No abstract

We present GRAMSES, a new pipeline for nonlinear cosmological N -body simulations in General Relativity (GR). This code adopts the Arnowitt-Deser-Misner (ADM) formalism of GR, with constant mean curvature and minimum distortion gauge fixings, which provides a fully nonlinear and background independent framework for relativistic cosmology. Employing a fully constrained formulation, the Einstein equations are reduced to a set of ten elliptical equations which are solved using multigrid relaxation with adaptive mesh refinements (AMR), and three hyperbolic equations for the evolution of tensor degrees of freedom. The current version of GRAMSES neglects the latter by using the conformal flatness approximation, which allows it to compute the two scalar and two vector degrees of freedom of the metric. In this paper we describe the methodology, implementation, code tests and first results for cosmological simulations in a ΛCDM universe, while the generation of initial conditions and physical results will be discussed elsewhere. Inheriting the efficient AMR and massive parallelisation infrastructure from the publicly-available N -body and hydrodynamic simulation code RAMSES, GRAMSES is ideal for studying the detailed behaviour of spacetime inside virialised cosmic structures and hence accurately quantifying the impact of backreaction effects on the cosmic expansion, as well as for investigating GR effects on cosmological observables using cosmic-volume simulations.

A number of codes for general-relativistic simulations of cosmological structure formation have been developed in recent years. Here we demonstrate that a sample of these codes produce consistent results beyond the Newtonian regime. We simulate solutions to Einstein’s equations dominated by gravitomagnetism—a vector-type gravitational field that does not exist in Newtonian gravity and produces frame-dragging, the leading-order post-Newtonian effect. We calculate the coordinate-invariant effect on intersecting null geodesics by performing ray tracing in each independent code. With this observable quantity, we assess and compare each code’s ability to compute relativistic effects.

We investigate the transverse modes of the gravitational and velocity fields in ΛCDM, based on a high-resolution simulation performed using the adaptive-mesh refinement general-relativistic N-body code gramses. We study the generation of vorticity in the dark matter velocity field at low redshift, providing fits to the shape and evolution of its power spectrum over a range of scales. By analysing the gravitomagnetic vector potential, which is absent in Newtonian simulations, in dark matter haloes with masses ranging from ∼1012.5 h−1M⊙ to ∼1015 h−1M⊙, we find that its magnitude correlates with the halo mass, peaking in the inner regions. Nevertheless, on average, its ratio against the scalar gravitational potential remains fairly constant, below percent level, decreasing roughly linearly with redshift and showing a weak dependence on halo mass. Furthermore, we show that the gravitomagnetic acceleration in haloes peaks towards the core and reaches almost 10−10  h cm/s2 in the most massive halo of the simulation. However, regardless of the halo mass, the ratio between the gravitomagnetic force and the standard gravitational force is typically at around the 10−5 level inside the haloes, again without significant radius dependence. This result confirms that the gravitomagnetic effects have negligible impact on structure formation, even for the most massive structures, although its behaviour in low density regions remains to be explored. Likewise, the impact on observations remains to be understood in the future.

We address the generation of initial conditions (ICs) for {\sc gramses}, a code for nonlinear general relativistic (GR) N-body cosmological simulations recently introduced in ref. [1]. {\sc gramses} adopts a constant mean curvature slicing with a minimal distortion gauge, where the linear growth rate is scale-dependent, and the standard method for realising initial particle data is not straightforwardly applicable. A new method is introduced, in which the initial positions of particles are generated from the displacement field realised for a matter power spectrum as usual, but the velocity is calculated by finite-differencing the displacement fields around the initial redshift. In this way, all the information required for setting up the initial conditions is drawn from three consecutive input matter power spectra, and additional assumptions such as scale-independence of the linear growth factor and growth rate are not needed. We implement this method in a modified {\sc 2LPTic} code, and demonstrate that in a Newtonian setting it can reproduce the velocity field given by the default {\sc 2LPTic} code with subpercent accuracy. We also show that the matter and velocity power spectra of the initial particle data generated for {\sc gramses} simulations using this method agree very well with the linear-theory predictions in the particular gauge used by {\sc gramses}. Finally, we discuss corrections to the finite difference calculation of the velocity when radiation is present, as well as additional corrections implemented in {\sc gramses} to ensure consistency. This method can be applied in ICs generation for GR simulations in generic gauges, and simulations of cosmological models with scale-dependent linear growth rate.