On cosmological evolution with the Lambda-term and any linear equation of state

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, in a proper compact representation. Various estimates of the expansion are presented in a general case when no closed-form solutions are available. 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 $\Omega_\Lambda/\Omega_M$ and age of the universe.