PHYSICAL REVIEW D 95, 064028 (2017) Inhomogeneous cosmology with numerical relativity Hayley J. Macpherson,* Paul D. Lasky, and Daniel J. Price Monash Centre for Astrophysics and School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia (Received 16 November 2016; published 17 March 2017) We perform three-dimensional numerical relativity simulations of homogeneous and inhomogeneous expanding spacetimes, with a view toward quantifying nonlinear effects from cosmological inhomoge- neities. We demonstrate fourth-order convergence with errors less than one part in 106 in evolving a flat, dust Friedmann-Lemaître-Roberston-Walker spacetime using the Einstein Toolkit within the Cactus framework. We also demonstrate agreement to within one part in 103 between the numerical relativity solution and the linear solution for density, velocity and metric perturbations in the Hubble flow over a factor of ∼350 change in scale factor (redshift). We simulate the growth of linear perturbations into the nonlinear regime, where effects such as gravitational slip and tensor perturbations appear. We therefore show that numerical relativity is a viable tool for investigating nonlinear effects in cosmology. DOI: 10.1103/PhysRevD.95.064028 I. INTRODUCTION spacetimes [41], the propagation and collision of gravita- tional wave perturbations [42,43] and linearized perturba- Modern cosmology relies on the cosmological principle— tions to a homogeneous spacetime [44,45]. More recent that the Universe is sufficiently homogeneous and isotropic work has continued to include symmetries to simplify the on large scales to be described by a Friedmann-Lemaître- numerical calculations, e.g. Refs. [46,47]. Robertson-Walker (FLRW) model. Cosmological N-body Simulations free of these symmetries have only emerged simulations, e.g. Refs. [1–3], encode these assumptions by within the last year. Giblin et al. [48] studied the evolution prescribing the expansion to be that of the FLRW model, of small perturbations to an FLRW spacetime, exploring governed by the Friedmann equations, while employing a observational implications in Ref. [49]. Bentivegna and Newtonian approximation for gravity. Bruni [50] showed differential expansion in an inhomo- The transition to cosmic homogeneity begins on scales ∼80 −1 geneous universe and quantified the backreaction param- h Mpc, e.g. Refs. [4,5], but is inhomogeneous and eter from Ref. [51] for a single mode perturbation. These anisotropic on smaller scales. Upcoming cosmological works all indicate that the effects of nonlinear inhomoge- surveys utilizing Euclid, the Square Kilometre Array and – neities may be significant. the Large Synoptic Survey Telescope [6 8] will reach a In this work, we perform a feasibility study of numerical precision at which nonlinear general relativistic effects solutions to the full Einstein equations for inhomogeneous from these inhomogeneities could be important. A more – cosmologies by simulating the growth of structure in a extreme hypothesis [9 20] is that such inhomogeneities model three-dimensional universe and comparing to known may provide an alternative explanation for the accelerating analytic solutions. Our approach is similar to Refs. [48–50], expansion of the Universe, via backreaction (see with differences in the generation of initial conditions and Refs. [21,22] for a review), replacing the role assigned numerical methods. We use the freely available Einstein Λ – to dark energy in the standard CDM model [23 26]. Toolkit, based on the Cactus infrastructure [52,53].We Quantifying the general relativistic effects associated benchmark our three-dimensional numerical implementa- with nonlinear structures ultimately requires solving tion on two analytic solutions of Einstein’s equations ’ Einstein s equations. Post-Newtonian approximations are relevant to cosmology: FLRW spacetime and the growth – a worthwhile approach [27 36]; however, the validity of of linear perturbations. We also present the growth of these must be checked against a more precise solution since perturbations in the nonlinear regime and analyze the the density perturbations themselves are highly nonlinear. resulting gravitational slip [54,55] and tensor perturbations. An alternative approach is to use numerical relativity, In Sec. II, we describe our numerical methods, including which has enjoyed tremendous success over the past gauge choices (Sec. II A) and an overview of the deriva- – decade [37 39]. Cosmological modeling with numerical tions of the linearly perturbed Einstein equations used for relativity began with evolutions of planar and spherically our initial conditions (Sec. II B). In Sec. III, we describe the symmetric spacetimes using the Arnowitt-Deser-Misner setup (Sec. III A) and results (Sec. III B) of our evolutions (ADM) formalism [40], including Kasner and matter-filled of a flat, dust FLRW universe. The derivation of initial conditions for linear perturbations to the FLRW model are *[email protected] described in Sec. IVA, with results presented in Sec. IV B. 