Post-inflationary Non-Gaussianities on
the Cosmic Microwave Background
Shi Chun SU
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
Trinity Hall May 2015

This thesis is dedicated to my family for their love. It is such a fortune to have the freedom
to make one’s decision based on one’s own interest. However, the opportunity does not
come for free. When I was young, my parents devoted their lives to the family and
sacrificed a lot. My siblings share the responsibility when they have grown up. As the
youngest child in my family, I am so lucky to explore whatever I want. There are certainly
plenty of challenges in life but I always feel so grateful for the selfless support from my
family.
I would also like to dedicate this thesis to my supervisors – Eugene Lim and Paul Shellard.
Eugene continuously guides and encourages me throughout the whole Ph.D. Paul gives me
lots of advices and support. Without their supervision, some of the key work done in this
thesis will not be possible. It is my great honor working with them.

Declaration
I declare that this dissertation is the result of my own work and contains no outcome of
work done in collaboration, except where specifically indicated in the text. The contents are
original and no parts have been submitted for consideration for any other qualification.
Shi Chun SU
May 2015

Acknowledgements
I would like to thank J. Fergusson for useful discussions and help with the bispectrum visu-
alization as well as A. Kaliazin for computational support and technical advice. I would also
like to thank D. Baumann, C. Bonvin, C. Fidler, D. Grin, B. Horn, W. Hu, Z. Huang, L. Hui,
J. Lesgourgues, K. Nakao, G. Pettinari, C. Pitrou, T. Tram, M. LoVerde and W. Yi for ad-
vices and discussions on the researches I worked throughout the Ph.D. Moreover, I want to
thank V. Assassi, T.-J. Chen, C. Lau, A. Lazanu, H. Lee, H. F. Gruetjen, M. M. Schmittfull
and K. Yamamoto for sharing lots of wonderful times of Ph.D. with me. Especially, I am
grateful for working on an interesting project [62] about the cosmic microwave background
(CMB) B-mode polarization with D. Baumann and H. Lee.
The numerical simulations were implemented on the COSMOS supercomputer, part of
the DiRAC HPC Facility which is funded by STFC and BIS.

Abstract
The cosmic microwave background (CMB) provides unprecedented details about the his-
tory of our universe and helps to establish the standard model in modern cosmology. With
the ongoing and future CMB observations, higher precision can be achieved and novel win-
dows will be opened for studying different phenomena. Non-Gaussianity is one of the
most exciting effects which fascinate many cosmologists. While numerous alternative infla-
tionary models predict detectable primordial non-Gaussianities generated during inflation,
the single-field slow-roll inflation of the standard model is known to produce negligible
non-Gaussianities. However, post-inflationary processes guarantee the generation of non-
Gaussianities through the nonlinear evolution of our universe after inflation, regardless of
the underlying inflationary theory. These non-Gaussianities not only may contaminate the
potential primordial non-Gaussian signals, but also may offer independent tests for late-
time physics (such as General Relativity). Therefore, it is of great interest to study them
quantitatively.
In this thesis, we will study the post-inflationary non-Gaussianities in two main aspects.
First, we calculate the CMB bispectrum imprinted by the 2nd-order perturbations during
recombination. We carry out a numerical calculation including all the dominant effects
at recombination and separate them consistently from the late-time effects. We find that
the recombination bispectrum is subdominant compared to the ISW-lensing bispectrum.
Although the effect will not be detectable for the Planck mission, its signal-to-noise is large
enough that they present themselves as systematics. Thus, it has to be taken into account
in future experiments. Second, we formulate the lensing, redshift and time-delay effects
through the Boltzmann equation. The new formalism allows us to explicitly list out all the
approximations implied in the canonical remapping approach. In particular, we quantify
the correction of the CMB temperature power spectrum from the lens-lens couplings and
confirm that the correction is small.

Table of contents
Table of contents xi
List of figures xv
List of tables xix
Nomenclature xx
1 Review of CMB Cosmology 1
2 Cosmological Perturbations and CMB Spectra 7
2.1 Coordinate System and Tetrad Basis . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Tetrad Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 CMB Temperature and Polarization . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Intensity Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Stokes’ Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Einstein Field Equations . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Cosmology in Background Order . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 Friedmann–Lemaître–Robertson–Walker Metric . . . . . . . . . . 16
2.4.2 ΛCDM Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.3 Friedmann Equations . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Cosmological Perturbations in First Order . . . . . . . . . . . . . . . . . . 18
2.5.1 Einstein Field Equations . . . . . . . . . . . . . . . . . . . . . . . 18
2.5.2 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.3 Adiabatic Initial Conditions . . . . . . . . . . . . . . . . . . . . . 21
xii Table of contents
2.6 Statistics of CMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6.1 CMB Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6.2 CMB Bispectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Cosmological Perturbations in Second Order 31
3.1 Second-order Einstein Field Equations . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Gauge Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.2 Multipole Decomposition of Stress-Energy Tensor . . . . . . . . . 34
3.1.3 Einstein Field Equations . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.4 Continuity and Euler Equations . . . . . . . . . . . . . . . . . . . 41
3.2 Second-order Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Liouville Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.2 Collision Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.3 Collision Terms for Baryons . . . . . . . . . . . . . . . . . . . . . 50
3.2.4 Perturbed Optical Depth . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.5 Truncation Scheme for the Boltzmann Hierarchy . . . . . . . . . . 55
3.3 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.1 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.2 Tight-coupling Approximation . . . . . . . . . . . . . . . . . . . . 58
3.3.3 Numerical Stability and Convergence . . . . . . . . . . . . . . . . 60
4 CMB Bispectrum during Recombination 65
4.1 Definition of CMB Temperature in Second Order . . . . . . . . . . . . . . 66
4.1.1 Brightness Temperature . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.2 Bolometric Temperature . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Formulating Second-order Bispectrum . . . . . . . . . . . . . . . . . . . . 68
4.2.1 Second-order Line-of-sight Approach . . . . . . . . . . . . . . . . 68
4.2.2 Separation between Recombination and Late Time . . . . . . . . . 69
4.2.3 Bispectrum from Second-order Perturbations . . . . . . . . . . . . 74
4.2.4 Quadratic Source Function with m ̸= 0 . . . . . . . . . . . . . . . . 79
4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3.1 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . 80
4.3.2 Validation with Analytical Bispectrum . . . . . . . . . . . . . . . . 86
4.3.3 fNL and Signal-to-noise . . . . . . . . . . . . . . . . . . . . . . . 90
4.4 Conclusions and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 Weak Lensing in Boltzmann Equation 93
Table of contents xiii
5.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.1.1 Lensing in Second Order . . . . . . . . . . . . . . . . . . . . . . . 97
5.1.2 Lensing in Arbitrarily High Orders . . . . . . . . . . . . . . . . . . 101
5.1.3 Lensing as Dyson Series . . . . . . . . . . . . . . . . . . . . . . . 108
5.1.4 Diagrammatic Approach . . . . . . . . . . . . . . . . . . . . . . . 111
5.1.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2 Power Spectrum from Lens-lens Couplings . . . . . . . . . . . . . . . . . 120
5.2.1 Diagrammatic Approach . . . . . . . . . . . . . . . . . . . . . . . 120
5.2.2 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . 122
5.2.3 Other Approximations . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 Discussion and Conclusions 131
6.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.1.1 Spectral Distortions . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.1.2 Modified Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.1.3 Perturbed Optical Depth . . . . . . . . . . . . . . . . . . . . . . . 132
6.1.4 Lensing in CMB Polarizations and Other Observables . . . . . . . 132
6.1.5 Validation of Newtonian Approximation . . . . . . . . . . . . . . . 133
6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
References 135
Appendix A Glossary of Notations and Symbols 143
Appendix B Table of Cosmological Parameters 149
Appendix C Spin-weighted Spherical Harmonics 151
C.1 Properties of Spin-weighted Spherical Harmonics . . . . . . . . . . . . . . 151
C.2 Properties of Wigner 3-j Symbols . . . . . . . . . . . . . . . . . . . . . . 153
C.3 Decomposition of Spherical Harmonics . . . . . . . . . . . . . . . . . . . 154
Appendix D Perturbations of Tetrads and Ricci Rotation Coefficients 157
D.1 Perturbations of Tetrads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
D.2 Perturbations of Ricci Rotation Coefficients . . . . . . . . . . . . . . . . . 158

List of figures
1.1 The CMB temperature anisotropies seen by Planck [1]. . . . . . . . . . . . 2
2.1 The illustration of the pullback φλ from the physical manifold M1 onto
the background manifold M0. The dashed gray grids represent the tetrad
bases. Due to the existing perturbations, the pulled-back tetrad bases on the
background manifold do not align with the bases of the coordinate system
(the misalignment is exaggerated in the graph). . . . . . . . . . . . . . . . 10
2.2 The full-sky unlensed power spectra of the CMB temperature from our code
(black line) and CAMB (dashed orange line) against ℓ. A perfect match is
achieved. Our code uses Newtonian gauge while CAMB uses synchronous
gauge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 The diagrammatic outline of relations between different perturbations. The
corresponding evolution equations are shown in the brackets. The waved
lines denotes couplings through GR while the two headed arrow denotes the
Compton scattering. We remark that the 1st-order perturbed optical depth
(δτ) is needed for the 2nd-order Compton scattering. . . . . . . . . . . . . 32
3.2 The graphs of transfer functions of the 1st-order differential optical depth
δ τ˙ [I] against the scale factor a for k = 2× 10−5Mpc−1 (upper panel) and
k = 0.045Mpc−1 (lower panel). Here, we ignore Helium atoms. . . . . . . 54
3.3 The graph of the absolute 2nd-order kernel of Ψ˙ (solid line) against the
scale factor a with k0 = 0.199Mpc−1, k1 = 0.2Mpc−1, k2 = 0.002Mpc−1
and kˆ1· kˆ2 =−0.684. The dashed line shows the absolute numerical differ-
ence in the constraint equation Eq. (3.28). . . . . . . . . . . . . . . . . . . 62
3.4 The graph of the absolute 2nd-order kernel of Ψ˙ (solid line) against the
scale factor a with k0 = 0.159Mpc−1, k1 = 0.2Mpc−1, k2 = 0.2Mpc−1 and
kˆ1· kˆ2 = −0.684. The dashed line shows the absolute numerical difference
in the constraint equation Eq. (3.28). . . . . . . . . . . . . . . . . . . . . . 63
xvi List of figures
3.5 The graph of the absolute 2nd-order kernel of Ψ˙ (solid line) against the
scale factor a with k0 = 0.4Mpc−1, k1 = 0.2Mpc−1, k2 = 0.2Mpc−1 and
kˆ1· kˆ2 = 1. The dashed line shows the absolute numerical difference in the
constraint equation Eq. (3.28). . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1 The graphs of the bispectra generated from the quadratic source function
S[I,I] against ℓ for equilateral (upper) and squeezed (lower) limit. The curves
correspond to different ℓ truncations of the multipoles S[I,I]ℓm . We can see that
the convergence occurs when ℓ goes up to 10. . . . . . . . . . . . . . . . . 81
4.2 The 3D plot of the reduced bispectrum generated around recombination.
The bispectrum is normalized by the coefficient D(ℓ1, ℓ2, ℓ3) defined in Eq. (4.51)
to remove an overall ℓ−4 scaling. The red regions represent positive values
while the blue regions represent negative values of the recombination bis-
pectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3 The cross sections of the normalized reduced bispectrum in Fig. 4.2 with
the conditions 12(ℓ1 +ℓ2 +ℓ3) = 400 (upper), 1100 (middle) and 1600 (lower). 85
4.4 The recombination bispectrum (gray solid line) calculated numerically us-
ing Eq. (4.45) and the analytical bispectrum (black solid line) are plotted
in the squeezed limit. We can see that they match with each other very
well. Here, we measure the bispectrum in bolometric temperature which can
be converted from the bispectrum in brightness temperature by Eq. (4.46).
The “All L.T.” curve contains all the purely 2nd-order perturbations of the
source function as shown in Eq. (4.10) while the “All Q.T.” curve contains
all the cross products of two 1st-order perturbations of the source function
as shown in Eq. (4.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.5 The bispectra of the linear terms (L.T.) and quadratic terms (Q.T.) of the
source function S2ND in squeezed limit are shown in the upper and lower
panel respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.1 The diagrammatic illustration for the time-ordering of the nested integral
in Eq. (5.37). The circle node indicates the source while the square nodes
indicate the lenses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
List of figures xvii
5.2 The contributions on the temperature power spectrum from the residual dia-
grams of Column C and D containing lens-lens couplings in Table 5.4. The
top and bottom panels show the corrections to the power spectrum from
those diagrams in Column C and D respectively. In the top panel, the solid
gray line corresponds to the diagram (1C) while the dashed line corresponds
to the diagrams (2C) and (3C). Diagrams (2C) and (3C) have identical con-
tributions and thus we sum them up. In the bottom panel, the solid gray
line corresponds to the diagram (1D) while the dashed line corresponds to
the diagram (2D). For both panels, the signs of contributions from the dia-
grams in Row ⟨Θ[4]Θ[2]*⟩ of Table 5.4, i.e. the dashed lines, are reversed to
illustrate the cancellations. . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.3 The overall correction to the temperature power spectrum from the lens-lens
couplings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

List of tables
4.1 The table of the effective fNLs, FNLs and S/Ns of the local and equilateral
templates correlated to the recombination bispectrum, as well as its total
signal (auto-correlation). For ISW-lensing bispectrum, we found that its
correlation with the recombination bispectrum is ∼ 30%. For ease of com-
parison, the FNL quantities normalize the integrated bispectrum signal for
any shape relative to the fNL = 1 local model. We have used ℓmax = 2000
throughout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2 The table for comparisons among the results from all the codes implement-
ing the full recombination bispectrum. . . . . . . . . . . . . . . . . . . . . 91
5.1 The diagrams with their corresponding represented equations and their phys-
ical meanings. In the “Term” column, the first choice assumes Approxima-
tion 7 while the second one does not. . . . . . . . . . . . . . . . . . . . . . 112
5.2 All the possible couplings of the weak lensing effects on the CMB anisotropies
in the third, forth and fifth orders. As an example, Eq. (5.62) demonstrates
how to construct the formulae from the middle diagram in the first row of
Table 5.2b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.3 The forbidden contractions for the lensing effects on the power spectrum.
θ is the Heaviside step function. The Heaviside step functions come from
replacing the integration
∫ ηN+1
ηLSS dηN with
∫ η0
ηLSS dηNθ(ηN+1−ηN). . . . . . 123
5.4 All the non-zero configurations for the temperature power spectrum from
the next-to-leading order of the weak lensing effect. The dotted lines denote
correlations while the arrows indicate the cancellations between some pairs
of the diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
A.1 Physical parameters and background quantities . . . . . . . . . . . . . . . 143
A.2 Mathematical notations and functions . . . . . . . . . . . . . . . . . . . . 144
A.3 Notations for power spectra and bispectra . . . . . . . . . . . . . . . . . . 144
xx List of tables
A.4 Coordinate-related notations and quantities . . . . . . . . . . . . . . . . . 145
A.5 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
A.6 Quantities for EFEs and CMB photon . . . . . . . . . . . . . . . . . . . . 147
A.7 Liouville-related notations . . . . . . . . . . . . . . . . . . . . . . . . . . 148
A.8 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
B.1 The best-fit cosmological parameters are listed based on the Planck+WP
dataset in [3]. For the primordial tensor modes, parameters come from the
BICEP2’s data in [2] with nt = 0 and k∗ = 0.05/Mpc. The definitions of
these parameters can be found in Table 1 of [3]. As mentioned in [9], the
tension between Planck and Bicep due to the tensor mode is mild. Thus,
we do not re-fit the cosmological parameters using the data from the Planck
and BICEP2 together. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Chapter 1
Review of CMB Cosmology
The origin of our universe is a question which we cannot but be lured into attempting to
comprehend. In ancient time, answers to the question were descriptive and qualitative with
various hypotheses. Although these answers provide innovative insights about the universe,
it was hard to validate them due to the lack of observations available at that time. Until the
last century, the technology became advanced enough to detect the distant signals from the
sky. Different models of our universe can then be studied with experiments. After that, the
study of universe thrives and the foundation of modern cosmology has been established.
In particular, after the discovery of the cosmic microwave background (CMB), cosmol-
ogy has entered into an era with unprecedented precision. The CMB signals was imprinted
on the sky due to the decoupling of photons from baryons when our universe was only about
380,000 years old. Even though the CMB photons traveled for 13.8 billion years before be-
ing captured by us, the contamination from foreground is acceptable. Thus, the information
of the early universe contained in the CMB signals can be preserved and restored. By an-
alyzing the CMB data, we are able to deduce the evolution history of our universe, the
abundance of different components and the origin of the large-scale structures. The CMB
temperature anisotropies observed by Planck is shown in Fig. 1.1.
The CMB data favour a model of universe which is almost perfectly homogeneous and
isotropic with very tiny fluctuations at the early stage. Since the fluctuations are so small,
the linear perturbation theory is usually sufficient for describing the dominant features of the
fluctuations embedded on the CMB. Without the nonlinear evolutions, the calculation for the
evolution of perturbations can be significantly simplified in the linear order. This facilitates
our understanding of the early universe and the development of the standard model - ΛCDM
(see [32, 105] for more details) in modern cosmology.
2 Review of CMB Cosmology
Figure 1.1 The CMB temperature anisotropies seen by Planck [1].
The idea of inflation is by far the most popular one to describe the very beginning of
our universe. Not only it solves some puzzles in cosmology (e.g. flatness, homogeneity and
isotropy of our universe), but it also explains the origin of the tiny perturbations as quantum
fluctuations expanded to cosmological scales during inflation. Although the standard model
stays consistent with current observations, it is crucial to explore alternative models and
keep challenging the standard model with further observations. Hence, numerous inflation-
ary models containing fruitful physics are proposed in literature. Identifying our universe
among these possibilities becomes essential. In addition, new properties discovered about
the inflationary era may deepen our understanding of fundamental physics.
Depending on the inflationary model under consideration, various distinguishable fea-
tures different from the standard model can be generated. In particular, many alternative
inflationary models predict significant primordial non-Gaussianities which may be detected
by the current and future CMB experiments through the CMB bispectra. Comprehensive
studies have been done to calculate the shapes of the bispectra from different inflationary
models. These shapes can be constrained with the observational data. If any primordial non-
Gaussianities are detected, it will provide us further clues about the mechanism of inflation.
For example, single-field inflationary models do not produce significant non-Gaussianities
of the local type [72]. A detection of the local-type bispectrum can rule out all the single-
field inflationary models.
However, non-Gaussianities can also be generated through the nonlinear evolution of
some physical mechanisms after inflation. Nonlinear evolution is responsible to the complex
3structure formation in the late-time universe. For instance, the nonlinearity of General Rela-
tivity (GR) guarantees post-inflationary non-Gaussianities. A rough estimation expects that
these non-Gaussianities should have a nonlinear coupling constant fNL of order 1 at recom-
bination. Thus, these post-inflationary non-Gaussianities can potentially contaminate the
primordial non-Gaussianities from inflation. It is of importance to study the nonlinear evo-
lutions of the cosmological perturbations quantitatively. Moreover, post-inflationary non-
Gaussianities can be used as an independent test for the underlying mechanisms (e.g. GR)
on cosmological scales.
One of the ways to probe non-Gaussianities is to measure the CMB bispectra. While
power spectra are mainly dominated by Gaussian perturbations, bispectra vanish statisti-
cally with perfectly Gaussian perturbations. This property makes bispectra an ideal tool
for detecting non-Gaussianities1. Non-Gaussianities in the CMB bispectra can provide us
extra information about the Universe. With the improving sensitivities from the current and
future observations [6, 8, 16], a comprehensive study of non-Gaussianities from different
mechanisms becomes necessary. In this thesis, we will focus on two different types of non-
Gaussianities generated after inflation. The first one comes from the 2nd-order perturbations
imprinted on the CMB at recombination while the second one is sourced by the nonlinear
distortions of the CMB anisotropies due to the late-time metric perturbations, e.g. CMB
lensing effect.
Calculating the CMB bispectrum from the 2nd-order perturbations at recombination is
not a new problem. There are already some attempts to estimate the CMB bispectrum in lit-
erature, such as [57, 75, 87, 94]. However, most of these studies consider only some specific
limits of bispectrum and particular effects2. That is, a thorough calculation including the
full bispectrum and all the effects at recombination is desired. More seriously, the signal-to-
noises3 for detection of the amplitude of the intrinsic bispectrum claimed by different work
vary within a broad range – from order of 0.1 to 10. A conclusive and precise value is yet
to be obtained and a systematic calculation should be carried out to constrain the effect for
future CMB experiments. This is one of the two main objectives of this thesis and the result
has been published in [100]. There are other teams aiming at the similar goal but working
with different and independent approaches, e.g. [52, 82].
1Nevertheless, it is possible to generate non-Gaussianities without significant bispectra [24]. In these cases,
trispectra may be used to study the non-Gaussianities.
2The exception is [87] but its calculated signal-to-noise appears to be significantly larger than those calcu-
lated afterwards.
3These signal-to-noises are calculated for cosmic-variance-limited data, assuming no primordial non-
Gaussianities.
4 Review of CMB Cosmology
We can split the problem into two parts: analytical and numerical part. In the analyt-
ical part, we derive all the evolution equations governing the 2nd-order perturbations and
formulate the CMB bispectrum for the 2nd-order perturbations imprinted at recombination.
Expanding the perturbations up to 2nd order is quite straightforward but there are several
techniques we need. First, we will work with both the global coordinate system on the
background manifold and the tetrad basis in the tangent planes. We have to develop a trans-
formation between them up to 2nd order. Second, to obtain the multipoles in the Boltzmann
equation, we have to decompose a product of two spherical harmonics. The decomposition
can be achieved by exploiting the Wigner 3-j symbols. Third, we will need to derive the
evolution equation of the optical depth to 1st order.
The second part is the numerical calculation. To obtain the full bispectrum and cap-
ture all the effects at recombination, numerical implementation is inevitable and it turns out
to be the most challenging part of the problem. First, the 2nd-order Einstein field equa-
tions (EFEs) are numerically unstable. It is because there exists some pairs of large terms
canceling with each other and the residuals are a few orders of magnitude smaller. This
undermines the numerical precisions achievable. By combining the EFEs, we can mitigate
the numerical instability. The result can be verified by using the constraint equations of
EFEs. Second, the original quadratic source function in 2nd order requires multipoles with
infinitely high ℓ and fails to separate the effects at recombination from the late-time effects
consistently. We will show that these two issues can be solved by performing integration by
parts to the line-of-sight (LOS) integration. We validate the result by comparing with the
analytical solution of the squeezed-limit bispectrum.
Another post-inflationary non-Gaussian effect we will study is the distortions of the
CMB anisotropies from the late-time metric perturbations, which include lensing, redshift
and time-delay effects. In particular, CMB lensing is the dominant effect among these dis-
tortions and will be our main focus. CMB lensing was first considered in 1987 [18] and,
since then (for a review, see [67]) its contributions to the power spectra [23, 46, 89, 109],
bispectra [39, 46, 69] and trispectra [27, 47] of the CMB temperature and polarizations
have been studied. Recently, the first detection of the CMB lensing was achieved by cross-
correlating the lensing potential reconstructed by the Wilkinson Microwave Anisotropy
Probe (WMAP) data and large scale structure surveys (as suggested in [42]) [41, 98]. The
detection of the power spectrum of the lensing potential was first obtained with the Atacama
Cosmology Telescope (ACT) data [31] and then with the South Pole Telescope (SPT) data
[102]. The first full-sky map of the lensing potential was reconstructed by the Planck data
[4] and the significance of the detection has been boosted to larger than 25σ . Planck also
5detected the ISW-lensing bispectrum generated by the non-Gaussian lensed CMB tempera-
ture fluctuations with significance of 2.6σ [6]. Finally, the B-mode polarization induced by
lensing was detected with the SPTpol data [40] at 7.7σ significance. Driven by these sophis-
ticated experiments, the study of the CMB lensing has entered a new era with unprecedented
precision.
Furthermore, since its major contribution comes from lenses at low redshifts (z . 10),
the lensing effect is very sensitive to the late-time evolution of the universe. By using the
non-Gaussian properties of the lensing effect on the CMB temperature and polarizations,
different estimators [43, 44, 56, 77] were proposed to reconstruct the lensing power spec-
trum from the CMB data. Measuring the lensing power spectrum with high angular resolu-
tion and high sensitivity imposes constraints [55, 64] on the neutrino masses mν with σ(mν)
as small as 0.035 eV [64], which are much tighter compared to the one using the CMB power
spectra alone and have fewer biasing issues compared to the constraints from the large-scale
structures. The lensing power spectrum can also break the degeneracy between the neutrino
masses and the equation of state parameter ω of the dark energy [55]. The cross-correlations
between the reconstructed lensing potential and other late-time observations4 have been an-
alyzed through various experiments [4, 5, 19, 45, 96]. These cross-correlations allow us to
constrain the dynamics of the dark matter and the dark energy [48, 92].
In view of these applications of the CMB lensing, it is clear that an accurate computation
of the lensing effect is required. In particular, it remains to be articulated the approximations
involved in the usual lensing calculation. For example, in the canonical approach [23, 46],
the lensed CMB temperature anisotropies Θ˜ are expressed in terms of the unlensed CMB
temperature anisotropies Θ in the remapping. The usual derivation uses a perturbed geodesic
and employs some implicit approximations. While there are assessments about the fidelity
of the corrections of the remapping approach especially in high orders [26, 33, 44], keeping
correction terms consistently in high orders is non-trivial [61, 95] and a systematic study
in arbitrarily high orders has not been undertaken. The main goal of our study is to derive
the CMB lensing (as well as redshift and time-delay) through the Liouville (free-streaming)
part of the Boltzmann equation.
The key step to derive lensing, redshift and time-delay effects through the Boltzmann
equation up to arbitrarily high order is to treat these effects as higher-order actions on the
CMB anisotropies. The effects can be expressed as a nested integral in the line-of-sight
approach. The derivation using the Boltzmann equation allows us to explicitly list out all the
4For example, the integrated Sachs-Wolfe (ISW) effect, the Sunyaev-Zeldovich (SZ) effect, the Cosmic
Infrared Background (CIB), galaxies and quasars.
6 Review of CMB Cosmology
approximations implied in the canonical remapping approach. In particular, we find some
new approximations which have not been identified previously in literature. We can then
assess the significance of these approximations and justify the robustness of the remapping
approach. To facilitate the calculation, we develop a diagrammatic approach to keep track
of the couplings of different effects in arbitrarily high orders.
This thesis is organized as follows. We review the linear cosmological perturbations
and their impact on CMB in Chapter 2. We derive the Boltzmann equation and EFEs up
to 2nd order and present the details about solving the 2nd-order perturbations numerically
in Chapter 3. In Chapter 4, we single out the effects at recombination from the late-time
effects and perform the numerical calculation for the full bispectrum at recombination. In
Chapter 5, we formulate the lensing, redshift and time-delay effects through the Boltzmann
equation and study the effect of lens-lens couplings on the CMB power spectrum. We
discuss some potential extensions of the work in this thesis and draw the conclusion in
Chapter 6. Because there are many notations and symbols used, we list out all of them
in Appendix A for clarification. A table of cosmological parameters used in this thesis is
shown in Appendix B. Some useful definitions and formulae of the spin-weighted spherical
harmonics are presented in Appendix C. Details of the perturbations of tetrads and Ricci
rotation coefficients up to 2nd order can be found in Appendix D.
Chapter 2
Cosmological Perturbations and CMB
Spectra
Since the cosmic microwave background (CMB) was discovered in 1964 [79], the study
of Cosmology has entered an era with unprecedentedly high precision. Properties (such as
flatness, homogeneity and isotropy) of the CMB signals provide clues to the development of
the big bang theory with an inflationary epoch. The homogeneity and isotropy, in particular,
also support the model of a uniformly-distributed universe at early time with tiny fluctu-
ations as seeds for the late-time structure formations. These fluctuations were imprinted
in the CMB anisotropies when photons decoupled with baryons during recombination. By
studying the power spectra of these CMB anisotropies obtained from observations (such as
COBE, WMAP and Planck), we ought to develop the standard model in Cosmology which
consists of baryon (∼ 5%), dark matter (∼ 27%) and dark energy (∼ 68%)1 as the main
components of our universe.
In this chapter, we review the cosmological perturbation theory in the CMB study up
to 1st order. We will extend the study to 2nd-order perturbations in Chapter 3 and their
contribution to the CMB bispectrum at recombination in Chapter 4. First, we introduce the
coordinate system and the tetrad basis used throughout this thesis in Section 2.1. Then, we
define the CMB temperature and polarizations in Section 2.2. In Section 2.3, we review
the governing equations of the CMB study – the Einstein field equations (EFEs) and the
Boltzmann equation. We derive the background- and 1st-order governing equations for
the evolution of our universe in Section 2.4 and 2.5 respectively. Finally, we discuss how
information from the universe can be extracted through power spectra and bispectra of the
1The values are based on the CMB data combination Planck+WP in [3].
8 Cosmological Perturbations and CMB Spectra
CMB anisotropies in Section 2.6.
2.1 Coordinate System and Tetrad Basis
In this section, we construct the coordinate system for the background manifold and the
tetrad basis for the tangent planes used in this thesis. The construction allows us to expand
the perturbations consistently to any orders.
2.1.1 Coordinate System
The homogeneity and isotropy of the universe allow us to map the physical manifold onto
a background manifold equipped with a Friedmann metric. By assuming a flat universe, we
choose the coordinate system {xA}= {η ,xI} for A = 0,1,2,3 and I = 1,2,3 such that
ds2 = gµνdxµdxν
= a2(η)
{
−(1+2Φ)dη2 +2BIdxIdη+[(1−2Ψ)δIJ +2HIJ]dxIdxJ
}
, (2.1)
where a(η) is the scale factor and Φ, Ψ,BI andHIJ are metric perturbations as functions of
xA. In this thesis, capital Roman indices (A, B, C, . . . ) label the chosen coordinate system in
the background manifold while Greek indices (µ , ν , ρ , . . . ) represent abstract indices2. All
the metric perturbations in Eq. (2.1) can be considered as fields living on the background
manifold. Moreover, we can further decompose BI and HIJ by using the scalar-vector-
tensor (SVT) decomposition such that
BI = ∂IB+BI, HIJ =HIJ +∂(IEJ)+∂I∂JE , (2.2)
where 2∂(IEJ) ≡ ∂IEJ +∂JEI and ∂ IBI = ∂ IEI = ∂ IHIJ = 0 andHI I = 0 with ∂I ≡ ∂/∂xI .
With the assumption of a flat Friedmann metric in the background order, the equations
above hold in general and can be expanded into perturbations of different orders in the
following manner
A=A[I]+A
[II]
2!
+A
[III]
3!
+ . . . , (2.3)
2A more familiar notation would be to write the metric in Eq. (2.1) as gABdxAdxB. That is, the indices A
and B do double duty as coordinate labels and abstract indices.
2.1 Coordinate System and Tetrad Basis 9
where the Roman numbers inside the square brackets of the superscripts denote the orders
of perturbations.
2.1.2 Tetrad Basis
At each point on the physical manifold, we construct the tangent basis to decompose the
photon momenta. It is convenient to introduce the tetrad fields eˆa which satisfy the or-
thonormality conditions
eˆa · eˆb ≡gµνea
µeb
ν = ηab,
eˆa · eˆb ≡gµνeaµebν = ηab, (2.4)
where ηab≡ diag(−1,1,1,1) is the Minkowski metric, and indices a and b are equal to 0 to 3
labeling the four tetrad vectors and forms. Since photons have null 4-momenta (pµ pµ = 0),
their momenta can be written as
p = p0(eˆ0 + nˆ), (2.5)
where p0 is the photon energy measured by an observer with the velocity eˆ0 and nˆ is a
spacelike vector on the hypersurface “screen” denoting the direction of the photon as seen
by the observer, such that nµe0µ = 0 and nµnµ = 1. Furthermore, it is useful to define the
screen projector
Sµν(eˆ0,p)≡ gµν + e0µe0ν −nµnν . (2.6)
By foliating spacetime into hypersurfaces threaded through by the orbit defined by the ob-
server’s velocity eˆ0, we can describe radiations by a screen-projected rank-2 tensor – the
intensity matrix which lives on these hypersurfaces. We will discuss the intensity matrix in
detail in Section 2.2.1.
We note that the tetrad fields eˆa lie on the physical manifold. Thus, we have to pull
them back onto the background manifold in order to study the Boltzmann equation on the
background manifold. This is achieved [21] through the pullback φ∗ with a gauge field ξ
defined by
ξ eˆa ≡ limλ→1
φ∗λ ,ξ (eˆa) = limλ→1
∞
∑
N=0
λN
N!
LNξ eˆa, (2.7)
where λ denotes the foliations Mλ of an embedding (4+1)-dimensional manifold N =
10 Cosmological Perturbations and CMB Spectra
M× [0,1]3 and L is the Lie derivative. The pulled-back tetrads ξ eˆa lie on the background
manifold. Using the following natural background basis for vectors and forms on the back-
ground manifold
e¯a =
1
a(η)
∂
∂xA , e¯
a = a(η)dxA, (2.8)
we can express the tetrads in the natural background basis
ξ eˆa = ξXabe¯b , ξ eˆa = ξYbae¯b, (2.9)
where the coefficients can be expanded into perturbations as
ξXab ≡
∞
∑
N=0
1
N!ξ
X [N]ab , ξYab ≡
∞
∑
N=0
1
N!ξ
Y [N]ab . (2.10)
After fixing the residual freedoms of the transformation, we can determine ξX
[N]
ab and ξY
[N]
ab
order by order. The details and the coefficients in 1st and 2nd order are shown in Appendix
D.1. In Fig. 2.1, we illustrate the pullback of tetrad bases onto the background manifold
M0.
M1
M0
Figure 2.1 The illustration of the pullback φλ from the physical manifold M1 onto the
background manifold M0. The dashed gray grids represent the tetrad bases. Due to the
existing perturbations, the pulled-back tetrad bases on the background manifold do not align
with the bases of the coordinate system (the misalignment is exaggerated in the graph).
3We define the background manifold asM0 and the physical manifold asM1 with λ ∈ [0,1].
2.2 CMB Temperature and Polarization 11
2.2 CMB Temperature and Polarization
Here, we define the intensity matrix which describes the intensity and the polarizations of
the CMB photons.
2.2.1 Intensity Matrix
To encode all the information about the intensity and the polarization of the CMB photons,
we define the screen-projected intensity matrix Pµν(xA, pa) which satisfies4
eµ0Pµν(xA, pa) = nµPµν(xA, pa) = pµPµν(xA, pa) = 0. (2.11)
The intensity matrix of the CMB radiation can be decomposed into
Pµν =
1
2
ISµν +Pµν , (2.12)
where I denotes the photon intensity and the symmetric traceless polarization tensor Pµν
encodes the linear polarizations of the CMB photons. Here, we ignore the circular polar-
ization because it is not induced by Compton scattering and thus is usually neglected in the
CMB studies. The polarization tensor can be further decomposed into two scalar fields [67]
– E-mode PE and B-mode PB, i.e.
Pµν = ∇˜⟨µ∇˜ν⟩PE + εγ (µ∇˜γ)∇˜νPB, (2.13)
where ∇˜µ ≡ Sµν∇ν is the screen-projected covariant derivative, (. . .) denotes the symmetric
part, ⟨. . .⟩ denotes the symmetric traceless part and εµν ≡−i(e′µ−e′ν+−e′ν−e′
µ
+)/2 with e′± ≡
eˆ′x± ieˆ
′
y
5.
The normalized energy-integrated photon intensity and polarization tensor are defined
as
Iˆ(xA, nˆ)≡ 1+ 1
2π2I¯(η)
∫
I(xA, p0, nˆ)(p0)3dp0, (2.14)
Pˆµν(xA, nˆ)≡
1
2π2I¯(η)
∫
Pµν(xA, p0, nˆ)(p0)3dp0, (2.15)
4For details about how to construct the intensity matrix, see [101].
5eˆ′x and eˆ
′
y, which are orthogonal to each other, lie on the plane perpendicular to the direction nˆ of the light
path and the observer’s velocity eˆ0.
12 Cosmological Perturbations and CMB Spectra
where
I¯(η)≡ 1
2π2
∫
I¯(η , p0)(p0)3dp0, (2.16)
with I¯(η , p0) as the homogeneous and isotropic black-body spectrum in the background
order. Up to 1st order, the CMB temperature can be related to the photon intensity by
Iˆ(xA, nˆ) = 1+4Θ(xA, nˆ), (2.17)
where Θ≡ (T − T¯ )/T¯ with the background CMB temperature T¯ . More details on defining
the CMB temperature can be found in Section 4.1.
2.2.2 Stokes’ Parameters
We define the normalized energy-integrated Stokes’ parameters Qˆ and Uˆ such that
(Qˆ+ iUˆ)(nˆ)≡ e′a+e′
b
+Pˆab(nˆ),
(Qˆ− iUˆ)(nˆ)≡ e′a−e′
b
−Pˆab(nˆ), (2.18)
where e′± are defined below Eq. (2.13)
6. We can decompose the Stokes’ parameters into
spin-2 spherical harmonics
(Qˆ± iUˆ)(nˆ) =∑
ℓm
a±2,ℓm ±2Yℓm(nˆ), (2.19)
where the spin-weighted spherical harmonics sYℓms are defined in Appendix C.1. By acting
the raising (ð) and lowering (ð¯) operators7 twice and making use of Eq. (C.7) and Eq. (C.8),
the decompositions become
ð¯2(Qˆ+ iUˆ)(nˆ) =∑
ℓm
√
(ℓ+2)!
(ℓ−2)! a2,ℓm Yℓm(nˆ),
ð2(Qˆ− iUˆ)(nˆ) =∑
ℓm
√
(ℓ+2)!
(ℓ−2)! a−2,ℓm Yℓm(nˆ). (2.20)
6We remark that e′x and e
′
y are not the same as e¯x and e¯y in Eq. (C.25). In this work, we choose e¯x and e¯y to
be orthogonal to the wave vector k of the 2nd-order perturbations.
7The raising and lowering operators are defined in Eq. (C.5) and (C.6) respectively.
2.3 Governing Equations 13
The multipoles of the normalized energy-integrated Eˆ-mode and Bˆ-mode polarizations are
defined
aEˆ,ℓm ≡−
1
2
(
a2,ℓm+a−2,ℓm
)
, aBˆ,ℓm ≡
i
2
(
a2,ℓm−a−2,ℓm
)
. (2.21)
One of the advantages to decompose the CMB polarizations into E- and B-mode is that
scalar perturbations do not produce B-mode polarization [54, 108]. Thus, the B-mode po-
larization is a crucial probe for primordial gravitational waves8.
2.3 Governing Equations
In this section, we review the governing equations which are essential in studying CMB – the
Boltzmann equation and the EFEs. In Section 2.4 and 2.5, we will expand the perturbations
to establish the evolution equations in background and 1st order respectively. Furthermore,
we will derive the 2nd-order Boltzmann equation and EFEs in Chapter 3.
2.3.1 Einstein Field Equations
To derive the governing equations, we need the EFEs
Gµν +Λgµν = κTµν , (2.22)
where Gµν is the Einstein tensor, Λ is the cosmological constant, gµν is the metric tensor,
Tµν is the stress-energy tensor and κ ≡ 8πG/c4 with Newton’s gravitational constant G
and the speed of light c. We set the gravitational constant and the speed of light as unity
throughout this thesis. Moreover, the Einstein tensor is defined as
Gµν ≡ Rµν −
R
2
gµν , (2.23)
where the Ricci scalar R≡ gµνRµν and the Ricci tensor9
RAB ≡ Γ
C
AB,C−Γ
C
AC,B+ΓCDCΓBAB−ΓCDBΓDAC, (2.24)
8We remark that vector perturbations generate B-mode polarization as well but the power spectrum pro-
duced is expected to be distinguishable from that of primordial gravitational waves.
9In this thesis, we use Greek letters for abstract indices and Roman alphabets for coordinates of the back-
ground Manifold.
14 Cosmological Perturbations and CMB Spectra
with the Christoffel symbols
ΓCAB ≡
1
2
gCD
[
gAD,B+gBD,A−gAB,D
]
. (2.25)
The commas in subscripts denote partial derivatives such that gAB,C≡ ∂gAB/∂xC. In general,
the stress-energy tensor for a fluid can be expressed as
Tµν = ρuµuν +P(gµν +uµuν)+πµν , (2.26)
where ρ and P are the energy density and the pressure of the fluid respectively, uµ is the
fluid velocity, and πµν is the anisotropic stress. The velocity and the anisotropic stress of
fluids are defined such that
uµ = dx
µ
ds
, uµuµ =−1, uνπµν = πµµ = 0, (2.27)
where the proper distance s is an affine parameter. Under the chosen coordinate system of
the background manifold specified in Eq. (2.1), we can apply the SVT decomposition on
spatial components uI and πIJ so that
uI ≡
1
a(η) (∂Iυ+υI) , (2.28)
πIJ ≡ ρ¯a(η)2
[
σ,I,J−
1
3
δIJσ,K ,K +
1
2
(σI,J +σJ,I)+σIJ
]
. (2.29)
where the vector modes (υI and σI) and the tensor modes (σIJ) satisfy
∂ IυI = 0, ∂ IσIJ = 0, σ I I = 0. (2.30)
Because of the Bianchi identity, we have the conservation of energy and momentum
∇µTµν = 0, (2.31)
where ∇ is the covariant derivative.
2.3.2 Boltzmann Equation
To formulate the Boltzmann equation, we stay with the tetrad basis (denoted by indices
a,b,c, . . .) and elaborate all the relevant equations onto the background manifold as what we
2.3 Governing Equations 15
have done for the EFEs. The Boltzmann equation consists of two operators, the Liouville
operator L describing the free-streaming of photons and the collision operator C describing
Compton scattering. That is,
L[Pab(xA, p0, nˆ)] = Cab(xA, p0, nˆ), (2.32)
where we have switched to the tetrad basis via
Pab =Pµνeaµebν , Cab = Cµνeaµebν , Sab = Sµνeaµebν . (2.33)
We remark that the Boltzmann equation as a whole is gauge-invariant.
We begin with the Liouville operator, which is defined as
L≡ SacSb
d d
ds
= SacSbd
[
pA∇A+
dpe
ds
∂pe
]
, (2.34)
where the screen projector Sab is defined in Eq. (2.6), ∇A is the covariant derivative with
respect to xA, ∂pe ≡ ∂/∂ pe and pA≡ dxA/ds. The covariant derivative acting on the intensity
matrix can be expanded into
∇APcd = ∂APcd− ecDe f BΓBDAP f d− edDe f BΓBDAPc f , (2.35)
where the Christoffel symbols ΓCAB are defined in Eq. (2.25) and coefficients ea
A are shown
in Appendix D.1. In this thesis, we use the conformal time η instead of the proper distance s.
We can do so by multiplying Eq. (2.32) with the Jacobian ds/dη . Linearity of the Liouville
operator preserves the decomposition of Pab in Eq. (2.12), i.e.
L[Pab] =
1
2
L[I]Sab+L[Pab]. (2.36)
That means we can write down the evolution equations of the intensity and polarizations of
photons separately10.
To elaborate the Boltzmann equation, we need to know the photon trajectory, which is
given by the geodesic equation
dpa
ds
+ωbac pbpc = 0, (2.37)
10However, the equations are coupled through Compton scattering.
16 Cosmological Perturbations and CMB Spectra
where the Ricci rotation coefficients are defined as
ωbac ≡ eaµebν∇νecµ = ΓIJKebKecJeaI + ebIeaJ∂IecJ, (2.38)
which are determined completely by the metric gAB and the tetrads eaA. The perturbations
of the Ricci rotation coefficients in 1st and 2nd order can be found in Appendix D.2. Mean-
while, the momentum pA can be expressed in the tetrad basis
pA = paeaA. (2.39)
2.4 Cosmology in Background Order
Because of the homogeneity and isotropy, the evolution of our universe in background order
is determined by the scale factor a(η), which quantifies the expansion of our universe and
depends on the abundances of different components.
2.4.1 Friedmann–Lemaître–Robertson–Walker Metric
The homogeneity and isotropy of our universe suggest that the background-order spacetime
has a Friedmann–Lemaître–Robertson–Walker (FLRW) metric, i.e.
ds2 = g¯µνdxµdxν = a2(η)
[
−dη2 + dr
2
1− kr2
+ r2dΩ2
]
, (2.40)
where the Greek letters label numbers from 0 to 3, g¯µν is the background-order metric
tensor11, a(η) is the scale factor with the conformal time η , dΩ2 = dθ 2 + sin2θdφ2 and
k =−1, 0 or 1 indicates an open, flat or close universe respectively.
2.4.2 ΛCDM Cosmology
In standard model, the universe consists of baryons, cold dark matter (CDM), dark energy,
neutrinos and photons12. All these components interact with the spacetime metric through
gravity while baryons and photons also couple with each other through Compton scatterings.
11We use bar to denote the background-order quantities.
12We denote them by the subscripts b, c, Λ, n and r respectively.
2.4 Cosmology in Background Order 17
The background energy density and pressure are related by the equation of state (EOS)
P¯ = ωρ¯, (2.41)
where
ωb ≈ 0, ωc = 0, ωΛ =−1, ωn ≈ 1/3, ωr = 1/3. (2.42)
The pressure of baryons can be neglected because baryons are non-relativistic at the times
of interest. In contrast, neutrinos are approximately massless because they are relativistic at
early time.
Due to the homogeneity and isotropy, there exist preferable coordinate systems which
appreciate the symmetries, e.g. (η ,r,θ ,φ) used in Eq. (2.40). Under the coordinate system
(η ,r,θ ,φ), we can find that the quantities of the stress-energy tensor defined in Eq. (2.26)
are
π¯µν = 0, (2.43)
u¯A = 1
a(η){1,0,0,0} (2.44)
for all components in background order.
2.4.3 Friedmann Equations
By expanding the EFEs in background order, we obtain the well-known Friedmann equa-
tions
1
a2
H2 = 8πρ¯tot
3
+ Λ
3
−
k
a2
(2.45)
1
a2
H˙=−4π
3
(
ρ¯tot +3P¯tot
)
+ Λ
3
(2.46)
where the dot denotes derivatives with respect to the conformal time η13, H ≡ a˙/a is the
conformal Hubble parameter and ρ¯tot and P¯tot are the total energy density and pressure of
baryons, CDM, neutrinos and photons respectively. Throughout this thesis, we assume a
flat universe, i.e. k = 0. In particular, we haveH= 1/η in radiation-dominated era.
13The physical time is related to the conformal time by t = ∫ a(η)dη .
18 Cosmological Perturbations and CMB Spectra
2.5 Cosmological Perturbations in First Order
Linear cosmological perturbation theory has been widely studied and provides us tremen-
dous information through the CMB power spectra. A complete calculation of the linear
theory requires thorough understanding of numerous physical processes, such as the nucle-
osynthesis, Compton scattering at recombination and reionization, General Relativity and
so on. The details can be found in many textbooks (e.g. [32, 105]).
The cosmological perturbations are imprinted in the CMB anisotropies. Although these
anisotropies are very tiny (of order 10−5), they are detectable and are used to study the early
evolution of our universe. In this section, we will review the cosmological perturbations
in linear order and develop the corresponding governing equations. Apart from extending
the EFEs to 1st order, we will also need the 1st-order Boltzmann equation which describes
Compton scatterings between photons and baryons.
2.5.1 Einstein Field Equations
To derive the 1st-order EFEs, we keep all the perturbations in Eq. (2.1) up to 1st order,
i.e. A ≈ A[I] for any perturbation A14. Moreover, we fix the gauge freedom by choosing
Newtonian gauge in 1st order. Thus, we have
E = B = EI = 0. (2.47)
Under the standard model, we ignore the 1st-order vector and tensor perturbations, i.e.
BI =HIJ = 0. (2.48)
In other words, we have only two metric perturbations in 1st order, i.e. Φ and Ψ. However,
it is important to emphasize that the vector and tensor perturbations in 2nd order cannot
be neglected even they vanish in 1st order. This is because 2nd-order vector and tensor
perturbations can be generated by 1st-order scalar perturbations due to the nonlinear effect
of General Relativity (GR). We will discuss this in detail in Chapter 3. Using the conditions
14Without confusion, we ignore the superscript [I] in the following.
2.5 Cosmological Perturbations in First Order 19
in Eq. (2.27), we can find that the 1st-order quantities of the stress-energy tensor obey
uO =−1
a
Φ, (2.49)
πOO = 0, π IO = 0, π I I = 0. (2.50)
There are four EFEs for 1st-order scalar perturbations:
Time-time: GOO
3HΨ˙+ k2Ψ+3H2Φ+ κa
2
2
( ∑
X=b,c
ρ¯XδX + ∑
X=n,r
ρ¯X IˆX ,00) = 0, (2.51)
Trace space-space: GI I
Ψ¨+2HΨ˙+HΦ˙+(H2 +2H˙)Φ+ 1
3
k2(Ψ−Φ)− κa
2
2 ∑X=n,r
c2s,X ρ¯X IˆX ,00 = 0, (2.52)
Traceless space-space:
(
3△−2∂ J∂I−△−1δ JI
)(
3GIJ−δ IJ GKK
)
Ψ−Φ−
κa2
5k2 ∑X=n,r
ρ¯X IˆX ,20 = 0, (2.53)
Time-space: △−1∂ IGOI
Ψ˙+HΦ+ κa
2
2 ∑X=b,c
ρ¯X(1+ωX)υX −
κa2
6k ∑X=n,r
ρ¯X IˆX ,10 = 0, (2.54)
where △ ≡ ∂ I∂I , the dots over symbols denote derivatives with respect to the conformal
time η , δ ≡ δρ/ρ¯ , cs ≡
√
δP/δρ is the speed of sound and the subscripts b, c, n and
r represent baryons, CDM, neutrinos and photons respectively. We have transformed the
quantities of stress-energy tensors into multipoles for neutrinos and photons15. We remark
that two of the four EFEs are redundant due to the gauge freedom. Only two EFEs are
adequate for determining the evolution of the two scalar metric perturbations, which are Φ
and Ψ. However, in practice, they are useful for numerical validation.
Furthermore, we need the evolution equations for energy densities and velocities of
baryons and CDM. These can be obtained by perturbing the conservation of energy and
15The details of the transformation will be presented in Section 3.1.2.
20 Cosmological Perturbations and CMB Spectra
momentum in Eq. (2.31). Thus, we have the continuity and Euler equation
δ˙X − k2υX −3Ψ˙= 0, (2.55)
kυ˙X + kHυX + kΦ= CυX , (2.56)
where X = b,c denotes baryons and CDM respectively and the collision term Cυc = 0 for
CDM and Cυb =−R ˙¯τ
(
Iˆ10/4+ kυb
)
for baryons16.
2.5.2 Boltzmann Equation
The Boltzmann equation describes the evolution of the multipoles of photons and neutri-
nos. To derive the Boltzmann equation, we need to know the differential optical depth of
Compton scattering, which is defined as
τ˙ ≡ σT a xe ne, (2.57)
where σT is the Thomson cross-section, xe is the fraction of free electrons and ne is the
number density of total electrons. In 1st order, we need only the background-order differ-
ential optical depth. i.e. ˙¯τ(η). In the following, we will present the 1st-order Boltzmann
equations for photon intensity and E-mode polarization17:
˙ˆIℓ0 + k
(
0U
0
ℓ0Iˆℓ+1,0− 0D
0
ℓ0Iˆℓ−1,0
)
−4δ 0ℓ Ψ˙−4δ
1
ℓ kΦ= Cℓ0[Iˆ], (2.58)
˙ˆEℓ0 + k
(
2U
0
ℓ0Eˆℓ+1,0− 2D
0
ℓ0Eˆℓ−1,0
)
= Cℓ0[Eˆ], (2.59)
where the coefficients sDnℓm, sT
n
ℓm and sU
n
ℓm are defined in Appendix C.3 and the collision
terms are
Cℓ0[Iˆ] = ˙¯τ
(
−Iˆℓ0 +δ 0ℓ Iˆ00 +4δ 1ℓ υ˜b,(0)+δ 2ℓ Πˆ
)
, (2.60)
Cℓ0[Eˆ] = ˙¯τ
(
−Eˆℓ0−
√
6δ 2ℓ Πˆ
)
, (2.61)
with iυ˜(0) = υ ie¯∗I(0) and
Πˆ≡
1
10
(
Iˆ20−
√
6Eˆ20
)
. (2.62)
16The density ratio is defined as R ≡ (4/3)ρ¯r/ρ¯b. More discussions of the collision term can be found in
Section 3.2.3.
17The detailed derivation can be found in [32, 105].
2.5 Cosmological Perturbations in First Order 21
For neutrinos, we only need to consider the intensity as in Eq. (2.58) but the collision term
equals to zero. We remark that only the m = 0 modes are required for solving 1st-order
perturbations. This is because we align the wavenumber k with the azimuthal direction of
the multipoles.
2.5.3 Adiabatic Initial Conditions
Throughout this thesis, we assume single-field slow-roll inflation. With the single field,
the initial conditions of perturbations are expected to be adiabatic. This implies two condi-
tions18: the ratios between ρ1/3b , ρ
1/3
c , ρ
1/4
n and ρ
1/4
r at each spatial point remain constant
and
υb = υc = υn = υr. (2.63)
In 1st order, the conditions become
1
3
δb =
1
3
δc =
1
4
Iˆn,00 =
1
4
Iˆr,00, (2.64)
−4kυb =−4kυc = Iˆn,10 = Iˆr,10. (2.65)
That means we only need to solve for the initial energy density and velocity of one compo-
nent. The adiabatic initial conditions can be determined by the comoving curvature pertur-
bation, which is expressed in 1st order as
R=Ψ+ 2
3(1+ω)H
(
Ψ˙+HΦ
)
, (2.66)
where ω is the equation of state of the total fluid. The comoving curvature perturbation is
conserved up to 2nd order for superhorizon scales [103].
To fix the initial conditions, we consider the initial conformal time ηi deep in the
radiation-dominated era such that the cosmological scales are out of horizon (kηi ≪ 1).
Therefore, the conformal Hubble parameter H = 1/ηi and the energy density of the total
fluid is dominated by neutrinos and photons, i.e. ρ¯tot ≈ ρ¯n + ρ¯r. By ignoring Ψ˙19 and terms
18These two conditions of adiabaticity also hold in 2nd order.
19By expanding the perturbations into powers of kηi, we can show that the leading term of Ψ is a constant
and thus Ψ˙≈ 0.
22 Cosmological Perturbations and CMB Spectra
with higher powers of kηi, the initial conditions satisfy
R=Ψ+ 1
2
Φ, ˙ˆIn,20 =
2k
3
Iˆn,10, (2.67)
3H2Φ= κa
2
2
(
ρ¯nIˆn,00 + ρ¯rIˆr,00
)
, (2.68)
HΦ= κa
2
6k
(
ρ¯nIˆn,10 + ρ¯rIˆr,10
)
, (2.69)
Ψ−Φ= κa
2
5k2
(
ρ¯nIˆn,20 + ρ¯rIˆr,20
)
, (2.70)
and Iˆr,ℓ0 = Eˆr,ℓ0 = 0 for ℓ ≥ 2 due to the tight coupling of Compton scattering. Moreover,
higher-order moments of neutrinos are suppressed, i.e. Iˆn,ℓ0 = 0 for ℓ ≥ 3. This is because
moments with ℓ are suppressed by (kη)ℓ. By making use of 3H2 = κa2(ρ¯n+ ρ¯r), Eq. (2.64)
and Eq. (2.65), the initial perturbations are related to the comoving curvature perturbation
by
Φ= 10
15+4 fn
R, Ψ=
(
1+ 2
5
fn
)
Φ, (2.71)
Iˆn,00 =−2Φ, Iˆn,10 = 2(kηi)Φ, Iˆn,20 =
2
3
(kηi)2Φ, (2.72)
where fn ≡ ρ¯n/(ρ¯n + ρ¯r).
2.6 Statistics of CMB
In this section, we construct power spectra and bispectra of the CMB anisotropies. In stan-
dard model, the quantum fluctuations generated during inflation have the primordial power
spectrum, which is in form of
PΦ(k) = 2π2As
kns−4
kns−1∗
, (2.73)
where As is the initial amplitude, k∗ is the pivot wavenumber and ns is the spectral index. The
power spectrum becomes scale-invariant when ns = 1. Here, we assume that the spectral
index is independent of k and ignore the higher-order Taylor expansion of the primordial
power spectrum, such as the running index αs. The quantum fluctuations set the initial
conditions for perturbations. Then, the perturbations evolve and are imprinted on the CMB
anisotropies. By studying the evolution history, we can infer the parameters in Eq. (2.73)
2.6 Statistics of CMB 23
by fitting the predicted CMB power spectrum with the observed one. From Planck mission
[3], As = 2.215×10−9 and ns = 0.9619 for k∗ = 0.05/Mpc.
2.6.1 CMB Power Spectrum
Unlensed Power Spectrum
First of all, we express the Boltzmann equation in 1st order as
˙ˆI+ ikµ Iˆ−4Ψ˙+4ikµΦ= ˙¯τ
[
−Iˆ+ Iˆ00−4iµυb +
1
20
(
1−3µ2
)(
Iˆ20−
√
6Eˆ20
)]
, (2.74)
where the overhead dots denote the derivatives with respect to the conformal time η , µ ≡
kˆ · nˆ and Iˆ(η ,k, nˆ) ≡ I[I]/I¯ is defined in Eq. (2.14)20. With the line of sight approach as
shown in [91], we find that the photon brightness defined as △≡ Iˆ/4 can be expressed as
an integral
△[I](η ,k, nˆ) = eτ¯(η)
∫ η
0
dη˜eikµ(η˜−η)−τ¯(η˜)
{
Ψ˙− ikµΦ+ ˙¯τ
[
△00− iµυb +
1
2
(1−3µ2)Π
]}
≡ eτ¯(η)
∫ η
0
dη˜eikµ(η˜−η)S˜T (η˜ ,k, nˆ), (2.75)
where the optical depth τ¯ ≡
∫ η0
η˜
˙¯τ(η ′)dη ′,△00 is the monopole of the fractional brightness
and Π≡ (Iˆ20−
√
6Eˆ20)/40. Here, we make use of the fact that ˙¯τ → ∞ when η → 0.
For any function X of η , we can perform the integration by parts such that
∫ η0
0
dηeikµηµX =
[
eikµη
X
ik
]η0
0
−
∫ η0
0
dηeikµη X˙
ik
. (2.76)
By applying the integration by parts to Eq. (2.75), we can simplify the photon brightness at
present η0 and obtain
△(η0,k, nˆ) =
∫ η0
0
dη˜eikµ(η˜−η0)ST (η˜ ,k)Φ(k), (2.77)
where Φ(k) is the primordial fluctuation of the gravitational potential and the new source
20We perform the Fourier transformation to the perturbations, i.e. X(η ,x) = ∫ d3k(2π)3/2 e
ik·xX(η ,k).
24 Cosmological Perturbations and CMB Spectra
function is defined as
ST (η˜ ,k)≡ g
(
△00 +Φ+
υ˙b
k
+ Π
2
+ 3Π¨
2k2
)
+ g˙
(
υb
k
+ 3Π˙
k2
)
+ 3g¨Π
2k2
+ e−τ¯
(
Φ˙+ Ψ˙
)
,
(2.78)
with g≡ ˙¯τe−τ¯ .
With the temperature definition in Eq. (2.17), the temperature anisotropies observed
today is
Θ(nˆ) =
∫
d3k
(2π)3/2
∫ η0
0
dη˜e−ikµ r˜ST (η˜ ,k)Φ(k), (2.79)
where the conformal distance r˜ ≡ η0− η˜ . The exponential function in Eq. (2.79) can be
expanded by
e−ikµr =∑
ℓ
(−i)ℓ(2ℓ+1) jℓ(kr)Pℓ(µ), (2.80)
where jℓ(kr) is the spherical Bessel function of the first kind and Pℓ(µ) is the Legendre
polynomial. Using this, we can rewrite the temperature anisotropies as
Θ(nˆ) =∑
ℓ
(−i)ℓ(2ℓ+1)
∫
d3k
(2π)3/2Pℓ(µ)△ℓ(k)Φ(k), (2.81)
with the transfer function
△ℓ(k) =
∫ η0
0
dηST (η ,k) jℓ(kr). (2.82)
Moreover, we can decompose the temperature anisotropies into spherical harmonics
aℓm =
∫
d2nˆ Θ(nˆ) Y ∗ℓm(nˆ). (2.83)
With Eq. (2.81) and Eq. (C.11), we can find that
aℓm = (−i)ℓ
√
2
π
∫
d3k Y ∗ℓm(kˆ)△ℓ(k)Φ(k). (2.84)
By assuming the statistical isotropy of the CMB signals, we have the correlation
⟨aℓmaℓ′m′⟩= (−1)mCℓδℓℓ′δ−mm′, (2.85)
2.6 Statistics of CMB 25
where ⟨. . .⟩ denotes the ensemble average andCℓ is the power spectrum. Using ⟨Φ(k)Φ(k′)⟩=
δ (k+k′)PΦ(k), the CMB power spectrum can be expressed as
Cℓ =
2
π
∫
dk k2PΦ(k) [△ℓ(k)]2 . (2.86)
In Fig. 2.2, we plot the CMB power spectrum with the best-fit parameters from Planck as
shown in Appendix B.
0 500 1000 1500 2000
1000
2000
3000
4000
5000
{
{
H
{
+
1
L
C
{

