Necessity of revisions for the cosmological matter perturbations from general relativity

The differential equations, which are used for matter perturbations, are usually derived from the Newton gravity and the Euler equation in the expanded universe. In this paper, by the explicit calculations of metric perturbation theory in $\Lambda \mathrm{CDM}$ model, we derive the general relativistic corrections exactly for the equations and show that the differential equations derived from the Newton gravity and the Euler equation are valid in the small scale region from nonrelativistic matter dominant era onwards. However, the corrections are not negligible in radiation dominant era.