We quantize to completion an inflationary universe with small inhomogeneities in the framework of loop quantum cosmology. The homogeneous setting consists of a massive scalar field propagating in a closed, homogeneous scenario. We provide a complete quantum description of the system employing loop quantization techniques. After introducing small inhomogeneities as scalar perturbations, we identify the true physical degrees of freedom by means of a partial gauge fixing, removing all the local degrees of freedom except the matter perturbations. We finally combine a Fock description for the inhomogeneities with the polymeric quantization of the homogeneous background, providing the quantum Hamiltonian constraint of the composed system. Its solutions are then completely characterized, owing to the suitable choice of quantum constraint, and the physical Hilbert space is constructed. Finally, we consider the analog description for an alternate gauge and, moreover, in terms of gauge-invariant quantities. In the deparametrized model, all these descriptions are unitarily equivalent at the quantum level.

We present a complete quantization of an approximately homogeneous and isotropic universe with small scalar perturbations. We consider the case in which the matter content is a minimally coupled scalar field and the spatial sections are flat and compact, with the topology of a three-torus. The quantization is carried out along the lines that were put forward by the authors in a previous work for spherical topology. The action of the system is truncated at second order in perturbations. The local gauge freedom is fixed at the classical level, although different gauges are discussed and shown to lead to equivalent conclusions. Moreover, descriptions in terms of gauge-invariant quantities are considered. The reduced system is proven to admit a symplectic structure, and its dynamical evolution is dictated by a Hamiltonian constraint. Then, the background geometry is polymerically quantized, while a Fock representation is adopted for the inhomogeneities. The latter is selected by uniqueness criteria adapted from quantum field theory in curved spacetimes, which determine a specific scaling of the perturbations. In our hybrid quantization, we promote the Hamiltonian constraint to an operator on the kinematical Hilbert space. If the zero mode of the scalar field is interpreted as a relational time, a suitable ansatz for the dependence of the physical states on the polymeric degrees of freedom leads to a quantum wave equation for the evolution of the perturbations. Alternatively, the solutions to the quantum constraint can be characterized by their initial data on the minimum-volume section of each superselection sector. The physical implications of this model will be addressed in a future work, in order to check whether they are compatible with observations.Comment: 20 pages, no figures. v2: minor changes, in particular, abstract shortened, final discussion improve

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 geometry. With this ansatz, we derive an approximate Schrödinger equation for the cosmological perturbations and study its range of validity. We also prove that, with an alternate factor ordering, the dynamics deduced for the perturbations is similar to the one found in the so-called dressed metric approach, apart from a possible scaling of the matter field in order to preserve its unitary evolution in the regime of Quantum Field Theory in a curved background and some quantization prescription issues. Finally, we obtain the effective equations that are naturally associated with the Mukhanov-Sasaki variables, both with and without introducing the Born-Oppenheimer ansatz, and with the different factor orderings that we have studied.

We investigate the ambiguities in the Fock quantization of the scalar perturbations of a FriedmannLemaître-Robertson-Walker model with a massive scalar field as matter content. We consider the case of compact spatial sections (thus avoiding infrared divergences), with the topology of a threesphere. After expanding the perturbations in series of eigenfunctions of the Laplace-Beltrami operator, the Hamiltonian of the system is written up to quadratic order in them. We fix the gauge of the local degrees of freedom in two different ways, reaching in both cases the same qualitative results. A canonical transformation, which includes the scaling of the matter field perturbations by the scale factor of the geometry, is performed in order to arrive at a convenient formulation of the system. We then study the quantization of these perturbations in the classical background determined by the homogeneous variables. Based on previous work, we introduce a Fock representation for the perturbations in which: (a) the complex structure is invariant under the isometries of the spatial sections and (b) the field dynamics is implemented as a unitary operator. These two properties select not only a unique unitary equivalence class of representations, but also a preferred field description, picking up a canonical pair of field variables among all those that can be obtained by means of a time-dependent scaling of the matter field (completed into a linear canonical transformation). Finally, we present an equivalent quantization constructed in terms of gauge-invariant quantities. We prove that this quantization can be attained by a mode-by-mode time-dependent linear canonical transformation which admits a unitary implementation, so that it is also uniquely determined.PACS numbers: 98.80. Qc, 04.62.+v, 98.80.Cq

We study the evolution of a homogeneous and isotropic spacetime for which the spatial sections have three-torus topology, coupled to a massless scalar field with small scalar perturbations within loop quantum cosmology. We consider a proposal for the effective dynamics based on a previous hybrid quantization completed by us. Consequently, we introduce a convenient gauge fixing and adopt reduced canonical variables adapted to that hybrid quantum description. Besides, we keep backreaction contributions on the background coming from terms quadratic in the perturbations in the action of the system. We carry out a numerical analysis assuming that the inhomogeneities were in a massless vacuum state at distant past (where the initial data are set). At distant future, we observe a statistical amplification of the modes amplitude in the infrared region, as well as a phase synchronization arising from quantum gravity phenomena. A description of the perturbations in terms of the Mukhanov-Sasaki gauge invariants provides the same qualitative results. Finally, we analyze some consequences of the backreaction in our effective description. I.INTRODUCTIONThe quantization of homogeneous and isotropic spacetimes has been extensively studied in loop quantum cosmology [1] by applying the techniques developed in loop quantum gravity [2] to this kind of symmetry reduced solutions of general relativity. The first model quantized to full completion along these lines was a flat FriedmannRobertson-Walker (FRW) spacetime coupled to a massless scalar field. The study of its dynamics [3,4] showed that the cosmological singularity is resolved and replaced by a quantum bounce (a phenomenon in loop quantum cosmology that can be traced back to the quantum properties of the geometry [5]) and that the evolution preserves remarkably well the semiclassicality of physical states of interest [6,7]. Several other cosmological spacetimes have also been studied in recent years, such as FRW spacetimes with a nonflat spatial topology and/or different kinds of matter content [8], anisotropic cosmologies of various Bianchi types [9], and even inhomogeneous cosmological models, like, e.g., the case of Gowdy spacetimes [10].Inflationary cosmologies with small inhomogeneities [11] are currently considered in loop quantum cosmology to reach a more accurate description of the early evolution of the Universe. In particular, one of the simplest scenarios consists of a perturbed flat FRW spacetime coupled to a massive scalar field. In the unperturbed (i.e., homogeneous) case, the effective dynamics argued to arise in loop quantum cosmology [12] has been studied in some detail ] 13 ], with interesting conclusions: the probability of having sufficient inflation (or, more precisely, a number of e-foldings compatible with the cosmological observations) is high enough so as to solve the fine-tuning problem present in general relativity. However, the inclusion of inhomogeneities is a necessary step to treat more realistic situations. A formal, complete, and rigorous qua...