Modelling cosmological structure formation in the non-linear regime
Lane, Francesca Carina
We are entering an age of precision cosmology with upcoming instruments and surveys such as Euclid, the Dark Energy Spectroscopic Instrument (DESI), the Large Synoptic Survey Telescope (LSST) or the Rubin Observatory and the James Webb Space Telescope (JWST). These will allow us to place even tighter statistical constraints on viable dark matter, dark energy and modified gravity theories. This could lead to discovering the answer to some of Astronomy’s biggest questions about structure formation in the Universe. In order to test these theories, we must have theoretical predictions to compare to observations; some of the most powerful observables are 2-point statistics. These include the matter correlation function and its Fourier transform, the power spectrum. These simple statistics could allow us to distinguish between cosmological models and modified gravity theories. Calculating these statistics on large-scales and in the linear regime is simple as we can apply the linear theory. However, on small scales (r < 100 Mpc), non-linear gravitational and baryonic effects become too important to ignore and our linear theories break down. The most successful theoretical technique, Standard Perturbation Theory (SPT), can be extended into the non-linear regime using techniques such as loop corrections and IR-resummation. Although these extensions have allowed us to compute the non-linear power spectrum on smaller scales, they can only push so far into the non-linear regime. This is especially the case for the correlation function where the most popular perturbation theory technique cannot accurately model even the mildly non-linear regime. Currently, the most accurate methods for calculating the non-linear regime are N-body and hydrodynamical simulations. These simulations must be run at high resolutions and with large box sizes to match the accuracy of upcoming observations. They are therefore very costly to run and if one wishes to test multiple dark matter theories, for example, a new simulation must usually be run for each theory. In order to produce a wide variety of predictions for observations economically, we must return to our perturbative methods and search for ways to extend their reach further. In this thesis, we will introduce a perturbative method for calculating the correlation function and power spectrum in the non-linear regime. We focus on these statistics as they are simplest to calculate and the most commonly used. This method is called the Cosmological Trajectories Method (CTM). We will present a formula for the power spectrum and show how an expanded version of the power spectrum can be calculated numerically. One of the main advantages of the CTM is that it can be applied to a wide range of cosmologies and redshifts. An approximation called Beyond Zel’dovich is then introduced and the CTM is used to compute its power spectrum and correlation function. This approximation aims to extend the Zel’dovich approximation (usually used to set the initial conditions of numerical simulations, study the non-linear regime and BAO reconstruction) into the non-linear regime. The Zel’dovich approximation can describe the formation of the cosmic web and is exact in 1D up until shell-crossing. In 3D the approximation performs well at high redshifts and mildly non-linear scales (k ≲ 0.1 h Mpc−1) until it breaks down due to shell-crossing and the formation of caustics. We compare the Beyond Zel’dovich approximation to other methods including SPT 1-loop and Convolution Lagrangian Perturbation Theory (CLPT). We find that the Beyond Zel’dovich approximation breaks down at low redshifts and for scales k > 0.1 h Mpc−1. This motivates the introduction of a Gaussian damped initial power spectrum in order to damp down this breakdown of the Beyond Zel’dovich approximation on small scales and at late times. With the introduction of a Gaussian cut-off, we conclude that the Beyond Zel’dovich approximation is best applied at redshifts z ≥ 2. At these redshifts, it outperforms the Zel’dovich approximation and at redshifts z ≥ 4 it also outperforms the Euclid Emulator. The Beyond Zel’dovich approximation performs well on mildly non-linear scales and at redshifts above z = 2. This implies that this approximation could be also used to model redshift-space distortions in this regime and applied to BAO and Lyman-α observations. We, therefore, extend the CTM into redshift space. We compute the redshift-space power spectrum and correlation function using the Beyond Zel’dovich approximation for a range of redshifts. Finally, we compare these results to a range of other methods including the Kaiser formula and SPT 1-loop as we did for the real-space statistics.