The dressed-metric approach is shown to violate general covariance by demonstrating that it cannot have an off-shell completion in which the correct infinitesimal relations of space-time hypersurface deformations are realized. The main underlying reason -- a separation of background degrees of freedom and modes of inhomogeneity that is incompatible with covariance -- is shared with other approaches such as hybrid loop quantum cosmology.
The hybrid and the dressed metric formalisms for the study of primordial perturbations in Loop Quantum Cosmology lead to dynamical equations for the modes of these perturbations that are of a generalized harmonic-oscillator type, with a mass that depends on the background but is the same for all modes. For quantum background states that are peaked on trajectories of the effective description of Loop Quantum Cosmology, the main difference between the two considered formalisms is found in the expression of this mass. The value of the mass at the bounce is especially important, since it is only i…
Loop quantum cosmology has recently been applied in order to extend the analysis of primordial perturbations to the Planck era and discuss the possible effects of quantum geometry on the cosmic microwave background. Two approaches to loop quantum cosmology with admissible ultraviolet behavior leading to predictions that are compatible with observations are the so-called hybrid and dressed metric approaches. In spite of their similarities and relations, we show in this work that the effective equations that they provide for the evolution of the tensor and scalar perturbations are somewhat diffe…
The dressed metric and the hybrid approach to perturbations are the two main approaches to capture the effects of quantum geometry in the primordial power spectrum in loop quantum cosmology. Both consider Fock quantized perturbations over a loop quantized background and result in very similar predictions except for the modes which exit the horizon in the effective spacetime in the Planck regime. Understanding precise relationship between both approaches has so far remained obscured due to differences in construction and technical assumptions. We explore this issue at the classical and effectiv…
Loop quantum cosmology tries to capture the main ideas of loop quantum gravity and to apply them to the Universe as a whole. Two main approaches within this framework have been considered to date for the study of cosmological perturbations: the dressed metric approach and the deformed algebra approach. They both have advantages and drawbacks. In this article, we accurately compare their predictions. In particular, we compute the associated primordial tensor power spectra. We show -- numerically and analytically -- that the large scale behavior is similar for both approaches and compatible with…
A general classical theorem is presented according to which all invariant relations among the space time metric scalars, when turned into functions on the Phase Space of full Pure Gravity (using the Canonical Equations of motion), become weakly vanishing functions of the Quadratic and Linear Constraints. The implication of this result is that (formal) Dirac consistency of the Quantum Operator Constraints (annihilating the wave Function) suffices to guarantee space time covariance f the ensuing quantum theory: An ordering for each invariant relation will always exist such that the emanating ope…
In the context of the averaging problem in relativistic cosmology, we provide a key to the interpretation of cosmological parameters by taking into account the actual inhomogeneous geometry of the Universe. We discuss the relation between `bare' cosmological parameters determining the cosmological model, and the parameters interpreted by observers with a ``Friedmannian bias'', which are `dressed' by the smoothed-out geometrical inhomogeneities of the surveyed spatial region.
We investigate the gravitational particle production in the bounce phase of Loop Quantum Cosmology (LQC). We perform both analytical and numerical analysis of the particle production process in a LQC scenario with Bunch-Davies vacuum initial condition in the contracting phase. We obtain that if we extend the validity of the dressed metric approach beyond the limit of small backreaction in which it is well justified, this process would lead to a radiation dominated phase in the pre-inflationary phase of LQC. Our results indicate that the test field approximation, which is required in the trunca…
Identifying an appropriate set of ``observables'' is a nontrivial task for most approaches to quantum gravity. We describe how it may be accomplished in the context of a recently proposed family of stochastic (but classical) dynamical laws for causal sets. The underlying idea should work equally well in the quantum case.
