Effective dynamics of scalar perturbations in a flat Friedmann-Robertson-Walker spacetime in loop quantum cosmology

Abstract

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...

Cosmological perturbations in hybrid loop quantum cosmology: Mukhanov-Sasaki variables

“…Recall, in this sense, that except for a different scaling of the inhomogeneous modes in the matter field and the associated MS variables, the quadratic contribution to the constraint Θ e + Θ o π φ is just the MS Hamiltonian for the inhomogeneities which generates their evolution in the time T with dt = 2V dT in the classical theory, with t being the proper time [see Eq. (27) and the definition of the homogeneous part of the lapse function in Eq. (3)].…”

“…In this section, we will provide the effective equations for the MS variables in the quantization schemes that we have been discussing, extrapolating the experience gained in homogeneous models and assuming a direct relation between the annihilation and creation (23) and (27)] and the definition of the homogeneous part of the lapse function, it is not difficult to realize thatĈ per /2 generates evolution in a timeT that, at leading perturbative order, is related with the proper one by dt = V dT (the factor of 1/2 in the constraint is introduced here for later convenience).…”

“…), and the introduction of a structure of Poisson brackets for the expectation values and moments of the basic variables, this scheme allows one to study the modified field equations for the perturbations. The other line of attack deals with the direct quantization of the FRW geometry and the perturbations [25][26][27][28][29][30]. In principle, both types of approaches are complementary, since some of the assumptions used in the derivation of effective equations from the closure of the algebra would ultimately be possible to check only when one has at his disposal a genuine quantum treatment.…”

Fermions in hybrid loop quantum cosmology

“…Another important issue is the backreaction that the quantum fermions produce onto the geometry. There is an increasing interest on this problem in LQC, and although some very preliminary discussions have been carried out in different contexts [56,57], the development of fully self-consistent formalisms to study this backreaction is a necessary step before dealing successfully with it. These are the questions and problems that we want to discuss in this work.…”

Quantum corrections to the Mukhanov-Sasaki equations

“…Many of the studies about the consequences of quantum cosmology in observational astrophysics and, in particular, in the CMB have been carried out within the framework of loop quantum cosmology (LQC) [7][8][9][10][11][12][13][14]. However, the discussion is not limited to that formalism by any means [15], and other formalisms like quantum geometrodynamics [16,17] have also been explored.…”