The geometry of large causal diamonds and the no-hair property of asymptotically de Sitter space-times

In a previous paper we obtained formulae for the volume of a causal diamond or Alexandrov open set I+(p) boolean AND I-(q) whose duration tau(p, q) is short compared with the curvature scale. In the present Letter we obtain asymptotic formulae valid when the point q recedes to the future boundary I+ of an asymptotically de Sitter space-time. The volume (at fixed tau) remains finite in this limit and is given by the universal formula V(T) = 4/3 pi(2lncosh tau/2 tanh(2) tau/2) plus corrections (given by a series in e(-t)q) which begin at order e(-4t)q. The coefficients of the corrections depend on the geometry of T+. This behaviour is shown to be consistent with the no-hair property of cosmological event horizons and with calculations of de Sitter quasi-normal modes in the literature. (c) 2007 Elsevier B.V. All rights reserved.