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The detection of relativistic corrections in cosmological $$N$$-body simulations. (English) Zbl 1448.70026
Summary: Cosmological $$N$$-body simulations are done on massively parallel computers. This necessitates the use of simple time integrators and, additionally, of mesh-grid approximations of the potentials. Recently, J. Adamek et al. [“General relativity and cosmic structure formation”, Nat. Phys. 12, No. 4, 346–349 (2016; doi:10.1038/nphys3673)] and C. B. Hinojosa and B. Li [“GRAMSES: a new route to general relativistic $$N$$-body simulations in cosmology. I: Methodology and code description”, J. Cosmol. Astropart. Phys. 2020, No. 01, Paper No. 7, 41 p. (2020; doi:10.1088/1475-7516/2020/01/007)] have developed general relativistic $$N$$-body simulations to capture relativistic effects mainly for cosmological purposes. We therefore ask whether, with the available technology, relativistic effects like perihelion advance can be detected numerically to a relevant precision. We first study the spurious perihelion shift in the Kepler problem, as a function of the integration method used and then as a function of an additional interpolation of forces on a two-dimensional lattice. This is done for several choices of eccentricities and semi-major axes. Using these results, we can predict which precisions and lattice constants allow for a detection of the relativistic perihelion advance in $$N$$-body simulation. We find that there are only small windows of parameters – such as eccentricity, distance from the central object and the Schwarzschild radius – for which the corrections can be detected in the numerics.
##### MSC:
 70F10 $$n$$-body problems 65Z05 Applications to the sciences 70F15 Celestial mechanics 70H40 Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics