# The Shape of Bouncing Universes

## TL;DR

This paper explores the evolution of closed, oscillating universes in general relativity, examining how entropy, anisotropy, and a cosmological constant influence their long-term behavior and ultimate fate.

## Contribution

It analyzes the cyclic evolution of the most general closed anisotropic universes, including effects of entropy increase and dark energy, extending classical models to modern cosmological scenarios.

## Key findings

- Cycles grow in size and age with entropy increase.
- Cycles become more anisotropic at their maxima.
- A positive cosmological constant leads to everlasting expansion.

## Abstract

What happens to the most general closed oscillating universes in general relativity? We sketch the development of interest in cyclic universes from the early work of Friedmann and Tolman to modern variations introduced by the presence of a cosmological constant. Then we show what happens in the cyclic evolution of the most general closed anisotropic universes provided by the Mixmaster universe. We show that in the presence of entropy increase its cycles grow in size and age, increasingly approaching flatness. But these cycles also grow increasingly anisotropic at their expansion maxima. If there is a positive cosmological constant, or dark energy, present then these oscillations always end and the last cycle evolves from an anisotropic inflexion point towards a de Sitter future of everlasting expansion.