Newtonian limit of fully nonlinear cosmological perturbations in Einstein's gravity

TL;DR

This paper proves that under certain limits, the fully nonlinear relativistic cosmological perturbation equations in Einstein's gravity reduce exactly to Newtonian hydrodynamic equations, confirming the consistency between relativistic and Newtonian descriptions.

Contribution

It demonstrates the exact correspondence between relativistic perturbation equations and Newtonian hydrodynamics in specific gauges and limits, clarifying the Newtonian limit in cosmology.

Findings

Relativistic equations reduce to Newtonian hydrodynamics in the infinite speed-of-light limit.

Exact correspondence in zero-shear and uniform-expansion gauges.

Newtonian equations are recovered in Minkowski background under the same conditions.

Abstract

We prove that in the infinite speed-of-light limit (i.e., non-relativistic and subhorizon limits), the relativistic fully nonlinear cosmological perturbation equations in two gauge conditions, the zero-shear gauge and the uniform-expansion gauge, exactly reproduce the Newtonian hydrodynamic perturbation equations in the cosmological background; as a consequence, in the same two gauge conditions, the Newtonian hydrodynamic equations are exactly recovered in the Minkowsky background.