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 SectionIII 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. numerical application. See also [27, 28] for a study of Having abandoned the Newtonian concept of absolute relativistic corrections to particle motion. space, index placement has become meaningful and must Notice that this weak field limit is enough to at least be treated accordingly. For convenience, and to avoid test for strong backreaction effects: as long as the metric ambiguity or confusion, we will always consider the per- perturbations remain small, we know that we are within turbations Bi and hij with covariant indices only, and ex- the domain of validity of our approximations. If however plicitly write the Euclidean metric wherever it is needed . they become large, then our scheme breaks down, but we for index contraction. We use the notation f = ∂f/∂xi, . ,i learn in this case that the standard approaches to cosmo- ∆f = δij∂2f/∂xi∂xj, and a prime 0 to denote ∂/∂τ. logical predictions break down as well. This enables us Latin indices take values 1, 2, 3, while Greek indices run to diagnose the failure in our approach, and to recognize from 0 to 3, and a sum is implied over repeated indices. whether we need to go beyond the weak-field limit. Except for very particular, symmetric cases, solving In this paper we mainly describe our formalism and Einstein’s equations can only be achieved in approxi- the numerical algorithms, and perform tests in a plane mation. The approximation scheme we use here is well 3 adapted for cosmological settings where one is interested to be of the order of ∼ 10−10. On intermediate and in the solution far away from compact sources (like black small scales, however, the scheme takes into account also holes) and one can therefore consider a weak field limit. the most important nonlinear relativistic terms. Since However, we clearly want to go beyond linear perturba- we stay in a relativistic framework throughout, we en- tion theory in order to allow for matter perturbations to sure that gauge issues are automatically addressed in the evolve fully into the nonlinear regime of structure forma- correct way such that we could easily construct all kinds tion. We will therefore use an approach that is equivalent of gauge invariant observables consistently. Such observ- to the formalism of [25, 26] who have adapted a short- ables are, however, not the focus of this paper, which is wave approximation in order to treat small scale inho- aimed at the development of the necessary tools. mogeneities in cosmology. In this approach it is assumed With the metric (3) and our choice of gauge, the com- that large (nonlinear) matter perturbations generally oc- ponents of the Einstein tensor read cur on small spatial scales, an assumption that is well H2 2 supported by observations. Accordingly, spatial deriva- G0 + 3 = 3HΦ0 + 3H2Ψ tives of metric perturbations should have a larger weight 0 a2 a2 in a perturbative expansion than the metric perturba- 3 ij tions themselves. In fact, gravitational potentials are of − (1 + 4Φ) ∆Φ − δ Φ,iΦ,j , (4a) −5 2 the order of ∼ 10 from galactic scales out to the scales of CMB observations. Gradients, on the other hand, need 0 2 1 0 Gi = − ∆Bi + Φ,i + HΨ,i , (4b) not be that small. They are related to peculiar veloci- a2 4 0 2 i ties, and these are typically observed to be of the order H H δj −3 Gi + δi 2 + = 2Φ00 + 4HΦ0 + 2HΨ0 of ∼ 10 on galactic and cluster scales, see [32] and j j a2 a2 a2 references therein. Second spatial derivatives are related 0 2 mn to the density contrast, which becomes non-perturbative, +2Ψ 2H + H − (1 + 4Φ) ∆Φ − 2δ Φ,mΦ,n i.e. larger than unity. In more detail, the approximation scheme1 we want to mn + (1 + 2Φ − 2Ψ) ∆Ψ − δ Ψ,mΨ,n implement amounts to giving every spatial derivative a −1/2 weight where is a weight characterizing the small- δik + (1 + 4Φ) Φ − (1 + 2Φ − 2Ψ) Ψ ness of the metric perturbations. We then consistently a2 ,jk ,jk keep all the terms up to order . This means that we keep terms which are linear in the metric perturbations 0 00 0 +B(j,k) + 2HB(j,k) + hjk + 2Hhjk − ∆hjk either by themselves or multiplied with second spatial derivatives which are considered to be of O(1), so that we are accurate to O(). In addition, we retain quadratic +Ψ,jΨ,k − 2Ψ(,j Φ ,k) + 3Φ,jΦ,k , (4c) terms of first spatial derivatives. As velocities are ex- pected to be of the same order as gradients, we go to up to terms which are higher order corrections to our quadratic order also here to remain self-consistent. Cor- approximation scheme (cf. Appendix A of [26]). Here respondingly, we have to keep density fluctuations, which H(τ) = H(τ)a(τ) = a0/a is the comoving Hubble rate. are of O(1) multiplied with metric perturbations or with For the time-time component (4a) and the spatial trace squared velocities and squared spatial derivatives of met- of (4c), the FLRW background terms were moved to the ric perturbations. Time derivatives are simply given the left hand side. Indices in parentheses are to be sym- same weight as the quantity they act upon. This means, metrized. however, that we do not take a quasi-static limit where The metric perturbations are sourced by the perturba- ν time derivatives are neglected altogether. tions of the total stress-energy tensor Tµ . It contains con- On very large scales, where all perturbations remain tributions from all types of matter, possibly a cosmolog- small, the scheme is equivalent to relativistic linear per- ical constant, and any other additional components one turbation theory since we do not, for instance, keep terms wants to include in a model. We will explicitly consider which are quadratic in the metric perturbations them- the contributions from non-relativistic matter (baryons selves. On these scales, linear perturbation theory is very and cold dark matter) and a cosmological constant Λ, accurate because second-order corrections are expected and keep an unspecified remainder to collectively account for other sources of stress-energy: Λ T ν = T ν − δν + T ν . (5) µ mµ 8πG µ Xµ 1 We present the scheme from a slightly different point of view than [26], whose approach may be conceptually more elaborate We do not distinguish between baryons and cold dark but also more difficult to grasp. In particular, the weight of matter. In a perturbed universe, physical quantities each term in the perturbative expansion is considered separately at small and large scales. The reader who is interested in a like e.g. the energy density of matter are affected by more detailed derivation of the equations is invited to study their the metric perturbations. Our aim is to take these cor- accounts. rections consistently into account when running a nu- 4 merical simulation of structure formation. For conve- fact, this correction can be interpreted as a “backreac- nience, we will therefore write the components of the tion” term. However, as long as the homogeneous mode matter stress-energy tensor in terms of “bare” quanti- remains small, the scale factor a still gives a meaningful ties which can easily be computed in any standard N- description of the background cosmology. body framework. In these frameworks, the phase space In short, for conceptual simplicity, we will use the distribution of dark matter particles is usually sampled “bare” ρ and δ which one would infer from a given par- by a collection of test particles whose positions and pe- ticle configuration assuming an unperturbed Friedmann culiar velocities are followed through a simulation. By model with scale factor a, but we have to keep in mind making a particle-to-mesh projection, it is possible to that these quantities have to be interpreted differently obtain average quantities like the (bare) bulk velocity in the perturbed universe and have to be dressed with . ui = hvii = h∂xi/∂τi, where h·i denotes the averaging appropriate corrections when they enter the perturbed procedure associated with the projection method. The Einstein equations. 0 0 method consists of a prescription of how to assign N- From G0 = 8πGT0 , and using Friedmann’s equation, body particle properties, like mass or velocity, to nearby we obtain a first evolution equation for the metric: grid points, thus constructing a matter density and veloc- ity field on the grid by means of a weighted average, see 3 (1 + 4Φ) ∆Φ − 3HΦ0 − 3H2Ψ + δijΦ Φ AppendixA4 for more details. This can be considered 2 ,i ,j as “the poor mans equivalent” of a phase space integral. 1 = 4πGa2ρ¯ δ + 3Φ (1 + δ) + (1 + δ) hv2i We assume here that this procedure is carried out in 2 the standard way without any knowledge about the per- 2 0 turbed metric, which is why we refer to “bare” quantities. − 4πGa δTX0 . (8) Therefore, the projection gives rise to the bare rest mass 0 . 0 0 0 density defined as Here, δTX0 = TX0 − T X0, where −T X0 and ρ¯ are the homogeneous modes of the energy densities as they occur . rest mass in Friedmann’s equation. In particular, ρ = × a−3 . (6) coordinate volume 3 2 3 4πGρ¯ = H0 Ωm (1 + z) , (9) In terms of the bare quantities, the physical (dressed) 2 energy density of matter reads with H0, Ωm and z being bare parameters of the FLRW background. The Poisson equation (2) is obtained from 1 . 0 2 Eq. (8) by dropping all terms except the first one on each ρphys = −Tm0 = 1 + 3Φ + hv i ρ , (7) 2 side. One can view Eq. (8) as a parabolic (diffusion-type) up to higher order corrections which we neglect. As one equation for Φ to which Eq. (2) is a first approximation can see, our approximation scheme takes into account the if the diffusion timescale tdiff is much shorter than the leading corrections coming from the perturbation of the dynamical timescale tdyn of the source term. To illus- volume and the kinetic energy of the particles. trate this statement, let us consider a structure of size It should also be noted that even the homogeneous r consisting of non-relativistic matter. A quick estimate modes of ρ and ρphys do not agree in general, because gives3 the kinetic energy is strictly positive and hence has a 2 positive average over the hypersurface of constant time. tdiff r √ It is therefore necessary to specify more precisely how ' 2 1 + δ , (10) tdyn r one wants to perform the split between background and H perturbations. We decide to evolve the scale factor ac- −1 where rH = aH is√ the Hubble radius, and we use the cording to Friedmann’s equation using an exactly pres- free-fall time ' rH / 1 + δ as an indicator for tdyn. sureless equation of state for cold dark matter (CDM), in This ratio is tiny for any realistic structure with r the spirit of Newtonian N-body simulations. This means rH . For instance, plugging in typical numbers for a stel- −20 that we treat all finite velocity effects entirely on the lar object yields tdiff /tdyn ∼ 10 , while for a struc- level of the perturbations. As we shall see later, this ture comparable to the local supercluster one obtains −5 leads to a non-vanishing homogeneous mode in Φ, which tdiff /tdyn ∼ 10 . This means that Φ will simply adjust accounts for the correction to the scale factor2 due to an adiabatically to its “instantaneous” equilibrium solution effective pressure and other quadratic contributions. In
