To appear in the Proceedings of the Bad Honef Workshop entitled \emph{Quantum Gravity: Challenges and Perspectives}, dedicated to the memory of John A. Wheeler
A brief overview of loop quantum cosmology of homogeneous isotropic models is presented with emphasis on the origin of and subtleties associated with the resolution of big bang and big crunch singularities. These results bear out the remarkable intuition that John Wheeler had. Discussion is organized at two levels. The the main text provides a bird's eye view of the subject that should be accessible to non-experts. Appendices address conceptual and technical issues that are often raised by experts in loop quantum gravity and string theory.
Among the available quantum gravity proposals, string theory, loop quantum gravity, non-commutative geometry, group field theory, causal sets, asymptotic safety, causal dynamical triangulation, emergent gravity are among the best motivated models. As an introductory summary to this special issue of Comptes Rendus Physique, I explain how those different theories can be tested or constrained by cosmological observations.
Applications of Riemannian quantum geometry to cosmology have had notable successes. In particular, the fundamental discreteness underlying quantum geometry has led to a natural resolution of the big bang singularity. However, the precise mathematical structure underlying loop quantum cosmology and the sense in which it implements the full quantization program in a symmetry reduced model has not been made explicit. The purpose of this paper is to address these issues, thereby providing a firmer mathematical and conceptual foundation to the subject.
We give a definition for the Wigner function for quantum mechanics on the Bohr compactification of the real line and prove a number of simple consequences of this definition. We then discuss how this formalism can be applied to loop quantum cosmology. As an example, we use the Wigner function to give a new quantization of an important building block of the Hamiltonian constraint.
Loop quantum cosmology in (b, v) variables, which is governed by a unit step size difference equation, is embedded into a full theory context based on similar variables. A full theory context here means a theory of quantum gravity arrived at using the quantisation techniques used in loop quantum gravity, however based on a different choice of elementary variables and classical gauge fixing suggested by loop quantum cosmology. From the full theory perspective, the symmetry reduction is characterised by the vanishing of certain phase space functions which are implemented as operator equations in…
A brief summary of the singularity resolution in loop quantum cosmology of homogeneous isotropic models is presented. The article is addressed to relativists who do not specialize in quantum gravity. For further details, and answers to more technical asked questions, the reader is directed to the original papers and to more comprehensive recent reviews. Dedicated to the memory of Basilis Xanthopoulos, a founder of this NEB series of conferences.
The new frontier of cosmology will be led by three-dimensional surveys of the large-scale structure of the Universe. Based on its all-sky surveys and redshift depth, the SKA is destined to revolutionize cosmology, in combination with future optical/ infrared surveys such as Euclid and LSST. Furthermore, we will not have to wait for the full deployment of the SKA in order to see transformational science. In the first phase of deployment (SKA1), all-sky HI intensity mapping surveys and all-sky continuum surveys are forecast to be at the forefront on the major questions of cosmology. We give a br…
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 set of homogeneous isotropic connections, as used in loop quantum cosmology, forms a line $l$ in the space of all connections $\cal A$. This embedding, however, does not continuously extend to an embedding of the configuration space $\overline l$ of homogeneous isotropic loop quantum cosmology into that of loop quantum gravity, $\overline{\cal A}$. This follows from the fact that the parallel transports for general, non-straight paths in the base manifold do not depend almost periodically on $l$. Analogous results are given for the anisotropic case.
