Cosmological non-linear hydrodynamics with post-Newtonian corrections

Abstract

The purpose of this paper is to present general relativistic cosmological hydrodynamicequations in Newtonian-like forms using the post-Newtonian (PN) method. The PNapproximation, based on the assumptions of weak gravitational fields and slowmotions, provides a way to estimate general relativistic effects in the fully non-linearevolution stage of the large-scale cosmic structures. We extend Chandrasekhar’sfirst-order PN (1PN) hydrodynamics based on the Minkowski background to theone based on the Robertson–Walker background. We assume the presence ofFriedmann’s cosmological spacetime as a background. In the background we includethe 3-space curvature, the cosmological constant and general pressure. In theNewtonian order and 1PN order we include general pressure, stress, and flux. We showthat the Newtonian hydrodynamic equations appear naturally in the 0PN order.The spatial gauge degree of freedom is fixed in a unique manner and the basicequations are arranged without taking the temporal gauge condition. In thisway we can conveniently try alternative temporal gauge conditions dependingon the mathematical convenience. We investigate a number of temporal gaugeconditions under which all the remaining variables are equivalently gauge invariant.We show that compared with the action-at-a-distance nature of the Newtoniangravitational potential, 1PN corrections make the propagation speed of a perturbedpotential dependent on the temporal gauge condition; we show that to 1PN orderthe physically relevant propagation speed of gravity is the same as the speed oflight. Our aim is to present the fully non-linear cosmological 1PN equations ina form suitable for implementation in conventional Newtonian hydrodynamicsimulations with minimal extensions. The 1PN terms can be considered as relativisticcorrections added to the well-known Newtonian equations. The proper arrangementof the variables and equations in combination with suitable gauge conditionswould allow for possible future 1PN cosmological simulations to become moretractable. Our equations and gauges are arranged for that purpose. We suggest waysof controlling the numerical accuracy. The typical 1PN order terms are about10−6–10−4 times smaller than the Newtonian terms. However, we cannot rule out the possible presenceof cumulative (secular) effects due to the time-delayed propagation of the relativisticgravitational field with finite speed, in contrast to the Newtonian case where changes in thegravitational field are felt instantaneously. The quantitative estimation of such effects is leftfor future numerical simulations. If the reader is interested in the applications of1PN equations, she/he may go directly to section 4 of the paper after reading theintroduction.