Hybrid quantum Gowdy cosmology: Combining loop and Fock quantizations

Abstract

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...

Loop quantum cosmology of diagonal Bianchi type I model: Simplifications and scaling problems

“…This motivates us to investigate similar simplifications in the LQC diagonal Bianchi I model. Similar results have also been obtained independently by the Madrid group [23,24].…”

Nonlinear (loop) quantum cosmology

“…One of the alternatives is so-called hybrid quantization [22], in which one combines a loop-quantized homogeneous background model with Fock quantized inhomogeneous modes. Wave functions can then be solved for and evolved, at least numerically and with certain truncations [23].…”

Loop Quantum Cosmology

“…Effective dynamics, which captures the key features of underlying quantum geometry, provides an excellent approximation of the quantum evolution and has been extensively used to extract novel physical predictions, such as, on bounds on energy density, expansion and shear scalars [45,47,50], resolution of strong singularities [45,46] and probability for inflation [72,73,74]. These have been also used to probe the physics beyond homogeneity approximation such as in Gowdy models [51] and imprint of quantum geometry on cosmological perturbations [52], which have also been derived using detailed properties of quantum spacetime [54]. It is hoped that the ongoing investigations on primordial perturbations will provide potential tests for LQC in near future astronomical experiments.…”

“…As an example, using effective spacetime description, singularity resolution in LQC can be understood occurring due to bounds on the growth of spacetime curvature, which plays a conjugate role to geometry [45,46,47], and insights on the generic resolution of strong singularities and geodesic extendability in flat isotropic [45,48] and Bianchi-I models [46] have been obtained. Effective equations have also been used to gain insights on constraining quantization ambiguities [49,50], and have been used to explore the physics of Gowdy models [51]. A lot of activity is devoted to applying effective equations to understand the signatures of quantum geometry in cosmological perturbations [52].…”