On the frontier lines between the regions of invariant solution type for solutions of the Friedmann equation satisfying the Hubble condition

TL;DR

This paper analyzes the boundary lines between different solution types of the Friedmann equation with a positive cosmological constant, deriving explicit functions that describe these frontiers based on matter and radiation densities.

Contribution

It provides a mathematical characterization of the frontier lines between solution types in the Friedmann equation, expressing the cosmological constant as functions of matter and radiation densities.

Findings

Frontier lines are smooth functions of matter and radiation densities.

These functions satisfy simple asymptotic relations.

The solution involves solving the equation σ=σ_cr.

Abstract

It is well-known that there are four distinct basic types (two BigBang types, Lemaitre and BigCrunch type) for solutions of the general Friedmann equation with positive cosmological constant, where radiation and matter do not couple. In the note it is shown that the "frontier lines" between BigBang/BigCrunch and BigBang/Lemaitre are given by two smooth function branches, expressing the cosmological constant as unique functions of the matter and radiation density, which satisfy a simple asymptotic relation w.r.t. the matter density. The proof is based on the solution of the equation $\sigma=\sigma_{cr}$, where $\sigma$ is the radiation invariant of the Friedmann equation and $\sigma_{cr}$ the "critical radiation parameter" (see [3]).