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 nature 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 again lead to local black holes, but overall the spacetime remains inflationary if the spacetime is open. It has recently been found [29] that a spectrum of Gaussian, linear, adiabatic, scalar, growing mode perturbations not only gives rise to acoustic oscillations in the primordial plasma, but also to shock waves causing departures from local thermal equilibrium as well as vorticity and gravitational waves.

4 3 Cosmological singularities and spikes

Historically CC has been very important in the study of cosmological singularities and the BKL conjectures. There have been several reviews of this topic recently (see, for example, [4]). CC will also be crucial for investigating spikes, but to date this topic is in its infancy, particularly from the computational point of view.

3.1 BKL Belinskii, Khalatnikov and Lifshitz (BKL) [30] have conjectured that within GR, for a generic inhomogeneous cosmology, the approach to the (past) spacelike singularity is vacuum dominated, local, and oscillatory (mixmaster). The associated dynamics is referred to as the BKL dynamics. In more detail, the BKL conjecture asserts that: (i) The initial big bang cosmological singularity is generally spacelike. (ii) The space derivatives can be neglected so that the dynamics are the same as those of a spatially homogeneous model where the space derivatives are zero; but at each local spatial point the dynamics can be of a different homogeneous spacetime. (iii) Conse- quently the dynamics behaves asymptotically like a Bianchi IX or VIII spatially homogeneous cosmological model, and the dynamics at each spatial point is thus oscillatory and consists of a series of Kasner epochs punctuated by short bounces. (iv) Due to the nonlinearity of the Einstein field equations, anisotropy plays a similar dynamical role in the equations to a stiff energy density. For massless scalar field matter the energy density grows at the same rate as the effective anisotropy energy density as the singularity is approached. However, if the matter is not a scalar field, then sufficiently close to the singularity all matter terms can be neglected in the field equations relative to the dynamical anisotropy, and we can simply consider the vacuum field equations. BKL checked the consistency of their assumptions, but that doesn’t necessarily imply that those assumptions are valid in general physical situations. CC has been used to check the BKL conjecture and the accuracy of the BKL dynamics. Efforts to numerically study the BKL conjecture were initiated in [31], and more recently numerical simulations have been performed [32] to show that the BKL conjecture is satisfied for special classes of spacetimes. In particular, spatial derivatives were shown to become less important and sensitivity to initial conditions was displayed as the singularity is approached. Recent work in mathematical GR has raised new questions concerning the BKL conjecture and generic initial conditions, and this may further stimulate renewed CC studies.

3.2 Spikes In particular, a new spike phenomenon that had not been anticipated by BKL was found in numerical simulations [31]. Since it is a general feature of solutions of partial diﬀerential equations that spikes occur, it is expected that they occur in solutions of Einstein’s ﬁeld equations in GR.

5 The mechanism of spike formation in GR is straightforward. The orbits of nearby worldlines approach a saddle point in state space; if this collection of orbits straddle the stable manifold of the saddle point, then one of the orbits becomes stuck on the stable manifold of the saddle point and heads towards the saddle point, while the neighbouring orbits leave the saddle point. In the case of spikes, the spatial derivative terms do have a significant effect at special points. In particular, in the approach to the singularity in the mixmaster regime, a spike occurs when a particular spatial point is stuck in an old Kasner epoch while its neighbours eventually bounce to the new one. Because spikes become arbitrarily narrow as the singularity is approached, they are a challenge to the numerical simulations. However, despite these chal- lenges, spikes were discovered in numerical simulations [31]. Spikes were orig- inally found in the context of vacuum orthogonally transitive (OT) spatially inhomogeneous G2 models [31, 33]. In [34], further improved numerical evidence was presented that spikes in the Mixmaster regime of G2 cosmologies are transient and recurring. Therefore, studies of G2 and more general cosmological models have pro- duced numerical evidence that the BKL conjecture generally holds except pos- sibly at isolated points (surfaces in the three-dimensional space) where spiky structures (‘spikes’) form [34, 35]. The presence of such spikes in G2 models violates the local part of the BKL conjecture, and it is believed that this recurring violation of BKL locality holds in more general spacetimes. Unfortunately, simulations do not have enough spatial resolution to treat the spikes in detail. CC in is its infancy in studying spikes in more general (less symmetric) models. The scale invariant equations of Uggla et al. [36] are suitable for numerical simulations [32] to be used to simulate the approach to the singularity for the general case of no symmetry. Spikes are also a challenge to the mathematical treatment of spacetimes. Mathematical justification has been presented in [37]. However, more success has been obtained in finding exact spike solutions. For example, an exact non- OT G2 vacuum spike solution was presented in [38], and a new exact G1 stiff fluid spike solution has also been found [39]. We note that the dynamics of these new stiff fluid spike solutions is qualitatively different from that of the vacuum spike solutions. Spikes naturally occur in a class of non-vacuum G2 models and, due to gravitational instability, leave small residual imprints on matter in the form of matter perturbations. Particular interest has been paid to spikes formed in the initial oscillatory regime, and their imprint on matter perturbations and structure formation has been studied numerically [40]. These residual matter perturbations occur naturally within generic cosmological models within GR and form on surfaces. In more general, less symmetric (than G2) models, it is expected that the spike surfaces will intersect along a curve, and this curve may intersect with a third spike surface at a point, leading to matter inhomogeneities forming on a web of surfaces, curves and points.

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