The Simplest Exact Solutions in the LTB Model

TL;DR

This paper investigates exact nonhomogeneous solutions in the LTB model with specific conditions, analyzing their relation to FRW models, initial conditions, and asymptotic behavior.

Contribution

It provides a detailed analysis of exact solutions in the LTB model with f^2=1 and zero cosmological constant, including the limit to FRW models and initial condition effects.

Findings

Derived the parameter ν(μ) distinguishing LTB from FRW models.

Demonstrated the limit transformation to FRW models.

Analyzed the asymptotic behavior as τ approaches infinity.

Abstract

The rules of calculating three undetermined functions which defined a solution in the LTB model are used to study the class of exact nonhomogene\-ous models with $f^2(\mu) = 1$, $\Lambda = 0$. The parameter $\nu(\mu)$ defined the difference between LTB and FRW models is found out and the limit transformation to the FRW model is shown. The initial conditions are present throught density and Habble function at the moment of time $\tau = 0$. Two criteria of homogeneous of matter distribution are studied. The asimptotic of the present solution for $\tau \rightarrow +\infty$ is studied.