The "improved dynamics" of loop quantum cosmology is extended to include anisotropies of the Bianchi I model. As in the isotropic case, a massless scalar field serves as a relational time parameter. However, the extension is non-trivial because one has to face several conceptual subtleties as well as technical difficulties. These include: a better understanding of the relation between loop quantum gravity (LQG) and loop quantum cosmology (LQC); handling novel features associated with the non-local field strength operator in presence of anisotropies; and finding dynamical variables that make the action of the Hamiltonian constraint manageable. Our analysis provides a conceptually complete description that overcomes limitations of earlier works. We again find that the big bang singularity is resolved by quantum geometry effects but, because of the presence of Weyl curvature, Planck scale physics is now much richer than in the isotropic case. Since the Bianchi I models play a key role in the Belinskii, Khalatnikov, Lifshitz (BKL) conjecture on the nature of generic space-like singularities in general relativity, the quantum dynamics of Bianchi I cosmologies is likely to provide considerable intuition about the fate of generic space-like singularities in quantum gravity. Finally, we show that the quantum dynamics of Bianchi I cosmologies projects down exactly to that of the Friedmann model. This opens a new avenue to relate more complicated models to simpler ones, thereby providing a new tool to relate the quantum dynamics of LQG to that of LQC.

We study the effective cosmological dynamics, emerging as the hydrodynamics of simple condensate states, of a group field theory model for quantum gravity coupled to a massless scalar field and reduced to its isotropic sector. The quantum equations of motion for these group field theory condensate states are given in relational terms with respect to the scalar field, from which effective dynamics for spatially flat, homogeneous and isotropic space-times can be extracted. The result is a generalization of the Friedmann equations, including quantum gravity modifications, in a specific regime of the theory corresponding to a Gross-Pitaevskii approximation where interactions are subdominant. The classical Friedmann equations of general relativity are recovered in a suitable semi-classical limit for some range of parameters of the microscopic dynamics. An important result is that the quantum geometries associated with these GFT condensate states are non-singular: a bounce generically occurs in the Planck regime. For some choices of condensate states, these modified Friedmann equations are very similar to those of loop quantum cosmology.

The improved dynamics of loop quantum cosmology is extended to include the Bianchi type II model. Because these space-times admit both anisotropies and non-zero spatial curvature, certain technical difficulties arise over and above those encountered in the analysis of the (anisotropic but spatially flat) Bianchi type I space-times, and of the (spatially curved but isotropic) k=+/-1 models. We address these and show that the big-bang singularity is resolved in the same precise sense as in the recent analysis of the Bianchi I model. Bianchi II space-times are of special interest to quantum cosmology because of the expected behavior of the gravitational field near generic space-like singularities in classical general relativity.Comment: 26 page

The loop quantum cosmology "improved dynamics" of the Bianchi type IX model are studied. The action of the Hamiltonian constraint operator is obtained via techniques developed for the Bianchi type I and type II models, no new input is required. It is shown that the big bang and big crunch singularities are resolved by quantum gravity effects. We also present the effective equations which provide modifications to the classical equations of motion due to quantum geometry effects. PACS numbers: 98.80.Qc,04.60.Pp,04.60.-m I. INTRODUCTIONLoop quantum cosmology (LQC) [1,2] is an approach to quantum cosmology following the ideas of loop quantum gravity (LQG) [3][4][5]. One of the major results of LQC in the homogeneous and isotropic Friedmann-Robertson-Walker (FRW) models is that, while general relativity approximates the dynamics very well in the low (with respect to the Planck scale) curvature regime, the classical big bang singularity is avoided: when the matter energy density approaches the Planck energy density, deviations from general relativity become significant and a "quantum bounce" due to quantum gravity effects occurs when the matter energy density reaches a critical energy density of the order of the the Planck density [6][7][8][9][10][11][12][13][14]. More recently, it has been shown that the singularity is also resolved in the improved dynamics approach of loop quantum cosmology in the anisotropic Bianchi type I and type II cosmological models [15,16] and in the hybrid loop-Fock quantization of the inhomogeneous Gowdy model [17]. The goal of this paper is to extend the LQC improved dynamics analysis of the Bianchi type I and type II models to the more complicated Bianchi type IX models.At the classical level, the Bianchi IX model has a much richer phenomenology than Bianchi I and II models as it displays Mixmaster dynamics as the singularity is approached [18]. In essence, a space-time which exhibits Mixmaster dynamics is one which can be described for long periods of time (known as epochs) as a Bianchi I space-time characterized by three anisotropic expansion rates. Such a space-time will occasionally undergo a "Mixmaster bounce" from one epoch to another where the three expansion rates change in a specific manner. Bianchi I models approach the singularity in a rather straightforward way as they do not undergo any Mixmaster bounces while Bianchi II models may undergo a single Mixmaster bounce between two epochs as the singularity is approached (see [18] and references therein). The Bianchi IX model, on the other hand, undergoes many Mixmaster bounces and this behaviour is chaotic [18,19]. Since much of this behaviour occurs when the curvature is of the Planck scale, quantum gravity effects cannot be neglected and the Mixmaster behaviour may be significantly modified when they are taken into account.

We show how the large-scale cosmological dynamics can be obtained from the hydrodynamics of isotropic group field theory condensate states in the Gross-Pitaevskii approximation. The correct Friedmann equations are recovered in the classical limit for some choices of the parameters in the action for the group field theory, and quantum gravity corrections arise in the high-curvature regime causing a bounce which generically resolves the big-bang and big-crunch singularities.

In the matter bounce scenario, a dust-dominated contracting space-time generates scale-invariant perturbations that, assuming a nonsingular bouncing cosmology, propagate to the expanding branch and set appropriate initial conditions for the radiation-dominated era. Since this scenario depends on the presence of a bounce, it seems appropriate to consider it in the context of loop quantum cosmology where a bouncing universe naturally arises. For a pressureless collapsing universe in loop quantum cosmology, the predicted power spectrum of the scalar perturbations after the bounce is scale-invariant and the tensor to scalar ratio is negligibly small. A slight red tilt can be given to the scale-invariance of the scalar perturbations by a scalar field whose equation of state is P = −ǫρ, where ǫ is a small positive number. Then, the power spectrum for tensor perturbations is also almost scale-invariant with the same red tilt as the scalar perturbations, and the tensor to scalar ratio is expected to be r ≈ 9 × 10 −4 . Finally, for the predicted amplitude of the scalar perturbations to agree with observations, the critical density in loop quantum cosmology must be of the order ρc ∼ 10 −9 ρ Pl .

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.

A non-singular bouncing cosmology is generically obtained in loop quantum cosmology due to non-perturbative quantum gravity effects. A similar picture can be achieved in standard general relativity in the presence of a scalar field with a non-standard kinetic term such that at high energy densities the field evolves into a ghost condensate and causes a non-singular bounce. During the bouncing phase, the perturbations can be stabilized by introducing a Horndeski operator. Taking the matter content to be a dust field and an ekpyrotic scalar field, we compare the dynamics in loop quantum cosmology and in a non-singular bouncing effective field model with a non-standard kinetic term at both the background and perturbative levels. We find that these two settings share many important properties, including the result that they both generate scale-invariant scalar perturbations. This shows that some quantum gravity effects of the very early universe may be mimicked by effective field models.