Effects of Nonlinear Inhomogeneity on the Cosmic Expansion with Numerical Relativity

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Eloisa Bentivegna1,2,* and Marco Bruni3 1Dipartimento di Fisica e Astronomia, Università degli Studi di Catania, Via Santa Sofia 64, 95123 Catania, Italy 2INFN, Sezione di Catania, Via Santa Sofia 64, 95123 Catania, Italy 3Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, United Kingdom (Received 7 December 2015; revised manuscript received 8 March 2016; published 24 June 2016) We construct a three-dimensional, fully relativistic numerical model of a universe filled with an inhomogeneous pressureless fluid, starting from initial data that represent a perturbation of the Einstein–de Sitter model. We then measure the departure of the average expansion rate with respect to this homogeneous and isotropic reference model, comparing local quantities to the predictions of linear perturbation theory. We find that collapsing perturbations reach the turnaround point much earlier than expected from the reference spherical top-hat collapse model and that the local deviation of the expansion rate from the homogeneous one can be as high as 28% at an underdensity, for an initial density contrast −2 of 10 . We then study, for the first time, the exact behavior of the backreaction term QD. We find that, for small values of the initial perturbations, this term exhibits a 1=a scaling, and that it is negative with a linearly growing absolute value for larger perturbation amplitudes, thereby contributing to an overall deceleration of the expansion. Its magnitude, on the other hand, remains very small even for relatively large perturbations.

DOI: 10.1103/PhysRevLett.116.251302

Cosmology as a physical theory of the Universe was one wishes to correctly interpret the data that will be born soon after the formulation of general relativity one produced by the upcoming precision surveys [8,9].While hundred years ago [1], yet the extent to which relativistic some approaches have been introduced to estimate the role of nonlinearity may affect structure formation remains largely relativistic corrections in N-body simulations [10–15],the unexplored. With the increasing volume of cosmological only viable avenue to an exact computation of the systematic data and their precision, more sophisticated modeling is errors resulting from the omission of these effects is the required, and thus it is becoming timely to quantify these direct numerical integration of Einstein’s equation in the relativistic effects. The current theoretical framework for corresponding scenarios. Integrating the equations of general cosmology is based on three main ingredients: a homo- relativity, possibly coupled to stress-energy sources, is the geneous and isotropic Friedmann-Lemaître-Robertson- field of numerical relativity, a framework strongly motivated Walker (FLRW) background, relativistic perturbation theory by gravitational-wave-source modeling, but which has, over to describe fluctuations in the early Universe and at very large the years, developed in a number of parallel areas such as scale, and Newtonian methods, notably N-body simulations, cosmology, mathematical relativity, and modified gravity to study the evolution of fluctuations into the nonlinear [16]. Some of this work has already been aimed at studying regime of structure formation. Reconciling this framework inhomogeneous cosmologies [17–21]. While these numeri- with the observations requires the existence of dark compo- cal-relativity studies do not yet aspire to the level of realism nents, cold dark matter (CDM) and a cosmological constant achieved by N-body simulations [22,23], they are useful Λ or some other form of dark energy. The resulting standard test beds to quantify the relativistic effects of nonlinear cosmological model, ΛCDM, satisfies a vast class of inhomogeneity on the cosmic expansion. observational constraints, in particular the high-precision In this Letter, we integrate Einstein’s equation coupled measurements of the cosmic microwave background anisot- to an inhomogeneous irrotational pressureless fluid (dust) ropies [2]. However, the existence and nature of these dark with a three-dimensional density profile and no continuous constituents are one of the most debated topics not only symmetries. We choose initial data corresponding to a in modern cosmology, but also in theoretical physics. perturbed Einstein–de Sitter (EdS) model, i.e., a flat FLRW One aspect that has been the subject of intense debate is model with dust, with the aim of measuring, with no the question whether nonlinear relativistic “backreaction” approximations, the departures of the fully nonlinear effects due to formation of structures may play an important numerical solution from the idealized FLRW background role in the average cosmic expansion [3–7]. and its perturbations. On the numerically generated space- Quantifying the systematic errors involved in the differ- times, we measure a number of local and average properties ent modeling approximations, such as the use of Newtonian of cosmological interest, such as the growth of over- gravity for structure formation, is a crucial undertaking if densities and the formation of voids, the inhomogeneous

