Dapor-Liegener formalism of loop quantum cosmology for Bianchi I spacetimes

Abstract

We discuss the quantization of vacuum Bianchi I spacetimes in the modified formalism of loop quantum cosmology recently proposed by Dapor and Liegener. This modification is based on a regularization procedure where both the Euclidean and Lorentzian terms of the Hamiltonian are treated independently. Whereas the Euclidean part has already been dealt with in the literature for Bianchi I spacetimes, the Lorentzian one has not yet been represented quantum mechanically. After a brief review of the quantum kinematics and the quantization of the Euclidean sector, we represent the Lorentzian part of the Hamiltonian constraint by an operator according to the factor ordering rules of the Martín-Benito-Mena Marugán-Olmedo prescription. We study the general properties of this quantum operator and the superselection rules derived therefrom, resulting in an action similar to that of the Euclidean operator except that the orientation of the densitized triad is not preserved, a fact which leads to a generic enlargement of the superselection sectors. We conclude with an explanation of the mechanism that prevents this enlargement in the isotropic case and a comment on the effect of alternative prescriptions for the implementation of the improved dynamics.

Symmetry restriction and its application to gravity

“…Together with the fiducial connection A a oI this allows to be lifted to the discrete setting by giving rise to the fiducial fluxes P I o (e) = P I (e)| Eo,Ao . We define the volume of the spatial manifold for said fiducial fluxes: (379) A similar version to this regularisation had already been under active investigation in the earlier papers [58,59], albeit using instead of µ o a slight modification, the μ-scheme which can not be derived via our restriction method. In our setting, for any observables we are interested in all Poisson brackets can be computed on the symmetry restricted level.…”

Primordial perturbations in the Dapor-Liegener model of hybrid loop quantum cosmology

“…This decoupling of the singular state, together with the fact that positive and negative volumes are not connected by the repeated action of the constraint, leads to the Hilbert subspaces spanned by the eigenstates with positive or negative volumes being left invariant. Furthermore, the action of the constraint superselects for the FLRW geometry Hilbert subspaces H ± ε with support on discrete semilattices of step four {±(ε + 4n), n ∈ N} [43,44], that have a strictly positive minimum ε or a strictly negative maximum −ε for the volume. For the sake of definiteness, from now on we restrict our discussion to H + ε with a fixed ε ∈ (0, 4].…”

“…This alternative for the construction of the Hamiltonian of LQC has been considered in several papers in the last two years [36][37][38][39][40][41][42][43]. The extension to anisotropic universes of the Bianchi type I has also been studied [44]. An especially appealing property of the new Hamiltonian constraint obtained in this manner for flat FLRW cosmologies is that the effective solutions typically present two branches with different cosmological behavior: one of them corresponding to an asymptotically de Sitter cosmology, even in the absence of a genuine cosmological constant, and the other one describing an FLRW universe.…”

The time-dependent mass of cosmological perturbations in loop quantum cosmology: Dapor-Liegener regularization

“…This objective has inspired a number of studies (see, for instance, Refs. [47][48][49][50][51][52][53][54]), the aim of which was to explore the features of the model, how they compare to the standard ones, and the implementability of the procedure in more complicated scenarios, among others. It has been found that, although the big bang singularity is still resolved by a bouncing mechanism, the resulting bounce is quantitatively and qualitatively different from the standard one.…”

“…The Dapor-Liegener regularization scheme has been implemented in isotropic and anisotropic scenarios [53]. The introduction of perturbations within the dressed metric approach has already been considered in Ref.…”