Cosmological-Horizon Entropy and Microstates

Cosmological-horizon entropy is a robust semiclassical thermodynamic quantity, but the state space it counts is unknown. A Euclidean saddle, an observer-algebra entropy, entanglement across a horizon, and the logarithm of microscopic degeneracy are not interchangeable definitions. This page fixes four-dimensional Einstein gravity with positive cosmological constant $\Lambda=3/L^2$, one static observer, and the Gibbons–Hawking ensemble before comparing proposed microscopic interpretations. No complete state count for four-dimensional de Sitter space is established as of 2026-08-20; every such identification remains proposal-level.

Required background. Gauge Theory and String-Theory Preparation provides ensembles, constrained states, and saddle-point methods.

Helpful background. Quantum Gravity frames microscopic entropy; Quantum Information distinguishes von Neumann, algebraic, and coarse-grained entropy; Causality and Global Structure supplies cosmological horizons; and Semiclassical Gravity supplies Euclidean thermodynamics.

Pure de Sitter thermodynamics

The four-dimensional static metric has $f(r)=1-r^2/L^2$. Its horizon lies at $r=L$, with

$$A_c=4\pi L^2,\qquad T_c=\frac{1}{2\pi L},\qquad S_c=\frac{A_c}{4G}=\frac{\pi L^2}{G}.$$

The temperature follows from Euclidean regularity, and the entropy follows from the on-shell gravitational action or horizon first law. Gibbons and Hawking 1977, pp. 2738–2745 established this semiclassical thermodynamic interpretation. Nothing in that calculation exhibits individual states, a protected index, or a Hilbert-space basis.

The observer dependence matters. Different geodesic observers have different horizons, although symmetry gives the same area in empty de Sitter. Entanglement language additionally requires a factorization or algebra and an ultraviolet regulator. In gravity, constraints obstruct a naïve tensor product between the two sides of a horizon.

Schwarzschild–de Sitter and the sign of the first law

For a static mass $M$,

$$f(r)=1-\frac{2GM}{r}-\frac{r^2}{L^2}.$$

For $0\lt M\lt L/(3\sqrt3G)$, the positive roots $r_b\lt r_c$ are the black-hole and cosmological horizons, and $f\gt0$ between them. Their temperatures with respect to the displayed static coordinate time, and their entropies, are

$$T_i=\frac{|f'(r_i)|}{4\pi},\qquad S_i=\frac{\pi r_i^2}{G}.$$

At fixed $L$, varying $f(r_c)=0$ gives

$$\frac{dS_c}{dM}=\frac{4\pi}{f'(r_c)}=-\frac{1}{T_c}, \qquad dM=-T_c\,dS_c.$$

Here $M$ is the metric mass parameter with physical mass units, and the normalization of $\partial_t$ is the one displayed in $f$; there is no asymptotically flat infinity selecting a unit Killing vector. A detector at fixed radius measures each redshifted temperature $T_i/\sqrt{f(r)}$. Adding positive mass shrinks the cosmological horizon. This sign is not the black-hole first law with labels changed; it records which energy and horizon are held fixed. Generic Schwarzschild–de Sitter has unequal horizon temperatures, so a single smooth Euclidean thermal ensemble for both horizons does not exist.

At the Nariai endpoint,

$$r_b=r_c=\frac{L}{\sqrt3},\qquad M_N=\frac{L}{3\sqrt3\,G}.$$

The roots merge, and static coordinates degenerate. A rescaled near-horizon limit yields $\mathrm{dS}_2\times S^2$, so reading the limit solely from the unrescaled surface gravity is misleading. Any microscopic comparison must declare whether it holds fixed the original static time, a near-horizon time, or a Euclidean ensemble.

Results and limitations through 2026-08-20

The formulation axes are observer, selected horizon, charges, ensemble, Euclidean saddle, subtraction, algebra, regulator, and whether “count” means degeneracy, index, entanglement, or generalized entropy.