First, we review the $C^\ast$-algebraic foundations of loop quantization, in particular, the construction of quantum configuration spaces and the implementation of symmetries. Then, we apply these results to loop quantum gravity, focusing on the space of generalized connections and on measures thereon. Finally, we study the realm of homogeneous isotropic loop quantum cosmology: once viewed as the loop quantization of classical cosmology, once seen as the symmetric sector of loop quantum gravity. It will turn out that both theories differ, i.e., quantization and symmetry reduction do not commut…
The purpose of this review is to provide a brief overview of some recent conceptual developments about possible criteria to guarantee the uniqueness of the quantization in a variety of situations that are found in cosmological systems. These criteria impose some conditions on the representation of a group of physically relevant linear transformations. Generally, this group contains any existing symmetry of the spatial sections. These symmetries may or may not be sufficient for the purpose of uniqueness and may have to be complemented with other remaining symmetries that affect the time directi…
A brief review of various numerical techniques used in loop quantum cosmology and results is presented. These include the way extensive numerical simulations shed insights on the resolution of classical singularities, resulting in the key prediction of the bounce at the Planck scale in different models, and the numerical methods used to analyze the properties of the quantum difference operator and the von Neumann stability issues. Using the quantization of a massless scalar field in an isotropic spacetime as a template, an attempt is made to highlight the complementarity of different methods t…
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 …
In this work, we present an overview of uniqueness results derived in recent years for the quantization of Gowdy cosmological models and for (test) Klein-Gordon fields minimally coupled to Friedmann-Lema\^ıtre-Robertson-Walker, de Sitter, and Bianchi I spacetimes. These results are attained by imposing the criteria of symmetry invariance and of unitary implementability of the dynamics. This powerful combination of criteria allows not only to address the ambiguity in the representation of the canonical commutation relations, but also to single out a preferred set of fundamental variables. For t…
Loop Quantum Gravity is a nonperturbative and background independent program for the quantization of General Relativity. Its underlying formalism has been applied successfully to the study of cosmological spacetimes, both to test the principles and techniques of the theory and to discuss its physical consequences. These applications have opened a new area of research known as Loop Quantum Cosmology. The hybrid approach addresses the quantization of cosmological systems that include fields. This proposal combines the description of a finite number of degrees of freedom using Loop Quantum Cosmol…
In loop quantum cosmology, a difference equation for the wave function describes the evolution of a universe model. This is different from the differential equations that arise in Wheeler-DeWitt quantizations, and some aspects of general properties of solutions can appear differently. Properties of particular interest are boundedness and the presence of small-scale oscillations. Continued fraction techniques are used to show in different matter models the presence of special initial conditions leading to bounded solutions, and an explicit expression for these initial values is derived.
This is an introduction to loop quantum cosmology (LQC) reviewing mini- and midisuperspace models as well as homogeneous and inhomogeneous effective dynamics.
In a paper of Ashtekar and Campiglia, residual diffeomorphisms have been used to single out the standard representation of the reduced holonomy-flux algebra in homogeneous loop quantum cosmology (LQC). We show that, in the homogeneous isotropic case, unitarity of the translations w.r.t. the extended $\mathbb{R}$-action (exponentiated reduced fluxes in the standard approach) singles out the Bohr measure on both the standard quantum configuration space $\mathbb{R}_\mathrm{Bohr}$ as well as on the Fleischhack one. Thus, in both situation, the same condition singles out the standard kinematical Hi…
Loop quantum cosmology is a conflicted field in which exuberant claims of observability coexist with serious objections against the conceptual and physical viability of its current formulations. This contribution presents a non-technical case study of the recent claim that loop quantum cosmology might alleviate anomalies in observations of the cosmic microwave background.
We review some recent progress in the extraction of inflationary observables in loop quantum cosmology. Inverse-volume quantum corrections induce a growth of power in the large-scale cosmological spectra and are constrained by observations.
Using general features of recent quantizations of the Hamiltonian constraint in loop quantum gravity and loop quantum cosmology, a dynamical interpretation of the constraint equation as evolution equation is presented. This involves a transformation from the connection to a dreibein representation and the selection of an internal time variable. Due to the discrete nature of geometrical quantities in loop quantum gravity also time turns out to be discrete leading to a difference rather than differential evolution equation. Furthermore, evolving observables are discussed in this framework which …
Loop quantum cosmology of the closed isotropic model is studied with a special emphasis on a comparison with traditional results obtained in the Wheeler-DeWitt approach. This includes the relation of the dynamical initial conditions with boundary conditions such as the no-boundary or the tunneling proposal and a discussion of inflation from quantum cosmology.