2
Π
H
Μ
K
2
L
Figure 2.2 The full-sky unlensed power spectra of the CMB temperature from our code
(black line) and CAMB (dashed orange line) against ℓ. A perfect match is achieved. Our
code uses Newtonian gauge while CAMB uses synchronous gauge.
CMB Lensing Effect
It is known that the CMB lensing has a significant contribution to the CMB anisotropies
and thus their correlations. In the canonical approach [67], the lensed CMB temperature
anisotropies Θ˜ are expressed in terms of the unlensed CMB temperature anisotropies Θ in
the remapping21
Θ˜(nˆ) =Θ(nˆ+α ). (2.87)
21When similar approaches apply to CMB polarizations, geometric corrections on the spherical sky have to
be considered to maintain statistical isotropy of the lensed fields [22].
26 Cosmological Perturbations and CMB Spectra
The deflection angle α is a perturbation defined as
α (nˆ) = ∇nˆψ(nˆ)≡ 2
∫ rLSS
0
dr
r− rLSS
r rLSS
∇nˆΨW(r,−rnˆ), (2.88)
where ψ(nˆ) is known as the lensing potential, ΨW ≡ (Φ+Ψ)/2 is the Weyl potential, r
is the conformal distance between the gravitational potential and the observer, rLSS is the
conformal distance between the LSS and the observer and ∇nˆ is the covariant derivative on
the sphere. We refer to [23, 46] and Chapter 5 for details about how lensing effects distort
the CMB power spectrum.
2.6.2 CMB Bispectrum
If the CMB signals are not perfectly Gaussian, non-Gaussianities exist and may induce CMB
bispectra. These non-Gaussianities can be produced during and after inflation and the treat-
ments can be very different. In this section, we discuss the CMB bispectra from the primor-
dial non-Gaussianities generated during inflation. The post-inflationary non-Gaussianities
will be studied in Chapter 4. We remark that although the bispectra involve non-linear
perturbations, the bispectra from different non-Gaussianities are separable22.
For the primordial non-Gaussianities, the leading-order effect to the CMB anisotropies
can be expressed as
Θ(nˆ) =
∫
d3k
(2π)3/2
∫ η0
0
dη˜eikµ(η˜−η0)ST (η˜ ,k)ΦNG(k), (2.89)
where ΦNG(k) denotes the non-Gaussian fluctuation of the gravitational potential. We note
that the 1st-order source function defined in Eq. (2.78) is used. That is, the leading order
captures how the primordial non-Gaussianities evolve linearly over time after inflation. In
Chapter 4, we will see that the nonlinear source function is necessary for studying the post-
inflationary non-Gaussianities.
By decomposing the temperature anisotropies into spherical harmonics, the CMB bis-
pectrum is the ensemble average of three coefficients aℓms
Bm1m2m3ℓ1ℓ2ℓ3 ≡ ⟨aℓ1m1aℓ2m2aℓ3m3⟩. (2.90)
22It is true at least for the leading terms of the bispectra.
2.6 Statistics of CMB 27
The CMB angular-averaged bispectrum Bℓ1ℓ2ℓ3 can then be defined as
Bℓ1ℓ2ℓ3 ≡ ∑
m1m2m3
(
ℓ1 ℓ2 ℓ3
m1 m2 m3
)
Bm1m2m3ℓ1ℓ2ℓ3 , (2.91)
where the Wigner 3j-symbol
(
ℓ1 ℓ2 ℓ3
m1 m2 m3
)
guarantees the satisfaction of the triangle
conditions |ℓ j− ℓk| ≤ ℓi ≤ ℓ j + ℓk for any permutations of the indices i, j,k = 1,2,3. More-
over, it is convenient to use the reduced bispectrum bℓ1ℓ2ℓ3 , which is related to the angular-
averaged bispectrum by
Bℓ1ℓ2ℓ3 ≡
√
(2ℓ1 +1)(2ℓ2 +1)(2ℓ3 +1)
4π
(
ℓ1 ℓ2 ℓ3
0 0 0
)
bℓ1ℓ2ℓ3. (2.92)
Using Eq. (C.24), we have the relation
Bm1m2m3ℓ1ℓ2ℓ3 = G
m1m2m3
ℓ1ℓ2ℓ3 bℓ1ℓ2ℓ3, (2.93)
where the Gaunt integral is defined in Eq. (C.14).
By expressing the coefficients using Eq. (2.84), the CMB bispectrum can be written as
Bm1m2m3ℓ1ℓ2ℓ3 =
[
3
∏
i=1
(−i)ℓi
√
2
π
∫
d3kiY ∗ℓimi(kˆi)△ℓ(ki)
]
⟨Φ(k1)Φ(k2)Φ(k3)⟩, (2.94)
where ⟨Φ(k1)Φ(k2)Φ(k3)⟩ is the Φ(k)-field bispectrum. For our interest, it is enough to
define the Φ(k)-field bispectrum in form of
⟨Φ(k1)Φ(k2)Φ(k3)⟩= (2π)−3/2δ 3(k1 +k2 +k3)B(k1,k2,k3), (2.95)
where the Dirac delta function δ 3 ensures the conservation of momentum and B(k1,k2,k3)
specifies the shape of the bispectrum.
Since the source function ST is independent on the direction of k (i.e. ST = ST (η ,k)),
we can simplify the bispectrum further by using Eq. (2.80), Eq. (C.11) and
∫
d3xei(k1+k2+k3)·x = (2π)3δ 3(k1 +k2 +k3). (2.96)
28 Cosmological Perturbations and CMB Spectra
In particular, the reduced bispectrum can be simplified as
bℓ1ℓ2ℓ3 =
8
π3
∫
r2dr (k1k2k3)2 dk1dk2dk3 jℓ1(k1r) jℓ2(k2r) jℓ3(k3r)
△ℓ1(k1)△ℓ2(k2)△ℓ3(k3)B(k1,k2,k3). (2.97)
In Chapter 4, we will see that the 2nd-order source function S[II]T has more dependences other
than η and k. Hence, the corresponding reduced bispectrum becomes more complicated.
We separate the temperature anisotropies into the Gaussian (G) and non-Gaussian (NG)
parts, i.e.
aℓm = aGℓm+aNGℓm , (2.98)
where aNGℓm can come from primordial non-Gaussianities or post-inflationary nonlinear evo-
lution. Since the Gaussian signals do not contribute to the bispectrum, the leading-order
CMB bispectrum is
Bm1m2m3ℓ1ℓ2ℓ3 = ⟨a
G
ℓ1m1a
G
ℓ2m2a
NG
ℓ3m3⟩+2 perm. (2.99)
In the standard model, the primordial fluctuations were originated from the Bunch-
Davies vacuum state and are thus Gaussian. It can be shown that significant non-Gaussianities
would not be produced from the single-field slow-roll inflation [72]. However, the question
of having primordial non-Gaussianities is still open and has long been studied. Numerous
mechanisms have been proposed to generate measurable non-Gaussianities [10]. The de-
tection of these non-Gaussianities will facilitate our understanding of the very early uni-
verse and even the fundamental physics taken place in that era. The CMB bispectrum
is of importance to probe the primordial non-Gaussianities from inflation. In particular,
non-Gaussianities generated from different mechanisms have distinguishable features in the
CMB bispectrum which can be classified into different types. Here, we review some of the
most popular types of bispectrum templates – local, equilateral and orthogonal type.
2.6 Statistics of CMB 29
Local-type Bispectrum
The primordial non-Gaussianity of the local type is a correction to the primordial Gaussian
Newtonian potential ΦG(x) expressed in real space as
Φ(x) =ΦG(x)+ f locNL
[
Φ2G(x)−⟨Φ2G(x)⟩
]
, (2.100)
where f locNL is the nonlinear coupling constant of the local type and the ensemble average
⟨ΦG(x)⟩= 0.
In Fourier space, we decompose Φ(k) =ΦG(k)+ΦNG(k) with the non-Gaussian part
ΦNG(k)≡ f locNL
[∫
d3q
(2π)3/2ΦG(k+q)Φ
∗
G(q)− (2π)3/2δ 3(k)⟨Φ2G(x)⟩
]
, (2.101)
where the superscript ∗ denotes the complex conjugate and δ 3(k) is the Dirac delta function
in three dimension23.
If the Newtonian potential is purely Gaussian with mean zero, the Φ(k)-field bispectrum
will simply vanish. However, due to the non-Gaussian part in Eq. (2.101), the bispectrum is
non-vanishing and the leading components are24
⟨ΦG(k1)ΦG(k2)ΦNG(k3)⟩+2 perm. = (2π)−3/2δ 3(k1 +k2 +k3)B(k1,k2,k3), (2.102)
where
B(k1,k2,k3) = 2 f locNL [PΦ(k1)PΦ(k2)+PΦ(k2)PΦ(k3)+PΦ(k1)PΦ(k3)] . (2.103)
The bispectrum of the local type peaks at the squeezed-limit triangle for which one of
the three modes kis in Fourier space is much smaller than the other two. Significant local-
type bispectrum can be generated by multifield inflationary models [60], inhomogeneous
reheating scenario [34] and etc. However, single-field inflationary models do not produce
any measurable non-Gaussianity of the local type. For example, f locNL = (5/12)(1−ns)≪ 1
for the single-field slow-roll inflation [72]. A convincing detection of f locNL from inflation
will thus rule out the single-field inflationary models, regardless of their dynamics [30].
23We make use of the fact that Φ(x) is a real quantity so that Φ∗(k) =Φ(−k).
24Because of the second term in Eq. (2.101), Eq. (2.102) is only valid for ki ̸= 0.
30 Cosmological Perturbations and CMB Spectra
The constraint on the local type from Planck 2013 [6] is
f locNL = 2.7±5.8, (2.104)
at 68% confidence level (CL).
Equilateral-type Bispectrum
Similar to the local type, the equilateral type is specified by Eq. (2.102) with
B(k1,k2,k3) = 6 f equNL
{
− [PΦ(k1)PΦ(k2)+2 perm.]−2 [PΦ(k1)PΦ(k2)PΦ(k3)]2/3
+
[
PΦ(k1)1/3PΦ(k2)2/3PΦ(k3)+5 perm.
]}
. (2.105)
Inflationary models, such as the Dirac-Born-Infeld (DBI) inflation, can generate significant
non-Gaussianity of this type [97]. The constraint on the equilateral type from Planck 2013
[6] is
f equNL =−42±75, (2.106)
at 68% CL.
Orthogonal-type Bispectrum
The Φ(k)-field bispectrum for the orthogonal type is specified by
B(k1,k2,k3) = 6 f orthNL
{
−3 [PΦ(k1)PΦ(k2)+2 perm.]−8 [PΦ(k1)PΦ(k2)PΦ(k3)]2/3
+3
[
PΦ(k1)1/3PΦ(k2)2/3PΦ(k3)+5 perm.
]}
. (2.107)
Significant non-Gaussianity of the orthogonal type can be produced by single-field infla-
tionary models [25]. The constraint on the orthogonal type from Planck 2013 [6] is
f orthNL =−25±39, (2.108)
at 68% CL.
Chapter 3
Cosmological Perturbations in Second
Order
The perturbation theory is essential for studying Cosmology. For example, linear pertur-
bations have been studied in depth for understanding the linear evolution of cosmological
perturbations and their imprint on the observations. With improving precisions of measure-
ments, higher-order perturbations from different mechanisms become important and can
play a crucial role in Cosmology.
Nonlinear evolution can occur during and after inflation. Usually, non-Gaussianities are
produced and can be probed by 3-point correlations of various observations, e.g. cosmic
microwave background (CMB) and large-scale surveys. Although the single-field slow-
roll inflation in standard Cosmology is known to have negligible non-Gaussianities [72],
alternative inflationary models can generate detectable non-Gaussianities as clues about the
underlying theory of inflation1. Nevertheless, whether these primordial non-Gaussianities
exist or not has to be verified by observations. In contrast, post-inflationary era guarantees
the existence of nonlinear evolution through some well-known processes. In particular, the
nonlinearity of General Relativity (GR) promotes the growth of nonlinear perturbations and
is responsible for the formation of complex structures at late time.
Nonlinear perturbations can have profound impacts on the CMB statistics, especially
bispectra and trispectra. They can contain fruitful information about the universe and thus
deserve a thorough study. In this chapter, we establish all the governing equations necessary
for solving the cosmological perturbations up to 2nd order, including the Boltzmann equa-
1See [24] for a review.
32 Cosmological Perturbations in Second Order
tion (BE) and the Einstein field equations (EFEs). Fig. 3.1 outlines the relations between
different perturbations and their corresponding evolution equations. As the leading non-
linear order, 2nd-order perturbations are enough for understanding the nonlinear evolution
around recombination when the perturbations in cosmological scales are still tiny compared
to the background quantities and higher-order perturbations are highly suppressed. We will
find that the derivation here is indispensable when we calculate the CMB bispectrum gen-
erated at recombination in the next chapter. In principle, the canonical perturbation theory
breaks down at low redshifts. Thus, other approaches, such as N-body simulation and ef-
fective field theory [14], are adopted when we study the late-time large-scale evolution.
Metric
Φ,Ψ,BI,HIJ
(EFEs)
Cold dark
matter
(CDM)
δc, υc
(∇νT
µν
c = 0)
Baryons
δb, υb
(∇νT
µν
b = C
µ
b )
Neutrinos
In,ℓm
(Collisionless
BE)
Photons
Ir,ℓm
(BE)
Cosmological
Constant
Λ
τ¯ , δτ
Figure 3.1 The diagrammatic outline of relations between different perturbations. The corre-
sponding evolution equations are shown in the brackets. The waved lines denotes couplings
through GR while the two headed arrow denotes the Compton scattering. We remark that
the 1st-order perturbed optical depth (δτ) is needed for the 2nd-order Compton scattering.
This chapter is organized as follows. In Section 3.1, we extend the discussion about
the coordinate system (see also Section 2.1.1) and fix the 2nd-order gauge choice to the
Newtonian gauge. We then derive the EFEs, continuity equation and Euler equation in
2nd order. In Section 3.2, we derive the hierarchy of the 2nd-order Boltzmann equation,
including the Liouville and collision parts. We also formulate the evolution equation of the
3.1 Second-order Einstein Field Equations 33
1st-order optical depth which is essential for the 2nd-order Compton scattering. Finally, we
present the numerical calculation of the 2nd-order perturbations in Section 3.3. In particular,
we set the initial conditions, apply the tight-coupling approximation (TCA) and validate the
numerical results with the constraint equations.
3.1 Second-order Einstein Field Equations
Deriving the 2nd-order EFEs is rather straightforward but messy. For simplicity, we will
first fix the gauge choice. Throughout this chapter, we use Newtonian gauge in both 1st and
2nd order2. Then, we expand the Einstein tensor and stress-energy tensor up to 2nd order
based on the equations in Section 2.3.1. The derivation is similar to what we have done in
Section 2.5.1 for the 1st-order EFEs. However, there is a subtlety worth being emphasized.
That is, the scalar-vector-tensor decomposition does not hold in 2nd order. For example,
1st-order scalar perturbations can induce 2nd-order vector and tensor perturbations. This is
because of the existing cross terms of two 1st-order perturbations in the 2nd-order EFEs.
3.1.1 Gauge Choices
We begin with the metric in Eq. (2.1) and expand it up to 2nd order, i.e.
ds2 = a2(η)
{
−(1+2Φ)dη2 +2BIdxIdη+[(1−2Ψ)δIJ +2HIJ]dxIdxJ
}
, (3.1)
where perturbations Φ, Ψ, BI and HIJ can be expanded in form of
A(η ,xI) =A[I](η ,xI)+ 1
2
A[II](η ,xI). (3.2)
With the decompositions ofBI andHIJ in Eq. (2.2), we fix the gauge freedom to Newtonian
gauge up to 2nd order so that
E = B = EI = 0. (3.3)
Different gauge choices are physically equivalent. We choose Newtonian gauge up to 2nd
order mainly because this allows us to cross-check the derivation of 2nd-order evolution
equations directly with other work in literature [52, 82, 87] which used Newtonian gauge
2It is possible to choose different gauges in different orders. See Section 2.1.1 for more discussions.
34 Cosmological Perturbations in Second Order
as well. However, different gauge choices might be different when we solve the evolution
equations numerically. It turns out that the 2nd-order EFEs in Newtonian gauge are numer-
ically unstable but the instability can be tackled by some tricks (see Section 3.3.3 for more
details).
Moreover, we assume no 1st-order vector and tensor perturbations, i.e. B[I]I = H
[I]
IJ =
0. Any initial vector modes will decay due to the expansion of our universe unless there
exist supports in the matters, such as cosmological defects [104], to continuously source
the vector modes. Hence, vector perturbations are usually negligible. Recently, BICEP
team detects B-mode polarization[2] which may originate from the primordial gravitational
waves. If it is confirmed, the 1st-order tensor perturbations should be taken into account in
our calculation. However, the 2nd-order vector and tensor perturbations are generally non-
negligible because they can be generated by the 1st-order scalar perturbations nonlinearly.
3.1.2 Multipole Decomposition of Stress-Energy Tensor
To derive the 2nd-order EFEs, we need to calculate the stress-energy tensors of different
components. For relativistic components like neutrinos and photons, we study their distri-
bution functions and corresponding multipoles through the Boltzmann equation, instead of
their stress-energy tensors through the continuity and Euler equations. That means we have
to relate the stress-energy tensor with the multipoles to elaborate the EFEs.
While the EFEs and thus the stress-energy tensors are constructed under the coordinate
system (η , xI) of the background manifold, the multipoles in the Boltzmann equation are
naturally expressed under the tetrad basis (eˆa) defined in the tangent planes of the physi-
cal manifold. When we relate the stress-energy tensor with the corresponding multipoles,
we work on the background manifold consistently and pull the tetrad basis back onto the
background manifold. Since the pulled-back tetrad basis does not necessarily align with the
basis of the coordinate system (η , xI), we have to take into account the perturbations of the
pulled-back tetrad3. In other words, we need to establish the following relations
TAB←→ T ab←→ Iˆℓm. (3.4)
For relativistic component with an intensity distribution I(η ,x, p0, nˆ), the stress-energy
3Details of the pullback are discussed in Section 2.1.2.
3.1 Second-order Einstein Field Equations 35
tensor can be expressed as
T µν =
1
(2π)3
∫
dnˆ
∫
p0dp0I(η ,x, p0, nˆ)pµ pν . (3.5)
The normalized energy-integrated photon intensity defined in Eq. (2.14) can be expanded
into multipoles of spherical harmonics
Iˆ(nˆ) =∑
ℓm
1
Nℓ
IˆℓmYℓm(nˆ), (3.6)
where Nℓ is defined in Eq. (C.2). First of all, we relate the stress-energy tensor with its
corresponding multipoles, both in tetrad basis. By using the fact that pi = p0ni in tetrad
basis as shown in Eq. (2.5) and the decomposition developed in Appendix C.3, we have4
T 00 =−T ii =−ρ¯ Iˆ00, (3.7)
e¯∗i(m)T
i
0 =−
i
3
ρ¯ Iˆ1m, (3.8)
[
nin j
]
2m
(
T i j−
1
3
δ i jT kk
)
= 2
15
ρ¯ Iˆ2m, (3.9)
where ρ¯ = I¯(η) is the background-order energy density of the radiation.
Now, we are ready to pull stress-energy tensors back onto the background manifold
using the transformation of tetrad basis discussed in Section 2.1.2, i.e.
T ab = TAB ξ eaA ξ ebB, (3.10)
where we simply write ξT as T for the pulled-back stress-energy tensors and the perturba-
tions of ξ e
a
A and ξ eb
B can be found in Eq. (D.2) to (D.12). In 2nd order, we obtain
T 0
[II]
0 = TO
[II]
O =−δρ [II]−2(ρ¯+ P¯)υ ,Iυ,I, (3.11)
T i
[II]
j = T I
[II]
J = π I
[II]
J +δP[II]δ IJ +2(ρ¯+ P¯)υ ,Iυ,J−4Ψπ IJ, (3.12)
T i
[II]
0 = T I
[II]
O − (ρ¯+ P¯)BI[II] +2(ρ¯+ P¯)(Φ+Ψ)υ ,I
=− (ρ¯+ P¯)(∂ Iυ [II] +υ I[II] +BI[II]−2Ψυ ,I)−2(δρ+δP)υ ,I−2π IJυ ,J
=− (ρ¯+ P¯)υ i[II]−2(δρ+δP)υ ,I−2π IJυ ,J, (3.13)
4In tetrad basis, the upper and lower indices for spatial components are related trivially by δi j and δ i j.
Thus, we treat them as the same set of indices.
36 Cosmological Perturbations in Second Order
where the fluid quantities defined in Eq. (2.26) are expanded up to 2nd order. i.e.
ρ = ρ¯+δρ [I] + 1
2
δρ [II], (3.14)
P = P¯+δP [I] + 1
2
δP [II], (3.15)
uA = a δOA+u[I]A +
1
2
u[II]A , (3.16)
πAB = π [I]AB+
1
2
π [II]AB , (3.17)
with the overhead bar denoting the background quantities and the superscript O denoting
the index corresponding to the conformal time η .
3.1.3 Einstein Field Equations
With the metric defined and the gauge fixed, the 2nd-order Einstein tensor can be calculated
straightforwardly by expanding Eq. (2.23). For the stress-energy tensor, we define the sound
speed cs of a fluid as
[N]c
2
s ≡
δP [N]
δρ [N]
, (3.18)
where [N] indicates the perturbation order. The reference frame is chosen so that the ob-
server’s velocity is orthonormal to the hypersurfaces of constant time (see Appendix D.1 for
the explicit definition of the reference frame). Using the conditions in Eq. (2.27), we find
that, for all components,5
uO[II] = 1
a
(
−Φ[II] +3ΦΦ+∂Iυ∂ Iυ
)
, (3.19)
πOO[II] = 0, π IO[II] = 2π IJ∂Jυ , π I
[II]
I = 0, (3.20)
where the scalar mode υ is defined in Eq. (2.28).
By substituting all these perturbations into Eq. (2.22), we can obtain the 2nd-order
EFEs. Before doing so, we can expect that the 2nd-order EFEs must be in form of
DX [X [II], X¯ ] = CX [X [I], X¯ ], (3.21)
5Henceforth, we omit the superscripts [I] denoting the 1st-order perturbations whenever it is clear in the
context.
3.1 Second-order Einstein Field Equations 37
where D is a linear differential operator on any 2nd-order perturbations X [II] and the op-
erator C contains only terms with products of two 1st-order perturbations. Clearly, these
are the only two ways to obtain terms in 2nd order. In particular, the linear operator D has
exactly the same form as the corresponding equations in 1st order. Mathematically, we can
consider the 2nd-order equations as differential equations sourced by the cross terms in the
operator C (or the inhomogeneous part).
Moreover, it is more convenient to express the perturbations in Fourier space by the
transformation
X [N](η ,x) =
∫
d3k
(2π)3/2 e
ik·xX [N](η ,k). (3.22)
In particular, we define the kernel X [II](η ,k1,k2) of any 2nd-order perturbations as
X [II](η ,k)≡
∫
d3k1d3k2
(2π)3/2 δ (k−k1−k2)X
[II](η ,k1,k2), (3.23)
and the convolution operator T as
Tk1,k2(k){. . .} ≡
∫
d3k1d3k2
(2π)3/2 δ (k−k1−k2) . . . . (3.24)
Scalar Modes
There are four EFEs for the 2nd-order scalar perturbations:6
Time-time: GOO
3HΨ˙+ k2Ψ+3H2Φ+ κa
2
2
( ∑
X=b,c
ρ¯XδX + ∑
X=n,r
ρ¯X IˆX ,00) =Tk1,k2(k){C1(η ,k1,k2)} ,
(3.25)
Trace space-space: GI I
Ψ¨+2HΨ˙+HΦ˙+(H2 +2H˙)Φ+ 1
3
k2(Ψ−Φ)
−
κa2
2 ∑X=n,r
[II]c
2
s,X ρ¯X IˆX ,00 =Tk1,k2(k){C2(η ,k1,k2)} , (3.26)
6For simplicity, we omit the label [II] for the 2nd-order quantities.
38 Cosmological Perturbations in Second Order
Traceless space-space:
(
3△−2∂ J∂I−△−1δ JI
)(
3GIJ−δ IJ GKK
)
Ψ−Φ−
κa2
5k2 ∑X=n,r
ρ¯X IˆX ,20 =Tk1,k2(k){C3(η ,k1,k2)} , (3.27)
Time-space: △−1∂ IGOI
Ψ˙+HΦ+ κa
2
2 ∑X=b,c
ρ¯X(1+ωX)υX −
κa2
6k ∑X=n,r
ρ¯X IˆX ,10 =Tk1,k2(k){C4(η ,k1,k2)} ,
(3.28)
where △ ≡ ∂ I∂I , the dots over symbols denote derivatives with respect to the conformal
time η , δ ≡ δρ/ρ¯ and the subscripts b, c, n and r represent baryons, CDM, neutrinos and
photons respectively. Two of the four EFEs are redundant and can be used for consistency
checks. We will discuss this further in Section 3.3.3. Cis for i = 1 to 4 contain only cross
terms of two 1st-order perturbations and are defined as7
C1 ≡−
[
4(k21 + k22)+3k1·k2
]
Ψ1Ψ2 +12H2Φ1Φ2 +12H(Φ1−Ψ1)Ψ˙2
+3Ψ˙1Ψ˙2 +κa2k1·k2 ∑
X=b,c
ρ¯X(1+ωX)υX ,1υX ,2, (3.29)
C2 ≡−
[
4
3
(k21 + k22)+k1·k2
]
Ψ1Ψ2 +
(
4H2 +8H˙− 2
3
k1·k2
)
Φ1Φ2− Ψ˙1Ψ˙2
+2Φ˙1Ψ˙2 +8HΦ1Φ˙2 +
(
8HΨ˙1 +4Ψ¨1−
4
3
k21Φ1
)
(Φ2−Ψ2)
−
2
3
k1·k2Φ1Ψ2−
κa2
3
k1·k2 ∑
X=b,c
ρ¯X(1+ωX)υX ,1υX ,2, (3.30)
C3 ≡−4Ψ1Ψ2−2Φ1Φ2
−
1
k2
[
k1·k2(Φ1 +Ψ1)(Φ2 +Ψ2)+2k22Ψ1Φ2
+κa2k1·k2 ∑
X=b,c
ρ¯X(1+ωX)υX ,1υX ,2
]
+ 3
k2
[
(kˆ·k1)(kˆ·k2)(Φ1 +Ψ1)(Φ2 +Ψ2)+2(kˆ·k2)2Ψ1Φ2
+κa2(kˆ·k1)(kˆ·k2) ∑
X=b,c
ρ¯X(1+ωX)υX ,1υX ,2
]
, (3.31)
7Again, we remove the superscript [1] from the 1st-order perturbations, e.g. Φ[I] is replaced by simply Φ.
3.1 Second-order Einstein Field Equations 39
C4 ≡ 4HΦ1Φ2−4Ψ1Ψ˙2 +2
k·k2
k2
Ψ˙1Φ2 +4
k·k2
k2
Φ1Ψ˙2
+ k·k2
k2
κa2 ∑
X=b,c
ρ¯X
[
(1+ωX)(Φ1 +Ψ1)−
(
1+ [I]c2s,X
)
δX ,1
]
υX ,2
−
κa2
3k ∑X=n,r
ρ¯X(Φ1 +Ψ1)IˆX ,10(k2), (3.32)
where kˆ ≡ k/k and the subscripts 1 and 2 on the 1st-order quantities indicate the k1- and
k2-dependence, e.g. Φ1 =Φ(k1). In Eq. (3.32), we make use of the 1st-order relation that
Iˆ1m(k′) = 3(1+ω)k′(m)υ(k′), (3.33)
with k′(0) =−kˆ ·k′. Throughout the thesis, we align z-axis with the direction of−k whenever
the decomposition of spherical harmonics is performed, i.e. k(±1) = 0 and k(0) = −k. To
derive Eq. (3.27), it is useful to know that the inverse
(
3∂ I∂J−△δ IJ
)−1 = 1
2
△−2∂ J∂I−
1
6
△−1δ JI, (3.34)
which corresponds to, in Fourier space,
(−3kIkJ + k2δ IJ)−1 =−
1
2k4
kJkI +
1
6k2
δ JI. (3.35)
Alternatively, the time-space EFE in Eq. (3.28) can be obtained by△−1∂IGIO such that
Ψ˙+HΦ+ κa
2
2 ∑X=b,c
ρ¯X(1+ωX)υX −
κa2
6k ∑X=n,r
ρ¯X IˆX ,10 =Tk1,k2(k)
{
C ′4(η ,k1,k2)
}
,
(3.36)
where
C ′4 ≡
k·k2
k2
[
4H(Φ1−Ψ1)Φ2 +2Ψ˙1Φ2−4Ψ˙1Ψ2−8Ψ1Ψ˙2
]
−
k·k2
k2
κa2 ∑
X=b,c
ρ¯X
[
(1+ωX)(Φ1 +Ψ1)+
(
1+ [I]c2s,X
)
δX ,1
]
υX ,2
+ κa
2
3k ∑X=n,r
