Dynamics Of Loop Quantum Cosmology Research Articles

Group theoretical quantization of isotropic loop cosmology

We achieve a group theoretical quantization of the flat Friedmann-Robertson-Walker model coupled to a massless scalar field adopting the improved dynamics of loop quantum cosmology. Deparemetrizing the system using the scalar field as internal time, we first identify a complete set of phase space observables whose Poisson algebra is isomorphic to the $\mathfrak{s}\mathfrak{u}(1,1)$ Lie algebra. It is generated by the volume observable and the Hamiltonian. These observables describe faithfully the regularized phase space underlying the loop quantization: they account for the polymerization of the variable conjugate to the volume and for the existence of a kinematical nonvanishing minimum volume. Since the Hamiltonian is an element in the $\mathfrak{s}\mathfrak{u}(1,1)$ Lie algebra, the dynamics is now implemented as SU(1, 1) transformations. At the quantum level, the system is quantized as a timelike irreducible representation of the group SU(1, 1). These representations are labeled by a half-integer spin, which gives the minimal volume. They provide superselection sectors without quantization anomalies and no factor ordering ambiguity arises when representing the Hamiltonian. We then explicitly construct SU(1, 1) coherent states to study the quantum evolution. They not only provide semiclassical states but truly dynamical coherent states. Their use further clarifies the nature of the bounce that resolves the big bang singularity.

Effective dynamics in Bianchi type II loop quantum cosmology

We numerically investigate the solutions to the effective equations of the Bianchi II model within the ``improved'' loop quantum cosmology dynamics. The matter source is a massless scalar field. We perform a systematic study of the space of solutions, and focus on the behavior of several geometrical observables. We show that the big bang singularity is replaced by a bounce and the pointlike singularities do not saturate the energy density bound. There are up to three directional bounces in the scale factors, one global bounce in the expansion, the shear presents up to four local maxima and can be zero at the bounce. This allows for solutions with density larger than the maximal density for the isotropic and Bianchi I cases. The asymptotic behavior is shown to behave like that of a Bianchi I model, and the effective solutions connect anisotropic solutions even when the shear is zero at the bounce. All known facts of Bianchi I are reproduced. In the ``vacuum limit,'' solutions are such that almost all the dynamics is due to the anisotropies. Since Bianchi II plays an important role in the Bianchi IX model and the Belinskii, Khalatnikov, Lifshitz conjecture, our results can provide an intuitive understanding of the behavior in the vicinity of general spacelike singularities, when loop-geometric corrections are present.

Palatini approach to bouncing cosmologies and DSR-like effects

It is shown that a quadratic gravitational Lagrangian in the Palatini formulation is able to capture different aspects of quantum gravity phenomenology in a single framework. In particular, in this theory field excitations propagating with different energy-densities perceive different background metrics, a fundamental characteristic of the DSR and Rainbow Gravity approaches. This theory, however, avoids the so-called soccer ball problem. Also, the resulting isotropic and anisotropic cosmologies are free from the big bang singularity. This singularity avoidance occurs non-perturbatively and shares some similitudes with the effective dynamics of loop quantum cosmology.

Path Integrals and Alternative Effective Dynamics in Loop Quantum Cosmology

The alternative dynamics of loop quantum cosmology is examined by the path integral formulation. We consider the spatially flat FRW models with a massless scalar field, where the alternative quantizations inherit more features from full loop quantum gravity. The path integrals can be formulated in both timeless and deparameterized frameworks. It turns out that the effective Hamiltonians derived from the two different viewpoints are equivalent to each other. Moreover, the first-order modified Friedmann equations are derived and predict quantum bounces for contracting universe, which coincide with those obtained in canonical theory.

Non-singular Universes à la Palatini

It has recently been shown that f(R) theories formulated in the Palatini variational formalism are able to avoid the big bang singularity yielding instead a bouncing solution. The mechanism responsible for this behavior is similar to that observed in the effective dynamics of loop quantum cosmology and an f(R) theory exactly reproducing that dynamics has been found. I will show here that considering more general actions, with quadratic contributions of the Ricci tensor, results in a much richer phenomenology that yields bouncing solutions even in anisotropic (Bianchi I) scenarios. Some implications of these results will be discussed.

Probability of inflation in loop quantum cosmology

Inflationary models of the early universe provide a natural mechanism for the formation of large scale structure. This success brings to forefront the question of naturalness: Does a sufficiently long slow roll inflation occur generically or does it require a careful fine tuning of initial parameters? In recent years there has been considerable controversy on this issue. In particular, for a quadratic potential, Kofman, Linde and Mukhanov have argued that the probability of inflation with at least 65 e-foldings is close to one, while Gibbons and Turok have argued that this probability is suppressed by a factor of ~ $\10^{-85}$. We first clarify that such dramatically different predictions can arise because the required measure on the space of solutions is intrinsically ambiguous in general relativity. We then show that this ambiguity can be naturally resolved in loop quantum cosmology (LQC) because the big bang is replaced by a big bounce and the bounce surface can be used to introduce the structure necessary to specify a satisfactory measure. The second goal of the paper is to present a detailed analysis of the inflationary dynamics of LQC using analytical and numerical methods. By combining this information with the measure on the space of solutions, we address a sharper question than those investigated in the literature: What is the probability of a sufficiently long slow roll inflation WHICH IS COMPATIBLE WITH THE SEVEN YEAR WMAP DATA? We show that the probability is very close to 1. The material is so organized that cosmologists who may be more interested in the inflationary dynamics in LQC than in the subtleties associated with measures can skip that material without loss of continuity.

