channel (the most sensitive of the Planck frequency channels) as given in [27]. We there-
fore assume a Gaussian beam with FWHM of 7.1 arcmin and assume uncorrelated isotropic noise
in the temperature and polarization maps with root-mean-square noise levels of σT and σP respec-
tively. For the nominal two-sky-survey mission (one year of observation) we adopt σT = 12.2 µK
and σP = 23.3 µK in 3.4 arcmin pixels (Healpix [137] resolution Nside = 1024), corresponding to
42µK-arcmin noise in temperature and 80µK-arcmin in Stokes Q and U polarization. We also
consider an extended mission of four complete sky surveys (two years of observation) with σT and
σP reduced by a factor of
√
2.
To give a feel for the nature of the anisotropy signal, Figs. 3.1 and 3.2 show simulated sky
maps for a noise-free realisation of a scale-invariant (q = 0) quadrupole-modulation model with
g20 = 0.1 (g2M = 0, M 6= 0). These maps have been generated using the prescription described
in [174], generalized to polarization. This uses an approximate square root of the anisotropic
covariance matrix, linear in the gLM , to simulate maps as the sum of a statistically isotropic part
and an anisotropic part. The isotropic component of a noise-free temperature map is shown on
the left in Fig. 3.1 and the anisotropic component, clearly showing a preferred direction along the
polar axis, is shown on the right.
The anisotropic contribution to the Stokes Q and U polarization maps is shown in Fig. 3.2.
Here, we have smoothed the polarization maps with a Gaussian of FWHM 3 deg. to enhance
the visual impact of the statistical anisotropy in the Q Stokes map. Because of the quadrupole
asymmetry in the primordial power, modes of the primordial perturbation with wavevectors along
the polar axis tend to have their amplitude enhanced. For such modes, the polarization generated
by Thomson scattering is pure Q in the polar basis and varies in amplitude with the polar angle as
sin2 θ [76]. The dominant effect of the statistical anisotropy is therefore observed in the Q Stokes
parameter and is concentrated toward the equatorial plane.
To illustrate the machinery summarised in Section 3.2.2, we have generated five simulations
of the scale-invariant quadrupole-modulation model with g2M = 0.1δ0M and added instrumental
noise appropriate to one year of observation with the Planck 143 GHz channel. We then estimate
g2M via Eq. (3.11) as a function of lmax, the maximum multipole retained in the analysis. Since
the only nonzero coefficient in these simulations is g20, and the survey is assumed isotropic, the
recovered estimates gˆ2M are statistically equivalent for M 6= 0 and so we show results only for
the two (real) components Gˆ20 and Gˆ2−2 in Fig. 3.3. We analyse temperature alone (top panels),
E-mode polarization alone (middle), and both jointly (bottom).
With temperature alone, the errors on G2M decrease approximately as 1/lmax over the range
79
3. TESTING A DIRECTION-DEPENDENT PRIMORDIAL POWER
SPECTRUM WITH CMB OBSERVATIONS
Figure 3.1: Noise-free simulation of a model with scale-invariant quadrupole asymmetry in the
primordial power with gLM = 0.1δL2δM0. The isotropic component of the temperature map is
shown on the left, and the anisotropic component on the right. The colour scales are in µK.
Figure 3.2: Anisotropic components of the Q (left) and U (right) polarization maps in a noise-free
simulation of the model in Fig. 3.1. The maps have been smoothed here with a Gaussian beam
of FWHM 3◦ to enhance the imprint of the preferred axis in the Q map. The colour scales are in
µK.
plotted in Fig. 3.3 (l ≤ 1500) reaching 0.005 by lmax (in agreement with the minimum-variance
estimators of [178]). This behaviour follows from simple mode-counting since the temperature
maps are signal-dominated over this multipole range.1 However, the polarization maps are noise-
dominated over much of this multipole range and so the errors approach constant values for
lmax & 600. Nevertheless, the Planck polarization maps alone can provide (almost) independent
constraints on an anisotropic modulation to the temperature maps. For two sky surveys, the errors
on the g2M from polarization are four times worse than in temperature. Consistency between
temperature and polarization constraints would provide an important test of systematic effects
should Planck show any evidence of an anisotropic power spectrum.
1For modes that are signal-dominated, the scale dependence of the trace term in the Fisher matrix, Eq. (3.13),
is weak. Treating the trace as constant gives Fisher information varying as l2max which is proportional to the number
of modes retained in the analysis.
80
Figure 3.3: Estimates of the G20 (left) and G2−2 (right) anisotropy parameters (shown with points)
and their (one-sigma) Fisher errors ([Red] solid lines) as a function of lmax from five simulations of
the model in Fig. 3.1 for one year of Planck data. The input parameters G20 = 0.1 and G2−2 = 0.0
are shown with horizontal [Blue] solid lines. From top to bottom we analyse temperature only,
E-mode polarization only and temperature plus polarization.
3.3.2 Constraints on scale-dependence
In Fig. 3.4, we compare the Fisher errors on the amplitude of the modulation for a scale-invariant
model (q = 0) and two models with scale-dependence (q = 1 and q = 2). For larger q, the
asymmetry in the variance of the Fourier modes is confined to larger scales and so relatively more
of the constraining power derives from low-l multipole moments. The low-l modes of polarization
are enhanced by scattering at reionization [179] and are expected to be signal-dominated in the
one-year Planck data. The polarization constraints on the g2M therefore become more comparable
81
3. TESTING A DIRECTION-DEPENDENT PRIMORDIAL POWER
SPECTRUM WITH CMB OBSERVATIONS
0 200 400 600 800 1000 1200 1400
0.00
0.01
0.02
0.03
0.04
0.05
0.06
lmax
D
G
20
2y TT+EE+TE
1y TT+EE+TE
2y E Only
1y E Only
2y T Only
1y T Only
200 400 600 800 1000 1200 1400
0.00
0.05
0.10
0.15
0.20
lmax
D
G
20
2y TT+EE+TE
1y TT+EE+TE
2y E Only
1y E Only
2y T Only
1y T Only
Power k-1 model
10 20 30 40 50
0.010
0.015
0.020
0.025
lmax
D
G
20
1y TT+EE+TE
1y E Only
1y T Only
Power k-2 model
Figure 3.4: Fisher errors for G20 from temperature, E-mode polarization, and temperature plus
polarization in models with power-asymmetry spectral indices q = 0 (left), q = 1 (middle) and
q = 2 (right). For q = 0 and q = 1 we show results for one and two years of observations; for q = 2
we show only the one-year errors since they improve very little with further observing time. Note
that the one- and two-year errors from temperature alone are indistinguishable when q ≥ 0.
to those from the temperature as q increases and the improvement from observing for longer in
polarization lessens.
If the amplitude of any primordial power asymmetry is high enough, it might be possible to
constrain the scale-dependence of the asymmetry with Planck. To forecast constraints on the
spectral index of the power asymmetry, q, we extend the Fisher matrix analysis of Sec. 3.2.2 to
include q as a parameter. We must now evaluate the Fisher matrix at nonzero gLM but we assume
that the asymmetry is still small enough that we can neglect asymmetry in the (Ctot)−1 terms.
The FLM,L′M ′ is then unchanged from Eq. (3.13) but the additional elements are
FLM,q =
1
2
∑
l1m1
∑
l2m2
Tr
[
(Ctotl1 )
−1∂Cl1m1,l2m2
∂g∗LM
(Ctotl2 )
−1∂Cl2m2,l1m1
∂q
]
=
∑
l1l2

Tr(C˜l1l2(Ctotl2 )−1∂qC˜l2l1(Ctotl1 )−1) gLM (2l1 + 1)(2l2 + 1)8π
(
l1 l2 L
0 0 0
)2 ,(3.14)
Fq,q =
1
2
∑
l1m1
∑
l2m2
Tr
[
(Ctotl1 )
−1∂Cl1m1,l2m2
∂q
(Ctotl2 )
−1∂Cl2m2,l1m1
∂q
]
=
∑
l1l2
[
Tr
(
∂qC˜l1l2(C
tot
l2 )
−1∂qC˜l2l1(C
tot
l1 )
−1
) (2l1 + 1)(2l2 + 1)
8π
×
∑
LM
|gLM |2
(
l1 l2 L
0 0 0
)2 . (3.15)
Note that these elements vanish if gLM = 0. From the scaling of the Fisher matrix elements with
gLM , we expect the error on q to scale as the inverse of the amplitude of the asymmetry.
As an example, we consider a fiducial model with scale-invariant quadrupole asymmetry with
G2M = 0.03δM0. Such a model is compatible with the constraints on quadrupole asymmetry from
the beam-corrected analysis of WMAP data in Ref. [176], but the amplitude should be detectable
82
∆q (1σ) T only E only T and E
One year 0.399 1.300 0.322
Two years 0.389 0.878 0.299
Table 3.1: Fisher errors on the scale-dependence of the power asymmetry assuming a scale-
invariant quadrupole asymmetry with g2M = 0.03δM0.
with Planck at the 6σ level (fixing q = 0). Forecasts for the (marginalised) errors on q for this
model are given in Table 3.1. With temperature and polarization, Planck should constrain the
spectral index to a 1σ accuracy of ∆q ∼ 0.3. The marginalised errors on the gLM are similar to
the case in which q is fixed.
3.3.3 Axisymmetric models
The preceding analysis makes no assumptions about axisymmetry of the primordial power asym-
metry. However, if we have good reason to expect axisymmetry, so the model is described by a
preferred axis mˆ and the M = 0 multipoles {g∗L} in a frame with the polar axis along mˆ, we can
constrain these parameters by post-processing our estimates gˆLM . We illustrate how this works,
assuming a fixed scale dependence for the primordial asymmetry.
For models with a nearly scale-invariant anisotropy spectrum, many small-scale modes con-
tribute to the gˆLM , so we might expect the statistics of the gˆLM to be approximately Gaussian.
Expressing mˆ in terms of its azimuthal angle α and polar angle β, mˆ = D(α, β, 0)zˆ (i.e. a rotation
of the zˆ direction through Euler angles α and β), in the Gaussian approximation we can write
Pr({gˆLM}|mˆ, {g∗L}) ∝ exp(−χ2/2) where
χ2 =
∑
LM
∑
L′M ′
(gˆ∗LM − g˜∗LM)FLM,L′M ′(gˆL′M ′ − g˜L′M ′). (3.16)
Here,
g˜LM ≡ DLM0(α, β, 0)g∗L (3.17)
are the multipoles of the primordial asymmetry rotated from their preferred frame. (DLMM ′(α, β, γ)
are the Wigner rotation matrices.) If we now assign a uniform prior on the direction mˆ [so that
Pr(α, β)dαdβ = (4π)−1dαd cosβ = (4π)−1dmˆ] and a flat prior on the g∗L, Bayes’ theorem gives
for the posterior
Pr(mˆ, {g∗L}|{gˆLM})dmˆ = Pr(α, β, {g∗L}|{gˆLM})dαdβ
∝ e−χ2/2dmˆ. (3.18)
For an isotropic survey (and assumed weak anisotropy), the Fisher matrix is isotropic and we
can write FLM,L′M ′ = δLL′δMM ′/σ
2
L. Substituting Eq. (3.17) into Eq. (3.16) and using the addition
83
3. TESTING A DIRECTION-DEPENDENT PRIMORDIAL POWER
SPECTRUM WITH CMB OBSERVATIONS
theorem for the rotation matrices, we have
χ2 =
∑
L
1
σ2L
(
g2∗L − 2g∗L
∑
M
ℜ [gˆLMDL∗M0(α, β, 0)]+∑
M
|gˆLM |2
)
. (3.19)
Noting that DL∗M0(α, β, 0) =
√
4π/(2L+ 1)YLM(mˆ), if we define
gˆL(mˆ) ≡
∑
M
gˆLMYLM(mˆ), (3.20)
(i.e. a map of the estimated gˆLM at multipole L), we can write the posterior as
Pr(mˆ, {g∗L}|{gˆLM})dmˆ ∝
∏
L
exp
(
− 2π
(2L+ 1)σ2L
[g∗LYL0(zˆ)− gˆL(mˆ)]2
)
× exp
(
2π
(2L+ 1)σ2L
gˆ2L(mˆ)
)
dmˆ. (3.21)
The marginal distribution for the direction of the axis is given by integrating over g∗L:
Pr(mˆ|{gˆLM})dmˆ ∝
∏
L
exp
(
2π
(2L+ 1)σ2L
gˆ2L(mˆ)
)
dmˆ, (3.22)
so that contours of constant density are given by the contours of
∑
L gˆ
2
L(mˆ)/[(2L+1)σ
2
L]. For the
simple case of a primordial power asymmetry at a single multipole L, e.g. a quadrupole asymmetry,
ln Pr(mˆ|{gˆLM}) is proportional to the square of the map of the reconstructed multipoles.
To illustrate these ideas, in Fig. 3.5 we plot the marginal distributions for the direction and
amplitude of the model analysed in Sec. 3.3.1 using one of the simulations shown in Fig. 3.3.
We parameterise the direction by the Cartesian components of the equal-area projection x =
2 sin(β/2) cos(α) and y = 2 sin(β/2) sin(α). With one year of simulated Planck temperature data,
the constraints on the direction of the axis are ∆x ≈ ∆y = 0.032 (i.e. 1.8◦) at 68% confidence
and the x and y components are almost uncorrelated as expected from isotropy. The amplitude
g∗2 = 0.1± 0.005 (68% confidence) with the error being very nearly σ2. With polarization alone,
these constraints weaken to ∆x ≈ ∆y = 0.10 (i.e. 5.7◦) and g∗2 = 0.11 ± 0.02. The maximum
of the posterior in this simulation is at x = 0.042, y = −0.035 and g∗2 = 0.099 (T only) and
x = −0.033, y = 0.062 and g∗2 = 0.11 (E only) and the χ2 at these values are 0.223 and 0.236
respectively, with 5− 3 = 2 degrees of freedom.
3.4 Conclusions
As summarized in Sec. 3.1, cosmological perturbations are usually assumed to satisfy statistical
isotropy, as expected in simple models of inflation. Yet there is some tentative observational
84
-0.4 -0.2 0.0 0.2 0.4
-0.4
-0.2
0.0
0.2
0.4
x
y
99% CL H1y E OnlyL
95% CL H1y E OnlyL
68% CL H1y E OnlyL
99% CL H1y T OnlyL
95% CL H1y T OnlyL
68% CL H1y T OnlyL
0.05 0.10 0.15
0
10
20
30
40
50
60
70
g* values
LH
g *
L Input g*=0.1
1y E Only
1y T Only
Figure 3.5: Marginal distributions for the direction (left) and amplitude (g∗2, right) from a sim-
ulation of the nominal (one-year) Planck survey for a model with an axisymmetric quadrupole
asymmetry aligned with the polar axis (gLM = 0.1δL2δM0). We parameterise the direction with
the equal-area projection x = 2 sin(β/2) cos(α) and y = 2 sin(β/2) sin(α) and show in solid lines
the 68% [Red], 95% middle [Blue] and 99% outer [Green] contours from the temperature alone;
dashed-line contours are from the E-mode polarization alone. For the amplitude (right), we plot
the marginal distributions from T alone (solid line) and E alone (dashed line).
evidence from the CMB suggesting possible departures from statistical isotropy. Here we have
shown that Planck should be able to set strong constraints on small departures from statistical
anisotropy and, under favourable circumstance, could set constraints on their scale dependence
and any preferred direction.
In this Chapter we have developed and applied quadratic estimators to test for an asymmetry
in the primordial power spectrum from temperature and polarization measurements of the CMB.
Our estimators are optimal in the limit of isotropic primordial power. We have tested our methods
against simulations that include a quadrupole power asymmetry.
We have analysed the ability of the Planck mission to constrain models with quadrupole power
asymmetry using temperature and polarization data. Using temperature data alone, Planck should
be able to constrain each multipole g2M of a scale-invariant quadrupole anisotropy at the 0.01 level
(2σ), well below the current constraints derived fromWMAP (|g2M | < 0.07 from Ref. [176]). Using
polarization data alone from an extended Planck mission (four sky surveys) such an anisotropy can
be constrained to an accuracy only about three times worse than from the temperature. This offers
the possibility of a consistency check on the existence on any observed departure from statistical
isotropy.
If the amplitude of a power asymmetry is large enough, it may be possible to constrain its scale
dependence. We have estimated the Fisher errors when additionally constraining a free spectral
index describing a power law modulation, gLM(k) = gLM(k0/k)
q. For a scale-free quadrupole
modulation with an amplitude of 1% (i.e. g20/
√
4π = 0.01 in an axi-symmetric model), we find
that an extended Planck mission can constrain the spectral index to a 1σ accuracy of ∆q ∼ 0.3.
85
3. TESTING A DIRECTION-DEPENDENT PRIMORDIAL POWER
SPECTRUM WITH CMB OBSERVATIONS
Finally, we have considered the constraints on a preferred direction in models with a purely
axisymmetric modulation of the primordial power spectrum. In a scale-free model with a 1%
quadrupole modulation, the direction of the preferred axis can be determined from Planck data to
a precision of about 2◦ using temperature observations alone and to about 6◦ using polarization
data alone.
The quadratic estimators developed here for isotropic surveys can, in principle, be straight-
forwardly extended to deal with real-world effects such as anisotropic noise and Galactic masks.
However, the calculation of the a¯Xlm in Eq. (3.11) requires inverse weighting the temperature and
polarization data with the full covariance matrix for the anisotropic survey. This has been done for
the WMAP temperature data at its native resolution [174; 180] but extending this to polarization
and the resolution required for Planck requires further work. In practice, fast estimators can still
be constructed for anisotropic surveys, with moderate loss of performance, by replacing a¯Xlm with
some heuristically-weighted pseudo multipoles (i.e. those computed directly on the masked sky)
following techniques used for CMB power spectrum estimation (e.g. see [136] for a review). In
either case, care must be taken to subtract the mean-field response – that obtained on average for
no primordial power asymmetry – from the quadratic estimator since this is no longer confined
to the L = 0 mode for an anisotropic survey. The mean-field and the estimator normalisation are
then generally best determined by Monte-Carlo simulations.
86
Chapter 4
Constraints on the standard and
non-standard models of the early
Universe from CMB B-mode
polarisation
4.1 Introduction
Observations of the Cosmic Microwave Background (CMB) radiation have proved a valuable tool
for studying the physics of the very early Universe. Scalar, vector and tensor perturbations
generated in the early Universe have left observable imprints in the temperature and polarization
anisotropies of the CMB. Recent experiments, including the Wilkinson Microwave Anisotropy
Probe (WMAP) satellite [61; 72; 155], QUaD [156], BICEP [182] and others [183; 184; 185;
186; 187], have led to a precise determination of the basic parameters of the standard ΛCDM
cosmological model, including the parameters describing the primordial density perturbations.
According to this concordance model, the Universe underwent a period of near-exponential
expansion, termed inflation, at very early times. The standard model of inflation is based on
the single field slow-roll scenario. In this scenario, the expansion is driven by a scalar field (the
inflaton) gradually rolling down a flat potential during the inflationary stage. Inflation ended
when the slow-roll conditions were broken, and the inflaton decayed into relativistic particles
which re-heated the Universe.
In spite of many phenomenological successes of inflation based on effective field theory, serious
problems remain concerning the origin of the scalar field driving inflation, namely the singularity
problem [188] and the trans-Plankian problem [189]. Consequently, efforts have been made to
realize inflation in a more natural way from some fundamental theory of microscopic physics.
Brane inflation [50; 190; 191] from high dimensional string theory is a typical example. In this
87
4. CONSTRAINTS ON THE STANDARD AND NON-STANDARD MODELS OF
THE EARLY UNIVERSE FROM CMB B-MODE POLARISATION
scenario, the Universe is embedded into a high dimensional warped space-time. The anti-brane
is fixed at the bottom of a warped throat, while the brane is mobile and experiences a small
attractive force towards the anti-brane. Inflation ends when the brane and the anti-brane collide
and annihilate, initiating the hot big bang epoch. During the brane collision, cosmic strings would
be copiously produced, and would leave an imprint on the CMB sky [47; 48]. Searching for this
cosmic string signal in the CMB is an important way to test the correctness of this scenario.
Another approach for realizing a period of inflation, based on Loop Quantum Gravity (LQG)
has been proposed recently (see [192] for instance). LQG is a non-perturbative and background-
independent quantization of General Relativity. Based on a canonical approach, it uses Ashtekar
variables, namely SU(2) valued connections and conjugate densitized triads. The quantization
is obtained through holonomies of the connections and fluxes of the densitized triads. More
importantly, when the energy density of the Universe was approaching the critical density ρc, the
Universe entered into a bouncing period due to repulsive quantum geometrical effects. Thus, the
big bang is replaced by a “big bounce”. This, to some extent, avoids the singularity problem in
the standard ΛCDM model.
Differentiating between these three classes of models (single field inflation, brane inflation and
loop quantum cosmology), which are motivated by different microscopic physics, is a crucially
important goal for modern cosmology. Since the primordial scalar, vector and tensor perturba-
tions produced in these models are quite different from each other, they will in general leave
different signatures in the CMB radiation. Numerous authors have constrained the parameters of
single field inflation models from CMB and large scale structure observations. (See for example
[72; 156; 193] for some recent analyses.) Most of these analyses have made use of both temper-
ature and polarization CMB measurements and their constraints have been dominated by the
temperature measurements. However, with the advent of a new generation of CMB polarization
measurements [156; 182], it is now possible to obtain meaningful constraints from measurements
of the polarization of the CMB. Since B-mode polarization on very large scale is generated only by
tensor perturbations in the early Universe, this is a particularly attractive technique: a detection
of B-mode polarization at large enough angular scales must be due to gravitational waves (tensor
perturbations) and the connection with early Universe physics is then very clear. In addition,
small scale B-mode polarization is possibly sourced by cosmic string. Note that on small scales,
B-modes are also generated by gravitational lensing of the dominant E-mode polarization signal.
Therefore, for testing models of the early Universe, large-angular scale measurements are required
in order to avoid confusion from gravitational lensing.
In this Chapter, we extend the investigation of inflationary constraints from CMB B-mode
polarization measurements alone by considering the constraints obtainable on the three classes of
models described above. The Chapter is organized as follows. In Section 4.2.1, we briefly review
the characterization of CMB polarization in terms of E- and B-modes. In Section 4.2.2, we present
the currently available B-mode constraints, and the predicted noise levels for some current and
88
future CMB experiments. In Section 4.2.3, we first present the likelihood and hyper-parameter
analysis methods, which we will use in the parameter estimation. We then describe the Fisher
information matrix formalism which we will use to forecast the constraints obtainable by using
future experiments. In Section 4.3.1, we discuss the tensor perturbations which arise in the single
field inflationary (hereafter SFI) model 1, and in Section 4.3.2, we constrain the parameters of the
SFI model using the BICEP and QUaD data. In Section 4.3.3, we calculate the signal-to-noise
ratio for a number of future CMB experiments and their combinations, and present forecasts for
the constraints obtainable on the tensor-to-scalar ratio r with future observations. In Section 4.3.4,
we discuss four types of single field slow-roll inflation models, and their detectability with future
experiments. We then follow a similar line of discussion for the LQG model in Section 4.4 and
the brane inflation/cosmic string model in Section 4.5. We summarize our results in Section 4.6.
4.2 B-mode polarization and its observations
4.2.1 B-mode polarization
Let us first briefly review the statistics of the CMB polarization field. The polarized radiation
field can be described by a 2×2 intensity matrix Iij(n) [194], where n denotes the direction on the
sky, and Iij(n) is defined with respect to the orthogonal basis (e1, e2) which is perpendicular to n.
Linear polarization is related to the two Stokes parameters, Q = 1
4
(I11−I22) and U = 12I12, whereas
the temperature anisotropy is T = 1
4
(I11 + I22). The polarization magnitude and orientation are
given by P =
√
Q2 + U2 and α = 1
2
tan−1(U/Q).