2 and average expansion rate, and the backreaction term γ¯ij ¼ aðtÞ δij, where the scale factor aðtÞ is a solution of defined in the averaging framework [3]. The main results of Friedmann’s equations this study are that (i) those perturbations that are large a_ 2 8πρ¯ ä 4π enough to collapse stop partaking in the cosmic expansion ¼ ¼ − ρ¯; ð Þ (i.e., reach the “turnaround” point) much earlier than a2 3 a 3 1 expected from a spherical top-hat collapse model with the same initial density contrast, (ii) locally, the effects of where the dot represents a time derivative, and we denote nonlinear inhomogeneities can be substantial, leading to a the EdS-background quantities with an overbar. The matter ρ¯ ∼ a−3 departure from the average expansion rate of over 28% at continuity equation gives for the background density. ρ the underdensities, and (iii) the average expansion rate is Starting from the inhomogeneous density , one can define δ ¼ðρ − ρ¯Þ=ρ¯ hardly affected by the inhomogeneities, with a backreaction the density contrast ; its growth in the term which is never larger than 10−8. synchronous-comoving gauge is governed, at first order, by Method.—We integrate Einstein’s equation and the fluid 3 3 δ00 þ δ0 − δ ¼ 0 ð Þ conservation equation using a variant of the Baumgarte- 2a 2a2 . 2 Shapiro-Shibata-Nakamura formulation [24–26], along with the Wilson formulation for the hydrodynamical The system of (1) and (2) is then solved by system [27] and the conformal transverse traceless formu- t 2=3 aðtÞ¼a ; ð Þ lation for the Einstein constraints [28,29], an approach i t 3 already used in cosmological settings [30–32]. We choose i −3=2 to represent the spacetime in the synchronous-comoving δðtÞ¼δþaðtÞþδ−aðtÞ ; ð4Þ gauge [33], popular in cosmological perturbation theory [22], which corresponds to the Lagrangian coordinates of where δþ and δ− are the so-called growing and decaying the observers at rest with the matter. modes. We will use these expressions below as a consistency To integrate this system, we use the Einstein Toolkit [34], check in the small-perturbation regime. a free, open-source community infrastructure for numerical Another useful framework is that of cosmological aver- relativity. In particular, we use the MCLACHLAN code [35] aging [3], where Einstein’s equation is reduced from a set of for the evolution of the gravitational variables, the CARPET partial differential equations for the fields to a set of ordinary [36] package for handling adaptive mesh refinement, and differential equations in time for some of their averages the multigrid elliptic solver CT_MULTILEVEL [37] to gen- over a given spatial region D. Defining its volume as erate initial data; this is then coupled to a new module Z 3 pffiffiffi 3 which evolves the hydrodynamical equations. All equa- aD ¼ γd x; ð5Þ tions are discretized using fourth-order finite differencing. D The Einstein Toolkit is routinely used for simulations in where γ is the determinant of the spatial metric γij, one finds relativistic astrophysics, and passes a variety of tests [34]. that the average scale factor aD satisfies a system similar to Likewise, as will be presented elsewhere, the new module Friedmann’s (1), and in particular that correctly reproduces several exact cosmological models ä 4π M Q with varying degrees of inhomogeneity. All results pre- D ¼ − D þ D ; ð Þ 3 6 sented are convergent at the correct rate as the grid spacing aD 3 aD 3 is decreased, and we use this fact to extrapolate the where continuum solution of the evolution system and to estimate the error bars resulting from its numerical integration at Z pffiffiffi 3 finite resolution. These are the quantities that appear in MD ¼ γρd x ð7Þ all plots. D Perturbations and averaging.—We recall two appro- 2 Q ¼ ðhK2i − hKi2 Þ − 2hA2i : ð Þ aches commonly used to solve the evolution system D 3 D D D 8 approximately, so that we can compare our solution to K K ≡ −γ_ =2 these schemes and check that we obtain the correct Here is the trace of the extrinsic curvature ij ij , 2 ij behavior in the appropriate regime. A ¼ AijA =2, Aij is the traceless part of Kij, and h·iD For irrotational dust in the synchronous-comoving gauge, denotes the average of a field over D. Note that −K represents the line element can be written (with no loss of generality the local expansion rate, and in the FLRW background H ¼ 2 2 i j ¯ [33])asds ¼ −dt þ γijdx dx , where γij is the spatial −K=3 is the Hubble parameter. While this setup is exact, the metric. For spacetimes that are close enough to a FLRW computation of QD itself requires tensorial quantities that do model, one can use perturbation theory to follow the not satisfy ordinary differential equations; i.e., the system of departures from the exact background solution. In the matter ordinary differential equations for the averaged quantities is era, this is the spatially flat EdS model, with metric not closed. To circumvent this problem, one typically closes

