Cosmological Evolution with \Lambda-Term and Any Linear equation of State

TL;DR

This paper derives exact solutions for cosmological models with a positive cosmological constant and linear equations of state, analyzing their behavior and implications for universe expansion and age estimation.

Contribution

It provides four new closed-form solutions for flat universes with various equations of state and establishes a relation between solutions with different parameters for open universes.

Findings

Exponential expansion at large times with parameter-dependent amplitude.

Amplitude independence from equation of state for small cosmological constant in open universe.

Estimates of cosmic age based on observable ratios and equation of state parameters.

Abstract

Recent observational indications of an accelerating universe enhance the interest in studying models with a cosmological constant. We investigate cosmological expansion (FRW metric) with $\Lambda>0$ for a general linear equation of state $p=w\rho$, $w>-1$, so that the interplay between cosmological vacuum and quintessence is allowed, as well. Four closed-form solutions (flat universe with any $w$, and $w=1/3$, $-1/3, -2/3$) are given, of which the last one appears to be new. For the open universe a simple relation between solutions with different parameters is established: it turns out that a solution with some $w$ and (properly scaled) $\Lambda$ is expressed algebraically via another solution with special different values of these parameters. The expansion becomes exponential at large times, and the amplitude at the exponent depends on the parameters. We study this dependence in detail, deriving various representations for the amplitude in terms of integrals and series. The closed-form solutions serve as benchmarks, and the solution transformation property noted above serves as a useful tool. Among the results obtained, one is that for the open universe with relatively small cosmological constant the amplitude is independent of the equation of state. Also, estimates of the cosmic age through the observable ratio $\Omega_\Lambda/\Omega_M$ and parameter $w$ are given; when inverted, they provide an estimate of $w$, i. e., the state equation, through the known age of the universe.