Microscopic origin of the entropy of de Sitter spacetime

TL;DR

This work addresses the microscopic origin of de Sitter entropy by constructing an infinite family of semiclassical dS microstates realized as backreacted geometries with a Dirichlet wall and a constant-tension thin-shell brane. It leverages nonperturbative wormhole contributions in the semiclassical Euclidean gravitational path integral to compute overlaps between microstates, and shows that the resulting microcanonical counting yields a Hilbert-space dimension of $e^{\mathbf{S}_+ + \mathbf{S}_-}$, where $\mathbf{S}_\pm$ are the horizon entropies of the two static patches. The key result is a state-counting derivation of the Gibbons-Hawking entropy for dS spacetime, linking bulk wormhole physics to the microscopic degrees of freedom. The findings offer a framework for understanding cosmological entropy in quantum gravity, suggest holographic interpretations and future directions, including connections to interface CFTs and possible quantum corrections to dS entropy.

Abstract

We construct an infinite family of semiclassical de Sitter (dS) microstates, realized as backreacted geometries of dS spacetime with a constant tension thin-shell brane located outside the dS event horizon. We further show that wormhole contributions to the semiclassical Euclidean gravitational path integral lead to universal nonperturbative overlaps between these microstates. By evaluating the nonperturbative overlaps, we count the dimension of the Hilbert space spanned by the semiclassical dS microstates and find that it precisely equals the exponential of the Gibbons-Hawking entropy of dS spacetime. Our construction thus provides a state-counting derivation for Gibbons-Hawking entropy of dS spacetime.

Figures (6)

• Figure 1: Left: Euclidean dS$_{d+1}$ saddle point geometry with a Dirichlet wall at $r = r_c$. Right: Penrose diagram of the corresponding Lorentzian spacetime obtained by Wick rotation. Each point in the figures represents a $\left(d-1\right)$-dimensional sphere of radius $r$.
• Figure 2: Penrose diagram of two dS$_{d+1}$ spacetimes glued along a dS$_d$ thin-shell brane.
• Figure 3: A plot of $\frac{w_{\pm}^*}{l_{\pm}}$ as a function of $T$ is shown for different values of $l_+$, where without loss of generality, we have set $l_{-}=1$.
• Figure 4: Left: The Euclidean geometry of the dS$_{d+1}$ solution with both a Dirichlet wall and a thin-shell brane. Right: The Euclidean gravitational path integral in the lower half of the Euclidean dS geometry defines a semiclassical dS microstate on the time-reflection symmetric surface.
• Figure 5: Left: The 4-boundary wormhole geometry is constructed by filling the Dirichlet walls $\bar{\Sigma}_{4,\pm,c}\equiv \bar{\Sigma}_{\pm,c}^{i_1^{\mathrm{bra}}i_2^\mathrm{ket}}\cup \bar{\Sigma}_{\pm,c}^{i_2^{\mathrm{bra}}i_3^\mathrm{ket}}\cup \bar{\Sigma}_{\pm,c}^{i_3^{\mathrm{bra}}i_4^\mathrm{ket}}\cup \bar{\Sigma}_{\pm,c}^{i_4^{\mathrm{bra}}i_1^\mathrm{ket}}$ with two Euclidean pure dS geometries $\bar{M}_{4,\pm,c}$, which are glued together along four thin-shell branes $Q_k$ ($k=1,\dots,4$) connecting $i_k^{\mathrm{bra}}$ to $i_k^{\mathrm{ket}}$. Right: The fully connected 4-boundary wormhole that contributes to the fourth moment of overlaps between large tension semiclassical dS microstates is pinched into two nearly complete Euclidean pure dS geometries $M_{4,\pm,c}$.