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)∩I−(q) whose duration τ(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 τ) remains finite in this limit and is given by the universal formula V(τ)=43π(2lncoshτ2−tanh2τ2) plus corrections (given by a series in e−tq) which begin at order e−4tq. The coefficients of the corrections depend on the geometry of I+. 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.