Researchers have addressed a question analogous to the black hole information paradox, but applied to cosmological horizons: when does an individual Hawking pair begin to carry information out of a de Sitter horizon? This study, employing two-dimensional flow geometries that smoothly interpolate between an asymptotic AdS₂ boundary and a dS₂ static patch, models the emission of a Hawking pair via a probe state constructed from local operators and their modular conjugates.

To achieve this, the scientists promoted the centaur algebra of observables to a Type II∞ factor through the crossed-product construction. This allowed them to compute the entropy difference between a thermofield-double reference state and the Hawking-pair state. The results show that this difference traces a characteristic "mini-Page curve" for the cosmological horizon: it starts near zero, reaches a minimum near τ ≈ β/8, and then increases again. The location of this minimum is interpreted as the time at which quantum information begins to escape the cosmological horizon.

Extending the analysis to the microcanonical ensemble, it was shown 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 λ = 2π/β. This finding identifies the scrambling time as the scale at which the information carried by the pair becomes accessible to a static-patch observer. This work represents a significant advance in understanding how quantum information behaves in extreme cosmological environments.