De Sitter entropy: on-shell versus off-shell

TL;DR

This work shows that de Sitter entropy can be understood both as a bulk (on-shell) volume contribution and as a boundary (off-shell) horizon contribution. For pure gravity, the on-shell bulk entropy in de Sitter space reproduces the Wald/area entropy $S_A=\frac{A}{4}$ of the Killing horizon, generalizing to $f(R)$ gravity as $S_V^{\text{on shell}}(\text{gravity})=\frac{A}{4}f'(R)$. Extending to matter, the renormalized entanglement entropy of a non-minimally coupled scalar field satisfies $S_V^{\text{on shell}}(\text{matter})=S_A^{\text{off shell}}(\text{matter})$, with logarithmic corrections in the conformal case matching the integrated conformal anomaly. The authors carefully renormalize the conical singularity contributions and address subtleties in the energy definitions, showing that, at the Gibbons-Hawking temperature, bulk and boundary entropies coincide. The results illuminate the dual bulk/boundary perspectives on generalized de Sitter entropy and clarify when and how the area law emerges in quantum-corrected settings.

Abstract

Attributing thermodynamic properties to the Bunch-Davies state in static patch of de Sitter space and setting the corresponding equations of state, we demonstrate that, for pure gravity, the bulk entropy computed on-shell as a volume integral in de Sitter space coincides with the Wald entropy (area law) in any spacetime dimension and for any theory of f(R) gravity. We extend this result to the renormalized entanglement entropy of a non-minimally coupled scalar field. From the on-shell perspective, entropy emerges as a bulk contribution, whereas from the off-shell viewpoint, it manifests as a boundary (horizon) contribution. As a result, in de Sitter space, generalized entropy can be understood in two distinct ways: either as a bulk or as a boundary contribution.