On the border lines between the regions of distinct solution type for solutions of the Friedmann equation satisfying the Hubble condition

TL;DR

This paper provides a detailed mathematical description of the boundary conditions between different solution types of the Friedmann equation with a positive cosmological constant, focusing on Hubble solutions and their dependence on matter and radiation densities.

Contribution

It introduces a constructive method to describe border lines between solution types of the Friedmann equation using smooth functions of matter and radiation densities.

Findings

Derived explicit functions for border lines between solution types.

Established asymptotic relations for these functions.

Provided a mathematical framework for classifying solution behaviors.

Abstract

It is well-known that there are four distinct basic types (two Big Bang types, Lemaitre and Big Crunch type) for solutions of the general Friedmann equation with positive cosmological constant, where radiation and matter do not couple (see e.g. [2, p.7]. In that paper the system of case distinction parameters contains a "critical radiation parameter" $\sigma_{cr}$. The present note contains the constructive description of the so-called {\em border lines} between Big Bang/Big Crunch type and Big Bang/Lemaitre type for so-called Hubble solutions of the Friedmann equation by two smooth function branches, expressing the cosmological constant as unique functions of the matter and radiation density (which is considered as a parameter). These functions satisfy simple asymptotic relations w.r.t. the matter density. They are constructed as the solutions of the equation $\sigma=\sigma_{cr}.