We construct a noncommutative extension of the Loop Quantum Cosmology effective scheme for the open FLRW model with a free scalar field via a theta deformation. Firstly, a deformation is implemented in the configuration sector, among the holonomy variable and the matter degree of freedom. We show that this type of noncommutativity retain, to some degree, key features of the Loop Quantum Cosmology paradigm for a free field. Secondly, a deformation is implemented in the momentum sector, among the momentum associated to the holonomy variable and the momentum associated to the matter field. We sho…
The era of precision cosmology has allowed us to accurately determine many important cosmological parameters, in particular via the CMB. Confronting Loop Quantum Cosmology with these observations provides us with a powerful test of the theory. For this to be possible we need a detailed understanding of the generation and evolution of inhomogeneous perturbations during the early, Quantum Gravity, phase of the universe. Here we describe how Loop Quantum Cosmology provides a completion of the inflationary paradigm, that is consistent with the observed power spectra of the CMB.
After introducing the basic ingredients of Loop Quantum Cosmology, I will briefly discuss some of its phenomenological aspects. Those can give some useful insight about the full Loop Quantum Gravity theory and provide an answer to some long-standing questions in early universe cosmology.
In this paper, we discuss the properties of one-parameter sequences that arise when solving the Hamiltonian constraint in Bianchi I loop quantum cosmology using a separation of variables method. In particular, we focus on finding an expression for the sequence for all real values of the parameter, and discuss the pre-classicality of this function. We find that the behavior of these preclassical sequences imply time asymmetry on either side of the classical singularity in Bianchi I cosmology.
We present a construction of the Hubble operator for the spatially flat isotropic loop quantum cosmology. This operator is a Dirac observable on a subspace of the space of physical solutions. This subspace gets selected dynamically, requiring that its action be invariant on the physical solution space. As a simple illustrative application of the expectation value of the operator, we do find a generic phase of (super)inflation, a feature shown by Bojowald from the analysis of effective Friedmann equation of loop quantum cosmology.
Loop Quantum Cosmology is an appealing quantum completion of classical cosmology, which brings along various theoretical features which in many cases offer remedy or modify various classical cosmology aspects. In this paper we address the gravitational baryogenesis mechanism in the context of Loop Quantum Cosmology. As we demonstrate, when Loop Quantum Cosmology effects are taken into account in the resulting Friedmann equations for a flat Friedmann-Robertson-Walker Universe, then even for a radiation dominated Universe, the predicted baryon-to-entropy ratio from the gravitational baryogenesis…
Recently, Chamseddine and Mukhanov introduced a higher-derivative scalar-tensor theory which leads to a modified Friedmann equation allowing for bouncing solutions. As we note in the present work, this Friedmann equation turns out to reproduce exactly the loop quantum cosmology effective dynamics for a flat isotropic and homogeneous space-time. We generalize this result to obtain a class of scalar-tensor theories, belonging to the family of mimetic gravity, which all reproduce the loop quantum cosmology effective dynamics for flat, closed and open isotropic and homogeneous space-times.
Loop quantum cosmology is a symmetry reduced quantization of cosmological spacetimes based on loop quantum gravity. While it has been successful in resolution of various cosmological singularities and connecting Planck scale physics to phenomenology, its connection with loop quantum gravity has remained elusive. It is therefore important to integrate more and more features of the full theory into this framework and understand the reliability of physical predictions. In particular, if one wishes to connect the effective Hamiltonian in loop quantum cosmology to an expectation value of the scalar…
Loop quantum cosmology is considered in inflation era. A slow roll scalar field solution with power law potential is presented in the neighborhood of transition time, i.e. when the universe enters inflation phase from super-inflation. The compatibility of the model with Planck 2013 data is discussed. The domain of validity of second and the generalized second laws of thermodynamics for this solution and some other examples is studied.
