Constructing an entangled state in Heisenberg picture for inflationary cosmology

TL;DR

This paper explores how time-dependent entangled transformations in the Heisenberg picture affect the initial entangled states of inflaton fields in inflationary cosmology, clarifying their role in observable predictions.

Contribution

It introduces a method to construct and analyze initial entangled states in the Heisenberg picture using time-dependent transformations, extending previous Schrödinger picture approaches.

Findings

Time-dependency of the entanglement parameter is essential.

First-order perturbation simplifies the entanglement parameter's time dependence.

Relation between vacuum states with transformed mode functions is established.

Abstract

The effects of the entanglement of the inflaton field which initially entangled with those of another field on observables like power spectrum are known in the context of the Schr\"{o}dinger field theory. To clarify this effect in Heisenberg picture, there were some attempts to construct the initial entangled state by making use of an entangled transformation (like Bogoliubov transformation) between the Bunch-Davies vacuums and squeezed states. We study the role of the time-dependent entangled transformation in the Schr\"{o}dinger field theory. We derive the relation between two vacuum states which their mode functions are transformed by the entangled transformation in Schr\"{o}dinger picture. We discuss that the time-dependency of the entanglement parameter is inevitable and only in the first order of the entanglement parameter perturbation this time-dependency vanishes. We study the entangled transformation in the Heisenberg picture in term of the entangled parameter which appears in the entangled state in Schr\"{o}dinger picture.