It was recently shown that the volume operator of loop quantum cosmology (LQC) and all its positive powers are ill-defined on physical states. In this paper, we investigate how it effects predictions of cosmic microwave background (CMB) power spectra obtained within dressed metric approach for which expectations values of $\hat{a}$ are the key element. We find that almost every step in the procedure is ill-defined and relies heavily upon a (seemingly premature) numerical truncation. Thus, it suggests that more care is needed in making predictions regarding pre-inflationary physics. We propose …
In this work, we compute the time-dependent masses that govern the dynamics of scalar and tensor perturbations propagating on an effective flat, homogeneous, and isotropic background within the framework of loop quantum cosmology, regularized according to the procedure put forward by Dapor and Liegener. To do so, we follow the two main approaches that, in the field of loop quantum cosmology, lead to hyperbolic equations for the perturbations in the ultraviolet sector: the hybrid and dressed metric formalisms. This allows us to compare the masses resulting from both proposals and analyze their …
Based on the observation that the exterior space-times of Schwarzschild-type solutions allow two symmetric slicings, a static spherically symmetric one and a timelike homogeneous one, modifications of gravitational dynamics suggested by symmetry-reduced models of quantum cosmology can be used to derive corresponding modified spherically symmetric equations. Generally covariant theories are much more restricted in spherical symmetry compared with homogeneous slicings, given by $1+1$-dimensional dilaton models if they are local. As shown here, modifications used in loop quantum cosmology do not …
A cosmological model with two global internal times shows that time reparameterization invariance, and therefore covariance, is not guaranteed by deparameterization. In particular, it is impossible to derive proper-time effective equations from a single deparameterized model if quantum corrections from fluctuations and higher moments are included. The framework of effective constraints shows how proper-time evolution can consistently be defined in quantum cosmological systems, such that it is time reparameterization invariant when compared with other choices of coordinate time. At the same tim…
A large number of models have been analyzed in loop quantum cosmology, using mainly minisuperspace constructions and perturbations. At the same time, general physics principles from effective field theory and covariance have often been ignored. A consistent introduction of these ingredients requires substantial modifications of existing scenarios. As a consequence, none of the broader claims made mainly by the Ashtekar school --- such as the genericness of bounces with astonishingly semiclassical dynamics, robustness with respect to quantization ambiguities, the realization of covariance, and …
This article is devoted to the study of scalar perturbations in loop quantum cosmology. It aims at clarifying the situation with respect to the way initial conditions are set and to the specific choice of an inflaton potential. Several monomial potentials are studied. Both the dressed metric and deformed algebra approaches are considered. We show that the calculation of the ultraviolet part of the spectrum, which is the physically relevant region for most background trajectories, is reliable, whereas the infrared and intermediate parts do depend on some specific choices that are made explicit.
This article reviews some aspects of local covariance and of the ambiguities and anomalies involved in the definition of the stress energy tensor of quantum field theory in curved spacetime. Then, a summary is given of the approach proposed by Buchholz et al. to define local thermal equilibrium states in quantum field theory, i.e., non-equilibrium states to which, locally, one can assign thermal parameters, such as temperature or thermal stress-energy. The extension of that concept to curved spacetime is discussed and some related results are presented. Finally, the recent approach to cosmolog…
In loop quantum cosmology, non-perturbative quantum gravity effects lead to the resolution of the big bang singularity by a quantum bounce without introducing any new degrees of freedom. Though fundamentally discrete, the theory admits a continuum description in terms of an effective Hamiltonian. Here we provide an algorithm to obtain the corresponding effective action, establishing in this way the covariance of the theory for the first time. This result provides new insights on the continuum properties of the discrete structure of quantum geometry and opens new avenues to extract physical pre…
We study cosmological perturbations in the framework of Loop Quantum Cosmology, using a hybrid quantization approach and Mukhanov-Sasaki variables. The formulation in terms of these gauge invariants allows one to clarify the independence of the results on choices of gauge and facilitates the comparison with other approaches proposed to deal with cosmological perturbations in the context of Loop Quantum Theory. A kind of Born-Oppenheimer ansatz is employed to extract the dynamics of the inhomogeneous perturbations, separating them from the degrees of freedom of the Friedmann-Robertson-Walker ge…
We introduce a generalization of the 4-dimensional averaging window function of Gasperini, Marozzi and Veneziano (2010) that may prove useful for a number of applications. The covariant nature of spatial scalar averaging schemes to address the averaging problem in relativistic cosmology is an important property that is implied by construction, but usually remains implicit. We employ here the approach of Gasperini et al. for two reasons. First, the formalism and its generalization presented here are manifestly covariant. Second, the formalism is convenient for disentangling the dependencies on …
