Real-time observables in de Sitter thermodynamics

TL;DR

The paper analyzes real-time finite-temperature correlators for free scalar fields in the de Sitter static patch across dimensions, revealing that when the inverse temperature β is a rational multiple of the de Sitter temperature, certain Matsubara poles cancel in the symmetric Wightman function, with complete cancellations at β = β_dS = 2π. It shows that Lorentzian correlators at β_dS can be obtained by analytic continuation from the round S^{d+1} correlator, and it establishes a direct dynamical link between the Harish-Chandra character χ(t) for SO(1,d+1) and the integrated spectral function via χ(t) = 2i d/dt ilde{tr} G^C(t). The work also analyzes exceptional non-positive masses, including the massless case where static patch quantization can be consistently defined and connected to a unique S^{d+1} correlator, and discusses extensions to spinning fields, broader backgrounds, and interacting theories. These results provide a concrete bridge between group-theoretic data and observable static-patch dynamics, with potential implications for microscopic models of de Sitter horizons and thermal behavior in curved spacetimes.

Abstract

We study real-time finite-temperature correlators for free scalars of any mass in a $dS_{d+1}$ static patch in any dimension. We show that whenever the inverse temperature is a rational multiple of the inverse de Sitter temperature, certain Matsubara poles of the symmetric Wightman function disappear. At the de Sitter temperature, we explicitly show how the Lorentzian thermal correlators can all be obtained by analytic continuations from the round $S^{d+1}$. We establish the precise relation between the Harish-Chandra character for $SO(1,d+1)$ and the integrated spectral function, providing a novel dynamical perspective on the former and enabling generalizations. Furthermore, we study scalars with exceptional non-positive masses. We provide a physical picture for the distinctive structures of their characters. For the massless case, we perform a consistent static patch quantization, and find the unique $S^{d+1}$ correlator that analytically continues to the correlators in the quantum theory.

Figures (8)

• Figure 2.1: $\left|\mathcal{G}^C_{l=1}(\omega)\right|$ for a massive scalar in $dS_4$ in complex $\omega$-plane. The white spots mark the poles.
• Figure 2.2: $\left|\mathcal{G}^{\rm Sym}_{l=1}(\beta,\omega)\right|$ for a principal-series scalar with $\Delta = \frac{3}{2} + 0.7 i$ in $dS_4$ in complex $\omega$-plane. The white spots mark the poles.
• Figure 3.1: Left: Penrose diagram for a static patch. The yellow dot indicates the bifurcation surface at $r=\ell$. Right: Upon the Wick rotation $t \to -i\tau$, $\tau\sim\tau+2\pi$, the static patch becomes the sphere, with $\tau$ parametrizing the thermal circle.
• Figure 4.1: Left: The Penrose diagram for a static patch, with the black arrows indicating incoming and outgoing waves. Right: The scattering potential \ref{['eq:potential']} for a principal-series scalar with $\Delta=\frac{3}{2}+0.1 i$ and $l=3$ in $dS_4$, which is hardly distinguishable from the Rindler potential (red) in \ref{['appeq:rindlerscatt']} as $x\to \infty$. In these diagrams, the brown lines indicate a brick wall regulator tHooft:1984kcu at $x=R=2.5$.
• Figure 4.2: Spectral densities for a principal scalar with $\Delta = \frac{3}{2} + 0.7 i$ in $dS_4$ in complex $\omega$-plane. The white spots mark the poles.