3 As diffusion√ is a random process, the diffusion length is propor- 2 A homogeneous mode in Ψ on the other hand can be gauged tional to κt where κ = a/(3H) is the diffusion coefficient that away by a global reparametrization of time. We will take the can be read off from the prefactors of ∆Φ and Φ˙ = aΦ0 in Eq. conventional definition of cosmic time by imposing that Ψ has a (8). The diffusion time scale for a structure of size r is then 2 vanishing homogeneous mode. tdiff = r /κ. 5
ν on these scales, which is why Eq. (2) is a reasonable ap- δTXµ = 0). The equations are, of course, still coupled ν proximation in this case. On cosmological scales r ' rH , indirectly by the evolution of Tµ . however, one may expect that retardation effects may be- The equation for vector perturbations is come relevant. Furthermore, gauge dependence typically 1 becomes an issue as well on these scales, even in the lin- 0 − ∆Bi − Φ,i − HΨ,i ear regime. In our case this leads, for instance, to the 4 2 H Ψ-term present in Eq. (8) which does not appear in = 4πGa2 ρ¯(1 + δ) δ uj − B + δT 0 . (13) Poisson’s equation. ij i Xi In order to obtain a second scalar equation, it is useful Note that the left hand side is already written in to combine the time-time components and spatial trace Helmholtz decomposition and that the longitudinal com- in the following way: ponent of the equation is completely determined by Φ 2 and Ψ. On the other hand, by taking the curl of it, we a i 0 1 00 2 i 0 1 00 Gi − 3G0 − G0 = 4πGa Ti − 3T0 − T0 could get rid of the scalar potentials. However, the non- 2 H H j linear term δ · u remains and the term δ · Bi prevents us (11) from writing an equation for the curl of B only. There- ν 00 i Since Tµ is covariantly conserved, one can replace T0 fore, we prefer to solve Eq. (13) directly after subtracting essentially by spatial derivatives of the stress-energy ten- its longitudinal component through the use of Φ and Ψ sor, which are more easily computed at every time step (see also SectionIIIB for more details on how we do this of a simulation. After some algebra and using again in practice). Friedmann’s equations, one arrives at Finally, in order to obtain an equation for the tensor perturbations, it is useful to consider the traceless com- ij (1 + 2Φ − 2Ψ) ∆Ψ − δ Ψ,i (Φ,j + Ψ,j) bination 1 3 0 0 0 ij 0 i 1 i k i 1 i k + (∆Φ + 4Φ∆Φ + 4Φ ∆Φ) + δ Φ,iΦ,j G − δ G = 8πG T − δ T , (14) H H j 3 j k j 3 j k a2 i i 4 = 4πG − ρ¯(1 + 3Φ) (1 + δ) hγv i − ρ¯(1 + δ) u Ψ,i which yields H ,i 00 0 0 0 i i 0 hij + 2Hhij − ∆hij + B(i,j) + 2HB(i,j) + (1 + 4Φ) Φ,ij + 3¯ρΦ δ + δTX0,i − δTX0 (3Φ,i − Ψ,i + Bi) 1 − (1 + 2Φ − 2Ψ) Ψ,ij + Ψ,iΨ,j − 2Φ(,i Ψ ,j) + 3Φ,iΦ,j − Φ0 3δT 0 − δT i − δikh0 δT j , (12) X0 Xi 2 jk Xi 1 mn − δij (1 + 4Φ) ∆Φ − (1 + 2Φ − 2Ψ) ∆Ψ + δ Ψ,mΨ,n up to terms which we neglect in our approximation 3 scheme. Here, because a spatial derivative acts on it, we − 2δmnΨ Φ + 3δmnΦ Φ = 8πGa2 ρ¯(1 + δ) π have to use the relativistic momentum up to cubic order ,m ,n ,m ,n ij in velocity, i.e. we take into account the Lorentz factor 2 k 1 k which we approximate to quadratic order, γ ' 1 + v /2. + δikδTXj − δijδTXk . (15) Dropping again all terms except the first one on either 3 side gives a Poisson equation for Ψ which, however, is Here, πij is the bare anisotropic stress of matter, defined . n m 2 sourced by the divergence of the momentum density in- as πij = δimδjnhv v i − δijhv i/3. Again, taking a dou- stead of the density perturbation which we had in Eq. (2). ble curl would remove the vector and the linear scalar Of course these two are related by the continuity equa- contributions. But since there are many nonlinear scalar tion, and hence the two expressions are the same to a first terms which survive, this does not seem promising. approximation, e.g. for linear perturbations of CDM. The system of equations is closed once the equations of In Eqs. (8) and (12) we make no assumptions about motion for all sources of stress-energy are specified. For ν the size of the perturbations δTXµ, so they can be used CDM particles the evolution of their phase space dis- to obtain the scalar metric perturbations Φ, Ψ in any tribution follows from the geodesic equation of massive setting where these stress-energy terms can be reliably (non-relativistic) particles, given by computed, whatever their origin may be. 2 i i k ∂ x ∂x ij 0 ∂x Note also that Eq. (8) is a purely scalar equa- + H + δ Ψ,j − Bj − HBj + 2B[j,k] = 0 , i 0 ∂τ 2 ∂τ ∂τ tion. Moreover, whenever the two terms δTX0Bi and ik 0 j (16) δ hjkδTXi can be neglected, also Eq. (12) decouples from vector and tensor perturbations. In a numerical simulation, one can then solve these equations first and use the solutions in order to solve for the vector and ten- 4 sor quantities separately. This is the approach we will In equations (A2) and (A4) of [26] they include a contribution from Bi on the right hand side, which to our understanding follow in SectionIII, where we discuss the implementa- should be dropped because it is multiplied by a velocity and tion of this scheme for a pure (Λ)CDM universe (where hence is of higher order in the approximation scheme. 6 up to corrections beyond our approximation. Indices in in any suitable existing N-body code, which takes care square brackets are to be anti-symmetrized. In a pure of all remaining tasks, like parallelization and mesh man- ν (Λ)CDM simulation, TXµ = 0, the tensor perturbations agement. µ ν hij do not feed back into the evolution of Tν within our From now on, we assume δTXµ = 0. That is, we approximation. Therefore, as long as one does not want develop a relativistic N-body algorithm for a standard to extract the tensor signal, the system of equations can ΛCDM model. In this context, it is expected that be closed without considering Eq. (15). Of course, not Newtonian codes already give accurate results. On small all observables can then be reconstructed consistently, as scales, there is a broad consensus on this. But on large hij occurs for instance in the null geodesic equation for scales, in particular scales which are outside the hori- light rays. zon when the initial conditions are laid down, this is still Interacting massive particles, like baryons, do not fol- under debate. Already within linear perturbation the- low exact geodesics due to additional forces acting on ory there are relativistic corrections and the gauge de- them. These forces have to be modeled according to the pendence of the variables is important on super horizon relevant microphysics. In this paper we do not concern scales [7,8, 37, 38]. ourselves with this problem which is mainly relevant on Large-volume simulations will be important for the small scales. future large cosmological surveys, and so, not surpris- ingly, the horizon scale has already been crossed in re- cent Newtonian simulations. Examples are the Millen- III. NUMERICAL IMPLEMENTATION nium simulation [35], the Hubble Volume Project [39, 40], the Marenostrum Numerical Cosmology Project [41] and In this section we discuss in some more detail what others [42]. It will be interesting to compare them with it takes to convert a state-of-the-art Newtonian N-body our semi-relativistic approach and we think that this will code into a general relativistic one in the sense of our provide a useful test of the validity of the Newtonian approximation scheme, which is particularly tuned for treatment on cosmological scales. Furthermore, we hope cosmological applications. N-body simulations have a to be able to give robust statements about the actual size long history alongside advancements in super computing of relativistic corrections. In order to study extensions or technology, and it is beyond the scope of this paper to alternatives to the standard model, the numerical scheme will have to be amended for an appropriate treatment of give a comprehensive review of the subject. Some useful ν references are [33], [34], and [35, 36]. Further references δTXµ 6= 0. can be found therein. Let us now briefly sketch the main components of our The N-body problem refers to the dynamics of N par- algorithm. At all times, the state of the system, repre- ticles which interact through long range forces. There senting Cauchy data on a hypersurface of constant time are essentially two ways to address the problem. One which we may call a “snapshot of the Universe”, is stored in two sets of data: a particle list containing positions xi possibility is to consider the force acting on each parti- i i cle as a sum of two-body forces, which leads to the so- and peculiar velocities v = ∂x /∂τ of N test particles which are used to sample the phase space distribution called “particle-to-particle” class of N-body algorithms. 5 The alternative, referred to as the “particle-to-mesh” ap- function of CDM, and a 3D grid (representing position proach, is to construct a force field which permeates the space) holding values of all relevant fields at each grid entire volume of the simulation. To this end, one projects point. A standard particle-to-mesh projection allows to the particle configuration onto a 3D grid (representing construct, from the particle list, the relevant moments of the CDM distribution function on the grid, which we position space) and thus generates a discretized density i field. This grid information is used to compute the grav- have defined as the “bare” quantities ρ, u , and πij. Some itational potential according to a discretized version of details on how this construction works can be found in Eq. (2). The particles are then accelerated by the gra- AppendixA. dient of the potential, again suitably discretized and in- Instead of merely solving for the Newtonian potential terpolated to the particle positions. Based on these two as in a Newtonian particle-to-mesh code, we solve dis- approaches, many highly sophisticated algorithms have cretized (grid-based) versions of equations (8), (12), (13), been developed. There also exist algorithms, P 3M for and possibly (15). The grid-based representation of the example, which use a combination of both approaches. metric, suitably interpolated to the particle positions (see AppendixA), is then used to evolve the particle list by In GR, gravitational interaction is mediated by the an infinitesimal time increment according to the geodesic space-time metric. Therefore, it fits naturally into the equation (16), after which a next particle-to-mesh projec- framework of the “particle-to-mesh” approach. Instead of constructing the Newtonian gravitational potential, how- ever, we have to deal with a multi-component field, the metric, which determines the gravitational acceleration 5 Unfortunately, representing the full distribution function of CDM according to Eq. (16). In this paper, we discuss numeri- in six-dimensional (discretized) phase space is prohibitive, which cal solvers which allow to obtain the metric components is why the concept of test particles has to be introduced in the on a structured mesh. These can then be implemented first place. 