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ρ=ρ¯ y ¼ z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FIG. 1. Profilepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the matter density ratio on the γ=γ¯ y ¼ z d ¼ y2 þ z2 2 2 −2 FIG. 2. Profile of on the plane ( ) for plane (d ¼ y þ z ) for δi ¼ 10 , when a ¼ ai (left) and −2 δi ¼ 10 , when a ¼ ai (left) and when a ∼ 96ai (right). when a ∼ 96ai (right). the system with a well-motivated ansatz for QD. One can, CT_MULTILEVEL [37], obtaining the initial profile for γ for instance, calculate its perturbative behavior: at first order, (normalized to the EdS value) shown in Fig. 2. this term is identically zero, while at second order it scales We then evolve the coupled gravitational and hydrody- −1 as QD ∼ a [7,38], but with a coefficient containing only namical equations, until the linear size of the domain has surface terms of the averaging volume, which vanish for increased by roughly 100 times. We measure the departure periodic domains. Beyond this order, the analytical approach of the volume expansion, represented by aD, from the EdS becomes exceedingly difficult. A main goal of this Letter background model, for different initial amplitudes of the is to present an exact measurement of this quantity on an density contrast δi; as clearly shown in Fig. 3, this differ- inhomogeneous spacetime. ence is always small. We also monitor the density contrast Results.—Our numerical investigation involves the evo- at the overdensities and underdensities. As expected from lution of a cubic domain of coordinate side L, with periodic linear perturbation theory, and shown in Fig. 4, for small boundary conditions (because we set G ¼ c ¼ 1, L will values of the initial δi the density contrast grows linearly serve as the unit in which all other quantities, including with a, with a well-behaved evolution through a=ai ¼ 100. −2 mass and time, are measured). We discretize this domain For δi ¼ 10 , there is a clear departure from this behavior, with 1603 points (running two lower resolutions with 803 with the overdensity becoming nonlinear already at 3 and 40 to quantify the error bars). We choose the initial a=ai ¼ 5, and eventually growing unbounded when density profile as that of the EdS model at the time when the a=ai ∼ 96. In Fig. 5 we plot the fractional difference of −1 K Hubble horizon Hi ¼ L=4, plus a superimposed pertur- (the local expansion rate) from the background value −6 K¯ ¼ −3H bation of initial amplitude δi (varying between 10 and at the overdensities and underdensities. As 10−2) and comoving wavelength L, expected, the expansion is larger at the underdensities −2 and smaller at the overdensities. For δi ¼ 10 , the depar- X3 2πxj ρ ¼ ρ¯ 1 þ δ : ð Þ ture from the expansion rate of the EdS background is i i i sin L 9 j¼1 substantial: again, the expansion is already visibly nonlinear at a=ai ¼ 5, and the overdensity reaches the −2 The ratio ρ=ρ¯ for δi ¼ 10 is shown in Fig. 1.Asδi decreases, we expect to recover a cubic domain of the EdS model. By increasing δi, we should then be able to observe the onset of nonperturbative effects. We first need to solve the Einstein constraints; to simplify them, we choose a vanishing traceless part of the extrinsic curvature and a spatially constant K. This corresponds, initially, to having a vanishing first-order perturbation of the expansion and a nonzero decaying mode δ− in (4) [22]. The momentum constraint is then identically satisfied, and the Hamiltonian constraint reduces to the nonlinear elliptic equation K2 i 5 FIG. 3. Fractional difference of the scale factor aD of the Δψ − − 2πρi ψ ¼ 0; ð10Þ 12 simulation domain with respect to the EdS scale factor a,asa −2 −3 −4 −5 function of the equal-time a, for δi ¼ 10 ; 10 ; 10 ; 10 , and 1=12 ¯ wherepffiffiffiffiffiffiffiffiffiffiffiffiψ ¼ γ . Using (9) and Ki ¼ Ki ¼ −3Hi ¼ 10−6 (top to bottom). The numerical error bars, where visible, are − 24πρ¯i ¼ −12=L, we solve this equation with included as shaded regions.

