Nonsingular bouncing universes in loop quantum cosmology

In isotropic loop quantum cosmology, non-perturbatively modified dynamics of a minimally coupled scalar field violates weak, strong and dominant energy conditions when they are stated in terms of the equation-of-state parameter. The violation of the strong energy condition helps to have non-singular evolution by evading singularity theorems thus leading to a generic inflationary phase. However, the violation of weak and dominant energy conditions raises concern, as in general relativity these conditions ensure causality of the system and stability of vacuum via the Hawking–Ellis conservation theorem. It is shown here that the non-perturbatively modified kinetic term contributes negative pressure but positive energy density. This crucial feature leads to the violation of energy conditions but ensures the positivity of energy density, as the scalar matter Hamiltonian remains bounded from below. It is also shown that the modified dynamics restricts group velocity for inhomogeneous modes to remain sub-luminal thus ensuring causal propagation across spatial distances.

We consider an isotropic and homogeneous universe in loop quantum cosmology (LQC). We assume that the matter content of the universe is dominated by dust matter in early time and a phantom matter at late time which constitutes the dark energy component. The quantum gravity modifications to the Friedmann equation in this model indicate that the classical big bang singularity and the future big rip singularity are resolved and are replaced by quantum bounce. It turns out that the big bounce and recollapse in the herein model contribute to a cyclic scenario for the universe. We then study the quantum theory of a massive, nonminimally coupled scalar field undergoing cosmological evolution from primordial bounce towards the late time bounce. In particular, we solve the Klein–Gordon equation for the scalar field in the primordial and late time regions, in order to investigate particle production phenomena at late time. By computing the energy density of created particles at late time, we show that this density is negligible in comparison to the quantum background density at Planck era. This indicates that the effects of quantum particle production do not influence the future bounce.

The matter bounce scenario is an alternative to inflation where scale-invariant perturbations are generated in a dust-dominated contracting space-time.In the context of loop quantum cosmology, the cosmological singularity is generically resolved and replaced by a bounce; furthermore, it is possible to deterministically evolve the perturbations from the contracting branch through the quantum bounce and calculate the form of the perturbations at the onset of the expanding branch.For the matter bounce scenario in loop quantum cosmology, scale-invariant perturbations and a small tensor-to-scalar ratio are predicted [1], in agreement with the latest observations of Planck and BICEP2 [2].

In this article, I mainly discuss the dynamics of the pre-inflationary Universe for the potential V ( ϕ ) ∝ ϕ n with n = 5 / 3 in the context of loop quantum cosmology, in which the big bang singularity is resolved by a non-singular quantum bounce. In the case of the kinetic energy-dominated initial conditions of the scalar field at the bounce, the numerical evolution of the Universe can be split up into three regimes: bouncing, transition, and slow-roll inflation. In the bouncing regime, the numerical evolution of the scale factor does not depend on a wide range of initial values, or on the inflationary potentials. I calculate the number of e-folds in the slow-roll regime, by which observationally identified initial conditions are obtained. Additionally, I display the phase portrait for the model under consideration.

Quantum gravity is expected to be necessary in order to understand situations in which classical general relativity breaks down. In particular in cosmology one has to deal with initial singularities, i.e., the fact that the backward evolution of a classical spacetime inevitably comes to an end after a finite amount of proper time. This presents a breakdown of the classical picture and requires an extended theory for a meaningful description. Since small length scales and high curvatures are involved, quantum effects must play a role. Not only the singularity itself but also the surrounding spacetime is then modified. One particular theory is loop quantum cosmology, an application of loop quantum gravity to homogeneous systems, which removes classical singularities. Its implications can be studied at different levels. The main effects are introduced into effective classical equations, which allow one to avoid the interpretational problems of quantum theory. They give rise to new kinds of early-universe phenomenology with applications to inflation and cyclic models. To resolve classical singularities and to understand the structure of geometry around them, the quantum description is necessary. Classical evolution is then replaced by a difference equation for a wave function, which allows an extension of quantum spacetime beyond classical singularities. One main question is how these homogeneous scenarios are related to full loop quantum gravity, which can be dealt with at the level of distributional symmetric states. Finally, the new structure of spacetime arising in loop quantum gravity and its application to cosmology sheds light on more general issues, such as the nature of time.Electronic Supplementary MaterialSupplementary material is available for this article at 10.12942/lrr-2008-4.

