Semiclassicality and Decoherence of Cosmological Perturbations

TL;DR

This paper demonstrates that quantum cosmological perturbations generated during inflation become effectively classical through decoherence, with both Heisenberg and Schrödinger formalisms leading to the same classical stochastic Gaussian predictions.

Contribution

It explicitly shows the equivalence of Heisenberg and Schrödinger approaches in describing decoherence and classicalization of inflationary perturbations, highlighting the role of the squeezing parameter.

Findings

Quantum perturbations become classical with stochastic Gaussian amplitudes.

Decoherence occurs once the decaying mode is neglected.

Standard inflationary predictions remain valid despite environmental interactions.

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 Schr\"odinger 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.