Abstract
Numerical simulations are a versatile tool for providing insight into the complicated process of structure formation in cosmology^{1}. This process is mainly governed by gravity, which is the dominant force on large scales. At present, a century after the formulation of general relativity^{2}, numerical codes for structure formation still employ Newton’s law of gravitation. This approximation relies on the two assumptions that gravitational fields are weak and that they originate from nonrelativistic matter. Whereas the former seems well justified on cosmological scales, the latter imposes restrictions on the nature of the ‘dark’ components of the Universe (dark matter and dark energy), which are, however, poorly understood. Here we present the first simulations of cosmic structure formation using equations consistently derived from general relativity. We study in detail the small relativistic effects for a standard lambda cold dark matter cosmology that cannot be obtained within a purely Newtonian framework. Our particlemesh Nbody code computes all six degrees of freedom of the metric and consistently solves the geodesic equation for particles, taking into account the relativistic potentials and the framedragging force. This conceptually clean approach is very general and can be applied to various settings where the Newtonian approximation fails or becomes inaccurate, ranging from simulations of models with dynamical dark energy^{3} or warm/hot dark matter^{4} to core collapse supernova explosions^{5}.
Main
The applicability of Newton’s law of gravitation in the context of cosmic structure formation has been discussed extensively in the recent literature^{6,7,8}. In particular, it is now well understood that this simplified description is fairly accurate when applied within standard lambda cold dark matter (ΛCDM) cosmology where perturbations come entirely from nonrelativistic matter. However, the situation is not satisfactory for two reasons. First, the quality of observational data is rapidly increasing, and upcoming galaxy surveys will eventually reach a precision where a naive treatment of the effects of spacetime geometry becomes insufficient^{9,10}. Second, the true nature of dark matter and dark energy is not yet established. To study models beyond ΛCDM, some of which may feature relativistic sources of stressenergy, employing the Newtonian approximation is not always justified. A number of numerical codes have been developed for particular models^{11,12,13,14}, yet a general framework would be desirable. Furthermore, Newtonian gravity is acausal and not sensitive to the presence of a cosmological horizon. Even if a judicious interpretation of the output of Newtonian simulations can cure this problem at the linear level, it comes back when one goes beyond linear perturbation theory, and it would be preferable to use the correct physics from the outset.
Moving from the absolute space and time of the Newtonian picture towards a general relativistic view where geometry is dynamical poses a significant conceptual challenge. Recent progress is owed to a suitable formulation of the relativistic setting in terms of a weakfield expansion which is well adapted for (but not restricted to) cosmological applications^{7,15,16}. Based on these ideas we have developed a numerical code, gevolution, which uses the LATfield2 library^{17} and is designed to perform cosmological Nbody simulations fully in the context of general relativity, evolving all six metric degrees of freedom. In brief, our approach can be summarized as follows.
We choose a suitable ansatz for the metric which is split into background and perturbations. We work in Poisson gauge where the perturbed Friedmann–Lemaître–Robertson–Walker metric is
where a denotes the scale factor of the background, x^{i} are comoving coordinates on the spacelike hypersurfaces, and τ is conformal time. ϕ and Ψ are the Bardeen potentials that describe the scalar metric perturbations; B_{i} (also denoted as B) is transverse and is responsible for frame dragging; h_{ij} is transverse and traceless, it contains the two spin2 degrees of freedom of gravitational waves.
We also assume that the perturbations of the metric, but not necessarily their derivatives, remain small on the scales of interest. This is a valid approximation for cosmological scales even when perturbations in the stressenergy tensor are large.
Einstein’s equations are then expanded in terms of the metric perturbations, but without a perturbative treatment of the stressenergy tensor. We include all terms linear in metric perturbations and go to quadratic order in terms which have two spatial derivatives acting on ϕ, Ψ. This weakfield expansion contains all six components of the metric correctly at leading order. To determine the evolution of the metric, we numerically solve the ‘00’ and the traceless part of the ‘ij’ Einstein equations:
Equation (2) is the generalization of the Newtonian Poisson equation (Δϕ = 4πGa^{2}δρ) and contains additional relativistic terms. Here Δϕ and are not required to be small—in fact, they become much larger than the background value inside dense regions. Equation (3) can be used to evolve all the nonNewtonian degrees of freedom of the metric: ϕ − Ψ, the two spin1 degrees of freedom, B, and the two spin2 helicities, h_{ij}. To do this, we decompose the equation into scalar, vector and tensor parts in Fourier space.
The metric is then used to solve the equations of motion for matter (and possibly other degrees of freedom); collisionless particles propagate along geodesics in the perturbed geometry. This determines the evolution of the stressenergy tensor.
As a first application we choose standard ΛCDM cosmology. We expect only small effects in this case, but we can use these simulations to gain confidence in our new approach. We generate^{18} two halo catalogues containing ∼500,000 halos each, one using our relativistic approach and one with the traditional Newtonian approach as reference, starting from identical linear initial conditions. A Kolmogorov–Smirnov test shows no significant disagreement in the distributions of some 25 different halo properties such as mass, spin or shape parameters.
