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

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.

22 pages with two new plots. Discussion on uniqueness added, and possible links with existing models expanded. Periodicity for 'generalized polymerized' theory and its comparison with standard polymerization discussed. References added. To appear in CQG

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• Quantum Nature of the Big Bang: Improved dynamics
• Corrections to the Friedmann Equations from LQG for a Universe with a Free Scalar Field
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• A geometric perspective on singularity resolution and uniqueness in loop quantum cosmology
• Jacobi's Last Multiplier and Lagrangians for Multidimensional Systems
• Lagrangian formulation for Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations

Cited by in corpus (15)

• Towards Cosmological Dynamics from Loop Quantum Gravity
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• Loop Quantum Cosmology Gravitational Baryogenesis
• Emergent universe in theories with natural UV cutoffs
• Generic absence of strong singularities and geodesic completeness in modified loop quantum cosmologies
• Implications of quantum ambiguities in k=1 loop quantum cosmology: distinct quantum turnarounds and the super-Planckian regime
• Loop Quantum Cosmology Corrected Gauss-Bonnet Singular Cosmology
• Inflation and Bounce from Classical and Loop Quantum Cosmology Imperfect Fluids
• A note on the AdS/CFT correspondence and the nature of spacetime in quantum gravity
• Spin networks and the big bang singularity avoidance in the AdS/CFT correspondence
• Reconstructing the Universe Evolution from Loop Quantum Cosmology Scalar Fields
• Effective Actions for Loop Quantum Cosmology in Fourth-Order Gravity
• Cosmological bounce and the cosmological constant problem
• Cosmologies in $f(R,{\mathcal{L}}_m)$ theory with non-minimal coupling between geometry and matter
• 2+1 dimensional loop quantum cosmology of Bianchi I models