The cosmological constant as a boundary term

TL;DR

This paper compares different formulations of unimodular gravity and general relativity, focusing on how the cosmological constant emerges and behaves as a boundary term, affecting the quantum wave functions and their equations.

Contribution

It clarifies the relationship between boundary conditions, the cosmological constant, and quantum wave functions in various unimodular gravity models.

Findings

Unimodular gravity with a fixed volume element has a time variable conjugate to the cosmological constant.

Wave functions in this model satisfy a Schrödinger equation, unlike in general relativity.

In the covariant model, wave functions are time-independent and satisfy a Wheeler-DeWitt equation.

Abstract

We compare the path integral for transition functions in unimodular gravity and in general relativity. In unimodular gravity the cosmological constant is a property of states that are specified at the boundaries whereas in general relativity the cosmological constant is a parameter of the action. Unimodular gravity with a nondynamical background spacetime volume element has a time variable that is canonically conjugate to the cosmological constant. Wave functions depend on time and satisfy a Schr\"odinger equation. On the contrary, in the covariant version of unimodular gravity with a 3-form gauge field, proposed by Henneaux and Teitelboim, wave functions are time independent and satisfy a Wheeler-DeWitt equation, as in general relativity. The 3-form gauge field integrated over spacelike hypersurfaces becomes a "cosmic time" only in the semiclassical approximation. In unimodular gravity the smallness of the observed cosmological constant has to be explained as a property of the initial state.