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

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]).