Unitary paradox of cosmological perturbations

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 [Formula: see text] (in Planck units), where [Formula: see text] is the first Hubble slow-roll parameter and [Formula: see text] 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 [Formula: see text], where [Formula: see text] is the Hubble rate dominated by dark energy, [Formula: see text] is the total number of e-folds of inflation and [Formula: see text] is a purification factor that takes the range [Formula: see text].