The Martin-Benito-Mena Marugan-Olmedo prescription for the Dapor-Liegener model of Loop Quantum Cosmology

Abstract

Recently, an alternative Hamiltonian constraint for Loop Quantum Cosmology has been put forward by Dapor and Liegener, inspired by previous work on regularization due to Thiemann. Here, we quantize this Hamiltonian following a prescription for cosmology proposed by Mart\'ın-Benito, Mena Marugán, and Olmedo. To this effect, we first regularize the Euclidean and Lorentzian parts of the Hamiltonian constraint separately in the case of a Bianchi I cosmology. This allows us to identify a natural symmetrization of the Hamiltonian which is apparent in anisotropic scenarios. Preserving this symmetrization in isotropic regimes, we then determine the Hamiltonian constraint corresponding to a Friedmann-Lemaître-Robertson-Walker cosmology, which we proceed to quantize. We compute the action of this Hamiltonian operator in the volume eigenbasis and show that it takes the form of a fourth-order difference equation, unlike in standard Loop Quantum Cosmology, where it is known to be of second order. We investigate the superselection sectors of our constraint operator, proving that they are semilattices supported only on either the positive or the negative semiaxis, depending on the triad orientation. Remarkably, the decoupling between semiaxes allows us to write a closed expression for the generalized eigenfunctions of the geometric part of the constraint. This expression is totally determined by the values at the two points of the semilattice that are closest to the origin, namely the two contributions with smallest eigenvolume. This is in clear contrast with the situation found for the standard Hamiltonian of Loop Quantum Cosmology, where only the smallest value is free. This result indicates that the degeneracy of the new geometric Hamiltonian operator is equal to two, doubling the possible number of solutions with respect to the conventional quantization considered until now.

15 pages, published in Physical Review D

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• The time-dependent mass of cosmological perturbations in loop quantum cosmology: Dapor-Liegener regularization
• Dapor-Liegener formalism of loop quantum cosmology for Bianchi I spacetimes
• The volume operator in loop quantum cosmology
• Loop Quantum Cosmology: Physics of Singularity Resolution and its Implications
• Symmetry restriction and its application to gravity
• Effective LQC model for k=+1 isotropic cosmologies from spatial discretisations
• Primordial perturbations in kinetically dominated regimes of general relativity and hybrid quantum cosmology
• Quantization ambiguities and the robustness of effective descriptions of primordial perturbations in hybrid Loop Quantum Cosmology
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