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It is expected that the quantum nature of spacetime leaves its imprint in all semiclassical gravitational systems, at least in certain regimes, including gravitational waves. In this paper we investigate such imprints on gravitational waves within a specific framework: space is assumed to be discrete (in the form of a regular cubic lattice), and this discrete geometry is quantised following Dirac's canonical quantisation scheme. The semiclassical behavior is then extracted by promoting the expectation value of the Hamiltonian operator on a semiclassical state to an effective Hamiltonian. Considering a family of semiclassical states representing small tensor perturbations to Minkowski background, we derive a quantum-corrected effective wave equation. The deviations from the classical gravitational wave equation are found to be encoded in a modified dispersion relation and controlled by the discreteness parameter of the underlying lattice. For finite discretisations, several interesting effects appear: we investigate the thermodynamical properties of these modified gravitons and, under certain assumptions, derive the tensor power spectrum of the cosmic microwave background. The latter is found to deviate from the classical prediction, in that an amplification of UV modes takes place. We discuss under what circumstances such effect can be in agreement with observations., Comment: 9 pages, 3 figures

We reexamine a set of existing procedures aimed at recovering the effective description of the dynamics of LQG in the context of cosmological solutions. In particular, the studies of those methods, to which the choice of cuboidal graphs and graph-preserving Hamiltonian is central, result in the formulation of a set of no-go statements, severely limiting the possibility of recovering a physically consistent effective dynamics this way., Comment: 9 pages

In the loop quantum gravity context, there have been numerous proposals to quantize the reduced phase space of a black hole, and develop a classical effective description for its interior which eventually resolves the singularity. However, little progress has been made towards understanding the relation between such quantum/effective minisuperspace models and what would be the spherically symmetric sector of loop quantum gravity. In particular, it is not clear whether one can extract the phenomenological predictions obtained in minisuperspace models, such as the singularity resolution and the spacetime continuation beyond the singularity, based on results in full loop quantum gravity. In this paper, we present an attempt in this direction in the context of Kantowski-Sachs spacetime, through the proposal of two new effective Hamiltonians for the reduced classical model. The first is derived using Thiemann classical identities for the regularized expressions, while the second is obtained as a first approximation of the expectation value of a Hamiltonian operator in loop quantum gravity in a semi-classical state peaked on the Kantowski-Sachs initial data. We then proceed with a detailed analysis of the dynamics they generate and compare them with the Hamiltonian derived in General Relativity and the common effective Hamiltonian proposed in earlier literature., Comment: 13 pages, 2 figures; revised version accepted for publication in Phys. Rev. D

We present a detailed analysis of a quantum model for Loop Quantum Cosmology based on strict application of the Thiemann regularization algorithm for the Hamiltonian in Loop Quantum Gravity, extending the results presented previously in our brief report. This construction leads to a qualitative modification of the bounce paradigm. Quantum gravity effects still lead to a quantum bounce connecting deterministically large classical Universes. However, the evolution features a large epoch of de Sitter Universe, with emergent cosmological constant of Planckian order, smoothly transiting into a spatially flat expanding Universe. Moreover, we present an effective Hamiltonian describing the quantum evolution to high accuracy and for which the dynamics can be solved analytically., Comment: 51 pages, 15 figures

The quantum nature of the Big Bang is reexamined in the framework of Loop Quantum Cosmology. The strict application of a regularization procedure to the Hamiltonian, originally developed for the Hamiltonian in loop quantum gravity, leads to a qualitative modification of the bounce paradigm. Quantum gravity effects still lead to a quantum bounce connecting deterministically large classical Universes. However, the evolution features a large epoch of de Sitter Universe, with emergent cosmological constant of Planckian order, smoothly transiting into a flat expanding Universe., Comment: 6 pages, 2 figures, revised version accepted for publication in Phys.Rev.Lett., with the addition of fig.2 and subsequent comments

This is the first paper of a series dedicated to LQG coherent states and cosmology. The concept is based on the effective dynamics program of Loop Quantum Cosmology, where the classical dynamics generated by the expectation value of the Hamiltonian on semiclassical states is found to be in agreement with the quantum evolution of such states. We ask the question of whether this expectation value agrees with the one obtained in the full theory. The answer is in the negative. This series of papers is dedicated to detailing the computations that lead to that surprising result. In the current paper, we construct the family of coherent states in LQG which represent flat ($k=0$) Robertson-Walker spacetimes, and present the tools needed to compute expectation values of polynomial operators in holonomy and flux on such states. These tools will be applied to the LQG Hamiltonian operator (in Thiemann regularization) in the second paper of the series. The third paper will present an extension to $k\neq0$ cosmologies and a comparison with alternative regularizations of the Hamiltonian., Comment: 43 pages, 1 figure

