A Grassmann representation of the Hubble parameter

TL;DR

This paper extends the Riccati equation for the Hubble parameter in cosmology to include odd Grassmannian time variables, resulting in a superfield formulation with explicit solutions, revealing novel aspects of cosmological evolution.

Contribution

It introduces a Grassmannian superfield approach to the Hubble parameter, generalizing classical equations to include supersymmetric components and providing explicit solutions.

Findings

Derived a system of differential equations for supercomponents of the Hubble superfield.

Explicitly solved the superfield equations.

Identified non-evolutionary behavior of the second even Hubble component.

Abstract

The Riccati equation for the Hubble parameter H of barotropic FRW cosmologies in conformal time for \kappa \neq 0 spatial geometries and in comoving time for the \kappa =0 geometry, respectively, is generalized to odd Grassmannian time parameters. We obtain a system of simple differential equations for the four supercomponents (two of even type and two of odd type) of the Hubble superfield function {\cal H} that is explicitly solved. The second even Hubble component does not have an evolution governed by general relativity although there are effects of the latter upon it