As spin ±2 fields, the Stokes parameters Q and U change under a rotation by an angle ψ as
(Q± iU)(n)→ e∓2iψ(Q± iU)(n). Thus, (Q± iU)(n) requires an expansion with spin ±2 spherical
harmonics [195]
(Q± iU)(n) =
∑
lm
a
(±2)
lm ±2Ylm(n). (4.1)
The multipole coefficients a
(±2)
lm can be calculated as
a
(±2)
lm =
∫
(Q± iU)(n)[±2Y ∗lm(n)]dn. (4.2)
The E- and B-mode multipoles are defined in terms of the coefficients a
(±2)
lm in the following
manner:
aElm = −
1
2
(a
(2)
lm + a
(−2)
lm ), a
B
lm = −
1
2i
(a
(2)
lm − a(−2)lm ). (4.3)
One can now define the E-mode polarization sky map E(n) and the B-mode polarization sky map
1Here we restrict our discussion to the single field slow-roll inflationary model. SFI models with non-trivial
sound speed are not covered here. Thanks to the discussion with Daniel Baumann.
89
4. CONSTRAINTS ON THE STANDARD AND NON-STANDARD MODELS OF
THE EARLY UNIVERSE FROM CMB B-MODE POLARISATION
B(n) as
E(n) =
∑
lm
[
(l + 2)!
(l − 2)!
] 1
2
aElmYlm(n), B(n) =
∑
lm
[
(l + 2)!
(l − 2)!
] 1
2
aBlmYlm(n). (4.4)
The scalar field E(n) and the pseudoscalar field B(n) completely describe the polarization
field. E-modes are curl-free modes and appear as symmetric radial and tangential polarization
patterns on the sky. B-modes are divergence-free modes with left-handed and right-handed vortical
polarization patterns on the sky.
One constructs the various CMB power spectra by correlating the T , E and B modes in
harmonic space. In the absence of parity-violating effects 1, there are only four non-zero cross-
correlations: TT , TE, EE and BB. The angular power spectra of the polarization fields are
defined as
CEEl ≡
1
2l + 1
∑
m
〈aElmaE∗lm〉, CBBl ≡
1
2l + 1
∑
m
〈aBlmaB∗lm 〉, (4.5)
where the brackets denote an ensemble average. If the fluctuations are Gaussian distributed, all
of the cosmological information is encoded in the angular power spectra.
4.2.2 Constraints on the B-mode signal
5 10 50 100 500 100010
-8
10-6
10-4
0.01
1
100
Multipole Moments l
lHl
+
1L
C l
B
B
2
Π
@Μ
K
2 D
QUaD data
BICEP data
WMAP 5 years data
Lensing effect
GΜ=10-7,Α=1.9 HCosStrL
r=0.03 HSFIL
m=10-8,k*=0.002 HLQGL
Figure 4.1: Comparison of different theoretical predictions, and the currently available data for
the B-mode power spectrum. The unit for m is Mpl and k∗ is Mpc
−1. Note that the error-bars
are plotted in log-log scale.
In Fig. 4.1, we plot the current constraints on the B-mode power spectrum along with some
representative B-mode signals from different theories. The red curve is the predicted CBBl calcu-
1Since both T and E have parity factor (−1)l under rotation, while B has parity (−1)l+1, the TB and EB
should vanish for symmetry reasons.
90
lated from LQG, with a mass parameter m = 10−8Mpl and k∗ = 0.002Mpc
−1 (see Section 4.4.1
for a discussion on LQG). The blue curve is the predicted CBBl from SFI for a tensor-to-scalar
ratio r = 0.03, and the pink curve is the BB power spectrum generated by cosmic strings for a
string tension Gµ = 10−7 and wiggling parameter α = 1.9 (see Section 4.5 for a discussion on
cosmic strings). The green curve is the B-mode signal from gravitational lensing which acts as a
source of confusion when attempting to measure the primordial B-mode signal. The black, purple
and orange points with associated error-bars are the currently available B-mode data from the
WMAP 5-year observations (l ≤ 20 [196]), the BICEP experiment (9 band powers [182; 197]),
and the QUaD experiment (23 band powers [156; 198]).
The WMAP constraints are relatively weak due to instrumental noise, cosmic variance and
residual foreground noise. In addition, the constraining power is further restricted by the un-
certainty in the optical depth τ to the last scattering surface. We will therefore not use the
WMAP data in the following likelihood analysis. The BICEP data probes intermediate scales
(21 ≤ l ≤ 335) around the recombination bump in the primordial B-mode spectrum. On these
scales, the primordial signal is less affected by cosmic variance and is comparable to or larger than
the lensing signal for tensor-to-scalar ratios r & 0.01. The QUaD experiment, whose primary aim
was a high resolution measurement of the E-mode signal, probes small scales (164 ≤ l ≤ 2026).
Its ability to constrain the primordial signal is thus severely restricted due to lensing confusion
and the rapid decline in the primordial signal with inverse scale. It may however be useful for
constraining the cosmic string signal which peaks on small scales.
5 10 50 100 500 100010
-12
10-9
10-6
0.001
1
1000
Multipole Moments l
lHl
+
1L
C l
B
B
2
Π
@Μ
K
2 D
Ideal experiment
CMBPol Noise Level
Planck Noise Level
Lensing effect
r=0.03 HSFIL
5 10 50 100 500 100010
-12
10-9
10-6
0.001
1
1000
Multipole Moments l
lHl
+
1L
C l
B
B
2
Π
@Μ
K
2 D
Spider Noise Level
QUIET Noise Level
PolarBear Noise Level
Lensing effect
r=0.03 HSFIL
Figure 4.2: Polarization noise power spectra for forthcoming experiments. Note that these curves
include uncertainties associated with the instrumental beam. The blue curves show the B-mode
power spectrum for the standard inflationary model with r = 0.03. In the left panel, we plot the
instrumental noise for the Planck and CMBPol satellites, as well as the lensing B-mode signal
and the noise level for the ideal experiment. In the right panel, we plot the instrumental noise for
the ground-based PolarBear and QUIET experiments, and the balloon-borne Spider experiment.
We fix the other parameters at WMAP 7-year best-fit values (Ωbh
2 = 0.02258, Ωch
2 = 0.1109,
ns = 0.963, As(k0) = 2.43× 10−9 (pivot scale k0 = 0.002,Mpc−1), h = 0.71, and τ = 0.088).
The constraints plotted in Fig. 4.1 are all consistent with zero signal at the ∼ 2σ level.
91
4. CONSTRAINTS ON THE STANDARD AND NON-STANDARD MODELS OF
THE EARLY UNIVERSE FROM CMB B-MODE POLARISATION
Detecting B-mode polarization therefore remains an outstanding experimental challenge, and
represents a key goal for current and future CMB experiments including ground-based (BICEP-
II [28], QUIJOTE [29], PolarBear [30], QUIET [31]), balloon-borne (EBEX [32], Spider [33],
PIPER) and satellite (Planck [27], B-Pol [199], litebird [200], CMBPol [201]) experiments. In what
follows, we will forecast the constraints potentially achievable with the following five representative
experiments: the Planck and CMBPol satellite missions, the ground-based PolarBear and QUIET
(Phase II) experiments, and the balloon-borne experiment, Spider. In addition, for reference,
we shall consider the ideal (but unrealistic) case where there is no foreground contamination
and no instrumental noise, and where the lensing signal can be cleaned to around 1 part in 40
[202]. The instrumental specifications which we use to model the various experiments are listed
in Appendix A. Fig. 4.2 shows the noise levels of these experiments compared to the SFI signal
for r = 0.03.
4.2.3 Data Analysis Methodology
In this subsection, we describe the methodology we use to constrain the models using current
data, and to forecast constraints for future experiments. The parameters of the standard ΛCDM
model have already been tightly constrained by CMB TT, EE and TE data [72] and the remaining
uncertainties in these parameters have little impact on the B-mode power spectrum, e.g. [181].
Therefore, consistent with the approach adopted by Ref. [182], throughout this Chapter, we only
vary those parameters which influence the level of primordial B-modes. We fix the other cosmo-
logical parameters at their WMAP 7-year best-fit values, which are derived under the assumption
r = 0 and constant ns across all wavelengths [72]: Ωbh
2 = 0.02258, Ωch
2 = 0.1109, ns = 0.963,
As(k0) = 2.43× 10−9 (pivot scale k0 = 0.002Mpc−1), h = 0.71, and τ = 0.088.
4.2.3.1 χ2 analysis and hyper-parameters
To constrain the three models using current data, we initially employ a conventional χ2 analysis
to obtain the likelihood function for each data set. For LQG and SFI, we use the CAMB code [25]
to output the transfer function for the B-mode power spectrum. CBBl can then be calculated as
CBBl =
π
4
∫
Pt(k)∆
B
l (k)
2d ln k, (4.6)
where ∆Bl (k) is the transfer function for each multipole l, and Pt(k) is the primordial tensor power
spectrum (see Eqs. (4.17) and (4.37)). For the cosmic string model, we use the publicly available
code CMBACT [203] to generate the B-mode power spectrum.
We then follow the procedure of likelihood construction in Refs. [156; 182] to construct the
expected bandpowers for each model, and we use the lognormal approximation (as illustrated in
92
e.g. [182]) to calculate χ2 according to
χ2(α) =
[
ZˆBB − ZBB(α)
]T
DBB(α)
−1
[
ZˆBB − ZBB(α)
]
, (4.7)
where α is the parameter we wish to constrain, and ZBB(α) and ZˆBB are the model and observed
band powers, transformed to the lognormal basis. DBB(α) is the covariance matrix of the observed
bandpowers, once again transformed to the lognormal basis. Minimizing the χ2 across all sampled
values of α yields the best-fit model.
To obtain joint constraints from BICEP and QUaD, we can simply add the χ2 values and
minimize the resulting joint χ2,
χ2tot = χ
2
BICEP + χ
2
QUaD. (4.8)
The goodness of fit for each model can be ascertained by comparing the minimum χ2 value
with the number of degrees of freedom n. If the value of χ2min/n is close to unity within the
range (1 − √2/n,1 + √2/n), we can say that the model provides a good fit to the data. If
χ2min/n≫ 1+
√
2/n, then the model is not a good fit to the data, while if χ2min/n≪ 1−
√
2/n, then
the model is overfitting the data which may happen if the model has redundant free parameters
and/or the errors on the data have been overestimated.
Note that in the conventional joint χ2 analysis of Eq. (4.8), we have weighted each data set
equally. This may be problematic if the two data sets are not mutually consistent, or if there
are unquantified systematics in the data [205; 206]. In such cases, one may wish to weight the
data appropriately. The assignment of weights often occurs when two or more of the data sets are
inconsistent, and is usually made in a somewhat ad-hoc manner [206]. Generally speaking, assign-
ing the weights for each data set is a somewhat subjective way of performing a joint analysis, but
one well-motivated approach to assigning weights is the “hyper-parameter” approach, formulated
within a Bayesian context, which can objectively allow the statistical properties of each data set
to determine its own weight in the analysis [205; 206].
In the hyper-parameter technique, the effective χ2 is defined as
χ2hyper =
∑
j
nj lnχ
2
j , (4.9)
where j sums over all of the data sets, χ2j is the χ
2 for each data set, and nj is the number of
degrees of freedom for each data set 1.
Once the χ2 values for the combined data set have been obtained, we can find the posterior
1There is a subtle difference between our definition and those in [205] and [206]. In [205], nj is the number of
data points in each data set ndata, while in [206], nj = ndata + 2. However, we prove in Appendix B that, when
considering several constraint equations on the random variables, the distribution function form is not changed
under the constraints, only if the hyper-parameter is defined as Eq. (4.9).
93
4. CONSTRAINTS ON THE STANDARD AND NON-STANDARD MODELS OF
THE EARLY UNIVERSE FROM CMB B-MODE POLARISATION
distribution for the parameter α using
− 2 lnP (α|D1, · · · , DN) = χ2, (4.10)
where χ2 can be either the conventional χ2tot or the hyper-parameter version χ
2
hyper [205].
In Appendix B, we calculate the expectation value and variance (see Eq. (B.17)) of χ2hyper.
This calculation shows that in the hyper-parameter case, a model can be said to be a good fit to
the data if the minimum χ2 is within the range (1−√V (n)/E(n), 1 +√V (n)/E(n)), where n is
the number of degrees of freedom for the data sets 1.
4.2.3.2 Fisher information matrix
In order to make forecasts for the constraints achievable with future experiments, one can calculate
the Fisher information matrix under the assumption that each parameter is Gaussian-distributed.
The standard Fisher matrix Fαβ is defined as [207; 208],
Fαβ =
1
2
Tr[C,αC
−1C,βC
−1], (4.11)
where C is the total covariance matrix, which includes both signal and noise contributions:
Cl1m1l2m2 = (C
BB
l1,sig
+NBBl1,tot)δl1l2δm1m2 . (4.12)
Here, NBBl,tot is the total noise contribution to the covariance matrix, which includes instrumental
noise, foreground contamination as an effective noise, and confusion noise from lensing B-modes
(see Appendix A for the details). In our case where we consider only B-mode polarization, the
Fisher matrix can be simplified as [208; 209]
Fαβ =
∑
l
(
2l + 1
2
fsky
)
(CBBl,sig),α(C
BB
l,sig),β
(CBBl,sig +N
BB
l,tot)
2
, (4.13)
where fsky is the fraction of sky observed. For Planck, CMBPol, Spider and the ideal experiment,
since these are nearly full-sky observations, we perform the summation in Eq. (4.13) from l = 2 to
l = 3000. For the ground-based PolarBear and QUIET experiments, the summation is performed
from l = 21 to l = 3000. We restrict the summation for these experiments to l > 20 since ground-
based experiments are insensitive to the largest angular scales because of their finite survey areas
(see [210] for instance).
The inverse of the Fisher matrix F−1 can, crudely speaking, be considered the best achiev-
able covariance matrix for the parameters given the experimental specification. The Cramer-Rao
1The Bayesian Evidence (BE) is another important technique for discriminating between different models.
However, the currently available data is clearly not constraining enough at present to discriminate between models.
We therefore defer any discussion of the BE until more precise data becomes available.
94
inequality means that no unbiased method can measure the ith parameter with an uncertainty
(standard deviation) less than 1/
√
Fii [207; 208]. If the other parameters are not known but are
also estimated from the data, the minimum standard deviation rises to (F−1)
1/2
ii [207; 208]. There-
fore we can estimate the best prospective signal-to-noise ratio as α/∆α, where ∆α = (F−1)
1/2
αα .
This formula will be used frequently in the following discussion.
4.3 Constraining the SFI model
4.3.1 Scalar and tensor primordial power spectra in the SFI model
In addition to nearly scale-invariant scalar perturbations, inflationary models also predict vector
and tensor perturbations [211]. However, the vector perturbations are expected to be negligible
since these modes decayed very rapidly once they entered the Hubble horizon. We will therefore
ignore any vector component in what follows.
We will work in the perturbed Friedmann-Lemaitre-Robertson-Walker Universe, for which the
metric can be written as
ds2 = −c2dt2 + a2(t)(δij + hij)dxidxj . (4.14)
The tensor perturbations hij are described by two transverse-traceless components. The power
spectrum for the two polarization modes of hij (hij = h
+e+ij + h
×e×ij , h = h
+ = h×) is
〈hkhk′〉 = (2π)3δ3(k− k′)2π
2
k3
Pt(k), (4.15)
where hk is the Fourier component of the perturbation field h. Standard inflationary models
predict a nearly scale-invariant tensor power spectrum Pt(k). In order to describe the weak scale-
dependence of Pt(k), we can define the tensor spectral index nt in the usual way:
nt ≡ d lnPt(k)
d ln k
. (4.16)
The tensor power spectrum can then be written in the following power-law form
Pt(k) = At(k0)
(
k
k0
)nt
. (4.17)
In the case where inflation is driven by a single inflaton field, the following calculation yields
the primordial power spectrum of the scalar perturbations for slow-roll inflation (see [42; 56] for
instance. For alternative calculations, see [212].)
Ps(k) =
H4
(2πφ˙)2
∣∣∣∣
k=aH
95
4. CONSTRAINTS ON THE STANDARD AND NON-STANDARD MODELS OF
THE EARLY UNIVERSE FROM CMB B-MODE POLARISATION
=
9
(2π)2
1
(3M2pl)
3
V 3
V ′2
∣∣∣∣∣
k=aH
=
8
3ǫ
(
V
1
4√
8πMpl
)4
, (4.18)
while the power spectrum of tensor perturbations is given by
Pt(k) =
8
M2pl
(
H
2π
)2∣∣∣∣∣
k=aH
=
2
3
V
π2M4pl
∣∣∣∣∣
k=aH
. (4.19)
Here, V (φ) is the inflaton potential, and H is the Hubble parameter at the time of inflation. It is
customary to define the tensor-to-scalar ratio r as
r ≡ Pt
Ps
= 8M2pl
(
V ′
V
)2
. (4.20)
The tensor-to-scalar ratio and the tensor spectral index are related to the slow-roll parameter ǫ
via [42; 56]
r = 16ǫ, nt = −2ǫ. (4.21)
These expressions lead to the so-called consistency relation for single field slow-roll inflation [213]:
nt = −r
8
. (4.22)
Unfortunately, this consistency relation is extremely difficult to constrain observationally because
of the small amplitude of the tensor power spectrum. We discuss the possibilities for testing this
relation with future observations in Section 4.3.3.
The normalization of the power spectrum of scalar perturbations (defined at the pivot wavenum-
ber k0 = 0.002 Mpc
−1) is Ps(k0) = (2.43 ± 0.11) × 10−9(1σ CL, WMAP 7-year data [72]). We
can use this normalization together with Eq. (4.18) to derive the relationship between the energy
scale of inflation and the value of r:
V
1
4 = 1.06× 1016GeV
( r
0.01
) 1
4
. (4.23)
That relation indicates that a detection of the tensor-to-scalar ratio at r ≈ 0.01 or greater would
indicate that inflation happened at an energy scale comparable to the Grand Unification Theory
(GUT) energy scale (O(1016)GeV).
We can also use the slow-roll approximation to derive the following relation, which characterizes
96
the distance in the field space from the end of inflation to the time when CMB scale fluctuations
were created, namely the Lyth bound [63; 64]
∆φ
Mpl
&
( r
0.01
) 1
2
. (4.24)
Thus, a tensor-to-scalar ratio greater than ∼ 0.01, would directly indicate a super-Planckian field
evolving from φCMB to φend. Such a detection could provide important observational clues about
the nature of quantum gravity. The boundary r ∼ 0.01 is therefore an important benchmark value
which can confirm or rule out a wide class of large field inflation models.
4.3.2 Constraints on SFI from current data
In this subsection, we present constraints on the tensor-to-scalar ratio r from BICEP and QUaD
data. To obtain the constraints, we follow the methodology outlined in Section 4.2.3.1.
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
r values
PH
rL
BICEP+QUaD,HyperParameter
BICEP+QUaD,conventional Χ2
QUaD Only
BICEP Only
r<0 excluded
Figure 4.3: The current constraints on r from BICEP and QUaD data.
The results of the analysis are shown in Fig. 4.3. It is clear from the figure that the BICEP
data provides a fairly strong upper limit on the value of r. The best-fit value is r = 0.01+0.31−0.26 (1σ
CL), which is very close to the result obtained by the BICEP team themselves [182]. As expected,
r is essentially unconstrained by QUaD, whose measurements are made at much smaller scales
(l & 200) than the scale at which the primordial B-mode signal peaks (l ∼ 80). In producing
joint constraints, we find that both the conventional χ2 and the hyper-parameter version (χ2hyper)
are completely dominated by the BICEP data. For the conventional χ2 analysis, the best-fit of
the joint analysis gives r = 0.03+0.32−0.27 (1σ CL), and for the hyper-parameter χ
2 analysis, we obtain
r = 0.02+0.31−0.26 (1σ CL). The details of the constraints are listed in the first row of Table 4.1.
Note that the tendency for the QUaD data to prefer larger values of r appears to be related to a
97
4. CONSTRAINTS ON THE STANDARD AND NON-STANDARD MODELS OF
THE EARLY UNIVERSE FROM CMB B-MODE POLARISATION
marginal excess of power in the QUaD measurements over the multipole range 300 < l < 500 (see
Fig. 4.1). After a careful investigation, we find that this shift to higher value of r is due to this
part of the data, specifically l ∼ 370, which might suffer from the unaccounted systematic errors
1.
Comparing the BICEP result to the WMAP 7-year results [72], the tightest upper-bound on r
that the WMAP team quote is r ≤ 0.24 (2σ). This constraint is derived from a combination of the
WMAP data with both large scale structure measurements and the HST key project constraint
on the Hubble constant. It is clear that measurements of the TT and TE CMB spectra in
combination with other astrophysical probes currently play a significant role in constraining the
value of r. However, we note that the constraints obtained from B-mode polarization alone are
already comparable to the combined constraints from all other cosmological probes and are likely
to overtake them with the next generation of CMB polarization experiments.
In Table 4.2 we quote the goodness-of-fits for the various analyses and we quote the weights
for the hyper-parameter analysis in Table 4.3. In Table 4.2, E(n) is the expectation value for
each fit, calculated using Eqs. (B.4) and (B.17). If the model provides a good fit to the data,
the minimum χ2 over the expectation value should be well within the range (1 −√V (n)/E(n),
1 +
√
V (n)/E(n)). Examining the table, we see that the SFI model can fit the BICEP data well,
but that the fit to the QUaD data is relatively poor. This poor fit to the QUaD data adds further
weight to our conclusion above regarding the anomalous power in the QUaD results in the range
300 < l < 500.
4.3.3 Prospects for future observations
In this section, we discuss the detection capabilities of future CMB experiments. In order to
forecast the error bars of the parameters r and nt in the fiducial models, we use the Fisher matrix
technique, introduced in Section 4.2.3.2.
In Fig. 4.4, we plot the signal-to-noise ratio (r/∆r) for a detection of tensors as a function of
the fiducial value of r, for a number of current and forthcoming experiments. In the left panel, we
only consider r as the free parameter, and keep nt fixed at nt = 0. We see that the Planck satellite
can potentially detect the signal of the tensor perturbations at more than 3σ confidence level if
r > 0.05. For r = 0.1, the value of r/∆r becomes 5 which would constitute a robust detection.
These results are consistent with those presented in [54; 55; 214]. The predicted constraints for
PolarBear, QUIET and Spider are somewhat tighter with r/∆r > 3 for models with r > 0.02.
The predicted constraints for the proposed CMBPol mission suggest that tensor perturbations
could be detected (at the 3σ level) for values of r as low as r ∼ 0.002. Such a measurement would
provide an excellent opportunity to differentiate between various inflationary models. We also find
that for the ideal CMB experiment which includes only a residual lensing noise contribution (after
1Thanks to the conversation with QUaD team member Michael L. Brown.
98
Models Sampling Conventional χ2 hyper-parameter χ2
Range BICEP QUaD (tot) BICEP+QUaD BICEP+QUaD
SFI: r (−1.0, 10.0) 0.01+0.31+0.68−0.26−0.49 10.0×−3.0−9.70 0.03+0.32+0.70−0.27−0.50 0.02+0.31+0.75−0.26−0.51
LQG: m [10−8Mpl] (0.01, 10
2) 0.18+1.16+5.74× 47.5
×
−27.7× 0.22
+1.14+5.72
× 0.20
+1.16+5.96
×
k∗ [10
−4Mpc−1] (0.1, 102) 1.07+1.36+5.88× 53.4
+131.63×
−23.3× 1.07
+1.37+5.89
× 1.07
+1.36+5.98
×
Cosmic String (10−3, 102) (0.001)+5.586+9.961× tot: 7.60
+1.38+2.63
−1.56−3.60
Gµ× 107 (l>500): 4.32+2.41+4.25× 2.98+2.82+4.74× 2.32+3.45+5.69×
Table 4.1: The best-fit values for the parameters, and the 1σ and 2σ CL for the scalar field inflation (SFI), Loop Quantum Gravity
(LQG) and cosmic string models. For the SFI and LQG models, we use all the QUaD data and combine these with BICEP using both
a conventional joint χ2 and the hyper-parameter approach (χ2hyper). For the cosmic strings model, in addition to the entire QUaD
data set, we also examine the effect of removing the l < 500 QUaD data points. When combining the QUaD and BICEP data for
the cosmic strings model, we also restrict the QUaD data to l > 500 (see the discussion in section 4.5.2). The notation “×” indicates
that the values of the parameters are out of sampling range.
