Cosmological non-linear hydrodynamics with post-Newtonian corrections

The purpose of this paper is to present general relativistic cosmological hydrodynamic equations in Newtonian-like forms using the post-Newtonian (PN) method. The PN approximation, based on the assumptions of weak gravitational fields and slow motions, provides a way to estimate general relativistic effects in the fully non-linear evolution stage of the large-scale cosmic structures. We extend Chandrasekhar's first-order PN (1PN) hydrodynamics based on the Minkowski background to the one based on the Robertson-Walker background. We assume the presence of Friedmann's cosmological spacetime as a background. In the background we include the 3-space curvature, the cosmological constant and general pressure. In the Newtonian order and 1PN order we include general pressure, stress, and flux. We show that the Newtonian hydrodynamic equations appear naturally in the 0PN order. The spatial gauge degree of freedom is fixed in a unique manner and the basic equations are arranged without taking the temporal gauge condition. In this way we can conveniently try alternative temporal gauge conditions depending on the mathematical convenience. We investigate a number of temporal gauge conditions under which all the remaining variables are equivalently gauge invariant. We show that compared with the action-at-a-distance nature of the Newtonian gravitational potential, 1PN corrections make the propagation speed of a perturbed potential dependent on the temporal gauge condition; we show that to 1PN order the physically relevant propagation speed of gravity is the same as the speed of light. Our aim is to present the fully non-linear cosmological 1PN equations in a form suitable for implementation in conventional Newtonian hydrodynamic simulations with minimal extensions. The 1PN terms can be considered as relativistic corrections added to the well-known Newtonian equations. The proper arrangement of the variables and equations in combination with suitable gauge conditions would allow for possible future 1PN cosmological simulations to become more tractable. Our equations and gauges are arranged for that purpose. We suggest ways of controlling the numerical accuracy. The typical 1PN order terms are about 10(-6)-10(-4) times smaller than the Newtonian terms. However, we cannot rule out the possible presence of cumulative ( secular) effects due to the time-delayed propagation of the relativistic gravitational field with infinite speed, in contrast to the Newtonian case where changes in the gravitational field are felt instantaneously. The quantitative estimation of such effects is left for future numerical simulations. If the reader is interested in the applications of 1PN equations, she/he may go directly to section 4 of the paper after reading the introduction.