Thermodynamic ensembles with cosmological horizons

TL;DR

This paper develops a rigorous statistical framework for de Sitter horizons by introducing a boundary in gravitational path integrals, analyzing thermodynamic phases, and connecting to the Gibbons-Hawking partition function.

Contribution

It introduces a boundary-based approach to define thermodynamic ensembles for de Sitter space, clarifying their statistical foundations and stability properties.

Findings

Formulation of path integrals over constrained phase space

Analysis of thermodynamic stability and phases

Derivation of Gibbons-Hawking partition function as a limit

Abstract

The entropy of a de Sitter horizon was derived long ago by Gibbons and Hawking via a gravitational partition function. Since there is no boundary at which to define the temperature or energy of the ensemble, the statistical foundation of their approach has remained obscure. To place the statistical ensemble on a firm footing we introduce an artificial "York boundary", with either canonical or microcanonical boundary conditions, as has been done previously for black hole ensembles. The partition function and the density of states are expressed as integrals over paths in the constrained, spherically reduced phase space of pure 3+1 dimensional gravity with a positive cosmological constant. Issues related to the domain and contour of integration are analyzed, and the adopted choices for those are justified as far as possible. The canonical ensemble includes a patch of spacetime without horizon, as well as configurations containing a black hole or a cosmological horizon. We study thermodynamic phases and (in)stability, and discuss an evolving reservoir model that can stabilize the cosmological horizon in the canonical ensemble. Finally, we explain how the Gibbons-Hawking partition function on the 4-sphere can be derived as a limit of well-defined thermodynamic ensembles and, from this viewpoint, why it computes the dimension of the Hilbert space of states within a cosmological horizon.