2470-0010=2017=95(6)=064028(13) 064028-1 © 2017 American Physical Society MACPHERSON, LASKY, and PRICE PHYSICAL REVIEW D 95, 064028 (2017) The growth of the perturbations to the nonlinear amplitude f ¼ const, while f ¼ 1=α corresponds to the “1 þ log” is presented in Sec. V, with analysis of results and higher- slicing common in black hole binary simulations. We order effects in Secs. VA and VB, respectively. We adopt choose harmonic slicing with f ¼ 0.25 to maintain the geometric units with G ¼ c ¼ 1; greek indices run from 0 stability of our evolutions, as in Ref. [47]. Harmonic slicing to 3, while Latin indices run from 1 to 3, with repeated also allows for longer evolutions for the same computa- indices implying summation. tional time, compared to 1 þ log slicing, due to the increased rate of change of the lapse. We adopt this gauge II. NUMERICAL METHOD for numerical convenience and acknowledge possible alternative methods include using synchronous gauge with We integrate Einstein’s equations with the Einstein adaptive time stepping. We use (2) for evolution only. We Toolkit, a free, open-source code for numerical relativity scale to the gauge described in the next section for analysis. [52]. This utilizes the Cactus infrastructure, consisting of a central core, or “flesh,” with application modules called “thorns” that communicate with this flesh [56]. The B. Perturbative initial conditions Einstein Toolkit is a collection of thorns for computational Bardeen’s formalism of cosmological perturbations [67] relativity, used extensively for simulations of binary neu- was developed with the intention of connecting metric tron star and black hole mergers (e.g. Refs. [57–59]). perturbations to physical perturbations in the Universe. Numerical cosmology with the Einstein Toolkit is a This connection is made clear by defining the perturbations new field [50]. We use the McLachlan code [60] to as gauge-invariant quantities in the longitudinal gauge. The evolve spacetime using the Baumgarte-Shapiro-Shibata- general line element of a perturbed, flat FLRW universe, Nakamura (BSSN) formalism [61,62] and the GRHydro including scalar (Φ, Ψ), vector (Bi) and tensor (hij) code to evolve the hydrodynamical system [59,63,64],a perturbations takes the form new setup for cosmology with the Einstein Toolkit. We use the fourth-order Runge-Kutta method, adopt the 2 2 2 i ds ¼ a ðηÞ½−ð1 þ 2ΨÞdη − 2Bidx dη Marquina Riemann solver and use the piecewise parabolic þð1 − 2ΦÞδ i j þ i j ð Þ method for reconstruction on cell interfaces. GRHydro is ijdx dx hijdx dx ; 3 globally second order in space due to the coupling of η ðηÞ hydrodynamics to the spacetime [64,65]. We therefore where is conformal time, a is the FLRW scale factor δ expect fourth-order convergence of our numerical solutions and ij is the identity matrix. We derive initial conditions for the spatially homogeneous FLRW model. Once per- from the linearly perturbed Einstein equations, implying turbations are introduced into this model, we expect our negligible vector and tensor perturbations [31]. This is valid solutions to be second-order accurate. as long as our simulations begin at sufficiently high enough We have developed a new thorn, FLRWSolver,to redshift that the Universe may be approximated by an initialize an FLRW cosmological setup with optional linear FLRW model with small perturbations. Considering only perturbations. We evolve our simulations in a cubic domain scalar perturbations, the metric becomes on a uniform grid with periodic boundary conditions with i 3 3 3 2 ¼ 2ðηÞ½−ð1 þ 2ΨÞ η2 þð1 − 2ΦÞδ i j ð Þ x in ½−240; 240. Our domain sizes are 20 , 40 and 80 , ds a d ijdx dx ; 4 respectively, using 70 (8 cores), 380 (8 cores) and 790 (16 cores) CPU hours. where Φ and Ψ are Bardeen’s gauge-invariant scalar potentials [67]. Here, we see that Ψ, the Newtonian A. Gauge potential, will largely influence the motion of nonrelativ- istic particles, where the time-time component of the metric The gauge choice corresponds to a choice of the lapse dominates the motion. The Newtonian potential plays the α βi function, , and shift vector, . The metric written in the dominant role in galaxy clustering. Relativistic particles 3 þ 1 ( ) formalism is will also be affected by the curvature potential Φ, and so ds2 ¼ −α2dt2 þ γ ðdxi þ βidtÞðdxj þ βjdtÞ; ð1Þ both potentials influence effects such as gravitational ij lensing [67,68]. where γij is the spatial metric. Previous cosmological The metric perturbations are coupled to perturbations in simulations with numerical relativity adopt the synchro- the matter distribution via the stress-energy tensor. We nous gauge, corresponding to α ¼ 1; βi ¼ 0 [48,50].We approximate the homogeneous and isotropic background as instead utilize the general spacetime foliation of Ref. [66], a perfect fluid in thermodynamic equilibrium, giving 2 ¼ðρ þ Þ þ ð Þ ∂tα ¼ −α fðαÞK; ð2Þ Tμν P uμuν Pgμν; 5 ij μ where fðαÞ > 0 is an arbitrary function and K ¼ γ Kij. where ρ is the total energy density, P is the pressure and u We set the shift vector βi ¼ 0.