We present a new cosmological model derived from Loop Quantum Gravity. The formulation is based on a projection of the kinematical Hilbert space of the full theory down to a subspace representing the proper arena for an inhomogeneous Bianchi I model. This procedure gives a direct link between the full theory and its cosmological sector. The emerging quantum cosmological model represents a simplified arena on which the complete canonical quantization program can be tested. The achievements of this analysis could also shed light on Loop Quantum Cosmology and its relation with the full theory.
Loop quantum cosmological methods are extended to homogeneous models in diagonalized form. It is shown that the diagonalization leads to a simplification of the volume operator such that its spectrum can be determined explicitly. This allows the calculation of composite operators, most importantly the Hamiltonian constraint. As an application the dynamics of the Bianchi I model is studied and it is shown that its loop quantization is free of singularities.
Loop quantum gravity is a mature theory. To proceed to explicit calculations in cosmology, it is necessary to make assumptions and simplifications based on the symmetries of the cosmological setting. Symmetry reduction is especially critical when dealing with cosmological perturbations. The present article reviews several approaches to the problem of building a consistent formalism that describes the dynamics of perturbations on a quantum spacetime and tries to address their respective strengths and weaknesses. We also review the main open issues in loop quantum cosmology.
We investigate the background dynamics when dark energy is coupled to dark matter in the universe described by Einstein cosmology and Loop Quantum Cosmology. We introduce a new general form of dark sector coupling, which presents us a more complicated dynamical phase space. Differences in the phase space in obtaining the accelerated scaling attractor in Einstein cosmology and Loop Quantum Cosmology are disclosed.
The dynamical behaviors of interacting dark energy in loop quantum cosmology are discussed in this paper. Based on defining three dimensionless variables, we simplify the equations of the fixed points. The fixed points for interacting dark energy can be determined by the Friedmann equation coupled with the dynamical equations {in Einstein cosmology}. But in loop quantum cosmology, besides the Friedmann equation, the conversation equation also give a constrain on the fixed points. The difference of stability properties for the fixed points in loop quantum cosmology and the ones in Einstein cosm…
The appearance of the big bounce (BB) in the evolution of the universe is analyzed in the setting of loop quantum cosmology (LQC). Making use of an idea of a minimum length turns classical Big Bang into BB. We argue why the spectrum of the kinematical area operator of loop quantum gravity cannot be used for the determination of this length. We find that the fundamental length, at the present stage of development of LQC, is a free parameter of this model.
Quantum Reduced Loop Gravity is a promising framework for linking Loop Quantum Gravity and the effective semiclassical dynamics of Loop Quantum Cosmology. We review its basic achievements and its main perspectives, outlining how it provides a quantum description of the Universe in terms of a cuboidal graph which constitutes the proper framework for applying loop techniques in a cosmological setting.
We develop a Gauss-Bonnet extension of Loop Quantum Cosmology, by introducing holonomy corrections in modified $F(\mathcal{G})$ theories of gravity. Within the context of our formalism, we provide a perturbative expansion in the critical density, a parameter characteristic of Loop Quantum Gravity theories, and we result in having leading order corrections to the classical $F(\mathcal{G})$ theories of gravity. After extensively discussing the formalism, we present a reconstruction method that makes possible to find the Loop Quantum Cosmology corrected $F(\mathcal{G})$ theory that can realize va…
We show that the effective dynamics of the recently proposed isotropic condensate state of group field theory with Laplacian kinetic operator can be equivalent to that of homogeneous and isotropic loop quantum cosmology in the improved dynamics quantization scheme, where the area of elementary holonomy plaquettes is constant. This constitutes a somewhat surprising example of a cosmological model of quantum gravity where the operations of minisuperspace symmetry reduction and quantization can actually commute.
