Third order perturbations of a zero-pressure cosmological medium: Pure general relativistic nonlinear effects

TL;DR

This paper derives third-order relativistic corrections to cosmological perturbations in a zero-pressure medium, showing they are small and can be neglected in large-scale Newtonian simulations near the horizon.

Contribution

It provides the first derivation of pure third-order relativistic correction terms in cosmological perturbation theory, extending previous second-order analyses.

Findings

Third-order relativistic corrections are higher order in the gravitational potential.

Corrections are independent of horizon scale and depend only on linear perturbation strength.

Relativistic effects are of order 10^{-5}, supporting the reliability of Newtonian simulations near the horizon.

Abstract

We consider a general relativistic zero-pressure irrotational cosmological medium perturbed to the third order. We assume a flat Friedmann background but include the cosmological constant. We ignore the rotational perturbation which decays in expanding phase. In our previous studies we discovered that, to the second-order perturbation, except for the gravitational wave contributions, the relativistic equations coincide exactly with the previously known Newtonian ones. Since the Newtonian second-order equations are fully nonlinear, any nonvanishing third and higher order terms in the relativistic analyses are supposed to be pure relativistic corrections. In this work we derive such correction terms appearing in the third order. Continuing our success in the second-order perturbations we take the comoving gauge. We discover that the third-order correction terms are of $\phi_v$-order higher than the second-order terms where $\phi_v$ is a gauge-invariant combination related to the three-space curvature perturbation in the comoving gauge; compared with the Newtonian potential we have $\delta \Phi \sim {3 \over 5} \phi_v$ to the linear order. Therefore, the pure general relativistic effects are of $varphi_v$-order higher than the Newtonian ones. The corrections terms are independent of the horizon scale and depend only on the linear order gravitational potential perturbation strength. From the temperature anisotropy of cosmic microwave background we have ${\delta T \over T} \sim {1 \over 3} \delta \Phi \sim {1 \over 5} \phi_v \sim 10^{-5}$. Therefore, our present result reinforces our previous important practical implication that near current era one can use the large-scale Newtonian numerical simulation more reliably even as the simulation scale approaches near the horizon.