ρ¯X(Φ1 +Ψ1)IˆX ,10(k2). (3.37)
It can be shown that Eq. (3.32) and Eq. (3.37) are equivalent. Throughout this thesis, we
40 Cosmological Perturbations in Second Order
assume that c2s = 0 and πAB = 0 for baryons8 and CDM, which are valid for the cosmological
scales we work on. Furthermore, we assume that c2s = 1/3 for neutrinos and photons.
Vector Modes
There are two sets of EFEs for the two divergenceless vector perturbations in 2nd order:
Time-space:
(
H2−H˙+ k
2
4
)
B˜(±1)+
κa2
2 ∑X=b,c
ρ¯X(1+ωX)υ˜X ,(±1)
+ κa
2
6 ∑X=n,r
ρ¯X IˆX ,1±1 =Tk1,k2(k)
{
C(1,±1)(η ,k1,k2)
}
, (3.38)
Space-space:
k
√
3
(
˙˜B(±1)+2HB˜(±1)
)
−
2κa2
15 ∑X=n,r
ρ¯X IˆX ,2±1 =Tk1,k2(k)
{
C(2,±1)(η ,k1,k2)
}
, (3.39)
where iB˜(±1) ≡ B(±1) = BI e¯∗I(±1), iυ˜(m) ≡ υ(m) = υ I e¯∗I(m) for m =−1,0,1and
C(1,±1) ≡ k1,(±1)
[
Φ1(2Ψ˙2−4HΨ2)−4Ψ˙1Ψ2
]
− k1,(±1)
{
κa2 ∑
X=b,c
ρ¯XυX ,1
[
(1+ωX)(Φ2 +Ψ2)+
(
1+ [I]c2s,X
)
δX ,2
]
}
−
κa2
3 ∑X=n,r
ρ¯X IˆX ,1±1(k1)(Φ2 +Ψ2)+
κa2
2 ∑X=b,c,n,r
ρ¯X(1+ωX)B˜(±1), (3.40)
C(2,±1) ≡ 2 [k1k2]2±1
[
Φ1Φ2−2Φ1Ψ2 +3Ψ1Ψ2− ∑
X=b,c
κa2ρ¯X (1+ωX)υX ,1υX ,2
]
+4 [k1k1]2±1 [Φ1Φ2−Φ1Ψ2 +2Ψ1Ψ2] , (3.41)
with k1,(±1) = kI1e¯∗I(±1) and [k′k′′]2m≡
[
nin j
]
2m k
′
I k
′′
J which are decomposed based on Eq. (C.31)
and Eq. (C.34) respectively. Similar to scalar modes, one set of the EFEs for vector modes
is redundant and can be used for consistency checks.
8Because of baryon pressure, the sound speed of baryons deviates from zero but the correction should be
tiny. See Section 5.6 of [71] for more details.
3.1 Second-order Einstein Field Equations 41
Tensor Modes
There is only one set of EFEs for the two tensor perturbations in 2nd order:
Space-space:
H¨(±2)+2HH˙(±2)+ k2H(±2)−
2κa2
15 ∑X=n,r
ρ¯X IˆX ,2±2 =Tk1,k2(k)
{
C(2,±2)(η ,k1,k2)
}
,
(3.42)
whereH(±2) ≡
[
nin j
]
2±2HIJ and
C(2,±2) ≡ 2 [k1k2]2±2
[
Φ1Φ2−2Φ1Ψ2 +3Ψ1Ψ2− ∑
X=b,c
κa2ρ¯X (1+ωX)υX ,1υX ,2
]
+4 [k1k1]2±2 [Φ1Φ2−Φ1Ψ2 +2Ψ1Ψ2] . (3.43)
For tensor modes, there is no redundant equation for consistency checks.
The EFEs derived above are used to determine the evolutions of the 2nd-order metric
perturbations (see the rectangular block in Fig. 3.1). In the next subsection, we derive the
continuity and Euler equations governing the energy density (δ ) and velocity (υ) of non-
relativistic fluids, e.g. baryons and CDM. In contrast, multipole moments are required to
completely describe the relativistic fluids, e.g. neutrinos and photons. We will study them
comprehensively in Section 3.2 when we formulate the Boltzmann equation.
3.1.4 Continuity and Euler Equations
Baryons and CDM are non-relativistic and have only the first two moments, i.e. the en-
ergy density and velocity. The corresponding evolution equations can be derived from the
conservation of stress–energy
∇νT
µν
X ≡ T
µν
X ,ν +Γ
µ
ανT
αν
X +ΓνανT
µα
X = C
µ
X , (3.44)
where X = b and c stand for baryons and CDM respectively, ∇ denotes the covariant deriva-
tive and C is the collision term. Since CDM is collisionless (i.e. Cµc = 0), its energy and
momentum are conserved. In contrast, baryons couple with photons through the Compton
scattering so we expect exchanges of energy and momentum between baryons and photons.
Working under the coordinate system (refer to Eq. (2.1)) of the background manifold,
42 Cosmological Perturbations in Second Order
we can derive the continuity equation for the 2nd-order energy density from the time com-
ponent of Eq. (3.44)9
CδX = δ˙ [II]X − k2υ
[II]
X −3Ψ˙
[II]
+Tk1,k2(k)
{
−6(2Ψ1 +δX ,1)Ψ˙2−2(k·k2)δX ,1υX ,2−2k22(Φ1 +Ψ1)υX ,2
+2k1·k2(2Ψ1−2Φ1−HυX ,1−2υ˙X ,1)υX ,2
}
, (3.45)
where the collision term CδX vanishes for CDM and is expressed as in Eq. (3.91) for baryons.
Similarly, we obtain the Euler equations for the 2nd-order scalar and vector velocities
(i.e. υ [II] and υ [II](±1) respectively) from the spatial components of Eq. (3.44)
10
CυX = kυ˙ [II]X +Hkυ
[II]
X + kΦ[II] (3.46)
+2Tk1,k2(k)
{
kˆ·k2
[
(δX ,1−Φ1 +Ψ1)υ˙X ,2 +δX ,1Φ2 +2(Ψ1−Φ1)Φ2
− (k1·k2)υX ,1υX ,2 +H(δX ,1−Φ1 +Ψ1)υX ,2− Ψ˙1υX ,2
]
}
,
CυX ,(±1) = ˙˜υ
[II]
X ,(±1)+Hυ˜
[II]
X ,(±1) (3.47)
+2Tk1,k2(k)
{
k2,(±1)
[
(δX ,1−Φ1 +Ψ1)υ˙X ,2 +δX ,1Φ2 +2(Ψ1−Φ1)Φ2
− (k1·k2)υX ,1υX ,2 +H(δX ,1−Φ1 +Ψ1)υX ,2− Ψ˙1υX ,2
]
}
,
where the collision term CυX vanishes for CDM and is shown as in Eq. (3.92) and Eq. (3.93)
for baryons. The discussion about the collision part of baryons will be postponed to Sec-
tion 3.2.3 after we study the Compton scattering through the Boltzmann equation in Sec-
tion 3.2.2. In the next section, we will develop the evolution equations for the multipoles of
neutrinos and photons using the Boltzmann equation.
3.2 Second-order Boltzmann Equation
Similar to the case in 1st order, the 2nd-order Boltzmann equation is needed for two main
purposes. First, we derive the governing equations of the 2nd-order multipoles from the
9Without confusion, we omit the notations for the 1st-order perturbations.
10For scalar mode, we elaborate△−1∂I
(
∇AT IAX
)
under the coordinate system in Eq. (2.1).
3.2 Second-order Boltzmann Equation 43
Boltzmann equation and use these equations to solve the evolution of the multipoles of neu-
trinos and photons11. Second, we construct the evolution equation of the photon intensity
matrix from the Boltzmann equation and perform the line-of-sight approach to calculate the
2nd-order CMB anisotropies during recombination. We will focus on the former in this
section and discuss the latter in Section 4.2. To begin, we classify terms in the 2nd-order
Boltzmann equation into four categories
L[II] +L[I,I] = C[II] +C[I,I] (3.48)
where L is the Liouville operator, C is the collision operator, the superscripts [II] and [I,I]
denote linear terms of purely 2nd-order perturbations and quadratic terms of two 1st-order
perturbations respectively. The Liouville operator conserves the phase-space distribution
function along the trajectories if there are no collisions while the collision operator specifies
how photons and electrons interact through the Compton scattering.
There are some subtleties when we study the 2nd-order Boltzmann equation. The
quadratic terms in Eq. (3.48) contain couplings between two multipoles. The Wigner 3-j
symbols or Clebsch–Gordan coefficients are thus required when we decompose multipoles
of spherical harmonics in the Boltzmann equation12. Moreover, when we solve the multi-
poles and perform the line-of-sight integration in Boltzmann equation, we choose the con-
formal time η as the unit of time. Thus, it is more convenient to redefine the Liouville and
collision operator such that
L→
(
dη
ds
)
L, C→
(
dη
ds
)
C. (3.49)
This transformation is equivalent to the transformation with simply the background-order
(dη/ds)[0]. When we consider Nth-order Boltzmann equation, (dη/ds)[L] will multiply
with (N−L)th-order Boltzmann equation, which vanishes[83] due to the lower-order Boltz-
mann equation. From now on, we will adopt this transformation in the derivation of the 2nd-
order Boltzmann equation. Throughout this section, we will derive the Boltzmann equation
in the tetrad basis.
11For neutrinos, the Boltzmann equation is collisionless and we are interested in its intensity only.
12Some useful formulae for the multipole decomposition can be found in Appendix C.
44 Cosmological Perturbations in Second Order
3.2.1 Liouville Operator
The Liouville (free-streaming) operator acting on the intensity matrixPab can be expressed
as13
L[Pab] = SacSbd
(
dxA
dη ∇APcd +
dp0
dη
∂
∂ p0Pcd +
dni
dη
∂
∂niPcd
)
, (3.50)
where the covariant derivative ∇A can be elaborated by using Eq. (2.35) and Sab is the screen
projector defined in Eq. (2.6). In Tetrad basis, the screen projector contains no perturbations
and is written as
Sab = ηab+δ 0aδ 0b−nanb, (3.51)
where the Minkowski metric ηab ≡ diag(−1,1,1,1), n0 = 0 and nini = 1.
Before we continue the derivation, we want to emphasize one thing. When we perform
the perturbation expansion, we decompose the photon momentum p into p0 and ni under the
tetrad basis as the momentum lives on the tangent plane. Similarly, the screen projector is
represented in tetrad basis, i.e. Si j instead of SIJ . In contrast, we express everything else in
the coordinate system {xA} of the background manifold, such as metric perturbations (BJ ,
HIJ , ...) and the spatial derivatives (∂/∂xI). We do so because of the following reason:
The photon momentum p at a point of the physical manifold is measured as the quanti-
ties p0 and ni under a chosen basis (tetrad basis in our case) of the tangent plane. These
quantities are unperturbed. On the other hand, the metric perturbations are expanded on
the background manifold. We perform all the calculations consistently on the background
manifold and thus we need to pull back the tetrad fields eˆa onto the background manifold
as shown in Eq. (2.7). Due to the existing perturbations, we do not expect the pulled-back
tetrads ξ eˆa to be perfectly aligned with the natural background basis e¯A (see Fig. 2.1 for an
illustration). Therefore, ξ eˆa
A, which are the coefficients of the pulled-back tetrads under
the natural background basis e¯A, are perturbative. After the perturbation expansion, implicit
summations Si j∂J and ni∂I stand for ξ e¯ jJSi j∂J and ξ e¯iIni∂I with the alignment ξ e¯a = e¯A.
To expand the 2nd-order Liouville terms, we need to solve the geodesic equation Eq. (2.37)
up to 2nd-order and obtain the following useful terms14
13Note that we multiply the Boltzmann equation with the Jacobian ds/dη .
14We ignore both vector and tensor perturbations in 1st order.
3.2 Second-order Boltzmann Equation 45
Background order:
(
dxI
dη
)[0]
= ni, (3.52)
(
dni
dη
)[0]
= 0, (3.53)
(
dp0
dη
)[0]
=−Hp0, (3.54)
1st order:
(
dxI
dη
)[I]
= ni
(
Ψ[I]+Φ[I]
)
, (3.55)
(
dni
dη
)[I]
=−Si j∂J
(
Ψ[I]+Φ[I]
)
, (3.56)
(
dp0
dη
)[I]
= p0
(
Ψ˙[I]−ni∂IΦ[I]
)
, (3.57)
2nd order:
(
dp0
dη
)[II]
= p0
[
Ψ˙[II]−ni∂IΦ[II]+
(
∂IB[II]J −H˙
[II]
IJ
)
nin j
+2
(
Φ[I]−Ψ[I]
)
ni∂IΦ[I]+4Ψ[I]Ψ˙[I]
]
. (3.58)
Using Eq. (2.36), we can separate Liouville parts of photon intensity and polarizations
and expand them in 2nd order such that
∂I[II]
∂η +
(
dxI
dη
)[0] ∂I[II]
∂xI +
(
dp0
dη
)[0] ∂I[II]
∂ p0 +
(
dp0
dη
)[II] ∂ I¯
∂ p0
+2
(
dp0
dη
)[I] ∂I[I]
∂ p0 +2
(
dxI
dη
)[I] ∂I[I]
∂xI +2
(
dni
dη
)[I] ∂I[I]
∂ni , (3.59)
(∇ηPab)[II] +
(
dxI
dη
)[0] ∂P [II]ab
∂xI +
(
dp0
dη
)[0] ∂P [II]ab
∂ p0
+2
(
dp0
dη
)[I] ∂P [I]ab
∂ p0 +2
(
dxI
dη
)[I] ∂P [I]ab
∂xI +2Sa
cSb
d
(
dni
dη
)[I] ∂P [I]cd
∂ni , (3.60)
where I¯ = I¯(η , p0) and P¯ab = 0 are homogeneous and isotropic. The covariant derivative
of polarizations in the first term of Eq. (3.60) can be expanded using Eq. (2.35). We do
46 Cosmological Perturbations in Second Order
not consider the terms with the Christoffel symbols in this thesis. However, they should
be taken into account in the future. We can interpret physically the terms in Eq. (3.59) as
follows. The first two terms in the first line corresponds to the 2nd-order photon intensity
propagating through the spacetime on the background metric. The third term is responsible
to the redshifting due to the Hubble expansion while the forth term is responsible to the
redshifting (nonlinear Sachs-Wolfe and integrated Sachs-Wolfe effects) due to the 2nd-order
metric perturbations on the background-order photon intensity. Similarly, the first term in
the second line is the redshifting (Sachs-Wolfe and integrated Sachs-Wolfe effects) of the
1st-order metric perturbations on the 1st-order photon intensity. The second term generates
to time delay effect while the third one generates lensing effect. Similar interpretation works
for polarization in Eq. (3.60). We will discuss these effects, especially the lensing effect, in
greater details in Chapter 5.
By using the decomposition of spherical harmonics in Appendix C.3 and the coefficients
in Appendix D, we can derive the 2nd-order Liouville operators of different multipoles as
follows:
Pure Second-order Liouville Operator L[II]
The pure 2nd-order Liouville operators have exactly the same form as the 1st-order ones
without neglecting the 1st-order vector and tensor perturbations
L[II]ℓm [Iˆ] =
˙ˆIℓm+ k
(
0U
0
ℓmIˆℓ+1,m− 0D
0
ℓmIˆℓ−1,m
)
−4δ 0ℓ δ
0
mΨ˙−4δ 1ℓ δ 0mkΦ
+ 4k√
3
δ 2ℓ
(
δ 1mB˜(1)+δ−1m B˜(−1)
)
+4δ 2ℓ
(
δ 2mH˙(2)+δ−2m H˙(−2)
)
, (3.61)
L[II]ℓm [Eˆ] =
˙ˆEℓm+ k
(
2U
0
ℓmEˆℓ+1,m− 2D
0
ℓmEˆℓ−1,m− 2T
0
ℓmBˆℓm
)
, (3.62)
L[II]ℓm [Bˆ] =
˙ˆBℓm+ k
(
2U
0
ℓmBˆℓ+1,m− 2D
0
ℓmBˆℓ−1,m+ 2T 0ℓmEˆℓm
)
, (3.63)
where the coefficients sDnℓm, sT
n
ℓm and sU
n
ℓm are defined in Appendix C.3.
Quadratic Second-order Liouville Operator L[I,I](k)
We define the kernel operator L[I,I](k1,k2) as
L[I,I][Xˆ ](k)≡Tk1,k2(k)
{
L[I,I][Xˆ ](k1,k2)
}
, (3.64)
3.2 Second-order Boltzmann Equation 47
where Xˆ stands for Iˆ, Eˆ and Bˆ. The quadratic 2nd-order Liouville operators for photon
intensity and polarizations are
L[I,I]ℓm [Iˆ] =−8
1
∑
n=−1
k2,(n)
(
0U
n
ℓmIˆ1,ℓ+1,m−n− 0D
n
ℓmIˆ1,ℓ−1,m−n
)
Φ2
−8
[
Iˆ1,ℓmΨ˙2 +2δ 0ℓΨ1Ψ˙2 +δ 1ℓ k1,(m)Φ1(Φ2−Ψ2)
]
+L[I,I]T,ℓm[Iˆ]+L
[I,I]
L,ℓm[Iˆ], (3.65)
L[I,I]ℓm [Eˆ] =−8
1
∑
n=−1
k2,(n)
(
2U
n
ℓmEˆ1,ℓ+1,m−n− 2D
n
ℓmEˆ1,ℓ−1,m−n
)
Φ2
−8Eˆ1,ℓmΨ˙2 +L[I,I]T,ℓm[Eˆ]+L
[I,I]
L,ℓm[Eˆ], (3.66)
L[I,I]ℓm [Bˆ] = 8
1
∑
n=−1
k2,(n) 2T
n
ℓmEˆ1,ℓ,m−nΦ2 +L[I,I]T,ℓm[Bˆ]+L
[I,I]
L,ℓm[Bˆ], (3.67)
where
L[I,I]T,ℓm[Iˆ] =−2
1
∑
n=−1
k1,(n)
(
0U
n
ℓmIˆ1,ℓ+1,m−n− 0D
n
ℓmIˆ1,ℓ−1,m−n
)
(Φ2 +Ψ2), (3.68)
L[I,I]L,ℓm[Iˆ] = 2
1
∑
n=−1
k2,(n)
[
0U
n
ℓm(ℓ+2)Iˆ1,ℓ+1,m−n+ 0Dnℓm(ℓ−1)Iˆ1,ℓ−1,m−n
]
(Φ2 +Ψ2),
(3.69)
L[I,I]T,ℓm[Eˆ] =−2
1
∑
n=−1
k1,(n)
(
2U
n
ℓmEˆ1,ℓ+1,m−n− 2D
n
ℓmEˆ1,ℓ−1,m−n
)
(Φ2 +Ψ2), (3.70)
L[I,I]L,ℓm[Eˆ] = 2
1
∑
n=−1
k2,(n)
[
2U
n
ℓm(ℓ+2)Eˆ1,ℓ+1,m−n+ 2Dnℓm(ℓ−1)Eˆ1,ℓ−1,m−n
]
(Φ2 +Ψ2),
(3.71)
L[I,I]T,ℓm[Bˆ] = 2
1
∑
n=−1
k1,(n) 2T
n
ℓmEˆ1,ℓ,m−n(Φ2 +Ψ2), (3.72)
L[I,I]L,ℓm[Bˆ] =−2
1
∑
n=−1
k2,(n) 2T
n
ℓmEˆ1,ℓ,m−n(Φ2 +Ψ2), (3.73)
with k1,(n) = kI1e¯∗I(n), k2,(n) = kI2e¯∗I(n) and the subscript 1 on Iˆ, Eˆ and Bˆ denoting the k1-
dependence, e.g. Iˆ1,ℓm = Iˆℓm(k1). Since we align the azimuthal direction of the multipoles
with the direction of -k, we need to rotate multipoles of any 1st-order quantity X corre-
spondingly such that
X [I]ℓm(k′) =
√
4π
2ℓ+1Y
∗
ℓm(k′)X [I]ℓ0 (k′) (3.74)
48 Cosmological Perturbations in Second Order
where k′ stands for k1 or k2 and X
[I]
ℓ0 (k′) is calculated with the azimuthal direction aligning
with the direction of -k′15.
3.2.2 Collision Operator
To complete the derivation of the 2nd-order Boltzmann equation, we need to study the
Compton scattering between baryons and photons
γ(p)+ e(q)←→ γ(p′)+ e(q′), (3.75)
where a photon with momentum p scatters with an electron with momentum q and the
outcome is a photon with momentum p′ and an electron with momentum q′. We note that
photons interact with protons as well through Compton scattering. However, the mass ratio
of electron to proton (i.e. me/mp ∼ 10−3) suppresses this type of scattering. Therefore,
we can safely ignore protons in the collision operator. Furthermore, the strong Coulomb
interaction between electrons and protons allows us to treat them as a single fluid. We will
only list out the 2nd-order collision terms in the following and refer to [15, 83] for detailed
derivations and discussions.
Pure Second-Order Collision Operator C[II]
Again, the pure 2nd-order collision terms have the same forms as the corresponding colli-
sion terms in 1st-order
C[II]ℓm [Iˆ] = ˙¯τ
(
−Iˆℓm+δ 0ℓ δ 0mIˆ00 +4δ 1ℓ υ˜b,(m)+δ 2ℓ Πˆm
)
, (3.76)
C[II]ℓm [Eˆ] = ˙¯τ
(
−Eˆℓm−
√
6δ 2ℓ Πˆm
)
, (3.77)
C[II]ℓm [Bˆ] =− ˙¯τBˆℓm, (3.78)
where ˙¯τ(η) is the differential optical depth in the background order, iυ˜(m) = υ I e¯∗I(m) and
Πˆm ≡
1
10
(
Iˆ2m−
√
6Eˆ2m
)
. (3.79)
15As mentioned in Section 2.5.2, only m = 0 multipoles of 1st-order quantities are needed.
3.2 Second-order Boltzmann Equation 49
Quadratic Second-order Collision Operator C[I,I](k)
We define the kernel operator C[I,I](k1,k2) as
C[I,I][Xˆ ](k)≡Tk1,k2(k)
{
C[I,I][Xˆ ](k1,k2)
}
, (3.80)
where Xˆ stands for Iˆ, Eˆ and Bˆ. The quadratic 2nd-order collision operators for photon
intensity and polarizations become
C[I,I]ℓm [Iˆ] = 2 ˙¯τ
[
1
∑
n=−1
(
−0U
n
ℓmIˆ1,l+1,m−n+ 0DnℓmIˆ1,l−1,m−n
)
k2,(n)υb,2
+δ 0ℓ
(
−
4
3
k1·k2υb,1 +2
1
∑
n=−1
0U
n
00Iˆ1,1,−n k2,(n)
)
υb,2
+3δ 1ℓ
1
∑
n=−1
0D
n
1nIˆ1,0,m−n k2,(n)υb,2
+δ 2ℓ
1
∑
n=−1
0D
n
2m
(
−
1
2
Iˆ1,1,m−n+7k1,(m−n)υb,1
)
k2,(n)υb,2
+5δ 3ℓ
1
∑
n=−1
0D
n
3mΠˆ1,m−n k2,(n)υb,2
]
+2(δ τ˙1 +Φ1)C2,ℓm[Iˆ], (3.81)
C[I,I]ℓm [Eˆ] = 2 ˙¯τ
[
1
∑
n=−1
(
−2U
n
ℓmEˆ1,ℓ+1,m−n+ 2DnℓmEˆ1,ℓ−1,m−n
)
k2,(n)υb,2
+
√
6δ 2ℓ
1
∑
n=−1
0D
n
2m
(
1
2
Iˆ1,1,m−n− k1,(m−n)υb,1
)
k2,(n)υb,2
−5
√
6δ 3ℓ
1
∑
n=−1
2D
n
3mΠˆ1,m−n k2,(n)υb,2
]
+2(δ τ˙1 +Φ1)C2,ℓm[Eˆ], (3.82)
C[I,I]ℓm [Bˆ] =−2 ˙¯τ
[
1
∑
n=−1
2T
n
ℓm
(
Eˆ1,ℓ,m−n−2
√
6δ 2ℓ Πˆ1,m−n
)
k2,(n)υb,2
]
, (3.83)
where δ τ˙ is the fraction of the 1st-order differential optical depth defined in Eq. (3.95).
It is worth pointing out that there are energy exchanges (the terms in the second line of
Eq. (3.81)) between electrons and photons in 2nd order, i.e. the Compton scattering is in-
elastic in 2nd order.
50 Cosmological Perturbations in Second Order
3.2.3 Collision Terms for Baryons
Here, we formulate the collision terms for baryons as shown in Eq. (3.45) to (3.47). Al-
though there can be exchanges of energy and momentum between baryons and photons
due to the Compton scattering, the stress-energy of baryons and photons as a whole has to
conserve, i.e.
∇µT
µν
b +∇µT
µν
r = 0, (3.84)
where the subscripts b and r denote baryons and photons respectively. Since the continuity
and Euler equations of baryons are derived under the coordinate system (η ,xI), the follow-
ing transformations are useful
a
(
∇AT
AO
)[II]
=
(
∇aT
a0)[II]−2Φ[I]
(
∇aT
a0)[I] , (3.85)
a
(
∇AT
AI
)[II]
=
(
∇aT
ai)[II] +2Ψ[I]
(
∇aT
ai)[I] . (3.86)
In tetrad basis, the covariant derivative of the stress-energy tensor of photons is related to
the collision term [93] by16
∇aT
ab
r =
1
(2π)3
∫
dnˆ
∫
p0dp0pbpaea
OC[I], (3.87)
where I = I(η ,x, p0, nˆ) is the intensity distribution and ea is defined in Section 2.1.2. In
second order, it can be shown that
1
I¯
(
∇aT
a0
r
)[II] = 1
a
(
C[II]00 [Iˆ]+C
[I,I]
00 [Iˆ]−2Φ[I]C
[I]
00[Iˆ]
)
, (3.88)
1
I¯
e¯∗i(m)
(
∇aT
ai
r
)[II] = i
3a
(
C[II]1m [Iˆ]+C
[I,I]
1m [Iˆ]−2Φ[I]C
[I]
1m[Iˆ]
)
, (3.89)
where
C[Iˆ]≡ 1
2π2I¯
∫
C[I](p0)3dp0. (3.90)
16We have used the redefinition in Eq. (3.49) so that our C[I] is equal to pOC[ f ] in [93].
3.2 Second-order Boltzmann Equation 51
With the equations above, we can find that the collision terms are
Cδb ≡−2R ˙¯τTk1,k2(k)
{
1
∑
n=−1
(−1)n
4
Iˆ1,1nk2,(n)υb,2− (k1·k2)υb,1υb,2
}
, (3.91)
Cυb ≡−R ˙¯τ
(
Iˆ[II]10
4
− υ˜ [II]b,(0)
)
−2R ˙¯τTk1,k2(k)
{
(δ τ˙1 +Ψ1)
(
Iˆ2,10
4
− υ˜b,2,(0)
)
− Iˆ1,00υ˜b,2,(0)+
1
4
1
∑
n=−1
0U
n
10Iˆ1,2,−nυ˜b,2,(n)
}
, (3.92)
Cυb,(m) ≡ R ˙¯τ
(
Iˆ[II]1m
4
− υ˜ [II]b,(m)
)
+2R ˙¯τTk1,k2(k)
{
(δ τ˙1 +Ψ1)
(
Iˆ2,1m
4
− υ˜b,2,(m)
)
− Iˆ1,00υ˜b,2,(m)+
1
4
1
∑
n=−1
0U
n
1mIˆ1,2,m−nυ˜b,2,(n)
}
, (3.93)
where m =±1, the density ratio R≡ (4/3)ρ¯r/ρ¯b, υ˜ [II]b,(0) =−kυb and υ˜b,2,(0) =−kˆ ·k2υb,2.
The formalism of the Boltzmann equation is almost finished. In the next subsection, we
will discuss the evolution equation of the 1st-order optical depth to complete the derivation.
3.2.4 Perturbed Optical Depth
In order to elaborate the 2nd-order Compton scattering, we need to know the evolution
of the optical depth up to 1st order as discussed in Section 3.2.2. Here, we will outline
the formalism of the evolution equation of the 1st-order optical depth17. For what we are
interested in, the perturbed optical depth is crucial for studying recombination era of the
CMB. However, it can also be used to study the 21cm absorption[68].
To start with, we expand the differential optical depth up to 1st order
τ˙ = ˙¯τ(1+δ τ˙ [I]). (3.94)
With the definition of differential optical depth in Eq. (2.57), its 1st-order perturbation can
be written as
δ τ˙ [I] = n
[I]
e
n¯e
+ x
[I]
e
x¯e
= δ [I]b +δx[I]e , (3.95)
17See [65, 76, 93] for more details.
52 Cosmological Perturbations in Second Order
where δx[I]e ≡ x[I]e /x¯e and the second equality holds due to the tight coupling between elec-
trons and baryons through Coulomb scattering. That means, we need to solve the fraction
of free electrons in 1st order. In principle, this can be done by formulating the Boltzmann
equation of free electrons. Practically, it is not straightforward at all because many atomic
transitions between many different states are involved during recombination. It was shown
in [78] that hydrogen atoms during recombination can be approximated as atoms with ef-
fectively 3 levels18, i.e. ground state, first excited state and continuum state.
In the background order, the fraction of free electrons satisfies
∂ x¯e
∂η =
a
n¯e
Q¯[n¯fe, n¯b, T¯M, T¯R, H¯], (3.96)
where the overhead bar denotes the background-order quantities, nfe = nexe is the number
density of free electrons, nb is the number density of total protons, TM is the kinetic tem-
perature of the electrons, TR is the radiation temperature and H is the physical-time Hubble
parameter. The function Q is defined as [93]
Q≡−
[
αB(TM)n2fe−βB(TR)(nb−nfe)e−B12/(kBTR)
]
CH, (3.97)
where kB is the Boltzmann constant and B12 ≡ E1−E2 with Ei as the ionization energy of
the i-th level. The function αB is defined as
αB(x)≡ 10−19FH
ayb
1+ cyd m
3s−1, (3.98)
where a = 4.309, b = −0.6166, c = 0.6703, d = 0.5300, y ≡ x/104K and the fudge factor
FH = 1.14 accounts for non-equilibrium populations of the excited states [88]. The function
βB is defined as
βB(TR)≡ αB(TR)e−E2/(kBTR)
(
2πmekBTR
h2P
)3/2
, (3.99)
where me is the electron mass and hP is the Planck constant. The function CH is
CH ≡ 1−
βB(TR)
3PA21 +Λ2S→1S+βB(TR)
, (3.100)
18The approximation can be extended to Helium atoms. However, we consider only Hydrogen throughout
this thesis.
3.2 Second-order Boltzmann Equation 53
where Λ2S→1S = 8.2245809s−1 and
3PA21 ≈
8πHν321
(nb−nfe)c3
, (3.101)
with ν21 as the frequency of the Lyα photons and c as the speed of light.
In 1st order, we have
∂
∂η δxe =
a
n¯fe
[
(Φ−δxe)Q¯+δQ
]
, (3.102)
where
Q¯+δQ = Q [n¯fe(1+δxe), n¯b(1+δb), T¯M +δTM, T¯R +δTR, H¯(1+δH)] , (3.103)
with
δTR =
Iˆ00
4
, δH =−Ψ−
δ˙b
3H
. (3.104)
If we are interested in recombination only, we can treat the temperatures of baryons and
photons as the same, i.e. TM = TR. The evolutions of the 1st-order differential optical depth
δ τ˙ [I] are shown in Fig. 3.2 for different scales. It can be shown [93] that the 1st-order
differential optical depth for large scales is well approximated by
δ τ˙ [I] ≈ δb
(
1−
x˙e
3Hxe
)
. (3.105)
We remark that the 1st-order optical depth here is obtained by perturbing the fitting
function of the background-order optical depth, which contains some fudge factors. This
will undermine the accuracy of the calculated optical depth in 1st order, especially for small
scales when the details of the underlying dynamics become important. For calculating the
CMB bispectrum, the problem can be ignored in the squeezed limit where the contribu-
tion from the 1st-order optical depth with small scales is suppressed. To understand this,
we note that the term in Boltzmann equation with the 1st-order optical depth is in form
of δ τ˙(k1)C(k2). In squeezed limit, the term is highly suppressed if k1 ≫ k2 because the
collision term C(k2) is negligible in large scales due to causality. However, this is not nec-
essarily true for non-squeezed configurations. Therefore, a more sophisticated calculation
for the perturbed optical depth should be considered in the future.
54 Cosmological Perturbations in Second Order
10
-5
10
-4
0.001 0.01 0.1
0.01
0.1
1
10
100
1000
a
∆
Τ
 
