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Letters Published Online: 7 March 2016 | Doi: 10.1038/Nphys3673 LETTERS PUBLISHED ONLINE: 7 MARCH 2016 | DOI: 10.1038/NPHYS3673 General relativity and cosmic structure formation Julian Adamek1*, David Daverio1*, Ruth Durrer1 and Martin Kunz1,2 Numerical simulations are a versatile tool for providing not restricted to) cosmological applications7,15,16. Based on these insight into the complicated process of structure formation ideas we have developed a numerical code, gevolution, which uses in cosmology1. This process is mainly governed by gravity, the LATfield2 library17 and is designed to perform cosmological which is the dominant force on large scales. At present, a N-body simulations fully in the context of general relativity, century after the formulation of general relativity2, numerical evolving all six metric degrees of freedom. In brief, our approach codes for structure formation still employ Newton’s law of can be summarized as follows. gravitation. This approximation relies on the two assumptions We choose a suitable ansatz for the metric which is split into that gravitational fields are weak and that they originate background and perturbations. We work in Poisson gauge where the from non-relativistic matter. Whereas the former seems well perturbed Friedmann–Lemaître–Robertson–Walker metric is justified on cosmological scales, the latter imposes restrictions 2 2(τ/ . Ψ/ τ 2 i τ on the nature of the ‘dark’ components of the Universe ds Da [− 1C2 d −2Bi dx d (dark matter and dark energy), which are, however, poorly . Φ)δ i j i j understood. Here we present the first simulations of cosmic C 1−2 ij dx dx Chij dx dx U (1) structure formation using equations consistently derived from general relativity. We study in detail the small relativistic where a denotes the scale factor of the background, xi are comoving eects for a standard lambda cold dark matter cosmology that coordinates on the spacelike hypersurfaces, and τ is conformal time. cannot be obtained within a purely Newtonian framework. Our Φ and Ψ are the Bardeen potentials that describe the scalar metric particle-mesh N-body code computes all six degrees of freedom perturbations; Bi (also denoted as B) is transverse and is responsible of the metric and consistently solves the geodesic equation for frame dragging; hij is transverse and traceless, it contains the two for particles, taking into account the relativistic potentials and spin-2 degrees of freedom of gravitational waves. the frame-dragging force. This conceptually clean approach is We also assume that the perturbations of the metric, but not very general and can be applied to various settings where the necessarily their derivatives, remain small on the scales of interest. Newtonian approximation fails or becomes inaccurate, ranging This is a valid approximation for cosmological scales even when from simulations of models with dynamical dark energy3 or perturbations in the stress-energy tensor are large. warm/hot dark matter4 to core collapse supernova explosions5. Einstein's equations are then expanded in terms of the metric The applicability of Newton's law of gravitation in the context perturbations, but without a perturbative treatment of the stress- of cosmic structure formation has been discussed extensively in energy tensor. We include all terms linear in metric perturbations the recent literature6–8. In particular, it is now well understood that and go to quadratic order in terms which have two spatial this simplified description is fairly accurate when applied within derivatives acting on Φ, Ψ . This weak-field expansion contains standard lambda cold dark matter (3CDM) cosmology where all six components of the metric correctly at leading order. To perturbations come entirely from non-relativistic matter. However, determine the evolution of the metric, we numerically solve the `00' the situation is not satisfactory for two reasons. First, the quality and the traceless part of the` ij' Einstein equations: of observational data is rapidly increasing, and upcoming galaxy 3 surveys will eventually reach a precision where a naive treatment of .1C4Φ)1Φ −3HΦ0 −3H2Ψ C .rΦ/2 D−4πGa2δT 0 (2) the effects of spacetime geometry becomes insufficient9,10. Second, 2 0 the true nature of dark matter and dark energy is not yet established. i j 1 ij 1 00 0 1 0 δ δ − δ δ T CH − 1 C C H . / C(Φ −Ψ/ To study models beyond 3CDM, some of which may feature k l 3 kl 2 hij hij 2 hij B.i,j/ 2 B i,j ,ij relativistic sources of stress-energy, employing the Newtonian − (Φ −Ψ )Φ C Φ Φ C ΦΦ UD π 2(δ i − 1 δ i/ approximation is not always justified. A number of numerical 2 ,ij 2 ,i ,j 4 ,ij 8 Ga ikTl 3 kl Ti (3) codes have been developed for particular models11–14, yet a general framework would be desirable. Furthermore, Newtonian gravity is Equation (2) is the generalization of the Newtonian Poisson acausal and not sensitive to the presence of a cosmological horizon. equation (1Φ D 4πGa2δρ) and contains additional relativistic 1Φ δ 0 D −δρ D 0 − N 0 Even if a judicious interpretation of the output of Newtonian terms. Here and T0 T0 T0 are not required to be simulations can cure this problem at the linear level, it comes back small—in fact, they become much larger than the background value N 0 when one goes beyond linear perturbation theory, and it would be T0 inside dense regions. Equation (3) can be used to evolve all the preferable to use the correct physics from the outset. non-Newtonian degrees of freedom of the metric: Φ − Ψ , the two Moving from the absolute space and time of the Newtonian spin-1 degrees of freedom, B, and the two spin-2 helicities, hij. To picture towards a general relativistic view where geometry is do this, we decompose the equation into scalar, vector and tensor dynamical poses a significant conceptual challenge. Recent progress parts in Fourier space. is owed to a suitable formulation of the relativistic setting in The metric is then used to solve the equations of motion terms of a weak-field expansion which is well adapted for (but for matter (and possibly other degrees of freedom); collisionless 1Département de Physique Théorique & Center for Astroparticle Physics, Université de Genève, 24 Quai E. Ansermet, 1211 Genève 4, Switzerland. 2African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg 7945, South Africa. *e-mail: [email protected]; developers@latfield.org 346 NATURE PHYSICS | VOL 12 | APRIL 2016 | www.nature.com/naturephysics © 2016 Macmillan Publishers Limited. All rights reserved NATURE PHYSICS DOI: 10.1038/NPHYS3673 LETTERS ab Figure 1 | Spin-1 and spin-2 metric perturbations. Visualization of a simulation volume of (512 Mpc=h)3 at redshift zD0. Dark matter halos are rendered as orange blobs scaled to the virial radius. a, Illustration of the spin-2 perturbation hij by applying it as ane transformation to spheroids of fixed radius; the shape and size of the resulting ellipsoids therefore indicates, respectively, the polarization and amplitude of the tensor perturbation. Note how the signal is dominated by long-wavelength perturbations that are significantly correlated with the matter distribution. b, Stream plot of the spin-1 perturbation B, indicating how ‘spacetime is dragged around’ by vortical matter flows. (The Supplementary Material contains movies that also show the time evolution of the simulation.) particles propagate along geodesics in the perturbed geometry. This To fully capture the amplitude of the non-Newtonian terms, it is determines the evolution of the stress-energy tensor. important that the scale of matter–radiation equality is represented As a first application we choose standard 3CDM cosmology. in the simulation volume. On the other hand, these terms are We expect only small effects in this case, but we can use generated by nonlinearities and are therefore amplified at small these simulations to gain confidence in our new approach. We scales. To obtain our results we had to cover at least three orders of generate18 two halo catalogues containing ∼500,000 halos each, magnitude in scale. Our largest simulation used a lattice with 4,0963 one using our relativistic approach and one with the traditional points, corresponding to 6.7×1010 particles as we always start with Newtonian approach as reference, starting from identical linear one particle per grid point. This simulation used 16384 CPUs on initial conditions. A Kolmogorov–Smirnov test shows no significant the Cray XC30 supercomputer Piz Daint; its 1088 time steps were disagreement in the distributions of some 25 different halo completed in about 7 h, therefore totalling about 115000 CPU hours. properties such as mass, spin or shape parameters. Figure2 shows the numerical power spectra at three redshifts. The largest non-Newtonian effect is frame dragging, which is For these spectra we used eight simulations with lattices of 2,0483 associated with the spin-1 perturbation B. This perturbation is points and two different box sizes, all starting at redshift z D 100. sourced by the curl part of the momentum density found, for Four simulations had a box size of (2,048 Mpc=h)3, the other four example, in rotating massive objects. As long as perturbations are were (512 Mpc=h)3. In the regime where it can be trusted we find small this is a second-order effect which has been studied using excellent agreement with second-order perturbation theory, which perturbation theory19. The power spectrum for B has also been demonstrates that our code produces valid results. At late times and computed in the non-perturbative regime of structure formation on small scales, perturbation theory breaks down. Nonlinearities using a post-Newtonian framework20 which is expected to give good enhance the frame dragging by more than an order of magnitude agreement with our approach as long as one considers 3CDM at redshifts z D 1 and z D 0 and on scales k & 1h/Mpc.
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