Nonlinear approximation schemes in relativistic cosmology

Abstract

In this thesis I present two main projects, based on the application of different nonlinear approximation schemes in relativistic cosmology. The first project comprises effects on the matter distribution on very large scales sourced by the nonlinearity of General Relativity, while the second project examines a weak lensing analysis considering all scales.
In the first project, the results of which were published in [55], we used two different approximation schemes, the gradient expansion and standard perturbation theory, to examine the effects the nonlinear nature of General Relativity (GR) has on the density field allowing primordial non-Gaussianity in the inital conditions. The gradient expansion restricts the computations to very large scales, where the gradients of the metric perturbations are significantly smaller than their time derivatives. In this work, we neglect contributions of the order O(∇4) in the gradient expansion. We show that at order O(3) and O(4) in standard perturbation theory the intrinsic nonlinearity of GR produces a mixing of the primordial non-Gaussianity in the density contrast. The main 1esult of this project is that at higher orders due to the nonliearity of GR a mixing of f_NL, g_NL, and h_NL occurs in the density field.
In the second project, of which the results will be published in the paper [54], we computed the convergence and the shear up to higher orders using the post-Friedmann approximation scheme. This approximation scheme is especially beneficial for the subject of weak lensing due to its validity on all scales. In weak lensing, we integrate along the line of sight and thereby couple large scales to small scales. The post-Friedmann approximation is a post-Newtonian-type approximation scheme in a cosmological setting that combines both the fully nonlinear Newtonian dynamics on small scales and the relativistic perturbations on large scales. It comprises scalar, vector, and tensor perturbations, whereas the lowest order of the vector perturbation is sourced by Newtonian quantities, yet its effects are purely relativistic. The vector potential does not influence matter dynamics but affects the photon geodesic and therefore the weak lensing analysis. We present the convergence and shear projected on a spherical screen space, which allows us to go beyond the thin-lens or small-angle approximation, in terms of the redshift z up to the order O 1/c⁴. The main reason for the investigation was to have a formalism for the weak lensing analysis that includes the effect of the gravimagnetic potential on nonlinear scales. It was shown in [33] that the magnitude of this vector potential is small but not negligible on nonlinear scales. Hence, the main result of this project is the computation of the convergence and shear up to higher orders using a formalism that includes the gravimagnetic potential and is valid on all scales including nonlinear, small scales. In particular, we show the contribution of the vector potential to the convergence and shear.

Date of Award	Apr 2019
Original language	English
Awarding Institution	University of Portsmouth
Supervisor	Marco Bruni (Supervisor), David Bacon (Supervisor) & David Wands (Supervisor)