Title:

The perturbed universe : dynamics, statistics and phenomenology

This thesis is broadly concerned with the dynamics, statistics and phenomenology of the perturbed Universe. By studying the perturbations to cosmological spacetimes, and the subsequent growth of large scale structure, we find that we can link both fundamentally and astrophysically interesting physics to cosmological observables. We use a healthy mix of statistical, analytical and numerical techniques throughout this thesis. In Chapter 2 we introduce and summarise the statistics of random fields, as these are fundamental objects used to model cosmological observables. We introduce the spherical FourierBessel expansion as a tool to perform genuine 3dimensional studies of cosmological random fields. In Chapter 3 we introduce the theory of inflation and discuss the basic machinery that allows us to calculate the statistical properties of the quantum mechanical flucatuations that seed large scale structure. What we see is that different fundamental physics in the early Universe leads to different statistical properties that we may test. The second half of Chapter 3 introduces the large scale structure of the Universe that describes the clustering of galaxies on cosmological scales. We discuss the growth and evolution of structure under gravitational collapse and the core observables that are predicted, such as the power spectrum, variance and skewness. Chapter 4 introduces the Minkowski functionals. These are a set of topological statistics that probe the morphological properties of random fields. In particular they may be used to quantify deviations from Gaussianity in the large scale structure of galaxies. The deviations from Gaussianity can be generated by two primary mechanisms: 1) The gravitational collapse of perturbations is a nonlinear process. Even if we have Gaussian initial conditions, gravitational collapse will induce nonGaussianity. 2) Different theories for the early Universe will imprint different non Gaussian features in the primordial perturbations that seed large scale structure, i.e. we have nonGaussian initial conditions. We can connect the amplitude and momentum dependence of the nonGaussianity to different fundamental interactions. We introduce a topological statistic based on the Minkowski functionals that retains the momentum dependence giving us greater distinguishing power between different contributions to nonGaussianity. In Chapter 5 we introduce the Baryon Acoustic Oscillations (BAOs) as described in the spherical FourierBessel formalism. The BAOs are a solid prediction in cosmology and should help us to constrain cosmological parameters. We implement a full 3dimensional study and study how redshift space distortions, induced by the motion of galaxies, and nonlinearities, induced by gravitational collapse, impact the characteristics of these BAOs. Chapter 6 extends the spherical FourierBessel theme by introducing the thermal Sunyaev Zelâ€™dovich (tSZ) effect and cosmological weak lensing (WL). It is thought that weak lensing will provide an unbiased probe of the dark Universe and that the tSZ effect will probe the thermal history of the Universe. Unfortunately, the tSZ effect loses redshift information as it is a line of sight projection. We study the crosscorrelation of the tSZ effect with WL in order to reconstruct the tSZ effect in a full 3dimensional study in an attmept to recover the lost distance information. We use the halo model, spectroscopic redshift surveys and suvery effects to understand how detailed modelling effects the tSZWL cross correlation. Chapter 7 marks a real change in theme and introduces the subject of relativistic cosmology. Inparticular we introduce the 1+3, 1+1+2 and 2+2 formalisms as tools to study cosmological perturbations. We provide rather selfcontained introductions and provide some minor corrections to the literature in the 1+1+2 formalism as well as introducing new results. In Chapter 8 we apply the 1+1+2 and 2+2 approaches to the Schwarzschild spacetime. Here we outline the full system of equations in both approaches and how they are related, setting up a correspondence between the two. Our aim is to construct closed, covariant, gaugeinvariant and frameinvariant wave equations that govern the gravitational perturbations of the Schwarzschild spacetime. We correct a result in the literature and derive two new equations. The first governs axial gravitational perturbations and is related to the magnetic Weyl scalar. The second is valid for both polar and axial perturbations and is given by a combination of the magnetic and electric Weyl 2tensors. We discuss their relation to the literature at large. Finally, in Chapter 9 we apply the 1+1+2 and 2+2 approaches the LTB spacetime. This inhomogeneous but spherically symmetric spacetime is the first stepping stone into genuinely inhomogeneous cosmological spacetimes. We seek a closed, covariant master equation for the gravitational perturbations of the LTB spacetime. We present an equation governing axial gravitational perturbations and a preliminary equation, valid for both the polar and axial sectors, that is constructed from the electric and magneticWeyl 2tensors but is coupled to the energymomentum content of the LTB spacetime. We discuss how auxilliary equations may be introduced in order to close the master equation for polar and axial perturbations. This last result leads to the identification of H as a master variable for axial perturbations of all vacuum LRSII spacetimes and the LTB spacetime. It is thought that these results can be extended to nonvacuum LRSII spacetimes. Likewise, the master variable constructed from Weyl variables constitutes a master variable for all vacuum LRSII spacetimes and it is thought that this will extend to the nonvacuum case.