In the framework of loop quantum cosmology anomaly free quantizations of the Hamiltonian constraint for Bianchi class A, locally rotationally symmetric and isotropic models are given. Basic ideas of the construction in (non-symmetric) loop quantum gravity can be used, but there are also further inputs because the special structure of symmetric models has to be respected by operators. In particular, the basic building blocks of the homogeneous models are point holonomies rather than holonomies necessitating a new regularization procedure. In this respect, our construction is applicable also for…
In this paper, we study the dynamics of k-essence in loop quantum cosmology (LQC). The study indicates that the loop quantum gravity (LQG) effect plays a key role only in the early epoch of the universe and is diluted at the later stage. The fixed points in LQC are basically consistent with that in standard Friedmann-Robertson-Walker (FRW) cosmology. For most of the attractor solutions, the stability conditions in LQC are in agreement with that for the standard FRW universe. But for some special fixed point, more tighter constraints are imposed thanks to the LQG effect.
A short review of recent results on exact solutions in multidimensional cosmology and overview of reports at workshop in MG8 (held by the author) are presented.
In this paper, we study the possibility of model building of cyclic universe with Quintom matter in the framework of Loop Quantum Cosmology. After a general demonstration, we provide two examples, one with double-fluid and another double-scalar field, to show how such a scenario is obtained. Analytical and numerical calculations are both presented in the paper.
We demonstrate that it is possible to produce different isotropic embeddings of anisotropic Loop Quantum Cosmology, resulting to "lattice refinement" in the isotropic system. To introduce the general approach, we first use a simple model with only two anisotropic directions. We then employ the specific case of a Bianchi I model, to show how the method extends to three-dimensional systems. To concisely calculate the step-size of the resulting isotropic state, we define the "symmetric dual" of states and operators, for the two- and three-dimensional systems, respectively.
The paradigmatic models often used to highlight cosmological features of loop quantum gravity and group field theory are shown to be equivalent, in the sense that they are different realizations of the same model given by harmonic cosmology. The loop version of harmonic cosmology is a canonical realization, while the group-field version is a bosonic realization. The existence of a large number of bosonic realizations suggests generalizations of models in group field cosmology.
Loop quantum cosmology predicts that quantum gravity effects resolve the big-bang singularity and replace it by a cosmic bounce. Furthermore, loop quantum cosmology can also modify the form of primordial cosmological perturbations, for example by reducing power at large scales in inflationary models or by suppressing the tensor-to-scalar ratio in the matter bounce scenario; these two effects are potential observational tests for loop quantum cosmology. In this article, I review these predictions and others, and also briefly discuss three open problems in loop quantum cosmology: its relation to…
A consistent combination of quantum geometry effects rules out a large class of models of loop quantum cosmology and their critical densities as they have been used in the recent literature. In particular, the critical density at which an isotropic universe filled with a free, massless scalar field would bounce must be well below the Planck density. In the presence of anisotropy, no model of the Schwarzschild black hole interior analyzed so far is consistent.
Isotropic models in loop quantum cosmology allow explicit calculations, thanks largely to a completely known volume spectrum, which is exploited in order to write down the evolution equation in a discrete internal time. Because of genuinely quantum geometrical effects the classical singularity is absent in those models in the sense that the evolution does not break down there, contrary to the classical situation where space-time is inextendible. This effect is generic and does not depend on matter violating energy conditions, but it does depend on the factor ordering of the Hamiltonian constra…
Volume operators measuring the total volume of space in a loop quantum theory of cosmological models are constructed. In the case of models with rotational symmetry an investigation of the Higgs constraint imposed on the reduced connection variables is necessary, a complete solution of which is given for isotropic models; in this case the volume spectrum can be calculated explicitly. It is observed that the stronger the symmetry conditions are the smaller is the volume spectrum, which can be interpreted as level splitting due to broken symmetries. Some implications for quantum cosmology are pr…
