The McVittie solution with a negative cosmological constant

TL;DR

This paper investigates the McVittie solution with a negative cosmological constant, revealing that such a universe inevitably collapses into a Big Crunch, contrasting with the positive case.

Contribution

It provides a systematic construction of conformal diagrams for the McVittie solution with b3<0, highlighting the collapse behavior and global structure.

Findings

b3<0 leads to a Big Crunch collapse

Conformal diagrams constructed via numerical integration

Global structure dominated by collapse

Abstract

Whereas current cosmological observations suggest that the universe is dominated by a positive cosmological constant ($\Lambda > 0$), the AdS/CFT correspondence tells us that the case $\Lambda<0$ is still worthy of consideration. In this paper we study the McVittie solution with $\Lambda<0$. Following a related study, the solution is understood here by way of a systematic construction of conformal diagrams based on detailed numerical integrations of the null geodesic equations. As in the pure Robertson - Walker case, we find that $\Lambda<0$ ensures collapse to a Big Crunch, a feature which completely dominates the global structure.