Unitary paradox of cosmological perturbations

TL;DR

This paper discusses a paradox in cosmology where the entanglement entropy of superhorizon modes exceeds the Bekenstein-Hawking bound, occurring during inflation or dark energy era depending on certain conditions.

Contribution

It introduces the concept of the unitary paradox of cosmological perturbations and identifies the critical times during inflation and dark energy era when the paradox occurs.

Findings

The paradox occurs during inflation at a specific critical time $t_c$.

If fine-tuned, the paradox shifts to the dark energy era at $t_c'$.

The paradox's occurrence depends on parameters like $eta$, $H_{inf}$, and $N$.

Abstract

If we interpret the Bekenstein-Hawking entropy of the Hubble horizon as thermodynamic entropy, then the entanglement entropy of the superhorizon modes of curvature perturbation entangled with the subhorizon modes will exceed the Bekenstein-Hawking bound at some point; we call this the unitary paradox of cosmological perturbations by analogy with black hole. In order to avoid a fine-tuned problem, the paradox must occur during the inflationary era at the critical time $t_c=\ln(3\sqrt{\pi}/\sqrt{2}\epsilon_HH_{inf})/2H_{inf}$ (in Planck units), where $\epsilon_H= -\dot{H}/H^2$ is the first Hubble slow-roll parameter and $H_{inf}$ is the Hubble rate during inflation. If we instead accept the fine-tuned problem, then the paradox will occur during the dark energy era at the critical time $t_c'=\ln(3\sqrt{\pi}H_{inf}/\sqrt{2}fe^{2N}H_\Lambda^2)/2H_\Lambda$, where $H_\Lambda$ is the Hubble rate dominated by dark energy, $N$ is the total number of e-folds of inflation, and $f$ is a purification factor that takes the range $0<f<3\sqrt{\pi}H_{inf}/\sqrt{2}e^{2N}H_\Lambda^2$.