The concept of effective dynamics has proven successful in LQC, a loop-inspired quantization of cosmological spacetimes. We apply the same idea of its derivation in LQC to the full theory, by computing the expectation value of the scalar constraint with respect to some coherent states peaked on the phase-space variables of flat Robertson-Walker spacetime. We comment on the relation with effective LQC and find a deviation stemming from the Lorentzian part of the Hamiltonian., Comment: 5 pages, 1 figure. Accepted version in PLB

In this paper we present a construction of effective cosmological models which describe the propagation of a massive quantum scalar field on a quantum anisotropic cosmological spacetime. Each obtained effective model is represented by a rainbow metric in which particles of distinct momenta propagate on different classical geometries. Our analysis shows that upon certain assumptions and conditions on the parameters determining such anisotropic models, we surprisingly obtain a unique deformation parameter $\beta$ in the modified dispersion relation of the modes. Hence inducing an isotropic deformation despite the general starting considerations. We then ensure the recovery of the dispersion relation realized in the isotropic case, studied in [arXiv:1412.6000], when some proper symmetry constraints are imposed, and we estimate the value of the deformation parameter for this case in loop quantum cosmology context., Comment: 14 pages, 1 figure

We present a new method for constructing operators in loop quantum gravity. The construction is an application of the general idea of "coherent state quantization", which allows one to associate a unique quantum operator to every function on a classical phase space. Using the heat kernel coherent states of Hall and Thiemann, we show how to construct operators corresponding to functions depending on holonomies and fluxes associated to a fixed graph. We construct the coherent state versions of the fundamental holonomy and flux operators, as well as the basic geometric operators of area, angle and volume. Our calculations show that the corresponding canonical operators are recovered from the coherent state operators in the limit of large spins., Comment: 34 pages, 4 figures (plus many pretty diagrams)

We present the affine coherent state quantization of the Bianchi I model. As in our previous paper on quantum theory of Friedmann models, we employ a variable associated with a perfect fluid to play a role of clock. Then we deparameterize the model. A distinctive feature, absent in isotropic models, is an extra non-holonomic constraint, which survives the deparameterization and constrains the range of physical variables. The appearance of the constraint reflects the `amplification' of singularity due to anisotropy. The quantization smoothes the extra constraint and allows quantum contracting trajectories to be smoothly transformed into expanding ones. Making use of affine coherent state we develop a semiclassical description. Figures are included to illustrate our result., Comment: 21 pages, 4 figures

In this letter, we describe a general mechanism for emergence of a rainbow metric from a quantum cosmological model. This idea is based on QFT on a quantum space-time. Under general assumptions, we discover that the quantum space-time on which the field propagates can be replaced by a classical space-time, whose metric depends explicitly on the energy of the field: as shown by an analysis of dispersion relations, quanta of different energy propagate on different metrics, similar to photons in a refractive material (hence the name "rainbow" used in the literature). In deriving this result, we do not consider any specific theory of quantum gravity: the qualitative behavior of high-energy particles on quantum space-time relies only on the assumption that the quantum space-time is described by a wave-function $\Psi_o$ in a Hilbert space $\mathcal{H}_G$., Comment: 4 pages, 2 figures. Accepted version in PLB

We study the gravitational collapse of a massless scalar field within the effective scenario of loop quantum gravity. Classical singularity is avoided and replaced by a quantum bounce in this model. It is shown that, quantum gravity effects predict a threshold scale below which no horizon can form as the collapse evolves towards the bounce., Comment: Contribution to the Spanish Relativity Meeting in Portugal 2012 (ERE2012), Guimaraes, Portugal

We perform a LQC-quantization of the FRW cosmological model with nonminimally coupled scalar field. Making use of a canonical transformation, we recast the theory in the minimally coupled form (Einstein frame), for which standard LQC techniques can be applied to find the physical Hilbert space and the dynamics. We then focus on the semiclassical sector, obtaining a classical effective Hamiltonian, which can be used to study the dynamics. We show that the classical singularity is replaced by a "mexican hat"-shaped bounce, joining the contracting and expanding branches. The model accommodates Higgs-driven inflation, with more than enough e-folding for any physically meaningful initial condition., Comment: 3 pages, 2 figures. To appear in Proceedings of the 13th Marcel Grossmann Meeting (MG13), Stockholm, Sweden, 1-7 July 2012

Quantum theory of a scalar field is developed on the LQC Bianchi I space-time. By comparing the the quantum field theory for a single mode on classical and quantum background geometries we find that an effective Bianchi I space-time emerges. We show that by disregarding the back-reaction no Lorentz-violation is present, despite the effective metric being different than the classical Bianchi I one., Comment: 3 pages. To appear in Proceedings of the 13th Marcel Grossmann Meeting (MG13), Stockholm, Sweden, 1-7 July 2012

