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