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General Relativistic N-Body Simulations in the Weak Field Limit

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General Relativistic N-Body Simulations in the Weak Field Limit General Relativistic N-body simulations in the weak field limit Julian Adamek,1, ∗ David Daverio,1, † Ruth Durrer,1, ‡ and Martin Kunz1, 2, § 1Département de Physique Théorique & Center for Astroparticle Physics, Université de Genève, Quai E. Ansermet 24, CH-1211 Genève 4, Switzerland 2African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg, 7945, South Africa We develop a formalism for General Relativistic N-body simulations in the weak field regime, suitable for cosmological applications. The problem is kept tractable by retaining the metric pertur- bations to first order, the first derivatives to second order and second derivatives to all orders, thus taking into account the most important nonlinear effects of Einstein gravity. It is also expected that any significant “backreaction” should appear at this order. We show that the simulation scheme is feasible in practice by implementing it for a plane-symmetric situation and running two test cases, one with only cold dark matter, and one which also includes a cosmological constant. For these plane-symmetric situations, the deviations from the usual Newtonian N-body simulations re- main small and, apart from a non-trivial correction to the background, can be accurately estimated within the Newtonian framework. The correction to the background scale factor, which is a genuine “backreaction” effect, can be robustly obtained with our algorithm. Our numerical approach is also naturally suited for the inclusion of extra relativistic fields and thus for dark energy or modified gravity simulations. I. INTRODUCTION distances it becomes necessary to take into account rel- ativistic effects [6–11]. Although small, the impact of Cosmological late-time calculations and simulations the perturbations, e.g., on distance measurements, is not are usually divided into two distinct approaches, lin- negligible and can either be used as an additional probe of ear perturbation theory of General Relativity (GR) and cosmology (e.g. [12, 13]) or lead to an additional noise in Newtonian N-body simulations. Linear perturbation the measurements [14, 15] that may already be relevant theory is used on large scales where perturbations are for e.g. the Planck satellite results [16, 17]. A general small, so that higher-order terms can safely be neglected. relativistic extension of N-body simulations will auto- On small scales on the other hand, Newtonian gravity matically include these effects, and since the metric is provides a very good approximation to the dynamics of fully known, this will in addition allow us to integrate non-relativistic massive particles. the geodesic equation of photons through the simulation volume to obtain accurate predictions for observations This distinction however breaks down in at least two, that include all relevant relativistic effects. and maybe three cases: If we implement a numerical scheme that is able to The first case concerns the addition of relativistic fields follow the relativistic evolution of the Universe, we are in a context where nonlinearities are important. An ex- then also able to test the third (and more speculative) ample is the impact of topological defects on the for- case for a general relativistic framework for cosmological mation of structure [1] or the effect of modified-gravity simulations in the nonlinear regime: GR is a nonlinear (MG) theories on gravitational clustering on small scales, theory, and in principle nonlinear effects on small scales see [2] and refs. therein. In this case we have to simulate can “leak” to larger scales and lead to unexpected non- the nonlinear dynamics at least for the matter perturba- perturbative behavior. This idea is often called “back- tions, and in general also for the fields, as MG theories reaction” (see e.g. [18–22] for some recent reviews). If generically need nonlinear screening mechanisms that be- it is realized in nature, it would link the recent onset come important on small scales. Newtonian gravity is of accelerated expansion to the beginning of nonlinear usually not appropriate as e.g. both topological defects structure formation and so provide a natural solution to [3,4] and MG theories [5] exhibit a non-trivial anisotropic the coincidence problem of dark energy. This cannot eas- stress that leads to a gravitational slip which cannot be ily be checked within Newtonian simulations as the terms included in standard N-body simulations. from backreaction e.g. in the Buchert formalism [19] be- The second case is due to the ongoing revolution in come total derivatives in the Newtonian approximation observational cosmology where surveys are now reaching and therefore do not contribute in a simulation with pe- arXiv:1308.6524v2 [astro-ph.CO] 16 Jan 2014 unprecedented sizes, mapping out a significant fraction riodic boundary conditions [23]. of the observable Universe. On large scales and at large However, it is very difficult to test backreaction analyt- ically due to the nonlinear nature of the problem, and a numerical GR simulation appears to be the most straight- forward way to rigorously test this possibility. This test ∗Electronic address: [email protected] †Electronic address: [email protected] of backreaction is in a sense a bonus on top of the impor- ‡Electronic address: [email protected] tant applications of such a code for precision cosmology §Electronic address: [email protected] in general and especially in the dark sector. 