The holonomy-flux algebra $\A$ of loop quantum gravity is known to admit a natural representation that is uniquely singled out by the requirement of covariance under spatial diffeomorphisms. In the cosmological context, the requirement of spatial homogeneity naturally reduces $\A$ to a much smaller algebra, $\A_{\rm Red}$, used in loop quantum cosmology. In Bianchi I models, it is shown that the requirement of covariance under \emph{residual} diffeomorphism symmetries again uniquely selects the representation of $\A_{\rm Red}$ that has been commonly used. We discuss the close parallel between …
We investigate the consequences of different regularizations and ambiguities in loop cosmological models on the predictions in the scalar and tensor primordial spectrum of the cosmic microwave background using the dressed metric approach. Three models, standard loop quantum cosmology (LQC), and two modified loop quantum cosmologies (mLQC-I and mLQC-II) arising from different regularizations of the Lorentzian term in the classical Hamiltonian constraint are explored for chaotic inflation in spatially-flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe. In each model, two different treatmen…
Our analysis shows how the covariant chaotic behavior characterizing the evolution of the Mixmaster cosmology near the initial singularity can be taken as the semiclassical limit in the canonical quantization performed by the corresponding Hamiltonian representation.
While loop quantum cosmology (LQC) predicts a robust quantum bounce of the background evolution of a Friedmann-Robertson-Walker (FRW) spacetime prior to the standard slow-roll inflation, whereby the big bang singularity is resolved, there are several different quantization procedures to cosmological perturbations, for instance, {\em the deformed algebra, dressed metric, and hybrid quantizations}. This paper devotes to study the quantum bounce effects of primordial perturbations in the hybrid approach. The main discrepancy of this approach is the effective positive mass at the quantum bounce fo…
The path-integral approach to cosmology consists in the computation of transition amplitudes between states of the quantum geometry of the universe. In the past, the concrete computation of these transitions amplitudes has been performed in a perturbative regime, breaking the full general covariance of the theory. Here I present how it is possible to define a general covariant path integral in quantum cosmology, by relying on the most recent results of the canonical and covariant formulations of Loop Quantum Gravity. I present two strategies that have been implemented. The first starts from th…
This thesis contains an analysis of the problem of time in quantum cosmology and its application to a cosmological minisuperspace model. In the first part, we introduce the problem of time and the theoretical foundations. In the second part, we focus on a specific minisuperspace universe, analyse it classically, and quantise it using the canonical quantisation method. The chosen model is a flat FLRW universe with a free massless scalar field and a perfect fluid. We extract the Wheeler--DeWitt equation, and calculate its solutions. There are three dynamical variables that may be used as clock p…
We provide the power spectrum of small scalar perturbations propagating in an inflationary scenario within loop quantum cosmology. We consider the hybrid quantization approach applied to a Friedmann--Robertson--Walker spacetime with flat spatial sections coupled to a massive scalar field. We study the quantum dynamics of scalar perturbations on an effective background within this hybrid approach. We consider in our study adiabatic states of different orders. For them, we find that the hybrid quantization is in good agreement with the predictions of the dressed metric approach. We also propose …
We consider cosmological perturbations around homogeneous and isotropic spacetimes minimally coupled to a scalar field and present a formulation which is designed to preserve covariance. We truncate the action at quadratic perturbative order and particularize our analysis to flat compact spatial sections and a field potential given by a mass term, although the formalism can be extended to other topologies and potentials. The perturbations are described in terms of Mukhanov-Sasaki gauge invariants, linear perturbative constraints, and variables canonically conjugate to them. This set is complet…
A set of diverse but mutually consistent results obtained in different settings has spawned a new view of loop quantum gravity and its physical implications, based on the interplay of operator calculations and effective theory: Quantum corrections modify, but do not destroy, space-time and the notion of covariance. Potentially observable effects much more promising than those of higher-curvature effective actions result; loop quantum gravity has turned into a falsifiable framework, with interesting ingredients for new cosmic world views. At Planckian densities, space-time disappears and is rep…
We show that the standard Hamiltonian of isotropic loop quantum cosmology is selected by physical criteria plus one choice: that it have a `minimal' number of terms. We also show the freedom, and boundedness of energy density, even when this choice is relaxed. A criterion used is covariance under dilations, the continuous diffeomorphisms remaining in this context, which are not canonical but conformally canonical transformations. We propose how to implement conformally canonical transformations in quantum theory. Removal of the infrared regulator yields independence of ordering ambiguities.
