Power-law expansion cosmology in Schr\"odinger-type formulation

TL;DR

This paper explores a Schr"odinger-type formulation of cosmology for FRLW universes with scalar fields and fluids, extending to phantom fields, and derives related potentials and wave functions under power-law expansion assumptions.

Contribution

It introduces a Schr"odinger-type framework for cosmology including phantom fields and derives explicit expressions for potentials and wave functions.

Findings

Schr"odinger wave function is generally non-normalizable.

Scalar field potential as a function of time is obtained.

Formulation includes extensions to phantom field cases.

Abstract

We investigate non-linear Schr\"{o}dinger-type formulation of cosmology of which our cosmological system is a general relativistic FRLW universe containing canonical scalar field under arbitrary potential and a barotropic fluid with arbitrary spatial curvatures. We extend the formulation to include phantom field case and we have found that Schr\"{o}dinger wave function in this formulation is generally non-normalizable. Assuming power-law expansion, $a \sim t^q$, we obtain scalar field potential as function of time. The corresponding quantities in Schr\"{o}dinger-type formulation such as Schr\"{o}dinger total energy, Schr\"{o}dinger potential and wave function are also presented.