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

Understand

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 ${\cal I}^+$.