Computational Issues in Mathematical Cosmology

Computational Issues in Mathematical Cosmology Alan Coley1, Luis Lehner2, Frans Pretorius3 & David Wiltshire4 March 9, 2017 1. Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada. 2. Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada. 3. Department of Physics, Princeton University, Princeton, New Jersey 08544, USA. 4. Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand. 1 Introduction Numerical computations have always played an important role in physical cosmology, particularly in conventional N-body Newtonian simulations [1]. Compu- tational Cosmology (CC) is playing an increasingly important role in exploring general relativistic effects, and numerical simulations are often an extremely useful way to gain new insights into mathematical cosmology [2–4]. CC codes were initially developed to address inhomogeneous cosmologies [5], and CC has always been important in the study of cosmological singularities. As we shall briefly discuss below, CC simulations of large scale dynamical processes in the early Universe have recently been reported. 1 More complex CC codes aimed at understanding cosmological evolutions are currently being developed. Recently numerical studies of cosmic bubble collisions and inflation in inhomogeneous geometries have been initiated. These issues will be addressed in this special Focus Issue of Classical and Quantum Gravity. 1There are a number of numerical packages used in computational cosmology. The Einstein Toolkit is a standard open-source software suite for relativistic astrophysics, and also contains modules for the evolution and analysis of cosmological spacetimes in full general relativity: http://einsteintoolkit.org (see also http://hyperspace.uni-frankfurt.de/category/codes/). Dedicated programs include: GRCHOMBO [6] (see also www.grchombo.org), CosmoGRAPH [7], gevolution [8] and DEFROST [9] (see also www.sfu.ca/physics/cosmology/defrost/). 1 2 Inhomogeneous cosmology Most studies of theoretical and observational cosmology have utilized analyt- ical methods. This is, in part, because the observed high degree of statistical spatial homogeneity and isotropy (and particularly observations of the cosmic microwave background radiation) supports the assumption that the large scale structure of the Universe is well-described by the exact Friedmann- Lemaˆıtre-Robertson-Walker (FLRW) metrics, with deviations from the FLRW solutions small and hence amenable to treatment by perturbation theory. How- ever, at the present epoch statistical spatial homogeneity (as estimated by the 2–point galaxy–galaxy correlation function [10]) only occurs on a mini- −1 mum spatial scale ∼> 100 h Mpc (where h is related to the Hubble constant −1 −1 by H0 = 100h km sec Mpc ). On smaller scales, matter in the Universe is strongly inhomogeneous at the present epoch, exhibiting a complex cosmic web [11] of hierarchical structure: the Universe is dominated in volume by expanding voids [12], with bound galaxy clusters forming sheets, filaments and knots that surround the voids and thread them. In fact, some 40% of the volume of the Universe is contained in voids of just one characteristic diameter, −1 30 h Mpc, and a density contrast δρ < −0.94 [13,14] very close to empty (δρ = −1). This means that on these scales we are in the “nonlinear regime” of cosmology with respect to perturbation theory. At present the nonlinear regime is still treated almost exclusively by N– body numerical simulations using Newtonian gravity [1]. The full relativistic problem is very complicated; after all, it took decades to solve even the two body problem in numerical relativity. A full theory of computational cosmology (CC) in general relativity that deals with the complexity of late epoch structure may well take decades of development. However, it is important that a start is made, especially given that the standard Newtonian approach to structure formation invokes two as yet unknown components – dark energy and dark matter – for most of the energy density of the Universe. Even if relativistic effects were only to lead to percent level corrections to a FLRW model plus Newtonian gravity, the observed inhomogeneity demands we confront basic unknowns. There is one crucial difference between Newtonian and general relativistic dynamics. In Newtonian gravity energy is simply defined and absolutely conserved. In general relativity this is not the case in the absence of time symmetry, as occurs in the actual Universe. Future computational cosmology must not only deal with subtle numerical issues but also confront fundamental physical questions relating to the coupling of gravitational energy and matter energy-momentum as one coarse grains in a hierarchy of structure as complex