Accuracy of Nonlinear Approximations in Spheroidal Collapse: Why Are Zeldovich‐Type Approximations Good?

Abstract

Among various analytic approximations for the growth of density fluctuations in the expanding universe, the Zeldovich approximation and its extensions in a Lagrangian scheme are known to be better than other approximations, even in mildly nonlinear regimes. We compare analytic approximations with true density evolution in the presence of spheroidal symmetry. We consider Eulerian and Lagrangian perturbation theories up to third order, and frozen-flow and linear potential approximation. We also introduce the Pade approximation in the Eulerian scheme which improves usual perturbation theories. In the course of comparison, we clarify the reason why these Zeldovich-type approximations effectively work beyond the linear regime, with reference to the two following aspects of the problem: (1) the dimensionality of the system and (2) the Lagrangian scheme on which the Zeldovich approximation is grounded. We mention which of these two aspects supports the validity of the Zeldovich-type approximations. We also give a suggestion for a better approximation method beyond the Zeldovich type.