7 tion is performed in order to update the metric, and so schemes, which can be shown to be unconditionally sta- on. ble, hence allowing for much larger time steps. Since we In principle, our N-body scheme does not look very violate the Courant–Friedrichs–Lewy condition of the ex- different from the Newtonian one. The equation for the plicit scheme, time resolution now becomes too coarse to particle acceleration is slightly modified, and instead of follow the time evolution of the small scales correctly, and an algorithm which solves the elliptic Poisson equation different schemes show different behavior on these scales. (2), we need to implement algorithms which solve equa- We will choose a scheme for which the small scales are ν 0 tions (8), (12), (13), and possibly (15). With δTXµ = 0, driven towards their equilibrium solution (Φ → 0), be- equations (8) and (12) do not depend on Bi or hij di- cause this corresponds to the behavior one expects from rectly, so a natural ordering is to use them to obtain so- our discussion following Eq. (10). lutions for the scalar perturbations Φ and Ψ, which can later be plugged into the vector equation (13). Similarly, Our choice is to use a slightly modified version of the the vector perturbation Bi can then be determined with- alternating direction implicit (ADI) method described in out any knowledge of hij, and all the solutions can finally [44]. There, an operator splitting method is used to con- be used in Eq. (15) to solve for the tensor perturbations. struct an update rule which consists of three steps, i.e. Let us now discuss possible algorithmic solutions of these one for each space dimension. In each step, an implicit set equations, one at a time. of equations has to be solved along a different spatial di- rection. For a linear parabolic operator, it can be shown that this algorithm is stable. We modify this method A. The scalar equations slightly by adding the nonlinear terms, similar to how it is done in [45]. Since the nonlinear terms are expected to remain subdominant, this will not affect the stability The two scalar equations (8) and (12) form a coupled of the algorithm. More details, including the discretized system which is first order in time for Φ, but contains version of Eq. (8), can be found in AppendixA. no time derivatives of Ψ. It is important to note that these equations are in fact a set of constraints, since Φ In order to perform a single update step from τ to and Ψ themselves are not independent dynamical degrees τ + dτ within the ADI framework, one has to invert a of freedom. From their constrained dynamics we expect tridiagonal matrix for each “line” of grid points along that they evolve only very slowly, much slower even than the implicit direction. This can be done efficiently using the motion of individual N-body particles, for instance. the Thomas algorithm, which is easily parallelized for It is therefore sufficient, as a first approach, to construct supercomputing applications [46]. In fact, the numerical a numerical update scheme which is only first order ac- solution of Eq. (8) with this method is computationally curate in time. This approach has the advantage that not more expensive than solving the Poisson equation (2) one can very easily decouple the update of Φ and Ψ: we with the Fourier method. Since the ADI method oper- will use Eq. (8) to update Φ, using the known Ψ of the ates in position space (rather than Fourier space), it is 0 previous time step. The new Φ and Φ obtained from this also compatible with adaptive mesh refinement, which is update step can then be used in Eq. (12) to find a new a widely used approach to effectively increase the resolu- solution for Ψ. tion of a simulation. We are therefore confident to obtain Under these premises, Eq. (8) is a nonlinear parabolic a high performance of the algorithm within any state-of- equation for Φ with a given source. If we forget about the-art particle-to-mesh framework, competitive with the the nonlinearity for a moment (which is anyway expected standard Poisson solvers which are used in Newtonian to remain very weak), it resembles a diffusion equation. codes. This type of mathematical problem is very well studied and can be solved with various standard methods found Next we construct a new solution for Ψ with the help in the literature, see e.g. [43–45]. of Eq. (12). Having already computed Φ and Φ0, this is a The methods basically differ in the way how the dif- nonlinear elliptic equation for Ψ with known coefficients ferential operators are discretized, and the choice can and source. It can be solved again by standard meth- have a tremendous impact on the performance of the ods, for instance using a multigrid algorithm coupled to algorithm. For instance, a fully explicit scheme is a nonlinear Gauß-Seidel relaxation scheme [43]. Again, unstable unless the time step dτ satisfies a so-called particular details can be found in AppendixA. This ap- Courant–Friedrichs–Lewy condition, which essentially proach should also compare well to the performance of a states that information may not travel by more than Newtonian code. one lattice unit within one time step. For our equa- tion, Φ-information will basically propagate at the speed Having to solve two equations, (8) and (12), both of light, while on the other hand, particle velocities are requiring a numerical effort comparable to the simple roughly three orders of magnitude smaller. Therefore, Poisson equation (2), the relativistic simulation will ac- it would take some thousand iterations before a particle cordingly be somewhat more expensive, but this is of would have travelled only across a single cell of the lat- course to be expected when we solve for two fields in- tice. This drawback can be overcome by using implicit stead of one. 8
B. The vector equation need. Here we have suppressed the tensor indices in h and in the source but these are trivial in Fourier space. Let us now discuss how to obtain a solution for the In the matter dominated era, at redshift z & 2, we can directly write down the homogeneous solutions. They vector perturbation Bi, for given solutions Φ and Ψ. We are simply given by spherical Bessel functions, h = use Eq. (13), which is a linear elliptic equation for Bi. (1) −1 −1 Because of the product with the density perturbation on (kτ) j1(kτ) and h(2) = (kτ) y1(kτ), with Wronskian the right hand side, we use a method which operates in W = k/(kτ)4. position space, multigrid relaxation being an obvious pos- As is evident from the above expression, the oscilla- sibility. We also have to avoid that numerical errors drive tions are only driven when the frequency of the source the solution away from the transverse gauge. This can be term approximately matches the one of the Green’s func- done by subtracting the spurious longitudinal component tion, which (for a non-relativistic matter source) happens from the solution. It is given by the gradient of a scalar only close to horizon crossing. At very much slower vari- ij function χ which solves ∆χ = δ B˜i,j, where B˜i is a nu- ation of the source, the behavior of the system becomes merical solution with spurious longitudinal component. that of a damped oscillator whose rest position adia- One then has Bi = B˜i − χ,i. All together, finding the so- batically follows a slowly varying external force. This lution Bi amounts to solving four linear elliptic equations means, as soon as the oscillations have died away, the (one for each component of B˜i and one for χ) which are displacement of the oscillator is simply proportional to of the same numerical difficulty as Poisson’s equation. the instantaneous force. In addition to this contribution If one is not interested in the signal from tensor pertur- which is sourced inside the horizon, we have free gravita- bations (gravitational waves), one can now close the loop tional waves which have been produced at horizon cross- −4 by implementing a particle update according to a suitable ing, with energy density scaling as ρh(k, τ) ∝ a . discretization of the geodesic equation (16). This is pos- sible because hij does not occur in the geodesic equation of massive particles at the level of our approximation. IV. PLANE-SYMMETRIC CLUSTERING Otherwise, one has to finally proceed with Eq. (15). In order to test essential parts of our algorithm with- out needing to run many simulations on a supercomputer, C. The tensor equation we restrict ourselves in this paper to the numerical sim- ulation of plane-symmetric configurations. The planar symmetry trivializes two of the three spatial dimensions, Having solutions for Φ, Ψ, and Bi in hand, Eq. (15) which reduces the computational requirements dramati- is a linear wave equation for the tensor modes hij with cally. However, it should be stressed that imposing this a given source. Since it is linear, the different modes symmetry is quite a strong restriction which precludes decouple in Fourier space where we are effectively dealing us from studying realistic models of cosmology. Further- with an ordinary differential equation (ODE) per Fourier- more, the allowed configurations do not contain vector grid point. In addition, the gauge condition can be easily or tensor modes by construction. The results presented implemented in Fourier space. However, the hyperbolic here should therefore be understood as numerical tests of nature of the equation means that in principle we would the algorithms, and can only be indicative of what will need a very high resolution in time in order to resolve happen in more realistic situations. We shall extend our the rapid oscillations of the high-momentum waves that investigation to the full three-dimensional case in forth- travel at the speed of light. coming work. This problem could be approached with the help of a As a first case, we consider an Einstein-de Sitter standard stiff ODE solver, but we can gain more insight background with an initial perturbation which is given into the behavior of the tensor modes by using a Green’s by a single plane wave with a comoving wavelength of function method: In a ΛCDM setup the source is given 100Mpc/h. only by non-relativistic matter. Therefore, the rapid os- We initialize the simulation at z ' 5000 using the lin- cillations come entirely from the homogeneous part of the ear solution with an amplitude of Φ = Ψ = 1.5 × 10−4, equation while the source term varies only slowly. Given which is quite large in the sense that it will finally result the homogeneous solutions in Fourier space, h(1)(τ) and in particle velocities which reach ∼ 1% of the speed of h(2)(τ), one easily obtains the solution with source term light. The generation of initial conditions in our general by performing the integral relativistic approach has to be done in a slightly different way than for Newtonian simulations, in order to correctly τ h (τ)h (τ 0) − h (τ 0)h (τ) (1) (2) (1) (2) 0 0 account for gauge dependencies. Details are given in Ap- h(τ) = 0 S(τ )dτ , ˆτin W (τ ) pendixB. (17) Results are shown in Fig.1. The fluctuation first grows . 0 0 where W = h(1)h(2)−h(1)h(2) is the Wronskian of the free linearly, but at z = 3 has reached a nonlinear density solutions h(1) and h(2), and h(τ) is the driven solution contrast of δ ∼ 10. Eventually, at z = 0, two shell- with vanishing initial values at τin, which is what we crossings have occurred, which are highly nonlinear ef- 9
