The Hilbert space of de Sitter quantum gravity

TL;DR

This work addresses the space of quantum states for gravity in asymptotically de Sitter spacetimes by solving the Wheeler-DeWitt equation in the large-volume limit. The authors show that physical states take the universal form $\Psi[g,\chi]\simeq e^{iS[g,\chi]}Z[g,\chi]$, where $Z$ is a diffeomorphism-invariant functional with Weyl transformation properties matching those of a CFT partition function, and its coefficient functions obey CFT-like Ward identities. The nongravitational limit recovers Higuchi’s group-averaged basis, while the full theory defines a broader "theory space" of possible $Z$'s, enabling a systematic exploration of cosmological correlators and holographic structure in de Sitter space. A second, complementary basis—the "small fluctuations" basis—renders the Hilbert space normalizable and clarifies how Higuchi’s construction generalizes to finite gravitational coupling. Altogether, the paper provides a bottom-up, symmetry-driven framework for understanding the Hilbert space of de Sitter quantum gravity and its relation to holography and cosmological observables.

Abstract

We obtain solutions of the Wheeler-DeWitt equation with positive cosmological constant for a closed universe in the large-volume limit. We argue that this space of solutions provides a complete basis for the Hilbert space of quantum gravity in an asymptotically de Sitter spacetime. Our solutions take the form of a universal phase factor multiplied by distinct diffeomorphism invariant functionals, with simple Weyl transformation properties, that obey the same Ward identities as a CFT partition function. The Euclidean vacuum corresponds to a specific choice of such a functional but other choices are equally valid. Each functional can be thought of as specifying a "theory" and, in this sense, the space of solutions is like "theory space". We describe another basis for the Hilbert space where all states are represented as excitations of the vacuum that have a specific constrained structure. This gives the finite $G_N$ generalization of the basis proposed by Higuchi in terms of group averaging, which we recover in the nongravitational limit.

Figures (1)

• Figure 1: We are considering a late time slice (in red) with topology $S^d$ in an asymptotically de Sitter spacetime. In the late time expansion, the Wheeler-DeWitt equation can be solved and, up to a universal phase factor, the space of solutions is the space of functionals that transform under diff $\times$ Weyl in the same way as CFT$_d$ partition functions.