@
I
D
10
-5
10
-4
0.001 0.01 0.1
0.01
0.1
1
10
100
1000
a
∆
Τ
 
@
I
D
Figure 3.2 The graphs of transfer functions of the 1st-order differential optical depth δ τ˙ [I]
against the scale factor a for k = 2×10−5Mpc−1 (upper panel) and k = 0.045Mpc−1 (lower
panel). Here, we ignore Helium atoms.
3.3 Numerical Implementation 55
3.2.5 Truncation Scheme for the Boltzmann Hierarchy
In principle, we need infinite multipoles (ℓ→ ∞) for solving the Boltzmann equation but
this is numerically infeasible. Instead, we truncate the hierarchy and solve the Boltzmann
equation up to ℓmax. However, this will induce some errors. To mitigate the truncation error,
we exploit the following scheme at ℓ= ℓmax according to Appendix D of [87]
˙ˆIℓm = k
[√
ℓ+ |m|
ℓ−|m|
2ℓ+1
2ℓ−1 Iˆℓ−1,m−
ℓ+1+ |m|
kη Iˆℓm
]
, (3.106)
˙ˆEℓm+ i ˙ˆBℓm = k
[√
1−
m2
ℓ2
√
ℓ+2
ℓ−1
2ℓ+1
2ℓ−1(Eˆℓ−1,m+ iBˆℓ−1,m)
+
(
−
ℓ+3
kη + i
m
ℓ
)
(Eˆℓm+ iBˆℓm)
]
. (3.107)
3.3 Numerical Implementation
In Section 3.1 and 3.2, we have derived the 2nd-order EFEs (Eqs. (3.25)-(3.28), (3.38)-
(3.39) and (3.42)), continuity and Euler equations (Eqs. (3.45)-(3.47)) and Boltzmann equa-
tion (Eqs. (3.61)-(3.63), (3.65)-(3.67), (3.76)-(3.78) and (3.81)-(3.83)), which govern the
evolution of the 2nd-order perturbations. In this section, we will present the details of how
to solve these evolution equations numerically. First of all, we will set the initial conditions
of the 2nd-order perturbations in Section 3.3.1. Then, we generalize the TCA to 2nd order
in Section 3.3.2. Finally, we discuss the numerical stability and convergence checking in
Section 3.3.3.
Numerically, we solve the kernels of 2nd-order perturbations defined in Eq. (3.23) for
different values of k1, k2 and k1 ·k219. Moreover, we rewrite all the governing equations so
that they are symmetric on k1 and k2, i.e.
F(k1,k2)→
1
2
[F(k1,k2)+F(k2,k1)] , (3.108)
where F is any kernel depending on k1 and k2. With this symmetry, we only need to solve
the cases when k1 ≥ k2. For the decomposition of 2nd-order multipoles, we align the az-
imuthal direction with the direction of -k. However, unlike solving 1st-order perturbations,
19For our interest in the CMB bispectrum, other dependences of k1 and k2 can be integrated out. We will
discuss this further in Section 4.2.
56 Cosmological Perturbations in Second Order
we need to solve the multipoles with m ̸= 0 in 2nd order. As we will see in Section 4.2, mul-
tipoles with m up to 2 are enough when we calculate the CMB bispectrum using the line-
of-sight approach. However, the main contribution should come from the scalar (m = 0)
modes. Therefore, in our numerical calculation, we skip the 2nd-order vector and tensor
perturbations which correspond to m = 1 and m = 2 modes respectively20. We point out
that evolution equations of 2nd-order perturbations with different m modes are decoupled
and can be solved separately.
3.3.1 Initial Conditions
We have established all the governing equations of the 2nd-order perturbations in Section
3.1 and 3.2. To solve the equations, we need to fix the initial conditions of the perturbations
up to 2nd order. The derivation of 2nd-order initial conditions is similar to that in 1st order,
which is discussed in Section 2.5.3 and [71].
Non-Gaussianities can be generated during inflation as the primordial non-Gaussianities21
and after inflation through non-linear evolutions of the cosmological perturbations. The ini-
tial conditions in 2nd order will depend on what non-Gaussianities we consider. In this
section, we consider the latter only. In particular, we assume the single-field slow-roll in-
flation in standard Cosmology. That is, we assume there are negligible primordial non-
Gaussianities in the non-perturbative ζ [72]. The primordial non-Gaussianities of alterna-
tive inflationary models, if exist, evolve with the linear transfer function and their bispectra
can be added linearly afterwards.
With the single field, the initial conditions are adiabatic and satisfy Eq. (2.64) and
Eq. (2.65). In 2nd order, the energy densities and velocities of different components22
1
3
δ [II]b =
1
3
δ [II]c =
1
4
Iˆ[II]n,00−
(
1
4
Iˆn,00
)2
= 1
4
Iˆ[II]r,00−
(
1
4
Iˆr,00
)2
, (3.109)
−
1
4
Iˆ[II]n,10 =−
1
4
Iˆ[II]r,10 = kυ
[II]
b +2(kˆ ·k2)δb,1υb,2 = kυ [II]c +2(kˆ ·k2)δc,1υc,2. (3.110)
20It has been found [52] that the vector and tensor perturbations have negligible effects on CMB bispectrum.
21See [24] for a review on primordial non-Gaussianities.
22We express the initial conditions for the kernels of 2nd-order perturbations.
3.3 Numerical Implementation 57
Moreover, the 2nd-order comoving curvature perturbation can be written as23 [73, 87, 103]
R[II] = Ψ[II] + 2
3(1+ω)H
[
Ψ˙[II] +HΦ[II]−4HΦ2− Ψ˙
2
H
−4(Φ−Ψ)Ψ˙
]
+(1+3c2s )
[
δ
3(1+ω)
]2
+ 4Ψδ
3(1+ω) , (3.111)
where ω , cs and δ are the equation of state, the speed of sound and the perturbed fractional
energy density of the total fluid respectively. We consider the initial conformal time ηi such
that kηi ≪ 1. That is, the universe is initially deep in the radiation-dominated era so that
ω = c2s = 1/3 and ρ¯tot = ρ¯n + ρ¯r.
By ignoring Ψ˙ up to 2nd order and terms with higher powers of kηi, the initial kernels
of 2nd-order perturbations fulfill24
R[II] =Ψ[II] + 1
2
Φ[II]−2Φ2 + 1
8
δ 2 +Ψδ , (3.112)
C1 = 3H2Φ[II] +
κa2
2
(
ρ¯nIˆ[II]n,00 + ρ¯rIˆ
[II]
r,00
)
, (3.113)
C3 =Ψ[II]−Φ[II]−
κa2
5k2
(
ρ¯nIˆ[II]n,20 + ρ¯rIˆ
[II]
r,20
)
, (3.114)
C4 =HΦ[II]−
κa2
6k
(
ρ¯nIˆ[II]n,10 + ρ¯rIˆ
[II]
r,10
)
, (3.115)
˙ˆI[II]n,2m =
2k
3
Iˆ[II]n,1m+L
[I,I]
2m
[
Iˆn
]
, (3.116)
where C1, C3 and C4 are defined in Eq. (3.29), (3.31) and (3.32) respectively and L
[I,I]
2m
[
Iˆ
]
can be inferred by Eq. (3.65). Multipoles of neutrinos with ℓ ≥ 3 are highly suppressed
by kηi and thus negligible. Due to the tight coupling of Compton scattering, multipoles of
photons with ℓ ≥ 2 are neglected except the quadrupoles of photon intensity and E-mode
polarization
Iˆ[II]r,2m =
1
10
(
Iˆ[II]r,2m−
√
6Eˆ[II]2m
)
+C[I,I]2m
[
Iˆr
]
, (3.117)
Eˆ[II]2m =−
√
6
10
(
Iˆ[II]r,2m−
√
6Eˆ[II]2m
)
+C[I,I]2m
[
Eˆ
]
, (3.118)
where C[I,I]2m
[
Iˆ
]
and C[I,I]2m
[
Eˆ
]
can be deduced by (3.81) and (3.82) respectively. Using these
23Due to the nonlinear relation ofR to the non-perturbative ζ , there are 2nd-order contributions to the initial
value ofR. Hence, even with subsequent linear evolution only, there is a non-zero bispectrum.
24Here, we use the 2nd-order EFEs and Boltzmann equations derived in Section 3.1 and 3.2.
58 Cosmological Perturbations in Second Order
two equations, we can find that the quadrupoles of photons are
Iˆ[II]r,2m =
5
4
1
∑
n=−1
0D
n
2mIˆ1,1(m−n)Iˆ2,1n, (3.119)
Eˆ[II]2m = 0. (3.120)
Furthermore, the solutions of Eq. (3.112) to (3.116) are
Φ[II] = 10
15+4 fn
[
R[II] + fn
(
2ηi
5
C4 +
3
10k2ηi
L[I,I]20 [Iˆn]
)
−
3(1− fn)
5(kηi)2
Iˆ[II]r,20
−C3 +2Φ1Φ2−
1
8
δ1δ2−Ψ1δ2
]
, (3.121)
Ψ[II] =Φ[II] + 3
5(kηi)2
[
fnIˆ
[II]
n,20 +(1− fn)Iˆ
[II]
r,20
]
+C3, (3.122)
Iˆ[II]n,00 =−2Φ[II] +
2C1
3H2
, (3.123)
Iˆ[II]n,10 = 2kηi(Φ[II]−ηiC4), (3.124)
Iˆ[II]n,20 =
2
3
(kηi)2(Φ[II]−ηiC4)−
ηi
2
L[I,I]20
[
Iˆn
]
, (3.125)
whereR[II] ≈−2R2 [72].
3.3.2 Tight-coupling Approximation
We develop the 2nd-order code based on CAMB[70] and use DVERK for solving the gov-
erning equations up to 2nd order25. Since DVERK is a non-stiff ODE solver, it breaks down
when the evolution equations are stiff. Physically, this happens when baryons and photons
are tightly coupled at early time, i.e. ˙¯τ ≫ 1. To tackle this numerical instability, we rewrite
and solve the equations with the TCA at early time and then switch back to the full equations
afterwards. In this subsection, we formulate the TCA for 1st- and 2nd-order perturbations26.
The key idea of TCA is to expand the Boltzmann equations into powers of the inverse 1/ ˙¯τ
up to a certain order depending on the precision required.
We start with the 1st-order perturbations. For energy densities of baryons and photons,
we keep the corresponding evolution equations unchanged because the collision is elastic
25We develop our own code for solving background- and 1st-order perturbations as well. In 1st order, we
use Newtonian gauge instead of synchronous gauge in CAMB.
26More details can be found in [71, 84].
3.3 Numerical Implementation 59
and thus no terms with ˙¯τ are involved. That is, we have from Eq. (2.55) and Eq. (2.58)
δ˙b− k2υb−3Ψ˙= 0, (3.126)
˙ˆI00 +
k
3
Iˆ10−4Ψ˙= 0. (3.127)
For velocities of baryons and photons, we combine Eq. (2.56) with Eq. (2.58) and eliminate
their collision terms to obtain two exact equations27
υ˙b =
1
1+R
[
−Hυb−R
(
1
4
Iˆ00−
1
10
Iˆ20
)
+R
(
1
4k
˙ˆI10 + υ˙b
)]
−Φ, (3.128)
˙ˆI10 =
4k
R
(υ˙b +Hυb)+
4k(1+R)
R
Φ+ k
(
Iˆ00−
2
5
Iˆ20
)
, (3.129)
where we approximate Eq. (3.128) by expanding the term
1
4k
˙ˆI10 + υ˙b ≈
2RH
1+R
(
1
4k
Iˆ10 +υb
)
−
1
(1+R) ˙¯τ
[
a¨
a
υb−H
(
1
2
Iˆ00 +Φ
)
−
1
4
˙ˆI00
]
+O( ˙¯τ−2).
(3.130)
The quadrupoles of photons can be estimated by
Iˆ20 ≈
8k
9 ˙¯τ
Iˆ10, (3.131)
Eˆ20 ≈−
√
6
4
Iˆ20, (3.132)
while higher multipoles are neglected due to the suppression of tight coupling.
For 2nd-order perturbations, we apply the TCA up to the leading order which is enough
[84] for our purpose. The evolution equations of the first two moments are
0 = δ˙ [II]b − k2υ
[II]
b −3Ψ˙
[II] +L[I,I][δb], (3.133)
0 = ˙ˆI[II]00 +
k
3
Iˆ[II]10 −4Ψ˙
[II] +L[I,I]00 [Iˆ], (3.134)
υ˙ [II]b =
1
1+R
[
−Hυ [II]b −R
(
1
4
Iˆ[II]00 −
1
10
Iˆ[II]20
)
−2R
kˆ ·k2
k
(
δ˙1υ2 +δ1υ˙2
)
−
1
k
L[I,I][υb]+
R
4k
L[I,I]10 [Iˆ]
]
−Φ[II], (3.135)
27We assume the speed of sound for baryons as zero, i.e. c2s = 0 in Section 5.7 of [71].
60 Cosmological Perturbations in Second Order
˙ˆI[II]10 =
4k
R
(
υ˙ [II]b +Hυ
[II]
b
)
+ 4k(1+R)
R
Φ[II] + k
(
Iˆ[II]00 −
2
5
Iˆ[II]20
)
−L[I,I]10 [Iˆ]+
4
R
L[I,I][υb],
(3.136)
where L[I,I]00 [Iˆ] and L
[I,I]
10 [Iˆ] can be evaluated by using Eq. (3.65) and
L[I,I][δb]≡−6(2Ψ1 +δX ,1)Ψ˙2−2(k·k2)δX ,1υX ,2−2k22(Φ1 +Ψ1)υX ,2
+2k1·k2(2Ψ1−2Φ1−HυX ,1−2υ˙X ,1)υX ,2, (3.137)
L[I,I][υb]≡ kˆ·k2
[
(δX ,1−Φ1 +Ψ1)υ˙X ,2 +δX ,1Φ2 +2(Ψ1−Φ1)Φ2
− (k1·k2)υX ,1υX ,2 +H(δX ,1−Φ1 +Ψ1)υX ,2− Ψ˙1υX ,2
]
. (3.138)
While higher multipoles are negligible, the quadrupoles of photons in 2nd order are
Iˆ[II]20 =
5
4
1
∑
n=−1
0D
n
20Iˆ1,1,nIˆ2,1,−n, (3.139)
Eˆ[II]20 = 0. (3.140)
3.3.3 Numerical Stability and Convergence
Technically, solving the 2nd-order perturbations is very similar to the case in 1st order,
except two main differences. First, the quadratic terms in EFEs and Boltzmann equation
require the solutions of 1st-order perturbations. Thus, we need to numerically evolve 1st-
order perturbations and store the values in memory. When we elaborate the quadratic terms
at a specific conformal time, we perform interpolation with the pre-calculated perturbations
in 1st order. Second, since we have to solve the kernels of 2nd-order perturbations for a
set of (k1, k2, kˆ1 · kˆ2, η) with k1 ≥ k2, it is much more time-consuming compared to the
1st-order calculation with a set of (k, η).
We will work with Newtonian gauge in the numerical calculation. However, a cru-
cial trick is required when we solve the perturbations in Newtonian gauge, in both 1st and
2nd order. In the following, we present the trick for 2nd-order perturbations only but it
is straightforward to apply the trick to 1st-order study. Instead of using either two of the
2nd-order EFEs in Eq. (3.25) to (3.28) directly28, we use Eq. (3.27) and a combination of
28For 1st order, the corresponding equations are Eq. (2.51) to (2.54).
3.3 Numerical Implementation 61
Eq. (3.25) and Eq. (3.26) in form of GI I +GOO/3, which is rewritten as
Ψ¨+3HΨ˙+HΦ˙+ k
2
3
Ψ+ κa
2
15 ∑n,r
ρ¯X IˆX ,20 +
κa2
6 ∑b,c
ρ¯XδX +
(
κa2
3 ∑b,c
ρ¯X +
4Λa2
3
)
Φ
= Tk1,k2(k)
{
1
3
C ′1 +C ′2 +
4
3
(
κa2∑
b,c
ρ¯X +4Λa2
)
Φ1Φ2
}
, (3.141)
where
C ′1 ≡ C1−12H
2Φ1Φ2, (3.142)
C ′2 ≡ C2− (4H2 +8H˙)Φ1Φ2. (3.143)
The reason why the trick is essential is that either Eq. (3.25) or Eq. (3.26) contains some
pairs of terms with large numerical values. However, the sum of each pair can be several
orders of magnitude smaller. The cancellations boost up the numerical errors such that the
solutions become unstable. Eq. (3.141) works well in the radiation-dominated era because
it replaces the sums of these pairs with small quantities explicitly. To understand this phys-
ically, we notice that quantities, such asH2 and H˙, are determined by ρ¯n and ρ¯r exclusively
during radiation-dominated era. Nevertheless, the sum H2 + H˙ is of order κa2(ρ¯b + ρ¯c)
which can be a few orders of magnitude smaller. Since we are interested in the evolution
of 2nd-order perturbations up to recombination, Eq. (3.141) is sufficient over the time cov-
ered. For matter-dominated era, Eq. (3.26) and Eq. (3.27) can be used directly with the
replacement
H2 +2H˙=−κa
2
3
(ρ¯n + ρ¯r)+Λa2, (3.144)
to tackle the large-value cancellations.
Solving the 2nd-order perturbations is numerically non-trivial. It is thus important to
verify the calculation. One way to do so is to make use of the redundant EFEs to check if
numerical errors are well controlled. In Fig. 3.3 to 3.5, we show the absolute numerical dif-
ference by elaborating the unused constraint equation Eq. (3.28). We can see that the differ-
ences are within percentage level for different configurations of k1, k2 and k1 ·k2. Moreover,
there are two leaps of the differences in each plot. The one at a ∼ 1.5× 10−5 corresponds
to the turning-off of TCA in 1st-order equations while the other one at a∼ 5×10−5 comes
from the turning-off of TCA in 2nd-order equations. We remark that we apply the TCA up
62 Cosmological Perturbations in Second Order
to the leading order only for 2nd-order perturbations29. Therefore, we can expect the errors
to be reduced by at least a factor of 100 if we include the next-to-leading-order correction
of the TCA. This can be done in the future if higher precision is needed. However, the
leading-order TCA is sufficient for calculating the CMB bispectrum at recombination. Fi-
nally, we point out that although a significant deviation of the constraint equation is shown
for a. 10−5 in Fig. 5.8 of [80], we manage to reproduce their curvature potential derivative
for a& 10−5. We confirm that the deviation does not affect the bispectrum calculation.
1 ´ 10
-5
5 ´ 10
-5
1 ´ 10
-4
5 ´ 10
-4
0.001 0.005
10
-15
10
-12
10
-9
10
-6
0.001
a HΗL
Y
 
Figure 3.3 The graph of the absolute 2nd-order kernel of Ψ˙ (solid line) against the scale fac-
tor a with k0 = 0.199Mpc−1, k1 = 0.2Mpc−1, k2 = 0.002Mpc−1 and kˆ1· kˆ2 =−0.684. The
dashed line shows the absolute numerical difference in the constraint equation Eq. (3.28).
29See Section 3.3.2 for more details about the TCA.
3.3 Numerical Implementation 63
1 ´ 10
-5
5 ´ 10
-5
1 ´ 10
-4
5 ´ 10
-4
0.001 0.005
10
-11
10
-8
10
-5
0.01
a HΗL
Y
 
Figure 3.4 The graph of the absolute 2nd-order kernel of Ψ˙ (solid line) against the scale
factor a with k0 = 0.159Mpc−1, k1 = 0.2Mpc−1, k2 = 0.2Mpc−1 and kˆ1· kˆ2 =−0.684. The
dashed line shows the absolute numerical difference in the constraint equation Eq. (3.28).
1 ´ 10
-5
5 ´ 10
-5
1 ´ 10
-4
5 ´ 10
-4
0.001 0.005
10
-14
10
-11
10
-8
10
-5
0.01
a HΗL
Y
 
Figure 3.5 The graph of the absolute 2nd-order kernel of Ψ˙ (solid line) against the scale
factor a with k0 = 0.4Mpc−1, k1 = 0.2Mpc−1, k2 = 0.2Mpc−1 and kˆ1· kˆ2 = 1. The dashed
line shows the absolute numerical difference in the constraint equation Eq. (3.28).

