Comments on the de Sitter Double Cone

TL;DR

The work extends the Saad–Shenker–Stanford double cone to the static patch of de Sitter space by incorporating two clocks in opposite static patches. It shows that, once the de Sitter isometries are properly gauged and the clock degrees of freedom are included, the resulting Lorentzian partition function exhibits a linear ramp in the spectral form factor, signaling random matrix statistics for the de Sitter horizon. This ramp arises from the relative clock time between the two observers and is sourced by a nonperturbative measure factor that cancels the one-loop decay of the observer sector, allowing late-time unitarity and a bulk trace interpretation. The analysis combines Brownian motion, coinvariant Hilbert space construction, and double-cone geometry to argue for a holographic-like interpretation of de Sitter horizons and their spectral rigidity, while highlighting open questions about signs, symmetry extensions, and the microscopic origin of the double-cone sector. Overall, the paper provides a concrete mechanism by which clocked observers induce random-matrix–style ramp behavior in de Sitter space, pointing toward a dynamical, ensemble-like structure of horizon microstates.

Abstract

We study the double cone geometry proposed by Saad, Shenker, and Stanford in de Sitter space. We demonstrate that with the inclusion of static patch observers, the double cone leads to a linear ramp consistent with random matrix behavior. This ramp arises from the relative time shift between two clocks located in opposite static patches.

Figures (3)

• Figure 1: A de Sitter double cone with the presence of two clocks.
• Figure 2: The gauge fixing condition for the two observers in Euclidean sphere (a) and de Sitter space (b), the only degree of freedom left is the relative time shift ($t$) between the two observers.
• Figure 3: The double cone geometry is obtained by taking a finite quotient of the static de Sitter spacetime.