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arXiv · 2607.08737

The mini-Page Curve in Cosmology

Ricardo Espíndola·Shoichiro Miyashita
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Abstract

The black hole information paradox has motivated extensive study of how and when information escapes from evaporating black holes. Here we address the analogous question for cosmological horizons: when does an individual Hawking pair begin to carry information out of a de Sitter horizon? We study this problem in a class of two-dimensional flow geometries that interpolate smoothly between an asymptotic AdS$ _2 $ boundary and a dS$ _2 $ static patch. Modeling the emission of a Hawking pair via a probe state constructed from local operators and their modular conjugates, we promote the centaur algebra of observables to a Type II$ _\infty $ factor through the crossed-product construction. This allows us to compute the entropy difference between the thermofield-double reference state and the Hawking-pair state. We find that this difference traces a characteristic mini-Page curve for the cosmological horizon: it starts near zero, reaches a minimum near $ \tau \approx \beta/8 $ before increasing again. We interpret the location of the minimum as the time at which quantum information begins to escape the cosmological horizon. Extending the analysis to the microcanonical ensemble, we show that the algebraic entropy coincides with the generalized entropy of an entanglement wedge cut that tracks the emitted particle along the horizon. Furthermore, the relative modular flow generated between the two states yields a Lyapunov exponent $\lambda =2\pi/\beta$, identifying the scrambling time as the scale at which the information carried by the pair becomes accessible to a static-patch observer.

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Ricardo Espíndola, Shoichiro Miyashita. 2026-07-09. The mini-Page Curve in Cosmology. https://arxiv.org/abs/2607.08737

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