Chapter 4
CMB Bispectrum during Recombination
Even if the quantum fluctuations from inflation are perfectly Gaussian, non-Gaussianities
can exist through the nonlinear evolution afterwards. These post-inflationary non-Gaussianities
can have significant impacts on cosmic microwave background (CMB) study. Depending on
when they are generated, we can classify them into two categories: recombination bispectra
if they are produced during the recombination and late-time bispectra if they are imprinted
from the late-time large-scale structures. Understanding the shapes of these bispectra is cru-
cial for eliminating their contaminations to the primordial non-Gaussianities and verifying
the gravitational theory in cosmological scales. While the late-time ISW-lensing1 bispectra
are known to be important [39, 46, 69], there are still discrepancies about the significance
of the recombination bispectra2. A complete analysis is thus desired for resolving those
discrepancies and predicts the detectability of the recombination bispectra for the upcoming
CMB experiments.
To study the recombination bispectra, it is necessary to calculate the cosmological per-
turbations up to 2nd order, which has been discussed in details in Chapter 3. In this chapter,
we will focus on how these perturbations are linked to the observation – CMB tempera-
ture bispectrum. We present the numerical calculation of the bispectrum produced around
recombination. More precisely, we include all the effects due to the Compton collision at
recombination and the early-time ISW-related couplings around recombination. In order to
do so, we have to distinguish the physics at recombination from the late-time mechanisms.
This separation leads to a 2nd-order line-of-sight (LOS) integrand which is different from
the one used in [52, 82] and provides an alternative way to calculate the CMB bispectrum
1ISW stands for the integrated Sachs-Wolfe effect.
2See, for example, [11, 13, 20, 29, 57, 58, 66, 75, 86, 87, 94].
66 CMB Bispectrum during Recombination
around recombination. Our approach interprets the quadratic 2nd-order effects as couplings
of two well-known 1st-order effects in Newtonian gauge 3. This provides a clear physical
picture about what effects are taken into account.
We start with several definitions of the CMB temperature in 2nd order in Section 4.1.
Then, we calculate the 2nd-order CMB temperature anisotropies through the LOS approach
and formulate the recombination bispectrum under the flat-sky and thin-shell approxima-
tions in Section 4.2. In particular, we present the trick to separate the effects during re-
combination from the late-time effects in 2nd-order in Section 4.2.2. This trick also solves
the hierarchy problem of 2nd-order LOS approach. In Section 4.3, the details about the
numerical calculations are given. We present the calculated bispectrum at recombination
and validate it with the analytical solution in squeezed limit. Moreover, we constrain fNLs
for different templates and forecast the signal-to-noise for the recombination bispectrum.
Finally, we discuss some potential further studies and draw the conclusion in Section 4.4.
4.1 Definition of CMB Temperature in Second Order
Since the inelastic 2nd-order Compton scattering induces the CMB spectral distortions [85],
there does not exist a unique way of defining the CMB temperature in 2nd order. In this
section, we will discuss several possible definitions. First, we assume that the CMB tem-
perature T can be related to the distribution function of the CMB photons in form of the
black-body spectrum
I(η ,x, p0, nˆ) =
[
ep
0/T (η ,x,p0,nˆ)−1
]−1
, (4.1)
where the CMB temperature anisotropies Θ can be expanded into perturbations as
T (η ,x, p0, nˆ) = T¯ (η)
[
1+Θ[I](η ,x, nˆ)+ 1
2
Θ[II](η ,x, p0, nˆ)
]
, (4.2)
with the background CMB temperature T¯ . We note that Θ[II] depends on the photon mo-
mentum p0 due to the spectral distortion. By applying Taylor expansion at Θ= 0, we obtain
I(η ,x, p0, nˆ) = I¯ − p0 ∂ I¯∂ p0Θ
[I] +
[(p0)2
2
∂ 2I¯
∂ p0∂ p0 + p
0 ∂ I¯
∂ p0
]
(
Θ[I]
)2
−
p0
2
∂ I¯
∂ p0Θ
[II], (4.3)
3See Section 4.2.2 for more details,
4.1 Definition of CMB Temperature in Second Order 67
where I¯(η , p0) is the photon intensity in background order. After performing the energy
integration as in Eq. (2.14), we have
Iˆ = 1+4Θ[I](η ,x, nˆ)+6
[
Θ[I](η ,x, nˆ)
]2
−
1
4π2I¯(η)
∫ ∂ I¯
∂ p0Θ
[II](η ,x, p0, nˆ)
(
p0
)4
dp0.
(4.4)
Depending on details of the spectral distortion, Θ[II] in the last integral of Eq. (4.4) can have
different dependences on p0.
4.1.1 Brightness Temperature
Because of the spectral distortion, we cannot simply fit the perturbed CMB photon spectrum
with black-body spectrum to define the CMB temperature. Instead of using Eq. (4.2), we
define the brightness temperature TI ≡ T¯ [1+ΘI(η ,x, nˆ)] such that
ΘI ≡
1
4
[
Iˆ(η ,x, nˆ)−1
]
. (4.5)
In particular, we have Θ[I]I =Θ[I] and
Θ[II]I =
1
4
Iˆ[II] = 3
[
Θ[I](η ,x, nˆ)
]2
−
1
8π2I¯(η)
∫ ∂ I¯
∂ p0Θ
[II](η ,x, p0, nˆ)
(
p0
)4
dp0. (4.6)
4.1.2 Bolometric Temperature
We define the bolometric temperature TE ≡ T¯ [1+ΘE(η ,x, nˆ)] by
(1+ΘE)4 ≡ Iˆ(η ,x, nˆ). (4.7)
In 1st order, bolometric temperature is identical to brightness temperature, i.e. Θ[I]E = Θ
[I]
I .
However, they are not the same in 2nd order
Θ[II]E =Θ
[II]
I −3
(
Θ[I]I
)2
= 1
4
Iˆ[II]−
3
16
(
Iˆ[I]
)2 . (4.8)
It can be proven that the black-body radiation with the bolometric temperature has the same
energy density as the actual distribution [87]. In other words, we define the CMB tem-
68 CMB Bispectrum during Recombination
perature such that the energy density of the actual distribution matches with that of the
black-body radiation with the bolometric temperature. It is worth pointing out that the
CMB bispectra calculated will be different when different definitions of temperature are
adopted. However, these bispectra are related analytically and can be transformed from one
into another straightforwardly. We will discuss this further in Section 4.2.
4.2 Formulating Second-order Bispectrum
In this section, we first extend the LOS approach to calculate the 2nd-order CMB temper-
ature anisotropies. Then, we apply the integration by parts (IBPs) to the LOS integrand in
order to separate the recombination effects from the late-time ones. Finally, we establish the
formalism for the temperature bispectrum generated from 2nd-order perturbations.
4.2.1 Second-order Line-of-sight Approach
To calculate the 2nd-order temperature anisotropies, we first rewrite the 2nd-order Boltz-
mann equation using Eq. (3.48), (3.61) and (3.76) such that
△˙[II] + ik · nˆ△[II] + ˙¯τ△[II] =∑
ℓm
1
Nℓ
eτ¯
(
S˜[II]ℓm + S˜
[I,I]
ℓm
)
Yℓm(nˆ), (4.9)
where 4△≡ Iˆ is the fractional brightness, τ¯ ≡
∫ η0
η ˙¯τ(η˜)dη˜ is the optical depth, Nℓ is defined
in Eq. (C.2) and multipoles of the purely 2nd-order and quadratic source functions (S˜[II] and
S˜[I,I] respectively) are defined as4
eτ¯ S˜[II]ℓm ≡δ
0
ℓ δ
0
mΨ˙+δ 1ℓ δ 0mkΦ−
k
√
3
δ 2ℓ
(
δ 1mB˜(1)+δ−1m B˜(−1)
)
−δ 2ℓ
(
δ 2mH˙(2)+δ−2m H˙(−2)
)
+ ˙¯τ
(
δ 0ℓ δ
0
m△00 +δ 1ℓ υ˜b,(m)+
1
4
δ 2ℓ Πˆm
)
, (4.10)
eτ¯ S˜[I,I]ℓm ≡−
1
4
L[I,I]ℓm [Iˆ]+
1
4
C[I,I]ℓm [Iˆ]. (4.11)
4We omit [II] denoting the 2nd-order perturbations.
4.2 Formulating Second-order Bispectrum 69
After integrating over the conformal time and wavenumbers, the 2nd-order fractional bright-
ness can be expressed as
△[II](η0,k, nˆ) =Tk1,k2(k)
{
∫ η0
0
dηe−ik·nˆr∑
ℓm
1
Nℓ
(
S˜[II]ℓm + S˜
[I,I]
ℓm
)
Yℓm(nˆ)
}
, (4.12)
where r ≡ η0−η .
As expected, the 2nd-order source function consists of two parts: purely 2nd-order part
and quadratic part. The purely 2nd-order part has the same form as the 1st-order source
function [50] and requires solutions of 2nd-order perturbations. However, purely 2nd-order
source terms exist up to ℓ= 2. In contrast, the quadratic part requires solutions of 1st-order
perturbations only but the source terms extend to infinitely high ℓ modes. Thus, there is a
hierarchy problem when we perform the LOS approach to the quadratic source. Moreover,
since we are interested in the 2nd-order effects imprinted during recombination, we have to
distinguish these effects from the late-time effects (e.g. CMB lensing) consistently. In the
next subsection, we will show how performing IBPs can resolve the hierarchy problem and
identify the effects during recombination.
4.2.2 Separation between Recombination and Late Time
In this subsection, we demonstrate a trick to distinguish between the effects during recom-
bination and late times. The trick also mitigates the hierarchy problem so that it is enough
to truncate the multipoles of the quadratic source function at low ℓs. Before we work on the
2nd-order source function, it is useful to review how we separate the early-time and late-
time ISW effect in 1st order. The 1st-order fractional brightness at time η can be expressed
in Fourier space as Eq. (2.75) with the 1st-order source function in form of5
S˜[I]T = e−τ¯
(
Ψ˙− ik · nˆΦ+ ˙¯τ△+ C
4
)
. (4.13)
The first two terms generate redshift effects on the background CMB due to the linear per-
turbations of the metric while the last two terms contain all the effects generated at recom-
bination due to the Compton scattering6. As shown in Section 2.6.1, we can use the IBPs
5For simplicity, we have omitted the superscript [I] on the right-hand side of the equation.
6We ignore the reionization and other late-time scatterings throughout this thesis.
70 CMB Bispectrum during Recombination
on the second term of Eq. (4.13) such that the source term becomes
e−τ¯
(
−δ (η˜−η)Φ+ ˙¯τΦ+ Φ˙+ Ψ˙+ ˙¯τ△+ C
4
)
, (4.14)
where δ (η) is the Dirac delta function. The first two terms encode the Sachs-Wolfe (SW)
effect while the next two terms are the ISW effect.
Integration by parts are not merely mathematical tricks. Since we are replacing deriva-
tive terms in the integrand with non-zero boundary terms, this means that it is important
where we evaluate the boundary terms at. Physically, performing the IBPs “re-distributes”
the physical effects along the LOS integration. In this case, Eq. (4.14) separates the red-
shift effects occurring at different times physically as follows. The first two terms are the
SW effect due to the difference of the gravitational potential between recombination and
time η , i.e. the SW effect measured by an observer at time η . In particular, Φ in the first
term is a constant when η = η0. The third and fourth terms (i.e. Φ˙+ Ψ˙) include the early-
time ISW effect from the time-varying gravitational potential due to the transition from
radiation-dominated to matter-dominated era and the late-time ISW effect from the time-
varying gravitational potential due to the domination of dark energy. In this paper, we cal-
culate the bispectrum at and around recombination. Explicitly, it means that we include SW
and early-time ISW but exclude late-time ISW. In the following, we extend the separation
to the redshift-related couplings in 2nd order.
The quadratic part of the 2nd-order source function can be read in configuration space
as7
S˜[I,I] = 2e−τ¯
[
Si j∂J(Φ+Ψ)
∂△
∂ni − (Φ+Ψ)n
i∂I△+4(−ni∂IΦ+ Ψ˙)△
+(Φ−Ψ)ni∂IΦ+2ΨΨ˙
]
+ 1
4
e−τ¯C[I,I][Iˆ], (4.15)
where Si j is the screen projector defined in Eq. (2.6) and C[I,I] is the quadratic collision
operator expressed in Eq. (3.81). The first three terms in Eq. (4.15), which contain frac-
tional brightness△, are numerically problematic. It is because the contributions of the LOS
approach from these terms extend to late times at which high multipoles of the brightness
are generated through propagation and projection onto the spherical harmonics. This means
that solving these terms is numerically expensive since truncating at low multipoles is not
7The notation for 1st-order perturbations is omitted for simplicity.
4.2 Formulating Second-order Bispectrum 71
enough. Because we focus on the contributions at and around recombination, we ignore
the first and second terms in Eq. (4.15) which correspond to the lensing and time-delay
effects respectively and contribute at late times. The third term in Eq. (4.15) is responsi-
ble for the 1st-order redshift effects, including the SW and ISW effects, on the 1st-order
temperature anisotropies. Intuitively, it includes contributions at and around recombination
(i.e. couplings with SW and early-time ISW effect) as well as late-time contributions (i.e.
couplings with late-time ISW effect). The key issue here is to physically distinguish be-
tween the early-time and late-time effects for those redshift-related couplings. Although the
quadratic collision operator C[I,I] contains high ℓ multipoles, its contribution is concentrated
at recombination when high ℓ multipoles are still highly suppressed8.
From now on, we focus on the third term of Eq. (4.15). We can express its contribution
(denoted by the subscript R) to the 2nd-order fractional brightness as
△[II]R (η0,k, nˆ) = 8Tk1,k2(k)
{∫ η0
0
dη e−ik·nˆr−τ¯△1(nˆ)
(
Ψ˙2− ik2 · nˆ Φ2
)
}
, (4.16)
where the subscripts 1 and 2 denote the dependence on k1 and k2 respectively. As mentioned
in Section 4.2 of [82], if we perform IBPs on the term with k2 · nˆ in Eq. (4.16), we will get
a time integration whose integrand contains the time derivative of the 1st-order fractional
brightness △˙. We can iteratively apply IBPs on that time integration but this approach
fails to single out a unique recombination contribution. More seriously, contributions from
higher multipoles grow when we integrate over the time further after recombination. That
is, the hierarchy problem remains unsolved.
However, there exists a unique way to separate the early-time and late-time redshift
effects which is physically reasonable and allows us to calculate the bispectrum around
recombination with a truncation at low ℓs. The key is to realize that Eq. (4.16) contains a
double time integration by substituting△1 with Eq. (2.75). We can then exchange the inner
and outer time integrations such that Eq. (4.16) is rewritten as
△[II]R = 8Tk1,k2(k)
{∫ η0
0
dη˜ e−ik1·nˆr˜S˜T (η˜ ,k1, nˆ)
∫ η0
η˜
dη e−ik2·nˆr
[
Ψ˙2(η)− ik2 · nˆ Φ2(η)
]
}
. (4.17)
8In Section 4.3, we find that ℓ up to about 10 is sufficient for getting a converging bispectrum for the
quadratic collision terms.
72 CMB Bispectrum during Recombination
Now, we perform IBPs on the integration over η and obtain
△[II]R = 8Tk1,k2(k)
{∫ η0
0
dη˜ e−ik·nˆr˜S˜T (η˜ ,k1, nˆ)
{
Φ2(η˜)+
∫ η0
η˜
dη eik2·nˆ(r˜−r)
[
Φ˙2(η)+ Ψ˙2(η)
]}
}
. (4.18)
We make use of the fact that k = k1 +k2 to get the exponential with k in the first line of
Eq. (4.18). By doing so, we do not produce the problematic term with △˙ through IBPs. Note
that we can ignore Φ evaluated at present η0 because it is a constant and does not contribute
to the bispectrum. The physical meaning of Eq. (4.18) is clear. Analogous to 1st-order
redshift effects which come from the background-order CMB signals redshifted by the 1st-
order perturbed metric, we have the 1st-order temperature anisotropies redshifted by the 1st-
order SW and ISW effects to generate 2nd-order temperature anisotropies, corresponding
to the first and second terms in Eq. (4.18) respectively. This trick allows us to distinguish
between the early-time and late-time redshift effects similar to what we have done typically
in 1st-order.
We now consistently separate the early-time and late-time redshift effects of the 1st-
order source function S˜T . For the first term of Eq. (4.18), we apply the IBPs on the term
with −ik1 · nˆΦ1 from the source function S˜T and obtain
−
∫ η0
0
dη˜ e−ik·nˆr˜−τ¯ ik1 · nˆ Φ1Φ2 =
∫ η0
0
dη˜ e−ik·nˆr˜−τ¯
(
Φ˙1Φ2 +
˙¯τ
2
Φ1Φ2
)
, (4.19)
for which we make use of the symmetry between the dummy variables k1 and k2 in the
convolution and ignore the constant boundary term evaluated at present η0. For the second
term of Eq. (4.18), we exchange the inner and outer time integrations and substitute the
integration over η˜ with Eq. (2.75). Putting all these together, we rewrite Eq. (4.16) as
△[II]R = 8Tk1,k2(k)
{∫ η0
0
dη e−ik·nˆr−τ¯
[
(△1 +Φ1)
(
Φ˙2 + Ψ˙2
)
+
˙¯τ
2
Φ1Φ2 +
(
˙¯τ△1 +
C1
4
)
Φ2
]}
, (4.20)
where
△1 +Φ1 = eτ¯(η)
∫ η
0
dη˜ e−ik1·nˆ(η−η˜)−τ¯(η˜)
(
˙¯τΦ1 + Φ˙1 + Ψ˙1 + ˙¯τ△1 +
C1
4
)
. (4.21)
Eq. (4.20) contains all possible 2nd-order couplings involving 1st-order redshift effects: the
4.2 Formulating Second-order Bispectrum 73
first term includes couplings between ISW and any 1st-order CMB effects (e.g. Doppler-
ISW, SW-ISW and even ISW-ISW couplings); the second term is the SW-SW coupling; the
third term corresponds to couplings between CMB signals generated by Compton scattering
and SW effect (e.g. Doppler-SW coupling).
At 2nd order, performing IBPs becomes tricky as mentioned in [82]. However, the ap-
proach we take uniquely separates SW, early-time ISW and late-time ISW effects along the
LOS integration, as the coupling terms shown in Eq. (4.20). In contrast, [87] also performs
IBPs to the third term of Eq. (4.15) but the integrand they obtained is different from that in
Eq. (4.20). Since different IBPs re-distribute the redshift effects differently along the time
integration, the SW and ISW effects are not manifestly separated in [87]. For example, any
residual couplings between the photon brightness and k · nˆΦ in the LOS integrand do not
damp out well after recombination because the term k · nˆΦ mixes up SW and ISW effects
throughout the LOS integration. In particular, it is problematic when a cutoff is used to
the time integration because early-time effects may be re-distributed to late times and vice
versa. In other words, integrating over the early-time regime does not necessarily calculate
the physical early-time effects. We speculate that this could be one of the reasons why [87]
concludes a much larger signal-to-noise.
Although we perform a similar cutoff to the time integration (explicitly from η = 230
to 1050 Mpc), it is numerically stable using Eq. (4.20). Indeed, the cutoff allows us to
capture the SW-related and early-time ISW-related couplings but exclude the late-time ISW-
related couplings. To understand how this works, we point out that only the first term in
Eq. (4.20) has non-negligible contribution (ignoring reionization) outside the chosen time
range because of the ISW term (Φ˙+ Ψ˙). However, ISW effect is highly suppressed after
η = 1050 Mpc when our universe is matter-dominated and only becomes significant again
when the dark energy dominates at late time. Indeed, the early-time ISW is suppressed well
before η = 1050 Mpc and thus we find it safe to simply use a sharp cutoff at η = 1050
Mpc with negligible impact to the bispectrum. However, the extended time integration
may raise some uncertainties on the early-time ISW-related couplings due to the thin-shell
approximation. We will discuss about this in details in Section 4.3.
With Eq. (4.20), we can replace the quadratic part S˜[I,I] with S[I,I] which is defined in
74 CMB Bispectrum during Recombination
configuration space as
S[I,I] ≡ 2e−τ¯
[
Si j∂J(Φ+Ψ)
∂△
∂ni − (Φ+Ψ)n
i∂I△+4(Φ˙+ Ψ˙)(△+Φ)
+(Φ−Ψ)ni∂IΦ+2ΨΨ˙+(4 ˙¯τ△+C)Φ+2 ˙¯τΦ2
]
+ 1
4
e−τ¯C[I,I][Iˆ]. (4.22)
Here, we emphasize that Eq. (4.22) uniquely separates the SW effect, the early-time and
late-time ISW effect and allows us to have a early-time cutoff on the time integration to
include the 2nd-order effects generated at and around recombination. Physically, the two
terms in the first line of Eq. (4.22) are the lensing and time-delay effects respectively and
will be ignored in our calculation. The IBPs performed help to distinguish SW effect from
ISW effect and clarify the physical meaning of each term. For example, we can interpret
(4 ˙¯τ△+C)Φ and 2 ˙¯τΦ2 as the photon-SW and SW-SW couplings respectively. The term
(Φ−Ψ)ni∂iΦ+ 2ΨΨ˙ is the quadratic part of the evolution equation of photon energy p0
in 2nd order, i.e. (dp0/dη)[I,I]. We will discuss the contributions of these effects on the
bispectrum in Section 4.3.
4.2.3 Bispectrum from Second-order Perturbations
By using the 2nd-order LOS approach in Eq. (4.12) with the purely (Eq. (4.10)) and quadratic
(Eq. (4.22)) 2nd-order source functions, we can calculate the 2nd-order temperature anisotropies
generated at recombination. Here, we formulate the bispectrum from the 2nd-order temper-
ature anisotropies at recombination under the flat-sky and thin-shell approximations9.
First of all, we define the new coefficients which are related to the reduced bispectrum
by
ξℓ1ℓ2M ≡∑
ℓ3
2L3
(
ℓ1 ℓ2 ℓ3
M −M 0
)(
ℓ1 ℓ2 ℓ3
0 0 0
)
bℓ1ℓ2ℓ3, (4.23)
where 2L≡ 2ℓ+1. Using Eq. (C.22), it can be shown that
bℓ1ℓ2ℓ3 =
(
ℓ1 ℓ2 ℓ3
0 0 0
)−1
∑
M
(
ℓ1 ℓ2 ℓ3
M −M 0
)
ξℓ1ℓ2M. (4.24)
Moreover, with Eq. (C.20) and Eq. (C.21), we can find that the new coefficients obey the
9The derivation here is based on [17].
4.2 Formulating Second-order Bispectrum 75
following properties
ξℓ2ℓ1M = ξℓ1ℓ2M = ξℓ1ℓ2−M. (4.25)
Due to the statistical isotropy of the CMB, the 3-point correlation function of the bright-
ness temperature ΘI defined in Eq. (4.5) depends merely on the relative angles with respect
to nˆ3, e.g. (θ1,φ1,θ2,φ2). Thus, we can align nˆ3 with the azimuthal direction and expand
the 3-point correlation function into
⟨Θ(nˆ1)Θ(nˆ2)Θ(nˆ3)⟩= ∑
ℓ1ℓ2m1m2
ϑℓ1ℓ2m1m2
√
L1L2
2π Yℓ1m1(θ1,φ1)Yℓ2m2(θ2,φ2)
= ∑
ℓ1ℓ2M
ϑℓ1ℓ2M−M
√
L1L2
2π Yℓ1M(θ1,0)Yℓ2−M(θ2,φ21). (4.26)
In the second equality of Eq. (4.26), we make use of the fact that the rotational invariance
guarantees the 3-point correlation function to be dependent on the relative phase φ21 ≡
φ2−φ1 instead of the absolute phases.
Now, we want to show that coefficients ξ and ϑ are identical. To do so, we perform a
rotationR to the two spherical harmonics in Eq. (4.26) so that nˆ3 points to (θ ,φ) instead of
the azimuthal direction, i.e.
Yℓm(R−1nˆ) =∑
m′
Yℓm′(nˆ)Dℓm′m(R), (4.27)
where the rotation matrices Dℓm′m can be expressed as
Dℓm′m(φ ,θ ,ψ) = (−1)m
′
√
4π
2Lm
Yℓ−m′(θ ,φ)e−imψ , (4.28)
with Euler angles φ , θ and ψ . Therefore, we have by using Eq. (C.13)
⟨
3
∏
i=1
aℓimi⟩= ϑℓ1ℓ2M−M
√
2L1L2L3
π
(
ℓ1 ℓ2 ℓ3
M −M 0
)(
ℓ1 ℓ2 ℓ3
−m1 −m2 −m3
)
. (4.29)
Applying Eq. (2.93), we can show that the reduced bispectrum becomes
bℓ1ℓ2ℓ3 =
(
ℓ1 ℓ2 ℓ3
0 0 0
)−1
∑
M
(
ℓ1 ℓ2 ℓ3
M −M 0
)
ϑℓ1ℓ2M−M. (4.30)
76 CMB Bispectrum during Recombination
By comparing with Eq. (4.24), we deduce
ξℓ1ℓ2M = ϑℓ1ℓ2M−M. (4.31)
Making use of ⟨Φ(k1)Φ(k2)Φ(k3)⟩= (2π)−3/2δ 3(k1 +k2 +k3)B(k1,k2), we obtain
⟨Θ(nˆ1)Θ(nˆ2)Θ(nˆ3)⟩=
∫
d3k1d3k2
(2π)6 dr1dr2dr3B(k1,k2)ST (η1,k1)ST (η2,k2)ST (η3,k3)
e−i[k1·r1+k2·r2−(k1+k2)·r3], (4.32)
where ri = rinˆi for i= 1, 2, 3. We emphasize that Eq. (4.32) is valid for both primordial and
post-inflationary non-Gaussianities. In flat-sky limit, θ1, θ2≪ 110 and thus the argument of
the exponential in Eq. (4.32) can be approximated as
−i
[
kz1(r1− r3)+ kz2(r2− r3)+ k⊥1 r1θ1 cosβ1 + k⊥2 r2θ2 cos(β2−φ21)
]
, (4.33)
where we align nˆ3 with z-axis and decompose
ki = (k⊥i cosβi,k⊥i sinβi,kzi ), (4.34)
for i = 1, 2, 3. With Eq. (4.26) and (4.31), the coefficients ξℓ1ℓ2M are
4π2
√
L1L2
∫
θ1dθ1
∫
θ2dθ2
∫
dφ21Yℓ1−M(θ ,0)Yℓ2M(θ2,φ21)⟨Θ(nˆ1)Θ(nˆ2)Θ(nˆ3)⟩. (4.35)
By employing Bessel functions of the first kind JM
JM(x) = (−1)MJ−M(x) =
1
2π
∫ π
−π
dy e−i(My−xsiny), (4.36)
∫
θdθJM(k⊥rθ)JM(Lθ) =
δ (L− k⊥r)
L
, (4.37)
and the flat-sky limit of spherical harmonics
Yℓm(θ ,φ)→
√
2L
4π (−1)
MJM(Lθ)eiMφ , (4.38)
10Roughly speaking, this is the condition when the three directions of the 3-point correlation are close to
each other so that the sphere of sky can be approximated by a flat plane.
4.2 Formulating Second-order Bispectrum 77
we can derive the coefficients in the flat-sky limit11
ξℓ1ℓ2M ≈
1
(2π)3
∫
dβ12 eiMβ12
∫
dkz1dk
z
2
∫
dr1dr2dr3
B(k1,k2)
r21r
2
2
ST (η1,k1)ST (η2,k2)ST (η3,k3)e−ik
z
1(r1−r3)−ikz2(r2−r3), (4.39)
where β12 ≡ β1−β2, k⊥1 = L1/r1 and k⊥2 = L2/r2.
To derive the reduced bispectrum, we notice that [17]
ξℓ1ℓ2M =
∫ 2π
0
dβ12
2π e
iMβ12 bℓ1ℓ2ℓ3, (4.40)
where we use the approximation
(
ℓ1 ℓ2 ℓ3
M −M 0
)
→
(
ℓ1 ℓ2 ℓ3
0 0 0
)
cos(Mβ12), (4.41)
and the Ponzano-Regge limit
(
ℓ1 ℓ2 ℓ3
0 0 0
)2
= 1πL1L2 sinβ12
. (4.42)
The reduced bispectrum can then be written in the flat-sky limit as
bℓ1ℓ2ℓ3 ≈
1
(2π)2
∫
dkz1dk
z
2
∫
dr1dr2dr3ST (η1,k1)ST (η2,k2)ST (η3,k3)
e−ik
z
1(r1−r3)−ikz2(r2−r3)B(k1,k2)
r21r
2
2
. (4.43)
So far, the bispectrum formulae developed are generic for both primordial and post-
inflationary non-Gaussianities. For the recombination bispectrum, we replace one of the
source functions by the 2nd-order source function as in Eq. (4.12) and keep the other two in
1st order
brecℓ1ℓ2ℓ3 ≈
1
(2π)2
∫
dkz1dk
z
2
∫
dr1dr2dr3ST (η1,k1)ST (η2,k2)S2NDT (η3,k1,k2, nˆ)
e−ik
z
1(r1−r3)−ikz2(r2−r3)PΦ(k1)
r21
PΦ(k2)
r22
+2 perm., (4.44)
11This imposes a requirement that B(k1,k2) depends solely on the relative phase β12 but not the absolute
ones.
78 CMB Bispectrum during Recombination
where µ ≡ kˆ1 · kˆ2, S2NDT contains both the purely (Eq. (4.10)) and quadratic (Eq. (4.22)) parts
of the 2nd-order source function. We emphasize that although the 2nd-order source function
has more complicated dependences, the derivation in this subsection still holds. Moreover,
the source function does not depend on all freedoms of (k1,k2, nˆ) but we express so in
Eq. (4.44) for simplicity. We will explicitly write down its dependences in Section 4.2.4.
Additionally, with the thin-shell approximation, the recombination bispectrum becomes
brecℓ1ℓ2ℓ3 ≈
1
(2π)2
∫
dkz1dk
z
2
[∫
dr1ST (η1,k1)e−ik
z
1r1
][∫
dr2ST (η2,k2)e−ik
z
2r2
]
[∫
dr3S
2ND
T (η3,k1,k2, nˆ)ei(k
z
1+kz2)r3
]
PΦ(k1)PΦ(k2)
r4LSS
+2 perm., (4.45)
where k⊥i = ℓi/rLSS for i = 1, 2, 3 with the distance rLSS from the last scattering surface.
The bispectrum formulae above are measured under the brightness temperatureΘI while
the bispectra in literature are commonly based on the bolometric temperature ΘE . However,
the bispectra of these two temperature definitions can be linked straightforwardly by
bΘEℓ1ℓ2ℓ3 = b
ΘI
ℓ1ℓ2ℓ3−3(Cℓ1Cℓ2 +Cℓ2Cℓ3 +Cℓ1Cℓ3), (4.46)
where the extra term comes from the second term in Eq. (4.8).
The flat-sky approximation avoids the use of the spherical Bessel function of the first
kind jℓ(x) which fluctuates rapidly for high ℓs Thus, lower resolutions of ks are sufficient
and this significantly reduces the computational power required. On the other hand, the thin-
shell approximation decouples the nested integrals and simplifies the numerical integrations.
However, there are drawbacks using these approximations. We expect that the bispectrum
in flat-sky limit will deviate from the exact one for low ℓs. Furthermore, since we include
the early-time (but not late-time) ISW-related effects by extending the LOS integration right
after recombination, the thin-shell approximation may break down for these effects. We will
discuss these problems further when we compare our numerically-calculated bispectrum
with the analytical one in Section 4.3.2.
Our ultimate goal is to calculate the bispectrum as observed today, which means that all
the effects post recombination till today are ideally included in the calculation. However,
since non-perturbative effects complicate the study at low redshifts (z . 10), an appropri-
ate milestone towards this goal is the computation of the bispectrum around recombination
which is what we undertake in this thesis. This means that we cut off the LOS integration
4.2 Formulating Second-order Bispectrum 79
early after recombination12, allowing us to make a definite statement about the amplitude
of the bispectrum at this time clearly. In particular, we ignore contributions from lensing,
time-delay and late-time ISW-related effects. The lensing and time-delay effects on bis-
pectra have been studied elsewhere [49, 69]. The effects from the 2nd-order vector and
tensor perturbations and the late-time ISW-related effects are studied in [52, 82] and are
sub-dominant.
4.2.4 Quadratic Source Function with m ̸= 0
When we decompose the purely 2nd-order source function, we need multipoles up to ℓ= 2
only to include all scalar, vector and tensor modes. In contrast, the decomposition of the
quadratic 2nd-order source function extends to arbitrarily high ℓ. This becomes a hierarchy
problem. With the trick discussed in Section 4.2.2, ℓ up to about 10 is sufficient for effects at
recombination. Even so, elaborating the quadratic source function is still subtle, especially
for multipoles with m ̸= 0.
The quadratic 2nd-order source function contains many terms. Here, we work only with
the term
S[I,I]Eg,ℓm =
1
Nℓ
Yℓm(nˆ)
1
∑
n=−1
0D
n
ℓmIˆ1,ℓ−1,m−nk2,(n)Φ2, (4.47)
where Nℓ is defined in Eq. (C.2) and 0Dnℓm can be found in Appendix C.3. We note that the
azimuthal direction is aligned with k3. By using Eq. (3.74) and the fact that ki for i = 1, 2
and 3 are in the same plane, the term becomes
S[I,I]Eg =− iℓ
√
(ℓ−m)!
(ℓ+m)!P
m
ℓ (cosθn3)eimφnk1 (4.48)
1
∑
n=−1
0D
n
ℓm
√
4π
2ℓ−1Y
∗
ℓ−1,m−n(kˆ1 · kˆ3,0)
√
4π
3
Y ∗1n(kˆ2 · kˆ3,π)Iˆℓ−1,0(k1)k2Φ2,
where Pmℓ (x) is the associated Legendre polynomial, cosθn3 = nˆ · kˆ3 and φnk1 ≡ φn−φk1 is
the relative phase between nˆ and k1. In particular, we can find that
cosφnk1 =
cosθn1 + cosθ13 cosθn3
sinθ13 sinθn3
, (4.49)
12See Section 4.2.2 for more details about how we separate the effects around recombination from the
late-time ones.
80 CMB Bispectrum during Recombination
where the subscript n denotes the observation direction nˆ and the numbers i = 1, 2 and 3
denote ki. In the flat-sky limit, we have
cosθn1 =
kz1
k1
, cosθn3 =
kz3
k3
, cosθ13 =
k21 + k23− k22
2k1k3
. (4.50)
Although we elaborate the term S[I,I]Eg,ℓm only, the result is generic for the quadratic source
function. Indeed, all terms in the quadratic source function have the same form as the
first line of Eq. (4.48), which can thus be singled out. When we calculate the 2nd-order
perturbations as in Chapter 3, we need to consider merely the second line of Eq. (4.48). This
explains why it is sufficient to solve the 2nd-order perturbations in a set of (k1,k2, kˆ1 · kˆ2).
4.3 Numerical Results
To calculate the bispectrum in Eq. (4.45), we solve the 1st- and 2nd-order perturbations
numerically. For both cases, we use the Newtonian gauge as discussed in Section 2.5 and
Chapter 3 respectively. In this section, we first describe the numerical implementations of
calculating the recombination bispectrum. Then, we verify the numerical result with the
analytical bispectrum in the squeezed limit. Finally, we present the constraints on fNLs and
the signal-to-noise of the recombination bispectrum.
4.3.1 Numerical Implementation
There are two points worth being emphasized when solving the 2nd-order perturbations
numerically. First, we solve and store the kernels of the 2nd-order perturbations as functions
of k1, k2, kˆ1· kˆ2 and η as we explained in Section 4.2.4. For conservative convergence, the
numbers of sampling points are chosen to be 180, 10, 120 for ks, µ and η respectively.
In particular, making use of the flat-sky approximation allows us to simplify the numerical
samplings in k-space13. We sample lnk uniformly with k from 2.4×10−6 to 0.86/Mpc and
lnη uniformly with η from 230 to 1050 Mpc. Moreover, we confirm the convergences of
the bispectra within a few percent of errors by varying the sample sizes of ks, µ and η as
well as the ranges of ks and integrated η .
13Because the highly-fluctuating Bessel functions are missing when calculating the bispectra with flat-sky
approximation.
4.3 Numerical Results 81
Second, there exist numerical instabilities in Newtonian gauge for both 1st and 2nd or-
der arising from two large terms canceling almost perfectly with each other with the residual
a few orders of magnitude smaller. This undermines the accuracies available in the numer-
ical calculation. To tackle this problem, we use two evolution equations which are linear
combinations14 of the original four scalar Einstein field equations (EFEs) as discussed in
Section 3.3.3.
600 800 1000 1200 1400 1600 1800 2000
{
- 20
- 15
- 10
- 5
5
b
{{{
 C
{
2
{ up to 5
{ up to 8
{ up to 10
600 800 1000 1200 1400 1600 1800 2000
{
20
25
30
35
40
b
20 {{
 H C
{
C
20
L
{ up to 5
{ up to 8
{ up to 10
Figure 4.1 The graphs of the bispectra generated from the quadratic source function S[I,I]
against ℓ for equilateral (upper) and squeezed (lower) limit. The curves correspond to dif-
ferent ℓ truncations of the multipoles S[I,I]ℓm . We can see that the convergence occurs when ℓ
goes up to 10.
14This technique is first employed in the CMBQuick code introduced by Pitrou [87].
82 CMB Bispectrum during Recombination
The solved 2nd-order perturbations are then fed into the purely 2nd-order source func-
tion S[II]. Although only 1st-order perturbations are needed to compute the quadratic source
function S[I,I], it has to be decomposed into multipoles S[I,I]ℓ,m . In principle, closing the hier-
archy requires multipoles up to infinite ℓ, but this is numerically not tractable. Hence, the
usual prescription is to truncate the hierarchy once convergence is reached. The LOS inte-
grand in Eq. (4.22) facilitates the convergence such that ℓ up to 10 is sufficient as shown in
Fig. 4.1. We include all the m ̸= 0 modes consistently up to ℓ= 10 for the quadratic source
function.
Numerical Results for Linear Perturbations
In order to calculate the 2nd-order perturbations and the corresponding CMB bispectrum,
we need the linear perturbations as sources. The idea is to store all the 1st-order perturba-
tions needed as functions of k and η into an huge array. Then, the values of those quadratic
terms at different k1, k2 and η are evaluated through the cubic spline interpolation with the
array. Unfortunately, many of the 1st-order quantities, such as multipoles of photons and
neutrinos, oscillate rapidly for small scales. Thus, high resolutions are required to make
sure that the oscillations are captured properly. Besides, we need the 1st-order quantities all
over the range of time from the initial time to the recombination.
Although the calculations on the linear perturbations have been well established in some
LOS solvers, such as CAMB [70] and CLASS [63], we reproduce the calculations with our
own code. One of the main reasons is that the sampling of k and η in these 1st-order
quantities for evaluating the 2nd-order equations is different from those used for calculating
the CMB power spectrum in these LOS solvers. For instance, the LOS solvers do not store
the 1st-order quantities well before the recombination because they have negligible effects
on the CMB power spectra15. In contrast, we need to store the 1st-order quantities from the
initial time to the recombination when we evaluate the 2nd-order Boltzmann equation and
EFEs. Thus, it is more efficient to develop our own code to control the sampling. Another
reason is that the 1st-order evolution equations are simply the homogeneous parts of the
2nd-order evolution equations (see Eq. (3.21)) and provide a decent check for the linear
parts of 2nd-order equations.
We developed our own code to calculate the full-sky CMB power spectrum with and
without the lensing effect16. The full-sky result without the lensing effect is shown in
15Photons are tightly coupled with baryons well before the recombination and any patterns of photons at
that time will be washed away by collisions.
16We also include the calculation of the unlensed power spectrum with the flat-sky approximation.
4.3 Numerical Results 83
Fig. 2.2 along with that from CAMB. We obtain a very nice agreement. Although the
calculation of the power spectrum is not necessary for studying the CMB bispectrum, the
calculation provides us some useful guides about how high the resolutions of k and η should
be used in bispectra.
Numerical Calculations of Second-Order Perturbations
Once the 1st-order perturbations are ready, the kernels of the 2nd-order perturbations can be
solved by the 2nd-order EFEs and Boltzmann equations. However, there are many quadratic
terms of 1st-order perturbations in these equations and it is not trivial to keep track of these
terms. Besides, the solutions obtained may subject to the issue of numerical instabilities.
Thus, it is important to establish a systematic check for the numerical 2nd-order solutions.
For the scalar perturbations, the verification can be achieved by using the redundancy
of the EFEs. There are four EFEs for the scalar perturbations and we only need to use two
of them to evolve the 2nd-order scalar metric perturbations, i.e. Φ[II] and Ψ[II] in Newtonian
gauge. However, the two redundant EFEs are consistent with the other two EFEs and can
thus be used to verify the numerical solutions. For example, if a term is missing in the EFEs
or the numerical instabilities grow significantly, the constraint equations will not be satisfied
well (see Figs. 3.3 to 3.5). We emphasize that the verification does not work well for the
collision terms of Boltzmann equation because the EFEs is sensitive only to the total energy
and momentum of all components. A missing collision term may change the energies and
momenta of baryons and photons but not the sum of their energies and momenta.
The 2nd-order perturbations were solved previously in [87]. However, there are some
typos for the quadratic terms in the EFEs. The third term 8H(Φ−Ψ)Ψ˙ of the first line in
Eq. (A.6) should be 8HΦΦ˙. The first term ∂I(Ψ+Ψ)∂ I(Ψ+Φ) in the square bracket of the
second line in Eq. (A.7) should be ∂I(Ψ+Φ)∂ I(Ψ+Φ). Moreover, the quadratic terms with
σS[I] in the EFEs are missing. These missing terms do not have overwhelming amplitudes
compared to all the other quadratic terms17. Thus, we do not expect that the large deviation
of the non-Gaussianities calculated in [87] comes from these terms.
17For example, the stress anisotropy σS[I] is usually subdominant except for very small scales.
84 CMB Bispectrum during Recombination
l
l
l
Figure 4.2 The 3D plot of the reduced bispectrum generated around recombination. The
bispectrum is normalized by the coefficient D(ℓ1, ℓ2, ℓ3) defined in Eq. (4.51) to remove
an overall ℓ−4 scaling. The red regions represent positive values while the blue regions
represent negative values of the recombination bispectrum.
4.3 Numerical Results 85
- 200
0
200
{
2
-{
3
100
200
300
{
1
- 1
0
1
b
{1{2{3
D@ {
1
, {
2
, {
3
D
- 1000
- 500
0
500
1000
{
2
-{
3
0
500
1000
{
1
- 1
0
1
b
{1{2{3
D@ {
1
, {
2
, {
3
D
- 1000
0
1000
{
2
-{
3
0
500
1000
1500
{
1
- 1
0
1
b
{1{2{3
D@ {
1
, {
2
, {
3
D
Figure 4.3 The cross sections of the normalized reduced bispectrum in Fig. 4.2 with the
conditions 12(ℓ1 + ℓ2 + ℓ3) = 400 (upper), 1100 (middle) and 1600 (lower).
86 CMB Bispectrum during Recombination
Recombination Bispectrum
We present the full recombination bispectrum in Fig. 4.2, showing isosurfaces of the bis-
pectrum density with the normalization factor [36]
D(ℓ1, ℓ2, ℓ3)≡
1
(2ℓ1 +1)(2ℓ2 +1)(2ℓ3 +1)
(
1
ℓ1 + ℓ2 + ℓ3 +3
+ 1ℓ1 + ℓ2 + ℓ3
)
. (4.51)
Although the main contribution to the recombination bispectrum is concentrated towards
the edges of the tetrahedron, the bispectrum fluctuates around zero in the squeezed limit
and hence its correlation with the local-type template is somewhat suppressed. Clearly from
Fig. 4.2, the bispectrum does not correlate particularly well with the popular templates –
local, equilateral or orthogonal. It possesses its own distinct shape. In Fig. 4.3, we show dif-
ferent tetrahedral cross-sections through the full recombination bispectrum taken at different
summations ℓ1 + ℓ2 + ℓ3 = const. On the other hand, it contains features in the squeezed
limit and along the edges which reflect those appearing in the ISW-lensing bispectrum - the
fluctuations along the edges of the tetrahedron. For the squeezed limit with ℓ1 ≪ ℓ2, ℓ3,
this is because they both contain a term with d(ℓ22Cℓ2)/dℓ2 (see Eq. (12) and (14) in [66])
which comes from modulations of large-scale fields. However, the ISW-lensing bispectrum
possesses an extra term which fluctuates according to the angle between the small ℓ1 and
the large ℓ2. More precisely, they have similar shapes when ℓ2 = ℓ3 but look very different
when ℓ1 + ℓ2 = ℓ3. This extra feature weakens the correlation between the ISW-lensing and
the recombination bispectra, which is only ∼ 30%.
4.3.2 Validation with Analytical Bispectrum
To verify the numerical bispectrum we calculated, we compare it with the analytical solu-
tion available in the squeezed limit. In the squeezed limit (ℓ1 ≪ ℓ2 and ℓ3), the reduced
bispectrum in the bolometric temperature has the simple form [12, 29]
bΘEℓ1ℓ2ℓ3 =Cℓ1Cℓ2 +Cℓ1Cℓ3 +Cℓ2Cℓ3−
CΘζℓ1
2
[
1
ℓ22
d(ℓ22Cℓ2)
dlnℓ2
+ 1ℓ23
d(ℓ23Cℓ3)
dlnℓ3
]
, (4.52)
where CΘζℓ1 is the correlation between the temperature anisotropies and the curvature per-
turbation ζ = △00−Ψ at recombination. Our bispectrum is in good agreement with the
analytical bispectrum as shown in Fig. 4.4. We note that the flat-sky approximation per-
forms well in the squeezed limit. The approximation breaks down only when ℓ1 . 200.
4.3 Numerical Results 87
The breakdown occurs earlier than mentioned in [17] for the flat-sky approximation. We
believe this is because the cutoff of the LOS integration is extended to η = 1050 Mpc in
order to include the couplings with the early-time ISW effect. In this case, the thin-shell
approximation may break down while the flat-sky approximation is still valid. However,
these ISW-related couplings should contribute mildly to the total bispectrum18.
500 1000 1500 2000
-6
-4
-2
0
2
4
{
b
{
{
1
0

H
2
C
{
1
0
+
C
{
{
L

6
All Q . T .
All L . T .
All
Analytic
Figure 4.4 The recombination bispectrum (gray solid line) calculated numerically using
Eq. (4.45) and the analytical bispectrum (black solid line) are plotted in the squeezed limit.
We can see that they match with each other very well. Here, we measure the bispectrum in
bolometric temperature which can be converted from the bispectrum in brightness temper-
ature by Eq. (4.46). The “All L.T.” curve contains all the purely 2nd-order perturbations of
the source function as shown in Eq. (4.10) while the “All Q.T.” curve contains all the cross
products of two 1st-order perturbations of the source function as shown in Eq. (4.11).
In addition, we remark that the numerical accuracy of the multipoles S[I,I]ℓ,m with m ̸= 0
cannot be checked with the analytical solution as they contribute negligibly in the squeezed
limit. The reason is as follows. In 1st-order perturbation theory, the source function
S[I]T (η ,k, nˆ) can be expressed as S˜
[I]
T (η ,k,µ)Φ(k) where µ ≡ nˆ · kˆ and Φ(k) is the pri-
mordial perturbation of the Newtonian potential. Since S˜[I]T is a function of µ , it can be
decomposed into spherical harmonics with only m = 0 modes. On the other hand, in 2nd-
order perturbation theory, the kernel of quadratic source function S[I,I] is a function of η and
nˆ as well as k1 and k2 which come from the couplings between two 1st-order perturbations.
The decomposition of S[I,I] into spherical harmonics will contain modes with m ̸= 0 because
k1 and k2 do not necessarily align with each other. In other words, the couplings break the
symmetry with m = 0. However, in the squeezed limit (e.g. k1≪ k2,k3), the conservation
18See Fig. 1 in [66], and the SW-ISW and γ-ISW effects in the lower panel of Fig. 4.5.
88 CMB Bispectrum during Recombination
law of momenta (k3 = k1 +k2) implies that k2 is almost parallel to k3. Thus, modes with
m ̸= 0 are suppressed because of the much smaller k1. In this case, we have only m = 0
modes when k3 is chosen to align with the azimuthal direction of the multipole decomposi-
tion. Similarly, the 2nd-order vector and tensor perturbations are negligible in the squeezed
limit as they are m = 1 and m = 2 modes respectively. Having said all these, we emphasize
that m ̸= 0 modes have to be considered for non-squeezed configurations.
In Fig. 4.5, we plot the bispectra of the terms in the 2nd-order source function. We
can see that the main contributions come from the effects of the Photon-SW, SW-SW and
quadratic Collisions as well as the purely 2nd-order SW, Doppler and anisotropic stress
effects. The sum of the Photon-SW and the SW-SW effects is roughly constant in the plot –
the total bispectrum from these two effects is approximately proportional to the product of
the power spectra of the long (ℓL) and short (ℓS) wavelength modes, i.e. CℓSCℓL . Moreover,
this constant offset also appears in Fig. 1 of [82]. This verifies that our IBPs approach
is consistent19 with the change-of-variables approach first introduced by [52]. Since we
decompose the quadratic source term S[I,I] into different physical couplings, the physical
explanation of the offset is clear: the power spectrum of the short wavelength mode comes
mainly from intrinsic intensity, SW and Doppler effects while the power spectrum of the
long wavelength mode is proportional to the square of the initial gravitational potential Φ
(SW effect). The offset shifts the bispectrum of the purely 2nd-order source terms up and
suppresses the correlation between the recombination bispectrum and that of the local type.
19The late-time ISW-related couplings are sub-dominant as expected.
4.3 Numerical Results 89
500 1000 1500 2000
-7
-6
-5
-4
-3
-2
-1
0
{
b
{
{
1
0

