# Cosmological Perturbation Theory

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## Abstract

Firstly, a dynamical analysis of adiabatic perturbations of a perfect fluid is performed to first-order about a general FLRW background using the 1 + 3 covariant and gauge-invariant formalism. The 1 + 3 covariant analog of the Mukhanov-Sasaki and Grishchuk variables needed to quantise the scalar and tensor perturbations respectively about a spatially-curved FLRW background space-time are identied. The dynamics of the vector perturbations is also discussed. Secondly, a dynamical analysis of an inhomogeneous and anisotropic effective Weyssenhoff fluid, which is a perfect fluid with spin in the Einstein-Cartan theory, is performed in a gauge-invariant manner using the 1+3 covariant and gauge-invariant approach. A verication of the dynamical equations is performed for the special case of irrotational flow with zero peculiar acceleration by evolving the constraints. Thirdly, a dynamical analysis of an effective homogeneous and irrotational Weyssenhoff fluid in general relativity is performed using the 1 + 3 covariant and gauge-invariant approach. The spin contributions to the field equations produce a bounce that averts an initial singularity, provided that the spin density exceeds the rate of shear. At later times, when the spin contribution can be neglected, a Weyssenhoff fluid reduces to a standard cosmological fluid in general relativity. Numerical solutions for the time evolution of the generalised scale factor in spatially-curved models are presented, some of which exhibit eternal oscillatory behaviour without any singularities. In spatially- at models, analytical solutions for particular values of the equation-of-state parameter are derived. Although the scale factor of a Weyssenhoff fluid generically has a positive temporal curvature near a bounce, it requires unreasonable fine tuning of the equation-of-state parameter to produce a sufficiently extended period of inflation to fit the current observational data. Fourthly, in order to determine numerically the background dynamics of general in inflationary models, suitable classical initial conditions have to be found from which to start the integration of the equations of motion. The method proposed by Boyanovsky, de Vega and Sanchez assumes a spatially-flat model with an inflaton potential typical of new inflation and determines the initial conditions at a time when the inflaton field is at the local maximum of the potential by assuming equipartition of the kinetic and potential energies of the inflaton field. This leaves the normalisation of the solutions undetermined. The procedure followed by Lasenby and Doran assumes a spatially closed model with a chaotic inflaton potential and determines the initial conditions as the model emerges from the initial singularity by performing a series expansion. We note that quite generically immediately after the initial singularity, the pre-inflationary dynamics of the universe is dominated by the kinetic energy of the inflaton, which naturally yields analytic solutions for any spatial curvature - provided there is no bounce - with no dependence on the potential. This therefore suggests a new generic way of setting initial conditions for inflation. Using this new procedure, we determine the initial conditions for chaotic and new inflation potentials and study both the subsequent background evolution and the spectrum of scalar perturbations produced in the spatially-flat case. See more