Classification of the FRW universe with a cosmological constant and a perfect fluid of the equation of state $p = w\rho$

TL;DR

This paper systematically classifies the evolution of FRW universes with a cosmological constant and perfect fluid, detailing different scenarios, singularities, and correcting previous classification errors.

Contribution

It provides a comprehensive classification of FRW universe evolutions with various parameters, correcting prior errors and clarifying solution behaviors.

Findings

Detailed solution classifications for different parameter regimes

Identification of conditions for singularities like big bang, big crunch, big rip

Corrections of errors in standard textbook classifications

Abstract

We systematically study the evolution of the Friedmann-Robertson-Walker (FRW) universe coupled with a cosmological constant $\Lambda$ and a perfect fluid that has the equation of state $p=w\rho$, where $p$ and $\rho$ denote, respectively, the pressure and energy density of the fluid, and $w$ is an arbitrary real constant. Depending on the specific values of $w,\; \Lambda$, and the curvature $k$ of 3-dimensional space, we separate all of the solutions into various cases. In each case the main properties of the evolution are given in detail, including the periods of deceleration and/or acceleration, and the existence of big bang, big crunch, and big rip singularities. In some cases, errors in classification and interpretation appearing in standard textbooks have been corrected.