We explore the extension of quantum cosmology outside the homogeneous approximation, using the formalism of loop quantum gravity. We introduce a model where some of the inhomogeneous degrees of freedom are present, providing a tool for describing general fluctuations of quantum geometry near the initial singularity. We show that the dynamical structure of the model reduces to that of loop quantum cosmology in the Born-Oppenheimer approximation. This result corroborates the assumptions that ground loop cosmology, sheds light on the physical and mathematical relation between loop cosmology and f…
In loop quantum gravity, modifications to the geometrical density cause a self-interacting scalar field to accelerate away from a minimum of its potential. In principle, this mechanism can generate the conditions that subsequently lead to slow-roll inflation. The consequences for this mechanism of various quantization ambiguities arising within loop quantum cosmology are considered. For the case of a quadratic potential, it is found that some quantization procedures are more likely to generate a phase of slow--roll inflation. In general, however, loop quantum cosmology is robust to ambiguities…
The cosmological singularities of the Bianchi I universe are analyzed in the setting of loop geometry underlying the loop quantum cosmology. We solve the Hamiltonian constraint of the theory and find the Lie algebra of elementary observables. Physical compound observables are defined in terms of elementary ones. Modification of classical theory by holonomy around a loop removes the singularities. However, our model has a free parameter that cannot be determined within our method. Testing the model by the data of observational cosmology may be possible after quantization of our modified classic…
We consider a flat cosmological model with a free massless scalar field and the cosmological constant $Λ$ in the framework of loop quantum cosmology. The scalar field plays the role of an intrinsic time. We apply the reduced phase space approach. The dynamics of the model is solved analytically. We identify elementary observables and their algebra. The compound physical observables like the volume and the energy density of matter field are analysed. Both compound observables are bounded and oscillate in the $Λ<0$ case. The energy density is bounded and oscillates in the $Λ>0$ case. However, th…
The two-loop (order $α'$) $β$-function equations, which are equivalent to the equations of motion of $α'$-corrected string effective action, are considered for anisotropic homogeneous space-times. These equations are solved for all Bianchi-type models in two schemes of effective action, namely $R^2$ and Gauss-Bonnet schemes with zero cosmological constant and then the metric, dilaton and $B$-field are found at $α'$ perturbative corrections.
We give a prescription to define in Loop Quantum Gravity the electric field operator related to the scale factor of an homogeneous and isotropic cosmological space-time. This procedure allows to link the fundamental theory with its cosmological implementation. In view of the conjugate relation existing between holonomies and fluxes, the edge length and the area of surfaces in the fiducial metric satisfy a duality condition. As a consequence, the area operator has a discrete spectrum also in Loop Quantum Cosmology. This feature makes the super-Hamiltonian regularization an open issue of the who…
Using a Liouville measure, similar to the one proposed recently by Gibbons and Turok, we investigate the probability that single-field inflation with a polynomial potential can last long enough to solve the shortcomings of the standard hot big bang model, within the semiclassical regime of loop quantum cosmology. We conclude that, for such a class of inflationary models and for natural values of the loop quantum cosmology parameters, a successful inflationary scenario is highly improbable.
This is a pedagogical guide to works on this subject which began in the 80s but has seen vibrant activities in the last decade. It aims to help orient readers, especially students, who wish to enter into research but bewildered by the vast and diverse literature on this subject. We describe the three main veins of activities: the Euclidean zero-mode dominance, the Lorentzian interacting quantum field theory and the classical stochastic field theory approaches in some detail, explaining the underlying physics and the technicalities of each. We show how these approaches are interconnected, and h…
In this paper we construct the Hamiltonian constraint operator of loop quantum cosmology using holonomies defined for arbitrary irreducible SU(2) representations labeled by spin J. We show that modifications to the effective semi-classical equations of motion arise both in the gravitational part of the constraint as well as matter terms. The modifications are important for phenomenological investigations of the cosmological imprints of loop quantum cosmology. We discuss the implications for the early universe evolution.
In this letter, we demonstrate that Loop Quantum Cosmology equations, with a near vanishing late time cosmological constant, can be induced on a brane in a Randall-Sundrum II scenario, corresponding to the holographic dual of a strongly coupled string regime in the bulk. Such results can establish a possible connection between Loop Quantum Gravity and string theory.