z = 100 z =3 z =0
0 50 100 0 50 100 0 50 100
-2 -2 1×10 1×10
v 0 0 v
-2 -2 -1×10 -1×10
1 1 10 10 ¯ ¯ ρ ρ / / phys 0 0 phys ρ 10 10 ρ
-1 -1 10 10
-4 -4 1×10 1×10
0 0 Φ, Ψ Φ, Ψ -4 -4 -1×10 -1×10 Ψ Ψ − 0 0 − Φ 10 10 10 Φ × × × 0 50 100 0 50 100 0 50 100 x [Mpc/h] x [Mpc/h] x [Mpc/h]
Figure 1: The plane wave example: the x-axis always refers to the non-trivial spatial direction and is given in comoving units. The three columns show snapshots at three different redshifts, z = 100 (perturbations are still linear), z = 3 (the density perturbation is becoming nonlinear) and z = 0 (today). The first row depicts the particle phase space of the simulation, it is easy to see how shell crossing leads to a spiral-like structure in (x, v). In the second row we plot the density. As the central region around x = 50Mpc/h is overdense, this region undergoes gravitational collapse, resulting eventually in shell crossing and the associated formation of caustics visible as sharp spikes at z = 0. The bottom row shows the behavior of the gravitational potentials Φ (dark blue dashed line) and Ψ (light blue dot-dashed line). We see that in this example the potentials always remain small, of the order of the initial value of 10−4. To display the difference between the potentials more clearly, we plot it multiplied by a factor of 10 in the bottom-most panel (using otherwise the same scale as for the potentials). This difference mainly consists of a homogeneous mode that can be interpreted as a correction to the background scale factor due to nonlinear effects. The simulation was carried out with a resolution of 1024 grid points and 16384 particles (a representative subset is shown in the phase space diagram). fects. As one can see on the bottom panel, the two rel- corresponding effective pressure and energy density. This ativistic potentials Φ and Ψ agree extremely well, up to is because other quadratic terms occur in the equations, ij 2 the homogeneous mode in Φ which reaches an amplitude like δ Φ,iΦ,j, which are of the same order as v and also of ∼ 5 × 10−6 at z = 0. contribute to the generation of the homogeneous mode.
The amplitude of the homogeneous mode of Φ agrees As a second case, we study a plane-symmetric setup well with our naive expectation that it should be gov- inspired by the ΛCDM cosmology. Since a cosmologi- erned by v2. However, its precise value could not have cal constant is a homogeneous source of stress-energy, been computed simply by averaging v2 from a Newton- its effect enters the dynamics of cosmological perturba- ian simulation and dressing Friedmann’s equations with a tions only through a modification of the background. We 10
z = 100 z =3 z =0
0 250 500 0 250 500 0 250 500 -3 -3 5×10 5×10
v 0 0 v
-3 -3 -5×10 -5×10
1 1 10 10 ¯ ¯ ρ ρ / / phys 0 0 phys ρ 10 10 ρ
-1 -1 10 10
-5 -5 5×10 5×10
0 0 Φ, Ψ Φ, Ψ
-5 -5 -5×10 -5×10 Ψ Ψ − 0 0 − Φ 100 100 100 Φ × × × 0 250 500 0 250 500 0 250 500 x [Mpc/h] x [Mpc/h] x [Mpc/h]
Figure 2: A ΛCDM toy setup with planar symmetry: ΩΛ ' 2/3, and the initial power spectrum for Φ is flat for k < 0.075 h/Mpc and scales like k−4 for higher wavenumbers. As in Fig.1 we show the phase space (top row), the density (middle row) and the gravitational potentials (bottom row), for three different redshifts. The last panel shows Φ − Ψ on the same scale as the potentials, but multiplied by a factor of 100 (as indicated in the panel). The difference consists again mostly of a homogeneous mode as in the plane wave example. The simulation was carried out with a resolution of 16384 grid points and contained 131072 particles.
choose a ΛCDM background with ΩΛ = 1 − Ωm ' 2/3 constant (scale invariant) at scales k < 0.075h/Mpc and and draw a Gaussian random sample of initial plane decays as k−4 on smaller scales. The simulation is ini- wave perturbations which are supposed to mimic a typ- tialized at z = 3900 using linear theory, which in the real ical linear power spectrum of standard cosmology. How- Universe would be at the transition between radiation ever, as opposed to the standard case of an isotropic and matter domination. However, in our toy setup we power spectrum, our plane symmetry forces us to only do not include any radiation. Linear theory guarantees include perturbations whose wave vector is perpendicular that the linear solution will be in the growing mode (of to the plane of symmetry. Therefore, in order to obtain the matter dominated solution) after a couple of Hubble the right amplitude of perturbations at each scale, we times, which means that radiation effects, for the purpose 2 3 choose kh|Ψk| i ∝ k P (k), where P (k) denotes the usual of CDM simulations, can be taken into account simply isotropic linear power spectrum. Conversely, when quot- by adjusting the amplitude of the growing mode. The ing results for the power spectra, we perform the angular physics of the radiation era and the transition to mat- integration over all wave vectors as if the perturbations ter domination are contained in the shape of the linear were statistically isotropic. power spectrum. We choose an initial power spectrum where k3P (k) is Results are plotted in Figs.2 and3. As in the previous 11
z = 100 z =3 z =0
0.01 0.1 1 0.01 0.1 1 0.01 0.1 1
6 6 10 10 ] ] 3 3 4 4 h h 10 10 / / 3 3 2 2 10 10 [Mpc [Mpc 0 0 10 10 2 2 /k /k -2 -2 ) ) 10 10 k k ( ( P P -4 ρphys/ρ¯ -4 10 v 10 -6 -6 10 10
-8 -8 10 10
-10 -10 10 10
-12 -12 ) ) 10 Φ 10 k k ( ( Φ Ψ -14 − -14 kP kP 10 Ψ 10 ξ -16 -16 10 10
-18 -18 10 10 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 k [h/Mpc] k [h/Mpc] k [h/Mpc]
Figure 3: Power spectra averaged over 50 realizations of the ΛCDM toy setup for three different redshifts. The upper panels show the power spectra of the density (solid line) and of the velocity perturbations (dashed line). In the lower panels we plot the power spectra of the gravitational potentials Φ (dark blue dashed line) and Ψ (light blue dot-dashed line) as well as their difference (orange solid line) compared to the second-order estimate (green dotted line) based on ξ from [26]. The red curves on the bottom of the plot show the truncation error (see text for definition) of the gravitational potentials and indicate the expected numerical accuracy, and hence the level to which we can trust Φ − Ψ. Each simulation of the ensemble was carried out with a resolution of 4096 grid points and contained 32768 particles. example, we see that the difference Φ − Ψ is dominated of the difference, the so-called truncation error, is seen by the homogeneous mode of Φ. The non-homogeneous at the bottom of each plot in the lower panel of Fig.3 (k 6= 0) component of Φ − Ψ has a very red spectrum. (for both Φ and Ψ). Certainly, the value of the poten- Since we are in a setup where the “dictionary” of [26], tials should only be trusted up to the level indicated by given by their equations (2.40)–(2.42), should be valid, this truncation error, and so should the value of Φ − Ψ. we can actually estimate this component by computing a Because the difference between the potentials is so small, . correction ξ = (Ψ−Φ)k6=0 with the help of their equation and has a red spectrum, it drops below the numerical (3.17), where ∆2ξ is given by a combination of quadratic accuracy on small scales. terms in the velocity and in gradients of the Newtonian As explained in SectionII, the homogeneous ( k = 0) potential. To this end, we first obtain the Newtonian mode of Φ can be regarded as a correction to the scale quantities present in the latter equation by applying the factor due to nonlinearities, and as such is a genuine back- “dictionary” in the reverse direction. As one can see, the reaction effect. Its amplitude is governed by v2 and hence result obtained by this prescription agrees extremely well grows during linear evolution roughly ∝ a, as can be seen with our numerical result on the scales where it can be for our two numerical examples in Fig.4. However, as trusted. mentioned earlier, its value can only be computed con- The limited resolution introduces an intrinsic uncer- sistently by taking into account all contributions at a 2 tainty in the numerical solutions for Φ and Ψ on small given order of approximation. At order v there are, for ij scales. A benefit of the multigrid scheme is that the size instance, also terms like δ Φ,iΦ,j which are relevant. of this uncertainty can easily be estimated: we compare In order to assess further the difference between the each solution at full resolution with the one at half the traditional Newtonian approach and our general rela- resolution, interpolated to full resolution. The spectrum tivistic one, we also run a purely Newtonian simulation 12
-5 z =0 10 -5 z =3 1×10 z = 100
-6 10 dx R
v 0 ∆ dx/ Φ
R -7 10
-5 -8 -1×10 10
1 10 100 -20 -10 0 10 20 z +1 ∆x [kpc/h]