99
4. CONSTRAINTS ON THE STANDARD AND NON-STANDARD MODELS OF
THE EARLY UNIVERSE FROM CMB B-MODE POLARISATION
Goodness Conventional χ2 hyper-parameter χ2
of fits BICEP QUaD (tot) BICEP+QUaD BICEP+QUaD
SFI: E(n) 8 22 31 82.58
χ2min/E(n) 1.00 1.64 1.56 1.26
Good-fits range (0.5,1.5) (0.70,1.30) (0.75,1.25) (0.89,1.11)
LQG: E(n) 7 21 30 75.49
χ2min/E(n) 1.10 1.76 1.60 1.22
Good-fits range (0.47,1.54) (0.69,1.31) (0.74,1.26) (0.90,1.10)
CosStr: E(n) 8 18 (l >500) 27 (QUaD l >500) 66.6 (QUaD l >500)
χ2min/E(n) 1.0 1.35 1.21 1.08
Good-fits range (0.5,1.5) (0.67,1.33) (0.73,1.27) (0.90,1.11)
Table 4.2: Reduced χ2 as an indication of the goodness-of-fit for each analysis. E(n) is the
expectation value for each fit. If the model provides a good fit to the data, the values of χ2min/E(n)
should be within the range (1−√V (n)/E(n), 1 +√V (n)/E(n)).
αeff = nA/χ
2
A BICEP QUaD
SFI 1.13 0.64
LQG 1.17 0.62
CosStr 1.13 0.95
Table 4.3: The values of the effective hyper-parameters αeff = nA/χ
2
A for the BICEP and QUaD
data which reflect the relative weights assigned to each data set.
10-6 10-5 10-4 0.001 0.01 0.1 1
1
5
10
50
100
500
r values
r
D
r
Hn
T
fix
ed
L
QUIET
PolarBear
Spider
Ideal
CMBPol
Planck
10-6 10-5 10-4 0.001 0.01 0.1 1
1
2
5
10
20
50
100
200
r values
r
D
r
Hn
T
fre
eL
QUIET
PolarBear
Spider
Ideal
CMBPol
Planck
Figure 4.4: The signal-to-noise ratio r/∆r for different experiments, calculated using the Fisher
matrix of Eq. (4.13). Left: The parameter r is treated as a free parameter but nt is kept fixed at
its fiducial value; Right: Both r and nt are treated as free parameters. The atmospheric noise in
PolarBear and QUIET is not included in the plot.
100
de-lensing), the SFI primordial signal could be detected (at > 3σ) only if r > 10−5 is satisfied.
In Fig. 4.5, we also plot the signal-to-noise ratio for the combination of Planck with the ground-
based experiments (PolarBear and QUIET). The former is sensitive to the B-mode signal at the
lowest multipoles ℓ < 20, while the latter are sensitive to the recombination peak of CBBl at
l ∼ 80. Similar to [215], we find that the combination of these experiments yields little formal
improvement in the signal-to-noise of the detection compared with the capabilities of the ground-
based experiments on their own. However, a detection of both the recombination bump (e.g. from
ground-based experiments) and the reionization bump (e.g. from Planck) would constitute much
more compelling evidence for tensors than either detection would constitute on its own.
10-6 10-5 10-4 0.001 0.01 0.1 1
1
5
10
50
100
500
r values
r
D
r
Hn
T
fix
ed
L
Planck+QUIET
Planck+PolarBear
Spider
Ideal
CMBPol
Planck
10-6 10-5 10-4 0.001 0.01 0.1 1
1
2
5
10
20
50
100
200
r values
r
D
r
Hn
T
fre
eL
Planck+QUIET
Planck+PolarBear
Spider
Ideal
CMBPol
Planck
Figure 4.5: The signal-to-noise ratio r/∆r for different experiments, and the combination of
Planck and ground-based experiments, calculated using the Fisher matrix of Eq. (4.13). Left:
The parameter r is treated as a free parameter but nt is kept fixed at its fiducial value; Right:
Both r and nt are treated as free parameters.
In the right hand panels of Fig. 4.4 and Fig. 4.5, we have plotted the results for the case
where we treat both r and nt as free parameters. Comparing with the corresponding results in
the left panels, we find (in agreement with previous works, e.g. [201; 216]) that the signal-to-noise
ratios become much smaller due to correlations between the parameters. These correlations were
investigated in some detail by [215] who also explored the optimal choice of pivot scale for which
the two parameters become decorrelated. In Fig. 4.6, we show separately the forecasted joint
constraints on r and nt parameters given the Planck+QUIET, Planck+PolarBear, Spider, and
CMBPol experiment.
In Fig. 4.7, we plot the values of ∆nt as a function of r for various cases. Here we find that,
for Planck, PolarBear, QUIET and Spider, the constraint on nt is relatively weak unless the value
of r is very large. For example, the predicted constraint for Planck is ∆nt = 0.13, and for QUIET
is ∆nt = 0.18 for a model with r = 0.1. The combination of Planck and QUIET could in principle
do somewhat better with ∆nt = 0.08. The proposed CMBPol mission could achieve ∆nt = 0.04
while the limiting value of the ideal experiment is ∆nt = 0.007, which is comparable to the current
constraint on the scalar spectral index ns [72]. For lower values of r, the predicted constraints are
101
4. CONSTRAINTS ON THE STANDARD AND NON-STANDARD MODELS OF
THE EARLY UNIVERSE FROM CMB B-MODE POLARISATION
correspondingly weaker. For example, for the CMBPol mission, the value of ∆nt increases from
0.04 to 0.1 if we replace the r = 0.1 model with r = 0.01.
0.000 0.005 0.010 0.015 0.020
-0.5
0.0
0.5
r
n
t
Planck+QUIET
Planck+PolarBear
Spider
CMBPol
1Σ CL:
0.000 0.002 0.004 0.006 0.008 0.010
-6
-4
-2
0
2
4
6
r
n
t
Planck+QUIET
Planck+PolarBear
Spider
CMBPol
1Σ CL:
Figure 4.6: Forecasted joint constraints on the parameters of the SFI model. The contours indicate
the 68% (1σ) confidence levels. The input models are indicated by the black points (Left: r = 0.01
and nt = 0, Right: r = 0.001 and nt = 0).
Testing the consistency relation nt = −r/8 (see Eq. (4.22)) is potentially one of the most
powerful ways to test the general SFI scenario. To assess whether future experiments might
achieve this goal, in Fig. 4.7 we compare the values of ∆nt with r/8. If ∆nt < r/8, then the
constraint on nt is tight enough to allow the consistency relation to be tested. We find that
∆nt < r/8 is satisfied only if r > 0.23 for the CMBPol experiment, and only if r > 0.06 for the
ideal experiment. An observational confirmation of the consistency relation is therefore extremely
unlikely to be achieved with any of the currently envisaged future experiments.
4.3.4 Single-field slow-roll inflationary models
In single field slow-roll inflationary models, the observables depend on three slow-roll parameters
[213]
ǫV ≡
M2pl
2
(
V ′
V
)2
, ηV ≡ M2pl
(
V ′′
V
)
, ξV ≡ M4pl
(
V ′V ′′′
V 2
)
, (4.25)
where V (φ) is the inflationary potential, and the prime denotes derivatives with respect to the field
φ. Here, ǫV quantifies the “steepness” of the slope of the potential, ηV measures the “curvature” of
the potential, and ξV quantifies the “jerk”. Since the potential is fairly flat in the slow-roll inflation
models, these three parameters must be much smaller than unity for inflation to occur. One of
102
10-6 10-5 10-4 0.001 0.01 0.1 1
1.
0.5
0.1
0.05
0.01
0.005
0.001
r values
D
n
T
Hr
fre
eL
DnT=r8
QUIET
PolarBear
Spider
Ideal
CMBPol
Planck
10-6 10-5 10-4 0.001 0.01 0.1 1
1.
0.5
0.1
0.05
0.01
0.005
0.001
r values
D
n
T
Hr
fre
eL
DnT=r8
Planck+QUIET
Planck+PolarBear
Spider
Ideal
CMBPol
Planck
Figure 4.7: Forecasted uncertainty on the tensor spectral index, ∆nt, for different experiments,
calculated using the Fisher matrix of Eq. (4.13). Here, both r and nt are treated as free parameters.
the important predictions of SFI models is that the scalar perturbations are nearly scale-invariant,
which has already been confirmed by WMAP results [72].
In SFI models, a standard slow-roll analysis yields the following relations
nt = −r
8
, ns = 1 + 2ηV − 6ǫV , r = 8
3
(1− ns) + 16
3
ηV , αs = −24ǫ2V + 16ǫV ηV − 2ξV , (4.26)
where ns is the tilt of primordial scalar power spectrum, and αs = dns/d ln k is the “running” of
ns. These formulae relate the tensor parameters nt and r to the scalar parameters ns and αs; the
latter can be constrained through CMB and large scale structure observations. As shown in Eq.
(4.26), the relation between r and ns involves the slow-roll parameter ηV which in turn depends
on the specific inflationary potential.
The strength of the primordial tensor perturbations depends on the value of r. Observations
have yielded quite tight constraints on ns, but we currently only have upper limits on the value
of r. The relation between ns and r depends on the specific inflationary model, and different
models predict very different values for r. In the following discussion, we categorize SFI models
into four classes based on different regimes for the curvature of the potential V (φ), and discuss
their individual constraints.
Case A: negative curvature models ηV < 0
The negative ηV models arise from a potential of spontaneous symmetry breaking. One type
of often-discussed potentials is the form V = Λ4 [1− (φ/µ)p] (p ≥ 2). This type of model predicts
a red tilt in the scalar spectrum (ns < 1), which is consistent with the WMAP 7-year results [72].
In addition, these models predict relatively small values for r. For the model with p = 2 in Ref.
[217],
r ≃ 8(1− ns)e−Ne(1−ns) , (4.27)
where Ne is the number of e-folds, taken to be in the range Ne ∈ [40, 70] based on current
103
4. CONSTRAINTS ON THE STANDARD AND NON-STANDARD MODELS OF
THE EARLY UNIVERSE FROM CMB B-MODE POLARISATION
observations of the CMB [72; 218]. Here we choose the value Ne = 60. Using the result ns =
0.963 ± 0.012 [72] yields the constraint r ∈ [0.021, 0.045]. From Fig. 4.4 (right panel), we see
that this is close to or even beyond the sensitivity range of the Planck satellite, but is within the
sensitivity ranges of PolarBear, QUIET, Spider and CMBPol. In other models with p > 2, the
predicted values of r are much smaller than that of the model with p = 2.
Case B: small positive curvature models 0 ≤ ηV ≤ 2ǫV
Two example potentials in this case are the monomial potentials V = Λ4(φ/µ)p with p ≥ 2 for
0 < ηV < 2ǫV and the exponential potential V = Λ
4 exp(φ/µ) for ηV = 2ǫV . In these models, to
first order in slow roll, the scalar index is always red ns < 1 and the following constraint on r is
satisfied
8
3
(1− ns) ≤ r ≤ 8(1− ns) . (4.28)
Using the result ns = 0.963 [72] , one finds that r ∈ [0.1, 0.3], which is within the sensitivity range
of the Planck satellite, as well as that of forthcoming CMB experiments. Thus, the Planck results
may provide some constraints on these type of models.
Case C: intermediate positive curvature models 2ǫV < ηV ≤ 3ǫV
The supergravity-motivated hybrid models have a potential of the form V ≃ Λ4[1+α ln(φ/Q)+
λ(φ/µ)4, up to one-loop correction during inflation. In this case,
ns < 1 , r > 8(1− ns) , (4.29)
are satisfied. Using the result ns = 0.963 [72], one finds that r > 0.3, which is slightly in conflict
with the current upper limit r < 0.24 (WMAP+BAO+H0, 2σ CL) [72]. Fig. 4.4 shows that this
model is also in the sensitivity range of the Planck satellite.
Case D: large positive curvature models ηV > 3ǫV
This class of models has a typical monomial potential similar to those of Case A, but with a
plus sign for the term (φ/µ)p: V = Λ4 [1 + (φ/µ)p]. This enables inflation to occur for small values
of φ < Mpl. The model predicts a blue tilt in the scalar power spectrum ns > 1 (Eq. (4.26)), which
is in conflict with current constraints on ns unless a running of ns is allowed [72]. When a running
in ns is included, the WMAP 7-year results suggest a blue power spectrum (ns = 1.008±0.042 for
1σ CL). Therefore, even though this model is not favoured by the WMAP 7-year results for the
constant of ns only, it is not excluded when a running of the spectral index is included. Planck
and future CMB experiments should constrain both ns and αs to high precision and so should be
able to rule out this model.
104
4.4 Loop Quantum Gravity and its observational probes
4.4.1 Primordial tensor perturbations in LQG models
Loop Quantum Gravity is a promising framework for constructing a quantum theory of gravity in
theoretical physics. Based on the reformulation of General Relativity as a kind of gauge theory
obtained by [219; 220], LQG is now a language and a dynamical framework which leads to a
mathematically coherent description of the physics of quantum spacetime [221]. Constraining
LQG theories experimentally is challenging because the quantum geometrical effect can only be
tested at very high energy scales, beyond the reach of current accelerator experiments. In this
section, we will calculate the possible observational signature of LQG in the CMB sky, which
opens a new window for cosmological tests of quantum gravity.
There are two main quantum corrections in the Hamiltonian of LQG when dealing with the
semi-classical approach, namely holonomy corrections and “inverse volume” corrections [221; 222].
The holonomy corrections lead to a dramatic modification of the Friedmann equation as [223]
H2 =
8πG
3
ρ
(
1− ρ
ρc
)
, (4.30)
where ρ is the energy density, and ρc is the critical energy density,
ρc =
4
√
3
γ3
M4pl ≃ 507.49M4pl. (4.31)
Here γ = 0.239 is the Barbero-Immirzi parameter, which is derived from the computation of the
black hole entropy [224]. Note that we use the reduced Planck mass in our calculation.
A generic picture for this model with the holonomy correction is the bouncing behavior ex-
hibited when the energy density of the Universe approaches ρc. The negative sign in Eq. (4.30) is
an appealing feature in the framework of LQG such that the repulsive quantum geometry effect
becomes dominant in the Planck region [221; 222]. This triggers a contraction period before the
bounce, during which time the Hubble parameter is negative and the Hubble radius is shrinking.
As a result, the perturbation modes on the largest scales crossed the Hubble horizon and froze
out during the contracting period, until the end of the contracting stage when the Hubble hori-
zon increased again. For the very large scale modes, this pre-inflationary bounce may imprint
distinctive features in the CMB sky, since they stretched out of the horizon at very early times
[221; 222].
Unfortunately, the power spectrum for the scalar perturbations is somewhat hard to obtain
because in the case of the holonomy correction, the anomaly free equations are still to be found
[225]. Therefore, in the following discussion, we will focus on the tensor power spectrum of LQG
and pursue the constraints obtainable on the model from B-mode observations.
105
4. CONSTRAINTS ON THE STANDARD AND NON-STANDARD MODELS OF
THE EARLY UNIVERSE FROM CMB B-MODE POLARISATION
Due to the pre-inflationary contracting period, the tensor power spectrum for LQG can be
calculated numerically. In [222], a simple parameterized form of the power spectrum is introduced
as follows
Pt =
2
π2
(
H
Mpl
)2
1
1 + (k∗/k)2
[
1 +
4R− 2
1 + (k/k∗)2
]
, (4.32)
where H is the Hubble parameter during the inflationary stage, k∗ is the position of the highest
peak in the power spectrum, and the quantity R is related to the mass of the scalar field as
R = (8π)0.32
[
Mpl
m
]0.64
. (4.33)
It is interesting to note that Eq. (4.32) reduces to the SFI result of Eq. (4.19) for k∗ → 0. In this
Chapter, for simplicity, we consider the tensor power spectrum (Eq. 4.32) with a constant H in
the early stage of inflation, which corresponds to the specific case of de Sitter inflation.
In this model, we assume that inflation is driven by the potential V (φ) = m2φ2/2. The Hubble
parameter is related to this potential via
H2 =
1
3M2pl
V (φ) =
1
6M2pl
m2φ2i , (4.34)
where φi is the initial value of the scalar field at the beginning of inflation. The number of e-folds
can be calculated as (using the slow-roll approximation 3Hφ˙ = −V ′)
Ne =
∫ f
i
Hdt
= − 1
M2pl
∫ f
i
V
V ′
dφ
= − 1
4M2pl
(
φ2f − φ2i
)
. (4.35)
Since φi ≫ φf , the above equation is approximately given by φ2i ≃ 4NeM2pl. The Hubble parameter
is then (from Eq. (4.34))
H2 ≃ 2
3
Nem
2. (4.36)
Substituting this into Eq. (4.32), we arrive at the following expression for the tensor power
spectrum:
Pt =
4Ne
3π
(
m
Mpl
)2
1
1 + (k∗/k)2
[
1 +
4× (8π)0.32 × (m/Mpl)−0.64 − 2
1 + (k/k∗)2
]
, (4.37)
where Ne is the number of e-folds which we fix at Ne ≃ 60. Finally, we use Eq. (4.6) to project
the perturbation modes onto the CMB sphere to find CBBl for the LQG model.
106
10-7 10-5 0.001 0.1 10 1000
10-19
10-17
10-15
10-13
10-11
10-9
10-7
k@hMpcD
P T
Hk
L
r=0.001 HSFIL
r=0.01 HSFIL
r=0.03 HSFIL
r=0.1 HSFIL
m=10-7,k*=0.002 HLQGL
m=10-8,k*=0.02 HLQGL
m=10-8,k*=0.002 HLQGL
5 10 50 100 500 100010
-9
10-7
10-5
0.001
0.1
Multipole Moments l
lHl
+
1L
C l
B
B
2
Π
@Μ
K
2 D
Lensing Effect
r=0.01 HSFIL
r=0.1 HSFIL
m=10-7,k*=0.002 HLQGL
m=10-8,k*=0.02 HLQGL
m=10-8,k*=0.002 HLQGL
Figure 4.8: Left: the primordial tensor power spectrum for different models of inflation. Right:
the corresponding B-mode angular power spectrum generated by different inflationary models.
Note the lensing B-mode signal which acts as an effective noise for detecting the primordial CBBl
signal. The units for m and k∗ are Mpl and Mpc
−1 respectively.
In the left panel of Fig. 4.8, for a number of representative sets of parameters, we plot the
primordial tensor power spectrum for the LQG model alongside the signal expected in an SFI
model for a number of different values of r. For the SFI model, since the power spectrum tilt
nt is very small (nt = 0 for de Sitter inflation), the power spectrum is very flat on all scales.
In comparison, the tensor power spectrum of LQG exhibits a bump feature, which is in fact the
signature of the pre-inflationary contraction period. Very large scale modes stretched out of the
Hubble horizon during the contracting period before inflation, and can be described by the solution
in the Minkowski vacuum fk = e
−ikη/
√
2k [221; 222]. Thus, the power spectrum at very large
scale takes the form Pt(k) ∼ k3 |fk|2 ∼ k2 [221; 222]. In contrast, the small scale modes are well
within the Hubble horizon and so the power on small scales is similar to the scale-invariant power
spectrum of the SFI model. The bump in the power spectrum on larger scales is characterized by
the magnitude of k∗. As k∗ increases, the bump is shifted to smaller scales and vice versa. The
amplitude of the spectrum and the width of the bump are determined by the mass parameter
m. We will link these two important parameters to the energy scale of inflation and the current
Hubble horizon scale in the next subsection.
The right panel of Fig. 4.8 shows that the bump in the primordial power spectrum results in
a peak in the CMB B-mode power spectrum, which is slightly different to that of the SFI model.
In addition, if the peak of the LQG spectrum is normalized to the same magnitude as that of SFI,
the small scale power will be suppressed in the LQG model, as compared to the SFI model.
4.4.2 Constraints from current data
In this subsection, we use the BICEP and QUaD data to constrain the parameters of LQG models.
Before we perform the parameter estimation, we link the two parametersm and k∗ with the energy
107
4. CONSTRAINTS ON THE STANDARD AND NON-STANDARD MODELS OF
THE EARLY UNIVERSE FROM CMB B-MODE POLARISATION
scale of inflation, and with the current Hubble horizon scale.
The parameter m relates to the energy scale of inflation in LQG as follows
V
1
4 = 3.02× 1015GeV
(
m
10−7Mpl
)1/2
, (4.38)
where 1015GeV is around the GUT energy scale. Therefore, a detection of m > 10−7Mpl would
strongly suggest that the energy scale of inflation is above the GUT scale.
The parameter k∗ describes the position of the peak in the primordial power spectrum. We
can compare it with the current Hubble wavenumber, which is kH ≡ H0 ≃ 2.33× 10−4Mpc−1. If
k∗ > kH , then modes with physical wavelengths (λ∗) equal to the Hubble horizon at the beginning
of inflation will have wavelengths less than the current Hubble horizon, whereas if k∗ < kH their
wavelengths will be larger than the current horizon scale. Thus, if k∗ > kH , we would expect to be
able to find pre-inflationary fluctuations within our current Hubble horizon [221; 222]. Conversely,
as k∗ → 0, the primordial tensor power spectrum (Eqs. (4.32) and (4.37)) reduces to the scale-
invariant tensor power spectrum as noted above. Therefore, a non-zero detection of k∗ would
strongly indicate the existence of a bounce and of a contracting period before inflation.
0 2.´10-7 4.´10-7 6.´10-7 8.´10-7 1.´10-6
1.
0.5
0.2
0.1
0.05
0.02
m
Ž
=mMpl
PH
mŽ
L
BICEP+QUaD,HyperParameter
BICEP+QUaD,conventional Χ2
QUaD Only
BICEP Only
GUT Scale
Figure 4.9: Probability distribution function (PDF) for the parameters in the LQG model. Left:
constraints on the mass parameter m˜ = m/Mpl. Right: constraints on k∗. The red, green, and
brown curves overlap with each other, indicating that the vast majority of the constraining power
for the combined data sets comes from the BICEP data.
The current constraints on the parameters k∗ and m are shown in Fig. 4.9. In the left panel,
we have marginalized over the parameter k∗ and we plot the PDF for the mass parameter m. The
1σ upper bound is m ≤ 1.36 × 10−8Mpl. The detailed results are listed in the second and third
rows of Table 4.1. We note that the GUT scale mass m ≃ 10−7Mpl is excluded at the 2σ level,
but is still well within 3σ. As was the case with the SFI models, the small-scale QUaD data is
unable to constrain the LQG models. Once again, the combined constraints are dominated by the
BICEP data as is clear from the figure and from the results listed in Table 1.
In the right panel of Fig. 4.9, we show the PDF for k∗ (marginalized over the m parameter).
108
Once again the results are dominated by BICEP. From this plot, wee see that there is a peak
in the PDF at k∗ = 1.07 × 10−4Mpc−1. Although it is not statistically significant, a detection
of such a feature would be an interesting result for the pre-inflationary bouncing behavior, since
the bounce of the primordial tensor power spectrum is characterized by a non-zero k∗ as we have
already discussed. We further note that the peak value of k∗ is only slightly smaller than the
current Hubble wavenumber kH , which would indicate that modes which had the same length
scale as the Hubble horizon at the beginning of inflation have not evolved into the Hubble horizon
yet. In that case, the bump in the tensor power spectrum of LQG is a super-horizon feature.
