On the phase of the de Sitter density of states

The one-loop gravitational path integral around Euclidean de Sitter space $S^D$ has a complex phase that casts doubt on a state counting interpretation. Recently, it was proposed to cancel this phase by including an observer. We explore this proposal in the case where the observer is a charged black hole in equilibrium with the de Sitter horizon. We compute the phase of the one-loop determinant within a two-dimensional dilaton gravity reduction, using both numerical and analytical methods. Our results interpolate between previous studies of a probe geodesic observer and the Nariai solution. We also revisit the prescription for going from the Euclidean path integral to the state-counting partition function, finding a positive sign in the final density of states.

💡 Research Summary

The paper addresses a long‑standing puzzle in de Sitter quantum gravity: the one‑loop Euclidean path integral around the round sphere S^D produces a complex phase i^{D+2} instead of a real, positive weight that would be expected for a state‑counting partition function. This phase originates from the conformal modes of the metric: while most of them must be Wick‑rotated to render the functional integral convergent, the ℓ = 0 (overall size) and ℓ = 1 (conformal Killing vectors) modes are not rotated, leading to an overall factor (−i)^{∞‑1‑(D+1)} ≈ i^{D+2}. Maldacena and collaborators suggested that including an “observer” in the path integral can cancel most of these problematic modes, because the observer breaks the conformal Killing symmetry and turns D‑1 of the D+1 vectors into physical unstable modes that must be Wick‑rotated. The resulting phase becomes i^{3}=−i, still not appropriate for a sum over states; an additional factor of i arises from the inverse Laplace transform over the inverse temperature β that enforces the Hamiltonian constraint H=0.

The present work makes this idea concrete by taking the observer to be a charged black hole in thermal equilibrium with the de Sitter horizon. In four‑dimensional Einstein‑Maxwell theory the relevant solutions are magnetically charged Reissner‑Nordström–de Sitter black holes. Their parameter space forms the familiar “shark‑fin” diagram in the (M,Q) plane, with two equilibrium lines: the lukewarm line (M=Q) where the black hole and cosmological horizons have the same temperature, and the charged Nariai curve where the two horizons coincide. Along these lines the Euclidean geometry is smooth (the Euclidean time circle shrinks at both horizons). By dimensional reduction on the spherical symmetry one obtains a two‑dimensional dilaton gravity model \