On the phase of the de Sitter density of states

TL;DR

This work extends the de Sitter phase problem by modeling an observer as a charged black hole in thermal equilibrium with the de Sitter horizon and analyzing the phase of the one-loop determinant via a two-dimensional dilaton gravity reduction. Numerically (and analytically in limiting cases) it shows the phase matches previous probe- and Nariai-limit results, with lukewarm solutions yielding a phase of -i and unstable Nariai branches yielding a real, positive contribution in certain regimes. By reordering the state-counting integral to impose the Hamiltonian constraint prior to the energy integral, the authors argue that the density of states is real and positive, reinforcing the viability of a quantum-mechanical description of the de Sitter horizon when an observer is present. The analysis thus supports Maldacena's proposal, while clarifying the sign structure and highlighting regimes (e.g., ultracold Nariai) where strong fluctuations require careful treatment and further nonperturbative understanding.

Abstract

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.

Figures (3)

• Figure 1: Left: The shark fin diagram of de Sitter Reissner Nordström black holes. Physical black hole solutions only exist in the shaded region of the $M-Q$ plane. Equilibrium black holes with smooth Euclidean geometries are along the lukewarm line and the Nariai curve. Along the dashed curve are solutions $\mathbb{H}^2\times S^2$. Right: Our results, computed below, for the phase of the corresponding dilaton gravity partition functions.
• Figure 2: For a dilaton potential $U(\phi)$ with a given $Q$, there are two solutions we can consider. On the upper right corner, we have the lukewarm solution where the dilaton $\phi$ varies from $\phi_b$ at the black hole horizon to $\phi_c$ at the cosmic horizon, where $\phi_b, \phi_c$ satisfy $\int_{\phi_b}^{\phi_c} U = 0$ and $U(\phi_c) =- U(\phi_b)$. We also have the charged Nariai solution with a constant dilaton $\phi_0$ at which the potential vanishes, i.e., $U(\phi_0) =0$.
• Figure 3: The spectrum of fluctuations (within the dilaton gravity reduction) for the charged Nariai background with various values of $Q$.