However, we stress again that all of our results are upper limits only and our formal constraint is
k∗ < 2.43× 10−4Mpc−1 (1σ CL).
-9.0 -8.5 -8.0 -7.5 -7.0 -6.5
-4.0
-3.5
-3.0
-2.5
Log10m
Ž
Lo
g 1
0k
*
QUaD Only,1Σ
BICEP+QUaDHHyperParameterL,3Σ
BICEP+QUaDHHyperParameterL,2Σ
BICEP+QUaDHHyperParameterL,1Σ
BICEP Only,3Σ
BICEP Only,2Σ
BICEP Only,1Σ
Best-fit valuesHBICEP+QUaDL
Figure 4.10: 2D constraints on the parameters m˜ = m/Mpl and k∗[Mpc
−1] of the LQG model.
The red line (BICEP 1σ CL) overlaps with the brown line (BICEP+QUaD, 1σ CL).
In Fig. 4.10, we plot the two-dimensional constraints on the parameters m and k∗ on a log
scale. Clearly, the current data is unable to provide strong constraints on the joint distribution of
these two parameters and can only provide upper limits. Once again, as expected, the constraints
are dominated by the BICEP data.
4.4.3 Prospects for future experiments
We now investigate the prospects for detection of LQG signatures with future CMB experiments
by studying the projected constraints on the two parameters m and k∗. Once again, we use the
Fisher matrix approach described in Section 4.2.3.2 and as in Section 4.3.3, we fix the background
parameters at their WMAP 7-year best-fit values [72].
In Fig. 4.11, we plot the projected signal-to-noise ratio for the parameters m (left panel) and
k∗ (right panel) for forthcoming experiments. In the left panel, we plot the signal-to-noise ratio
m/∆m as a function of m. For this plot, we have kept k∗ fixed at k∗ = 0.002 Mpc
−1. We find
109
4. CONSTRAINTS ON THE STANDARD AND NON-STANDARD MODELS OF
THE EARLY UNIVERSE FROM CMB B-MODE POLARISATION
1´10-9 5´10-9 1´10-8 5´10-8 1´10-7 5´10-7 1´10-6
1
2
5
10
20
50
m
Ž
=mMpl
mŽ
D
mŽ
QUIET
PolarBear
Spider
Ideal
CMBPol
Planck
k* = 0.002 Mpc-1
GUT Scale
1´10-4 5´10-4 0.10.050.010.0050.001
1
5
10
50
100
500
k*@Mpc-1D
k *
D
k *
QUIET
PolarBear
Spider
Ideal
CMBPol
Planck
Hubble Horizon Scale
m
Ž
=10-8
Figure 4.11: Predicted signal-to-noise ratios for the parameters in LQG for forthcoming and future
experiments. Here m˜ is the mass parameter, and k∗ is the position of the bump in the BB power
spectrum.
that, if m > 10−7 Mpl, i.e. the energy scale of inflation is higher than the GUT scale, then Planck,
Spider and CMBPol could potentially detect the LQG signal at more than 10σ due to their large
sky coverage. In contrast, the smaller scale experiments (PolarBear and QUIET) would only
detect the signal at the 2—3σ level. For m = 10−8 Mpl, close to the upper bound of the current
1σ confidence level, the large scale survey experiments (Planck, Spider and CMBPol) can still
detect the signal at more than 5σ. For this mass parameter and value of k∗, the ground-based
experiments, PolarBear and QUIET, would be insensitive to the signal since the B-mode power
spectrum from LQG falls off extremely rapidly with increasing l. However, for larger values of k∗,
the peak of the LQG CBBl power spectrum moves to smaller scales. We therefore expect a general
trend whereby the large-scale experiments will be sensitive to models with small values of k∗ and
small-scale experiments will be sensitive to models with larger values of k∗.
This is illustrated in the right-hand panel of Fig. 4.11 where we plot forecasts for k∗/∆k∗ as
a function of k∗. For these results, we have fixed m = 10
−8 Mpl. We find that the signal-to-noise
ratio of k∗ does not monotonically increase with increasing k∗ for any single experiment. Since
this parameter controls the angular scale at which the LQG B-mode signal peaks, as we vary
k∗ we move between the sensitivity ranges of different experiments. For example at k∗ ≈ 0.002
Mpc−1, both Planck and Spider could detect the signal at more than 5σ whereas PolarBear and
QUIET would achieve only a marginal detection. Conversely, if k∗ ≈ 0.02 Mpc−1, the reverse is
true: PolarBear and QUIET would make strong (& 5σ) detections while Planck and Spider would
struggle to detect a signal.
In Fig. 4.12, we plot the two-dimensional constraints on the parameters m and k∗ for two
typical models. The left panel shows the forecasted constraints for a model with k∗ = 0.002
Mpc−1 and the right panel shows the constraints for a model with k∗ = 0.0002 Mpc
−1. In both
cases, a fiducial value of m = 10−8 Mpl was adopted. We find that the former case can be well
110
6.´10-9 8.´10-9 1.´10-8 1.2´10-8
0.0016
0.0018
0.0020
0.0022
0.0024
m
Ž
=mMpl
k *
@M
pc
-
1 D
Planck+QUIET
Planck+PolarBear
Spider
CMBPol
1Σ CL:
0 5.´10-9 1.´10-8 1.5´10-8 2.´10-8 2.5´10-8
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
m
Ž
=mMpl
k *
@M
pc
-
1 D
Planck+QUIET
Planck+PolarBear
Spider
CMBPol
1Σ CL:
Figure 4.12: Forecasted 2D constraints for the LQG model. The input models are indicated
with the black points. (Left: m˜ = 10−8 and k∗ = 0.002Mpc
−1. Right: m˜ = 10−8 and k∗ =
0.0002Mpc−1).
constrained by either Spider, CMBPol, Planck+PolarBear or Planck+QUIET while the latter case
can only be meaningfully constrained by the CMBPol mission.
4.5 Cosmic strings and their detection
4.5.1 B-mode polarization from cosmic strings
Cosmic strings have been proposed as a possible source of the inhomogeneities in the Universe
[226]. Although current observations of the CMB temperature and polarization power spectra
prefer inflation than cosmic strings as the main source of the primordial density perturbations
[72], there is still significant motivation to search for the signature of cosmic strings from both
theoretical and observational considerations.
Cosmic strings can be formed in several inflationary pictures, and particularly in Brane inflation
models [50; 190; 191; 227]. Brane inflation arises from the framework of high dimensional string
theory and is a further important model for sourcing the dynamics of inflation. In this model, the
high dimensional Brane and anti-Brane collided and annihilated and cosmic strings were produced
at the end of the inflationary epoch. If this scenario is correct, the resulting strings would have
made an observable imprint on the CMB sky by way of the Kaiser-Stebbins effect [228].
Although current observations suggest that inflation sources the majority of the CMB anisotropy,
one cannot rule out a significant (up to ∼ 10%) contribution from cosmic strings [229]. In this sec-
tion, we will use CMB data to constrain the cosmic strings tension. In contrast to other works (see
111
4. CONSTRAINTS ON THE STANDARD AND NON-STANDARD MODELS OF
THE EARLY UNIVERSE FROM CMB B-MODE POLARISATION
e.g. [229]), we focus on the possible detection of cosmic strings through the B-mode power spec-
trum alone. B-mode polarization can only be generated in the early Universe by vector and tensor
perturbations, which provides a complementary route for detecting cosmic strings. Although the
contribution of scalar perturbations from cosmic strings is subdominant (< 10%) compared to the
contribution from SFI, the contributions of vector and tensor perturbations from cosmic strings
may constitute a very significant fraction of the B-mode polarization power on small angular scale
(high multipoles) [230].
To predict the CBBl power spectrum generated by cosmic strings, one must understand the
evolution of a cosmic string network and it is important to know the characteristics of the scaling
regime which needs to be assumed in the numerical simulation. There are two popular ways of
making progress. One is to solve the Nambu equations of motion for a string in an expanding
universe and ignore the effects of radiation backreaction (hereafter the Nambu-String model); the
other is to solve the equations of motion for the Abelian-Higgs (AH) model, but to limit the
dynamical range of the simulation. In this work, we will consider only the Nambu-String model
but we note that the Abelian-Higgs model can be constrained in a similar way [229].
In order to calculate the B-mode power spectrum, including the contributions of both vector
and tensor perturbations, we use the publicly available code CMBACT [203; 204] to generate
a fiducial CBBl for cosmic strings with the tension of the strings set to Gµ0 = 10
−7. Since the
amplitude of the B-mode power spectrum generated by cosmic strings is simply proportional to
the square of the cosmic string tension, we can scale the fiducial spectrum to any other value for
the cosmic string tension using
CBBl = C
BB,0
l
(
Gµ
Gµ0
)2
, (4.39)
where CBB,0l is the power spectrum normalized at Gµ0 = 10
−7.
4.5.2 Constraints from current data
In Fig. 4.13, we show the current constraints on the cosmic string tension Gµ from the BICEP and
QUaD CBBl data. The upper bound from BICEP alone is Gµ ≤ 9.961× 10−7 (2σ CL). Using the
QUaD data alone the result is Gµ = 7.60+2.63−3.60×10−7(2σ CL), shown as the blue curve in Fig. 4.13.
Taken at face value, the QUaD result represents a 2.8σ detection of the cosmic string tension.
Referring back to Fig. 4.1 and the discussion in Section 4.3.2, this result is likely coming from the
apparent excess of power seen in the QUaD data on scales 300 < l < 500. To investigate further,
we have repeated the analysis with the l < 500 QUaD data removed. This results in the likelihood
function for Gµ shown as the brown curve in Fig. 4.13. Excluding the l < 500 QUaD data shifts
the peak of the likelihood significantly towards a smaller value (best-fit Gµ = 4.32×10−7), and the
constraint is now consistent with zero at the 1σ CL. The 2σ upper limit becomes Gµ < 8.57×10−7
(see Table 4.1 for the full set of results). The fact that the QUaD result changes drastically when
112
0 5.´10-7 1.´10-6 1.5´10-6 2.´10-6
0.0
0.2
0.4
0.6
0.8
1.0
GΜ values
PH
G
Μ
L
BICEP+QUaDHl>500L,HyperParameter
BICEP+QUaDHl>500L,conventional Χ2
QUaD OnlyHl>500L
QUaD Only
BICEP Only
Figure 4.13: Current constraints from B-mode polarization on the cosmic string tension. To
obtain the constraints, we have fixed the wiggling parameter at α = 1.9.
we remove the l < 500 measurements suggests a problem with the l < 500 data. As in the case
of the SFI constraints presented earlier, we note that the shape of the QUaD data at l < 500 is
clearly inconsistent with the expected B-mode signal for cosmic strings. Once again, we suspect
that the anomalous signal seen in the QUaD data between l ≈ 300 and l ≈ 500 is likely due to
unquantified systematics. We therefore consider the QUaD result restricted to l > 500 to be a
more robust constraint and consequently we quote this as our main result.
We now examine the constraints obtained from combining the BICEP data with the l > 500
QUaD data. Since the peaks of the two likelihood functions do not overlap, we will carefully
consider both the conventional χ2 analysis and the hyper-parameter χ2. When we use the conven-
tional χ2, the peak of the combined likelihood lies midway between the peaks of the two individual
likelihoods (green line in Fig. 4.13), and the resulting constraint is Gµ < 7.72× 10−7 (2σ CL). If
we use the hyper-parameter χ2, the peak of the joint distribution moves slightly further towards
zero, and the best-fit is Gµ < 8.01× 10−7 (2σ CL). Since the conventional and hyper-parameter
approaches give very similar results, this suggests that the BICEP and l > 500 QUaD data are
mutually consistent which adds confidence to the joint constraint.
A number of previous works have also attempted to constrain cosmic strings through their
imprint on CMB (TT, TE,EE), large scale structure and gravitational waves data [231; 232; 233;
234; 235; 236]. In a recent work [229], constraints on the cosmic string tension were obtained from a
combination of CMB data (including the WMAP 5-year, ACBAR, BOOMERGANG, CBI, QUAD
and BIMA observations), matter power spectrum data from the SDSS Luminous Red Galaxies
sample, and Big Bang Nucleosynthesis constraints on the baryon fraction from measurements of
deuterium at high redshift. They obtained a combined upper limit of Gµ < 2.2×10−7 (2σ) in the
Nambu-String case.
113
4. CONSTRAINTS ON THE STANDARD AND NON-STANDARD MODELS OF
THE EARLY UNIVERSE FROM CMB B-MODE POLARISATION
As expected, this limit is much tighter than what we have obtained from B-modes alone since
current measurements of the TT, TE and EE power spectra are much stronger than the BB data
that we have used here. However, we note that in the analysis of Ref. [229], the main constraining
power comes from the CMB TT and SDSS data and ultimately constraints from such data will be
limited by degeneracies with other parameters (most notably, the spectral index ns). As is the case
for SFI models, the advantage of using the BB power spectrum to constrain the cosmic string
tension is that CBBl is only very weakly dependent on the other cosmological parameters, e.g.
ns. Our constraint is therefore an independent check of other constraints obtained on Gµ using
different data and our result, although weaker, is consistent with previous analyses. Cosmic string
constraints from forthcoming CMB polarization experiments will likely close the gap with other
techniques in terms of constraining power. We now turn to examining the constraints achievable
with these forthcoming experiments.
4.5.3 Prospects for future experiments
1.´10.-9 5.´10.-91.´10.-8 5.´10.-81.´10.-7 5.´10.-71.´10.-6
1
10
100
1000
104
HGΜL values
HΜ
L
D
Μ QUIET
PolarBear
Spider
Ideal
CMBPol
Planck+QUIET
Planck+PolarBear
Planck Α=1.9HwiggL
Figure 4.14: Forecasts of the signal-to-noise ratio for the cosmic string tension Gµ potentially
achievable with future CMB polarization experiments.
In this subsection, we forecast the detectability of the cosmic string tension Gµ for future
experiments. Once again, we use the Fisher matrix formalism described in Section 4.2.3.2. In
Fig. 4.14, we plot the signal-to-noise ratio for a measurement of Gµ for the various experiments.
As one might expect, the Planck and Spider experiments are unable to tightly constrain the
cosmic string tension since neither experiment will produce sensitive B-mode measurements on
small scales where the string signal peaks.
For example, the Planck satellite can detect a cosmic string signal at the 3σ level only if
Gµ & 2× 10−7 is satisfied. The ground-based experiments, PolarBear and QUIET, will obviously
be better at constraining Gµ than Planck and Spider. PolarBear should be able to detect Gµ >
114
5.0× 10−8 (at the 3σ level) while QUIET should be able to detect Gµ > 4.0× 10−8 (again at 3σ).
Adding Planck to either of these experiments does not change the results significantly. CMBPol
is much more sensitive than the other experiments for all fiducial Gµ values, and its 3σ detection
limit is Gµ & 1.5 × 10−8. Finally, we find that the ideal CMB experiment can detect cosmic
strings at the 3σ level if Gµ & 1.5× 10−9 is satisfied, which represents the fundamental detection
limit for CMB B-mode polarization experiments.
4.6 Conclusion
In this Chapter, we have examined the observational signatures of three different models of the
early Universe related to the inflationary process. These three models are in turn motivated by
three different aspects of microscopic physics: single field slow-roll inflation from effective field
theory (SFI), loop quantum cosmology from loop quantum gravity (LQG) and cosmic strings
from Brane inflation and/or high dimensional string theory. We have discussed their potential
observational signatures in the B-mode polarization of the CMB, and we have constrained the
parameters of each model using the latest CMB polarization data from the BICEP and QUaD
experiments. Using a Fisher matrix formalism we have forecasted the constraints achievable on
these models using future CMB polarization observations from a number of experiments including
Planck (space), PolarBear (ground), QUIET (ground), Spider (balloon), CMBPol (space) and an
idealized experiment.
We first discussed the SFI model. From the Lyth bound relation, we know that r ∼ 0.01 is an
important benchmark value to link the energy scale of inflation with GUT scale. The constraints
we obtained from current B-mode measurements are shown in Fig. 4.3. Using the BICEP data
alone, we find r = 0.01+0.31−0.26 (1σ CL) in close agreement with the BICEP team’s own analysis [182].
As expected, this constraint does not change significantly on the addition of the small-scale QUaD
data. Looking to the future, we find that the Planck satellite may be able to detect r ∼ 0.05 (at
the 3σ CL), while the Spider, QUIET and PolarBear experiments all have the potential to make
a 5σ detection of r ∼ 0.05. The possible future satellite mission, CMBPol could detect r ∼ 0.01
at the 20σ CL, and could even detect r ∼ 0.002 at the ∼ 3σ CL. (All of these forecasts are for
the case where the tensor spectral index is held fixed at nt = 0.) In summary, we find that all
future experiments can potentially constrain the value of r with sufficient sensitivity to allow the
SFI model from effective field theory to be tested in an interesting way.
We have also discussed the LQG model, which predicts a pre-inflationary bouncing era. Before
the bounce, the Universe was contracting and dominated by vacuum energy. Its tensor power
spectrum is characterized by two parameters: m and k∗, where the mass parameter m controls
the magnitude of the B-mode power spectrum, and k∗ controls the scale of the peak in the B-
mode spectrum. Our joint likelihood analysis using current data yields m < 6.16× 10−8Mpl and
k∗ < 7.05 × 10−4 Mpc−1 (both 2σ CL upper limits). The PDF for the parameter k∗ exhibits
115
4. CONSTRAINTS ON THE STANDARD AND NON-STANDARD MODELS OF
THE EARLY UNIVERSE FROM CMB B-MODE POLARISATION
a peak at position k∗ = 1.07 × 10−4 Mpc−1. Although this peak is not statistically significant,
we note that were the value of k∗ to be around this value, this would constitute evidence for a
pre-inflationary bounce. We have also presented forecasts for constraining the LQG model using
future CMB experiments. We find that if the value of k∗ is as large as 0.002 Mpc
−1, a number of
ongoing and future experiments (Planck, Spider and CMBPol) could potentially detect the signal
as long as m > 2.5× 10−9 Mpl. However, if the value of k∗ were to be as low as 0.0002 Mpc−1 (as
is mildly indicated by current data), then the signature of LQG becomes quite difficult to detect.
We find that, for a typical choice of m = 10−8 Mpl, only CMBPol and the ideal experiment could
detect the signal for such a low value of k∗.
Finally, we have presented current and prospective constraints on the cosmic string tension. We
find that the BICEP and QUaD data constrain the cosmic string tension to be Gµ ≤ 9.96× 10−7
and Gµ ≤ 8.57 × 10−7 respectively (both 2σ CL upper limits). The combined constraint is
Gµ < 8.01×10−7, which is weaker, but comparable to the constraints from the CMB temperature
anisotropy power spectrum. In terms of forecasts for the future, we find that the high-resolution
ground-based experiments, PolarBear and QUIET, are more useful for constraining Gµ (as com-
pared to e.g. Planck and Spider) since they are much more sensitive to the B-mode power spectrum
on small angular scales. These two experiments could detect Gµ ∼ 10−7 at more than 10σ CL.
We also find that CMBPol can detect the signal of cosmic strings if Gµ > 1.5 × 10−8 at 3σ CL,
and the ideal CMB experiment can detect the signal at the 3σ level even if the tension of cosmic
string is as low as Gµ = 1.5× 10−9.
Although the B-mode polarization derived constraints which we have presented in this Chapter
are currently only upper limits, they are nevertheless already reached strong constraining power.
In terms of constraining the parameters of early Universe models, B-mode polarization is clearly a
very powerful tool and will likely overtake other early Universe probes with the advent of the next
generation of CMB polarization experiments. In this Chapter, we have not directly considered the
issue of model selection. However, it is likely that, in addition to constraining model parameters,
future sensitive B-mode observations will also allow us to distinguish between models of the early
Universe such as those considered in this Chapter.
116
Chapter 5
Peculiar Velocity Field: Constraining
the tilt of the Universe
5.1 Introduction
The bulk flow, i.e. the streaming motion of galaxies or clusters, is a sensitive probe of the density
fluctuation on very large scales. Recently there have been several observations of a large amplitude
of the bulk flow on hundred Mpc scales, which are in conflict with the predictions of the Lambda
cold dark matter (ΛCDM) model [244; 245]. The bulk flow is measured with respect to a frame
in which the cosmic microwave background (CMB) temperature dipole vanishes. We define this
as the CMB rest frame. It is usually assumed that the CMB rest frame coincides with the matter
rest frame which we define to be the frame in which the velocities of matter in our horizon volume
are isotropic.
It is possible that the two reference frames actually do not coincide with each other, resulting
in a “tilted universe” [246; 247; 248]. If the inflationary epoch lasted just a little more than the 60
or so e-folds needed to solve the horizon problem, the observable “remnants” of the preinflationary
Universe may still exist on very large scales of the CMB. As a result, there could be some fraction of
the CMB dipole due to the intrinsic fluctuations rather than observer’s kinetic motion. Therefore,
when the observed dipole is subtracted from the galaxy peculiar velocity data, the subtraction
induces a mismatch between the CMB rest frame and matter rest frame. We define the intrinsic
CMB dipole to be the CMB dipole measured in the matter rest frame. The recent finding of such
a mismatch between the directions of the observed CMB dipole and reconstructed velocity dipole
[249; 250] suggested the possible existence of the intrinsic CMB dipole on the sky.
Preinflationary fluctuations in a scalar field may produce an intrinsic dipole anisotropy. In this
scenario, the observed CMB dipole would be a sum of the dipole from our motion in the matter rest
frame and the intrinsic CMB dipole caused by a large scale perturbation. This intrinsic dipole
117
5. PECULIAR VELOCITY FIELD: CONSTRAINING THE TILT OF THE
UNIVERSE
can be produced by a large scale isocurvature perturbation 1, but not a large scale adiabatic
perturbation [251]. Note that our local motion relative to the matter rest frame is caused by small
scale inhomogeneity (up to about the 100 Mpc scale) and will be negligibly affected by the very
large scale (≫ 10 Gpc) perturbation that would cause a tilt effect.
In this Chapter, we estimate the tilted velocity (u) between the two rest frames using galaxy
peculiar velocity data. This opens a new window on testing early-Universe models from observa-
tions of large scale structure.
5.2 Likelihood and mock catalogues
In order to use the galaxy peculiar velocity catalogues (see the following descriptions of the data),
we need to model the velocity of galaxies in different rest frames. For each galaxy velocity survey,
we first subtract off our local motion with respect to the CMB as estimated from the CMB dipole.
However, if there is a non-negligible intrinsic CMB dipole, the CMB defined rest frame will not
correspond to the matter rest frame and thus there will be a residual dipole in the galaxy peculiar
velocity survey. To test this we estimate the line-of-sight velocity Sn of each galaxy n in the CMB
rest frame with measurement noise σn. Suppose the CMB rest frame has a tilt velocity u with
respect to the matter rest frame, then the line-of-sight velocity of each galaxy with respect to the
matter rest frame becomes pn(u) = Sn − rˆn,iui, where rˆn,iui is the projected component of the
3-D Cartesian coordinate u onto the line-of-sight direction of galaxy n. After subtracting out the
“tilted velocity”, we model the galaxy line-of-sight velocity with respect to the matter rest frame
as pn = vn + δn, where vn is the galaxy line-of-sight velocity in the matter rest frame, and δn is
a superimposed Gaussian random motion with variance σ2n + σ
2
∗ , where σ∗ accounts for the 1-D
small scale nonlinear velocity. It can also compensate for an incorrect estimation measurement
noise σn [244; 245]. Therefore, the covariance matrix of pn(u) becomes
Gmn = 〈vmvn〉+ δmn(σ2n + σ2∗)
= 〈(ˆrm·v(rm)) (rˆn·v(rn))〉+ δmn(σ2n + σ2∗), (5.1)
in which the cosmic variance term is [252]
〈(rˆm·v(rm)) (ˆrn·v(rn))〉 =
Ω1.1m H
2
0
2π2
∫
dkP (k)fmn(k), (5.2)
where
fmn(k) =
∫
d2kˆ
4π
(
rˆm·kˆ
)(
rˆn·kˆ
)
× exp
(
ikkˆ· (rm − rn)
)
, (5.3)
1In our context, an isocurvature perturbation is distinguished from an adiabatic perturbation in that the ratios
of the number of photons to baryons and cold dark matter particles are not spatially invariant.