Figure 4: The homogeneous mode of Φ for the plane Figure 5: Difference of the phase space coordinates for N- wave setup (green, dashed) and the ΛCDM setup (blue, dot- body particles in the relativistic simulation of Fig.2 and a dashed). This homogeneous mode shows the size of the purely Newtonian one, initialized on the same linear solution. “backreaction” effect for the plane-symmetric case, i.e. the The three colors yellow, orange and red correspond to red- amount by which the average evolution of the perturbed uni- shifts z = 100, z = 3 and z = 0, respectively. As long as verse differs from the exact FLRW solution. More precisely, the solution is in the linear regime (yellow, z = 100), the it can be absorbed into the background by a redefinition difference in particle positions is essentially due to gauge de- a → a 1 + Φdx/ dx. pendence. At later times, however, particle positions and ve- ´ ´ locities become affected by relativistic corrections. Neverthe- less, the phase space distribution of particles for the two dif- ferent simulation techniques remains highly correlated, with which is initialized on the identical linear solution as the velocities in agreement to well within a percent, and particle example of Fig.2. We then perform following comparison positions within a few kpc/h. between the two simulations: on the initial data, before applying the initial infinitesimal displacements, each N- body particle of the Newtonian simulation is paired with particle displacements can be computed perturbatively, the corresponding N-body particle of the relativistic sim- a similar level of agreement was found in [28]. Our re- ulation. Since the scalar velocity perturbation is gauge sults give a quantitative indication that it extends to all invariant, their initial velocities match perfectly; on the scales on which gravitational fields remain weak, even if other hand, as explained in AppendixB, the initial par- the particle distribution becomes highly nonlinear. This ticle displacement is gauge dependent and consequently underlines the sound performance of the Newtonian ap- does not show perfect agreement. At z = 0, however, the proximation. Note that the grid resolution of the two entire simulation box is well inside the particle horizon, simulations was ∼ 30 kpc/h, which means that corre- and one therefore expects gauge dependence to be weak. sponding particles of the Newtonian and relativistic sim- Preserving the initial pairing of particles, they are fol- ulation are basically all found within the same grid cell. lowed through the two separate simulations. Fig.5 shows the separation of particle pairs in terms of phase space coordinates at z = 100, z = 3, and z = 0. Evidently, the V. SUMMARY AND OUTLOOK N-body particles remain highly correlated between the two simulations. The rms separation in position space In this paper we develop and discuss the techniques for rises from ∼ 2.5 kpc/h (comoving) in the initial data to relativistic N-body simulations in the weak field limit, ∼ 3.2 kpc/h at z = 0, where the rms velocity difference keeping gravitational perturbations to first order, their −6 reaches ∼ 10 . spatial gradients to second order and their second spa- This excellent agreement demonstrates that nonlinear tial derivatives to all orders. This corresponds to keep- effects of GR, at least when confined to the scalar sec- ing velocities to second order and the density contrast tor, have an almost negligible effect on the evolution of (which alone is expected to become large) to all orders. the N-body particle ensemble. On large scales where In this way we are able to simulate structure formation 13 in a General Relativistic context, taking along the rel- premise that our plane-symmetric setups are limited to ativistic effects that can become relevant with current scalar perturbations. However, recent quantitative anal- and future large cosmological surveys which will go out yses strongly suggest [53] that the inclusion of vector to redshifts of z & 2. Relativistic simulations are also a modes will also only have a small effect on the N-body natural approach to include relativistic fields like those dynamics, although there may be some potential for ob- needed for modified gravity simulations [47–51] or topo- serving Bi through its effect on photon propagation. The logical defects [1]. In addition, these simulations allow to same holds true for the tensor perturbations. A detailed study whether backreaction effects become large and so quantitative comparison with 3-dimensional Newtonian to test the backreaction scenario. N-body simulations is in preparation. There we shall The numerical challenges in implementing our formal- also calculate the induced vector and tensor perturba- ism as an extension of a standard N-body code are man- tion spectra. ageable, but a full implementation still requires consid- While observing relativistic effects within the context erable effort. To test the scheme and to obtain initial re- of ΛCDM standard cosmology will be challenging, we sults we have started with an effectively one-dimensional should keep in mind that relativistic simulations of struc- implementation which allows to treat plane-symmetric ture formation have a great potential in testing possible situations. Because plane symmetry does not admit extensions to the standard model. For instance, cosmic vector and tensor perturbations, and precludes us from neutrinos or a warm component of Dark Matter may con- studying a realistic cosmological setup, we can only draw stitute a semi-relativistic source which is relevant during limited conclusions at this point. Our results show that, the process of structure formation. In the sector of Dark within this constrained setup, relativistic corrections to Energy it is also very important to be able to test vari- the traditional Newtonian treatment remain very small. ous alternatives to a cosmological constant, and again it We highlight three different types of corrections: seems most promising to use large scale structure as a Firstly, fixing the background FLRW solution by as- probe. Once the numerical scheme for General Relativis- suming an exactly pressureless equation of state for tic simulations is fully implemented, the challenge may CDM, the scalar potential Φ dynamically acquires a ho- lie in the modelling and evolution of the stress-energy mogeneous mode. The homogeneous mode quantifies the tensor of the additional sources one wants to consider. backreaction and shows how the evolution of the averaged “background” slightly differs from the reference FLRW solution. With a relative amplitude of ∼ 10−7 in the Acknowledgments ΛCDM example, which is commensurate to the expected order of magnitude ∼ v2, it remains observationally ir- It is a pleasure to thank Chris Clarkson and Roy relevant, but we want to make the point that its exact Maartens for interesting comments. J.A. acknowledges value can only be calculated consistently by taking into funding from the German Research Foundation (DFG) account all relevant terms at a given order. At order through the research fellowship AD 439/1-1. This work 2 ij v there are, for instance, terms like δ Φ,iΦ,j which are is supported by the Swiss National Science Foundation. important. A second sign of relativistic corrections comes from considering the inhomogeneous component of Φ−Ψ. As is Appendix A: Lattice equations well known, in linear perturbation theory Φ−Ψ is sourced by anisotropic stress, which vanishes for a CDM source In this appendix, we explicitly present the finite- at linear order. It is therefore generated only at the non- difference operations which define the various algorithms linear level (e.g. [52]). However, as explained in [26], it presented in this paper. We assume that the code repre- can be accurately estimated from Newtonian quantities. sents continuous fields (like, e.g., the metric) on a struc- A comparison of our numerical results with this estimate tured mesh of rank three, which can basically be identi- is an excellent test for our algorithms. We demonstrate fied with a comoving coordinate grid covering a spacelike that the numerical scheme is able to give highly accu- hypersurface. We make use of the convenient notation rate results for this second order quantity, down to the n . i fi,j,k = f(τn, xi,j,k), where n is a discrete index labelling truncation error which is introduced by discretization. the time steps, while i, j, k are discrete indices labelling Finally, as a third way to quantify the difference be- the grid points of the mesh. In some cases, a quantity tween relativistic and purely Newtonian simulations, we may be defined with a fractional index whose meaning look at the pairwise phase space correlation of individual should be clear from the context. The grid spacing is particles between both simulation types when initialized given in units of dx1, dx2, dx3, and the size of a time on the same linear solution. We find that relativistic ef- step is denoted as dτ. fects on the phase space trajectory of particles remain The evolution of dark matter will be given by the no- completely tolerable. In fact, the coordinates of individ- tion of test particles which move on time-like geodesics. ual N-body particles remain correlated to within a few Their coordinates xi and velocities vi = ∂xi/∂τ are kpc/h (comoving), and their velocities to within 1%. stored in a list and can take continuous values. Like All these observations have to be considered under the in a standard particle-to-mesh N-body framework, the 14 code therefore has to provide appropriate projection and one step for each spatial dimension. The idea is that each interpolation operators in order to connect between grid- step amounts to only solving a linear tridiagonal system, based and particle-based information. If dark matter is which can be done very efficiently using the Thomas al- represented in a different way, or if one wants to include gorithm. There is no unique way of performing the split. baryons or other types of interacting matter, it should We simply follow [44] and add the nonlinear terms in the still be more or less straightforward to implement the cor- most straightforward fashion. There may be better split- rect gravitational acceleration. We leave it as an exercise ting procedures, and it may be worthwhile to investigate to the reader to figure out the appropriate prescription in this direction. With three spatial dimensions, the up- n+ 1 for their favorite hydrodynamical scheme. date makes use of two intermediate solutions, Φ 3 and n+ 2 Φ 3 . These should not be interpreted as solutions at fractional time steps, but rather as auxiliary surrogates 1. ADI scheme for Φn+1. They obey following finite difference equations:
The Alternate-Direction-Implicit (ADI) update for Φ is carried out by splitting the operation into three steps,
n+ 1 n+ 1 n+ 1 Φ 3 + Φ 3 − 2Φ 3 Φn + Φn − 2Φn Φn + Φn − 2Φn n i−1,j,k i+1,j,k i,j,k i,j−1,k i,j+1,k i,j,k i,j,k−1 i,j,k+1 i,j,k 1 + 4Φi,j,k + + (dx1)2 (dx2)2 (dx3)2
2 2 2 n+ 1 n n n n n n Φ 3 − Φn Φ − Φ Φ − Φ Φ − Φ i,j,k i,j,k 2 n 3 i+1,j,k i−1,j,k i,j+1,k i,j−1,k i,j,k+1 i,j,k−1 − 3H − 3H Ψi,j,k + + + dτ 2 4 (dx1)2 4 (dx2)2 4 (dx3)2 1 n+ 1 n+ 1 = 4πGa2ρ¯ δn + 3Φn 1 + δn + 1 + δ 2 hv2i 2 , (A1a) i,j,k i,j,k i,j,k 2 i,j,k i,j,k
n+ 2 n+ 2 n+ 2 n+ 2 n+ 1 Φ 3 + Φ 3 − 2Φ 3 Φn + Φn − 2Φn Φ 3 − Φ 3 n i,j−1,k i,j+1,k i,j,k n i,j−1,k i,j+1,k i,j,k i,j,k i,j,k 1 + 4Φi,j,k = 1 + 4Φi,j,k + 3H , (A1b) (dx2)2 (dx2)2 dτ
n+ 2 Φn+1 + Φn+1 − 2Φn+1 Φn + Φn − 2Φn Φn+1 − Φ 3 n i,j,k−1 i,j,k+1 i,j,k n i,j,k−1 i,j,k+1 i,j,k i,j,k i,j,k 1 + 4Φi,j,k = 1 + 4Φi,j,k + 3H . (A1c) (dx3)2 (dx3)2 dτ
Indeed, each equation is a linear tridiagonal problem. The stability of the scheme for a linear parabolic operator has been proven in [44]. This property can in principle be lost when nonlinear terms are present in the equation. However, we expect that in our case the nonlinearities remain sufficiently subdominant that this is not an issue. n+ 1 n+ 2 Note that in the plane symmetric setup, the last two equations become trivial and simply imply Φ 3 = Φ 3 = Φn+1, i.e. the auxiliary solutions are redundant.
