Microstate counting from defects in de Sitter

TL;DR

This work probes the microscopic origin of de Sitter entropy using a Lorentzian path integral framework in which microstates are labeled by defects such as end-of-the-world branes or thin shells. The authors show that the variance of microstate overlaps is governed by Lorentzian wormhole configurations with conical singularities (crotch geometries), yielding an area-law for the de Sitter entropy and its Schwarzschild–de Sitter generalization, $\,\log\dim\mathcal{H}_{\text{dS}} = \frac{A_c}{4G} + O(\log G)$ and $\log\dim\mathcal{H}_{\text{SdS}} = \frac{A_c + A_b}{4G} + O(\log G)$. They analyze two microscopic consistency conditions—the Null Energy Condition (NEC) and the matching background condition—and find that no configuration in their class satisfies both simultaneously, highlighting tensions between energy conditions and background matching in a UV-complete picture. The results provide a concrete link between de Sitter entropy and the topology of Lorentzian wormholes, suggesting physically meaningful microstate counting that relies on defect-induced geometries near the cosmological horizon. The framework offers a route to UV completions via multiple de Sitter vacua or SdS-like constructions and clarifies the role of observer-dependent causal structure in gravitational microstate counting.

Abstract

We explore the microscopic origin of de Sitter entropy using a Lorentzian path-integral approach. We construct a Hilbert space whose states are associated with configurations of thin shells or end-of-the-world branes, with state overlaps defined by the gravitational path integral. By considering states which are indistinguishable to an observer, we find that the variance of microstate overlaps is dominated by Lorentzian wormhole topologies with conical singularities. Evaluating these overlaps, we recover the expected area law for the entropy, relating the dimension of the de Sitter Hilbert space to the area of the cosmological horizon. Extending this analysis to Schwarzschild-de Sitter spacetime, we show that both the cosmological and black hole horizons contribute to the total entropy. Along the way, we present an explicit construction of the shell and brane configurations and examine their compatibility with relevant consistency conditions, including the null energy condition.

Figures (4)

• Figure 1: Left, a Lorentzian manifold $M$ with boundary $\Sigma$ where the space ends (Scenario I). Right, two Lorentzian manifolds $M_{\text{L}}$ and $M_\text{R}$ joined along a timelike codimension-one defect $\Sigma$ (Scenario II).
• Figure 2: Illustration of the two possible configurations for the defect, with the observer located at the south pole. Left, after introducing the defect, we retain the small portion (shown in grey) of the manifold. We refer to this configuration as the "small" cut, corresponding to $\gamma=1$. Right, after introducing the defect, we retain the large portion of the manifold. We call this type of configuration the "large" cut, which corresponds to $\gamma=-1$.
• Figure 3: Numerical solutions of the equation of motion (\ref{['eq: EofM']}) for the positions of the end-of-the-world brane ($w=-1$) in de Sitter and Schwarzschild–de Sitter spacetimes for various values of the defect energy parameter $c$. Here we set $\ell=1$, and $d=3$. Red and blue curves correspond to collapsing and runaway defects, respectively. The left plot shows brane solutions in de Sitter space where the dotted line indicates $r_\text{c}$, the radius of the cosmological horizon. The right plot shows end-of-the-world brane solutions in Schwarzschild–de Sitter space for $\mathcal{M}=0.25$. The dotted horizontal lines indicate, in order of increasing radius: the black hole horizon $r_{\text{b}}$, the splitting point between collapsing and runaway shells, and $r_{\text{c}}$. For this set of variables, $c=4\pi G \sigma$.
• Figure 4: Numerical solutions for the positions of shells of dust ($w=0$) gluing two Schwarzschild-de Sitter spaces for several values of the energy parameter $c$ in spacetime dimension $d=3$. The dotted lines are in increasing order: $r_{\text{b}}^{\text{R}}$, $r_{\text{b}}^{\text{L}}$, the splitting point, $r_{\text{c}}^{\text{R}}$ and $r_{\text{c}}^{\text{L}}$. The left plot shows solutions that satisfy the NEC. Here, the corresponding mass parameters are $\mathcal{M}_\text{L}=0.3$ and $\mathcal{M}_\text{R}=0.2$. To satisfy the NEC, the cosmological constants of the two sides have to be different: $\ell_\text{L}=1.5$ and $\ell_\text{R}=1$. In the right plot shells in red violate the NEC, while shells in blue satisfy the NEC. Here, the corresponding mass parameters are $\mathcal{M}_\text{L}=0.2$ and $\mathcal{M}_\text{R}=0.3$, and the cosmological constants are identical, $\ell_\text{L}=\ell_\text{R}=1$.