Second-Order Linear Differential Equations for Effective Barotropic FRW Cosmologies

In 1999, Faraoni wrote a simple second-order linear differential equation for FRW cosmologies with barotropic fluids. His results have been extended by Rosu, who employed techniques belonging to nonrelativistic supersymmetry to obtain time-dependent effective adiabatic indices. Further extensions are presented here using the known connection between the linear second-order differential equations and Dirac-like equations in the same supersymmetric context. These extensions are equivalent to adding an imaginary part to the effective adiabatic index, which is proportional to the mass parameter of the Dirac spinor. The natural physical interpretation of the imaginary part is related to the particular dissipation and instabilities of the effective barotropic FRW hydrodynamics that are introduced by means of this supersymmetric scheme.