The effective approach in Loop Quantum Cosmology (LQC) has provided means to obtain predictions for observable quantities in LQC models. While an effective dynamics in LQC has been extensively considered in different scenarios, a robust demonstration of the validity of effective descriptions for the perturbative level still requires further attention. The consistency of the description adopted in most approaches requires the assumption of a test field approximation, which is limited to the cases in which the backreaction of the particles gravitationally produced can be safely neglected. Within…
Quantum cosmology has traditionally been studied at the level of symmetry-reduced minisuperspace models, analyzing the behavior of wave functions. However, in the absence of a complete full setting of quantum gravity and detailed knowledge of specific properties of quantum states, it remained difficult to make testable predictions. For quantum cosmology to be part of empirical science, it must allow for a systematic framework in which corrections to well-tested classical equations can be derived, with any ambiguities and ignorance sufficiently parameterized. As in particle and condensed-matter…
In quantum cosmology, one applies quantum physics to the whole universe. While no unique version and no completely well-defined theory is available yet, the framework gives rise to interesting conceptual, mathematical and physical questions. This review presents quantum cosmology in a new picture that tries to incorporate the importance of inhomogeneity: De-emphasizing the traditional minisuperspace view, the dynamics is rather formulated in terms of the interplay of many interacting "microscopic" degrees of freedom that describe the space-time geometry. There is thus a close relationship with…
We propose a new derivation from the full Loop Quantum Gravity (LQG) to the Loop Quantum Cosmology (LQC) improved $\barμ$-scheme effective dynamics, based on the reduced phase space formulation of LQG and a proposal of effective Hamiltonian/action in the full LQG. A key step of our program is an improved regularization of the full LQG Hamiltonian on a cubic lattice. The improved Hamiltonian uses a set of "dressed holonomies" $h_Δ(\mathfrak{s})$ which not only depend on the connection $A$ but also depend on the length of the curve $\mathfrak{s}$. With the improved Hamiltonian, we propose a quan…
A union of matter bounce and Ekpyrotic scenarios is often studied in an attempt to combine the most promising features of these two models. Since non-perturbative quantum geometric effects in loop quantum cosmology (LQC) result in natural bouncing scenarios without any violation of energy conditions or fine tuning, an investigation of matter-Ekpyrotic bounce scenario is interesting to explore in this quantum gravitational setting. In this work, we explore this unified phenomenological model for a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe in LQC filled with dust and a s…
Gravity theory based on current algebra is formulated. The gauge principle rather than the general covariance combined with the equivalence principle plays the pivotal role in the formalism, and the latter principles are derived as a consequence of the theory. In this approach, it turns out that gauging the Poincaré algebra is not appropriate but gauging the $SO(N,M)$ algebra gives a consistent theory. This makes it possible to have Anti-de Sitter and de Sitter space-time by adopting a relation between the spin connection and the tetrad field. The Einstein equation is a part of our basic equat…
We present results concerning propagation of the Gaussian state across the cosmological quantum bounce. The reduced phase space quantization of loop quantum cosmology is applied to the Friedman-Robertson-Walker universe with a free massless scalar field. Evolution of quantum moments of the canonical variables is investigated. The covariance turns out to be a monotonic function so it may be used as an evolution parameter having quantum origin. We show that for the Gaussian state the Universe is least quantum at the bounce. We propose explanation of this counter-intuitive feature using the entro…