118
which can be calculated analytically (Appendix C).
Therefore, the likelihood of the tilted vector and the small scale velocity dispersion σ∗ can be
written as
L(u, σ∗) =
1
(detGmn)
1
2
exp
(
−1
2
pm(u)G
−1
mnpn(u)
)
, (5.4)
where we fix the cosmological parameters at the WMAP 7-yr best-fit values (Ωb = 0.0449, Ωc =
0.222, h = 0.71, and σ8 = 0.801 [72]).
We parameterize the velocity as u = {u, cos(θ), φ} where in Galactic coordinates φ = l and
θ = π/2− b. In order for our marginalized prior on u to be uniform we set Prior(u, σ∗) ∝ 1/u2.
Then the posterior distribution of the parameters (u, σ∗) given the data (D) is Pr(u, σ∗|D) ∝
Prior(u, σ∗)L(u, σ∗) .
x0 180 360
Figure 5.1: The SN data plotted in Galactic coordinates.. The red points are moving away from us
and the blue ones are moving towards us. The size of the points is proportional to the magnitude
of the line-of-sight peculiar velocity. ”X” is our estimate of the direction of the tilted velocity
estimated from the SN data.
Before we perform the likelihood analysis for the real data, we have tested this likelihood from
mock catalogues. We input a set of fiducial values of (σ∗,u) and generate 300 simulations of the
underlying velocity field (Pv(k) ∼ P (k)/k2). The average likelihood of these simulations (Fig.
5.2) for each parameter exactly peaks at the input values, with the appropriate width determined
by cosmic variance, instrumental noise and small scale velocity dispersion.
5.3 Data Analysis
We studied five different catalogues of galaxy peculiar velocities to constrain the tilted velocity
u and the velocity dispersion σ∗. The five data sets are (see [244] for the procedure of excluding
outliers):
• ENEAR is a survey of fundamental plane (FP) [269] distances to nearby early-type galaxies
[253]. After the exclusion of 4 outliers, there are distances to 698 field galaxies or groups.
119
5. PECULIAR VELOCITY FIELD: CONSTRAINING THE TILT OF THE
UNIVERSE
2 3 4 5 6 7 8
0.0
0.2
0.4
0.6
0.8
1.0
Σ* @100kmsD
LH
Σ
*
L Input Σ*=4.0
simulation 4
simulation 3
simulation 2
simulation 1
Averge 300 simulations
0 2 4 6 8 10
0.00
0.05
0.10
0.15
0.20
0.25
u @100kmsD
LH
u
L
Input u=5.0
-1.0 -0.5 0.0 0.5 1.0
0.0
0.5
1.0
1.5
Cos@ΘD
LH
Co
s@
Θ
DL Input Cos@ΘD=0.5
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
Φ
LH
Φ
L
Input Φ=4.0
Figure 5.2: The simulation of the bulk flow velocity and test of likelihood (5.4). We input a
particular set of parameters (σ∗, u) and do 300 simulations of the underlying density field with
the mean value of this set of parameters. One can see that the peaks of the average likelihood are
very close to the input values ([Blue] dashed lines).
For single galaxies, the typical distance error is 20%. The characteristic depth of the sample
is 29 h−1Mpc.
• SN are 103 Type Ia supernovae distances [254], limited to a distance of .150 h−1Mpc. SN
distances are typically precise to 8%. The characteristic depth is 32 h−1Mpc. See Fig. 5.1.
• SFI++ [255], based on the Tully-Fisher (TF) [269] relation, is the largest and densest peculiar
velocity survey considered here. After rejection of 38 (1.4%) field and 10 (1.3%) group
outliers, our sample consist of 2720 field galaxies and 736 groups, so we divide it into two
subsamples, field samples SFI++F , and group samples SFI++G. The characteristic depth
is 34 h−1Mpc.
• SMAC [256] is an all-sky fundamental plane (FP) [269] survey of 56 clusters, with charac-
teristic depth 65 h−1Mpc.
• COMPOSITE is the combined catalogues (4536 data in total, compiled in [244; 245]) which
has the characteristic depth of 33 h−1Mpc. It is a combination of SN, SFI++, SMAC,
120
0 2 4 6 8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Σ* @100kmsD
PH
Σ
*
ÈD
L
COMPO
SMAC
SFI++F
SFI++G
SN
ENEAR
HaL
HbL
0 2 4 6 8 10 12 14
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
u @100kmsD
PH
u
ÈD
L
HcL
-1.0 -0.5 0.0 0.5 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
CosHΘL values
PH
Co
sH
Θ
LÈ
D
L
HdL
2 3 4 5 6 7 8
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Φ values
PH
Φ
ÈD
L
Figure 5.3: The 1-D marginalized posterior probability distribution functions of the parameters :
(a) σ∗, (b) magnitude of u, (c) Cos(θ), (d) φ.
ENEAR and also the samples from SBF [257] (69 field and 23 group galaxies, characteristic
depth 17 h−1Mpc), EFAR [258] (85 clusters and groups, characteristic depth 93 h−1Mpc), SC
[259] (TF-based survey of spiral galaxies in 70 clusters, characteristic depth 57 h−1Mpc), and
Willick [260] (Tully-Fisher based survey of 15 clusters, characteristic depth 111 h−1Mpc).
In Fig. 5.3, we show the marginalized 1-D posterior probability distribution functions of
the small scale velocity and intrinsic dispersion σ∗, and the velocity vector u. The best-fit and
marginalized error bars are listed in Table 5.1. In panel (a) of Fig. 5.3, we see that different
surveys prefer different σ∗. For the SN catalogue, there is a smaller σ∗, because the supernovae
light curves can be used to calibrate the distance measurement very precisely, so the scatter of
the line-of-sight velocity is small. In the COMPOSITE catalogue, σ∗ ∼ 450 km/s reflects the
average value of σ∗ of all of the catalogues, which can be treated as the average value of the small
scale velocity and the intrinsic dispersion. Using a single σ∗ should not have a significant effect,
in general, because the catalogues are of typical depth > 40 h−1Mpc or in better units > 4000
km/s, and the distance indicators have typically 20% error in the measurement which means that
typically σn > 800 km/s. So whether σ∗ is 400 km/s or 600 km/s is not of much consequence if
σn > σ∗.
121
5. PECULIAR VELOCITY FIELD: CONSTRAINING THE TILT OF THE
UNIVERSE
0 2 4 6 8
0
2
4
6
8
10
12
14
Σ* @100kmsD
u
@1
00
km
s
D
COMPO
SMAC
SFI++F
SFI++G
SN
ENEAR
HaL HbL
Found by Watkins et al. 2009
-1.0 -0.5 0.0 0.5 1.0
2
3
4
5
6
7
8
CosHΘL
Φ
Figure 5.4: (a) The 68%, 95% and 99.7% Bayesian confidence interval contours for parameters
σ∗-u. (b) The 68% contours for Cos(θ)-φ. The black dot is the direction of the bulk flow found
by [244].
In panel (b) of Fig. 5.3, we show the marginalized 1-D distribution of the magnitude of the u.
Also, from Table 5.1 we can see that the SFI++ and ENEAR catalogues can provide fairly tight
upper bounds on u, and ENEAR peaks at zero while the SFI++ catalogues contain zero velocity
within 2σ. It should be noticed that the SN and the larger COMPOSITE catalogue can provide
a nonzero 2σ lower bound on u, in which the latter provides the tightest constraints. In panels
(c) and (d) of Fig. 5.3, we show the marginalized 1-D probability distribution of the direction
of the relative velocity (cos(θ), φ). We see that the distribution of cos(θ) and φ are very close to
Gaussian, and the four catalogues predict a very similar direction of the velocity (Fig. 5.4 and
Table 5.1). We should also notice that the various catalogues give consistent constraints on the
tilted velocity.
In panel (a) of Fig. 5.4, we plot 2-D contours of σ∗ and u, and we find that, in the posterior
distribution, σ∗ and u are not correlated. The reason is easy to understand: the “tilted Universe”
velocity u describes superhorizon dipole modulation of CMB photons, whereas σ∗ describes the
subhorizon modes for small scale velocities and the intrinsic dispersion of each individual galaxy,
so they come from completely different origins therefore without much correlation. Thus, even
though there might be a possibility that the instrumental noise σn has been underestimated, the
distribution of σ∗ would be shifted to smaller values, but that will not change the constraints on
u significantly. Thus, our constraint on u is pretty robust with respect to the estimate of small
scale instrumental noise. In panel (b) of Fig. 5.4, we plot the 1σ contour of Cos(θ) and φ. We can
see that the directions of tilted velocity found by different surveys are very consistent with each
122
other, and also very consistent with the results from Watkins et al. [244].
5.4 Discussion
We should notice that, the direction of the large bulk flow velocity found by [244] is within the
1σ confidence level of the tilted velocity here, which therefore can provide a physical origin of the
large bulk flow in [244]. In [245] they evaluated that the probability of getting a higher bulk flow
within ΛCDM with WMAP7 parameter values to be less than 2%. In our analysis, the bulk flow
of a galaxy survey should be accounted for in the error bars by the cosmic variance term (Eq. 5.2).
So an alternative explanation might be that there is a feature in the matter power spectrum which
would increase the cosmic variance. However, as the shear and octupole moments of the peculiar
velocity field are not anomalously large, this is disfavored by the data [261].
In addition, the direction we find is consistent with that found in [262] which used the dipole
of the kinetic Sunyaev-Zeldovich (kSZ) measurements to estimate the bulk flow on Gpc scales, but
our magnitude is lower than what they found. However, there is an additional level of uncertainty
in converting the kSZ dipole into a bulk flow which makes it difficult to estimate the magnitude
of the bulk flow from the kSZ dipole [262]. In the bulk flow approach, Ref. [245] suggests that the
bulk flow comes from scales > 300 h−1Mpc. If [262] proves correct it will be strong support for
the tilt explanation since the tilted velocity should be the same regardless of the scale probed to
measure it. A tilted Universe scenario was also proposed in [248] to explain the kSZ measurement.
An intrinsic dipole on the CMB sky caused by a tilted Universe may be explained by a pre-
inflationary relic isocurvature inhomogeneity. If inflation lasts for only a few e-folds longer than
required to solve the horizon problem, the scales that were superhorizon at the initial point of
inflation are not very far outside our current horizon today. Inflation requires that there was
initially a fairly smooth region of order of the inflationary Hubble horizon. A physical scale of
such a region at the onset of inflation l = epH−1i will have the physical scale L = e
PH−10 today,
where P = p+N −Nmin (Nmin is the minimal number of e-folds to solve the horizon problem and
is generally assumed to be around 60) [246]. In the short inflationary case where N is not much
larger than Nmin, a remnant of a preinflationary Universe still exists on superhorizon scales which
may be detectable. The quadrupole effect, aka the Grishchuk-Zel’dovich effect [263], is also part
of the effect of this large scale inhomogeneity.
An isocurvature perturbation can produce the “tilted Universe” effect, which arises in infla-
tionary models due to the perturbations of quantum fields other than the inflaton, such as the
axion, whose energy density is subdominant compared to that of the inflaton.
There are strong constraints on subhorizon isocurvature perturbations and thus if they are
responsible for the tilt it would require them to be much larger on very large scales than they are
on smaller scales. Although multifield models will not generically produce such a sharp breaking
in scale invariance or result in isocurvature modes in the later Universe, there are double inflation
123
5
.
P
E
C
U
L
IA
R
V
E
L
O
C
IT
Y
F
IE
L
D
:
C
O
N
S
T
R
A
IN
IN
G
T
H
E
T
IL
T
O
F
T
H
E
U
N
IV
E
R
S
E
Catalogues Characteristic Depth (h−1Mpc) σ∗ [100km/s] u [100km/s] l (degrees) b (degrees) ∆N (2σ)
ENEAR 29 2.8±0.3 0+2.2× 287.2
+68.9
−68.3 −3.8+37.5−36.0 ∆N ≥ 7
SN 32 1.8± 0.4 4.5+1.8−1.9 284.9+22.9−22.1 −1.0+18.8−18.3 6 ≤ ∆N ≤ 9
SFI++G 34 5.10
+0.7
−0.3 3.1
+1.6
−1.9 289.7± 34.9 10.3+26.8−25.5 ∆N ≥ 6
SFI++F 34 6.6
+0.3
−0.2 2.3
+1.2
−1.6 276.8
+40.1
−39.0 15.8
+28.4
−27.1 ∆N ≥ 6
SMAC 65 0.0+1.7× 5.9± 2.4 263.8+23.6−18.5 1.1+18.6−16.0 6 ≤ ∆N ≤ 8
COMPOSITE 33 4.8+0.2−0.1 3.4± 1.3 285.1+23.9−19.5 9.1+18.5−17.8 6 ≤ ∆N ≤ 8
Table 5.1: The best-fit and 1σ confidence level for the velocity dispersion σ∗, the magnitude (u) and the direction (l, b) of the tilted
velocity, and the 2σ constraints on the number of e-folds of inflation. The “×” means that the value is less than zero.
12
4
models which can achieve this [247].
The dipole anisotropy is associated with the isocurvature fluctuation as follows [246; 247] 1
u
c
≃ H
−1
0
L
δϕ
ϕ0
, (5.5)
where δϕ is the fluctuation of the quantum field, and ϕ0 is the background field. An inflationary
scenario that results in isocurvature perturbations in the later Universe, such as in Ref. [247],
is required. Assume that at the onset of inflation, the isocurvature quantum fluctuation satisfies
δϕ/ϕ0 ≃ 1 [Constants of O(1) will not affect the results much], thus the Hubble horizon at the
onset of the inflation is p = 0, l = H−1i , and L = e
∆NH−10 , where ∆N = N − Nmin is the excess
number of inflationary e-folds beyond Nmin. The constraints on a “tilted Universe” lead to the
constraints on the number of e-folds of inflation u/c ≃ H−10 /L ≃ e−∆N . Therefore, an observable
“tilted Universe” velocity requires that inflation should last a modest number of e-folds, which
should not be too long nor too short— if inflation lasts too long, such perturbation effects would
be washed out, if inflation is too short, the dipole effect will be too large, making the Universe
over-tilted. We show the constraints on the number of e-folds in the last column of Table 5.1. We
find that the required extra number of e-folds is at least 6 for all catalogues, and SN, SMAC and
COMPOSITE can also provide a 2σ upper bound on ∆N given their data.
Note that our study provides a similar constraint to the Grishchuk-Zel’dovich effect (which can
arise from either adiabatic or isocurvature perturbations) which requires that the extra number
of e-folds should be greater than about 7 at the 95% confidence interval [264]. Also, correlations
in the CMB multipoles may be used to estimate, from the Planck data, the tilt with an error
bar similar to what we obtained here [265]. In the future, there is the potential to use the large
number of SNe measured by the Large Synoptic Survey Telescope to constrain u with a standard
deviation of about 30 km/s [266]. So it may be possible to explore the duration of inflation and
the preinflationary quantum state at quite a precise level. The peculiar velocity field, is therefore
a powerful tool to probe the very early-Universe in a manner not accessible by CMB observations
alone.
5.5 Conclusion
In this Chapter, we developed a model and a statistical method to justify whether the apparent
bulk flow motion of galaxies in our surveys is due to the subtraction of the intrinsic dipole on the
CMB sky. In the conventional bulk flow scenario, the galaxies in the local region (. 150 h−1Mpc)
are moving toward a direction (l = 287o ± 9o, b = 8o ± 6o) in which v = 407 ± 81 km/s seems
in tension with the ΛCDM predictions. However, we point out that some fraction of the CMB
1There may be some factors of order unity in this equation for different models, but they have negligibly small
effect on the ∆N constraints due to the exponential. The same holds for δϕ/ϕ0 ≃ 1.
125
5. PECULIAR VELOCITY FIELD: CONSTRAINING THE TILT OF THE
UNIVERSE
dipole can be intrinsic due to large scale inhomogeneities generated by preinflationary isocurvature
fluctuations, so that in the CMB rest frame, all of the galaxies have streaming velocity towards
the particular direction, resulting in the tilted Universe.
We modeled an intrinsic CMB dipole as a tilted velocity u and developed a statistical tool to
constrain its magnitude and direction. We found that (1) the magnitude of the tilted velocity u
is around 400 km/s, and its direction is close to what was found in previous bulk flow studies.
For SN, SMAC and COMPOSITE catalogues, u = 0 is excluded at about the 2.5σ level, which
can explain the apparent flow; (2) there is little correlation between the tilted velocity u and the
galaxy’s small scale velocity and intrinsic dispersion σ∗, which confirms our assumption.
Furthermore, assuming that primordial isocurvature modes lead to an intrinsic dipole anisotropy,
constraints on the magnitude of the tilted velocity can result in constraints on the duration of
inflation, due to the fact that inflation can neither be too long (no dipole effect) nor too short
(very large dipole effect). Under this assumption, the constraints on the tilted velocity require
that inflation lasts at least 6 e-folds longer (at the 95% confidence interval) than that required to
solve the horizon problem.
Finally, we should point out that if there is an intrinsic fluctuation, the free electrons in clusters
should be able to ”see” the modulation on the sky, which can be tested through the kSZ effect.
Therefore, the results from the South Pole telescope [158] and the Atacama cosmology telescope
[267; 268] may shed some light on the intrinsic fluctuations. Such work is in progress.
126
Chapter 6
Cosmic Mach Number as A Sensitive
Test of Growth of Structure
6.1 Introduction
The Cosmic Mach Number (hereafter CMN) can provide a robust measure of the shape and
growth rate of the peculiar velocity power spectrum of the galaxies in the universe. One considers
a region of a given size r in the universe, and compares the bulk motion of the sphere with the
random velocities within that region. The bulk motion provides a measurement of the forces on
the region from irregularities external to it, thereby measuring the amplitude of perturbations
on scales much larger than the region. The random motions within the comoving region reflect
gravitational perturbations on scales smaller than r. Thus their ratio, the CMN M(r), depends
only on the shape of the perturbation spectrum and not on its amplitude.
The concept was introduced by Ostriker and Suto in 1990 [270] as a cosmological metric that
would be relatively independent of the “bias” of the test particles being observed and also relatively
independent of the then quite uncertain perturbation amplitude. They concluded that although
the existing data were poor, they gave estimates for the CMN that appeared to be inconsistent
with the then popular CDM model with Ωm = 1 but seemed to prefer the open universe model
instead. In a certain sense, the application of this test, correctly predicted the currently best
validated cosmological models which have a value for Ωm in the range 0.2− 0.3.
Subsequent to the original paper, Strauss et al (1993) [271] found again that the CDM models
with Ωm = 1 remained inconsistent with the better data they used, but some non-standard models
passed the test (see also [272]). Nagamine et al (2001) [273] looked at ΛCDM models and found
better agreement, but the then ΛCDM model with Ωm = 0.37 again produced too high values of
M over the range of 3 − 40Mpc/h, whereas a model with Ωm ∼ 0.2 (actually closer to WMAP
7-yr constraint [72]) was more consistent with the observations. In addition, there are various
other papers discussing the issues around the CMN, such as non-linear clustering properties of
127
6. COSMIC MACH NUMBER AS A SENSITIVE TEST OF GROWTH OF
STRUCTURE
dark matter on the CMN measurement [274], and constraint on the CMN from Sunyaev-Zeldovich
effect [275].
This history leads us to re-examine the issue in light of the much better knowledge now available
of both the new peculiar velocity data and the range of cosmological models remaining plausible
given the current cosmological constraints. In this Chapter, we will develop a new statistical tool
to measure the CMN from the peculiar velocity surveys, and investigate the power of the CMN
to distinguish various cosmological models, especially in the aspects of differentiating the ΛCDM
model from variance with non-trivial growth function provided by Modified Gravity (hereafter
MG) models, and from models with massive neutrinos.
6.2 Statistics of Cosmic Mach Number
In linear perturbation theory, the power spectrum of the velocity divergence (θ ≡ ∇·v) is related
to the power spectrum of density fluctuations via Pθθ(k, z) = f
2(k, z)P (k, z), where f(k, z) ≡
−d ln δ/d ln (1 + z), and δ is the density perturbation of matter. Since the data in our application
are at very low-redshift, we assume that they have the same redshift z = 0 throughout and drop
the z dependence for brevity 1. The mean square velocity dispersion 〈σ2(r)〉 and mean square
bulk flow 〈V 2(r)〉 in a window of size r can be calculated as [270; 273; 277]
〈
V 2(r)
〉
=
H20
2π2
∫ ∞
0
dkPθθ(k)W (kr),
〈
σ2(r)
〉
=
H20
2π2
∫ ∞
0
dkPθθ(k) [1−W (kr)] ,
where W (x) is a top-hat window function W (x) = [3(sin x − x cosx)/x3]2. Note that W (x) ∼
1(x . 1) and W drops to 0 quickly when x > 1. Thus W effectively changes the integral limits of
the above formula,
〈
V 2(r)
〉 ≃ H20
2π2
∫ 1/r
0
dkPθθ(k),
〈
σ2(r)
〉 ≃ H20
2π2
∫ ∞
1/r
dkPθθ(k), (6.1)
The CMN on different scales of the patches is defined as
M(r) ≡
(〈V 2(r)〉
〈σ2(r)〉
) 1
2
. (6.2)
Thus it basically measures the shape of Pθθ by contrasting
∫
dkPθθ(k) on large, and small scales.
1If the CMN technique is used on structure formation at high redshift, one needs to consider the contribution
from the relative velocity between baryons and dark matter [276].
128
It is important to notice that this derivation assumes that one studies many patches of size r
and separately observes 〈V 2(r)〉 and 〈σ2(r)〉. In fact our observational data is restricted to our
neighborhood galaxies so one can only use V (r0), i.e. our local bulk motion. Therefore, one needs
to take into account the cosmic variance when comparing our local CMN with the global definition
(6.2).
For a particular catalogue with N objects, the CMN M can be written as M = |u|/σ∗, where
u denotes the bulk flow velocity, which is a streaming motion of galaxies towards some particular
direction, and σ∗ stands for the small scale velocity dispersion. Unfortunately, neither u nor σ∗
is a direct observable. For each galaxy peculiar velocity catalogue, what we observe is the line of
sight velocity Sn with measurement error σn for the nth galaxy. Then one can construct a joint
likelihood function for u and M by contrasting Sn with the line-of-sight projection of the bulk
flow rˆn,iui. The uncertainty in (Sn − rˆn,iui) is simply (σ2n + σ2∗)
1
2 , where σ∗ = |u|/M is given by
the definition of the CMN. Therefore, the likelihood function takes the form of,
L(u,M) =
N∏
n=1
1
[σ2n + (|u| /M)2]
1
2
exp
(
− (Sn − rˆn,iui)
2
2(σ2n + (|u| /M)2)
)
. (6.3)
One can then marginalise over the 3D bulk flow vector u to obtain the distribution of M for
each survey.
