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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
Software:
GADGET ; RAMSES
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[1] Adamek, J.; Daverio, D.; Durrer, R.; Kunz, M., Gevolution: a cosmological n-body code based on general relativity, J. Cosmol. Astropart. Phys., 2016053, 7, 1-43 (2016)
[2] Adamek, J.; Daverio, D.; Durrer, R.; Kunz, M., General relativity and cosmic structure formation, Nat. Phys., 12, 346 (2016)
[3] Arnold, D.N.: Differential complexes and numerical stability. In: Tatsien, L. (Ed.) Proceedings of the International Congress of Mathematicians, Vol. I, pp. 137-157. Higher Ed. Press, Beijing (2002) · Zbl 1023.65113
[4] Back, A.: Runge-Kutta Behavior in the Presence of a Jump Discontinuity. (Preprint) (2005)
[5] Barrera-Hinojosa, C., Li, B.: GRAMSES: a new route to general relativistic N-body simulations in cosmology—I. Methodology and code description. arXiv:1905.08890 [astro-ph.CO]
[6] Butcher, Jc, Coefficients for the study of Runge-Kutta integration processes, J. Aust. Math. Soc., 3, 2, 185-201 (1963) · Zbl 0223.65031
[7] Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations. I. Nonstiff problems, volume 8 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (1993) · Zbl 0789.65048
[8] Hairer, E.; Lubich, C.; Wanner, G., Geometric numerical integration illustrated by the Störmer-Verlet method, Acta Numer., 12, 399-450 (2003) · Zbl 1046.65110
[9] Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, volume 31 of Springer Series in Computational Mathematics. Springer, Heidelberg (2010). Structure-Preserving Algorithms for Ordinary Differential Equations, Reprint of the second edition (2006) · Zbl 1094.65125
[10] Parsa, M.; Eckart, A.; Shahzamanian, B.; Karas, V.; Zajacek, M.; Zensus, J., Investigating the relativistic motion of the stars near the supermassive black hole in the galactic center, Astrophys. J., 419, 845, 22 (2017)
[11] Preto, M.; Saha, P., On post-Newtonian orbits and the Galactic-center stars, Astrophys. J., 703, 1743 (2009)
[12] Springel, V., The cosmological simulation code gadget-2, Mon. Not. R. Astron. Soc., 364, 4, 1105-1134 (2005)
[13] Stephani, H.: Relativity: An Introduction to Special and General Relativity. Cambridge University Press (Transl. from German By M. Pollock and J. Stewart) (1982) · Zbl 0494.53026
[14] Teyssier, R., Cosmological hydrodynamics with adaptive mesh refinement: a new high resolution code called ramses, Astron. Astrophys., 385, 337-364 (2002)
[15] Yu, H., Emberson, J., Inman, D., et al.: Differential neutrino condensation onto cosmic structure. Nature Astronomy 1(143), (2017)
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