Abstract
We compare relativistic approximation methods, which describe gravitational instability in the expanding universe, in a spherically symmetric model. Linear perturbation theory, second-order perturbation theory, the relativistic Zel'dovich approximation, and the relativistic post-Zel'dovich approximation are considered and compared with the Lema\^{\i}tre-Tolman-Bondi solution in order to examine the accuracy of these approximations. We consider some cases of inhomogeneous matter distribution while the homogeneous top-hat model has been usually taken in the previous Newtonian works. It is found that the Zel'dovich-type approximations are generally more accurate than the conventional perturbation theories in the weakly nonlinear regime. The applicable range of the Zel'dovich-type approximations is also discussed.
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