On the relationship between the modifications to the Raychaudhuri equation and the canonical Hamiltonian structures

TL;DR

This paper explores how modifications to the Raychaudhuri equation influence the canonical Hamiltonian structures in cosmological models, revealing connections to loop quantum cosmology and singularity avoidance.

Contribution

It introduces a method to derive canonical Hamiltonian structures from modified Raychaudhuri equations without an action, linking modifications to polymerization and singularity behavior.

Findings

Repulsive quadratic modifications lead to polymerized phase space similar to loop quantum cosmology.

Repulsive cubic modifications result in a generalized polymerized phase space.

Both repulsive modifications avoid singularities, unlike attractive ones.

Abstract

The problem of obtaining canonical Hamiltonian structures from the equations of motion, without any knowledge of the action, is studied in the context of the spatially flat Friedmann-Robertson-Walker models. Modifications to Raychaudhuri equation are implemented independently as quadratic and cubic terms of energy density without introducing additional degrees of freedom. Depending on their sign, modifications make gravity repulsive above a curvature scale for matter satisfying strong energy condition, or more attractive than in the classical theory. Canonical structure of the modified theories is determined demanding that the total Hamiltonian be a linear combination of gravity and matter Hamiltonians. In the quadratic repulsive case, the modified canonical phase space of gravity is a polymerized phase space with canonical momentum as inverse trigonometric function of Hubble rate; the canonical Hamiltonian can be identified with the effective Hamiltonian in loop quantum cosmology. The repulsive cubic modification results in a `generalized polymerized' canonical phase space. Both of the repulsive modifications are found to yield singularity avoidance. In contrast, the quadratic and cubic attractive modifications result in a canonical phase space in which canonical momentum is non-trigonometric and singularities persist. Our results hint on connections between repulsive/attractive nature of modifications to gravity arising from gravitational sector and polymerized/non-polymerized gravitational phase space.