Quantum geometry effects in loop quantum cosmology are known to resolve various singularities in the isotropic and anisotropic models. Previous investigations have shown that singularity resolution is tied to boundedness of physical quantities such as the energy density and the expansion and shear scalars in the spacetime. Using effective dynamics, we show that these do not necessarily lead to a generic boundedness of spacetime curvature in the Bianchi-I model and find the potential possibilities where the curvature invariants can diverge. These include cases where the pressure of the matter content diverges at a finite volume and energy density. By analyzing the strength of singularities, we show that such pressure singularities are weak. All known types of strong singularities of the classical theory are resolved. This analysis brings us closer to generalize the results on generic resolution of singularities in the isotropic models in loop quantum cosmology, to the anisotropic spacetimes.

We have studied homogeneous isotropic FRW model having dynamical dark energy DBI-essence with scalar field. The existence of cosmological scaling solutions restricts the Lagrangian of the scalar field $\phi$. Choosing $p=X g(X e^{\lambda \phi})$, where $X=-g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi /2$ with $g$ is any function of $X e^{\lambda \phi}$ and defining some suitable transformations, we have constructed the dynamical system in different gravity: (i) Loop Quantum Cosmology (LQC), (ii) DGP BraneWorld and (iii) RS-II Brane World. We have investigated the stability of this dynamical system around the critical point for three gravity models and investigated the scalar field dominated attractor solution in support of accelerated universe. The role of physical parameters have also been shown graphically during accelerating phase of the universe.

Recently, considerable progress has been made in understanding the early universe by loop quantum cosmology. Modesto et al. investigated the loop quantum black hole (LQBH)using improved semiclassical analysis and they found that the LQBH has two horizons, an event horizon and a Cauchy horizon, just like the Reissner—Nordström black hole. This paper focuses on the dynamical evolution of a massless scalar wave in the LQBH background. By investigating the relation between the complex frequencies of the massless scalar field and the LQBH parameters using the numerical method, we find that the polymeric parameter P makes the massless scalar field decay more quickly and makes the ground scalar wave oscillate slowly. However, the polymeric parameter P causes the frequency of the high harmonic massless scalar wave to shift according to its value. We also find that the loop quantum gravity area gap parameter α0 causes the massless scalar field to decay more slowly and makes the period of the massless scalar field wave become longer. In the complex ω plane, the frequency curves move counterclockwise when the polymeric parameter P increases and this spiral effect is more obvious for a higher harmonic scalar wave.

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.

It is stated that holonomy corrections inloop quantum cosmology introduce a modification in Friedmann's equation which prevent the big rip singularity. Recently in [1] it has been proved that this modified Friedmann equation isobtained in an inconsistent way, whatmeans that the results deduced from it, in particular the big rip singularity avoidance, are not justified. The problem isthat holonomy corrections modify the gravitational part of the Hamiltonian of the system leading, after Legendre's transformation, toa non covariant Lagrangian which is in contradiction with one of the main principles of General Relativity. A more consistent way to deal with the bigrip singularity avoidance isto disregard modification in the gravitational part of the Hamiltonian, andonly consider inverse volume effects [2]. In this case we will see that, not like the big bang singularity, thebig rip singularity survives in loop quantum cosmology.Another way to deal with the big rip avoidance is to take into account geometric quantum effectsgiven by the the Wheeler-De Witt equation. In that case, even though the wave packets spread, theexpectation values satisfy the same equations as their classical analogues. Then, following the viewpoint adopted in loop quantum cosmology, onecan conclude that the big rip singularity survives when one takes into account these quantum effects.However, the spreading of the wave packets prevents the recover of the semiclassical time, and thus, one might conclude thatthe classicalevolution of the universe come to and end before the big rip is reached. This is not conclusive because. as we will see, it always exists other external timesthat allows us to define the classical andquantum evolution of the universe up to the big rip singularity.

