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 completed into a canonical one for the entire system, including the homogeneous degrees of freedom. We find the global Hamiltonian constraint of the model, in which the contribution of the homogeneous sector is corrected with a term quadratic in the perturbations, that can be identified as the Mukhanov-Sasaki Hamiltonian in our formulation. We then adopt a hybrid approach to quantize the model, combining a quantum representation of the homogeneous sector with a more standard field quantization of the perturbations. Covariance is guaranteed in this approach inasmuch as no gauge fixing is adopted. Next, we adopt a Born-Oppenheimer ansatz for physical states and show how to obtain a Schrödinger-like equation for the quantum evolution of the perturbations. This evolution is governed by the Mukhanov-Sasaki Hamiltonian, with the dependence on the homogeneous geometry evaluated at quantum expectation values, and with a time parameter defined also in terms of suitable expectation values on that geometry. Finally, we derive effective equations for the dynamics of the Mukhanov-Sasaki gauge invariants, that include quantum contributions, but have the same ultraviolet limit as the classical equations. They provide the master equation to extract predictions about the power spectrum of primordial scalar perturbations.

The flat, homogeneous, and isotropic universe with a massless scalar field is a paradigmatic model in Loop Quantum Cosmology. In spite of the prominent role that the model has played in the development of this branch of physics, there still remain some aspects of its quantization which deserve a more detailed discussion. These aspects include the kinematical resolution of the cosmological singularity, the precise relation between the solutions of the densitized and non-densitized versions of the quantum Hamiltonian constraint, the possibility of identifying superselection sectors which are as simple as possible, and a clear comprehension of the Wheeler-DeWitt (WDW) limit associated with the theory in those sectors. We propose an alternative operator to represent the Hamiltonian constraint which is specially suitable to deal with all these issues in a detailed and satisfactory way. In particular, with our constraint operator, the singularity decouples in the kinematical Hilbert space and can be removed already at this level. Thanks to this fact, we can densitize the quantum Hamiltonian constraint in a well-controlled manner. Besides, together with the physical observables, this constraint superselects simple sectors for the universe volume, with a discrete support contained in a single semiaxis of the real line and for which the basic functions that encode the information about the geometry possess optimal physical properties. Namely, they provide a no-boundary description around the cosmological singularity and admit a well-defined WDW limit in terms of standing waves. Both properties explain the presence of a generic quantum bounce replacing the classical singularity at a fundamental level, in contrast with previous studies where the bounce was proved in concrete regimes -focusing on states with a marked semiclassical behavior-or for a simplified model.

We analyze the loop quantization of the family of vacuum Bianchi I spacetimes, a gravitational system whose classical solutions describe homogeneous anisotropic cosmologies. We rigorously construct the operator that represents the Hamiltonian constraint, showing that the states of zero volume completely decouple from the rest of quantum states. This fact ensures that the classical cosmological singularity is resolved in the quantum theory. In addition, this allows us to adopt an equivalent quantum description in terms of a well defined densitized Hamiltonian constraint. This latter constraint can be regarded in a certain sense as a difference evolution equation in an internal time provided by one of the triad components, which is polymerically quantized. Generically, this evolution equation is a relation between the projection of the quantum states in three different sections of constant internal time. Nevertheless, around the initial singularity the equation involves only the two closest sections with the same orientation of the triad. This has a double effect: on the one hand, physical states are determined just by the data on one section, on the other hand, the evolution defined in this way never crosses the singularity, without the need of any special boundary condition. Finally, we determine the inner product and the physical Hilbert space employing group averaging techniques, and we specify a complete algebra of Dirac observables. This completes the quantization program.

Abstract. This is an introduction to loop quantum cosmology (LQC) reviewing mini-and midisuperspace models as well as homogeneous and inhomogeneous effective dynamics.

The Gowdy cosmologies provide a suitable arena to further develop Loop Quantum Cosmology, allowing the presence of inhomogeneities. For the particular case of Gowdy spacetimes with the spatial topology of a three-torus and a content of linearly polarized gravitational waves, we detail a hybrid quantum theory in which we combine a loop quantization of the degrees of freedom that parametrize the subfamily of homogeneous solutions, which represent Bianchi I spacetimes, and a Fock quantization of the inhomogeneities. Two different theories are constructed and compared, corresponding to two different schemes for the quantization of the Bianchi I model within the improved dynamics formalism of Loop Quantum Cosmology. One of these schemes has been recently put forward by Ashtekar and Wilson-Ewing. We address several issues including the quantum resolution of the cosmological singularity, the structure of the superselection sectors in the quantum system, or the construction of the Hilbert space of physical states.

