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 tc = ln(3 root pi/root v 2 epsilon H-H(inf))/2Hinf (in Planck units), where epsilon H = - H/H-2 is the first Hubble slow-roll parameter and Hinf 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) = In(3 root pi H-inf/root 2fe(2N) H-Delta(2)) where H Delta 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 root pi H-inf/v 2(e2N)H(Delta)(2).