Note on the super-inflation in loop quantum cosmology

The k=0 Friedmann Lemaitre Robertson Walker model with a positive cosmological constant and a massless scalar field is analyzed in detail. If one uses the scalar field as relational time, new features arise already in the Hamiltonian framework of classical general relativity: In a finite interval of relational time, the universe expands out to infinite proper time and zero matter density. In the deparameterized quantum theory, the true Hamiltonian now fails to be essentially self-adjoint both in the Wheeler DeWitt (WDW) approach and in LQC. Irrespective of the choice of the self-adjoint extension, the big bang singularity persists in the WDW theory while it is resolved and replaced by a big bounce in loop quantum cosmology (LQC). Furthermore, the quantum evolution is surprisingly insensitive to the choice of the self-adjoint extension. This may be a special case of an yet to be discovered general property of a certain class of symmetric operators that fail to be essentially self-adjoint.

Wormhole and time machine are very interesting objects in general relativity. However, they need exotic matters which are impossible in classical level to support them. But if we introduce the quantum effects of gravity into the stress-energy tensor, these peculiar objects can be constructed self-consistently. Fortunately, loop quantum cosmology (LQC) has the potential to serve as a bridge connecting the classical theory and quantum gravity. Therefore it provides a simple way for the study of quantum effect in the semiclassical case. As is well known, loop quantum cosmology is very successful to deal with the behavior of early universe. In the early stage, if taken the quantum effect into consideration, inflation is natural because of the violation of every kind of local energy conditions. Similar to the inflationary universe, the violation of the averaged null energy condition is the necessary condition for the traversable wormholes. In this paper, we investigate the averaged null energy condition in LQC in the framework of effective Hamiltonian, and find out that LQC do violate the averaged null energy condition in the massless scalar field coupled model.

One of the greatest problems of primordial inflation is that the inflationary space-time is past-incomplete. This is mainly because Einstein's GR suffers from a space-like Big Bang singularity. It has recently been shown that ghost-free, non-local higher-derivative ultra-violet modifications of Einstein's gravity may be able to resolve the cosmological Big Bang singularity via a non-singular bounce. Within the framework of such non-local cosmological models, we are going to study both sub- and super-Hubble perturbations around an inflationary trajectory which is preceded by the Big Bounce in the past, and demonstrate that the inflationary trajectory has an ultra-violet completion and that perturbations do not suffer from any pathologies.

We provide a simple analytical solution of the Friedmann equations for a universe made of stiff matter, dust matter, and dark energy. A stiff matter era is present in the cosmological model of Zel'dovich (1972) where the primordial universe is assumed to be made of a cold gas of baryons. It also occurs in certain cosmological models where dark matter is made of relativistic self-gravitating Bose-Einstein condensates (BECs). When the energy density of the stiff matter is positive, the primordial universe is singular. It starts from a state with a vanishing scale factor and an infinite density. We consider the possibility that the energy density of the stiff matter is negative (anti-stiff matter). This happens, for example, when the BECs have an attractive self-interaction. In that case, the primordial universe is non-singular. It starts from a state in which the scale factor is finite and the energy density is equal to zero. For the sake of generality, we consider a cosmological constant of arbitrary sign. When the cosmological constant is positive, the universe asymptotically reaches a de Sitter phase where the scale factor increases exponentially rapidly. This can account for the accelerating expansion of the universe that we observe at present. When the cosmological constant is negative (anti-de Sitter), the evolution of the universe is cyclic. Therefore, depending on the sign of the energy density of the stiff matter and of the dark energy, we obtain singular and non-singular expanding or cyclic universes.