  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 $0<f<3\sqrt{\pi}H_{inf}/\sqrt{2}e^{2N}H_\Lambda^2$.
