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# Unitary paradox of cosmological perturbations

Dec 2022

If we interpret the Bekenstein-Hawking entropy of the Hubble horizon asthermodynamic entropy, then the entanglement entropy of the superhorizon modesof curvature perturbation entangled with the subhorizon modes will exceed theBekenstein-Hawking bound at some point; we call this the unitary paradox ofcosmological perturbations by analogy with black hole. In order to avoid afine-tuned problem, the paradox must occur during the inflationary era at thecritical time $t_c = \ln(3\sqrt{\pi}/\sqrt{2} \epsilon_H H_{inf})/2H_{inf}$ (inPlanck units), where $\epsilon_H$ is the first Hubble slow-roll parameter and$H_{inf}$ is the Hubble rate during inflation. If we instead accept thefine-tuned problem, then the paradox will occur during the darkenergy-dominated 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 number ofe-folds, and $f$ is a purification factor that takes the range$0<f<3\sqrt{\pi}H_{inf}/\sqrt{2}e^{2N}H_\Lambda^2$.

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