Perturbed Newtonian description of the Lemaître model with non-negligible pressure

We study the validity of the Newtonian description of cosmological perturbations using the Lemaître 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=ρ/3). We find that, even when the density contrast has a nonlinear amplitude, the Newtonian description of the cosmological perturbations using the gravitational potential ψ and the curvature potential φ is valid as long as we consider sub-horizon inhomogeneities. However, the relation ψ+φ=(φ²) - which holds for the case of a dust fluid - is not valid for a relativistic fluid, and an effective anisotropic stress is generated. This demonstrates the usefulness of the Lemaître 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 ψ+φ= [(φ²),(c_s^2φ δ)] for an inhomogeneity with density contrast δ 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 ψ+φ=(φ²) 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.