We use five different catalogues of galaxy peculiar velocity surveys, namely, SBF [257], ENEAR
[253], Type Ia Supernovae (SN) [254], SC [259], and SFI++F [255], to constrain the CMN by using
Eq. (6.3) 1. We marginalise over the ‘bulk flow’ velocity u in the 4-D parameter space and obtain
the 1-D posterior distribution of M as shown in panel (a) of Fig. 6.1. In the panel (a), one can
see that the distribution of the CMN is very Gaussian, and the width depends primarily on the
number of data entries in each catalogue. In addition, since each catalogue probes the CMN on
various depths, different catalogues form a complimentary set of tests of cosmic structures.
In panel (b) of Fig. 6.1, we put together the old (1990) and current CMN data with various
predictions computed from theoretic models: The [Blue] triangle data points are the current CMN
values computed from likelihood (6.3) by using the five-catalogues. The [Red] square data are the
ones used by Ostriker and Suto [270] in 1990, whose depths have been emulated by the current
data.
The ΛCDM model with WMAP 7-yr best-fit parameters (Ω = 0.271, h = 0.704, ns = 0.967,
Ωb = 0.0455, black solid line) is mildly consistent with the current CMN data at 3σ CL. In
comparison, we overplot the theoretical M(r) ([Green] dashed line) by using 1990’s ‘popular’
CDM parameters. One can immediately understand the reason why Refs. [270; 271] claimed
that there was a strong conflict between the data ([red] points) and popular CDM model ([Green]
1These catalogues are described in each individual paper in detail, also see [244]. To calculate the characteristic
depth of each catalogue, we use error-weighted depth as r =
∑
n wnrn/
∑
n wn, where wn = 1/(σ
2
n + σ
2
∗
).
129
6. COSMIC MACH NUMBER AS A SENSITIVE TEST OF GROWTH OF
STRUCTURE
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0
1
2
3
4
5
6
Cosmic Mach Number M
PH
M
ÈD
L
SFI++F, r=50.5
SC, r=32.8
SN, r=30.7
ENEAR, r=30.5
SBF, r=16.7HaL
ò
ò
ò
ò
ò
à
à
à
10 1005020 3015 70
1.
0.5
2.
0.2
5.
r @MpchD
Co
sm
ic
M
ac
h
N
um
be
rM
Hr
L
à O&S 1990 Data, 3Σ
ò Current Data, 3Σ
CDM with Wm=ns=1HO&S 1990L
Neutrino,ÚΝmΝ=0.2eV
f HRL, B0=10-4
WMAP7 LCDM
HbL
Figure 6.1: (a): The 1-D posterior distribution of the CMN M(r) from 5 different catalogues.
The unit of r is [Mpc/h]. (b): The comparison between the CMN data (3σ CL.) and theoretic
prediction: blue data points are the CMN data from posterior distributions shown in panel (a); red
data points are the data used in Ostriker and Suto 1990 [270]; the black line is the CMN prediction
from WAMP 7-yr best-fit values [72]; the green dashed line is calculated by using ‘popular’ CDM
cosmological parameters ns = Ωm = 1 and h = 0.5 in 1990s [270]; the brown dashed and purple
dot-dashed lines are the CMN from f(R) model with B0 = 10
−4 and from ΛCDM model with
neutrino mass
∑
ν mν = 0.2eV.
dashed line). There was in fact no inconsistency between data and a model with Ω ≈ 0.25 and
ns = 1. However, with the up-to-date data and strongly constrained ΛCDM cosmology, one can
see that the CMN data is consistent with the CMN prediction out to scale around 50Mpc/h.
In addition, we plot the CMN for f(R) theory with B0 = 10
−4 as brown dashed line, and the
ΛCDM model with massive neutrino
∑
ν mν = 0.2eV as purple dot-dashed line. These variations
of growth factor exhibit some substantial difference from the standard ΛCDM model. We should
notice that due to the large uncertainty of small scale non-linear velocity, we plot linear Pθθ on
scales k & 1h/Mpc in panel (b), but in reality, the nonlinear growth of structures on these scales
may suppress the CMN values over r . 15Mpc/h.
To investigate the prospective accuracy of the CMN achievable in future surveys, we perform
a forecast for the on-going 6dF peculiar velocity survey [278]. The redshift distribution of galaxies
for the 6dF survey is ng(z) = Az
γe−(z/zp)
γ
, where zp ≃ 0.0446, γ ≃ 1.6154 and A ≃ 622978 [278].
It peaks around z ≈ 0.05 and extends till z ≈ 0.15. We make an optimistic assumption that
the sub-catalogue of peculiar velocity field 1, which comprises about 12000 brightest early-type
galaxies, are located at z . 0.05, corresponding to the depth r . 150Mpc/h. We further assume
that the measurement error for line-of-sight velocity is around 20%, which is a typical error for the
fundamental plane distance measurement. We divide these data into different redshift bins, and
in each shell (r, r + dr), we calculate the Fisher Matrix value for the CMN FMM from Eq.(6.3)
1Although the total number of 6dF galaxies is 125071, the sub-catalogue galaxies with peculiar velocities
comprises 10% of the whole sample [278].
130
ìì
ì
ì
ì
ì
ì
ì
10 1005020 3015 15070
1.
0.5
2.
0.3
1.5
0.7
r @MpchD
M
Hr
L
ì 6dF Forecast
ÚΝmΝ=0.3 eV
ÚΝmΝ=0.2 eV
f HRL, B0=10-3
f HRL, B0=10-4
LCDM
HaL
Figure 6.2: (a): Forecast 1σ errors of the CMN from 6dF survey, in which we assume the data
is around our local region of the universe. Beside ΛCDM model, we plot B0 = 10
−4 ([Brown]
dashed line) and 10−3 ([Red] dashed line) for the f(R) model, and
∑
ν mν = 0.2eV ([Purple] dot-
dashed line) and 0.3eV ([Green] dashed line). (b): Pθθ [Black], and f(k, a) [Red] in ΛCDM (real
line), f(R) (dashed line) and massive neutrino (dot-dashed line) models. (c): fractional difference
between the f(R) and massive neutrino models with ΛCDM.
which leads to the forecasted error of the CMN as
σM(r) ≃ M(r)√
2N(r)
[
1 +
σ2n
(u(r)/M(r))2
]
, (6.4)
where N(r) is the number of data points in the shell (r,r+ dr), and u(r) is the average bulk flow
magnitude on depth r. We plot these forecast data in the panel (a) of Fig. 6.2. Comparing with
panel (b) of Fig. 6.1, we find the full range of the CMN data on scales [10,150] (Mpc/h) from 6dF
can improve the constraint on the variation of the scale-dependent growth factor significantly.
We summarize the experimental conditions for future experiment that can sharpen the CMN
test: (1) there should be considerably more galaxy samples (& 104) on scales [10, 150]Mpc/h; (2)
the smaller the measurement error σn is, and more homogeneous sky it can cover, the better it
can reduce the overall error of M . One should also notice that, when the CMN data is used to
constrain cosmology, the cosmic variance should be taken into account, since the bulk flow in our
local patch is greater than expected from ΛCDM model [244], i.e. as notes earlier, there can be
significant variance between a specific measurement of bulk flow V 2(r0) and its global average
〈V 2(r)〉.
6.3 A sensitive test of growth of structure
Since the CMN measures the shape of the peculiar velocity power spectrum Pθθ by design, it is
sensitive to any distortion of Pθθ. In the ΛCDM model, the growth is scale-independent. However,
131
6. COSMIC MACH NUMBER AS A SENSITIVE TEST OF GROWTH OF
STRUCTURE
in the modified gravity models, and models with massive neutrinos, the growth is generically scale-
dependent, thus Pθθ for these models is a distorted version of that for the ΛCDM model, making
the CMN an ideal tool to distinguish these models from ΛCDM. Let us consider the f(R) model
and ΛCDM model with massive neutrinos as examples.
In a viable f(R) model, where the Einstein-Hilbert action is extended to be a general function
of the Ricci scalar R, the effective value of Newton’s constant Geff has both time and scale
dependence, namely, Geff = µ(a, k)G, where µ(a, k) = (1 +
4
3
λ2k2a4)/(1 + λ2k2a4), and G is the
Newton’s constant in general relativity (GR) [279; 280]. The only free parameter is λ2, which
quantifies the Compton wavelength of the scalar field fR ≡ df/dR and characterises the scale-
dependence of the growth. It is more convenient to use a redefination of λ2, which is B0 =
2H20λ
2/c2, and B0 = 0 for GR [280]. The current constraint on B0 is B0 < 0.4 (95% CL.) using
the combined data of CMB, Integrated Sach-Wolfe (ISW) effect and Type Ia Supernovae [280].
In panels (b) and (c) of Fig. 6.2, we show Pθθ and f for a f(R) models with B0 = 10
−4 (dashed
lines), in comparison with that of ΛCDM model (solid lines). As we see, the growth is enhanced
at k & 0.01 h/Mpc, which can be seen by the CMN. In panel (a), we compare the CMN (f(R))
to the CMN (ΛCDM), with the forecasted 6dF data points overplotted. As discussed before, for
a survey with depth r ∈ [10, 150] Mpc/h, the growth on scales k ∼ 1/r ∈ [0.01, 0.1] h/Mpc can be
efficiently probed by the CMN. Also note that the CMN is determined by the integral of Pθθ (see
Eq. (6.1)), thus even a small distortion in Pθθ can leave an amplified imprints on the CMN. This
is why a 0.01% change in B0 from B0 = 0 can produce a 20% suppression in the CMN, which is
potentially observable by 6dF.
Similarly, the CMN is sensitive to the neutrino mass. When neutrinos became non-relativistic
and the Universe was deeply in the matter dominated era, neutrino thermal velocities damped out
the perturbation under the characteristic scale knr ≃ 0.018Ω1/2m (
∑
ν mν/1eV)h/Mpc, suppressing
the power spectrum on small scales. On scales greater than knr, neutrinos affect the overall
expansion of the Universe and therefore shift the peak of power spectrum to larger scales. We
normalize power spectrum on very large scale from WMAP 7-yr result [72] and plot the Pθθ(k)
and f(k, a) in panels (b) and (c) of Fig. 6.2. We can see that a neutrino mass of 0.2eV can
suppress the power spectrum on scales of k & 0.01h/Mpc quite significantly, which exactly falls
in the detection window of the CMN. Therefore, the cumulative ‘integral’ effect of the CMN can
manifest the neutrino free-streaming effect by enhancing its value on all scales. Comparing with
panel (b) of Fig. 6.1 and panel (a) of Fig. 6.2, we find that future CMN data is potentially able to
distinguish the nonzero neutrino mass. It is true that the parameter degeneracy, such as among
B0,
∑
ν mν , ns,Ωm might dilute the constraints, but the degeneracy can be easily broken with the
aid of other data. For example, we find that to mimic the same effect on CMN produced by
B0 = 10
−4, the spectral index ns has to be changed by 10% from its best fit value of WMAP7[72]
, which is not allowed by CMB data at 5σ level.
132
6.4 Conclusion
In this Chapter, we provide a statistical tool to measure the CMN, and further demonstrate that
it is a sensitive probe of the structure growth. By design, the CMN is immune to the uncertainty
in the overall amplitude of the density perturbation, and to linear galaxy bias. Also it is highly
sensitive to any scale-dependent distortion of Pθθ since any small distortion can be amplified via
the integral effect. Therefore it is an excellent tool to test alternative theories of gravity, and
models with a neutrino mass.
We first perform a likelihood analysis of the CMN from the current peculiar velocity field
data, and further confront the ΛCDM model with WMAP 7-yr parameters and popular CDM
model with parameters in 1990 with the current data and old data from [270]. We confirm that
the ΛCDM model prediction is consistent with current CMN data at 3σ CL. level. Based on
our forecast for 6dF, we find that the CMN can improve the constraints on the modified gravity
parameter B0 by orders of magnitude, and it can also tighten the present constraints on the
neutrino mass.
133
6. COSMIC MACH NUMBER AS A SENSITIVE TEST OF GROWTH OF
STRUCTURE
134
Chapter 7
Conclusions and Outlook
Over the last several decades, cosmology has been transformed into precision science. The standard
hot Big-Bang scenario, which was originally supported by three fundamental observations (Hubble
expansion, abundance of light elements, and CMB radiation), has been tested and confirmed by a
wealth of new precise observations, such as the cosmic microwave background anisotropies, Type-
Ia supernovae, and the large scale clustering of galaxies. The observational data provide strong
support for a Universe which is spatially flat and accelerating at the present day, consisting of
approximately 73% dark energy, 23% percent cold dark matter, and 4% baryons, which leads to
the concordance model ΛCDM.
Testing this concordance model and exploring unknown physics beyond this model (for exam-
ple, the nature of dark energy) have become important tasks for cosmologists. This thesis has
addressed four important questions in order to test ΛCDM model from different perspectives.
• The inflation paradigm predicts a nearly scale-invariant and statistically isotropic primordial
power spectrum. Do the current WMAP data favour statistical isotropy, and to what extent
will the more accurate data from the Planck satellite improve on WMAP?
• Simple inflationary models, such as single-field slow-roll V ∼ φ2 inflation, predict a nearly
scale-invariant tensor power spectrum with a tensor-to-scalar ratio of about 0.01. How
well, will the Planck satellite, as well as sub-orbital and ground-based experiment (such as
SPIDER, PolarBear and QUIET) be able to constrain this signal? In addition, what is the
detection limit for these experiments?
• Measurements of the peculiar velocities of various samples of galaxies suggest a very large
bulk flow on scales of 50 Mpc/h or more. It has been claimed that these observations are in
conflict with the prediction of the concordance ΛCDM model. If true, what are the physical
implications of the bulk flows?
• The peculiar velocity field is an important prediction from the linear perturbation theory.
How can we use current galaxy peculiar velocity data to test the growth of structure on
different scales, and to constrain alternative theories of gravity and neutrino masses?
135
7. CONCLUSIONS AND OUTLOOK
We have tackled the above questions in Chapters 2 to 6. The results of this thesis suggest the
following directions for future research.
Although we have studied large angular correlations in the WMAP data in some detail, several
authors have cast doubt on the accuracy of the low-multipoles measured by WMAP [281; 282; 283].
Refs. [281; 282; 283] argue that quadrupole is not cosmological, but due to a small error in the
dipole direction and antenna pointing direction in the WMAP scanning scheme. If their concerns
were to be confirmed, it could clearly be very difficult to accommodate a vanishing quadrupole
within the statistically-isotropic ΛCDM scenario.
Fortunately, we live at a time when results from complimentary experiments will soon be
available. The Planck satellite was launched by the European Space Agency in the 14th May, 2009,
and the collaboration team will release the first full-sky CMB map in January 2013. Since Planck
has 9 frequency bands, extending over a wide frequency range than WMAP’s 5 bands, it provides
much more information on Galactic foregrounds. One should therefore be able to reconstruct the
CMB reliably to lower Galactic latitude, therefore obtain a more stable reconstruction of the all
sky CMB.
So far, people have been searching for the statistical anisotropic power spectrum in the WMAP
temperature maps [173]. The WMAP polarization maps cannot be used to test departures from
statistical isotropy because they have very large noise and beam asymmetries. As Planck will
release high precision polarization maps, a direct extension of the work from Chapter 3 would be
to investigate the anisotropic power spectrum in the polarization map, and provide complimentary
checks of the temperature analysis, particularly if the temperature data suggest a nonzero result.
Distinguishing between different scenarios of the early Universe requires new experiments that
can give more powerful constraints on the primordial fluctuations. The Square Kilometer Array
[284] is such an example. This is a radio telescope array with an effective collecting area more
than 30 times greater than the largest array ever built. The HI redshift survey of SKA will lead to
significant improvements in the galaxy power spectrum at large scales over current observations.
A precise determination of the power spectrum on large scales may, for example, reveal sharp
features in the primordial power spectrum that are not possible to detect in CMB experiments.
Since different models of the early Universe may produce imprints on the cosmic structure at
large scales (e.g. models in light of trans-Planckian physics [285], thermal initial states [286; 287],
as well as particle production [288]), the SKA has the potential to discriminate between them.
By applying the Fisher matrix technique as used in Chapter 4, one can forecast the possible
constraints on these early Universe models from SKA.
As an extension of our analysis of large scale bulk flows, one needs to carefully test the con-
sistency of bulk flows with ΛCDM model. One of the major concerns with the analysis described
in [244; 245] is that different catalogues with very different depths and observational errors are
directly combined to obtain the posterior distribution of cosmological parameters. This approach
would be sensitive to systematics in the data, because each of the catalogues may prefer very dif-
136
òò ò ò
ò
à
à
à
à
à
20 40 60 80 100
1
2
3
4
5
6
7
R @MpchD
v
HR
L
@1
00
km
s
D à SFI++,Distance
ò SFI++,Redshift
LCDM with WMAP7 parameters
HaL
ì
-1.0 -0.5 0.0 0.5 1.0
0
1
2
3
4
5
6
CosHΘL
Φ
ì CMB Dipole
R=90 Mpch
R=70 Mpch
R=50 Mpch
R=30 Mpch
R=10 Mpch
SFI++
1Σ  CL
HbL
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
Σ8 values
PH
Σ
8È
D
L
COMPO:Redshift, Top-hat Func
COMPO:Distance, Gaussian Func
SFI++F:Redshift, Top-hat Func
SFI++F:Distance, Gaussian Func
SFI++G:Redshift, Top-hat Func
SFI++G:Distance, Gaussian FuncHcL
Figure 7.1: Demonstration of potential effects associated with Malmquist bias. Panel (a): bulk
flow magnitude as a function of depth R. By placing the galaxies at their redshifts ([Blue]
triangle points) rather than distance ([Red] square points), the bulk flow magnitude is consistent
with ΛCDM prediction. However, if we place the galaxies at their TF distances, we find large
flows on scales & 50Mpc/h. Panel (b): If we place the galaxies at their redshifts, the directions
of the observed bulk flow at different depths is consistent with CMB dipole direction. Panel
(c): Parameter estimation of σ8 by using the COMPOSITE catalogue, SFI++ group galaxies
(SFI++G), and SFI++ field galaxies (SFI++F) (see description of the catalogues in [244] for
details). By using COMPOSITE catalogue, placing the galaxies at TF distance, and generating
mock catalogues from Gaussian window function, Ref.[244] obtains the distribution ([Black] dashed
line) which excludes WMAP 5-yr best-fit σ8 = 0.796 at 3σ CL. However, by using the redshift as
the true distances and adopting a top-hat window function, one finds a consistency with ΛCDM
([Black] solid line). Here we fix the cosmological parameters to be WMAP 5-yr results (Ωm =
0.258, h = 0.719, ns = 0.963, Ωb = 0.0441) in order to compare with the results in [244].
ferent parameters. In addition, since Malmquist bias can also be a problem for peculiar velocity
surveys, it is important to model the Malmquist bias and to ensure that the distance estimations
137
7. CONCLUSIONS AND OUTLOOK
are accurate. Figure 7.1 shows some preliminary results of an investigation of these effects.
In Fig. 7.1, we show the potential problem of Malmquist bias in the Tully-Fisher (TF) distance
of SFI++ catalogues. In panel (a), we separate the SFI++ catalogue (see descriptions of the
catalogue in Sec.5.3 in Chapter 5) into different radial distance bins, and calculate the bulk flow
magnitude in each of them. If we place the galaxies at their TF distances, there are clearly large
flows on scales of & 50Mpc/h. However, if we place the galaxies at their redshifts, the magnitude
of the flows are consistent with ΛCDM predictions on different scales. This indicates that the
potential problem of the inhomogeneous Malmquist bias gives spurious flows. In panel (b) of Fig.
7.1, we show that if we place the galaxies at their redshifts, the direction of the bulk flows on
different depths are consistent with the CMB dipole direction.
Besides the potential problem of Malmquist bias, there is another potential issue around the
analysis in [244]. In order to weight the data sets and calculate the bulk flow velocity, Ref. [244]
uses a ‘Minimum Variance’ method. This method is to calculate the weight function of the data
catalogues by minimizing the variance between it and ideal surveys of different depths. Therefore,
Ref. [244] needs to first simulate ideal surveys at different depths. The galaxy redshift distribution
of an ideal survey in [244] is assumed to be Gaussian and isotropic n(r) ∝ exp(−r2/2R2). However,
this would make the bulk flow at 50Mpc/h ‘contaminated’ by galaxies at much larger radii (perhaps
& 100Mpc/h), for which the observational errors may be significant and observational errors
themselves may be underestimated. As an alternative, one could instead use top-hat window
function, i.e. calculate the bulk flow within some radius r as
B(r) =
3
4πr3
∫
x<r
v(x)d3x. (7.1)
In panel (c), we show the comparison between the TF distance+ Gaussian window function
(used in [244]) and our corrections for redshifts+ top-hat window function. In this panel, the
[Black] dashed line is the COMPOSITE catalogue if using the convention of [244], which excludes
the WMAP 5-yr best-fit value σ8 = 0.796 at 3σ confidence level. After correcting for both the
top-hat window function, and the galaxy redshifts, one can see that the distribution of σ8 from
COMPOSITE catalogue is shifted to lower value of σ8 ([Black] solid line) which is consistent with
WMAP 5-yr best-fit value. Other catalogues also have similar situations, such as SFI++F and
SFI++G, whose likelihoods, after the corrections, are all consistent with WMAP 5-yr σ8.
In Chapter 4, we have developed the hyper-parameter technique to combine different catalogues
with different systematics. This technique should be applied to the combination of peculiar velocity
catalogues. This project is now in progress.
In addition to the above analysis, it is also important to test whether the peculiar velocity
field is consistent with density field. The density field at different depths can be reconstructed
from infrared surveys, such as IRAS/PSCz [289] and future surveys based on WISE [290]. Once
138
the density field is reconstructed, one can apply linear perturbation theory to predict velocity
field. Therefore, one can compare the reconstructed velocity field from linear perturbation theory,
with the smoothed velocity field from SFI++ survey. This would give a direct test of whether
the observed peculiar velocity data is consistent with the linear perturbation theory with the
hyper-parameter analysis described above [291].
In the near future, we would like to use the 6dF data 1 to study the Cosmic Mach Number on
different scales. By applying this technique, with the combination of CMB and Type-Ia Supernovae
data, it may be possible to improve on the results of Chapter 6.
1The analysis of 6dF is on-going in the collaboration team, and the peculiar velocity catalogue for the brightest
∼ 12000 galaxies will be released soon.
139
7. CONCLUSIONS AND OUTLOOK
140
Appendix A
Instrumental Characteristics of CMB
Experiments
To calculate the total noise power spectrum NBBl (Section 4.2.3.2), we require the experimen-
tal specifications of each experiment, including the levels of both residual foreground noise and
instrumental noise. We list the instrumental noise for each frequency channel the experiments
we have considered in Table A.1-A.5. The effective noise power spectrum NBBl is given by the
optimal combination of the channels [201]
[NBBl ]
−2 =
∑
i≥j
[
(NBBfg,l (i) +N
BB
ins,l(i))(N
BB
fg,l (j) +N
BB
ins,l(j))
1
2
(1 + δij)
]−1
, (A.1)
where NBBins,l(i) and N
BB
fg,l (i) are the instrumental and residual foreground noise power spectra,
respectively. Note that the noise power spectra NBBins,l(i) listed in the tables do not include the
window function of the instrumental beam exp[l(l + 1)θ2F/(8 ln 2)].
To model polarized foregrounds, we focus on diffuse synchrotron (S) and dust (D) emission.
The foreground contamination can be quantified by the parameter σfg which multiplies the power
spectra CBBS,l (i), C
BB
D,l (i) of the foreground models. The smaller the value of σ
fg the deeper the
Band center [GHz] 30 44 70 100 143 217 353
FWHM [arcmin] 33 24 14 10.0 7.1 5.0 5.0
NBBins,l(i) [10
−6µK2] 2683 2753 2764 504 279 754 6975
fsky 0.65
Table A.1: Instrumental parameters for the Planck satellite (space-based experiment) [27]. Here
we have assumed 4 sky surveys (28 months).