The largest nonNewtonian effect is frame dragging, which is associated with the spin1 perturbation B. This perturbation is sourced by the curl part of the momentum density found, for example, in rotating massive objects. As long as perturbations are small this is a secondorder effect which has been studied using perturbation theory^{19}. The power spectrum for B has also been computed in the nonperturbative regime of structure formation using a postNewtonian framework^{20} which is expected to give good agreement with our approach as long as one considers ΛCDM (ref. 16). These results are useful benchmarks for our code, but we now go beyond: our simulations track the full threedimensional realization of B (see Fig. 1). We can therefore measure the actual framedragging force on individual particles.
Whereas the typical gravitational acceleration on Mpc scales is of the order of 10^{−9} cm s^{−2}, frame dragging contributes only at the level of 10^{−14} cm s^{−2} (both numbers are massweighted r.m.s. values from our simulations at redshift zero), and the highest value we measure is some 10^{−12} cm s^{−2}. Thus, for objects moving at 1,000 km s^{−1} (a typical peculiar velocity at those scales) the Δv due to frame dragging is no more than 1 km s^{−1}. It should be noted that these numbers are scale dependent, as the acceleration is larger on smaller scales.
We compare our simulation results also with predictions^{21,22} from secondorder perturbation theory for the power spectra of h_{ij}, of B and of ϕ − Ψ. The latter requires regularization in the infrared, which is implemented by the finite volume of the simulation.
To fully capture the amplitude of the nonNewtonian terms, it is important that the scale of matter–radiation equality is represented in the simulation volume. On the other hand, these terms are generated by nonlinearities and are therefore amplified at small scales. To obtain our results we had to cover at least three orders of magnitude in scale. Our largest simulation used a lattice with 4,096^{3} points, corresponding to 6.7 × 10^{10} particles as we always start with one particle per grid point. This simulation used 16384 CPUs on the Cray XC30 supercomputer Piz Daint; its 1088 time steps were completed in about 7 h, therefore totalling about 115 000 CPU hours.
Figure 2 shows the numerical power spectra at three redshifts. For these spectra we used eight simulations with lattices of 2,048^{3} points and two different box sizes, all starting at redshift z = 100. Four simulations had a box size of (2,048 Mpc/h)^{3}, the other four were (512 Mpc/h)^{3}. In the regime where it can be trusted we find excellent agreement with secondorder perturbation theory, which demonstrates that our code produces valid results. At late times and on small scales, perturbation theory breaks down. Nonlinearities enhance the frame dragging by more than an order of magnitude at redshifts z = 1 and z = 0 and on scales k ≳ 1h/Mpc. In this regime its power spectrum is only a factor 10^{−4} times smaller than the one of the Newtonian potential. This means that frame dragging is potentially an order 1% effect. Tensor perturbations on the other hand, even though they are enhanced by more than three orders of magnitude, remain a feeble contribution. Of course these power spectra are not directly observable: when comparing calculations with observations a projection onto the light cone has to be performed. This has been studied within perturbation theory^{23}, and one could implement this procedure also in our Nbody code, for example, by raytracing photons through the simulation box.
It should finally be noted that our framework jointly solves for background and perturbations in a selfconsistent way. Therefore, we confirm that clustering has only a small effect on the expansion rate of the Universe. The nonNewtonian effects which we have quantified in the nonlinear regime of structure formation remain small within a ΛCDM Universe, but they may nevertheless be measurable in the future. The vorticity, which is related to frame dragging, is substantial on small scales and remains at the per cent level relative to the gradient velocity field even on large scales. This can lead to observable corrections in redshift–space distortions. A detailed analysis of this possibility is still missing.
For the first time, general relativity has been implemented as the theory of gravity in a cosmological Nbody code, making it possible to feed these effects back into the dynamics. Our numerical framework will be particularly useful in scenarios where relativistic sources are present, such as models of dynamical dark energy, topological defects, or with relativistic particles such as neutrinos or warm dark matter. Such scenarios are expected to exhibit larger relativistic effects. Contrary to Newtonian schemes, we are also able to solve the full geodesic equation for arbitrary velocities, allowing for a realistic propagation of radiation or highvelocity particles.
Methods
A weakfield expansion is useful in situations where the metric perturbation variables ϕ, Ψ, B_{i}, h_{ij} defined in equation (1) are small, as is the case on cosmological scales. To get a tractable set of equations, we retain all terms which are linear in metric perturbations, but of the quadratic terms we keep only the most relevant ones. These are built from the scalar perturbations ϕ and Ψ only, and contain the highest possible number of spatial derivatives (two in this case, as all the partial differential equations are second order). This is enough to ensure that all six metric degrees of freedom are treated correctly at leading order, even in cases where some of them are strongly suppressed.
