I use mathematical models and numerical simulations to constrain cosmological inflation, the seeds of structure, and the mass of the neutrino. I revisit arguments that simple models of inflation with a small red tilt in the scalar power spectrum generically yield an observable tensor spectrum. I show that criteria for fine-tuning based upon the algebraic simplicity of the potential depend strongly upon the assumptions they incorporate about the potential. Furthermore, several models with algebraically simple potentials require carefully tuned initial field configurations. I demonstrate the existence of potentials with vanishingly small tensor amplitudes which are natural in terms of both their algebraic form and initial conditions. I thus argue that proposed experiments which make highly sensitive measurements of the tensor amplitude cannot definitively rule out the inflationary paradigm.

The overshoot problem is the need to make the initial kinetic energy of the inflaton small enough to ensure slow-roll. I investigate claims that brane inflation solves the overshoot problem through microphysical restrictions on the phase space of initial conditions. By carrying out a comprehensive analysis of the parameter space allowed by the latest advances in brane inflation model-building, I find that the vast majority of the phase space of initial conditions is still dominated by overshoot trajectories.

Current results from the Lyman-α forest assume that the primordial power spectrum of density perturbations follows a simple power-law form with running. I perform a large suite of numerical simulations, using them to calibrate a minimally parametric framework for describing the power spectrum. Combined with cross-validation this framework allows me to directly reconstruct the power spectrum shape, a consistency check on the standard model. I find no evidence for deviation from scale-invariance, but current Lyman-α data do not have sufficient statistical power to robustly probe the shape of the power spectrum at these scales. In contrast, the ongoing Baryon Oscillation Sky Survey will be able to do so with high precision.

I perform an extensive suite of N-body simulations of the matter power spectrum, probing deep into the non-linear regime while incorporating massive neutrinos. I compare my results to the widely used HALOFIT approximation, and find that in the strongly nonlinear regime it significantly over-predicts the suppression due to the free-streaming of neutrinos. Most published constraints are not affected, as they have used HALOFIT only in the linear or mildly non-linear regime. However, my results are important for future galaxy and weak lensing surveys.