Loop quantum cosmology of Bianchi type II models

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.

Bouncing cosmologies in Palatinif(R)gravity

We consider the early time cosmology of $f(R)$ theories in Palatini formalism and study the conditions that guarantee the existence of homogeneous and isotropic models that avoid the big bang singularity. We show that for such models the big bang singularity can be replaced by a cosmic bounce without violating any energy condition. In fact, the bounce is possible even for pressureless dust. We give a characterization of such models and discuss their dynamics in the region near the bounce. We also find that power-law Lagrangians with a finite number of terms may lead to nonsingular universes, which contrasts with the infinite-series Palatini $f(R)$ Lagrangian that one needs to fully capture the effective dynamics of loop quantum cosmology. We argue that these models could also avoid the formation of singularities during stellar gravitational collapse.

Modification of Heisenberg uncertainty relations in noncommutative Snyder space-time geometry

We show that the Euclidean Snyder non-commutative space implies infinitely many different physical predictions. The distinct frameworks are specified by generalized uncertainty relations underlying deformed Heisenberg algebras. Considering the one-dimensional case in the minisuperspace arena, the bouncing Universe dynamics of loop quantum cosmology can be recovered.

Singularities in Loop Quantum Cosmology

We show that simple scalar field models can give rise to curvature singularities in the effective Friedmann dynamics of loop quantum cosmology (LQC). We find singular solutions for spatially flat Friedmann-Robertson-Walker cosmologies with a canonical scalar field and a negative exponential potential, or with a phantom scalar field and a positive potential. While LQC avoids big bang or big rip type singularities, we find sudden singularities where the Hubble rate is bounded, but the Ricci curvature scalar diverges. We conclude that the effective equations of LQC are not in themselves sufficient to avoid the occurrence of curvature singularities.

Is loop quantization in cosmology unique?

We reexamine the process of loop quantization for flat isotropic models in cosmology. In particular, we contrast different inequivalent ``loop quantizations'' of these simple models through their respective successes and limitations and assess whether they can lead to any viable physical description. We propose three simple requirements which any such admissible quantum model should satisfy: (i) independence from any auxiliary structure, such as a fiducial interval/cell introduced to define the phase space when integrating over noncompact manifolds; (ii) existence of a well defined classical limit, and (iii) a sensible ``Planck scale'' where quantum gravitational effects become manifest. We show that even when it may seem that one can have several possible loop quantizations, these physical requirements considerably narrow down the consistent choices. Apart from the so-called improved dynamics of loop quantum cosmology, none of the other available inequivalent loop quantizations pass the above tests, showing the limitations of lattice refinement models to approximate the homogeneous sector and loop modified quantum geometrodynamics. We conclude that amongst a large class of loop quantizations in isotropic cosmology, there is a unique consistent choice.

Anti-de Sitter universe dynamics in loop quantum cosmology

A model for a flat isotropic universe with a negative cosmological constant $\ensuremath{\Lambda}$ and a massless scalar field as sole matter content is studied within the framework of loop quantum cosmology. By application of the methods introduced for the model with $\ensuremath{\Lambda}=0$, the physical Hilbert space and the set of Dirac observables are constructed. As in that case, the scalar field plays here the role of an emergent time. The properties of the system are found to be similar to those of the $k=1$ Friedmann-Robertson-Walker (FRW) model: for small energy densities, the quantum dynamics reproduces the classical one, whereas, due to modifications at near-Planckian densities, the big bang and big crunch singularities are replaced by a quantum bounce connecting deterministically the large semiclassical epochs. Thus in loop quantum cosmology the evolution is qualitatively cyclic.

Phantom field dynamics in loop quantum cosmology

We consider a dynamical system of phantom scalar field under exponential potential in the background of loop quantum cosmology. In our analysis, there is neither stable node nor repeller unstable node but only two saddle points, hence no big rip singularity. Physical solutions always possess potential energy greater than the magnitude of the negative kinetic energy. We found that the universe bounces after accelerating even in the domination of the phantom field. After bouncing, the universe finally enters the oscillatory regime.

Avoidance of future singularities in loop quantum cosmology

We consider the fate of future singularities in the effective dynamics of loop quantum cosmology. Nonperturbative quantum geometric effects which lead to ${\ensuremath{\rho}}^{2}$ modification of the Friedmann equation at high energies result in generic resolution of singularities whenever energy density $\ensuremath{\rho}$ diverges at future singularities of Friedmann dynamics. Such quantum effects lead to the avoidance of a big rip, which is followed by a recollapsing universe stable against perturbations. Resolution of sudden singularity, the case when pressure diverges but energy density approaches a finite value depends on the ratio of the latter to a critical energy density of the order of the Planck value. If the value of this ratio is greater than unity, the universe escapes the sudden future singularity and becomes oscillatory.