H
2
C
{
1
0
+
C
{
{
L

6
P
ISW
SW
Doppler
I
0
All L . T .
500 1000 1500 2000
-1
0
1
2
3
{
b
{
{
1
0

H
2
C
{
1
0
+
C
{
{
L

6
Collision
H
dp
d Τ
L
@ I , I D
SW -ISW
Γ-ISW
SW - SW
Γ- SW
All Q . T .
Figure 4.5 The bispectra of the linear terms (L.T.) and quadratic terms (Q.T.) of the source
function S2ND in squeezed limit are shown in the upper and lower panel respectively.
90 CMB Bispectrum during Recombination
4.3.3 fNL and Signal-to-noise
In Table 4.1, we summarize the effective fNLs, the normalized FNLs [35] and the signal-to-
noise ratios S/N which are defined as
fANL =
FA,rec
FA,A
(4.53)
FANL =
√
FA,A
Floc,loc
fANL (4.54)
(S/N)A =
√
FA,A (4.55)
where
FA,B ≡∑
ℓi
h2ℓ1ℓ2ℓ3
6Cℓ1Cℓ2Cℓ3
bAℓ1ℓ2ℓ3b
B
ℓ1ℓ2ℓ3 (4.56)
hℓ1ℓ2ℓ3 ≡
√
(2ℓ1 +1)(2ℓ2 +1)(2ℓ3 +1)
4π
(
ℓ1 ℓ2 ℓ3
0 0 0
)
. (4.57)
Physically, fANL measures the bias in estimating the amplitude of the A bispectrum due to the
CMB bispectrum during recombination. The FNL renormalizes fNL with respect to the local-
type bispectrum such that equitable comparisons among different templates are possible.
The signal-to-noise ratio (S/N)A measures the detectability of the A bispectrum for unit
fNL. The local-type F locNL is small because of the fluctuating bispectrum in squeezed limit
as we explained previously. The equilateral-type FequNL is small due to the lack of support in
the interior of the recombination bispectrum. Thus, the recombination bispectrum does not
correlate well with local and equilateral templates.
Model fNL FNL (S/N)
Equilateral 5.11 0.66 0.028
Local 0.88 0.88 0.22
ISW-Lensing 7.36
Auto-correlation — 3.19 0.69
Table 4.1 The table of the effective fNLs, FNLs and S/Ns of the local and equilateral
templates correlated to the recombination bispectrum, as well as its total signal (auto-
correlation). For ISW-lensing bispectrum, we found that its correlation with the recom-
bination bispectrum is ∼ 30%. For ease of comparison, the FNL quantities normalize the
integrated bispectrum signal for any shape relative to the fNL = 1 local model. We have
used ℓmax = 2000 throughout.
4.3 Numerical Results 91
These results are consistent in the squeezed limit with [12, 29]. Moreover, we find that
the equilateral type f equNL = 5.11 (F
equ
NL = 0.66) which is modest but consistent with [13].
Table 4.2 compares all the codes which are capable of implementing the full recombination
bispectrum. While there is a relatively good agreement on the values of f equNL , the values of
f locNL and S/N from different codes appear to be quite different. As we mentioned, the LOS
integrand used in [87] may mix up early-time and late-time redshift couplings and fail to
converge at low-ℓ truncation. This can result in the larger S/N and f locNL in their calculation.
Although we have made some approximations, we do not think that they lead to significant
uncertainties. The flat-sky approximation is validated by matching with the analytical solu-
tion in the squeezed limit as shown in Fig. 4.4. Although the thin-shell approximation may
add uncertainties on the bispectrum generated by the couplings with the early-time ISW
effect, these couplings are sub-dominant. The cutoff on the LOS integration allows us to
separate the early-time and late-time effects consistently. In particular, we confirm that the
m ̸= 0 modes, which are included in our calculation but not in [52, 82], increase the S/N
from 0.65 to 0.69. Their contribution to the bispectrum is mild.
Since the numerical calculation for the recombination bispectrum is very complicated
and can depend on the details of the implementations, a comparison between different codes
is essential to resolve the discrepancies among different teams in the future. This will be
important for the quantitative analysis of the Planck data where this contribution should be
incorporated in debiasing local and equilateral signals and in determining whether there is an
overall primordial non-Gaussian signature in the data. The recombination bispectrum will
combine with the ISW-lensing bispectrum at about 10% level and its correlation can affect
the significance of this determination in the Planck data (recall that ISW-lensing effect can
bias the local signal by as much as fNL = 9.5 [69]). For this reason, we shall continue
to incorporate more physical effects in our numerical pipeline to improve this quantitative
analysis further.
This work Pitrou [87] Huang [52] Pettinari [82]
S/N 0.69 ∼ 1 0.47 0.47
f equNL 5.11 ∼ 5 - 4.3
f locNL 0.88 ∼ 5 0.82 0.5
Table 4.2 The table for comparisons among the results from all the codes implementing the
full recombination bispectrum.
92 CMB Bispectrum during Recombination
4.4 Conclusions and Discussion
In this chapter, we focus on the bispectrum generated around recombination across the full
range of multipole combinations. We find that the effective fNLs of the equilateral and local
types are 5.11 and 0.88 respectively, while the overall signal-to-noise is (S/N)rec = 0.69, all
calculated using ℓmax = 2000. We note from Fig. 4.2 that the bispectrum possesses its own
distinct features differentiating it from well-known templates, such as local, equilateral and
ISW-lensing. To complete the full calculation of this bispectrum will require the inclusion
of the time-delay and lensing effects [49, 69], the addition of the 2nd-order vector and tensor
perturbations20, the late-time ISW effects, and finally the terms with the Christoffel symbols
in Eq. (2.35). With improving precision, this recombination bispectrum should be included
in the analysis of future CMB experiments.
20The 2nd-order vector and tensor perturbations have been included in [52, 82]. Their contributions are
believed to be subdominant.
Chapter 5
Weak Lensing in Boltzmann Equation
Cosmic microwave background (CMB) anisotropies generated at the last scattering surface
(LSS) propagating towards the present observer are inevitably distorted by the perturbed
metric of the intervening spacetime. Such distortions generate higher-order fluctuations
which include lensing, redshift and time-delay effects. Among these nonlinear effects, weak
lensing has the most significant impact on the CMB.
The Planck mission has conclusively detected lensing of the CMB radiation from fore-
ground sources to an overall significance of greater than 25σ [4]. The high precision of this
measurement motivates the development of a more complete formulation of the calculation
of this effect. While most effects on the CMB anisotropies are widely studied through di-
rect solutions of the Boltzmann equation, the nonlinear effect of CMB lensing is formulated
through the solutions of the geodesic equation. For the theoretical completeness, it is thus
of interest to include CMB lensing in the framework of the Boltzmann equation. Practi-
cally, the new formalism can be used as a consistency checking of the canonical remapping
approach1.
In this chapter, we aim to fill this gap. We present a new formalism to the calculation
of the lensing effect by directly solving the Boltzmann equation, as we did in the calcula-
tion of the CMB anisotropies at recombination. In addition to encoding the geometrical
information implicit in the use of the geodesic equations, our approach using the Boltzmann
equation also manifestly contains additional interactions, such as redshift effects and Comp-
ton scattering. We will discuss how these effects couple with lensing in high orders. From
our formalism, we explicitly re-derive the remapping approach in Eq. (2.88) and identify
1See Section 2.6.1 for a brief review or [67] for details.
94 Weak Lensing in Boltzmann Equation
the implicit approximations used in the remapping approach. Moreover, we developed a
diagrammatic approach to efficiently keep track of all the interaction terms and calculate all
possible non-trivial correlations to arbitrarily high orders. One of the primary benefits of our
approach is that the meaning of each physical term is clear and unambiguous. This allows
us to point out additional unexplored corrections implied in the remapping approach. To
demonstrate the formalism, we calculate explicitly the corrections to the temperature power
spectrum from lens-lens couplings, which are neglected in standard calculations. Without
going through the quantitative calculations of other approximations, we assess their contri-
butions to the CMB power spectrum. The work in this chapter is mainly based on [99].
This chapter is organized as follows. In section 5.1, we first derive the CMB weak
lensing effect from the 2nd-order Boltzmann equation. Then, we generalize the derivation
to arbitrarily high orders to establish the complete formalism and recover the remapping
approach in literature from our formalism by identifying the approximations required. After
that, we show how to express the lensing effect in form of Dyson Series and represent
the formalism through a diagrammatic approach. Finally, we discuss how the formalism
can be extended to study time-delay and redshift effects. In Section 5.2, we extend the
diagrammatic approach to calculations of the lensed CMB power spectrum. Focusing on
the corrections from lens-lens couplings, we illustrate how these diagrams can facilitate the
calculation of the lensing effects on the CMB power spectrum. We find that the correction
is. 0.1% of the CMB temperature power spectrum for ℓ up to 3000 and thus is comparable
to the cosmic variance.
5.1 Formalism
To study the CMB lensing effect, we start with the Boltzmann equation as in Eq. (2.32)2
L[Pab(xA, p0, nˆ)] = Cab(xA, p0, nˆ), (5.1)
with the Liouville operator L, the collision operator C and the screen-projected intensity
matrix Pab. The Liouville operator, which describes the free-streaming of the CMB pho-
tons, can be expressed under the chosen coordinate system and tetrad basis3 as Eq. (2.34),
2See Section 2.3.2 for a brief review on Boltzmann equation.
3See Section 2.1 for how we choose the coordinate system and the tetrad basis.
5.1 Formalism 95
i.e.
L[Pab] = SacSbd
(
dxA
dη ∇APcd +
dp0
dη
∂
∂ p0Pcd +
dni
dη
∂
∂niPcd
)
, (5.2)
where the screen projector Sab is defined in Eq. (2.6). As we will soon see, the first term
contains time-delay effect and Born approximation, the second term corresponds to redshift
effects including Sachs-Wolfe (SW) and integrated Sachs-Wolfe (ISW) effects and the third
term corresponds to lensing effect. To elaborate these terms, we need to know the geodesic
equation given in Eq. (2.37) and the metric of spacetime in Eq. (2.1). Hereafter, we use the
conformal time η instead of the affine parameter s by redefining the Liouville and collision
operators
L→
(
dη
ds
)
L, C→
(
dη
ds
)
C. (5.3)
That means, we restrict the calculation to the particular choice of coordinate system. How-
ever, this transformation is found convenient when we perform the line of sight integration
over the conformal time, e.g. in Eq. (5.14). As mentioned in Section 3.2.1, the transforma-
tion is equivalent to the transformation with simply the background-order (dη/ds)[0].
To solve the dynamics of the CMB photons, we perturb the Boltzmann equation and
the Einstein field equations (EFEs), and then solve these differential equations. As demon-
strated in Chapter 3 for the 2nd-order perturbations, quantities in lower orders are treated as
sources (or inhomogeneous parts) of the higher-order differential equations. That is, we can
start from the lowest (background) order and iteratively solve the equations order by order
until we get the precision required. Explicitly, we expand the quantities into perturbations
as
dX
ds
=
(
dX
ds
)[0]
+
(
dX
ds
)[I]
+ 1
2!
(
dX
ds
)[II]
+ 1
3!
(
dX
ds
)[III]
+ . . . , (5.4)
Pab = P¯ab+P [I]ab+
1
2!
P
[II]
ab +
1
3!
P
[III]
ab + . . . , (5.5)
where X denotes η , xI , p0 or ni.
The same expansion can be done for the collision terms Cab but we will focus on the
damping term − ˙¯τPab in Compton scattering with the residual terms collected in Dab, i.e.
Cab(xA, p0, nˆ) =− ˙¯τ(η)Pab(xA, p0, nˆ)+Dab(xA, p0, nˆ), (5.6)
96 Weak Lensing in Boltzmann Equation
where ˙¯τ(η) is the differential optical depth in the background order. The term − ˙¯τPab
is responsible for the damping effect on the CMB anisotropies. The Nth-order residual
term D[N]ab contains only contributions from the low multipoles (at most ℓ = 2) of the Nth-
order intensity matrix and cross terms of lower-order perturbations. For example, the low
multipoles contribute to intrinsic intensity and Doppler effect in Nth order. Pragmatically,
these low multipoles are calculated through the Boltzmann equation and the EFEs in Nth
order with a truncated value of ℓ. Then, they are fed back into the Boltzmann equation,
which we will then solve using the line of sight approach. Moreover, the residual term D[N]ab
also contains the perturbed differential optical depth δ τ˙(xA). This is only important when
we care about the nonlinear dynamics of Compton scattering. Since our focus is the CMB
weak lensing, the linear-order D[I]ab is sufficient. See Approximation 1 in Section 5.1.2 for a
more detailed discussion. Nevertheless, we emphasize that there is no dependence of P [N]ab
inside the residual term D[N]ab so it can be treated as a source term in the evolution equations
of P [N]ab .
Putting together both the Liouville terms Eq. (5.2) and the collision terms Eq. (5.6), we
can formulate the solution of the Nth-order intensity matrix P [N]ab by using the line of sight
approach as follows. The Boltzmann equation in Nth order can be written as
Sa
cSb
d
{
N
∑
L=0
(
N
L
)[(
dxI
dη
)[L] ∂P [N−L]cd
∂xI +
(
dp0
dη
)[L] ∂P [N−L]cd
∂ p0 +
(
dni
dη
)[L] ∂P [N−L]cd
∂ni
]}
+(∇ηPab)[N] =− ˙¯τP [N]ab +D
[N]
ab , (5.7)
where
(N
L
)
is the Binomial coefficient. The covariant derivative in Eq. (5.7) can be expanded
using Eq. (2.35). Here, we make an approximation by ignoring the terms with the Christoffel
symbols in Eq. (2.35) as Approximation 0, i.e. ∇ηPab≈ ∂Pab/∂η . We rearrange Eq. (5.7)
by putting all the terms with P [N]ab on the left-hand side, such that
∂P [N]ab
∂η +n
i∂P
[N]
ab
∂xI −Hp
0∂P
[N]
ab
∂ p0 +
˙¯τP [N]ab = SacSbdQ
[N]
cd , (5.8)
where we have collected all the source terms in Q[N]ab which is defined as
D[N]ab −
N
∑
L=1
(
N
L
)[(
dxI
dη
)[L] ∂P [N−L]ab
∂xI +
(
dp0
dη
)[L] ∂P [N−L]ab
∂ p0 +
(
dni
dη
)[L] ∂P [N−L]ab
∂ni
]
.
(5.9)
5.1 Formalism 97
Henceforth, the term ξ e¯
I
i
will be omitted in all contractions between i and I, e.g. ni∂/∂xI in
the second term of Eq. (5.8). Performing the Fourier transform such that any function of x
transforms as
f (x) =
∫
d3k
(2π)3/2 e
ik·x f (k), (5.10)
we obtain
∂Pˆ [N]ab
∂η + ik · nˆPˆ
[N]
ab + ˙¯τPˆ
[N]
ab = SacSbdQˆ
[N]
cd , (5.11)
where Pˆab and Qˆab are defined as
Pˆab(η ,k, nˆ)≡
1
I¯(η)
∫
dp0(p0)3Pab(η ,k, p0, nˆ), (5.12)
Qˆab(η ,k, nˆ)≡
1
I¯(η)
∫
dp0(p0)3Qab(η ,k, p0, nˆ), (5.13)
with the background-order energy-integrated photon intensity I¯(η) defined in Eq. (2.16).
Strictly speaking, we note that k · nˆ = nikI ξ e¯I i and contains no perturbations.
By integrating Eq. (5.11) over the line of sight, the Nth-order intensity matrix at present
(η = η0) can be expressed as
Pˆ
[N]
ab (η0,k, nˆ) =
∫ η0
0
dηe−ik·nˆr−τ¯(η)SacSbdQˆ
[N]
cd (η ,k, nˆ), (5.14)
with the background-order optical depth τ¯(η) and the conformal distance r ≡ η0−η . We
remark that all the quantities in Qˆ[N]ab can be determined either by solving the Einstein field
equations and the Boltzmann equation with a low truncated ℓ for up to Nth order or by per-
forming the line of sight integral for Pˆ [M]ab with M < N [83]. In other words, we can extend
the calculation to any orders we want iteratively, limited only by computational power and
human frailty.
5.1.1 Lensing in Second Order
In this subsection, we warm up by deriving the weak lensing effect on the CMB from the
2nd-order Boltzmann equation. We will match our result with the previous approach using
the remapping approach, i.e. Eq. (2.87), explicitly stating the assumptions needed for the
98 Weak Lensing in Boltzmann Equation
matching, and explaining their physical meanings. We will generalize the derivation to
arbitrarily high orders in the next subsection. For an alternative approach at formulating the
2nd-order lensing effect through the 2nd-order Boltzmann equation, see [53].
First of all, we expand the 2nd-order source term Q[II]ab of Eq. (5.9) into
D[II]ab −2
(
dp0
dη
)[I] ∂P [I]ab
∂ p0 −
(
dp0
dη
)[II] ∂P¯ab
∂ p0 −2
(
dxI
dη
)[I] ∂P [I]ab
∂xI −2
(
dni
dη
)[I] ∂P [I]ab
∂ni .
(5.15)
The first term in Eq. (5.15) corresponds to Compton scattering, the second and third terms
are related to the redshifts due to the perturbed metric along the light path, the fourth term
is responsible with the time-delay effect, and the last term is the weak lensing effect. Phys-
ically, the second and third terms can be interpreted as redshift effects of the 1st-order
perturbed metric on the 1st-order CMB anisotropies and the 2nd-order perturbed metric on
the background-order CMB respectively.
Dropping all the other terms except the weak lensing term, considering only the 1st-
order Compton scattering4, and after performing Fourier transformation and energy integra-
tion, we have5
Qˆ[II]ab (η ,k, nˆ) =Tk1,k2(k)
{
2iSi jk1, j [Φ(η ,k1)+Ψ(η ,k1)]
∂
∂niPˆab(η ,k2, nˆ)
}
, (5.16)
where T is the convolution operator defined in Eq. (3.24) and Si j is the screen projector in
tetrad basis. Here, we express (dni/dη)[I] with Eq. (3.56) and replace the background-order
k1,J = ξ e¯ jJk1, j with k1, j. We refer to Section 2.1 for a general review on coordinate system,
tetrad basis and manipulations of their corresponding indices.
From now on, we focus on the photon intensity only. This can be done by the contraction
Iˆ(η ,k, nˆ) = SabPˆab(η ,k, nˆ). (5.17)
Moreover, we define the CMB temperature anisotropies Θ as
1+4Θ(η ,k, nˆ)≡ Iˆ(η ,k, nˆ). (5.18)
4The correction of the direction nˆ in the Doppler term nˆ · vb is neglected when we ignore the 2nd-order
Compton scattering.
5From now on, we omit the superscript [I] denoting 1st-order perturbations whenever it is clear in the
context.
5.1 Formalism 99
In general, the definition of temperature is not unique, especially in nonlinear order with ex-
isting CMB spectral distortions. However, if we care about only lensing effects in nonlinear
order, there are no spectral distortions induced and thus the lensed CMB temperature can
be defined as usual in linear order. For more discussions on the temperature definition, see
Section 4.1.
With the 1st-order expression of Θ in Eq. (2.75), and after several integration by parts,
we obtain
∂Θ[I]
∂ni (η ,k, nˆ) = iki
[
eτ¯(η)
∫ η
0
dη˜eikµ(η˜−η)(η˜−η)ST (η˜ ,k)+
3
2k2
˙¯τ(η)Π(η ,k)
]
, (5.19)
where the source term ST is defined in Eq. (2.78).
To solve the 2nd-order lensed temperature anisotropies Θ˜[II](nˆ), we substitute Eq. (5.16)
into Eq. (5.14) and replace Pˆab with temperature anisotropies Θ. Then, by using Eq. (5.19),
the lensed temperature anisotropies can be written as
Θ˜[II](nˆ) =
∫
d3k1d3k2
(2π)3
∫ η0
0
dr 2k1k2(µ−µ1µ2)[Φ(η ,k1)+Ψ(η ,k1)]e−ik1µ1r
×
[∫ η0
r
dr˜e−ik2µ2r˜(r˜− r)ST (η˜ ,k2)−
3
2k22
e−ik2µ2rg(η)Π(η ,k2)
]
, (5.20)
where
µ ≡ kˆ1 · kˆ2 , µ1 ≡ nˆ · kˆ1 , µ2 ≡ nˆ · kˆ2 , r ≡ η0−η , r˜ ≡ η0− η˜ . (5.21)
We will now re-derive the remapping formula Eq. (2.87) and Eq. (2.88) of the weak
lensing from the Boltzmann equation. Unlike the familiar approach [23] where the lensing
deflection is calculated via solving the geodesic equations given the metric perturbations,
we explicitly solve the 2nd-order Boltzmann equation using Green’s function method. This
means that we have full control over the entire evolution of the photon distribution from
the LSS to today, allowing us to make clear the exact approximations required to recover
Eq. (2.87) and Eq. (2.88) as follows:
1. We neglect the term with Π in Eq. (5.20) and re-write the equation as
Θ˜[II](nˆ) =−2Si j
∫ η0
0
dr
∂
∂ni
{∫
d3k1
(2π)3/2 e
−ik1·nˆr [Φ(η ,k1)+Ψ(η ,k1)]
}
×
∫ η0
r
dr˜
r˜− r
rr˜
∂
∂n j
[∫
d3k2
(2π)3/2 e
−ik2·nˆr˜ST (η˜ ,k2)
]
. (5.22)
100 Weak Lensing in Boltzmann Equation
From the integration limits of the double integral, we can see that the lenses at r
can only distort the signals of sources further away, i.e. from r to η0, as expected.
The neglected term here comes from the boundary condition when we perform the
integration by parts in Eq. (5.19). In general, these boundary terms appear only for
those terms with modes ℓ≥ 2 in the source function S˜T defined in Eq. (2.75).
2. Then, we replace (r˜−r)/r˜ with (rLSS−r)/rLSS, contract
∫ η0
0 dr to
∫ rLSS
0 dr and extend∫ η0
r dr˜ to
∫ η0
0 dr˜ (i.e. effectively uncoupling the two integrals). We get
Θ˜[II](nˆ) = 2Si j ∂∂niψ(nˆ)
∂
∂n jΘ(nˆ) = 2∇nˆψ(nˆ) ·∇nˆΘ(nˆ), (5.23)
where ψ(nˆ) is the lensing potential defined in Eq. (2.88) and we have used the fact
that the screen-projected directional derivative
Si j
∂
∂n j = (∇nˆ)
i (5.24)
in the second equality. This approximation implies that we first integrate all the un-
lensed CMB signals – including late-time effects, such as the ISW effect and Compton
scattering at reionization – and then we treat those unlensed signals as a single source
at the LSS and distort them by lenses between the LSS and the observer. This ap-
proximation is motivated by the fact that most unlensed CMB signals are generated
at recombination and thus can be treated as a single source. Moreover, the lensing on
the late-time effects can be ignored. For example, the ISW effect is dominant at low
ℓ where the lensing effect is negligible.
If we do not perform the integration by parts to replace S˜T (η ,k, nˆ) with ST (η ,k), the
first approximation will not be needed6. In other words, the first approximation can be
folded into the second approximation. Here, we separate them in order to compare our
result to previous studies. This “single-source approximation” has been evaluated in [107]
for the weak lensing effect on the CMB power spectra. At 2nd order, their equation (Eq. (7))
for redshift-varying sources is exactly the same as Eq. (5.22). However, the term with Π is
missing in their study. In principle, we have to take into account the missing boundary terms
when we assume the single-source approximation.
6See the generalized derivation for high orders in Section 5.1.2.
5.1 Formalism 101
5.1.2 Lensing in Arbitrarily High Orders
We will now extend the calculation of the lensing effect to arbitrarily high orders and com-
plete the formalism. At 3rd and higher orders, the lensing effect can couple with other
effects non-trivially. For example, the 2nd-order lensed photons can be redshifted by linear
metric perturbations included in the term
(
dp0
dη
)[I] ∂P [II]ab
∂ p0 . (5.25)
Thus, it can be ambiguous to distinguish the lensing effect from other effects in high orders.
We will demonstrate how to derive the usual remapping approach [23] by explicitly
solving the Boltzmann equation. Throughout the derivation, we will clarify all the assump-
tions needed. Validations of these approximations can be found in Section 5.2. We start
with the source term Eq. (5.9) and make the following approximations:
1. Ignore nonlinear collision terms: We include only the 1st-order D[I]ab from Compton
scattering in Eq. (5.6). Nevertheless, we remark that other collision-related effects
can be important, such as the SZ effect on small scales at late time. However, the
lensing effects on them are believed to be negligibly small [106] and thus these effects
can be studied separately7. Without these collision terms, there are no distortions on
the frequency spectra of Pab. In other words, there exists a unique definition of
temperature anisotropies (see Section 4.1 for more details).
2. No time-delay: We drop all the perturbations on dxI/dη , i.e.
dxI
dη ≈ n
i. (5.26)
This means that when we accumulate the lensing effects, we perform the time integra-
tion along straight lines in the spacetime of the background manifold. This approxi-
mation is responsible for ignoring the time-delay effect and the Born approximation.
7See [28, 39] for the SZ effect on bispectrum.
102 Weak Lensing in Boltzmann Equation
The leading time-delay effect is a 2nd-order effect and corresponds to the term8
(
dxI
dη
)[I] ∂P [I]ab
∂xI . (5.27)
Physically, it is the effect of the 1st-order perturbed light paths on the 1st-order CMB
anisotropies. In contrast, the corrections from the Born approximation are at least
3rd-order [67] and include terms like
(
dxI
dη
)[I] ∂P˜ [II]ab
∂xI , (5.28)
where P˜ [II]ab is the 2nd-order lensed CMB signals. This is the effect of the 1st-order
perturbed light paths on the lensed CMB signals. Here, we emphasize that couplings
between different effects (e.g. lensing, redshift and time-delay effects) are possible
in higher orders and thus it can be ambiguous to distinguish between the time-delay
effect and Born corrections beyond their leading orders. Therefore, we instead clarify
what terms are neglected as in Eq. (5.26).
3. No redshifting by metric perturbations at nonlinear orders: Similarly, we drop all the
perturbations on dp0/dη except
(
dp0
dη
)[I] ∂P¯ab
∂ p0 , (5.29)
which is responsible for the SW and ISW effects in linear order. In other words,
we ignore any contributions due to redshifting beyond the linear order. In principle,
redshift-related late-time effects, such as the Rees-Sciama (RS) effect, may be impor-
tant9. In Section 5.2.3, we will see that the high-order couplings between the lensing
effect and the late-time redshifting are subdominant compared to the pure lensing
effect.
4. Ignore cross terms between metric perturbations: We approximate the term dni/dη
8In this chapter, we do not re-derive the previous approach [49] of the time-delay effect from the Boltzmann
equation explicitly as we will for the remapping approach of the lensing effect in Section 5.1.1 and 5.1.2.
Instead, we will argue that the interaction operator Eq. (5.69) responsible for the time-delay effect and the
Born approximation is negligible compared to that of the lensing effect, which is consistent with [49].
9See [90] and [74] for the RS effect on the CMB power spectrum and bispectrum respectively.
5.1 Formalism 103
as
dni
dη =
∞
∑
N=1
1
N!
(
dni
dη
)[N]
≈−
∞
∑
N=1
Si j
N!
∂J(Ψ[N]+Φ[N])≡−Si j∂J(ΨNL +ΦNL). (5.30)
At Nth order, it means that we ignore all the cross terms of lower-order (less than N)
perturbations. We call this the Newtonian approximation because we drop the cross
terms and linearize General Relativity (GR) as if in Newtonian gravity.
With these approximations, the source term Qab from Eq. (5.9) is simplified to
Qab =D[I]ab−
(
dp0
dη
)[I] ∂P¯ab
∂ p0 +2S
i j∂JΨNLW
∂Pab
∂ni , (5.31)
where the nonlinear Weyl potential ΨNLW ≡ (ΨNL +ΦNL)/2. Physically, the first three ap-
proximations imply that we consider only the pure lensing effects acting on the 1st-order
intensity matrix. In principle, we have to consider the nonlinear intensity matrix, especially
when we calculate the N-point correlation at large N. Having said that, we expect that the
lensing effects on the 2nd-order (and higher-order) intensity matrix generated at recombi-
nation are negligible in the CMB temperature power spectrum and bispectrum. For the
power spectrum, it is because |Θ[I]| ≫ |Θ[II]|. On the other hand, the 2nd-order tempera-
ture anisotropies Θ[II] at recombination generate a mild bispectrum [52, 82, 100]. Thus, the
lensing effect on Θ[II] at recombination is expected to be subdominant.
We will solve Eq. (5.31) order by order in perturbation theory as follows. First, we
expand the intensity matrix order by order
Pˆab(η0,k, nˆ) =
∞
∑
N=1
1
N!
Pˆ
[N]
ab (η0,k, nˆ). (5.32)
Now we impose the condition, which follows from Approximations 1 and 3, that the 1st-
order intensity matrix is sourced by the 1st-order collision termD[I]ab and redshifting
(
dp0
dη
)[I]
,
i.e. using Eq. (5.11)
∂Pˆ [I]ab
∂η + ik · nˆPˆ
[I]
ab+ ˙¯τPˆ
[I]
ab = SacSbdDˆ
[I]
cd +2Sab(Ψ˙Do− ik · nˆΦ). (5.33)
On the right hand side of Eq. (5.33), we have made use of the fact that there are no polar-
izations in the background order, i.e. P¯ab = (1/2)I¯Sab. Reinserting the solution Pˆ [I]ab back
into Eq. (5.11) with Approximations 1 to 4, and omitting for simplicity the k dependence of
104 Weak Lensing in Boltzmann Equation
Pˆ [II]ab , we get
∂Pˆ [II]ab
∂η + ik · nˆPˆ
[II]
ab + ˙¯τPˆ
[II]
ab = 2SacSbdTk′,k′′(k)
{
2iSi jk′jΨ
NL
W (k′)
∂Pˆ [I]cd(k′′)
∂ni
}
. (5.34)
Iterating this, we get at each N ≥ 1
∂Pˆ [N+1]ab
∂η + ik · nˆPˆ
[N+1]
ab + ˙¯τPˆ
[N+1]
ab
= (N+1)!
N!
Sa
cSb
dTk′,k′′(k)
{
2iSi jk′jΨ
NL
W (k′)
∂Pˆ [N]cd (k′′)
∂ni
}
. (5.35)
From now on, we focus on the temperature anisotropies. Similar to Eq. (5.14), we use
the line of sight approach and Eq. (5.31) to solve Eq. (5.35) as
1
(N+1)! Iˆ
[N+1](ηN+1,kN+1, nˆ) (5.36)
= eikN+1·nˆrN+1+τ¯(ηN+1)
∫ ηN+1
0
dηN
(
−
2
rN
)
Tk′N ,kN (kN+1)
{
∇iNnˆ
[
e−ik
′
N ·nˆrNΨNLW (ηN ,k′N)
]
×
(
∂
∂niN + ikN,iN rN
)[
e−ikN ·nˆrN−τ¯(ηN)
1
N!
Iˆ[N](ηN ,kN , nˆ)
]}
,
where N ≥ 1 denotes the Nth time of iteration, rN ≡ η0−ηN and Iˆ[1](η1,k1, nˆ) is the 1st-
order photon intensity, i.e. 4Θ[I](η1,k1, nˆ) in Eq. (2.75). For ηN+1 = η0 at present, we can
write the (N+1)th iteration explicitly as the following N-nested integral
1
(N+1)! Iˆ
[N+1](η0,kN+1, nˆ) =
∫ η0
0
dη˜
{
∫ η0
η˜
dηN
(
−
2
rN
)
Tk′N ,kN (kN+1)
{
∇iNnˆ
[
e−ik
′
N ·nˆrNΨNLW (ηN ,k′N)
]( ∂
∂niN + ikN,iN rN
){
. . .
∫ η3
η˜
dη2
(
−
2
r2
)
Tk′2,k2(k3)
{
∇i2nˆ
[
e−ik
′
2·nˆr2ΨNLW (η2,k′2)
]( ∂
∂ni2 + ik2,i2r2
){
∫ η2
η˜
dη1
(
−
2
r1
)
Tk′1,k1(k2)
{
∇i1nˆ
[
e−ik
′
1·nˆr1ΨNLW (η1,k′1)
]( ∂
∂ni1 + ik1,i1r1
){
e−ik1·nˆr˜4S˜T (η˜ ,k1, nˆ)
}}}}
. . .
}}
}
. (5.37)
In Eq. (5.37), we have used a familiar strategy often encountered in quantum field theory to
5.1 Formalism 105
rearrange the order of the integration limits
∫ ηM+1
0
dηM
∫ ηM
0
dη˜ =
∫ ηM+1
0
dη˜
∫ ηM+1
η˜
dηM (5.38)
to pull out the innermost integration with respect to η˜ which integrates over the sources.
This trick allows us to explicitly expose the physical meaning of the nested integral – the
lens at ηN , which generates the (N+1)th-order intensity from the Nth-order intensity, must
be located after the lenses at ηM (M < N) but before the observer at η0. The time-ordering
is illustrated in Fig. 5.1.
η˜ < η1 < η2 < . . . < ηN < η0
Figure 5.1 The diagrammatic illustration for the time-ordering of the nested integral in
Eq. (5.37). The circle node indicates the source while the square nodes indicate the lenses.
Following Eq. (5.32), the lensed photon intensity at present is the sum of all possible
iterations, i.e.
Iˆ(η0,k, nˆ) =
∞
∑
N=1
1
N!
Iˆ[N](η0,k, nˆ). (5.39)
Eq. (5.37) and Eq. (5.39) form the complete formula for the pure lensing effect on the 1st-
order photon intensity and include all possible ways to distort the 1st-order photon intensity