141
A. INSTRUMENTAL CHARACTERISTICS OF CMB EXPERIMENTS
Band center [GHz] 90 150 220
FWHM [arcmin] 6.7 4.0 2.7
NBBins,l(i) [10
−6µK2] 5.2 4.3 44.0
fsky 0.024
Table A.2: Instrumental parameters for the ground-based PolarBear experiment [30].
Band center [GHz] 40 90
FWHM [arcmin] 23 10
NBBins,l(i) [10
−6µK2] 0.26 0.64
fsky 0.04
Table A.3: Instrumental parameters for the ground-based QUIET experiment [31]. Here we have
assumed the phase-2 experiment.
Band center [GHz] 100 145 225 275
FWHM [arcmin] 58 40 26 21
NBBins,l(i) [10
−6µK2] 84.4 47.4 395 1170
fsky 0.5
Table A.4: Instrumental parameters for the balloon-borne Spider experiment [33]. Here we have
assumed a 30-day LDB flight.
Band center [GHz] 30 45 70 100 150 220 340
FWHM [arcmin] 26 17 11 8 5 3.5 2.3
NBBins,l(i) [10
−6µK2] 31.21 5.79 1.48 0.89 0.83 1.95 39.46
fsky 0.8
Table A.5: Instrumental parameters for the mid-cost (EPIC-2m) CMBPol satellite mission [201].
Parameter Synchrotron Dust
AS,D 4.7× 10−5 µK2 1.2× 10−4 µK2
ν0 30 GHz 94 GHz
l0 350 900
α -3 2.2
βBB -2.6 -1.4
Table A.6: Assumptions for foregrounds parameters [201]
142
foreground cleaning. Throughout this paper, we adopt σfg = 0.1. The residual foreground noise
is given by (see [201] for instance)
NBBfg,l (i) =
∑
f=S,D
[
CBBf,l (i)σ
fg + NBBf,l (i)
]
, (A.2)
where NBBf,l (i) is the noise power spectrum arising from the cleaning procedure itself in the presence
of instrumental noise.
Following [201; 237; 238], we model the scale (l) and frequency (νi) dependence of the syn-
chrotron and dust emission as
CBBS,l (i) = AS
(
νi
ν0
)2αS ( l
l0
)βS
(A.3)
and
CBBD,l (i) = p
2AD
(
νi
ν0
)2αD ( l
l0
)βBBD [ehν0/kT − 1
ehνi/kT − 1
]2
. (A.4)
In Eq. (A.4), p is the dust polarization fraction, estimated to be 5% [237], and T is the temperature
of the dust grains, assumed to be constant across the sky with T = 18K [237]. Other parameters
in Eqs. (A.3), (A.4) are specified in Table A.6 taken from [201].
The noise term NBBf,l (i) (f = S,D) entering Eq. (A.2) is calculated in [201; 237]
N
BB
f,l (i) =
NBBins,l(i)
nchan(nchan − 1)/4
(
νi
νref
)2α
. (A.5)
Here, nchan is the total number of frequency channels used in making the foreground template
map, and νref is the frequency of the reference channel. In the case of dust, νref is the highest
frequency channel included in the template making, while in the case of synchrotron, νref is the
lowest frequency channel. The value of α is given in Table A.6 for different foreground models.
We note that the ground-based experiments are insensitive to the largest angular scales, so when
calculating the Fisher matrix using Eq. (4.13), we sum over the l from 21 to 3000. In addition,
for ground-based experiments, the small scale fluctuations are not very sensitive to the residual
foreground noise, and we can also pick out relatively clean patchs of sky where the foreground
contamination is minimal. Therefore, to forecast the results for PolarBear and QUIET, we have
not included a residual foreground noise term.
In addition to instrumental and residual foreground noise, gravitational lensing converts E-
mode polarization into B-modes on small angular scales, contaminating the primordial B-mode
signal [239; 240; 241; 242]. The lensed CBBl (lens) will also contribute to the total noise power
143
A. INSTRUMENTAL CHARACTERISTICS OF CMB EXPERIMENTS
spectrum NBBl . The total noise power spectrum therefore becomes
NBBl,tot ≡ NBBl + CBBl (lens). (A.6)
For the ideal case, we assume that there is no instrumental or foreground noise, and that we
can successfully de-lens the CMB observations to a level of about 1/40 of the lensing signal [202].
In this case, the total effective noise power spectrum is
NBBl,tot = 1/40× CBBl (lens). (A.7)
Finally, we adopt fsky = 0.8 for the ideal experiment, which is the same as that used to model
CMBPol.
144
Appendix B
Statistics of the conventional χ2 and the
hyper-parameter χ2
In this appendix, we first review the basic results of conventional χ2 statistics and then generalize
the analysis to the hyper-parameter technique.
B.1 χ2 statistics
A conventional joint χ2 analysis will minimize the following combined χ2
χ2tot =
∑
j
χ2j , (B.1)
where each χ2n follows the chi-square distribution
f(χ2n) =
1
2
n
2 Γ
(
n
2
)(χ2n)n2−1 exp(−12χ2n). (B.2)
It is easy to show that this chi-square distribution is properly normalized, i.e.
∫ ∞
0
f(χ2n)dχ
2
n = 1, (B.3)
and the expectation value and the variance are
E(χ2n) = n, V (χ
2
n) = 2n. (B.4)
145
B. STATISTICS OF THE CONVENTIONAL χ2 AND THE
HYPER-PARAMETER χ2
Therefore, the minimum χ2 value of a properly constrained model should satisfy the following
relation
1−
√
V (n)
E(n)
≤ χ
2
min
E(χ2n)
≤ 1 +
√
V (n)
E(n)
. (B.5)
For the χ2 with order n, this is
1−
√
2
n
≤ χ
2
min
n
≤ 1 +
√
2
n
. (B.6)
If the χ2min/n ≥ 1 +
√
2
n
, we can say that the model does not provide a good fit to the data,
whereas if χ2min/n ≤ 1−
√
2
n
, we say that the model overfits the data, which may mean that the
model has redundant free parameters.
If there are m constraints on the n random variables, then χ2n still follows the chi-square
distribution, but with order n−m [243]
f(χ2n) =
1
2
n−m
2 Γ
(
n−m
2
)(χ2n)n−m2 −1 exp(−12χ2n). (B.7)
It is straightforward to verify, that the shape of the distribution does not change, but the expec-
tation value and the variance are changed simply as n→ n−m.
B.2 Hyper-parameter χ2
The hyper-parameter approach to combining the constraints from different data sets can be useful
in the case where the different data sets have different levels of systematics. To weight each data
set, one should multiply each χ2 by a free parameter,
χ2hyper =
∑
j
αjχ
2
j , (B.8)
where αj is the weight parameter for each data set. One can marginalize these weight parameters
in a Bayesian analysis, and in [205], the authors found that instead of minimizing the combined
χ2, one should instead minimize the following quantity
χ2hyper =
∑
j
nj lnχ
2
j , (B.9)
146
where nj is the number of degrees of freedom for each data set. If we only consider one data set,
the hyper-parameter statistic becomes
χ2hyper = n lnχ
2
n. (B.10)
From Eq. (B.2), using the following transformation
fY = fX
∣∣∣∣dYdX
∣∣∣∣ , (B.11)
one can show that the distribution of the hyper-parameter statistic is given by
g(χ2hyper) =
1
n · 2n2Γ (n
2
) exp(1
2
χ2hyper) exp(−
1
2
exp(
1
n
χ2hyper)), (B.12)
which has already been properly normalized. (When calculating the integral, one should use the
transformation exp(x
n
) = y). One can then show that the expectation value and variance of the
Hyper-parameter distribution is
E(χ2hyper) = n
(
ln 2 + ψ0
(n
2
))
, V (χ2hyper) = n
2ψ1
(n
2
)
, (B.13)
where ψn(x) is the “digamma function” defined as derivatives of the log Gamma function
ψn(x) =
dn+1
dxn+1
ln Γ(x). (B.14)
If there are m constraints on the n random variables (e.g. m parameters), then the distribution
of the conventional χ2n follows Eq. (B.7). It is then easy to show that the form of the distribution
is unchanged only if the χ2hyper is defined as
χ2hyper = (n−m) lnχ2n. (B.15)
Proof From Eqs. (B.7) and (B.11), one can derive the following PDF
g(χ2hyper) = f(χ
2
n)
dχ2n
dχ2hyper
= f
(
exp
(
χ2hyper
n−m
))
1
n−m exp
(
χ2hyper
n−m
)
147
B. STATISTICS OF THE CONVENTIONAL χ2 AND THE
HYPER-PARAMETER χ2
=
1
n−m exp
(
χ2hyper
n−m
)
1
2
n−m
2 Γ
(
n−m
2
)
×
[
exp
(
χ2hyper
n−m
)]n−m
2
−1
exp
(
−1
2
exp
(
χ2hyper
n−m
))
=
1
n−m
1
2
n−m
2 Γ
(
n−m
2
) exp(χ2hyper
2
)
exp
(
−1
2
exp
(
χ2hyper
n−m
))
. (B.16)
Therefore, in the case of m constraints, the distribution keeps its form, and the expectation value
and variance become
E(χ2hyper) = (n−m)
(
ln 2 + ψ0
(
n−m
2
))
, V (χ2hyper) = (n−m)2ψ1
(
n−m
2
)
. (B.17)
Thus, to ensure that the form of the χ2hyper distribution function is unchanged, the χ
2
hyper needs to
be defined as
χ2hyper =
∑
j
nj lnχ
2
nj
, (B.18)
where nj = ndata −m is the number of degree of freedom.
The hyper-parameter approach is an objective way to weight each data set when producing
joint constraints. The value of the weight is simply the value of the effective hyper-parameter,
which is defined as [205]
αA =
nA
χ2A
, (B.19)
where A specifies a particular data set. Therefore, the larger the value of α, the larger the weight
that the particular data set takes (see Table 4.3).
148
Appendix C
Analytic formulas of the correlated
Window Function f12(k)
The correlated window function f12(k)
f12(k) =
∫
d2kˆ
4π
(
rˆ1·kˆ
)(
rˆ2·kˆ
)
× exp
(
ikkˆ· (r1 − r2)
)
, (C.1)
can be calculated analytically, by transforming it into harmonic space and using the property of
spherical harmonics. The final integral should only depend on: (a) the angle α between r1 and
r2 (therefore r1 and r2 should be symmetric); (b) r1 and r2 (amplitude of the vector); (c) k (k’s
amplitude). Therefore, we can specify r1 = (0, 0, 1), r2 = (0, sinα, cosα), where α is the relative
angle between r1 and r2. Therefore,
A = r1 − r2 = (0,−r2 sinα, r1 − r2 cosα). (C.2)
So its direction and amplitude become
Aˆ =
1
A
(0,−r2 sinα, r1 − r2 cosα), (C.3)
and
A = [r21 + r
2
2 − 2r1r2 cosα]
1
2 . (C.4)
149
C. ANALYTIC FORMULAS OF THE CORRELATED WINDOW FUNCTION
F12(K)
We can set
kˆ =(sin θ cosφ, sin θ sinφ, cos θ), (C.5)
therefore,
kˆ · Aˆ = 1
A
((−r2 sinα) sin θ sinφ+ (r1 − r2 cosα) cos θ) . (C.6)
Now we can use spherical harmonic function Ylm(θ, φ) to decompose the integrand as follows.
(
rˆ1·kˆ
)(
rˆ2·kˆ
)
= cos θ(sinα sin θ sinφ+ cosα cos θ)
= i
√
2π
15
sinα (Y2,1 + Y2,−1) +
4
3
√
π
5
cosαY2,0 +
1
3
cosα, (C.7)
exp(ikkˆ · (r1 − r2)) =
∑
l
il(2l + 1)jl(kA)Pl(kˆ · Aˆ), (C.8)
in which we can just consider l = 0, 2 two terms in the summation. The reason is that the
mixing angle between k and A just causes the mixing between different m modes in the spherical
harmonics in Eq. (C.8), so the final non l = 0 and l = 2 terms vanish due to the orthogonality.
Therefore, Eq. (C.8) becomes
exp(ikkˆ · (r1 − r2)) = j0(kA)− 5j2(kA)P2(kˆ · Aˆ), (C.9)
where jl(kA) is spherical bessel function.
P2(kˆ · Aˆ) = 1
2
(
3(kˆ · Aˆ)2 − 1
)
= − 3
2A2
√
2π
15
(r2 sinα)
2 (Y2,2 + Y2,−2)
+
2
A2
√
π
5
(
(r1 − r2 cosα)2 − 1
2
(r2 sinα)
2
)
Y20
− 3
A2
(r2 sinα)(r1 − r2 cosα)
× i
√
2π
15
(Y2,1 + Y2,−1) . (C.10)
Note that there is no zero order term in P2(kˆ · Aˆ). Then we use the orthogonality property of Ylm∫
d2kˆYlmY
∗
l′m′ = δll′δmm′ and Y
∗
lm = Yl,−m(−1)m and get the final result
f12(k) =
1
3
cosα(j0(kA)− 2j2(kA)) + 1
A2
j2(kA)r1r2 sin
2 α. (C.11)
150
It is clear that this integration has the three properties we listed above and the window function
f12 depends only on (r1,r2,k,α). This is an independent and simplified but equivalent result to
Eq.(9.32) in [44].
151
C. ANALYTIC FORMULAS OF THE CORRELATED WINDOW FUNCTION
F12(K)
152
References
[1] D. H. Spergel et al., ApJS.148 (2003) 175-194. 1, 55, 56, 58, 72
[2] D. N. Spergel et al., ApJS, 170 (2007) 377. 1, 4
[3] E. Komatsu et al., ApJS. 180 (2009) 330. x, xii, 1, 4, 20, 27, 28, 33, 70, 71, 74
[4] S. Cole et al., MNRAS.362 (2005) 505. 1, 2
[5] M. Tegmark et al. ApJ.606 (2004) 702. x, 1, 2, 30
[6] D. Eisenstein et al. ApJ. 633 (2005) 560. 1, 2
[7] S. Perlmutter et al., ApJ.517 (1999) 565. 1
[8] A. Riess, ApJ. 607 (2004) 665. 1
[9] P. Astier, Astron.Astrophys. 447 (2006) 31. 1
[10] C. Reichardt et al. ApJ. 694 (2009) 1200. 1, 75
[11] D. Bacon, R. Massey, A. Refregier, R. Ellis, MNRAS. 344 (2003) 673. 1
[12] C. Heymans et al., MNRAS. 361 (2005) 160. 1
[13] E. Semboloni et al., Astron.Astrophys., 452 (2006) 51. 1
[14] S. Allen, R. Schmidt, A. Fabian, H. Ebeling, MNRAS. 342 (2003) 287. 1
[15] N. Bahcall, P. Bode, ApJ. 588 (2003) L1-L4. 1
[16] A. Voevodkin, A. Vikhlinin, ApJ. 601 (2004) 610. 1
[17] S. Allen et al., MNRAS. 383 (2008) 879. 1
[18] R. Croft et al., ApJ. 581 (2002) 20. 1
[19] M. Viel, M. Haehnelt, V. Springel, MNRAS. 354 (2004) 684. 1
153
REFERENCES
[20] T. Jena et al., MNRAS. 361 (2005) 70. 1
[21] U. Seljak et al., Phys. Rev. D71 (2005) 103515. 1
[22] A. Guth, Phys. Rev. D 23 (1981) 347. 1
[23] A. Linde, Reports of Progress in Physics, 47 (1984) 925. 1
[24] U. Seljak and M. Zaldarriaga, ApJ 469 (1996) 437. 2, 33
[25] http://camb.info; A. Lewis, A. Challinor and A. Lasenby, ApJ, 538 (2000) 473. 2, 33, 92
[26] P. J. E. Peebles, AJ, 75 (1970) 13. 1
[27] The Planck Collaboration, 2005, ‘The Scientific Programme of Planck’, eds. G. Efstathiou,
C. Lawrence, and J. Tauber, ESA-SCI(2005), ESA Publications. 1, 17, 28, 29, 30, 50, 51, 76,
79, 92, 141
[28] H. T. Nguyen. et al., Proc. SPIE, 7020 (2008) 70201F. 1, 92
[29] J. A. Rubino-Martin et al., arXiv: 0810.314; http://www.iac.es/project/cmb/quijote/. 1, 92
[30] http://bolo.berkeley.edu/POLARBEAR/index.html. 1, 50, 92, 142
[31] D. Samtleben, arXiv: 0802.2657; http://quiet.uchicago.edu/. 1, 50, 92, 142
[32] http://groups.physics.umn.edu/cosmology/ebex/index.html. 2, 30, 92
[33] B. P. Crill, et al., arXiv:0807.1548. 2, 30, 92, 142
[34] E. Hubble, Proceedings of the National Academy of Science, 15 (1929) 168. 4
[35] W. Freedman et al., ApJ, 553 (2001) 47. 4
[36] G. Gamov, Physical Review, 70 (1946) 572. 4
[37] M. Pettini and D. Browen, ApJ, 560 (2001) 41. 4
[38] K. Olive and E. Skillman, ApJ, 617 (2004) 29. 4
[39] A. Penzias and R. Wilson, ApJ, 142 (1965) 419. 4
[40] R. Dicke, P. Peebles, R. Roll, and D. Wilkinson, ApJ, 142 (1965) 414. 4
[41] G. Smoot et al., ApJ, 396 (1992) L1. 4, 55
[42] A. Riotto, astro-ph/0210162. ix, 4, 8, 11, 12, 14, 15, 95, 96
[43] H. Peiris, Cosmology Lecture Notes, http://camd05.ast.cam.ac.uk/Cosmology/. 6
154
REFERENCES
[44] S. Dodelson, Modern Cosmology, Academic Press 2003. x, 6, 16, 19, 22, 23, 31, 34, 36, 43,
45, 151
[45] N. Kaloper, A. Linde and R. Bousso, Phys. Rev. D 59 (1999) 043508. 6
[46] D. Baumann and H. Peiris, 0810.3022 [astro-ph]. 7, 19, 30
[47] N. T. Jones, H. Stoica and S. H. Tye, JHEP 0207, (2002) 0513. 16, 88
[48] S. Sarangi and S. H. Tye, Phys. Lett. B 536, (2002) 185. 16, 88
[49] L. McAllister and E. Silverstein, Gen. Rel. Grav. 40 (2008) 565. 16
[50] Y.Z. Ma and X. Zhang, JCAP 03 (2009) 006. 16, 87, 111
[51] M. Nolta et al., ApJS, 180 (2009) 296. 27
[52] D. Larson et al., ApJS.192 (2011) 16. ix, x, 19, 27
[53] S. Dodelson, AIP Conf. Proc. 689 (2003) 184-196. x, 27
[54] G. Efstathiou and S. Gratton, JCAP 01 (2009) 011, 0903.0345 [astro-ph] 30, 98
[55] G. Efstathiou, S. Gratton and F. Paci, MNRAS 397 (2009) 1355-1373, 0902.4803 [astro-ph].
30, 98
[56] D. Baumann et al., AIP Conf.Proc.1141 (2009) 10-120. 10, 11, 12, 25, 27, 29, 30, 95, 96
[57] J. M. Bardeen, Phys. Rev. D 22 (1980) 1882. 10
[58] D. Fixsen et al., ApJ. 473 (1996) 576, astro-ph/9605054. 19
[59] D. Fixsen, J. Mather, ApJ, 581 (2002) 817. 19
[60] J. Mather et al., ApJ, 420 (1994) 439. 19
[61] C. Bennett et al., ApJS. 148 (2003) 1, astro-ph/0302207. 55, 75, 87
[62] A. Challinor, Large Scale Structure formation Notes, Cambridge, 2008. ix, 10, 11, 12, 16, 17,
20, 21, 22, 23, 24, 25, 34, 35, 37
[63] D. H. Lyth, Phys. Rev. Lett. 78 (1997) 1861, hep-ph/9606387. 13, 97
[64] G. Efstathiou and K. J. Mack, JCAP 0505 (2005) 008, astro-ph/0503360. 13, 97
[65] J. Silk, Astrophys. J. 151 (1968) 459. 25
155
REFERENCES
[66] S. Dodelson, W. H. Kinney and E. W. Kolb, Phys. Rev. D. 56 (1997) 3207, astro-ph/9702166.
ix, 15
[67] A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett 48 (1982) 1220. 14, 15
[68] A. D. Linde, Phys. Lett. B 129 (1983) 177. 14, 15
[69] K. Freese, J. Frieman and A. Olinto, Phys. Rev. Lett. 65 (1990) 3233. 15
[70] A. Riotto, Nucl. Phys. B 515 (1998) 413. 15
[71] A. D. Linde and A. Riotto, Phys. Rev. D 56 (1997) 1841. 15
[72] E. Komatsu et al., ApJS, 192 (2011) 18. ix, x, xv, 11, 15, 18, 20, 28, 29, 30, 59, 87, 88, 92,
96, 98, 101, 103, 104, 109, 111, 119, 127, 130, 132
[73] M. Liguori et al., Phys. Rev. D 73 (2006) 043505. 17
[74] G. I. Rigopoulos, E. P. S. Shellard, and B. J. W. van Tent. Phys. Rev., D 73 (2006 )083522.
