Critical Robertson–Walker universes

The integral of the mass-energy density m of a closed Robertson–Walker (RW) spacetime with source a perfect fluid and cosmological constant Λ 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 (m-critical RW universes). A complete classification of m-critical RW universes 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 m-critical RW universes are discussed.