Yamamoto, Kazuhiro; Marra, Valerio; Mukhanov, Viatcheslav; Sasaki, Misao
(2016):
Perturbed Newtonian description of the Lemaitre model with nonnegligible pressure.
In: Journal of Cosmology and Astroparticle Physics, No. 3, 30

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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 nonnegligible 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 subhorizon inhomogeneities. However, the relation psi + phi = O(phi(2))  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 Lemaitre model which allows us to study in an exact nonlinear fashion the onset of anisotropic stress in fluids with nonnegligible 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[O(phi(2)), 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 = 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.