2. Multigrid
In order to solve Eq. (12), we follow [43] and use a nonlinear multigrid scheme (“FAS algorithm”) coupled to a Newton-Gauß-Seidel relaxation method. The key equation is (20.6.43) of [43], identifying ui with our sought-after 15
n Ψi,j,k and using the discretized operator
Ψn + Ψn − 2Ψn ! n . n n i−1,j,k i+1,j,k i,j,k L(Ψi,j,k) = 1 + 2Φi,j,k − 2Ψi,j,k + ... (dx1)2
n n n n n n Ψi+1,j,k − Ψi−1,j,k Φi+1,j,k − Φi−1,j,k + Ψi+1,j,k − Ψi−1,j,k − − ... 4 (dx1)2 " Φn − Φn−1 + Φn − Φn−1 − 2Φn + 2Φn−1 ! 1 n i−1,j,k i−1,j,k i+1,j,k i+1,j,k i,j,k i,j,k + 1 + 4Φi,j,k + ... H dτ (dx1)2 Φn − Φn−1 Φn + Φn − 2Φn ! + 4 i,j,k i,j,k i−1,j,k i+1,j,k i,j,k + ... dτ (dx1)2 n n−1 n n−1 n n # Φi+1,j,k − Φi+1,j,k − Φi−1,j,k + Φi−1,j,k Φi+1,j,k − Φi−1,j,k + 3 + ... 4dτ (dx1)2
1 1 1 n− 2 1 n− 2 2 " 4πGa ρ¯ (1 + δ) hγv i i+ 1 ,j,k − (1 + δ) hγv i i− 1 ,j,k + 1 + 3Φn 2 2 + ... H i,j,k dx1
n− 1 n− 1 1 2 1 2 n n n n−1 # (1 + δ) hγv i i+ 1 ,j,k + (1 + δ) hγv i i− 1 ,j,k Ψ − Ψ Φ − Φ + 2 2 × i+1,j,k i−1,j,k + ... − 3δn i,j,k i,j,k . (A2) 2 2dx1 i,j,k dτ
Here, the ellipsis indicates that the previous term has to step of our algorithm can be sketched as follows: be written also for the other two directions x2 and x3 accordingly. Note that our operator L is only accurate • update velocities: to first order in time although one could easily modify it 1 in− 2 dτ n such that it would be accurate to second order. However, 1 v 1 − H − dτ∇Ψ in+ 2 2 since the ADI algorithm used for the evolution of Φ is a v = (A3) 1 + dτ H first order scheme, the implementation of a second order 2 scheme for Ψ would probably not increase the overall accuracy. • update positions by half a step: We perform the relaxation sweep in the “checkerboard” n+ 1 n dτ n+ 1 fashion, such that updated values on neighboring sites xi 2 = xi + vi 2 (A4) are already available when updating site (i, j, k). We also 2 store the approximate solution for all multigrid levels be- tween time steps, such that the previous solution can be • do particle-to-mesh projection for used as an initial guess in the next step. Since the poten- n+ 1 (1 + δ) hγvii 2 (face-centered, i.e. onto a tial Ψ varies slowly in time, this initial guess is already mesh shifted by half a grid unit in direction xi) very close to the final solution. Tracking the solution in n+ 1 2 n+ 1 such a way, we found that the multigrid algorithm gen- and 1 + δ 2 hv i 2 (cell-centered) erally needs no more than a single V-cycle to meet the convergence criterion again after one time step. Note • update Φn → Φn+1 that the additional memory to store the coarse-grid ap- proximations, in 3D, is bounded by 1/7 ' 15% of the • update positions by half a step: memory consumed by the full resolution. n+1 n+ 1 dτ n+ 1 xi = xi 2 + vi 2 (A5) 2 3. Loop structure • do particle-to-mesh projection for δn+1 As usual, we perform the particle updates in a leap-frog fashion, i.e. we associate particle positions and accelera- • compute Ψn+1 tion to integer time steps, while velocities are associated to half-integer time steps. A complete loop of one time • n + + ... 16
The gradient of Ψ, and other metric terms, have to be Appendix B: Initial conditions for Newtonian and interpolated to the particle positions when updating the relativistic simulations velocities, as will be specified in the next subsection. The loop shown here only includes the scalar degrees of free- The smallness of perturbations in the early Universe dom and is sufficient for the plane-symmetric case. When allow us to pose the initial conditions at a time when per- vector modes are taken into account, the update of the turbation theory is valid. Traditionally, the well-known particle velocities has to be modified accordingly, and the solutions of linear perturbation theory have been used for vector component of the metric has to be computed at this purpose, although people have also started to con- an appropriate instance within the loop. Finally, if de- sider results from second order perturbation theory [55]. sired, one can also insert the evolution step of the tensor We shall discuss only the former approach, although ulti- component. mately, implementing the latter one is certainly desirable as it would guarantee that all second order terms within our framework are accurate. However, for the purpose of this work, we simply always chose the initial redshift high enough that second order terms can safely be ne- 4. Particle-to-mesh projection and force glected, giving our simulation enough time to evolve to interpolation the nonlinear solution on its own. As shown by Bardeen [56], linear cosmological per- In order to establish the connection between grid-based turbations can be characterized completely in terms of and particle-based information, we make use of some gauge-invariant quantities. Once the linear solutions for standard approaches which are detailed in [54]. There, a these quantities have been determined, it is just a matter systematic hierarchy of prescriptions which assign par- of relating gauge-dependent quantities to these solutions ticle properties to grid points is developed, explicitly in order to obtain the linear solutions in any gauge. In discussing nearest-grid-point (NGP), cloud-in-cell (CIC) the longitudinal gauge which we use in our relativistic and triangular-shaped-particle (TSP) as the first three framework, these relations are given by members of this hierarchy. Generally speaking, one can trade a reduction of discretization errors for a more com- Ψ = ΨQ(S) , (B1a) plicated assignment scheme. Φ = ΦQ(S) , (B1b) As a guiding principle, since we are explicitly using (V ) B = σ(V )Q , (B1c) conservation of stress-energy to arrive at Eq. (12), we aim i i (T ) (T ) for an assignment scheme which guarantees to satisfy the hij = 2H Qij , (B1d) discrete version of the continuity equation, ui = VQ(S)i + V (V )Q(V )i , (B1e) (S) δ + 3Φ = DgQ , (B1f) 3 0 3 i a ρ + a ρu ,i = 0 , (A6) (S) (V ) (T ) where the functions Q , Qi , Qij denote the scalar, where the divergence of the momentum density is dis- vector and tensor Fourier modes, respectively, and the cretized precisely in the same way as in Eq. (A2). amplitudes are given in the notation of [57]. In partic- ular, we use the same notation for Φ, Ψ and its Fourier 3 Choosing to construct a ρ with the TSP assignment transform. V and V (V ) denote the gauge-invariant am- scheme, one can analytically check that above equation plitudes of the scalar (curl-free) and vector (divergence- 3 i holds identically if we construct a ρu using CIC assign- free) parts of a Helmholtz decomposition of the velocity ment in direction xi (on the face-centered grid), and TSP field, respectively, and Dg is a possible gauge-invariant assignment in the remaining directions. We trivially add amplitude of the density perturbation. The left hand 2 the v -corrections by multiplying the result for each par- side of the last line is obtained by truncating our expres- 2 ticle by 1 + v /2, which accounts for the kinetic energy sion for the energy density, Eq. (7), at linear order. At density or Lorentz factor, respectively. this level, the anisotropic stress of non-relativistic matter The interpolation of the grid-based field information is negligible, as is its effective pressure. to the particle positions must be done in such a way that It is usually assumed that vector modes are not sig- the resulting force does not include a relevant self-force. nificantly excited in the early Universe or have had time According to [54], it is generally sufficient if the force to decay until the onset of matter domination. One can interpolation scheme is at most of the same polynomial then consistently set σ(V ) = V (V ) = 0 during linear evo- i order as the particle-to-mesh assignment scheme, at least lution, which sets Bi = 0 and implies that u is given by when all equations are linear. We do not expect that the gradient of a scalar function. nonlinearities introduce an instability in our case, and so At linear order, scalar, vector and tensor equations de- far our numerical simulations have shown no indication couple and can be analyzed separately. Furthermore, in for such an instability. In practice, we used CIC force Fourier space, Einstein’s equations reduce to a system interpolation. of ordinary differential equations for the gauge-invariant 17 amplitudes in each sector. In the scalar sector, this sys- Assuming a negligible initial velocity dispersion, the tem is second order in time, giving two independent solu- initial peculiar velocities of the particles can be worked tions. For the matter dominated era, these can be found out from Eqs. (B1) and (B2) as well. Ignoring again the analytically, decaying solution, one finds
