• N-body simulation
• gevolution
• curvature
• cosmology

In the last few years, the observed tension between late-time and early-time measurement of Hubble rate constant has gradually increased. This fact has called for new theories to explain it. Recently many cosmologists have opened to the possibility of introducing an intrinsic curvature term in standard model which would decrease or even eliminate Hubble rate tension. To further investigate this possibility we created a new formalism which allows to efficiently simulate a curved Friedmann-Roberson-Walker universe using a cosmological N-body simulation code with periodic boundary conditions.
This formalism is applied in gevolution, an N-body simulation code which implements full General Relativity using weak field approximation. Validation tests are performed using gevolution code time-slice and light-cone outputs. Raytracer code is used to process data obtained from the light-cone output.
It is possible to simulate a curved Friedemann-Roberson-Walker metric as a small perturbation of a flat Friedemann-Roberson-Walker metric; by doing so the simulation code can evolve it, as long as the perturbation and therefore the curvature value is small enough. The main challenge arises from the fact that the code uses periodic boundary conditions which are not compatible with a curved universe. To go around this problem we make use of a “Swiss cheese” model; the idea behind it is to divide the cube box in which the simulation takes place in three different areas:
– Area 1: A big sphere located in the middle of the simulation box. This is the area where the perturbations mentioned above are applied; consequently, this area appears and behaves like a portion of a curved Friedemann-Roberson-Walker universe.
– Area 2: A thin layer of space around Area 1, in this area the matter density is zero and the metric is Schwarzschild-De Sitter. The function of this area is to smoothly connect Area 1 with Area 3.
– Area 3: The area that goes from Area 2 to the sides of the simulation box in which there is an unperturbed flat Friedemann-Roberson-Walker space, that is the unperturbed simulation background. This area cause no issues with periodic boundary conditions.
It can be demonstrated, through Israel Condition, that this configuration maintains the same shape during the time evolution.
The code only takes input in the form of small perturbations in Newtonian gauge. A lot of effort was made to identify the correct coordinate transformation that brings the metric of Area 1 and Area 2 in a form that can be used as an input of the code.
After implementing the described modifications three tests have been performed to check the consistency of results with expectations in a curved universe.
– Test 1: Checking the value of gravitational potential and matter density at different temporal slices.
– Test 2: Checking the evolution of scale factor in different points of Area 1.
– Test 3: Fitting data of angular distance and redshift of all the particles in a lightcone that resides inside Area 1.
The first two tests suffer from a systematic error which we cannot quantify, however, these tests give valid indications about the goodness of the simulation; they have also been especially helpful in the debugging phase. The last test is instead totally reliable and can be used as final proof of simulation accuracy. Our simulations brilliantly passed all the tests, in conclusion, we can safely assert that by using our formalism it is possible to simulate a 3 Gpc curved universe with a negative curvature parameter in range 0⩾Ωk⩾−0.03 .