Perturbed Newtonian description of the Lema\^itre model with non-negligible pressure

TL;DR

This paper investigates the validity of the Newtonian approximation in cosmological perturbations within the Lemaitre model, especially for fluids with pressure like radiation, revealing conditions under which anisotropic stress arises and the approximation holds.

Contribution

It extends previous Newtonian analyses from dust to relativistic fluids with pressure, providing exact nonlinear insights into anisotropic stress onset and perturbation validity.

Findings

Newtonian description valid for sub-horizon inhomogeneities even with nonlinear density contrast.

Anisotropic stress arises when inhomogeneity scale is smaller than the sound horizon due to nonlinear fluid effects.

Relation + differs from dust case, depending on scale, amplitude, and fluid properties.

Abstract

We study the validity of the Newtonian description of cosmological perturbations using the Lemaitre model, an exact spherically symmetric solution of Einstein's equation. This problem has been investigated in the past for the case of a dust fluid. Here, we extend the previous analysis to the more general case of a fluid with non-negligible pressure, and, for the numerical examples, we consider the case of radiation (P=\rho/3). We find that, even when the density contrast has a nonlinear amplitude, the Newtonian description of the cosmological perturbations using the gravitational potential \psi and the curvature potential \phi is valid as long as we consider sub-horizon inhomogeneities. However, the relation \psi+\phi={\cal O}(\phi^2), which holds for the case of a dust fluid, is not valid for a relativistic fluid and effective anisotropic stress is generated. This demonstrates the usefulness of the Lemaitre model which allows us to study in an exact nonlinear fashion the onset of anisotropic stress in fluids with non-negligible pressure. We show that this happens when the characteristic scale of the inhomogeneity is smaller than the sound horizon and that the deviation is caused by the nonlinear effect of the fluid's fast motion. We also find that \psi+\phi= \max[{\cal O}(\phi^2),{\cal O}(c_s^2\phi \, \delta)] for an inhomogeneity with density contrast \delta whose characteristic scale is smaller than the sound horizon, unless w is close to -1, where w and c_s are the equation of state parameter and the sound speed of the fluid, respectively. On the other hand, we expect \psi+\phi={\cal O}(\phi^2) to hold for an inhomogeneity whose characteristic scale is larger than the sound horizon, unless the amplitude of the inhomogeneity is large and w is close to -1.