Semiclassicality and Decoherence of Cosmological Perturbations

TL;DR

This paper analyzes the quantum-to-classical transition and decoherence of inflationary cosmological perturbations by comparing the Heisenberg and Schrödinger representations. It establishes that both formalisms yield the same prediction: perturbations generated from vacuum fluctuations become equivalent to classical stochastic Gaussian fields in the quasi-isotropic, standing-wave mode, particularly in the limit $|r_k|\to \infty$, where mode functions can be made real. Crucially, decoherence is shown to arise by neglecting the exponentially small decaying mode, effectively achieving "decoherence without decoherence" and leaving present-day perturbations with unchanged rms values despite environmental interactions. The work clarifies the role of squeezing, provides explicit relations between Bogolubov transformations and squeezing parameters, and discusses observational implications for the primordial gravitational-wave background, including a potential but small quantum signature and entropy considerations.

Abstract

Transition to the semiclassical behaviour and the decoherence process for inhomogeneous perturbations generated from the vacuum state during an inflationary stage in the early Universe are considered both in the Heisenberg and the Schrödinger representations to show explicitly that both approaches lead to the same prediction: the equivalence of these quantum perturbations to classical perturbations having stochastic Gaussian amplitudes and belonging to the quasi-isotropic mode. This equivalence and the decoherence are achieved once the exponentially small (in terms of the squeezing parameter $r_k$) decaying mode is neglected. In the quasi-classical limit $|r_k|\to \infty$, the perturbation mode functions can be made real by a time-independent phase rotation, this is shown to be equivalent to a fixed relation between squeezing angle and phase for all modes in the squeezed-state formalism. Though the present state of the gravitational wave background is not a squeezed quantum state in the rigid sense and the squeezing parameters loose their direct meaning due to interaction with the environment and other processes, the standard predictions for the rms values of the perturbations generated during inflation are not affected by these mechanisms (at least, for scales of interest in cosmological applications). This stochastic background still occupies a small part of phase space.