Integration of the Friedmann equation for universes of arbitrary complexity

TL;DR

This paper develops an algorithmic method to construct constants of motion for complex FLRW cosmological models and explores the existence of a monotonic gravitational epoch function across different models.

Contribution

It introduces a systematic way to derive constants of motion for FLRW models with multiple species and analyzes the conditions for a universal epoch function.

Findings

Constants of motion are constructible for arbitrary non-interacting species.

A unique gravitational epoch function exists for models with positive cosmological constant.

The epoch function is absent in most models with non-positive cosmological constant.

Abstract

An explicit and complete set of constants of the motion are constructed algorithmically for Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) models consisting of an arbitrary number of non-interacting species. The inheritance of constants of the motion from simpler models as more species are added is stressed. It is then argued that all FLRW models admit what amounts to a unique candidate for a gravitational epoch function (a dimensionless scalar invariant derivable from the Riemann tensor without differentiation which is monotone throughout the evolution of the universe). The same relations that lead to the construction of constants of the motion allow an explicit evaluation of this function. In the simplest of all models, the $\Lambda$CDM model, it is shown that the epoch function exists for all models with $\Lambda > 0$, but for almost no models with $\Lambda \leq 0$.