Features of the Partition Function of a $\Lambda>0$ Universe

TL;DR

This paper analyzes the gravitational path integral in a positive cosmological constant universe, exploring contributions from various geometries, including Einstein and non-Einstein instantons, with detailed one-loop calculations on complex projective space.

Contribution

It provides a detailed one-loop analysis of the gravitational path integral including non-Einstein geometries, especially on $ ext{CP}^2$, expanding understanding of quantum gravity in de Sitter space.

Findings

Dominance of four-sphere saddle at leading order

Identification of non-Einstein instantons in Einstein-Maxwell theory

Exact one-loop results on $ ext{CP}^2$ for various fields

Abstract

We consider properties of the gravitational path integral, ${Z}_{\text{grav}}$, of a four-dimensional gravitational effective field theory with $\Lambda>0$ at the quantum level. To leading order, ${Z}_{\text{grav}}$ is dominated by a four-sphere saddle subject to small fluctuations. Beyond this, ${Z}_{\text{grav}}$ receives contributions from additional geometries that may include Einstein metrics of positive curvature. We discuss how a general positive curvature Einstein metric contributes to ${Z}_{\text{grav}}$ at one-loop level. Along the way, we discuss Einstein-Maxwell theory with $\Lambda>0$, and identify an interesting class of closed non-Einstein gravitational instantons. We provide a detailed study for the specific case of $\mathbb{C}P^2$ which is distinguished as the saddle with second largest volume and positive definite tensor eigenspectrum. We present exact one-loop results for scalar particles, Maxwell theory, and Einstein gravity about the Fubini-Study metric on $\mathbb{C}P^2$.