EMP and linear Schrodinger models for a conformally Bianchi I cosmology

TL;DR

This paper reformulates a conformal Bianchi I cosmological model using Einstein equations as a generalized Ermakov-Milne-Pinney equation and constructs a linear Schrödinger formulation, revealing new insights into the model's structure.

Contribution

It introduces a novel linear Schrödinger formulation for a conformal Bianchi I cosmology based on a generalized Ermakov-Milne-Pinney equation, extending previous classical-quantum correspondences.

Findings

Einstein equations are equivalent to a generalized EMP equation.

Constructed a Schrödinger formulation for the cosmological model.

Identified an extra term when including additional energy-momentum components.

Abstract

Using a conformal version of the Bianchi I metric and a perfect fluid energy-momentum tensor, we show that the resulting Einstein field equations are equivalent to a generalized Ermakov-Milne-Pinney equation. Using a transformation analogous to the known correspondence between the classical EMP and a linear Schr\"odinger equation, we then construct the linear Schr\"odinger formulation for this cosmological model. Also, a note is included to show the extra term that arises when the energy-momentum tensor is taken to include a second term in addition to the scalar field.