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.

arXiv comment13 pages, 2 figures, Latex; references added

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• Initial Conditions for Bubble Universes
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• A Scale-Dependent Distance Functional between Past Light Cones in Cosmology
• The volume of causal diamonds, asymptotically de Sitter space-times and irreversibility
• Energy Density Bounds in Cubic Quasi-Topological Cosmology
• Inhomogeneous post-inflationary Lambda-CDM cosmology as a moduli space expansion
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