Thermodynamic ensembles with cosmological horizons

TL;DR

This work constructs well-defined thermodynamic ensembles for spacetimes with a positive cosmological constant by introducing a York boundary that fixes quasilocal data. Using a spherically reduced phase space and imposing initial-value constraints, the authors derive a reduced action and analyze canonical and microcanonical ensembles, uncovering a rich phase structure including de Sitter, Schwarzschild–de Sitter, and Nariai configurations. They show how the Gibbons–Hawking de Sitter partition function emerges as a limiting case and argue that the cosmological horizon can be thermodynamically stabilized by reservoir dynamics in the canonical setting. The study clarifies foundational issues about the Euclidean path integral, stability of stationary points, and the physical interpretation of horizon entropy, reinforcing the link between gravitational thermodynamics and horizon microstates while highlighting open questions about negative temperatures and holographic descriptions.

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.

Figures (13)

• Figure 1: The function $f(r)$ for different values of the mass parameter. Real solutions for $r'$ (and therefore for the Riemannian spatial metric) exist only where $f(r)\ge0$, and a horizon can only exist where $f(r)=0$.
• Figure 2: The oscillation of radius $r$ in the constant-$\tau$ slice. The 2-spheres are shown by circles. Starting from the boundary at $R$, the radius may initially increase (decrease), reach $r_c$ ($r_b$) and continue past that. A horizon or an RP$^2$ may lie on any neck or waist with radius $r_b$ or $r_c$, respectively. Four possible cases involving the least number of oscillations are indicated.
• Figure 3: The stationary point "thermal de Sitter" as part of the static patch of de Sitter space, shaded on a space diagram (left) and on the Penrose diagram for the corresponding Lorentzian solution (right). The cosmological horizon lies outside the system.
• Figure 4: The black hole configuration. The system is the shaded region that extends from the black hole horizon $r_b$ to the system boundary $R\, \, (\neq \ell/\sqrt{3})$, on a space diagram (left) and a Penrose diagram for the corresponding Lorentzian solution (right). The cosmological horizon lies outside the system.
• Figure 5: The cosmological configurations, when the mass parameter is positive (a) or zero (b). The system is the shaded region, in either the space diagram (left) or the corresponding Lorentzian Penrose diagram (right). For a given temperature, the mass parameter of the stationary point in (a) is different from the one in figure \ref{['b-system']}, so a given SdS space is not both a b-horizon and a c-horizon stationary point.