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 = \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 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 \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 .