The structure of cosmic time

TL;DR

This paper investigates the mathematical structure of cosmic time in models with a cosmological constant, revealing a torus-based integral structure and a natural periodicity in imaginary time, with implications for fundamental physics.

Contribution

It demonstrates that cosmic time generally corresponds to an abelian integral on a torus and explores its implications for understanding the universe's fundamental properties.

Findings

Time is given by an abelian integral on a torus in most models.

Imaginary time periodicity suggests a fundamental mass or temperature scale.

The structure of time may be an order parameter from early universe phase transitions.

Abstract

Following the approach of Julien Lesgourgues [astro-ph/0409426], we analyze the mathematical structure of the time co-ordinate of present day cosmological models, where these models include a cosmological constant term to account for the observed acceleration of the universe: we find that in all cases, except for a set of measure zero in the parameter space, the time is given by an (abelian) integral on a torus; the imaginary period of this integral then gives a natural periodicity in imaginary time for the universe; following Stephen Hawking, this periodicity may be interpreted either as giving a fundamental mass scale for the universe, or (using Planck's constant) a fundamental temperature, or both. The precise structure that emerges suggests that the structure of time can be regarded as an order parameter arising perhaps in a phase transition in the early universe; one might hope that this structure would be predictable in any fundamental theory.