Approximate analytical solutions of the homologous collapse’s radial evolution in time

Abstract

ABSTRACT The homologous collapse of a uniform density sphere under its self-gravity is a well-known model in cosmological studies. In the analytical integration, the evolution of the radius of the sphere is defined in terms of a parameter that is related to the physical time through a transcendental equation. The aim of this note is to construct new approximate solutions of this transcendental equation. After close examination of the piecewise Padé approximation introduced in a recent paper by Zhou and collaborators, several modifications that improve the accuracy of this approximation with the same computational cost are proposed. This new approximation shows that the use of piecewise Hermite interpolation leads to approximate solutions with continuity and much higher accuracy than Padé interpolants. Numerical experiments that confirm the above results are also presented.