We undertake first steps in making a class of discrete models of quantum gravity, spin foams, accessible to a large-scale analysis by numerical and computational methods. In particular, we apply the Migdal-Kadanoff and tensor network renormalization (TNR) schemes to spin net and spin foam models based on finite Abelian groups and introduce 'cutoff models' to probe the fate of gauge symmetries under various such approximated renormalization group flows. For the TNR analysis, a new Gauß constraint preserving algorithm is introduced to improve numerical stability and aid physical interpretation. We also describe the fixed point structure and establish the equivalence of certain models.2 For 3D gravity, which is a topological theory, that is, without propagating degrees of freedom, discretization does not necessarily lead to a breaking of diffeomorphism symmetry [20,31,32]. This holds also for 3D gravity with cosmological constant [33]. One can, however, consider discretization or quantization methods, which a priori break diffeomorphism symmetry and look for methods to restore these symmetries, see [25,34,35].

We quantize an inhomogeneous cosmological model using techniques that include polymeric quantization. More explicitly, we construct well defined operators to represent the constraints and find the physical Hilbert space formed by their solutions, which reproduces the conventional Fock quantization for the inhomogeneities. The initial singularity is resolved in this inhomogeneous model in an extremely simple way and without imposing special boundary conditions, thus ensuring the robustness and generality of this resolution. Furthermore this quantization constitutes a well founded step towards the extraction of physical results and consequences from loop quantum cosmology, given the central role of the inhomogeneities in modern cosmology.PACS numbers: 04.60. Pp, 98.80.Qc, 04.62.+v Loop quantum cosmology (LQC) [1] has recently undergone important developments. The study of the physical consequences of loop quantum gravity (LQG) [2] for cosmology has motivated the analysis of symmetry reduced models that can be fully quantized by techniques that closely resemble those of LQG [3]. Various homogeneous and isotropic models have been successfully quantized [4]. Other less symmetric systems have been also analyzed [5,6,7] with partially satisfactory results.Any realistic attempt to extract physical results from LQC must consider inhomogeneities, which play a central role in current cosmology. As a most suitable arena to progress in this direction, we will study the quantization of the linearly polarized Gowdy T 3 model [8]. This family of spacetimes is highly symmetric, but preserves one important feature of the full theory: it has an infinite number of degrees of freedom. Its classical solutions are indeed well known and generically present an initial singularity [9]. Besides, its quantization has been addressed since long time ago by means of standard non-polymeric techniques [10] (see also [11] for a related system, namely the cylindrical waves). More recently, a rigorous Fock quantization of this model has been accomplished, which is shown to be essentially unique [12]. Nonetheless, even this Fock quantization (performed after deparametrizing the system) fails to resolve the cosmological singularity.To overcome this problem, we will carry out a hybrid quantization, that combines the polymeric quantization characteristic of LQC applied to the homogeneous solutions (which describe Bianchi I spacetimes) with Fock quantization of the inhomogeneities. This approach investigates the effects on quantum geometry underlying LQC only on the homogeneous sector, while disregards the discreteness of the geometry encoded by the inhomogeneities. A most natural treatment for the inhomogeneities is then the Fock quantization. Indeed, one would expect that a quantum field theory for the inhomogeneities, which can be regarded as a field living on a homogeneous (Bianchi I) background, be approximately valid on the polymerically quantized background. An additional motivation for this approach is the possibility that the inhomogeneities...

We complete the quantization of the vacuum Bianchi I model within the framework of loop quantum cosmology adopting a new improved dynamics scheme put forward recently. In addition, we revisit the hybrid quantization of the Gowdy T 3 cosmologies with linear polarization using that scheme, proving with rigor some steps that remained unconcluded. The family of Gowdy T 3 cosmologies is an inhomogeneous model whose subset of homogeneous solutions is given precisely by the vacuum Bianchi I model. Our hybrid approach combines the new loop quantum cosmology description of this homogeneous sector with a Fock quantization of the inhomogeneities. Both in the Bianchi I model and in the Gowdy model the Hamiltonian constraint provides an evolution equation with respect to the volume of the Bianchi I universe, which is a discrete variable with a strictly positive minimum. We show that, in vacuo, this evolution is well defined inasmuch as the associated initial value problem is well posed: physical solutions are completely determined by the data on the initial section of constant Bianchi I volume. This fact allows us first to carry out to completion the quantization of the vacuum Bianchi I model which had not yet been achieved and then to confirm the feasibility of the hybrid procedure when the homogeneous sector is quantized with the new improved dynamics scheme.