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

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\'{\i}n-Benito, Mena Marug\'an, 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-Lema\^itre-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.Comment: 15 pages, published in Physical Review

Efectos de la cosmología cuántica de lazos en perturbaciones primordiales y en espaciotiempos de agujero negro

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Física Teórica. Fecha de Lectura: 10-03-202

Space of solutions of the Ashtekar-Olmedo-Singh effective black hole model

We consider a general choice of integration constants in the resolution of the dynamical equations derived from a recently proposed effective model that describes black hole spacetimes in the context of loop quantum cosmology. The interest of our analysis is twofold. On the one hand, it allows for a study of the entire space of solutions of the model, which is absent in the literature and is fundamental for understanding the relation with any underlying quantum theory. On the other hand, choices of integration constants that generalize the type of solutions considered so far may lead to exotic behaviors in the effective black hole geometry, as well as modified thermodynamical properties. With these motivations in mind, we discuss the interior and exterior geometries, and present the conditions that a satisfactory matching at the horizons imposes. Then, we turn our attention to the Hawking temperature associated with the black horizon of the model, which we find to be affected by the freedom of choice of integration constants. Finally, we briefly comment on the asymptotic structure of the general solution and compare different notions of mass.Comment: 11 pages, accepted for publication in Physical Review

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

In this work, we extend the formalism of hybrid loop quantum cosmology for primordial perturbations around a flat, homogeneous, and isotropic universe to the new treatment of Friedmann-Lema\^itre-Robertson-Walker geometries proposed recently by Dapor and Liegener, based on an alternative regularization of the Hamiltonian constraint. In fact, our discussion is applicable also to other possible regularization schemes for loop quantum cosmology, although we specialize our analysis to the Dapor-Liegener proposal and construct explicitly all involved quantum operators for that case.Comment: 17 page

Two-time alternative to the Ashtekar-Olmedo-Singh black hole interior

21 pags., 5 figs. -- PACS numbers: 98.80.Qc, 04.70.Dy, 04.60.Ds, 04.60.−m.We investigate the viability of a recently proposed generalization of the Ashtekar-Olmedo-Singh spacetime for the effective description of the interior region of a Schwarzschild black hole within the framework of loop quantum cosmology. The approach is based on a choice of polymerization parameters that is more general than the ones previously considered in the literature and that results in the natural appearance of two times to describe the solutions. If one is interested in examining the physics derived from this model, it is fundamental to ensure that one can attain a well-defined effective geometry in the whole region under consideration, in particular as regards the redundancy of the two times, which one needs to express in terms of a single time coordinate. In order to determine whether this requirement is met, we analyze the definition of these times and their relation. We show that one can reach an acceptable interior spacetime geometry by exploiting the freedom to define the origins of the two times independently.This work has been supported by Project. No. MICINN PID2020-118159GBC41. The project that gave rise to these results received the support of a fellowship from “la Caixa” Foundation (ID 20100010434). The fellowship code is LCF/BQ/DR19/11740028

The MMO prescription for the Dapor-Liegener model

Conferencia invitada. -- LOOPS'19, University Park campus of Penn State (USA), June 17 - 21, 2019

Exploring Alternatives to the Hamiltonian Calculation of the Ashtekar-Olmedo-Singh Black Hole Solution

10 pags.In this article, we reexamine the derivation of the dynamical equations of the Ashtekar-Olmedo-Singh black hole model in order to determine whether it is possible to construct a Hamiltonian formalism where the parameters that regulate the introduction of quantum geometry effects are treated as true constants of motion. After arguing that these parameters should capture contributions from two distinct sectors of the phase space that had been considered independent in previous analyses in the literature, we proceed to obtain the corresponding equations of motion and analyze the consequences of this more general choice. We restrict our discussion exclusively to these dynamical issues. We also investigate whether the proposed procedure can be reconciled with the results of Ashtekar, Olmedo, and Singh, at least in some appropriate limit.This work has been supported by Project. No. MICINN FIS 2017–86497-C2-2-P from Spain (with extension Project. No. MICINN PID 2020-118159GB-C41 under evaluation). The project that gave rise to these results received the support of a fellowship from “la Caixa” Foundation (ID 100010434). The fellowship code is LCF/BQ/DR19/11740028. Partial funds for open access publication have been received from CSI

Loop quantum dynamics of the primordial Universe

XI CPAN, Oviedo, 21-23 octubre 201

Revisiting the Hamiltonian Formalism of the Ashtekar-Olmedo-Singh Black Hole Model

17 pags. -- The Sixteenth Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic Field TheoriesProceedings of the MG16 Meeting on General Relativity Online; 5–10 July 2021This work has been supported by Project. No. MICINN FIS2017-86497-C2-2-P and Project.No. MICINN PID2020-118159GB-C41 from Spain. The projects that gave rise tothese results received the support of a fellowship from “la Caixa” Foundation (ID100010434). The fellowship code is LCF/BQ/DR19/11740028.Peer reviewe

Hamiltonian formulation and loop quantization of a recent extension of the Kruskal spacetime

18 pags.We study the Hamiltonian formulation of the Ashtekar-Olmedo-Singh model for the description of the interior geometry of nonrotating, uncharged black holes. This model incorporates loop quantum effects through the introduction of two regularization parameters. We consider an extended phase space formalism proposed by the creators of the model that includes such parameters as configuration variables, constrained to be functions of the black hole mass. We generalize this restriction, allowing for an off shell phase space dependence. We then introduce a gauge fixing procedure and reduce the system, proving that the reduced symplectic structure cannot reproduce the standard relativistic one in terms of the densitized triad and the Ashtekar-Barbero connection. Actually, the reduced structure precisely compensates the modifications that arise in the Hamilton equations when the regularization parameters are treated as phase space functions, rather than as numbers, attaining a consistent Hamiltonian derivation of the dynamics. We then choose the extended phase space formalism as starting point to address the loop quantization of the model. Taking the definition of certain geometric operators as the only basic ingredient and adopting prescriptions that have proven successful in loop quantum cosmology, we construct a polymer representation of all the constraints and deduce the formal expression of the physical states, assuming reasonable spectral properties for the constraint operators. The physical states turn out to be characterized by a wave function of the black hole mass with support on a very specific set. We finally discuss conditions that guarantee the existence of physical states in the region of large black hole masses. This is a first step in the development of a new loop quantum theory of black holes.This work was partially supported by Project No. MICINN PID2020-118159GB-C41 from Spain. B.E.N. acknowledges financial support from the Standard Program of JSPS Postdoctoral Fellowships for Research in Japan. A. G.-Q. acknowledges that the project that gave rise to these results received the support of a fellowship from “La Caixa” Foundation (ID 100010434). The fellowship code is LCF/BQ/DR19/11740028