The Geometry of Large Causal Diamonds and the No Hair Property of Asymptotically de-Sitter Spacetimes

TL;DR

This paper derives asymptotic volume formulas for large causal diamonds in asymptotically de-Sitter spacetimes, revealing universal behavior linked to the no-hair property and horizon geometry.

Contribution

It provides new asymptotic volume formulas for causal diamonds approaching the future boundary in de-Sitter space, connecting geometry with horizon properties.

Findings

Volume remains finite as points approach ${ m I}^+$

Universal volume formula involving hyperbolic functions

Corrections depend on the geometry of ${ m I}^+$

Abstract

In a previous paper we obtained formulae for the volume of a causal diamond or Alexandrov open set $I^+(p) \cap I^-(q)$ whose duration $\tau(p,q) $ is short compared with the curvature scale. In the present paper we obtain asymptotic formulae valid when the point $q$ recedes to the future boundary ${\cal I}^+$ of an asymptotically de-Sitter spacetime. The volume (at fixed $\tau$) remains finite in this limit and is given by the universal formula $V(\tau) = {4\over 3}\pi (2\ln \cosh{\tau\over 2}-\tanh^2{\tau\over 2})$ plus corrections (given by a series in $e^{-t_q}$) which begin at order $e^{-4t_q}$. The coefficents of the corrections depend on the geometry of ${\cal I}^+$. This behaviour is shown to be consistent with the no-hair property of cosmological event horizons and with calculations of de-Sitter quasinormal modes in the literature.