2 While it would be desirable to simulate cosmology in symmetric situation. The application to realistic cosmo- a fully general relativistic way, this is technically very logical models in three spatial dimensions and a detailed challenging, as demonstrated by the efforts necessary to comparison with Newtonian N-body simulations will be simulate, for example, black hole mergers over just a presented in a forthcoming paper. An application to ob- short period of time, see [24] and refs. therein. We there- servations in plane symmetric universes, and a compar- fore need a scheme that captures the features of General ison with exact relativistic solutions is discussed in an Relativity that are relevant for cosmology, without the accompanying paper [29]. overhead of using full GR. This can be done by first as- The outline of this paper is as follows: in the next sec- suming that, as in linear perturbation theory, our Uni- tion we describe our approximation scheme and present verse on very large scales is close to being isotropic and the basic equations. In Section III we outline the nu- homogeneous, i.e., has a metric close to the Friedmann- merical implementation. In SectionIV we apply it to Lemaître-Robertson-Walker (FLRW) class and can thus a simple, plane-symmetric case. We discuss the results be taken to have a line element (keeping for the moment and conclude in SectionV. Some details of the numeri- only the scalar perturbations, and assuming a flat FLRW cal implementation and the generation of initial data are background universe) of the form deferred to two appendices. ds2 = a2(τ) −(1 + 2Ψ)dτ 2 + (1 − 2Φ)dx2 . (1) II. APPROXIMATION SCHEME AND The success of linear perturbation theory for the analy- FUNDAMENTAL EQUATIONS sis of perturbations in the cosmic microwave background (CMB) indicates that the gravitational potentials as well . When we go beyond linear perturbations, the line ele- as perturbations in the matter distribution like δ = δρ/ρ¯ ment (1) does not allow for a self-consistent description are small on large scales. However, the existence of galax- of cosmology, as scalar, vector and tensor perturbations ies, suns, planets and cosmological observers requires start to mix. We therefore use in this paper the more that at least the matter perturbations become large on general line element which admits also vector and tensor small scales, with δ 1. On small scales we expect that modes, the matter perturbations and the gravitational potential 2 2 2 i are connected to a high accuracy by Poisson’s equation, ds = a (τ) − (1 + 2Ψ)dτ − 2Bidx dτ + i j i j (1 − 2Φ)δijdx dx + hijdx dx . (3) ∆Φ = 4πGa2ρ¯ δ . (2) In order to remove the gauge freedom we restrict Bi to 2 In Fourier space ∆Φ becomes −k Φ, and on small scales be a pure vector mode and hij to be a pure tensor per- the wavenumber k is large, so that δ can become large turbation. To this end we choose the transverse/traceless ij ik ij even if the potential Φ remains small on all scales. In- gauge conditions δ Bi,j = δ hij,k = δ hij = 0, as was deed, the potential is expected to remain small since it done in [26]. The above line element and gauge condi- is small initially and it does not grow within linear per- tions can be imposed without loss of generality, i.e. we turbation theory. are not restricting the class of solutions, as long as we are The approach that we adopt is then to keep the grav- interested in solutions which are not too far away from itational potentials always only to first order, but to FLRW and hence have a cosmological interpretation. On be more careful with spatial derivatives. We keep first the other hand, even if the perturbations become large, derivatives of Φ and Ψ (and therefore also velocities) to e.g. |Ψ| & 1, it is not clear that deviations from the second order, and second and higher spatial derivatives Friedmann background are significant. It might just be of the potentials (and therefore also δ) to all orders. The that the longitudinal gauge becomes badly adapted. This theoretical foundations of this approach have been laid has been observed in some models of the early Universe, out recently in [25, 26]. Here, we want to take it an im- see e.g. [30, 31], but is not expected to occur in a situation portant step further and develop the technology for its which is close to Newtonian gravity.
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