Evidence or position	Controlled result	Limitation or adverse discriminator	Date
Semiclassical horizon entropy, Gibbons and Hawking 1977, pp. 2738–2751	Euclidean gravity gives $S=A/(4G)$ and thermal response for a cosmological horizon	The calculation supplies thermodynamics but no explicit microscopic degeneracy	1977
Static-patch observer algebra, Chandrasekaran, Longo, Penington, and Witten 2023, §§3–4, pp. 19–34	A type-II$_1$ algebra relates state entropy, up to a state-independent additive constant, to semiclassical generalized entropy	An infinite-dimensional factor is not a derivation of a finite Hilbert space of dimension $e^{S_c}$	2023
Doubled-SYK proposal, Narovlansky and Verlinde 2023, §§1.1, 2, and 4, pp. 5–19	A constrained double-scaled model matches selected entropy differences and observer two-point functions in a three-dimensional de Sitter interpretation	The total microscopic entropy, observer-accessible entropy, and geometric entropy do not coincide automatically; the higher-dimensional dictionary is absent	2023
Finite-state or complementarity interpretation	The finiteness of $A/(4G)$ motivates roughly $e^{S_c}$ states for one patch	Recurrences, nested observers, gravitational dressing, and the identity of the counted sector remain unresolved	2026 cutoff

Competing positions interpret $S_c$ as a microscopic state count, horizon entanglement, an observer-complementarity bound, or an entropy of a gravitational observable algebra. The established result is semiclassical thermodynamics. A literal degeneracy and its state space remain conjectural and model-dependent.

What would qualify as a microstate count

A convincing count must first specify an exact quantum system and a sector selected by de Sitter charges or observer data. It must then calculate a degeneracy $d$ or density $\rho$ whose logarithm has leading term $A/(4G)$, with the same ensemble and corrections as the gravitational saddle. A protected index would not suffice unless cancellations are controlled and the relation between index and absolute degeneracy is proved.

It must also reproduce variations. For Schwarzschild–de Sitter, the count should explain why inserting energy reduces $S_c$, how black-hole and cosmological contributions combine, and what becomes of states in the Nariai limit. Matching one leading number after choosing an additive entropy constant is weaker than matching the spectrum, relative entropies, and finite corrections.

Algebraic entropy addresses a different but precise question: how states restricted to an observer algebra differ. It may be the correct microscopic language, but that conclusion needs a nonperturbative algebra and ultraviolet completion. The controlled semiclassical construction does not settle whether an exact finite number of states exists.

What could change these conclusions

The conclusions could change if a protected count is shown to equal an absolute degeneracy, if an independent microscopic calculation reproduces the de Sitter and Schwarzschild–de Sitter variations including the Nariai limit, or if a no-go theorem excludes a proposed state space. The conclusions also depend on whether a corrected first law, new observer-algebra theorem, or controlled model changes the ensemble or additive entropy convention.

Exercises

Derive $dM=-T_c\,dS_c$ from $f(r_c)=0$ and compare pure de Sitter with the Nariai endpoint. Why does the calculation not count microstates?

Solution

Differentiating $f(r_c,M)=0$ at fixed $L$ gives $f'(r_c)\,dr_c-2G\,dM/r_c=0$. Since $dS_c=2\pi r_c\,dr_c/G$ and $f'(r_c)\lt0$, one obtains $dS_c/dM=4\pi/f'(r_c)=-1/T_c$. Pure de Sitter has $r_c=L$ and $S_c=\pi L^2/G$, whereas each coincident Nariai horizon has radius $L/\sqrt3$. This checks the thermodynamic variation and limiting geometry because it uses only field equations and horizon regularity. It does not identify states, an algebra, or a degeneracy, so it cannot decide which microscopic quantity the entropy counts.

References

• Chandrasekaran, Venkatesa, Roberto Longo, Geoff Penington, and Edward Witten. “An Algebra of Observables for de Sitter Space.” Journal of High Energy Physics 02 (2023) 082. arXiv:2206.10780.
• Gibbons, G. W., and S. W. Hawking. “Cosmological Event Horizons, Thermodynamics, and Particle Creation.” Physical Review D 15 (1977) 2738–2751. DOI: 10.1103/PhysRevD.15.2738.
• Narovlansky, Vladimir, and Herman Verlinde. “Double-Scaled SYK and de Sitter Holography.” Preprint (2023). arXiv:2310.16994.