Real scalar field, non-relativistic limit, and cosmological expansion

TL;DR

This paper extends a known transformation from relativistic to non-relativistic scalar fields to curved spacetime, applying it to cosmological models and analyzing the resulting non-relativistic dynamics and symmetries.

Contribution

It generalizes the relativistic to non-relativistic scalar field transformation to curved backgrounds and explores its implications in cosmology.

Findings

Transformation determined by a differential equation in curved spacetime

Derived non-relativistic action up to second order in small parameters

Interpreted transformation as a Bogoliubov transformation and discussed symmetries

Abstract

The existing transformation from a relativistic real scalar field to a complex non-relativistic scalar field by Namjoo, Guth, and Kaiser is generalized from Minkowski space to a more general background metric. In that case the transformation is not purely algebraic any more but determined by a differential equation. We apply the generalized transformation to a real scalar with $\phi^4$ interaction on an Friedmann-Lema\^itre-Robertson-Walker cosmologically expanding background and calculate the resulting non-relativistic action up to second order in small parameters. We also show that the transformation can be interpreted as a Bogoliubov transformation between relativistic and non-relativistic creation and annihilation operators and comment on emerging symmetries in the non-relativistic theory.