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is constrained to remain spatially constant, an inhomogeneous density and expansion accelerate the approach to turnaround at the peak, just like they do in the spherical Newtonian case [41,42]. The shear also gives a small −2 correction, which for δi ¼ 10 is non-negligible even in the initial perturbative regime [22,43,44]. These effects combine, leading to a negative contribution to the evolution of the local expansion rate −K, pushing it towards the turnaround (K ¼ 0), and accelerating the collapse. The difference with the top-hat collapse is an important issue which we will investigate in detail in future work. δ FIG. 4. Growth of the density contrast OD at the overdensities We then proceed to measure the backreaction quantity QD; −δ (solid lines) and its negative UD at the underdensities (dashed −2 −3 −4 −5 −6 the results are shown in Fig. 6. We extract a few relevant facts: lines), for δi ¼ 10 ; 10 ; 10 ; 10 , and 10 (top to bottom). ¯ first, given our initial conditions Ki ¼ Ki ¼ −3Hi,itfollows The linear-perturbation behavior is indicated by dotted lines. from the definition (8) that QD vanishes on the initial time slice. We also notice that, for smaller perturbations, QD K ¼ 0 a=a ∼ 60 turnaround point (signaled by OD )at i .At remains zero within our error bars; for larger perturbations, it turnaround, the linearly extrapolated density contrast is is clear from Fig. 6 that QD goes through a short transient only δT ¼ 0.6, much smaller than the standard value from −1 phase before following the scaling QD ∼ a for a period spherical top-hat collapse, δT ¼ 1.06 [39]. For the same which is shorter for higher δi. Given that the only second- initial density contrast, the underdensity asymptotically order contributions to QD are boundary terms that vanish on approaches the expansion of the Milne model periodic domains like the one we used [38], we conjecture (a vacuum FLRW model with negative spatial curvature, that only higher-order terms are contributing to QD. represented by the solid gray line in Fig. 5), as predicted in Finally, QD enters the nonperturbative regime, where it is [40], with a fractional departure from EdS of over 28% at a=a ∼ 96 negative and its absolute value increases linearly with the i . These are the first two important results of our scale factor. The effect is a very small deceleration of the calculations: even in this simple setup, with a perturbation expansion with respect to the EdS model. We conclude wavelength initially four times larger than the EdS Hubble that the absolute value of QD remains generally quite small, horizon, the onset of nonlinearity can occur very early, and but is not identically zero, as would follow from the inhomogeneities can affect the local expansion rate in a assumptions of [45]. substantial, nonperturbative way. Measuring the sign and scaling of the backreaction QD is In particular, the observed difference with respect to a particularly relevant task, as many speculations on the the spherical homogeneous top-hat collapse is due to the effect of inhomogeneities on the average cosmic expansion interplay of several factors, most notably the inhomo- rate are based on conjectures on these two properties. A geneous character of the density, expansion rate, and back-of-the-envelope estimate involves the comparison of σ 3-curvature, and the nonvanishing shear , absent in the two competing effects, as quantities like the matter density top-hat case. Whereas in the latter case the perturbation at the overdensities quickly depart from the background

1 − K =K¯ FIG. 5. Fractional expansion rate OD at the over- K =K¯ − 1 densities (solid lines) and its negative UD at the under- FIG. 6. Absolute value of the backreaction QD as a function −2 −3 −4 −5 −6 densities (dashed lines), for δi ¼ 10 ; 10 ; 10 ; 10 , and 10 of the equal-time scale factor in Einstein–de Sitter space, −2 −2 −3 −4 −5 −6 (top to bottom). For δi ¼ 10 the overdensity starts collapsing at for δi ¼ 10 ; 10 ; 10 ; 10 , and 10 (top to bottom). The −2 a ∼ 60ai. The underdensity with δi ¼ 10 expands much faster numerical error bars, where visible, are included as shaded than the background, asymptotically approaching the expansion regions. For comparison, we have superimposed dashed lines −1 of the Milne model (horizontal dark-gray line). representing the QD ∼ aD scaling.

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