17
[75] P. P. Avelino, E. P. S. Shellard, J. H. P. Wu, and B. Allen, ApJ., 507 (1998) L101. 17
[76] W. Hu and M. J. White, New Astrono. 2 (1997) 323, astro-ph/9706147. ix, 25, 26, 79
[77] A. G. Riess, et al. ApJ, 708 (2010) 645. 29
[78] W. J. Percival, et al. MNRAS 401 (2010) 2148. 29
[79] T. Padmanabhan, Structure formation in the universe, Cambridge University Press 1993. 31,
34, 36, 38, 39, 41
[80] H. J. Mo, F. V. D. Bosch and S. White, Galaxy Formation and Evolution, Cambridge Uni-
versity Press, 2010. 39
[81] D. Eisenstein and W. Hu, ApJ, 496 (1998) 605. 33
[82] R. K. Sheth and G. Tormen, MNRAS, 308 (1999) 119. 45
[83] A. Loeb, How did the first stars and galaxies form?, Princeton University Press 2010. 41, 43,
45, 46, 49, 50
[84] M. J. Rees and J. P. Ostriker, MNRAS 179 (1977) 541-559. 46
[85] D. Lynden-Bell, MNRAS 136 (1967) 101. 41
[86] W.H.Press and P.Schechter, ApJ 187 (1974) 425. 43
156
REFERENCES
[87] http://adsabs.harvard.edu/abs/2007hsct.confE..69V xi, 44
[88] J. R. Bond, S. Cole, G. Efstathiou and N. Kaiser, ApJ 379 (1991) 440. 45
[89] S. White, G. Efstathiou and C. Frenk, MNRAS, 262 (1993) 1023. 45
[90] J. Primack, 0909.2021 [astro-ph]. 46, 47, 48, 49
[91] M. Viel et al., Phys. Rev. Lett, 100 (2008) 041304. 48
[92] A. Kravtsov, 0906.3295 [astro-ph]. 46
[93] H. Pagels and J. R. Primack, Phys. Rev. Lett, 48 (1982) 223-226. 47
[94] F. Zwicky, Helvetica Physica Acta 6 (1933) 110C127; F. Zwicky, ApJ 86 (1937) 217. 46
[95] V. Rubin, W. K. Ford, ApJ 159 (1970) 379. 46
[96] S. Dodelson and L. Widrow, Phys. Rev. Lett. 72 (1994) 17-20. 48
[97] T. Asaka, M. Laine and M. Shaposhnikov, JHEP 0701 (2007) 091. 48
[98] A. Boyarsky et al., Phys. Rev. Lett. 97 (2006) 261302. 48
[99] A. Boyarsky, J. Nevalainen and O. Ruchayskiy, Astron. Astrophys. 471 (2007) 51-57. 48
[100] J. Pritchard and A. Loeb, Phys. Rev. D 78 (2008) 103511. 50
[101] G. Steigman and M. S. Turner, Nucl. Phys. B 253 (1985) 375-386. 47
[102] M. Dine, W. Fiscler and M. Srednicki, Phys. Lett. B, 104 (1981) 199. 47
[103] M. Kustler, G. Raffelt and B. Beltran, editors, Axions: Theory, cosmology and experimental
searches, Springer Lecture Notes in Physics 741, Berlin: Springer 2008. 47
[104] J. Simon and M. Geha, ApJ 670 (2007) 313-331. xi, 47
[105] L. Strigari et al., Nature 454 (2008) 1096-1097. 47
[106] W.J.G. de Blok et al.,AJ, 136 (2008) 2648-2719. 48
[107] E. Ramano-Diaz, I. Sholsman, C. Heller, and Y. Hoffman, ApJ, 685 (2008) L105-L108. 48
[108] E. Ramano-Diaz, I. Sholsman, C. Heller, and Y. Hoffman, ApJ 702 (2009) 1250-1267. 48
[109] O. Y. Gnedin and H. Zhao,MNRAS, 333 (2002) 299-306. 49
[110] S. Mashchenko, J. Wadsley, and H. M. P. Couchman, Science, 319 (2008) 174-177. 49
157
REFERENCES
[111] J.E. Gunn and B.A. Peterson, ApJ, 142 (1965) 1633-1641. 50
[112] R.H. Becker et al., AJ 122 (2001) 2850-2857. 50
[113] http://www.lofar.org/ 50
[114] P. Steinhardt and N. Turok, Science, 296 (2002) 1436-1439. 50
[115] LHC: http://lhc.web.cern.ch 51
[116] CDMS: http://cdms.berkeley.edu 51
[117] http://bigboss.lbl.gov 51
[118] G. Efstathiou, Nuovo Cim. B 122 (2007) 1423-1435, 0712.1513 [astro-ph]. 51
[119] E. L. Wright et al., ApJ, 396 (1992) L13. 55
[120] G. Hinshaw et al., ApJ, 464 (1996) L25. 55
[121] G. Efstathiou, MNRAS, 348 (2004) 885. 55, 56, 58, 59, 60, 61, 68
[122] G. Hinshaw et al., ApJS, 148 (2003) 135. 55
[123] G. Hinshaw et al., ApJS, 180 (2009) 225. 56
[124] C. J. Copi, D. Huterer, D. J. Schwarz, G. D. Starkman, Phys. Rev. D, 75 (2007) 3507. 56,
60, 69
[125] C. J. Copi, D. Huterer, D. J. Schwarz, G. D. Starkman, MNRAS, 399 (2009) 295. 56, 57,
59, 60, 69, 72, 73, 74
[126] M. Tegmark, A. de Oliveira-Costa, A. J. Hamilton, Phys. Rev. D, 68 (2003) 13523. 56, 75
[127] D. J. Schwarz, G. D. Starkman, D. Huterer, C. J. Copi, Phys. Rev. Lett. 93 (2004) 1301. 56
[128] K. Land, J. Magueijo, Phys. Rev. Lett. 95 (2005) 071301. 56, 75
[129] K. Land, J. Magueijo, MNRAS, 362 (2005) 838. 56
[130] A. de Oliveira-Costa, M. Tegmark, M. Zaldarriaga, A. Hamilton, Phys. Rev. D, 69 (2004)
063516. 56
[131] A. de Oliveira-Costa, M. Tegmark, Phys. Rev. D, 74 (2006) 023005. 56, 63, 67
[132] C. L. Francis, J. A. Peacock, MNRAS 406 (2010) 14, 0909.2495 [arXiv:astro-ph]. xii, 56, 72,
73
158
REFERENCES
[133] C. L. Bennett et al., ApJS 192 (2010) 17, 1001.4758 [arXiv:astro-ph]. 56, 76
[134] A. Pontzen, H. V. Peiris, Phys. Rev. D, 81 (2010) 103008, 1004.2706 [arXiv:astro-ph]. 56,
58, 61, 74
[135] S. Basak, A. Hajian, T. Souradeep, Phys. Rev. D, 74 (2006) 021301. 57
[136] G. Efstathiou, MNRAS, 349 (2004) 603. 58, 61, 67, 86
[137] K. M. Gorski, ApJ, 622 (2005) 759. 58, 79
[138] B. Gold et al., ApJS 180 (2009) 265. 59
[139] E. J. Groth, P. J. E. Peebles, ApJ, 310 (1986) 499. 61
[140] S. D. Landy, A. Szalay, ApJ, 412 (1993) 64L. 61
[141] A. Hamilton, ApJ, 417 (1993) 19. 61
[142] S. J. Maddox, G. Efstathiou, W. J. Sutherland, MNRAS, 283 (1996) 1227. 61
[143] P. Bielwicz, K. M. Gorski, A. J. Banday, MNRAS, 355 (2004) 1283. 62
[144] B. D. Wandelt, D. L. Larson, A. Lakshminarayanan, Phys. Rev. D, 70 (2004) 3511. 62
[145] H. K. Eriksen et al., ApJS, 155 (2004) 227. 62
[146] K. T. Inoue, P. Cabella, E. Komatsu, Phys. Rev. D, 77 (2008) 123539. 62
[147] K. Gorski, ApJ, 430 (1994) L85. 62
[148] K. Gorski et al., ApJ, 430 (1994) L89. 62
[149] D. J. Mortlock, A. D. Challinor, M. P. Hobson, MNRAS, 330 (2002) 405. 62
[150] M. Tegmark, Phys. Rev. D, 55 (1997) 5985. 67
[151] A. Hajian, 2007, arXIv:astro-ph/0702723. 69
[152] R. K. Sachs, A. M. Wolfe, ApJ, 147 (1967) 73. 73
[153] V. Mukhanov, Physical Foundations of Cosmology, Cambridge University Press 2005. 73
[154] T. Jarrett, Publications of the Astronomical Society of Australia, 21 (2004) 396. 73
[155] G. Hinshaw et al., ApJS, 170, 288 (2007), astro-ph/0603451. 75, 87
[156] M.L. Brown et al., ApJ, 705, 978 (2009), arXiv:0906.1003 [astro-ph]. 75, 87, 88, 91, 92
159
REFERENCES
[157] The ACT Collaboration: J.W. Fowler et al., ApJ 722 (2010) 1148 , arXiv:1001.2934 [astro-
ph]. 75
[158] N.R. Hall et al., ApJ, 718, 632 (2010), arXiv:0912.4315 [astro-ph]. 75, 126
[159] P. Bielewicz, K. M. Gorski and A. J. Banday, Mon. Not. Roy. Astron. Soc. 355, 1283 (2004),
astro-ph/0405007. 75
[160] C. J. Copi, D. Huterer, D. J. Schwarz and G. D. Starkman, Mon. Not. Roy. Astron. Soc.
367, 79 (2006), astro-ph/0508047. 75
[161] F. K. Hansen et al., ApJ 704 (2009) 1448, arXiv: 0812.3795 [astro-ph]. 75
[162] H. K. Eriksen et al., Astrophys. J. 605, 14 (2004), astro-ph/0307507. 75
[163] M. Cruz et al., ApJ 655, 11 (2007), astro-ph/0603859. 75
[164] M. Cruz et al., Science 318, 1612 (2007), arXiv: 0710.5737. 75
[165] C. J. Copi et al., Adv. Astron. Astrophys. 2010 2010 847541, arXiv: 1004.5602 [astro-ph].
76
[166] L. Ackerman, S. M. Carroll and M. B. Wise, Phys. Rev. D 75, 083502 (2007), astro-
ph/0701357. 76
[167] B. Himmetoglu, C.R. Contaldi, and M. Peloso, Phys. Rev. D 79, 063517 (2009),
arXiv:0812.1231 [astro-ph]. 76
[168] B. Himmetoglu, C.R. Contaldi, and M. Peloso, Phys. Rev. Lett. 102, 111301 (2009), arXiv:
0809.2779 [astro-ph]. 76
[169] A. L. Erickcek, M. Kamionkowski and S. M. Carroll, Phys. Rev. D. 78, 123520 (2008),
arXiv: 0806.0377 [astro-ph]. 76
[170] A. L. Erickcek, C.M. Hirata, and M. Kamionkowski Phys. Rev. D. 80, 083507 (2009),
arXiv:0907.0705 [astro-ph]. 76
[171] C. Gordon, W. Hu, D. Huterer and T. Crawford, Phys. Rev. D 72, 103002 (2005), astro-
ph/0509301. 76
[172] C. Dvorkin, H. V. Peiris and W. Hu, Phys. Rev. D 77, 063008 (2008), arXiv: 0711.2321
[astro-ph]. 76
[173] N. E. Groeneboom and H. K. Eriksen, ApJ 690 (2009) 1807, arXiv: 0807.2242 [astro-ph].
76, 136
160
REFERENCES
[174] D. Hanson and A. Lewis, Phys. Rev. D, 80, 063004 (2009), arXiv: 0908.0963 [astro-ph]. 76,
77, 78, 79, 86
[175] N. E. Groeneboom, L. Ackerman, I. K. Wehus, and H. K. Eriksen, ApJ 722 (2010) 452,
arXiv: 0911.0150 [astro-ph]. 76
[176] D. Hanson, A. Lewis, and A. Challinor, Phys. Rev. D., 81, 103003 (2010), arXiv:1003.0198
[astro-ph]. 76, 82, 85
[177] K.M. Huffenberger, B.P. Crill, A.E. Lange, K.M. Gorski, C.R. Lawrence, Astron. Astrophys.
510, A58, 2010, arXiv: 1007.3468 [astro-ph]. 76
[178] A. R. Pullen and M. Kamionkowski, Phys. Rev. D 76 103529 (2007), arXiv:0709.1144 [astro-
ph]. 76, 80
[179] M. Zaldarriaga, Phys. Rev. D., 55, 1822 (1997), astro-ph/9608050. 81
[180] K. M. Smith, O. Zahn, O. Dore´, Phys. Rev. D., 76, 043510 (2007), arXiv:0705.3980 [astro-
ph]. 86
[181] A. Lewis and A, Challinor, Phys. Rept. 429 (2006) 1-65. 92
[182] H. C. Chiang et al., Astrophys. J. 711 (2010) 1123. 87, 88, 91, 92, 93, 97, 115
[183] J. Kovac et al., Nature (London) 420 (2002) 772. 87
[184] A. C. S. Readhead et al., Science 306 (2004) 836. 87
[185] C. Bischoff et al. (CAPMAP Collaboration), ApJ. 684 (2008) 771. 87
[186] W. C. Jones et al., ApJ. 647 (2006) 823. 87
[187] J. H. P. Wu et al., ApJ. 665 (2007) 55. 87
[188] A. Borde and A. Vilenkin, Phys. Rev. Lett. 72 (2004) 3305. 87
[189] V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Phys. Rept. 215 (1992) 203; R.
H. Brandenberger, Lect. Notes Phys. 646 (2004) 127. 87
[190] S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, Phys. Rev. D, 68 (2003) 046005. 87, 111
[191] S. Kachru, R. Kallosh, A. Linde, J. M. Maldacena, L. McAllister and S. P. Trivedi, JCAP
0310 (2003) 013. 87, 111
[192] M. Bojowald, Living Rev. Relativity, 11 (2008) 4. 88
161
REFERENCES
[193] W. Zhao, D. Baskaran and L. P. Grishchuk, Phys. Rev. D 79 (2009) 023002; 80 (2009)
083005 (2009); W. Zhao, Phys. Rev. D 79 (2009) 063003. 88
[194] S. Chandrasekhar, in Radiative Transfer (Dover: New York, 1960). 89
[195] M. Zaldarriaga and U. Seljak, Phys. Rev. D, 55 (1997) 1830. 89
[196] http://lambda.gsfc.nasa.gov/. ix, 5, 91
[197] http://bicep.caltech.edu/. 91
[198] http://find.uchicago.edu/quad/. 91
[199] B-Pol Collaboration, Exper. Astron. 23 (2009) 5; http://www.b-pol.org/index.php. 92
[200] http://cmbpol.kek.jp/litebird/. 92
[201] D. Baumann et al., AIP Conf. Proc. 1141 (2009) 10. 92, 101, 141, 142, 143
[202] U. Seljak and C. M. Hirata, Phys. Rev. D, 69 (2004) 043005. 92, 144
[203] http://www.sfu.ca/ levon/cmbact.html. 92, 112
[204] L. Pogosian and T. Vachaspati, Phys. Rev. D, 60 (1999) 083504. 112
[205] O. Lahav et.al, Mon. Not. Roy. Astron. Soc. 315 (2000) L45. 93, 94, 146, 148
[206] M. P. Hobson, S. L. Bridle, O. Lahav, Mon. Not. Roy. Astron. Soc. 335 (2002) 377. 93
[207] M. G. Kendall and A. Stuart, The Advanced Theory of Statistics, Volume II (Griffin, Lon-
don, 1969). 94, 95
[208] M. Tegmark, A. Taylor and A. Heavens, Astrophys. J. 480 (1997) 22; M. Tegmark, Phys.
Rev. Lett. 79 (1997) 3806. 94, 95
[209] M. Zaldarriaga, D. Spergel and U. Seljak, Astrophys. J. 488 (1997) 1. 94
[210] A. H. Jaffe, M. Kamionkowski and L. Wang, Phys. Rev. D, 61 (2000) 083501. 94
[211] L. P. Grishchuk, Zh. Eksp. Teor. Fiz. 67 (1974) 825 (1974) [Sov. Phys. JETP 40 (1975) 409
(1975)]. 95
[212] L. P. Grishchuk, ‘Discovering Relic Gravitational Waves in Cosmic Microwave Background
Radiation’, chapter in General Relativity and John Archibald Wheeler, Eds. I. Ciufolini and R.
Matzner (Springer, 2010, 151-199) [arXiv: 0707.3319]. 95
[213] A. R. Liddle, D. H. Lyth, Phys. Rep. 231 (1993) 1. 96, 102
162
REFERENCES
[214] W. Zhao, D. Baskaran and L. P. Grishchuk, Phys. Rev. D, 82 (2010) 043003, 1005.4549
[arXiv:astro-ph]. 98
[215] W. Zhao and D. Baskaran, Phys. Rev. D 79 (2009) 083003; W. Zhao and W. Zhang, Phys.
Lett. B 677 (2009) 16. 101
[216] L. P. L. Colombo, E. Pierpaoli and J. R. Pritchard, arXiv: 0811.2622. 101
[217] H. V. Peiris et al., Astrophys. J. Suppl. 148 (2003) 213. 103
[218] L. Alabidi and D. Lyth, JCAP, 0605 (2006) 016; L. Alabidi and D. Lyth, JCAP, 0608 (2006)
013. 104
[219] A. Sen, J. Math. Phys. 22 (1981) 1781. 105
[220] A. Ashtekar, Phys. Rev. Lett. 57 (1986) 2244. 105
[221] J. Grain and A. Barrau, Phys. Rev. Lett. 102 (2009) 081301. 105, 107, 108
[222] J. Mielczarek, T. Cailleteau, J. Grain and A. Barrau, Phys. Rev. D, 81 (2010) 104049. 105,
106, 107, 108
[223] P. Singh, K. Vandersloot and G. V. Vereshchagin, Phys. Rev. D, 74 (2006) 043510. 105
[224] K. A. Meissner, Class. Quant. Grav. 21 (2004) 5245. 105
[225] J. Mielczarek, M. Kamionka, A. Kurek and M. Szydlowski, arXiv: 1005.0814. 105
[226] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects, Cambridge
University Press 2000. 111
[227] G. R. Dvali and S. H. H. Tye, Phys. Lett. B, 450 (1999) 72. 111
[228] N. Kaiser and A. Stebbins, Nature, 310 (1984) 391. 111
[229] R. Battye and A. Moss, arXiv: 1005.0479. 111, 112, 113, 114
[230] L. Pogosian and M. Wyman, Phys. Rev. D, 77 (2008) 083509. 112
[231] R. R. Caldwell and B. Allen, Phys. Rev. D, 45 (1992) 3447. 113
[232] R. R. Caldwell, R. A. Battye and E. P. S. Shellard, Phys. Rev. D, 54 (1996) 7146. 113
[233] B. Abbott et al. [LIGO Scientific Collaboration], Phys. Rev. D, 80 (2009) 062002. 113
[234] M. Wyman, L. Pogosian and I. Wasserman, Phys. Rev. D, 72 (2005) 023513; L. Pogosian,
I. Wasserman and M. Wyman, astro-ph/0604141. 113
163
REFERENCES
[235] N. Bevis, M. Hindmarsh, M. Kunz and J. Urrestilla, Phys. Rev. Lett. 100 (2008) 021301.
113
[236] J. Sievers et al., arXiv: 0901.4540. 113
[237] L. Verde, H. Peiris and R. Jimenez, J. Cosmol. Astropart. Phys. 0601 (2006) 019. 143
[238] M. Tucci, E. Martinez-Gonzalez, P. Vielva and J. Delabrouille, Mon. Not. R. Astron. Soc.
360 (2005) 926. 143
[239] M. Zaldarriaga and U. Seljak, Phys. Rev. D, 58 (1998) 023003. 143
[240] M. Kesden, A. Cooray and M. Kamionkowski, Phys. Rev. Lett. 89 (2002) 011304. 143
[241] L. Knox and Y. S. Song, Phys. Rev. Lett. 89 (2002) 011303. 143
[242] W.Hu, M. M. Hedman and M. Zaldarriaga, Phys. Rev. D, 67 (2003) 043004. 143
[243] K. Riley, M. P. Hobson and S. J. Bence, Mathematica Methods for Physics and Engineering,
Cambridge University Press 2005. 146
[244] R. Watkins, H. Feldman and M. Hudson, MNRAS 392, 743 (2009). xv, 117, 118, 119, 120,
122, 123, 129, 131, 136, 137, 138
[245] H. Feldman, R. Watkins and M. Hudson, MNRAS, 407, 2328 (2010). 117, 118, 120, 123,
136
[246] M. Turner, Phys. Rev. D. 44 (1991) 3737. 117, 123, 125
[247] D. Langlois and T. Piran, Phys.Rev. D 53 (1996) 2908-2919; D. Langlois, Phys. Rev. D54
(1996) 2447-2450. 117, 125
[248] A. Kashlinsky et al., Astrophys.J.,686, L49 (2008). 117, 123
[249] P. Erdogdu et al., MNRAS 373 (2006) 45-64, [astro-ph/0610005]. 117
[250] G. Lavaux, R. B. Tully, R. Mohayaee, S. Colombi, ApJ.709 (2010) 483-498, 0810.3658 [astro-
ph]. 117
[251] J. P. Zibin and D. Scott, Phys. Rev. D 78, 123529 (2008); A. L. Erickcek, S. M. Carroll and
M. Kamionkowski, Phys. Rev. D 78, 083012 (2008). 118
[252] K. Gorski, Astrophys.J., 332 (1988) L7. 118
[253] L. N. da Costa et al., Astron. J, 120 (2000) 95; M. Bernardi, et al., Astron. J, 123 (2002)
2990.; G. Wenger et al., Astron. J, 126 (2003) 2268. 119, 129
164
REFERENCES
[254] J. L. Tonry, et al., Astrophys.J., 594 (2003) 1. 120, 129
[255] C. M. Springob, et al., ApJ Supp, 172 (2007) 599. 120, 129
[256] M. J. Hudson, PASP, 111 (1999) 57; M. J. Hudson, MNRAS, 352 (2004) 61. 120
[257] J. L. Tonry et al., Astrophys.J., 546 (2001) 681. 121, 129
[258] M. Colless et al., MNRAS 321 (2001) 277. 121
[259] R. Giovanelli et al., Astron. J, 116 (1998) 2632; D. A. Dale, Astron. J, 118 (1999) 1489. 121,
129
[260] J. A. Willick et al., ApJ 522 (1999) 647. 121
[261] E. Macaulay et al., arXiv:1010.2651 [astro-ph.CO]. 123
[262] A. Kashlinsky, et al., Astrophys.J. 712, L81 (2010). 123
[263] L. P. Grishchuk, Ya. B. Zel’dovich, Astron. Zh. 55, 209 (1978). 123
[264] P. G. Castro, M. Douspis and P. G. Ferreira, Phys. Rev. D 68, 127301 (2003). 125
[265] A. Kosowsky and T. Kahniashvili, arXiv:1007.4539 [astro-ph.CO]; L. Amendola et al.,
arXiv:1008.1183 [astro-ph.CO]. 125
[266] C. Gordon, K. Land and A. Slosar, MNRAS, 387, (2008), 371. 125
[267] S. Das, et al. arXiv:1009.0847 [astro-ph]. 126
[268] J. Dunkley, et al. arXiv:1009.0866 [astro-ph]. 126
[269] Fundamental Plane (FP) and Tully-Fisher (TF) are phenomenological procedures to deter-
mine the galaxies’ intrinsic brightness and thus their distance independently of their redshift.
119, 120
[270] J. P. Ostriker and Y. Suto, ApJ, 348 (1990) 378-382. xv, 127, 128, 129, 130, 133
[271] M.A. Strauss, R. Cen and J.P. Ostriker, ApJ, 408 (1993) 389-402. 127, 129
[272] Y. Suto, N. Gouda and N. Sugiyma, ApJS, 74 (1990) 665-674; Y. Suto, R. Cen and J.P.
Ostriker, ApJ, 395 (1992) 1-20. 127
[273] K. Nagamine, J.P. Ostriker and R. Cen, ApJ, 553 (2001) 513-527. 127, 128
[274] M. Gramann, N. A. Bahcall, R. Cen and J. R. Gott, ApJ, 441 (1995) 449-457. 128
[275] F. Atrio-Barandela, A. Kashlinsky and J. P. Mucket, ApJ, 601 (2004) 111-114. 128
165
REFERENCES
[276] D. Tseliakhovich and C. Hirata, Phys. Rev. D 82 (2010) 083520; D. Tseliakhovich, R.
Barkana and C. Hirata,1012.2574 [astro-ph.CO] 128
[277] J. Peebles, Principles of Physical Cosmology, 1993. Princeton. Univ. Press. 128
[278] D. H. Jones et al., MNRAS 399 (2009) 683-698. 130
[279] G. B. Zhao, L. Pogosian, A. Silvestri, and J. Zylberberg, Phys. Rev. D 79 (2009) 083513.
132
[280] T. Giannantonio, M. Martinelli, A. Silvestri, A. Melchiorri, JCAP, 04 (2010) 030. 132
[281] H. Liu and T. P. Li, ApJ, 732 (2011) 125. 136
[282] H. Liu, S. L. Xiong and T. P. Li, MNRAS 413 (2011) L96-L100. 136
[283] H. Liu, S. L. Xiong and T. P. Li, 1003.1073 [arXiv:astro-ph.CO]. 136
[284] C. A. Blake, F. B. Abdalla, S. L. Bridle and S. Rawlings, New Astron. Rev. 48 (2004)
1063-1077. 136
[285] J. Martin and C. Ringeval, Phys. Rev. D 69 (2004) 083515. 136
[286] K. Bhattacharya, S. Mohanty and R. Rangarajan, Phys. Rev. Lett. 96 (2006) 121302. 136
[287] Pedro G. Ferreira and J. Magueijo, Phys. Rev. D 78 (2008) 061301. 136
[288] N. Barnaby, Z. Huang, L. Kofman, and D. Pogosyan, Phys. Rev. D., 80 (2009) 043501. 136
[289] M. I. Schmoldt et al., Astron. J. 118 (1999) 1146-1160. 138
[290] E. L. Wright et al., Astron. J. 140 (2010) 1868. 138
[291] H. Wang et al., MNRAS, 394 (2009) 398-414. 139
166