with the weak lensing effect.
In the following, we proceed to recover the remapping approach of the lensing effect
in Eq. (2.87) and Eq. (2.88) from the formulae shown in Eq. (5.37) and Eq. (5.39). Several
further approximations have to be made and we will clarify them explicitly. Before we do
that, we first commute the terms ∂/∂ni + ikir with the terms ∇inˆ[e−ik
′·nˆrΨNLW (η ,k′)] and
rearrange the former to the right hand side of the latter in Eq. (5.37). The commutation can
be achieved by using10
(
∂
∂niN + ikN,iN rN
){
∇iMnˆ
[
e−ik
′
M ·nˆrMΨNLW (ηM,k′M)
]
F(·)
}
(5.40)
= ∂S
iM jM
∂niN
∂e−ik′M ·nˆrM
∂n jM Ψ
NL
W F(·)+∇iMnˆ
[
e−ik
′
M ·nˆrMΨNLW
]( ∂
∂niN − ik
′
M,iN rM + ikN,iN rN
)
F(·)
iteratively for M < N, with F(·) as some arbitrary function and we have used Eq. (5.24) to
10We omit the dependences of ΨNLW on the right hand side of Eq. (5.40).
106 Weak Lensing in Boltzmann Equation
commute the two derivatives
∂
∂niN (∇nˆ)
iM = ∂∂niN
(
SiM jM
∂
∂n jM
)
= ∂S
iM jM
∂niN
∂
∂n jM +S
iM jM ∂
∂niN
∂
∂n jM . (5.41)
Now, we list out the extra approximations needed:
5. Neglect the directional derivative of the screen projector: This means that
∂SiM jM
∂niN ≈ 0. (5.42)
We can then move all the partial derivatives into the innermost integrand to act on
S˜T (η˜), and rewrite Eq. (5.37) as
1
(N+1)! Iˆ
[N+1](η0,kN+1, nˆ)
=
∫ η0
0
dη˜
{
∫ η0
η˜
dηN
(
−
2
rN
)
Tk′N ,kN (kN+1)
{
∇iNnˆ
[
e−ik
′
N ·nˆrNΨNLW (ηN ,k′N)
]{
. . .
∫ η3
η˜
dη2
(
−
2
r2
)
Tk′2,k2(k3)
{
∇i2nˆ
[
e−ik
′
2·nˆr2ΨNLW (η2,k′2)
]{
∫ η2
η˜
dη1
(
−
2
r1
)
Tk′1,k1(k2)
{
∇i1nˆ
[
e−ik
′
1·nˆr1ΨNLW (η1,k′1)
]{
[
∂
∂niN + i
N−1
∑
M=1
k′M,iN (rN− rM)+ ik1,iN rN
]
. . .
[
∂
∂ni2 + ik
′
1,i2(r2− r1)+ ik1,i2r2
]
×
[
∂
∂ni1 + ik1,i1r1
]
e−ik1·nˆr˜4S˜T (η˜ ,k1, nˆ)
}}}}
. . .
}}
}
. (5.43)
This approximation is related to the fact that the argument (nˆ+α ) in Eq. (2.87) is
not a unit vector. This approximation is implied in the remapping approach and has
been clarified in [22]. Nevertheless, the leading correction from this approximation
is at 3rd order11 and thus we do not expect significant corrections on the CMB power
spectra and bispectra. In Section 5.2.1, we will develop a set of diagrams to represent
different couplings in Eq. (5.43). In addition, we show that Eq. (5.43) can formally
be expressed as a Dyson series in Section 5.1.3.
6. Ignore lens-lens couplings: Operationally, this means that we drop all the terms with
11The directional derivative of the screen projector requires ∂/∂ni from a lens acting on ∇ jnˆ of another lens
as shown in Eq. (5.37). Therefore, the photon intensity is at least in 3rd order including the 1st-order source
term S˜T .
5.1 Formalism 107
k′M for any M in the second last line of Eq. (5.43). Physically, we ignore all the lensing
effects on lenses, i.e. lens-lens couplings [26]. We will consider the effects of the lens-
lens couplings in Section 5.2, but for the moment we continue our re-derivation of the
remapping approach. This approximation decouples all the Fourier integrals, allowing
us to immediately write down the photon intensity in configuration space as
1
(N+1)! Iˆ
[N+1](η0, nˆ)
=
∫ η0
0
dη˜
{
∫ η0
η˜
dηN
(
−
2
rN
)
∇iNnˆ
[
ΨNLW (ηN ,−nˆrN)
]
{
. . .
∫ η3
η˜
dη2
(
−
2
r2
)
∇i2nˆ
[
ΨNLW (η2,−nˆr2)
]
{
∫ η2
η˜
dη1
(
−
2
r1
)
∇i1nˆ
[
ΨNLW (η1,−nˆr1)
]
{
∫
d3k1
(2π)3/2
(
∂
∂niN + ik1,iN rN
)
. . .
(
∂
∂ni2 + ik1,i2r2
)(
∂
∂ni1 + ik1,i1r1
)
× e−ik1·nˆr˜4S˜T (η˜ ,k1, nˆ)
}}
. . .
}}
, (5.44)
where
ΨNLW (ηN ,−nˆrN)≡
∫
d3k′N
(2π)3/2 e
−ik′N ·nˆrNΨNLW (ηN ,k′N). (5.45)
7. Place all unlensed CMB signals at the LSS: Explicitly, we replace
∫ ηM+1
η˜
dηM −→
∫ ηM+1
ηLSS
dηM, (5.46)
(
∂
∂niM + ik1,iMrM
)
e−ik1·nˆr˜ =− rM− r˜
r˜
∂
∂niM e
−ik1·nˆr˜ −→−
rM− rLSS
rLSS
∂
∂niM e
−ik1·nˆr˜,
(5.47)
for any M. Notice that in order to recover the remapping approach, we do not replace r˜
with rLSS in the exponential. It means that when we utilize the line of sight approach
for the unlensed temperature anisotropies, we consider the effects at various times
properly. In contrast, when we calculate the lensing effects on the source term, we
treat the sources as if they are all located at the LSS. This approximation decouples
all the time integrations such that the nested integral is deconvolved into a product of
108 Weak Lensing in Boltzmann Equation
integrals as follows
1
(N+1)! Iˆ
[N+1](η0, nˆ)
= 2
∫ η0
ηLSS
dηN
(
rN− rLSS
rN rLSS
)
∇iNnˆ
[
ΨNLW (ηN ,−nˆrN)
]
{
. . .
2
∫ η3
ηLSS
dη2
(
r2− rLSS
r2 rLSS
)
∇i2nˆ
[
ΨNLW (η2,−nˆr2)
]
{
2
∫ η2
ηLSS
dη1
(
r1− rLSS
r1 rLSS
)
∇i1nˆ
[
ΨNLW (η1,−nˆr1)
]
{
∂
∂niN . . .
∂
∂ni2
∂
∂ni1
[∫
d3k1
(2π)3/2
∫ η0
0
dη˜ e−ik1·nˆr˜4ST (η˜ ,k1)
] }}
. . .
}
= 4
N!
α iN . . .α i2α i1 ∂∂niN . . .
∂
∂ni2
∂
∂ni1 Θ(nˆ), (5.48)
where the deflection angle α is expressed as
α (nˆ) = 2
∫ rLSS
0
dr
r− rLSS
r rLSS
∇nˆΨ
NL
W (η ,−nˆr) (5.49)
and the unlensed temperature anisotropies are
Θ(nˆ) =
∫
d3k1
(2π)3/2
∫ η0
0
dη˜ e−ik1·nˆr˜ST (η˜ ,k1). (5.50)
Finally, summing up all orders, we recover the remapping approach of the weak lensing
effect on the photon intensity [23], i.e.
Iˆ(η0, nˆ)−1 =
∞
∑
N=1
1
N!
Iˆ[N](η0, nˆ) =
∞
∑
N=0
4
N!
α iN . . .α i2α i1 ∂∂niN . . .
∂
∂ni2
∂
∂ni1 Θ(nˆ)
=4Θ(nˆ+α ) = 4Θ˜(nˆ), (5.51)
where Θ˜ denotes the lensed temperature anisotropies.
5.1.3 Lensing as Dyson Series
The Dyson series is a perturbative series which is used often to expand perturbations in
different orders when we study scattering theory. Here, we demonstrate how to formulate
the CMB lensing effect into a Dyson series. In this way, we start with the Boltzmann equa-
5.1 Formalism 109
tion to study the lensing effect which is a more complete form compared to the remapping
approach. In particular, the Dyson series allows us to take into account the fact that the
unlensed CMB effects occur at different times instead of treating them as a single source in
the remapping approach. If we ignore the Π term in Eq. (5.19), we can replace S˜T (η˜ ,k1, nˆ)
with ST (η˜ ,k1) and rewrite Eq. (5.43) as
1
(N+1)! Iˆ
[N+1](η0,kN+1, nˆ)
=
∫ η0
0
dη˜
{
∫ η0
η˜
dηN
(
−
2
rN
)
Tk′N ,kN (kN+1)
{
∇iNnˆ
[
e−ik
′
N ·nˆrNΨNLW (ηN ,k′N)
]{
. . .
∫ η3
η˜
dη2
(
−
2
r2
)
Tk′2,k2(k3)
{
∇i2nˆ
[
e−ik
′
2·nˆr2ΨNLW (η2,k′2)
]{
∫ η2
η˜
dη1
(
−
2
r1
)
Tk′1,k1(k2)
{
∇i1nˆ
[
e−ik
′
1·nˆr1ΨNLW (η1,k′1)
]{
[
i
N−1
∑
M=1
k′M,iN (rN− rM)+ ik1,iN (rN− r˜)
]
. . .
[
ik′1,i2(r2− r1)+ ik1,i2(r2− r˜)
]
×
[
ik1,i1(r1− r˜)
]
e−ik1·nˆr˜4ST (η˜ ,k1)
}}}}
. . .
}}
}
. (5.52)
With the assumption as shown in Eq. (5.42), we can express the photon intensity in
configuration space as
1
(N+1)! Iˆ
[N+1](η0, nˆ)
=
∫ η0
0
dη˜
{
∫ η0
η˜
dηN∇iNnˆ
[
ΨNLW (ηN ,−nˆrN)
]
(ˆnˆ,rN )iN
{
. . .
∫ η3
η˜
dη2∇i2nˆ
[
ΨNLW (η2,−nˆr2)
]
(ˆnˆ,r2)i2
{
∫ η2
η˜
dη1∇i1nˆ
[
ΨNLW (η1,−nˆr1)
]
(ˆnˆ,r1)i1
{
4ST (η˜ ,−nˆr˜)
}}
. . .
}}
, (5.53)
where
ST (η˜ ,−nˆr˜)≡
∫
d3k1
(2π)3/2 e
−ik1·nˆr˜ST (η˜ ,k1) (5.54)
110 Weak Lensing in Boltzmann Equation
and the vector operator ˆnˆ,r is defined such that it is non-zero only when it acts on X
(ˆnˆ,r)i X (η ′,−nˆr′)≡ 2
r− r′
r r′
∂
∂niX (η
′,−nˆr′), (5.55)
with X denoting ΨNLW or ST . Physically, acting ˆnˆ,r on Ψ
NL
W corresponds to lens-lens cou-
plings while acting on ST corresponds to lensing the sources. The diagrammatic approach
in Section 5.1.4 immediately applies to Eq. (5.53) by representing 0 as ST instead of Θ.
Now, we introduce the interaction operator Vˆ and the evolution operator Uˆ as
Vˆ (η , nˆ)≡ ∇inˆ
[
ΨNLW (η ,−nˆr)
]
(ˆnˆ,r)i, (5.56)
Uˆ(η0, η˜ , nˆ)≡ 1+
∞
∑
N=1
UˆN(η0, η˜ , nˆ) = T
[
e
∫ η0
η˜ dηVˆ (η ,nˆ)
]
, (5.57)
where r ≡ η0−η , T is the time-ordering operator, and
UˆN(η0, η˜ , nˆ)≡
1
N!
∫ η0
η˜
dηN . . .
∫ η0
η˜
dη2
∫ η0
η˜
dη1T
[
Vˆ (ηN , nˆ) . . .Vˆ (η2, nˆ)Vˆ (η1, nˆ)
]
.
(5.58)
Formally, this means that the evolution operator is the solution to the following equation
d
dη Uˆ(η , η˜ , nˆ) = Vˆ (η , nˆ)Uˆ(η , η˜ , nˆ). (5.59)
Using the evolution operator, Eq. (5.53) can be expressed as the following integral
1
(N+1)! Iˆ
[N+1](η0, nˆ) = 4
∫ η0
0
dη˜ UˆN(η0, η˜ , nˆ)ST (η˜ ,−nˆr˜). (5.60)
Thus, the lensed CMB anisotropies observed today are
Θ˜(nˆ) = Iˆ(η0, nˆ)
4
=
∫ η0
0
dη˜ Uˆ(η0, η˜ , nˆ)ST (η˜ ,−nˆr˜). (5.61)
Cast in this form, it is clear that we can interpret Eq. (5.61) as the accumulated lensing effect
on the CMB sources at different time η˜ .
5.1 Formalism 111
5.1.4 Diagrammatic Approach
In this subsection, we develop a set of rules as a book-keeping tool to compute any Nth-order
lensed CMB anisotropies. This facilitates the expansion of the lensing effect to higher orders
for calculations of the corresponding CMB power spectra and bispectra. In Section 5.2.1,
we will demonstrate the calculation of the lensed CMB temperature power spectrum by
using the diagrammatic approach. Although we develop the diagrammatic approach based
on Eq. (5.53), we note that the diagrams are compatible with Eq. (5.78) which considers
fixed sources at the LSS by applying Approximation 7 in Section 5.1.2. Moreover, similar
diagrams can be developed to include the redshift and time-delay effects which will be
discussed in Section 5.1.5.
As we can see in Eq. (5.53), the (N+1)th-order photon intensity contains time-ordered
integrals over two types of interaction terms – the unlensed source term ST (diagrammati-
cally 0 where the 0 below denotes the ordering)
12 and the Nth potential term ∇nˆ[ΨNLW (ηN)]
(diagrammatically N where the label N indicates the time-ordering of the potential terms).
With N > M, it means that the Nth potential term is located later in time compared to the
Mth potential term. We write down the time-ordered terms from left to right in a row. For
example, the 4th-order photon intensity Iˆ[4] is expressed as
0 1 2 3
which is simply a simplified expression of Fig. 5.1.
In addition to the interaction terms, there is an “action” term in Eq. (5.55), i.e. the
vector operator ˆnˆ,r acting on either interaction term ST or ∇nˆ[ΨNLW ]. We denote the action
by an over-arc, i.e.
0
. . . N
for action of ˆnˆ,rN on ST and
M . . . N
for action
of ˆnˆ,rN on ∇nˆ[ΨNLW (ηM)]. For each Nth action, there is an Nth time integral
∫ ηN+1
η˜ dηN
associated with it13. We summarize these rules in Table 5.1.
Hence, the prescription to writing down the integrals in Eq. (5.53) is to construct all
possible actions from the left to the right. Physically, an Nth node acting on an Mth node
for N > M means that the object closer to us at N is lensing the object further away at M.
In our formalism, is a lens and thus an overarc between two lenses denotes a lens-lens
12With Approximation 7 in Section 5.1.2, the circle node represents Θ instead of
∫ η0
0 dη˜ ST (η˜).
13Here, η˜ locates the source term ST and integrates from 0 to η0. With Approximation 7 in Section 5.1.2,
we replace η˜ by ηLSS.
112 Weak Lensing in Boltzmann Equation
Diagram Term Physical Meaning
0 Θ(nˆ) or
∫ η0
0 dη˜ ST (η˜ ,−nˆr˜) Unlensed source
generated at node
0.
N ∇
iN
nˆ
[
ΨNLW (ηN ,−nˆrN)
]
Lens at node N.
0
... N
∫ ηN+1
ηLSS
dηN ∇iNnˆ
[
ΨNLW (ηN)
] rN− rLSS
rNrLSS
∂Θ(nˆ)
∂niN
or
∫ η0
0
dη˜
∫ ηN+1
η˜
dηN ∇iNnˆ
[
ΨNLW (ηN)
] rN− r˜
rN r˜
∂ST (η˜)
∂niN
The lens at node
N distorts the
unlensed source,
i.e. lens-source
coupling.
M ... N
∫ ηN+1
ηLSS
dηN ∇iNnˆ
[
ΨNLW (ηN)
] rN− rM
rNrM
∇iMnˆ
∂ΨNLW (ηM)
∂niN
or
∫ ηN+1
η˜
dηN ∇iNnˆ
[
ΨNLW (ηN)
] rN− rM
rNrM
∇iMnˆ
∂ΨNLW (ηM)
∂niN
The lens at node N
distorts the lens at
node M, i.e. lens-
lens coupling.
Table 5.1 The diagrams with their corresponding represented equations and their physical
meanings. In the “Term” column, the first choice assumes Approximation 7 while the sec-
ond one does not.
5.1 Formalism 113
coupling. We emphasize that, although we have considered only the lensing effect here, it is
easy to generalize our diagrammatic approach to other types of interaction terms such as the
redshift and time-delay effects as will be discussed in Section 5.1.5 by adding more node
types.
As an example, we list out all the possible diagrams for the third-, forth- and fifth-order
terms of the lensed temperature anisotropies Θ˜ in Table 5.2. In fact, except the upper left
diagram in each of Table 5.2a, 5.2b and 5.2c, all the other diagrams involve the lens-lens
couplings. Ignoring these couplings is equivalent to applying Approximation 6 in Sec-
tion 5.1.2.
To illustrate the rules summarized in Table 5.1, we demonstrate how to construct the
formulae for the middle diagram in the first row of Table 5.2b. From the diagram, there are
one circle node and three square nodes, i.e. one source and three lenses. The lenses at node
1 and 2 distort the source while the lens at node 3 distorts the lens at node 1. Based on the
rules in Table 5.1, the integral associated with this diagram is
Θ˜[4](nˆ)⊃ 0 1 2 3
=
∫ η0
0
dη˜
∫ η0
η˜
dη3∇i3nˆ
[
ΨNLW (η3,−nˆr3)
]
{
∫ η3
η˜
dη2∇i2nˆ
[
ΨNLW (η2,−nˆr2)
]
{
∫ η2
η˜
dη1
(
2
r3− r1
r3 r1
)
∇i1nˆ
[
∂
∂ni3 Ψ
NL
W (η1,−nˆr1)
]{
(
2
r2− r˜
r2 r˜
)(
2
r1− r˜
r1 r˜
)
∂
∂ni2
∂
∂ni1 ST (η˜ ,−nˆr˜)
}}}
. (5.62)
While these diagrams may seem frivolous, its power arises when we want to compute
the two-point correlation functions, e.g. ⟨Θ[N]Θ[M]⟩ to all possible actions in the Nth and
Mth orders. To see that, we need a new rule to encapsulate contractions between all possible
interaction terms (either or ), which will be introduced in Section 5.2.1.
5.1.5 Extensions
In this subsection, we extend the formalism and the diagrammatic approach developed for
the weak lensing effect to study the redshift and time-delay effects. We will derive their
114 Weak Lensing in Boltzmann Equation
0 1 2 0 1 2
(a) The third iteration
0 1 2 3 0 1 2 3 0 1 2 3
0 1 2 3 0 1 2 3 0 1 2 3
(b) The fourth iteration
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
(c) The fifth iteration
Table 5.2 All the possible couplings of the weak lensing effects on the CMB anisotropies in
the third, forth and fifth orders. As an example, Eq. (5.62) demonstrates how to construct
the formulae from the middle diagram in the first row of Table 5.2b.
5.1 Formalism 115
corresponding interaction operators, following what we have done in Section 5.1.3. We will
also show how the formalism can be used to systematically estimate their contributions to
the CMB signals, especially for higher orders in which the redshift and time-delay effects
can couple with the lensing effect. The extension here can be treated as the relaxation of
Approximations 2 and 3 in Section 5.1.2. Nonetheless, we argue that the pure lensing effect
dominates in arbitrarily high orders. With the extension to the redshift and time-delay ef-
fects, the developed formalism becomes a comprehensive solution of the Liouville operator
(collisionless part) of the Boltzmann equation in arbitrarily high orders. We emphasize that
the formalism is generic and can be used to study the lensing, redshift and time-delay effects
of other observations – such as galaxy surveys.
Time Delay Effect and Born Correction
By ignoring dxI/dη (Approximation 2 in Section 5.1.2) in the Boltzmann equation, we
apply the Born approximation [44, 95] and neglect the time-delay effect [49]. Here, we
relax this approximation in our formalism. For simplicity, we first consider it without the
lensing terms. That is, Eq. (5.31) becomes
Qab =D[I]ab−
(
dp0
dη
)[I] ∂P¯ab
∂ p0 −2n
iΨNLW
∂Pab
∂xI , (5.63)
with the Newtonian approximation (Approximation 4 in Section 5.1.2) modified as
dxI
dη =
∞
∑
N=0
1
N!
(
dxI
dη
)[N]
≈ ni+
∞
∑
N=1
ni
N!
(Ψ[N]+Φ[N])≡ ni+2niΨNLW . (5.64)
Combining Eq. (5.11) and Eq. (5.63), we can find the recursive solution of the normalized
intensity matrix, by the line of sight approach
1
(N+1)!Pˆ
[N+1]
ab (ηN+1,kN+1, nˆ) = eikN+1·nˆrN+1+τ¯(ηN+1)
∫ ηN+1
0
dηNTk′N ,kN (kN+1)
{
−2ikN · nˆ
[
e−ik
′
N ·nˆrNΨNLW (ηN ,k′N)
][
e−ikN ·nˆrN−τ¯(ηN)
1
N!
Pˆ
[N]
ab (ηN ,kN , nˆ)
]}
, (5.65)
where N ≥ 1.
Now, we focus on the temperature anisotropies only. The (N + 1)th iteration of the
116 Weak Lensing in Boltzmann Equation
photon intensity at present (η0) is
1
(N+1)! Iˆ
[N+1](η0,kN+1, nˆ)
=
∫ η0
0
dη˜
{
∫ η0
η˜
dηNTk′N ,kN (kN+1)
{[
e−ik
′
N ·nˆrNΨNLW (ηN ,k′N)
]
(−2ikN · nˆ)
{
. . .
∫ η3
η˜
dη2Tk′2,k2(k3)
{[
e−ik
′
2·nˆr2ΨNLW (η2,k′2)
]
(−2ik2 · nˆ)
{
∫ η2
η˜
dη1Tk′1,k1(k2)
{[
e−ik
′
1·nˆr1ΨNLW (η1,k′1)
]
(−2ik1 · nˆ)
{
e−ik1·nˆr˜4S˜T (η˜ ,k1, nˆ)
}}}}
. . .
}}
}
. (5.66)
Similar to Approximation 1 in Section 5.1.1, we replace S˜T (η˜ ,k1, nˆ) with ST (η˜ ,k1)
by ignoring the boundary terms of Θ[I](η ,k1, nˆ) generated by integration by parts when
η ̸= η0. Following the derivation in Section 5.1.3, the (N + 1)th iteration of the photon
intensity from the time-delay effect can be written in configuration space as
1
(N+1)! Iˆ
[N+1](η0, nˆ)
=
∫ η0
0
dη˜
{
∫ η0
η˜
dηN
[
niNΨNLW (ηN ,−nˆrN)
]
(♦ˆnˆ,rN )iN
{
. . .
∫ η3
η˜
dη2
[
ni2ΨNLW (η2,−nˆr2)
]
(♦ˆnˆ,r2)i2
{
∫ η2
η˜
dη1
[
ni1ΨNLW (η1,−nˆr1)
]
(♦ˆnˆ,r1)i1
{
4ST (η˜ ,−nˆr˜)
}}
. . .
}}
, (5.67)
where ΨNLW (ηN ,−nˆrN) and ST (η˜ ,−nˆr˜) are defined in Eq. (5.45) and Eq. (5.54) respectively,
and the vector operator ♦ˆnˆ,r is defined such that it is non-zero only when it acts on X
(♦ˆnˆ,r)i X (η ′,−nˆr′)≡
2
r′
∂
∂niX (η
′,−nˆr′), (5.68)
with X denoting ΨNLW or ST . We notice that the r-dependence of the vector operator ♦ˆ is
redundant but we keep it as a consistent notation with Eq. (5.55). Furthermore, we can
define the interaction operator of the time-delay effect as
Vˆx(η , nˆ)≡ ni ΨNLW (η ,−nˆr)(♦ˆnˆ,r)i, (5.69)
5.1 Formalism 117
where the subscript x denotes that this term comes from the derivative with respect to space-
like coordinates xI of the background manifold in the Liouville operator. To include the
lensing effect, we can simply add the interaction operators of the lensing and time-delay
effects together to the Dyson series in Eq. (5.61). Moreover, the diagrammatic approach in
Section 5.1.4 can be extended straightforwardly by introducing a new node N to represent
the interactions of the time-delay effect.
To understand why the lensing effect dominates over the corrections from Born approx-
imation and the time-delay effect, we consider the ratio of the norms of the two interaction
operators in Eq. (5.56) and Eq. (5.69), i.e. R≡ ||Vˆx||/||Vˆ ||. With the flat-sky approximation,
we can replace ∇nˆ with ℓ in Fourier space and have R ∼ r/(r′− r)/ℓ where ℓ = |ℓ| and
|ni| is of order 1. For lens-source couplings which dominate the lensing effect, r/(r′− r)
is of order 1 and thus R ∼ 1/ℓ with ℓ ∼ 40 at the peak of the power spectrum of the lens-
ing potential14. This estimation is consistent with [49]. The argument here is made with
the interaction operator and is valid throughout the hierarchy, not just in the lowest orders
as verified previously in literature. That is, we can safely ignore any high-order couplings
involving the redshift and time-delay effects as the pure lensing effect dominates.
Redshift
Similarly, we relax Approximation 3 in Section 5.1.2 by including the redshift effects in
nonlinear orders so that
Qab =D[I]ab− p0
(
∂ηΨNL−ni∂IΦNL
) ∂Pab
∂ p0 , (5.70)
where the Newtonian approximation (Approximation 4 in Section 5.1.2) becomes
dp0
dη =
∞
∑
N=0
1
N!
(
dp0
dη
)[N]
≈−Hp0 +
∞
∑
N=1
p0
N!
(
∂ηΨ[N]−ni∂IΦ[N]
)
≡−Hp0 + p0
(
∂ηΨNL−ni∂IΦNL
)
. (5.71)
Here, we ignore the lensing and time-delay effects for simplicity but it is straightforward to
include them. Substituting Eq. (5.70) into Eq. (5.11), the recursive solution of the normal-
14More precisely, lenses at redshifts z. 10 dominate the lensing effect [67]. For these lenses, r/(r′−r). 2
and the argument holds as well.
118 Weak Lensing in Boltzmann Equation
ized intensity matrix can be written as
1
(N+1)!Pˆ
[N+1]
ab (ηN+1,kN+1, nˆ) = eikN+1·nˆrN+1+τ¯(ηN+1)
∫ ηN+1
0
dηNTk′N ,kN (kN+1)
{
4
×
[
∂ηN
(
e−ik
′
N ·nˆrNΦNL(ηN ,k′N)
)
−2e−ik
′
N ·nˆrN∂ηNΨNLW (ηN ,k′N)
]
×
[
e−ikN ·nˆrN−τ¯(ηN)
1
N!
Pˆ
[N]
ab (ηN ,kN , nˆ)
]}
, (5.72)
where N ≥ 1 and ΨNLW is the nonlinear Weyl potential. In Eq. (5.72), the term with Φ
NL cor-
responds to the SW effect while the term with ΨNLW corresponds to the ISW effect. Similar
trick is used in Section 4.2.2 when we separate the SW and ISW effects in the 2nd-order
Boltzmann equation.
The (N+1)th iteration of the photon intensity at present is
1
(N+1)! Iˆ
[N+1](η0,kN+1, nˆ) =
∫ η0
0
dη˜
{
∫ η0
η˜
dηNTk′N ,kN (kN+1)
{[
∂ηN
(
e−ik
′
N ·nˆrNΦNL(ηN ,k′N)
)
−2e−ik
′
N ·nˆrN∂ηNΨNLW (ηN ,k′N)
]{
. . .
∫ η3
η˜
dη2Tk′2,k2(k3)
{[
∂η2
(
e−ik
′
2·nˆr2ΦNL(η2,k′2)
)
−2e−ik
′
2·nˆr2∂η2Ψ
NL
W (η2,k′2)
]{
∫ η2
η˜
dη1Tk′1,k1(k2)
{[
∂η1
(
e−ik
′
1·nˆr1ΦNL(η1,k′1)
)
−2e−ik
′
1·nˆr1∂η1Ψ
NL
W (η1,k′1)
]{
e−ik1·nˆr˜4S˜T (η˜ ,k1, nˆ)
}}}}
. . .
}}
}
. (5.73)
In configuration space, the (N+1)th iteration of the photon intensity from the redshift effect
5.1 Formalism 119
can be written as15
1
(N+1)! Iˆ
[N+1](η0, nˆ) (5.74)
=
∫ η0
0
dη˜
{
∫ η0
η˜
dηN
[
∂ηNΦNL(ηN ,−nˆrN)−2 ˙˜Ψ
NL
W (ηN ,−nˆrN)
]{
. . .
∫ η3
η˜
dη2
[
∂η2Φ
NL(η2,−nˆr2)−2 ˙˜Ψ
NL
W (η2,−nˆr2)
]{
∫ η2
η˜
dη1
[
∂η1Φ
NL(η1,−nˆr1)−2 ˙˜Ψ
NL
W (η1,−nˆr1)
]{
4ST (η˜ ,−nˆr˜)
}}
. . .
}}
,
where
ΦNL(ηN ,−nˆrN)≡
∫
d3k′N
(2π)3/2 e
−ik′N ·nˆrNΦNL(ηN ,k′N), (5.75)
˙˜Ψ
NL
W (ηN ,−nˆrN)≡
∫
d3k′N
(2π)3/2 e
−ik′N ·nˆrN∂ηNΨNLW (ηN ,k′N). (5.76)
Moreover, we define the interaction operator of the redshift effect as
Vˆr(η , nˆ)≡ ∂ηΦNL(η ,−nˆr)−2 ˙˜Ψ
NL
W (η ,−nˆr), (5.77)
with the subscript r denoting the redshift effect. Here, we note that the interaction operator of
the redshift effect is just a scalar quantity. For the diagrammatic approach in Section 5.1.4,
it does not have the “action” term represented by an over-arc and can be indicated by solely
a new node N .
We point out that the interaction potential ||Vˆr|| suppresses the redshift couplings due
to an extra order of perturbations (see the discussion below Eq. (5.20)). This is expected
because Vˆr is exactly the source term for the SW and ISW effects when we perform the
line of sight approach for 1st-order perturbations. The only difference is the range of the
conformal time integration - we consider some subsets of the full range from 0 to η0 here.
Thus, the amplitude of CMB anisotropies will be suppressed by roughly an order of 10−5
when coupling with the redshift effect to one higher order. This explains why the redshift
effect is subdominant compared to the lensing effect.
15We replace S˜T (η˜ ,k1, nˆ) with ST (η˜ ,k1) in order to keep the notation consistent with Eq. (5.53) and
Eq. (5.67). However, unlike lensing and time-delay effects, it is not necessary for pure redshift effects.
120 Weak Lensing in Boltzmann Equation
5.2 Power Spectrum from Lens-lens Couplings
In Section 5.1.2, we develop a new formalism to study the lensing effect through the Boltz-
mann equation and explicitly enumerate all the approximations (Approximations 1 to 7)
required to recover the canonical remapping approach reviewed in Section 2.6.1. The for-
malism is then extended to the redshift and time-delay effects in Section 5.1.5 in order to
validate Approximation 2 and 3. In this section, we discuss the legitimacy of other approx-
imations. In particular, we calculate quantitatively the correction of the lens-lens couplings
on the temperature power spectrum and demonstrate how the diagrammatic approach in
Section 5.1.4 can facilitate the calculations of the power spectra.
We start with Eq. (5.53) and impose Approximation 7 to single out the lens-lens cou-
plings from other approximations. We can then express the (N+1)th-order lensed temper-
ature anisotropies in configuration space as16
Θ˜[N+1](nˆ) =
∫ η0
ηLSS
dηN∇iNnˆ
[
ΨNLW (ηN ,−nˆrN)
]
(ˆnˆ,rN )iN
{
. . .
∫ η3
ηLSS
dη2∇i2nˆ
[
ΨNLW (η2,−nˆr2)
]
(ˆnˆ,r2)i2
{
∫ η2
ηLSS
dη1∇i1nˆ
[
ΨNLW (η1,−nˆr1)
]
(ˆnˆ,r1)i1 Θ(nˆ)
}
. . .
}
, (5.78)
where the unlensed temperature anisotropies Θ(nˆ) are given by Eq. (5.50) and the vector
operator ˆnˆ,r is redefined such that it is non-zero only when it acts on X
(ˆnˆ,r)i X (η ′,−nˆr′)≡ 2
r− r′
r r′
∂
∂niX (η
′,−nˆr′), (5.79)
with X denoting ΨNLW or Θ
17. Physically, acting ˆnˆ,r on ΨNLW corresponds to lens-lens
couplings while acting on Θ corresponds to lensing the sources. We will develop a set of
diagrams to calculate the corresponding power spectrum in the following.
5.2.1 Diagrammatic Approach
To simplify the numerical calculation, we work in the limit of flat-sky, use the Limber ap-
proximation and consider only the linear part of the Weyl potential18 (i.e. ΨNLW → Ψ
[I]
W).
16Here, the lensed temperature anisotropies are expanded as Θ˜(nˆ) = ∑∞N=1 Θ˜[N](nˆ).
17With Approximation 7, r′ is replaced by rLSS when ˆnˆ,r acts on Θ.
18For simplicity, we omit the superscript [I] from now on.
5.2 Power Spectrum from Lens-lens Couplings 121
However, we remark that these extra approximations are not necessary to the diagrammatic
approach. Moreover, we do not expect these approximations to change the results signif-
icantly. Operationally, the approximations work as follows. Recall that for any function
f (nˆ), we can perform the Fourier transformation
f (nˆ) =
∫
dℓ
2π f (ℓ)e
iℓ·nˆ. (5.80)
With the assumption of statistical isotropy, it can be shown that
⟨Θ(ℓ)Θ∗(ℓ′)⟩ = δ (ℓ− ℓ′)CΘℓ , (5.81)
⟨ΨW(η , ℓ)Ψ∗W(η ′, ℓ′)⟩ = δ (ℓ− ℓ′)δ (η−η ′)CΨWℓ (η), (5.82)
where the angle brackets denote the ensemble averages. CΘℓ is the unlensed temperature
power spectrum expressed in Eq. (2.86) while CΨWℓ (η) is the equal-time power spectrum of
the Weyl potential.
With the flat-sky and Limber approximations, we can find
CΨWℓ (η) =
1
r2
PΦ
(ℓ
r
)[
TΨW
(
η , ℓ
r
)]2
, (5.83)
where r≡η0−η , PΦ(k) is the primordial power spectrum defined in Eq. (2.73) and TΨW(η ,k)
is the transfer function of the Weyl potential. Particularly, the Dirac delta function δ (η−η ′)
in Eq. (5.82) comes from the Limber approximation using
∫
dk k2 jℓ(kr) jℓ(kr′) =
π
2r2
δ (r− r′), (5.84)
where jℓ(x) is the spherical Bessel function of the first kind. This is valid for high ℓs when
the term PΦ(k)[TΨW(η ,k)]2 varies slowly compared to the spherical Bessel functions [67].
As mentioned in [67], the Limber approximation is rather good for lensing potential with
lens-source couplings. Thus, we expect the approximation does well for lens-lens couplings
because at least one ℓ-mode of source is replaced by a ℓ-mode of lenses. It means that the
main contribution of the ℓ-mode integration of lenses in the lensed power spectrum will
shift to even higher ℓ. Additionally, we ignore the correlation between the temperature
anisotropies and the Weyl potential, i.e.
⟨ΨW(η , ℓ)Θ∗(ℓ′)⟩ ≈ 0. (5.85)
It is a very good approximation because only the ISW effect of the temperature anisotropies
122 Weak Lensing in Boltzmann Equation
correlates well with the Weyl potential at low-ℓ modes (ℓ . 100) for which the cosmic
variance is overwhelming due to the limited modes available.
Now, we introduce some extra rules to extend the diagrammatic approach for the cal-
culation of power spectra. Diagrammatically, we represent the correlation function ⟨. . .⟩
by a dotted line connecting two nodes. For example, denotes ⟨ΨWΨ∗W⟩. Due to
the presence of the Dirac delta function δ (η −η ′) in Eq. (5.82), many possible configura-
tions are forbidden because we cannot place the lens further than the lensed object from the
observer. In Table 5.3, we summarize all the forbidden contractions in the diagrams, the
corresponding terms in the formula and their physical meanings. Because of the forbidden
contractions, the lens-lens couplings do not contribute to the 4th-order power spectrum. In
order to assess effects from the lens-lens couplings, we study the 6th-order power spectrum.
We itemize all the non-zero configurations for the 6th-order power spectrum as shown in
Table 5.4. There exist cancellations between some pairs of the diagrams which are crossed
out by the arrows. To understand this, we perform the Fourier transformation such that
(∇nˆ)i ≈
∂
∂ni → iℓi. (5.86)
In Fourier space, each pair of these cancelled diagrams looks exactly the same except that
one contains (ℓM · ℓX)δ (ℓM + ℓN) while another one contains (ℓN · ℓX)δ (ℓM + ℓN). The
difference comes from an overarc acting on either one of two correlated lenses. The pairs
are eliminated because ℓM =−ℓN .
5.2.2 Numerical Implementation
Among the rest of the diagrams in Table 5.4, diagrams (1A), (1B), (2A), (2B) and (4A) do
not involve the lens-lens couplings. They correspond to the expanded terms in the Taylor
series of the remapping approach in Eq. (2.87) which can be calculated non-perturbatively
[23]. In contrast, other residual terms, which are in Column C and D, contain the lens-lens
couplings. To do the numerical calculation, we write down the temperature power spectrum
5.2 Power Spectrum from Lens-lens Couplings 123
Forbidden Con-
tractions
Term Physical Meaning
⟨ΨW(η , ℓ)Θ∗(ℓ′)⟩ The CMB temperature
anisotropies correlate
weakly with the Weyl
potential.
M N
M′ N′
. . .
. . .
θ(ηN − ηM)θ(ηN′ − ηM′)δ (ηN −
ηM′)δ (ηN′−ηM)
Lens at ηN must be
placed after any lens at
ηM(< ηN).
M X N. . . . . . θ(ηN−ηX)θ(ηX −ηM)δ (ηN−ηM) Lens at ηX cannot be
placed before and after
the lenses at ηM = ηN
simultaneously.
N N+1
(rN+1− rN)δ (ηN+1−ηN) No lensing effect if the
lens is placed at the
same location as the
lensed object.
Table 5.3 The forbidden contractions for the lensing effects on the power spectrum. θ is the
Heaviside step function. The Heaviside step functions come from replacing the integration
∫ ηN+1
ηLSS dηN with
∫ η0
ηLSS dηNθ(ηN+1−ηN).
124 Weak Lensing in Boltzmann Equation
Correlations Diagrams
⟨Θ[3]Θ[3]∗⟩
0 1 2
0’ 1’ 2’
(1A)
0 1 2
0’ 1’ 2’
(1B)
0 1 2
0’ 1’ 2’
(1C)
0 1 2
0’ 1’ 2’
(1D)
⟨Θ[4]Θ[2]∗⟩
0 1 2 3
0’ 1’
(2A)
0 1 2 3
0’ 1’
(2B)
0 1 2 3
0’ 1’
(2C)
0 1 2 3
0’ 1’
(2D)
0 1 2 3
0’ 1’
(3A)
0 1 2 3
0’ 1’
(3B)
0
0 1 2 3
0’ 1’
(3C)
⟨Θ[5]Θ[1]∗⟩
0 1 2 3 4
0’
(4A)
0 1 2 3 4
0’
(4B)
0 1 2 3 4
0’
(4C)
0
0 1 2 3 4
0’
(4D)
0 1 2 3 4
0’
(4E)
0
0 1 2 3 4
0’
(5A)
0 1 2 3 4
0’
(5B)
0
0 1 2 3 4
0’
(5C)
0 1 2 3 4
0’
(5D)
0
Table 5.4 All the non-zero configurations for the temperature power spectrum from the next-
to-leading order of the weak lensing effect. The dotted lines denote correlations while the
arrows indicate the cancellations between some pairs of the diagrams.
5.2 Power Spectrum from Lens-lens Couplings 125
represented by all the residual diagrams
C˜Θ(6)ℓ =
∫
dℓ1dℓ2
(2π)4
{
K0(ℓ1, ℓ2)
{
1
2
(ℓ1 · ℓ)2(ℓ2 · ℓ)2CΘℓ
−[ℓ1 · (ℓ− ℓ1)]2[ℓ2 · (ℓ− ℓ1)]2CΘ|ℓ−ℓ1|
+1
2
(ℓ1 · ℓ)2(ℓ2 · ℓ)2CΘℓ
}
(5.87)
+K0(ℓ1, ℓ2)
{
[ℓ1 · (ℓ− ℓ1− ℓ2)]2[ℓ2 · (ℓ− ℓ1− ℓ2)]2CΘ|ℓ−ℓ1−ℓ2|
−[ℓ1 · (ℓ− ℓ1)]2[ℓ2 · (ℓ− ℓ1)]2CΘ|ℓ−ℓ1|
}
(5.88)
+K1(ℓ1, ℓ2)
{
2[ℓ1 · (ℓ− ℓ1− ℓ2)]2[ℓ2 · (ℓ− ℓ1− ℓ2)](ℓ2 · ℓ1)CΘ|ℓ−ℓ1−ℓ2|
−[ℓ1 · (ℓ− ℓ1)]2[ℓ2 · (ℓ− ℓ1)](ℓ2 · ℓ1)CΘ|ℓ−ℓ1|
−[ℓ1 · (ℓ− ℓ1)]2[ℓ2 · (ℓ− ℓ1)](ℓ2 · ℓ1)CΘ|ℓ−ℓ1|
}
(5.89)
+K2(ℓ1, ℓ2)
{
[ℓ1 · (ℓ− ℓ1− ℓ2)]2(ℓ2 · ℓ1)2CΘ|ℓ−ℓ1−ℓ2|
−[ℓ1 · (ℓ− ℓ1)]2(ℓ2 · ℓ1)2CΘ|ℓ−ℓ1|
} }
, (5.90)
where the functions Ki(ℓ1, ℓ2) are defined as
∫ η0
ηLSS
dη1 4CΨWℓ1 (η1)
(
rLSS− r1
rLSS r1
)2 ∫ η0
η1
dη2 4CΨWℓ2 (η2)