as that of the Universe we observe. −1 The related debate about the backreaction of small scale (∼< 100 h Mpc) inhomogeneities on average cosmic expansion, cosmic acceleration and the na- ture of dark energy (c.f., [15, 16]), cannot be resolved purely by mathematical analysis. The issue at stake is one of starting assumptions in response to an unsolved physical question: What is the largest scale that we can coarse-grain matter and geometry obeying Einstein’s equations on one scale while expecting 2 the average evolution to still be an exact solution of Einstein’s equations on a larger scale? CC within full general relativity is now beginning to address these fundamental issues [6–8]. To date the emphasis has been on studying the properties of the spacetimes. Recently, these CC studies have been extended by including photons traversing these spacetimes, consequently investigating the impact of inhomogeneities on cosmological observations [17]. −1 Separately, the first ray tracing studies of small scale (∼< 100 h Mpc) inhomogeneous exact solutions of Einstein’s equations – Szekeres models – are being performed, which are constrained by actual large galaxy surveys [18]. This pro- vides an independent route to testing the possible differences of full general relativity from Newtonian gravity in its nonlinear regime, in realistic models constrained by data. 2.1 Black hole lattices CC studies of whether an effective expansion rate of an inhomogeneous model is consistent with that of an equivalent homogeneous dust filled universe have been considered in the context of the evolution of a periodic lattice of black holes (BH) (the BHs mimicking strong, self-gravitating inhomogeneities, which are the most extreme example of non-linearity possible in GR) [19–21]. In [20] the authors explicitly constructed and evolved a three-dimensional, fully relativistic, eight-BH lattice with spherical topology; these quantities were then compared against a reference closed FLRW model. The effects of local inhomogeneities have been investigated in [21] using different initial data, describing an expanding inhomogeneous universe model composed of regularly aligned BHs of identical mass. In this limit, their simulations are always dominated by the expanding underdense regions, hence the correction to the energy density is negative and the effective pressure is positive. 2.2 Bouncing models Using exotic matter or alternative modified theories of gravity can lead to the initial singularity being replaced by a bounce to an expanding universe. For example, CC methods have been applied to the study of bouncing cosmologies in the ekpyrotic cosmological scenario by studying the evolution of adiabatic perturbations in a nonsingular bounce [22]; the results show that the bounce is dis- rupted in regions of the Universe with significant inhomogeneity and anisotropy over the background energy density, but is achieved in regions that are rela- tively homogeneous and isotropic. Sufficiently small perturbations, consistent with observational constraints, can pass through the nonsingular bounce with negligible alteration from the nonlinearity. It is also possible to study the persistence of black holes through a bounce [23] (see also [24]). Bounces can also be realized by quantum effects in regions of extreme energies, and Loop Quantum Gravity inspired cosmology has also been studied in [25]. 3 2.3 Very early Universe Studies of “bubble universes”, in which our Universe is one of many nucleating and growing inside an ever-expanding false vacuum, have also been undertaken with CC tools. In particular, [26] investigated the collisions between bubbles, by computing the cosmological observables arising from bubble collisions directly from the Lagrangian of a single scalar field. It is expected that initial conditions will contain some measure of inhomogeneities, and random initial conditions will not necessarily give rise to an inflationary spacetime; indeed, it is often stated that inflation requires a homogeneous patch of size roughly the inverse Hubble radius to begin. CC has been used to consider the effects of inhomogeneous initial conditions on inflation. It has been shown that large field inflation is robust to simple inhomogeneous (and anisotropic) initial conditions with large initial gradient energies in situations in which the field is initially confined to the part of the potential that supports inflation, while it is also known that small field inflation is much less robust to inhomogeneities than its large field counterpart [27]. Recently, the large field case with a varying extrinsic curvature has been investigated [28]; it was found that regions which are initially collapsing may

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