Self Similar Solutions of the Evolution Equation of a Scalar Field in an Expanding Geometry

TL;DR

This paper derives a time-independent functional Schrödinger equation for a self-interacting scalar field in an expanding universe, using a scale transformation, and explores solutions in the mean field approximation extending previous free field results.

Contribution

It introduces a novel scale transformation approach to obtain a time-independent Hamiltonian for interacting scalar fields in expanding geometries and generalizes known free field solutions.

Findings

Derived a time-independent Schrödinger equation with a time-odd term.

Extended Bunch-Davies solutions to interacting fields.

Analyzed mean field approximation in expanding spacetime.

Abstract

We consider the functional Schrodinger equation for a self interacting scalar field in an expanding geometry. By performing a time dependent scale transformation on the argument of the field we derive a functional Schrodinger equation whose hamiltonian is time independent but involves a time-odd term associated to a constraint on the expansion current. We study the mean field approximation to this equation and generalize in this case, for interacting fields, the solutions worked out by Bunch and Davies for free fields.