(
rLSS−r2
rLSS r2
)2
, if i = 0,
(
rLSS−r2
rLSS r2
)(
r1−r2
r1 r2
)
, if i = 1,
(
r1−r2
r1 r2
)2
, if i = 2.
(5.91)
Eq. (5.87) to Eq. (5.90) correspond to Column A, B, C and D of the residual diagrams in
Table 5.4 respectively.
126 Weak Lensing in Boltzmann Equation
0 500 1000 1500 2000 2500 3000
0
1
2
3
4
{
DC
{
C {
H%
L
0 500 1000 1500 2000 2500 3000
0
1
2
3
4
{
DC
{
C {
H%
L
Figure 5.2 The contributions on the temperature power spectrum from the residual diagrams
of Column C and D containing lens-lens couplings in Table 5.4. The top and bottom pan-
els show the corrections to the power spectrum from those diagrams in Column C and D
respectively. In the top panel, the solid gray line corresponds to the diagram (1C) while
the dashed line corresponds to the diagrams (2C) and (3C). Diagrams (2C) and (3C) have
identical contributions and thus we sum them up. In the bottom panel, the solid gray line
corresponds to the diagram (1D) while the dashed line corresponds to the diagram (2D). For
both panels, the signs of contributions from the diagrams in Row ⟨Θ[4]Θ[2]*⟩ of Table 5.4,
i.e. the dashed lines, are reversed to illustrate the cancellations.
Since we want to study the effects of lens-lens couplings, we now focus on Column
C and D of Table 5.4. In Fig. 5.2 and 5.3, we plot the contributions to the temperature
5.2 Power Spectrum from Lens-lens Couplings 127
power spectrum from the residual diagrams in Column C and D. From Fig. 5.2, we see
that the total contribution of each column is suppressed by cancellations and is at least
an order of magnitude smaller than that from an individual diagram in the same column.
The cancellations are due to the conservation of the power spectrum from the weak lensing
effect. Physically, the positive terms in Eq. (5.87) to Eq. (5.90) are the power redistributed
to the mode ℓ while the negative terms are the power redistributed away from the mode ℓ.
In summary, as shown in Fig. 5.3, the overall correction from the lens-lens couplings is
of order 0.1% for ℓ up to 3000 and thus is comparable to the cosmic variance. It can be a
systematic effect on CMB studies. We remark that the correction calculated here is for the
power spectrum of the CMB temperature anisotropies. This is different from the correction
of the cosmic shear power spectrum calculated in [61] which quantifies the correction of
the deflection angle in the remapping approach. Moreover, the order of magnitude for the
correction from the lens-lens coupling can be understood as follows: we can check that the
lensing correction of the CMB temperature power spectrum from the lens-source coupling
(captured by the canonical remapping approach) can exceed 20% and 2% of the unlensed
power spectrum in the leading and next-to-leading orders respectively[23] for ℓ up to 3000.
From Table 5.4, we can see that the difference to the corrections of CMB power spectrum
from lens-source coupling and lens-lens coupling in the next-to-leading order comes mainly
from replacing one of ℓ modes of the source with ℓi (i.e. ℓ modes of the lenses)19. With the
peak of the lensing power spectrum at ℓi ∼ 40, it means that the correction of CMB power
spectrum from lens-lens coupling can be a few of 0.01%.
19For example, compare (2B) and (2C) in Table 5.4.
128 Weak Lensing in Boltzmann Equation
0 500 1000 1500 2000 2500 3000
-0.08
-0.06
-0.04
-0.02
0.00
0.02
{
D
C {
C
{H
%
L
Figure 5.3 The overall correction to the temperature power spectrum from the lens-lens
couplings.
5.2.3 Other Approximations
In Section 5.1.5, we qualitatively validate Approximation 2 and 3 of Section 5.1.2. In Sec-
tion 5.2, we also calculate the correction from the lens-lens couplings corresponding to
Approximation 6 of Section 5.1.2. In this section, we will discuss the other approximations
implied when we recover the remapping approach.
Some of these approximations have been studied and their impacts on the CMB are
believed to be small. The single-source approximation, which is Approximation 2 in Sec-
tion 5.1.1 and Approximation 7 in Section 5.1.2, was evaluated in [107] using the flat-sky
approach. Its corrections to the TT temperature power, the EE polarization power and the
TE cross power spectra are of order 0.01% while the correction to the BB polarization power
spectrum is about 0.4%, for ℓ up to 2000. The pure time-delay effect, included in Approxi-
mation 2 of Section 5.1.2, has been studied in [49] where the effect is estimated to be of order
0.1% correction to the TE cross power spectrum and . 0.01% corrections to the TT , EE
and BB power spectra, for ℓ ∼ 1000. The 2nd-order Compton scattering at recombination,
which is ignored in Approximation 1 of Section 5.1.2, is studied in [37, 52, 53, 81, 82, 100].
5.3 Discussion and Conclusions 129
It have negligible effects on the CMB power spectra but systematic contaminations on the
CMB bispectra. Thus, it is expected that higher-order couplings of the scattering with lens-
ing or other effects are negligible. More details can be found in Chapter 3 and 4.
Approximation 1 in Section 5.1.1 has not been discussed previously and is first identi-
fied in [99]. However, we expect the contribution on the CMB power spectra to be negligi-
ble. It is because the correction to Approximation 1 in Section 5.1.1 is proportional to ℓ= 2
multipoles (the term with Π in Eq. (5.20)) which is very tiny compared to other intrinsic
CMB effects. Thus, the correction should be highly suppressed.
Among all the approximations, the Newtonian approximation as shown in Approxima-
tion 4 of Section 5.1.2 is the most difficult one to be assessed. On one hand, it relies heavily
on the accuracy of the large-scale studies to obtain the nonlinear gravitational potentials.
On the other hand, the Newtonian approximation performs well in small scales where the
evolution is local and the GR effect is negligible. Thus, we can linearize Eq. (5.30) and ig-
nore the cross terms. However, the scale of the time integration in the line of sight approach
along the lensing effects is comparable to the Hubble radius. The CMB lensing is clearly
not a local effect and its GR corrections can be important. Further assessment is needed in
the future.
5.3 Discussion and Conclusions
In this chapter, we formulate the lensing, redshift and time-delay effects from the Liouville
terms of the Boltzmann equation and express them as interaction operators in Dyson series.
Focusing on the lensing effect, we demonstrate how the new formalism helps to explicitly
enumerate and assess the required approximations for recovering the canonical remapping
approach. Moreover, we develop a diagrammatic approach to keep track of the couplings
of different effects for arbitrarily high orders. In particular, we found that the correction
from the lens-lens couplings to the temperature power spectrum is . 0.1% for ℓ up to 3000
and thus is comparable to the cosmic variance. We emphasize that the formalism can be
treated as a comprehensive solution of the collisionless Boltzmann equation in arbitrarily
high orders. In principle, it can be used to study the lensing, redshift and time-delay effects
of observations other than the CMB – such as galaxy surveys.

Chapter 6
Discussion and Conclusions
6.1 Discussion
Here, we discuss some potential extensions of our current work.
6.1.1 Spectral Distortions
As we mentioned in Section 4.1, 2nd-order Compton scattering induces spectral distortion
from the blackbody spectrum such that there is no longer a unique definition of cosmic mi-
crowave background (CMB) temperature. Although the spectral distortion complicates the
calculation of CMB bispectrum at recombination, it provides extra information about the
evolution history of our universe. While tightest constraints on the spectral distortion come
from the two-decade-old COBE/FIRAS [38], the future experiment - Pixie [59] can im-
prove the measurement by 2-3 orders of magnitude. In particular, the 2nd-order Boltzmann
equation developed in Section 3.2 can be used straightforwardly to study the y-distortion
at recombination. This has been studied in [85] and the effect is subdominant compared to
the y-distortion generated by the thermal Sunyaev-Zeldovich effect. The similar effect also
occurs at reionization and is worth to be considered. However, since reionization happened
at low redshift (z ∼ 10) when the evolution is highly nonlinear, it is uncertain how well
the 2nd-order equations perform during reionization. Furthermore, there are other types of
spectral distortions which can be produced before recombination. It may be worth to for-
mulate them through the 2nd-order Boltzmann equation with collisions other than Compton
scattering. This might provide an alternative way to study these distortions along with the
132 Discussion and Conclusions
typical energy-injection approach.
6.1.2 Modified Gravity
In Chapter 4, we find that the CMB bispectrum generated by 2nd-order perturbations at re-
combination is mild and will be treated as systematics in future CMB experiments. Since
the calculation is based on General Relativity (GR), the result can be very different if mod-
ified gravity models are considered. In other words, the recombination bispectrum can be
used to provide independent constraints for the alternative gravity models during recombi-
nation. There are some remarks for this application. To have an appealing gravity model, it
has to be compatible with other observations, e.g. CMB power spectra. This can be done by
refitting the cosmological parameters. However, if we assume single-field inflation, the re-
combination bispectrum in squeezed limit will be fixed and determined by power spectrum
(see Eq. (4.52)), which is tightly constrained by CMB observations. Nevertheless, alter-
native gravity models can contribute to the non-squeezed configurations of recombination
bispectrum. This should not be a concern if multi-field inflations are considered.
6.1.3 Perturbed Optical Depth
As we mentioned in Section 3.2.4, the 1st-order optical depth calculated is based on per-
turbing the fitting function of the background-order optical depth. Even though the approx-
imation works well for large scales, it is not necessarily true for small scales. At small
scales, the details of the underlying dynamics, which are not captured by the three-level
approximation, can be important. Thus, it is of interest to study the perturbed optical depth
in more comprehensive ways. One possibility is to consider atoms with multi-levels [7]
and to perturb their corresponding evolution equations. This can be achieved by numerical
calculations.
6.1.4 Lensing in CMB Polarizations and Other Observables
In Chapter 5, we formulate the lensing, redshift and time-delay effects on the CMB temper-
ature anisotropies through the Boltzmann equation. The derivation can be easily extended
to study the same effects on the CMB polarizations. Since the unlensed power spectra of
polarizations have very different shapes compared to the unlensed power spectrum of tem-
perature anisotropies. The corrections from the implied approximations in the remapping
6.2 Conclusions 133
approach can behave differently. Especially, the corrections are expected to be relatively
stronger for B-mode polarization because there is no ‘intrinsic’ power spectrum of B-mode
generated at recombination. The dominant B-mode polarization1 comes from the lensed E-
mode which is 2nd-order effects at least. On the other hand, it is of interest to check if there
exist any new approximations which have not been discovered and validate the remapping
approach. Similarly, it may be worth to extend the derivation to study lensing, redshift and
time-delay effects on other observations, e.g. galaxy surveys. This can be done by replacing
the CMB source terms with the background galaxy power spectrum and hence generalizing
the approach of [61].
6.1.5 Validation of Newtonian Approximation
In Approximation 4 of Section 5.1.2, we drop the cross terms and linearize GR as if in
Newtonian gravity. However, the scale of the time integration in the line of sight approach
along the lensing effects is comparable to the Hubble radius. The CMB lensing is clearly
not a local effect and its GR corrections can be important. Further assessment is needed in
the future. To tackle the problem, highly nonlinear evolution may have to be considered but
this is practically too difficult. A less rigorous checking can be achieved by considering the
term in Eq. (5.30) up to 2nd order and studying its effect on the linear-order temperature
anisotropies. This will give us some insights on how the cross terms affect the lensing effect
in the leading third order.
6.2 Conclusions
In this thesis, we study cosmological perturbations by solving the governing Boltzmann and
EFEs up to 2nd order, and calculate the corresponding CMB bispectrum during recombi-
nation. We include all the 2nd-order Liouville and collision terms, truncating the multipole
hierarchy at ℓ = 10, consistently including m ̸= 0 terms when calculating the bispectrum
with the flat-sky and thin-shell approximation. At this stage, we focus on contributions
at and around recombination and thus neglect 2nd-order vector and tensor perturbations,
lensing effects, and late-time non-linear ISW effects. We find that the signal-to-noise for
the bispectrum is 0.69 for ℓmax = 2000, yielding an overall signal FNL = 3.19 (normalized
relative to the local model). The effective fNLs of the equilateral and local type are 5.11
1Here, we ignore the vector and tensor perturbations in linear order.
134 Discussion and Conclusions
and 0.88 respectively. The recombination bispectrum should be included in the analysis of
future CMB experiments as systematics.
Moreover, we present a new formalism to calculate the lensing effect by directly solv-
ing the Boltzmann equation. This allows us to explicitly keep track of all known physical
effects through the entire time of flight of a CMB photon from recombination to the present
observer. Using this formalism and focusing on temperature anisotropies, we explicitly
articulate the approximations required to recover the usual remapping approach used in cur-
rent studies of the CMB lensing. We discover two new approximations which have not
been studied previously although we do not expect that they contribute significantly under
the current limits from observations. In addition, we calculate the correction to the CMB
temperature power spectrum for the lens-lens coupling effects and find that the correction is
. 0.1% for ℓ up to 3000. It is comparable to the cosmic variance and should be taken into
account as systematics.
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Appendix A
Glossary of Notations and Symbols
Table A.1 Physical parameters and background quantities
Symbol Description Equation
c Speed of light
G Gravitational constant
hP Planck constant
kB Boltzmann constant
me Electron mass
κ κ ≡ 8πG/c4
Λ Cosmological constant Eq. (2.22)
As Amplitude of initial power spectrum Eq. (2.73)
k∗ Pivot wavenumber Eq. (2.73)
ns Spectral index Eq. (2.73)
σT Thomson cross section
ω Equation of state of fluid Eq. (2.41)
cs Speed of sound of fluid Eq. (3.18)
a or a(η) Scale factor Eq. (2.1)
H Conformal Hubble function Below Eq. (2.45)
T¯ Background CMB temperature
g ˙¯τe−τ¯
R Density ratio (4/3)ρ¯r/ρ¯b
fn Neutrino fraction of total radiations ρ¯n/(ρ¯n + ρ¯r)
144 Glossary of Notations and Symbols
Table A.2 Mathematical notations and functions
Symbol Description Equation
∇ Covariant derivative
∇nˆ Screen-projected directional derivative Eq. (5.24)
θ(x) Heaviside step function
δi j, δ i j, δ i j The Kronecker delta
δ 3(k) The Dirac delta function
JM(x) Bessel function of the first kind
jℓ(x) Spherical Bessel function of the first kind
Pℓ(x) Legendre polynomial
sYℓm(nˆ) Spin-weighted spherical harmonics
Dℓm′m rotation matrices Eq. (4.28)
Gm1m2m3ℓ1ℓ2ℓ3 The Gaunt integral Eq. (C.14)
hℓ1ℓ2ℓ3 Eq. (4.57)(
ℓ1 ℓ2 ℓ3
m1 m2 m3
)
The Wigner 3-j symbols Eq. (C.13)
Table A.3 Notations for power spectra and bispectra
Symbol Description Equation
⟨...⟩ Ensemble average
aℓm Components of spherical harmonics on
temperature anisotropies
Eq. (2.83)
PΦ(k) Primordial power spectrum of Newtonian
potential
Above Eq. (2.86)
Cℓ CMB power spectrum Eq. (2.85)
Bm1m2m3ℓ1ℓ2ℓ3 3-point correlator of aℓms Eq. (2.90)
Bℓ1ℓ2ℓ3 CMB angular-averaged bispectrum Eq. (2.91)
bℓ1ℓ2ℓ3 Reduced bispectrum Eq. (2.92)
B(k1,k2,k3),
B(k1,k2)
Shape of bispectrum Eq. (2.95), above
Eq. (4.32)
f locNL , f
equ
NL , f
orth
NL Nonlinear coupling constants for local,
equilateral and orthogonal type
Eq. (2.103), (2.105),
(2.107), (4.53)
FNL Normalized nonlinear coupling constants Eq. (4.54)
(S/N) Signal-to-noise of bispectra Eq. (4.55)
FA,B Eq. (4.56)
145
Table A.4 Coordinate-related notations and quantities
Symbol Description Equation
Sub/Superscripts
Greek letters
Abstract indices for tensors
Sub/Superscripts
A, B, C, ...
Coordinates in background Manifold
Sub/Superscripts
I, J, K, ...
Spacelike coordinates in background
Manifold
Sub/Superscripts
a, b, c, ...
Indices for tetrad basis
Sub/Superscripts
i, j, k, ...
Spacelike indices for tetrad basis
Superscripts [I],
[II], ... , [N]
Orders of perturbations
η Conformal time Eq. (2.1)
ηi Initial conformal time See Section 2.5.3,
3.3.1
t Physical time
∫
a(η)dη
r Conformal distance from present η0−η
rLSS Conformal distance to last scattering sur-
face
p0 Photon energy in tetrad basis Eq. (2.5)
nˆ Direction of photon path Eq. (2.5)
eˆ0 Observer’s velocity Eq. (2.5)
eˆa, eˆa Tetrad fields Eq. (2.4)
e¯(m) For m =−1, 0, 1 Eq. (C.25)
[nˆnˆ]ℓm Decomposition into spherical harmonics Eq. (C.34)
k(±1) Multipole decomposition of wavenumber k
I e¯∗I(±1)
[k′k′′]2m Multipole decomposition of two
wavenumbers
[
nin j
]
2m k
′
I k
′′
J
ηµν Minkowski metric diag(−1,1,1,1)
gµν Metric Eq. (2.1)
Sµν Screen projector Eq. (2.6)
146 Glossary of Notations and Symbols
Table A.5 Perturbations
Symbol Description Equation
R Comoving curvature perturbations Eq. (2.66), (3.111)
Φ Newtonian gravitational potential Eq. (2.1)
Ψ Another scalar metric perturbation in
Newtonian gauge
Eq. (2.1)
BI Vector metric perturbation in Newtonian
gauge
Eq. (2.1), (2.2)
B˜(±1) Multipole decomposition of BI iB˜(±1) = BI e¯∗I(m)
HIJ Tensor metric perturbation in Newtonian
gauge
Eq. (2.1), (2.2)
H(±2) Multipole decomposition ofHIJ
[
nin j
]
2±2HIJ
δ Density perturbation δρ/ρ¯
Iˆℓm Intensity multipoles for photons and neu-
trinos
Eˆℓm, Bˆℓm Polarization multipoles
υ , υI Scalar and vector perturbation of fluid ve-
locity
Eq. (2.28)
υ˜(m) Multipole decomposition of fluid velocity iυ˜(m) = υ I e¯∗I(m)
σ , σI , σIJ Scalar, vector and tensor perturbation of
anisotropic stress
Eq. (2.29)
δ τ˙ Fraction of 1st-order differential optical
depth
Eq. (3.95)
δx[I]e Fraction of 1st-order free electrons x[I]e /x¯e
Θ Temperature anisotropies Eq. (4.5), (4.7)
△ Photon brightness Eq. (2.75)
S˜T , ST Source functions Eq. (2.75), (2.78),
(4.10), (4.11)
△ℓ(k) Transfer function Eq. (2.82)
α Deflection angle Eq. (2.88)
ψ(nˆ) Lensing potential Eq. (2.88)
ΨW Weyl potential (Φ+Ψ)/2
147
Table A.6 Quantities for EFEs and CMB photon
Symbol Description Equation
Gµν Einstein tensor Eq. (2.23)
Rµν , R Ricci tensor and scalar Eq. (2.24)
Γµαβ Christoffel symbols Eq. (2.25)
Tµν Stress-energy tensor Eq. (2.26)
ρ Energy density of fluid Eq. (2.26)
P Pressure of fluid Eq. (2.26)
uµ Fluid velocity Eq. (2.26)
πµν Anisotropies stress Eq. (2.26)
Πˆ A quantity defined with ℓ= 2 multipoles Eq. (3.79)
Π Π= Πˆ/4
ωbac Ricci rotation coefficients Eq. (2.38)
C1, C2, C3, C4,
C ′4,
Quadratic terms in 2nd-order scalar EFEs Eq. (3.29), (3.30),
(3.31), (3.32), (3.37)
C(1,±1), C(2,±1),
C(2,±2)
Quadratic terms in 2nd-order vector and
tensor EFEs
Eq. (3.40), (3.41),
(3.43)
L Liouville operator Eq. (2.34)
Cµν Collision operator Eq. (2.32)
Cδ Collision terms for continuity equation Eq. (3.45)
Cυ(m) Collision terms for Euler equation Eq. (2.56), (3.46),
(3.47)
Dab Residual collision operator Eq. (5.6)
Pµν Screen-projected intensity matrix Eq. (2.11)
I Photon intensity Eq. (2.12)
Pµν Symmetric traceless polarization tensor Eq. (2.12)
Iˆ Normalized energy-integrated photon in-
tensity
Eq. (2.14)
Pˆµν Normalized energy-integrated polariza-
tion tensor
Eq. (2.15)
Qˆ, Uˆ Normalized energy-integrated Stokes’ pa-
rameters
Eq. (2.18)
τ˙ Differential optical depth Eq. (2.57)
ne Number density of electrons
xe Fraction of free electrons
148 Glossary of Notations and Symbols
Table A.7 Liouville-related notations
Symbol Description Equation
Qab Source terms for line of sight approach Eq. (5.9)
T Time-ordering operator
ˆnˆ,r Vector operator for lensing effect Eq. (5.55) and (5.79)
♦ˆnˆ,r Vector operator for time-delay effect Eq. (5.68)
Vˆ (η , nˆ) Interaction operator for lensing effect Eq. (5.56)
Vˆx(η , nˆ) Interaction operator for time-delay effect Eq. (5.69)
Vˆr(η , nˆ) Interaction operator for redshift effect Eq. (5.77)
Uˆ(η0, η˜ , nˆ) Evolution operator for lensing effect Eq. (5.57)
CΨWℓ (η) Equal-time power spectrum of Weyl po-
tential
Eq. (5.82)
Ki(ℓ1, ℓ2) Nested double integrals of lensing effect Eq. (5.91)
Table A.8 Miscellaneous
Symbol Description Equation
Subscripts b, c, n,
r
Denoting baryons, CDM, neutrinos and
photons
Subscripts 1, 2 Acting on a first-order perturbation X
means Xi = X(ki,τ) for i = 1, 2
Header bar ‘ − ’ Denoting zeroth order perturbations
Header dot ‘ · ’ Partial derivative on conformal time
L 2L≡ 2ℓ+1
Nℓ Eq. (C.2)
Tk1,k2(k) Convolution operator Eq. (3.24)
sDnℓm, sT
n
ℓm, sU
n
ℓm The symbols used in second-order Boltz-
mann equations
End of Appendix
C.3
Appendix B
Table of Cosmological Parameters
Parameter Best fit
T¯ 2.7255 K
Ωbh2 0.022032
Ωch2 0.12038
Ωνh2 0.00064
ns 0.9619
As 2.215×10−9
H0 67.04 km s−1Mpc−1
YP 0.247695
Neff 3.046
w0 -1
τ 0.0925
zre 11.37
∆zre 0.5
Table B.1 The best-fit cosmological parameters are listed based on the Planck+WP dataset
in [3]. For the primordial tensor modes, parameters come from the BICEP2’s data in [2]
with nt = 0 and k∗ = 0.05/Mpc. The definitions of these parameters can be found in Table 1
of [3]. As mentioned in [9], the tension between Planck and Bicep due to the tensor mode
is mild. Thus, we do not re-fit the cosmological parameters using the data from the Planck
and BICEP2 together.

Appendix C
Spin-weighted Spherical Harmonics
C.1 Properties of Spin-weighted Spherical Harmonics
Spherical harmonics, which are used to decompose functions on the surface of a sphere, are
essential tools for studying the CMB signals projected in the full sky. Any function spin-s
s f depending on the direction nˆ can be expanded into multipoles of spin-weighted spherical
harmonics
s f (nˆ) =∑
ℓm
1
Nℓ
s fℓm sYℓm(nˆ), (C.1)
where we have, by following the normal modes in [51],
Nℓ ≡ i
ℓ
√
2ℓ+1
4π , (C.2)
and the spin-weighted spherical harmonic sYℓm(nˆ) is defined as
sYℓm(nˆ) =
√
(ℓ− s)!
(ℓ+ s)! ð
sYℓm(nˆ), 0≤ s≤ ℓ, (C.3)
sYℓm(nˆ) =
√
(ℓ+ s)!
(ℓ− s)! (−1)
sð¯−sYℓm(nˆ), − ℓ≤ s≤ 0, (C.4)
152 Spin-weighted Spherical Harmonics
with |m|, |s| ≤ ℓ, ℓ≥ 0 (otherwise equal to zero) and the Laplace’s spherical harmonics Yℓm.
The raising (ð) and lowering (ð¯) operators are defined as
ðs f (nˆ) =−(sinθ)s
[
∂
∂θ +
i
sinθ
∂
∂φ
]
(sinθ)−s s f (nˆ), (C.5)
ð¯s f (nˆ) =−(sinθ)−s
[
∂
∂θ −
i
sinθ
∂
∂φ
]
(sinθ)s s f (nˆ), (C.6)
for a spin-s function s f . By acting the operators on the spin-weighted spherical harmonics,
we obtain
ðsYℓm =
√
(ℓ− s)(ℓ+ s+1) s+1Yℓm, (C.7)
ð¯sYℓm =−
√
(ℓ+ s)(ℓ− s+1) s−1Yℓm. (C.8)
Here, we align the z-axis with the zenith.
There are many useful properties of the spherical harmonics. For example, the orthonor-
mality and completeness of the spin-weighted spherical harmonics are
∫
dnˆ sYℓm(nˆ)sY ∗ℓ′m′(nˆ) = δℓℓ′δmm′, (C.9)
∑
ℓm
sYℓm(nˆ)sY ∗ℓm(nˆ′) = δ (cosθ − cosθ ′)δ (φ −φ ′), (C.10)
where the complex conjugation of spherical harmonic is sY ∗ℓm(nˆ) = (−1)m+s−sYℓ,−m(nˆ). The
Legendre polynomial of degree ℓ is related to the spin-0 spherical harmonics
Pℓ(nˆ · nˆ′) =
4π
2ℓ+1
ℓ
∑
m=−ℓ
Yℓm(nˆ)Y ∗ℓm(nˆ′). (C.11)
In particular, when nˆ = nˆ′, we have Unsold’s theorem
ℓ
∑
m=−ℓ
Yℓm(nˆ)Y ∗ℓm(nˆ) =
2ℓ+1
4π . (C.12)
C.2 Properties of Wigner 3-j Symbols 153
C.2 Properties of Wigner 3-j Symbols
The integral of three spin-weighted spherical harmonics can be expressed as
∫
dnˆ s1Yℓ1m1(nˆ) s2Yℓ2m2(nˆ) s3Yℓ3m3(nˆ)
=
√
(2ℓ1 +1)(2ℓ2 +1)(2ℓ3 +1)
4π
(
ℓ1 ℓ2 ℓ3
−s1 −s2 −s3
)(
ℓ1 ℓ2 ℓ3
m1 m2 m3
)
, (C.13)
where the brackets are the Wigner 3-j symbols. In spin-0 case, the integral is defined as the
Gaunt coefficient
Gℓ1ℓ2ℓ3m1m2m3 ≡
∫
dnˆ Yℓ1m1(nˆ) Yℓ2m2(nˆ) Yℓ3m3(nˆ). (C.14)
Wigner 3-j symbols
(
ℓ1 ℓ2 ℓ3
m1 m2 m3
)
are non-zero only when the following conditions are
satisfied
m1 +m2 +m3 = 0, (C.15)
ℓ1 + ℓ2 + ℓ3 = N, (C.16)
|ℓ1− ℓ2| ≤ ℓ3 ≤ |ℓ1 + ℓ2|, (C.17)
where N is a positive integer. In case of m1 = m2 = m3 = 0, the symbols are non-zero if
ℓ1 + ℓ2 + ℓ3 is an even positive integer. Moreover, the Wigner 3-j symbols are related to
Clebsch–Gordan coefficients by
(
ℓ1 ℓ2 ℓ3
m1 m2 m3
)
= (−1)
ℓ1−ℓ2−m3
√
2ℓ3 +1
⟨ℓ1m1ℓ2m2|ℓ3−m3⟩. (C.18)
The Wigner 3-j symbols possess some symmetries which can be used to simplify the
calculation of its coefficients. For example, it is invariant under an even permutation of its
columns
(
ℓ1 ℓ2 ℓ3
m1 m2 m3
)
=
(
ℓ2 ℓ3 ℓ1
m2 m3 m1
)
=
(
ℓ3 ℓ1 ℓ2
m3 m1 m2
)
. (C.19)
154 Spin-weighted Spherical Harmonics
In contrast, there is a phase factor under an odd permutation of its columns
(
ℓ1 ℓ2 ℓ3
m1 m2 m3
)
= (−1)L
(
ℓ2 ℓ1 ℓ3
m2 m1 m3
)
= (−1)L
(
ℓ1 ℓ3 ℓ2
m1 m3 m2
)
, (C.20)
where L≡ ℓ1 + ℓ2 + ℓ3. Similarly, a phase factor appears when changing the signs of ms
(
ℓ1 ℓ2 ℓ3
−m1 −m2 −m3
)
= (−1)L
(
ℓ1 ℓ2 ℓ3
m1 m2 m3
)
. (C.21)
Furthermore, we have the following orthogonality relations
(2ℓ+1) ∑
m1m2
(
ℓ1 ℓ2 ℓ
m1 m2 m
)(
ℓ1 ℓ2 ℓ′
m1 m2 m′
)
= δℓℓ′δmm′, (C.22)
∑
ℓm
(2ℓ+1)
(
ℓ1 ℓ2 ℓ
m1 m2 m
)(
ℓ1 ℓ2 ℓ
m′1 m
′
2 m
)
= δm1m′1δm2m′2. (C.23)
In particular, we have
∑
m1m2m3
(
ℓ1 ℓ2 ℓ3
m1 m2 m3
)2
= 1. (C.24)
C.3 Decomposition of Spherical Harmonics
To perform the decomposition of spherical harmonics, it is more convenient to use another
basis which is related to that of a Cartesian coordinate system by1
e¯(0) ≡−e¯z, e¯(1) ≡
1
√
2
(e¯x+ ie¯y), e¯(−1) ≡
1
√
2
(−e¯x+ ie¯y), (C.25)
where the overhead bars denote the flat background space-time2. It can be shown that the
bases obey the orthonormality
3
∑
I=1
e¯I(m)e¯
∗I
(m′) = δmm′,
1
∑
m=−1
e¯I(m)e¯
∗J
(m) = δIJ, (C.26)
1Our definition is the same as that in [83]. See also [80] with a different definition.
2In the CMB study, all these bases live on the background manifold which has a trivial metric δIJ under
the Cartesian coordinate system.
C.3 Decomposition of Spherical Harmonics 155
where e¯∗(m) = (−1)me¯(−m). Any 3-vector v with spin-0 can then be decomposed into
v =∑
m
v(m)e¯(m) =−
iv
N1
1
∑
m=−1
Y ∗1,m(vˆ)e¯(m), (C.27)
where the components are
v(0) =−vz =−
iv
N1
Y ∗1,0(vˆ), (C.28)
v(1) =
1
√
2
(vx− ivy) =−
iv
N1
Y ∗1,1(vˆ), (C.29)
v(−1) =−
1
√
2
(vx+ ivy) =−
iv
N1
Y ∗1,−1(vˆ), (C.30)
with v = v vˆ. Using Eq. (C.1) and Eq. (C.27), we can perform the decomposition of the
spin-0 spherical harmonics with respect to nˆ such that
[nˆ]ℓm =−iδℓ1 e¯∗(m), (C.31)
[nˆ ·v]ℓm =−iδℓ1 v(m). (C.32)
Now, we consider expanding a product of two nˆ-dependent functions into spherical
harmonics, e.g. sh(nˆ) = s1 f (nˆ)s2g(nˆ) with s = s1 + s2. Using Eq. (C.1) and Eq. (C.14), we
obtain the multipoles shℓm as
∑
ℓ1m1
∑
ℓ2m2
i−ℓ1−ℓ2+ℓ(−1)m+s(2ℓ+1)s1 fℓ1m1 s2gℓ2m2
(
ℓ1 ℓ2 ℓ
−s1 −s2 s
)(
ℓ1 ℓ2 ℓ
m1 m2 −m
)
.
(C.33)
For example, if s1 f (nˆ) = s2g(nˆ) = nˆ, we have3
[nˆnˆ]ℓm =



1
3 ∑m1 e¯(m1)e¯
∗
(m1), if ℓ= 0,
−(−1)m∑m1m2
√
10
3


1 1 2
m1 m2 −m

 e¯∗(m1)e¯
∗
(m2), if ℓ= 2.
(C.34)
3This decomposition is used when we derive the evolution equations of 2nd-order scalar, vector and tensor
perturbations from the EFEs in Section 3.1.2.
156 Spin-weighted Spherical Harmonics
It obeys the orthogonality that
∑
i j
[nin j]2m[nin j]∗2m′ =
2
3
δmm′. (C.35)
Moreover, if s1 f (nˆ) = nˆ ·v, we obtain4
s [(nˆ ·v)sg(nˆ)]ℓm =



−i∑m1 sD
m1
ℓm v(m1) sgℓ−1,m−m1, if ℓ2 = ℓ−1,
−∑m1 sT
m1
ℓm v(m1) sgℓ,m−m1, if ℓ2 = ℓ,
i∑m1 sU
m1
ℓm v(m1) sgℓ+1,m−m1, if ℓ2 = ℓ+1,
(C.36)
where the coefficients are defined as
sD
0
ℓm =
1
ℓ(2ℓ−1)
√
(ℓ2−m2)(ℓ2− s2), (C.37)
sD
±1
ℓm =
1
ℓ(2ℓ−1)
√
(ℓ±m)(ℓ±m−1)(ℓ2− s2)
2
, (C.38)
sT
0
ℓm =−
ms
ℓ(ℓ+1) , (C.39)
sT
±1
ℓm =±
s
ℓ(ℓ+1)
√
(ℓ±m)(ℓ∓m+1)
2
, (C.40)
sU
0
ℓm =
1
(ℓ+1)(2ℓ+3)
√
[(ℓ+1)2−m2] [(ℓ+1)2− s2], (C.41)
sU
±1
ℓm =
1
(ℓ+1)(2ℓ+3)
√
(ℓ∓m+1)(ℓ∓m+2) [(ℓ+1)2− s2]
2
, (C.42)
4This is useful when we derive the 2nd-order Boltzmann equation with s2g(nˆ) as the photon intensity I(nˆ)
Iˆ(nˆ) and the Stokes parameters (Q± iU)(nˆ).
Appendix D
Perturbations of Tetrads and Ricci
Rotation Coefficients
In this appendix, we calculate the perturbations of tetrads and Ricci rotation coefficients up
to 2nd order. We will fix to the Newtonian gauge for 1st-order and 2nd-order perturbations
and neglect the 1st-order vector and tensor perturbations.
D.1 Perturbations of Tetrads
Using Eq. (2.4), we can calculate the coefficients ξXab order by order, i.e.
ηab = lim
λ→1
φ∗λ ,ξ (ηab) = limλ→1φ
∗
λ ,ξ (eaA)φ∗λ ,ξ (ebB)φ∗λ ,ξ (gAB)
=ξXac e¯Ac ξXbd e¯Bd ξgAB, (D.1)
where ξgAB is the pulled-back physical metric on the background manifold, e.g. Eq. (2.1).
Similarly, the coefficients ξYab can be solved by ηab. We remark that the upper and lower
indices of the coefficients are obtained by ηab and ηab.
There remain residual freedoms in determining the anti-symmetric part of the coeffi-
cients ξXab and ξYab due to boost and rotation. In particular, we align ξ eˆ
0 with dη , i.e. ξYi0 =
0. Physically, it means that the velocity of the chosen observer is orthonormal to the hyper-
surfaces of constant time. This fixes the boost freedom such that ξY[i0] =−ξY[0i] =−ξY(i0)
with (. . .) and [. . .] denoting the symmetric and anti-symmetric parts. On the other hand, the
rotation freedom can be fixed by setting ξY[i j] = 0. With these conditions, we can determine
158 Perturbations of Tetrads and Ricci Rotation Coefficients
the coefficients uniquely. We can find that1, for 1st order,
ξX
i[I]
0
=− ξY i
[I]
0
= 0, (D.2)
ξX
0[I]
0
=− ξY 0
[I]
0
=−Φ, (D.3)
ξX
0[I]
i
=− ξY 0
[I]
i
= 0, (D.4)
ξX
i[I]
j
=− ξY i
[I]
j
=Ψδ IJ, (D.5)
and for 2nd order,
ξX
i[II]
0
=− ξY i
[II]
0
= 0, (D.6)
ξX
0[II]
0
=−Φ[II] +3Φ2, (D.7)
ξY
0[II]
0
= Φ[II]−Φ2, (D.8)
ξX
0[II]
i
= B[II]I , (D.9)
ξY
0[II]
i
=−B[II]I , (D.10)
ξX
i[II]
j
= Ψ[II]δ IJ−HI
[II]
J +3Ψ2δ IJ, (D.11)
ξY
i[II]
j
=−Ψ[II]δ IJ +HI
[II]
J −Ψ
2δ IJ. (D.12)
D.2 Perturbations of Ricci Rotation Coefficients
Using the definition of the Ricci rotation coefficients in Eq. (2.38), we can show that the
coefficients are antisymmetric in their last two indices, i.e. ωabc = −ωacb2. This implies
that, for any order,
ω000 = ωi00 = 0. (D.13)
In background order, all the coefficients are zero except
ω¯i j0 =−ω¯i0 j =
H
a
δi j. (D.14)
1We omit the the superscripts [I] denoting 1st-order perturbations.
2Here, the Ricci rotation coefficients are expressed in tetrad basis. Hence, the indices can be raised and
lowered by multiplying with the Minkowski metric ηab and ηab respectively.
D.2 Perturbations of Ricci Rotation Coefficients 159
In 1st order, the non-zero coefficients are
ω [I]0i0 =−ω
[I]
00i =
1
a
∂IΦ, (D.15)
ω [I]i0 j =−ω
[I]
i j0 =
1
a
(
HΦ+ Ψ˙
)
δIJ, (D.16)
ω [I]i jk =−ω
[I]
ik j =
1
a
(∂JΨδIK−∂KΨδIJ) . (D.17)
For 2nd-order Liouville terms, only the following coefficients are used
ω [II]0i0 =−ω
[II]
00i =
1
a
(
∂IΦ[II]−4Φ∂IΦ+2Ψ∂IΦ
)
, (D.18)
ω [II]i0 j =−ω
[II]
i j0 =
1
a
[
1
2
(
∂IB[II]J +∂JB
[II]
I
)
−H˙[II]IJ +
(
Ψ˙[II] +HΦ[II]
)
δIJ
−3HΦ2δIJ−2ΦΨ˙δIJ +4ΨΨ˙δIJ
]
. (D.19)