We examine the possibility of dealing with gravitational singularities on a quantum level through the use of coherent state or wavelet quantization instead of canonical quantization. We consider the Robertson-Walker metric coupled to a perfect fluid. It is the simplest model of a gravitational collapse and the results obtained here may serve as a useful starting point for more complex investigations in future. We follow a quantization procedure based on affine coherent states or wavelets built from the unitary irreducible representation of the affine group of the real line with positive dilation. The main issue of our approach is the appearance of a quantum centrifugal potential allowing for regularization of the singularity, essential self-adjointness of the Hamiltonian, and unambiguous quantum dynamical evolution., Comment: 22+9 pages, 3 figures

Evolution of systems in which Hamiltonians are generators of gauge transformations is a notion that requires more structure than the canonical theory provides. We identify and study this additional structure in the framework of relational observables ("partial observables"). We formulate necessary and sufficient conditions for the resulting evolution in the physical phase space to be a symplectomorphism. We give examples which satisfy those conditions and examples which do not. We point out that several classic positions in the literature on relational observables contain an incomplete approach to the issue of evolution and false statements. Our work provides useful clarification and opens the door to studying correctly formulated definitions., Comment: 8 pages, 1 figure. Version published in Phys. Rev. D

In this work, we consider a semiclassical description of the spherically symmetric gravitational collapse with a massless scalar field. In particular, we employ an effective scenario provided by holonomy corrections from loop quantum gravity, to the homogeneous interior spacetime. The singularity that would arise at the final stage of the corresponding classical collapse, is resolved in this context and is replaced by a bounce. Our main purpose is to investigate the evolution of trapped surfaces during this semiclassical collapse. Within this setting, we obtain a threshold radius for the collapsing shells in order to have horizons formation. In addition, we study the final state of the collapse by employing a suitable matching at the boundary shell from which quantum gravity effects are carried to the exterior geometry., Comment: Version accepted in IJMPD

We develop a systematic classical framework to accommodate canonical quantization of geometric and matter perturbations on a quantum homogeneous isotropic flat spacetime. The existing approach of standard cosmological perturbations is indeed proved to be good only up to first order in the inhomogeneities, and only if the background is treated classically. To consistently quantize the perturbations \emph{and} the background, a new set of classical phase space variables is required. We show that, in a natural gauge, a set of such Dirac observables exists, and their algebra is of the canonical form. Finally, we compute the physical Hamiltonian that generates the dynamics of such observables with respect to the homogeneous part of a K-G "clock" field $T$. The results of this work provide a good starting point to understanding and calculating effects that quantum cosmological spacetime in the background has on the quantum perturbations of the metric tensor and of matter fields., Comment: 29 pages, v3. Replaced with revised version accepted in PRD

We consider a massive quantum test Klein-Gordon field probing an isotropic quantum cosmological space-time in the background. The result obtained is surprising. It turns out, that despite the isotropy of the quantum gravitational field, the semi-classical metric experienced by a mode of the K-G field is non-isotropic. The anisotropy depends on the direction of the momentum of the mode. Specifically, what we do is to derive a semi-classical space-time which emerges to a mode of the field. The method amounts to a comparison between QFT on a quantum background and QFT on a classical curved space-time, giving rise to an emergent metric tensor. The components of the semi-classical metric tensor are calculated from the equation of propagation of the quantum K-G field in the test field approximation. The anisotropies are of a quantum nature: they are proportional to Planck constant and "dress" the isotropic classical space-time obtained in the classical limit., Comment: 7 pages, v2. Refereed version (previously called "Emergent Isotropy-Breaking in Quantum Cosmology")

The FRW model with non-minimally coupled massive scalar field has been investigated in LQC framework. Considered form of the potential and coupling allows applications to Higgs driven inflation. Out of two frames used in the literature to describe such systems: Jordan and Einstein frame, the latter one is applied. Specifically, we explore the idea of the Einstein frame being the natural 'environment' for quantization and the Jordan picture having an emergent nature. The resulting dynamics qualitatively modifies the standard bounce paradigm in LQC in two ways: (i) the bounce point is no longer marked by critical matter energy density, (ii) the Planck scale physics features the 'mexican hat' trajectory with two consecutive bounces and rapid expansion and recollapse between them. Furthermore, for physically viable coupling strength and initial data the subsequent inflation exceeds 60 e-foldings., Comment: Clarity improved. Replaced with revised version accepted in JCAP

We develop the quantum theory of a scalar field on LQC Bianchi I geometry. In particular, we focus on single modes of the field: the evolution equation is derived from the quantum scalar constraint, and it is shown that the same equation can be obtained from QFT on an "classical" effective geometry. We investigate the dependence of this effective space-time on the wavevector of the mode (which could in principle generate a deformation in local Lorentz-symmetry), focusing our attention on the dispersion relation. We prove that when we disregard backreaction no Lorentz-violation is present, despite the effective metric being different than the classical Bianchi I one. A preliminary analysis of the correction due to inclusion of backreaction is briefly discussed in the context of Born-Oppenheimer approximation., Comment: 14 pages, v3. Corrected a reference in the bibliography