The implementation in gevolution adopts a particlemesh scheme where the metric field is represented on a regular cubic lattice whereas matter takes the form of an Nbody ensemble of particles that samples the sixdimensional phase space. The stressenergy tensor on the lattice is obtained by a particletomesh projection and is used in Einstein’s equations to solve for the metric perturbations. Finally, the particles are evolved by interpolating the metric to the particle positions and integrating the geodesic equation. gevolution is built on top of the LATfield2 C++ framework^{17}. LATfield2 distributes the fields and the particles over a twodimensional process grid (using MPI). It also provides the Fast Fourier Transform which gevolution needs for solving equations (2) and (3).
Einstein’s equations.
We determine the metric perturbations using the time–time as well as the traceless space–space part of Einstein’s equations given in equations (2) and (3). The remaining four equations are redundant, but we can use them as consistency check. The stressenergy tensor is constructed in the perturbed geometry and may hence also contain some terms which are linear in the metric perturbations—these terms have to be taken into account to maintain consistency. For a ΛCDM cosmology, such as the one studied here, the stressenergy tensor is obtained numerically by appropriate particletomesh projections which are ‘dressed’ by these geometric corrections.
We then solve for the metric variables by treating the quadratic terms in equations (2) and (3) as small perturbations. To this end, we simply estimate their value using the solutions taken from the previous time step and move them to the righthand side. As the equations are then approximated as linear in the metric perturbations, we use Fourier methods^{16} to obtain the new solutions. This is convenient in particular as the gauge conditions can easily be implemented in Fourier space, where they reduce to local projection operations. One can check that the new solutions are stable by reinserting them into the quadratic terms and iterating the procedure.
Particle trajectories.
As dark matter is assumed to be collisionless, the particles move along geodesics. For nonrelativistic velocities v = dx/dτ, the geodesic equation reads
where the Bdependent terms account for frame dragging. The integration of this equation can be simplified by writing , which transforms equation (4) to
This new equation actually describes the acceleration as seen in a Gaussian orthonormal coordinate frame. Frame dragging, characterized by the last term, has to compete with the ‘Newtonian’ force ∇Ψ. We measure both forces individually in our simulations.
Supplementary Fig. 1 shows the massweighted r.m.s. value of the frame dragging acceleration for simulations with different spatial resolutions. The simulation volume was (512 Mpc/h)^{3} in all cases, but we used lattice sizes from 512^{3} to 4096^{3} points, reaching a best resolution of 125 kpc/h. The r.m.s. acceleration depends on the scale probed and therefore also on resolution—the framedragging acceleration is larger on smaller scales, but the main contribution comes from large scales. Supplementary Fig. 2 shows a more detailed stream plot of the spin1 perturbation, revealing its longrange correlation and its correlation with the distribution of matter. At our best resolution we remain slightly above galactic scales. At smaller scales we expect baryonic effects to become important.
Power spectra.
In Fig. 2 we show the power spectra of the metric perturbations which are defined by
Here P_{ij} = δ_{ij} − k_{i}k_{j}/k^{2} is the transverse projector and the spin2 projection operator is given by M_{ijlm} = P_{il}P_{jm} + P_{im}P_{jl} − P_{ij}P_{lm}. These dimensionless power spectra measure the amplitude square of the metric perturbations at scale k per logk interval. To suppress finitevolume and resolution effects we measure only scales which are at least five times smaller than the box size and have a wavenumber which is at least five times smaller than the Nyquist frequency.
The secondorder power spectra of ϕ − Ψ and of B decay as ∼k^{−3.7} on small scales, whereas the tensor spectrum decays as ∼k^{−7.3}. Nonlinear evolution considerably flattens all the spectra around k ∼ 1h/ Mpc to slopes between −2 and −3.
Initial conditions.
Initial data is generated using a linear input power spectrum at initial redshift (we start at z = 100) which can be obtained by running a Boltzmann code^{24,25} for the model. We use CLASS with the default cosmological parameters which describe a ΛCDM model (Ω_{m} = Ω_{c} + Ω_{b} = 0.312, our code at present has no special treatment for baryonic matter). The initial displacement of the particles is also computed at linear order, taking into account our choice of gauge^{15}.
Code availability.
The gevolution code is available on a public Git repository: https://github.com/gevolutioncode/gevolution1.0.git. The required extension of the LATfield2 library is developed independently and is available on the repository: https://github.com/daverio/LATfield2.git.
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Acknowledgements
We thank R. Teyssier and M. Bruni for discussions. This work was supported by the Swiss National Supercomputing Centre (CSCS) under project ID s546. The numerical simulations were carried out on Piz Daint at the CSCS and on the Baobab cluster of the University of Geneva. Financial support was provided by the Swiss National Science Foundation.
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J.A. worked out the equations in our approximation scheme and implemented the cosmological code gevolution. He also produced the figures. D.D. developed and implemented the particle handler for the LATfield2 framework. R.D. contributed to the development of the approximation scheme and the derivation of the equations. M.K. proposed the original idea. All authors discussed the research and helped with writing the paper.
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Adamek, J., Daverio, D., Durrer, R. et al. General relativity and cosmic structure formation. Nature Phys 12, 346–349 (2016). https://doi.org/10.1038/nphys3673
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