Post-Newtonian Equations of Motion in the Flat Universe

Abstract

Using the (3 + 1) formalism, we derive the post-Newtonian (PN) equations of motion in a flat unvierse. To derive the equations of motion, we must carefully consider two points, one being the choice of the density in the Newtonian order (ρN) and the other the choice of the gauge condition. In choosing ρN, we require that the density fluctuation ρN - ρ0 agrees with a gauge invariant quantity in the linear approximation theory. As a gauge condition, we propose the cosmological post-Newtonian (CPN) slice condition with the pseudo transverse-traceless gauge condition, by which the evolution of the geometric variables derived in the PN approximation in the early stage of universe agrees with that of the gauge invariant quantities in the linear approximation. In the derived equations of motion, the force is calculated from six potentials which satisfy the Poisson equations. Hence, our formalism can be easily applied to numerical simulations in which the standard technique (e.g., particle-mesh method) is used. We apply the PN formula to the one-dimensional (1D) Zel'dovich solution to demonstrate that our strategy works well, and also to determine the effect of the PN forces on the evolution of the large-scale structure. It is found that the behavior of the density fluctuation and metric quantities in the early stage obtained by the present formalism agrees with that of the gauge invariant quantities in the linear approximation, although they do not always agree within the previous formalism due to the appearance of spurious gauge modes. We also discuss the evolution of the non-linear density fluctuation with very large scale, which may be affected by the PN correction in the last stage of the evolution.