Matching of Stephani and de Sitter solutions on the hypersurface of constant time

TL;DR

This paper investigates how the Stephani and de Sitter solutions, representing different cosmological models, can be matched on a constant-time hypersurface, revealing conditions on densities, pressures, and arbitrary functions.

Contribution

It provides a detailed analysis of matching Stephani and de Sitter solutions, including general and specific universe cases, and derives conditions on physical quantities and functions.

Findings

Matching conditions for densities and pressures on the hypersurface

Restrictions on arbitrary functions in the solutions

Applicability to flat, closed, and open universe models

Abstract

The spherically symmetric solution for perfect fluid with homogeneous density and inhomogeneous pressure has been considered. This solution is known as Stephani solution. The matching of this solution and de Sitter solution has been done on a hypersurface of constant time. The matching has been done for general case and for particular cases (flat, closed, open universe). An equality of the densities and a bound of the pressures have been shown on the matching hypersurface. Also, restrictions on some arbitrary functions have been found.