Shifted Riccati Procedure: Application to Conformal Barotropic FRW Cosmologies

TL;DR

This paper introduces a shifted Riccati procedure within a supersymmetric framework to generate and analyze cyclic, singular barotropic Stephani cosmologies with PT symmetry, expanding the understanding of inhomogeneous universe models.

Contribution

It develops a novel shifted Riccati method applied to supersymmetric cosmologies, producing new cyclic Stephani solutions with periodic singularities and PT symmetry.

Findings

Derived explicit solutions for shifted Riccati cosmologies.

Identified the shift parameter as purely imaginary for physical acceptability.

Established PT symmetry in the curvature indices of the solutions.

Abstract

In the case of barotropic FRW cosmologies, the Hubble parameter in conformal time is the solution of a simple Riccati equation of constant coefficients. We consider these cosmologies in the framework of nonrelativistic supersymmetry that has been effective in the area of supersymmetric quantum mechanics. Recalling that Faraoni [Amer. J. Phys. 67 (1999), 732-734] showed how to reduce the barotropic FRW system of differential equations to simple harmonic oscillator differential equations, we set the latter equations in the supersymmetric approach and divide their solutions into two classes of 'bosonic' (nonsingular) and 'fermionic' (singular) cosmological zero-mode solutions. The fermionic equations can be considered as representing cosmologies of Stephani type, i.e., inhomogeneous and curvature-changing in the conformal time. We next apply the so-called shifted Riccati procedure by introducing a constant additive parameter, denoted by S, in the common Riccati solution of these supersymmetric partner cosmologies. This leads to barotropic Stephani cosmologies with periodic singularities in their spatial curvature indices that we call U and V cosmologies, the first being of bosonic type and the latter of fermionic type. We solve completely these cyclic singular cosmologies at the level of their zero modes showing that an acceptable shift parameter should be purely imaginary, which in turn introduces a parity-time (PT) property of the partner curvature indices.