# Critical Robertson-Walker universes

## TL;DR

This paper classifies critical closed Robertson-Walker universes by analyzing an action functional derived from the energy density, providing explicit solutions using elliptic functions and discussing their physical energy conditions.

## Contribution

It offers a complete classification of critical RW spacetimes under volume-preserving variations, with explicit solutions in elliptic functions and analysis of energy conditions.

## Key findings

- Explicit solutions in terms of Weierstrass elliptic functions.
- Complete classification of critical RW spacetimes.
- Discussion of energy conditions and cyclic properties.

## Abstract

The integral of the energy density function $\mathfrak m$ of a closed Robertson-Walker (RW) spacetime with source a perfect fluid and cosmological constant $\Lambda$ gives rise to an action functional on the space of scale functions of RW spacetime metrics. This paper studies closed RW spacetimes which are critical for this functional, subject to volume-preserving variations (critical RW spacetimes). A complete classification of critical RW spacetimes is given and explicit solutions in terms of Weierstrass elliptic functions and their degenerate forms are computed. The standard energy conditions (weak, dominant, and strong) as well as the cyclic property of critical RW spacetimes are discussed.