Despite its importance in general relativity, a quantum notion of general covariance has not yet been established in quantum gravity and cosmology, where, given the a priori absence of coordinates, it is necessary to replace classical frames with dynamical quantum reference systems. As such, quantum general covariance bears on the ability to consistently switch between the descriptions of the same physics relative to arbitrary choices of quantum reference system. Recently, a systematic approach for such switches has been developed (arXiv:1809.00556, 1809.05093, 1810.04153). It links the descri…
It is believed that a primary principle of the theory of quantum gravity is the Holographic Principle according to which a physical system can be described only by degrees of freedom living on its boundary. The generalized covariant formulation of the principle considers the entropy content on truncated light-sheets: light-like hypersurfaces of non-positive expansion orthogonally generated from a boundary. When we construct the truncated light-sheets for cosmological observers we find a general expression for the minimum cosmic time interval from the apparent horizon to verify the holographic …
Relational path integral, effective actions and quantum frame covariance in gravity
Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo（东京大学宇宙线研究所（WPI） Kavli宇宙物理数学研究所）
In this Letter, we study analytically the evolutions of the flat Friedmann-Lemaitre-Robertson-Walker (FLRW) universe and its linear perturbations in the framework of {\em the dressed metric approach} in loop quantum cosmology (LQC). Assuming that the evolution of the background is dominated by the kinetic energy of the inflaton at the quantum bounce, we find that both evolutions of the background and its perturbations are independent of the inflationary potentials during the pre-inflationary phase. During this period the effective potentials of the perturbations can be well approximated by a P…
In an attempt to generalize general relativity, we propose a new Hermitian theory of gravity. Space-time is generalized to space-time-momentum-energy and both the principles of general covariance and equivalence are extended. The theory is endowed with a Hermitian metric on a complex manifold. The Hermitian metric contains, apart from the symmetric metric, an anti-symmetric part, which describes dynamical torsion. The causality structure is changed in a way such that there is a minimal time for events to be in causal contact and a maximal radius for a non-local instantaneous causally related v…
In this paper, we first provide a brief review of the effective dynamics of two recently well-studied models of modified loop quantum cosmologies (mLQCs), which arise from different regularizations of the Hamiltonian constraint and show the robustness of a generic resolution of the big bang singularity, replaced by a quantum bounce due to non-perturbative Planck scale effects. As in loop quantum cosmology (LQC), in these modified models the slow-roll inflation happens generically. We consider the cosmological perturbations following the dressed and hybrid approaches and clarify some subtle iss…
Fractional calculus is a couple of centuries old, but its development has been less embraced and it was only within the last century that a program of applications for physics started. Regarding quantum physics, it has been only in the previous decade or so that the corresponding literature resulted in a set of defying papers. In such a context, this manuscript constitutes a cordial invitation, whose purpose is simply to suggest, mostly through a heuristic and unpretentious presentation, the extension of fractional quantum mechanics to cosmological settings. Being more specific, we start by ou…
The evolutions of the flat FLRW universe and its linear perturbations are studied systematically in the dressed metric approach of LQC. When it is dominated by the kinetic energy of the inflaton at the quantum bounce, the evolution of the background can be divided into three different phases prior to the preheating, {\em bouncing, transition and slow-roll inflation}. During the bouncing phase, the evolution is independent of not only the initial conditions, but also the inflationary potentials. In particular, the expansion factor can be well described by the same exact solution in all the case…
