Asymptotic solution for expanding universe with matter-dominated evolution

Abstract

We applied the theory of regularly varying functions to the analysis of the cosmological parameters for the $\Lambda$CDM model with the matter dominated evolution. Carroll et al. proved in 1992 that for this type of universe with the curvature $k=0, -1$, the expression $H(t)t$\, ($H(t)$ is the Hubble parameter) depends solely on the density parameter $\Omega(t)$. Using this result and the theory of regular variation we infer for such universe the complete asymptotics of all main cosmological parameters. More specifically, the following is derived. If the limit $\omega= \lim_{t\to\infty} \Omega(t)$ does exist and $\omega \not= 0$ then the cosmological constant $\Lambda$ is equal to $0$. If $\omega=0$ then for the expansion scale factor $a(t)$ we have $a(t)\sim e^{\sqrt{\Lambda/3}}$. On the other hand, if the limit $\lim_{t\to\infty} \Omega(t)$ does not exist then $a(t)$ bounces between two power functions and therefore has infinitely many flexion points. Hence, the deceleration parameter in this case changes the sign infinitely many times.