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

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 ${\mathcal{I}}^{+}$ of an asymptotically de-Sitter spacetime. The volume (at fixed $\tau$) remains finite in this limit and is given by the universal formula $V(\tau )=\frac{4}{3}\pi (2\ln \cosh \frac{\tau }{2}-{\tanh }^2\frac{\tau }{2})$ 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 ${\mathcal{I}}^{+}$. 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.

13 pages, 2 figures, Latex; references added

References in corpus (3)

• Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School)
• Predicting the Cosmological Constant from the Causal Entropic Principle
• Quasinormal modes of D-dimensional de Sitter spacetime

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