Unitary paradox of cosmological perturbations

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