Unravelling cosmological perturbations

Timothy J Hollowood

doi:10.1016/j.physletb.2018.08.073

Abstract

We explain in detail the quantum-to-classical transition for the cosmological perturbations using only the standard rules of quantum mechanics: the Schrödinger equation and Born's rule applied to a subsystem. We show that the conditioned, i.e. intrinsic, pure state of the perturbations, is driven by the interactions with a generic environment, to become increasingly localized in field space as a mode exists the horizon during inflation. With a favourable coupling to the environment, the conditioned state of the perturbations becomes highly localized in field space due to the expansion of spacetime by a factor of roughly exp⁡(−cΔN), where ΔN∼50 and c is a model dependent number of order 1. Effectively the state rapidly becomes specified completely by a point in phase space and an effective, classical, stochastic process emerges described by a classical Langevin equation. The statistics of the stochastic process is described by the solution of the master equation that describes the perturbations coupled to the environment.

Highlights

It is breathtaking that quantum fluctuations [1,2,3,4,5,6] in the inflating universe [7,8,9,10,11,12] became the seeds of the structure in the universe and were imprinted as small fluctuations of the CMB
The formalism decides whether the quantum-to-classical transition happens: does the state |ψ becomes localized in phase space? The goal of this paper is to show that the conditioned state of the perturbations does become classical driven by interaction with the environment and the inflationary expansion
An important role is played by observers, or more precisely frames of reference associated to subsystems

Summary

Introduction

It is breathtaking that quantum fluctuations [1,2,3,4,5,6] in the inflating universe [7,8,9,10,11,12] became the seeds of the structure in the universe and were imprinted as small fluctuations of the CMB. If one wants to describe how a single perturbation becomes classical, we need a description of the trajectory of the individual mode, in other words the state from the frame of the reference of the mode itself This is the conditioned state constructed via the Born rule to satisfy (1.2). Fokker-Planck equation master equation Born’s rule: Born unravelling conditioned state quantum stochastic process localization of state point in phase space. The conditioned state |ψ (the state from the frame of reference of the subsystem) satisfies a particular unravelling of this master equation which takes the form of deterministic evolution with a non-linear, non-Hermitian, Hamiltonian, interspersed with stochastic jumps into orthogonal states (arising from applying the Born rule to each coherent interaction of the system with the environment). The perceived problem is that the state, spread out in the field direction, is still a pure state, so how can |ψ(ν, τ )|2 be interpreted as a probability distribution in field space even though this is phenomenologically the right thing to do?

Born unravelling

Unravelling the perturbations

Discussion