Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 013, 9 pages      arXiv:1012.1920      https://doi.org/10.3842/SIGMA.2011.013
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Shifted Riccati Procedure: Application to Conformal Barotropic FRW Cosmologies

Haret C. Rosu a and Kira V. Khmelnytskaya b
a) IPICyT, Instituto Potosino de Investigacion Cientifica y Tecnologica, Apdo Postal 3-74 Tangamanga, 78231 San Luis Potosí, Mexico
b) Universidad Autónoma de Querétaro, Centro Universitario, Cerro de las Campanas s/n C.P. 76010 Santiago de Querétaro, Qro. Mexico

Received November 30, 2010, in final form January 28, 2011; Published online February 02, 2011

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.

Key words: factorization; shifted Riccati procedure; barotropic FRW cosmologies; cosmological zero-modes.

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