Φ = Ψ = c + c τ −5 , (B2a) 1 2 ∂xi 2 k k = − δijΨ . (B4) V = c τ − c τ −4 , (B2b) ∂τ 3H ,j 3 1 2 2 2 2 2 2 k H 2 k 9H −3 Dg = − 1 + 3 c1τ − 1 − c2τ , In fact, Eqs. (B3) and (B4) can simply be regarded as 6 k2 6 2k2 the Zel’dovich approximation in longitudinal gauge. It is (B2c) worth noting the important difference to the correspond- ing approximation which is used to generate initial data where c and c are the two constants of integration. The 1 2 for Newtonian N-body simulations. In Newtonian grav- solution proportional to c decays rapidly and is usually 2 ity, the concept of a horizon is absent, and one wants discarded. One then arrives at the well-known result that Eq. (2) to hold on all scales. It is therefore reasonable the gauge-invariant Bardeen potentials Ψ and Φ are con- to identify the initial density perturbation δ with an- stant (and equal) during matter domination. The lin- other gauge-invariant measure thereof, namely the den- ear solution for the other gauge-invariant quantities is sity perturbation in comoving gauge, which is denoted uniquely determined once these potentials are specified. by D in [57] and is related to the gauge-invariant poten- The vector equations are only first order in time, and tial Φ precisely by Eq. (2). The Newtonian potential ψ the solution for the “frame-dragging potential” reads N is then again identified with Φ = Ψ. These identifica- σ(V ) ∝ τ −4 in the matter dominated era. Since this tions can be done consistently because there is an exact solution decays, B = 0 as initial condition remains a i correspondence between the Newtonian and relativistic reasonable choice. solutions (taken in an appropriate gauge and assuming a The tensor equations are again second order in time. pressureless equation of state) at the level of scalar linear As already mentioned in SectionIII, the two independent perturbations. This correspondence breaks down at the free solutions for H(T ) are given by (kτ)−1j (kτ) and 1 nonlinear level (or if the pressureless assumption is not (kτ)−1y (kτ), with j and y spherical Bessel functions. 1 1 1 valid), and the Newtonian and relativistic solutions live Both solutions oscillate and decay inside the horizon, in completely different worlds. Most importantly, there kτ 1. However, outside the horizon, only the latter is no particular gauge where Newtonian quantities can be one decays, while the former one remains approximately identified with relativistic ones. All one can hope is that constant. Its amplitude is determined by early Universe there is an approximate correspondence which is reason- physics. Usually, one assumes that these super-horizon able for practical purposes, and this is the idea behind modes are generated during inflation, at the time when the “dictionary” which was proposed in [26]. they exit the horizon. They can be constrained, e.g., by observations of the cosmic microwave background. Thus Owing to the gauge-dependence of the density contrast far, only upper limits have been obtained, and observa- δ, the initial particle configuration for the same linear so- lution (B2) is different for a relativistic simulation com- tions are therefore compatible with the choice hij = 0 as initial condition on all scales. pared to a Newtonian one. In particular, the Newtonian 2 Our strategy to generate initial data is then the fol- Zel’dovich approximation simply has ζ = −2ψN /3H , lowing. First, we specify the initial potentials Φ = Ψ. which corresponds to the relativistic expression in co- Choosing the constant solution, we also have Φ0 = 0. moving gauge. In the longitudinal gauge employed in Next, we have to generate initial data for the particle our framework, ζ is given by Eq. (B3). Evidently, on very list. Starting from a uniform particle distribution, the long wavelengths where k H, we have that ∆ζ ∝ Φ initial positions of the particles can be assigned by an in- and thus ζ can become large. This is, however, not a finitesimal displacement, given by the gradient of a scalar practical problem as the density contrast δ (and there- i ij fore the perturbation of the particle number density) is of function, δx = δ ζ,j. The resulting “bare” mass density contrast will be δ = −∆ζ. The scalar function ζ is then order Φ by construction and thus small. The long wave- uniquely (up to an irrelevant homogeneous piece) deter- length part of ζ,j is just a constant time-independent shift mined by the linearized version of Eq. (8), acting on a “fictitious” regular particle configuration that has no physical significance. In particular, this configu- ∆Φ − 3H2Ψ = 4πGa2ρ¯(−∆ζ + 3Φ) . (B3) ration is never physically realized, even if one follows the particle trajectories backwards in time. Note also that This linear relation can be solved algebraically with the on super-horizon scales, where perturbations are always Fourier method. Equivalently, one could determine ζ us- linear, one can always translate between different gauges ing the relations (B1) and the linear solutions (B2). by employing a linear gauge transformation. 18
[1] M. Obradovic, M. Kunz, M. Hindmarsh, and I. T. Iliev, [17] P. Fleury, H. Dupuy, and J.-P. Uzan, “Can all “Particle motion in weak relativistic gravitational cosmological observations be accurately interpreted fields,” Phys.Rev. D86 (2012) 064018, arXiv:1106.5866 with a unique geometry?,” arXiv:1304.7791 [astro-ph.CO]. [astro-ph.CO]. [2] P. Brax, C. Burrage, and A.-C. Davis, “Shining Light [18] S. Räsänen, “Accelerated expansion from structure on Modifications of Gravity,” JCAP 1210 (2012) 016, formation,” JCAP 0611 (2006) 003, arXiv:1206.1809 [hep-th]. arXiv:astro-ph/0607626 [astro-ph]. [3] J. Magueijo, A. Albrecht, P. Ferreira, and D. Coulson, [19] T. Buchert, “Dark Energy from Structure: A Status “The Structure of Doppler peaks induced by active Report,” Gen.Rel.Grav. 40 (2008) 467–527, perturbations,” Phys.Rev. D54 (1996) 3727–3744, arXiv:0707.2153 [gr-qc]. arXiv:astro-ph/9605047 [astro-ph]. [20] S. Räsaäen, “Backreaction: directions of progress,” [4] R. Durrer and M. Kunz, “Cosmic microwave Class.Quant.Grav. 28 (2011) 164008, arXiv:1102.0408 background anisotropies from scaling seeds: Generic [astro-ph.CO]. properties of the correlation functions,” Phys.Rev. D57 [21] C. Clarkson, G. Ellis, J. Larena, and O. Umeh, “Does (1998) R3199–R3203, arXiv:astro-ph/9711133 the growth of structure affect our dynamical models of [astro-ph]. the universe? The averaging, backreaction and fitting [5] I. D. Saltas and M. Kunz, “Anisotropic stress and problems in cosmology,” Rept.Prog.Phys. 74 (2011) stability in modified gravity models,” Phys.Rev. D83 112901, arXiv:1109.2314 [astro-ph.CO]. (2011) 064042, arXiv:1012.3171 [gr-qc]. [22] T. Buchert and S. Räsänen, “Backreaction in late-time [6] C. Bonvin, R. Durrer, and M. A. Gasparini, cosmology,” Ann.Rev.Nucl.Part.Sci. 62 (2012) 57–79, “Fluctuations of the luminosity distance,” Phys.Rev. arXiv:1112.5335 [astro-ph.CO]. D73 (2006) 023523, arXiv:astro-ph/0511183 [23] S. Räsänen, “Applicability of the linearly perturbed [astro-ph]. FRW metric and Newtonian cosmology,” Phys.Rev. [7] C. Bonvin and R. Durrer, “What galaxy surveys really D81 (2010) 103512, arXiv:1002.4779 [astro-ph.CO]. measure,” Phys.Rev. D84 (2011) 063505, [24] M. Shibata and K. Taniguchi, “Coalescence of Black arXiv:1105.5280 [astro-ph.CO]. Hole-Neutron Star Binaries,” Living Reviews in [8] A. Challinor and A. Lewis, “The linear power spectrum Relativity 14 no. 6, (2011). of observed source number counts,” Phys.Rev. D84 [25] S. R. Green and R. M. Wald, “A new framework for (2011) 043516, arXiv:1105.5292 [astro-ph.CO]. analyzing the effects of small scale inhomogeneities in [9] D. Bertacca, R. Maartens, A. Raccanelli, and cosmology,” Phys.Rev. D83 (2011) 084020, C. Clarkson, “Beyond the plane-parallel and Newtonian arXiv:1011.4920 [gr-qc]. approach: Wide-angle redshift distortions and [26] S. R. Green and R. M. Wald, “Newtonian and convergence in general relativity,” JCAP 1210 (2012) Relativistic Cosmologies,” Phys.Rev. D85 (2012) 025, arXiv:1205.5221 [astro-ph.CO]. 063512, arXiv:1111.2997 [gr-qc]. [10] O. Umeh, C. Clarkson, and R. Maartens, “Nonlinear [27] C. Rampf and G. Rigopoulos, “Initial conditions for general relativistic corrections to redshift space cold dark matter particles and General Relativity,” distortions, gravitational lensing magnification and Phys.Rev. D87 (2013) 123525, arXiv:1305.0010 cosmological distances,” arXiv:1207.2109 [astro-ph.CO]. [astro-ph.CO]. [28] G. Rigopoulos and W. Valkenburg, “On the accuracy of [11] I. Ben-Dayan, G. Marozzi, F. Nugier, and G. Veneziano, N-body simulations at very large scales,” “The second-order luminosity-redshift relation in a arXiv:1308.0057 [astro-ph.CO]. generic inhomogeneous cosmology,” JCAP 1211 (2012) [29] J. Adamek, E. Di Dio, R. Durrer, and M. Kunz, “The 045, arXiv:1209.4326 [astro-ph.CO]. distance-redshift relation in plane symmetric universes,” [12] C. Bonvin, R. Durrer, and M. Kunz, “The dipole of the arXiv:1401.3634 [astro-ph.CO]. luminosity distance: a direct measure of h(z),” [30] R. Brustein, M. Gasperini, M. Giovannini, V. F. Phys.Rev.Lett. 96 (2006) 191302, Mukhanov, and G. Veneziano, “Metric perturbations in arXiv:astro-ph/0603240 [astro-ph]. dilaton driven inflation,” Phys.Rev. D51 (1995) [13] A. Abate and O. Lahav, “The Three Faces of 6744–6756, arXiv:hep-th/9501066 [hep-th]. Omega_m: Testing Gravity with Low and High [31] C. Cartier, R. Durrer, and E. J. Copeland, Redshift SN Ia Surveys,” Mon.Not.Roy.Astron.Soc. 389 “Cosmological perturbations and the transition from (2008) L47–L51, arXiv:0805.3160 [astro-ph]. contraction to expansion,” Phys.Rev. D67 (2003) [14] W. Valkenburg, M. Kunz, and V. Marra, “Intrinsic 103517, arXiv:hep-th/0301198 [hep-th]. uncertainty on the nature of dark energy,” [32] S. J. Turnbull, M. J. Hudson, H. A. Feldman, arXiv:1302.6588 [astro-ph.CO]. M. Hicken, R. P. Kirshner, et al., “Cosmic flows in the [15] I. Ben-Dayan, M. Gasperini, G. Marozzi, F. Nugier, and nearby universe from Type Ia Supernovae,” G. Veneziano, “Average and dispersion of the Mon.Not.Roy.Astron.Soc. 420 (2012) 447–454, luminosity-redshift relation in the concordance model,” arXiv:1111.0631 [astro-ph.CO]. arXiv:1302.0740 [astro-ph.CO]. [33] G. Efstathiou, M. Davis, C. Frenk, and S. D. White, [16] V. Marra, L. Amendola, I. Sawicki, and W. Valkenburg, “Numerical Techniques for Large Cosmological N-Body “Cosmic variance and the measurement of the local Simulations,” Astrophys.J.Suppl. 57 (1985) 241–260. Hubble parameter,” arXiv:1303.3121 [astro-ph.CO]. [34] S. J. Aarseth, Gravitational N-body Simulations: Tools 19
and Algorithms. Cambridge University Press, 2003. space variables,” Num.Math. 4 (1962) 41–63. [35] V. Springel, S. D. White, A. Jenkins, C. S. Frenk, [46] N. Mattor, T. J. Williams, and D. W. Hewett, N. Yoshida, et al., “Simulating the joint evolution of “Algorithm for Solving Tridiagonal Matrix Problems in quasars, galaxies and their large-scale distribution,” Parallel,” Parallel Computing 21 (1995) 1769–1782. Nature 435 (2005) 629–636, arXiv:astro-ph/0504097 [47] K. Chan and R. Scoccimarro, “Large-Scale Structure in [astro-ph]. Brane-Induced Gravity II. Numerical Simulations,” [36] V. Springel, J. Wang, M. Vogelsberger, A. Ludlow, Phys.Rev. D80 (2009) 104005, arXiv:0906.4548 A. Jenkins, et al., “The Aquarius Project: the subhalos [astro-ph.CO]. of galactic halos,” Mon.Not.Roy.Astron.Soc. 391 (2008) [48] B. Li, G.-B. Zhao, R. Teyssier, and K. Koyama, 1685–1711, arXiv:0809.0898 [astro-ph]. “ECOSMOG: An Efficient Code for Simulating [37] T. Matsubara, “The gravitational lensing in Modified Gravity,” JCAP 1201 (2012) 051, redshift-space correlation functions of galaxies and arXiv:1110.1379 [astro-ph.CO]. quasars,” Astrophys.J.Lett. 537 (2000) L77, [49] J. Lee, G.-B. Zhao, B. Li, and K. Koyama, “Modified arXiv:astro-ph/0004392 [astro-ph]. Gravity Spins Up Galactic Halos,” Astrophys.J. 763 [38] N. E. Chisari and M. Zaldarriaga, “Connection between (2013) 28, arXiv:1204.6608 [astro-ph.CO]. Newtonian simulations and general relativity,” [50] P. Brax, A.-C. Davis, B. Li, H. A. Winther, and G.-B. Phys.Rev. D83 (2011) 123505, arXiv:1101.3555 Zhao, “Systematic simulations of modified gravity: [astro-ph.CO]. chameleon models,” JCAP 1304 (2013) 029, [39] VIRGO Collaboration, J. Colberg et al., “Clustering of arXiv:1303.0007 [astro-ph.CO]. galaxy clusters in CDM universes,” [51] E. Puchwein, M. Baldi, and V. Springel, “Modified Mon.Not.Roy.Astron.Soc. 319 (2000) 209, Gravity-GADGET: A new code for cosmological arXiv:astro-ph/0005259 [astro-ph]. hydrodynamical simulations of modified gravity [40] VIRGO Collaboration, A. Evrard et al., “Galaxy models,” arXiv:1305.2418 [astro-ph.CO]. clusters in Hubble volume simulations: Cosmological [52] G. Ballesteros, L. Hollenstein, R. K. Jain, and M. Kunz, constraints from sky survey populations,” Astrophys.J. “Nonlinear cosmological consistency relations and 573 (2002) 7–36, arXiv:astro-ph/0110246 effective matter stresses,” JCAP 1205 (2012) 038, [astro-ph]. arXiv:1112.4837 [astro-ph.CO]. [41] S. Gottlober, G. Yepes, A. Khalatyan, R. Sevilla, and [53] M. Bruni, D. B. Thomas, and D. Wands, “Computing V. Turchaninov, “Dark and baryonic matter in the General Relativistic effects from Newtonian N-body MareNostrum Universe,” AIP Conf.Proc. 878 (2006) simulations: Frame dragging in the post-Friedmann 3–9, arXiv:astro-ph/0610622 [astro-ph]. approach,” arXiv:1306.1562 [astro-ph.CO]. [42] C. Park, J. Kim, and J. R. Gott III, “Effects of [54] R. W. Hockney and J. W. Eastwood, Computer gravitational evolution, biasing, and redshift space Simulation Using Particles. Institute of Physics Publ., distortion on topology,” Astrophys.J. 633 (2005) 1–10, 1999. arXiv:astro-ph/0503584 [astro-ph]. [55] M. Crocce, S. Pueblas, and R. Scoccimarro, “Transients [43] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and from Initial Conditions in Cosmological Simulations,” B. P. Flannery, Numerical Recipes, ch. 20. Cambridge Mon.Not.Roy.Astron.Soc. 373 (2006) 369–381, University Press, 3rd ed., 2007. arXiv:astro-ph/0606505 [astro-ph]. [44] J. Douglas and H. H. Rachford, “On the Numerical [56] J. M. Bardeen, “Gauge Invariant Cosmological Solution of Heat Conduction Problems in Two and Perturbations,” Phys.Rev. D22 (1980) 1882–1905. Three Space Variables,” Trans.Amer.Math.Soc. 82 [57] R. Durrer, The Cosmic Microwave Background. (1956) 421–439. Cambridge University Press, 2008. [45] J. Douglas, “Alternating direction methods for three