The volume of causal diamonds, asymptotically de Sitter space-times and irreversibility

TL;DR

This paper proves that the volume of causal diamonds in asymptotically de Sitter space-times increases monotonically over cosmological time, reflecting irreversibility in the universe's evolution with a positive cosmological constant.

Contribution

It establishes the monotonic increase of causal diamond volume in asymptotically de Sitter space-times and links this to the irreversibility of cosmological evolution.

Findings

Volume of causal diamonds increases monotonically over time.

Asymptotic volume matches that of maximally symmetric de Sitter space.

Volume flow vanishes only in special constant curvature cases.

Abstract

In this note we prove that the volume of a causal diamond associated with an inertial observer in asymptotically de Sitter 4-dimensional space-time is monotonically increasing function of cosmological time. The asymptotic value of the volume is that of in maximally symmetric de Sitter space-time. The monotonic property of the volume is checked in two cases: in vacuum and in the presence of a massless scalar field. In vacuum, the volume flow (with respect to cosmological time) asymptotically vanishes if and only if future space-like infinity is 3-manifold of constant curvature. The volume flow thus represents irreversibility of asymptotic evolution in spacetimes with positive cosmological constant.