Loop Cosmology Research Articles

Loop quantum gravity effects might restrict a cyclic evolution

It is generally expected that in a non-singular cosmological model a cyclic evolution is straightforward to obtain on introduction of a suitable choice of a scalar field with a negative potential or a negative cosmological constant which causes a recollapse at some time in the evolution. We present a counter example to this conventional wisdom. Working in the realm of loop cosmological models with non-perturbative quantum gravity modifications we show that a modified version of standard loop quantum cosmology based on Thiemann's regularization of the Hamiltonian constraint while generically non-singular does not allow a cyclic evolution unless some highly restrictive conditions hold. Irrespective of the energy density of other matter fields, a recollapse and hence a cyclic evolution is only possible if one chooses an almost Planck sized negative potential of the scalar field or a negative cosmological constant. Further, cycles when present do not occur in the classical regime. Surprisingly, a necessary condition for a cyclic evolution, not singularity resolution, turns out to be a violation of the weak energy condition. These results are in a striking contrast to standard loop quantum cosmology where obtaining a recollapse at large volumes and a cyclic evolution is straightforward, and, there is no violation of weak energy condition. On one hand our work shows that some quantum cosmological models even though non-singular and bouncing are incompatible with a cyclic evolution, and on the other hand demonstrates that differences in various quantization prescriptions in loop cosmology need not be faint and buried in the pre-bounce regime, but can be striking and profound even in the post-bounce regime.

Bounce models within teleparallel modified gravity

In this paper, working in a Friedman-Lemaitre-Robertson-Walker (FLRW), first, in the flat case, we recover the generalized Friedman equation of Quantum Loop cosmology, and therefore the cosmological bounce, in the framework of modified teleparallel gravity $f(T)$-model, $T$ being the torsion scalar introduced in teleparallel gravity approach, without invoking unconventional exotic matter. Furthermore we study the associated perturbations again in a flat FLRW space-time. Then, we generalize the results to the curved FLRW space-time, where some issues related to the choice of tetrad exist, by using an appropriate formulation. In this context, the results of Born-Infeld model are also investigated.

Tachyon field in loop cosmology

The evolutionary pictures of tachyon field in modified loop cosmology have been investigated. We present the dynamical behavior of the tachyon field associated with an exponential potential and find that the pre-inflation dynamics are very similar in both modified loop cosmology and standard loop quantum cosmology. In addition, tachyonic inflation in modified loop cosmology models are discussed, and we find that the probability of inflation in modified loop cosmology is very closer to 1.

Note on nonsingular Einstein-Aether cosmologies

An effective Lagrangian approach based on an extended Einstein-Aether (EA) model is presented and formulated in a generic non flat Fridman-Lemaitre-Robertson-Walker (FLRW) space-time. For a flat FLRW space-time, a Friedmann equation similar to the one obtained in Quantum Loop Cosmology (QLC) is reproduced, with related non singular bounce solution. Finally, the Static Spherically Symmetric (SSS) solutions are investigated.

Higher derivative and mimetic models on non flat FLRW space–times

Abstract An effective Lagrangian approach, partly inspired by Quantum Loop Cosmology (QLC), is presented and formulated in a non flat FLRW space–times, making use of modified gravitational models. The models considered are non generic, and their choice is dictated by the necessity to have at least second order differential equations of motion in a non flat FLRW space–time. This is accomplished by a class of Lagrangian which are not analytic in the curvature invariants, or making use of a mimetic gravitational scalar field. In this paper we want to show that, for some effective models, although the associated generalized Friedmann equation may admit non singular metrics as solutions, the de Sitter space–time is present only for a restricted class, which includes General Relativity and Lovelock gravity. The other models admit pseudo de Sitter solutions, namely FLRW metrics, such that for vanishing spatial curvature looks like flat de Sitter patch, but for non vanishing spatial curvature are regular bounce metrics or, when the spatial curvature is negative, Big-Bang singular metrics.

Qualitative dynamics and inflationary attractors in loop cosmology

Qualitative dynamics of three different loop quantizations of spatially flat isotropic and homogeneous models is studied using effective spacetime description of the underlying quantum geometry. These include the standard loop quantum cosmology (LQC), its recently revived modification (referred to as mLQC-I), and another related modification of LQC (mLQC-II) whose dynamics is studied in detail for the first time. Various features of LQC, including quantum bounce and pre-inflationary dynamics, are found to be shared with the mLQC-I and mLQC-II models. We study universal properties of dynamics for chaotic inflation, fractional monodromy inflation, Starobinsky potential, non-minimal Higgs inflation, and an exponential potential. We find various critical points and study their stability, which reveal various qualitative similarities in the post-bounce phase for all these models. The pre-bounce qualitative dynamics of LQC and mLQC-II turns out to be very similar, but is strikingly different from that of mLQC-I. In the dynamical analysis, some of the fixed points turn out to be degenerate for which center manifold theory is used. For all these potentials, non-perturbative quantum gravitational effects always result in a non-singular inflationary scenario with a phase of super-inflation succeeded by the conventional inflation. We show the existence of inflationary attractors, and obtain scaling solutions in the case of the exponential potential. Since all of the models agree with general relativity at late times, our results are also of use in classical theory where qualitative dynamics of some of the potentials has not been studied earlier.

Brane-world and loop cosmology from a gravity–matter coupling perspective

We show that the effective brane-world and the loop quantum cosmology background expansion histories can be reproduced from a modified gravity perspective in terms of an f(R) gravity action plus a g(R) term non-minimally coupled with the matter Lagrangian. The reconstruction algorithm that we provide depends on a free function of the matter density that must be specified in each case and allows to obtain analytical solutions always. In the simplest cases, the function f(R) is quadratic in the Ricci scalar, R, whereas g(R) is linear. Our approach is compared with recent results in the literature. We show that working in the Palatini formalism there is no need to impose any constraint that keeps the equations second-order, which is a key requirement for the successful implementation of the reconstruction algorithm.

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.

Evolution in bouncing quantum cosmology

We present the method of describing an evolution in quantum cosmology in the framework of the reduced phase space quantization of loop cosmology. We apply our method to the flat Friedmann–Robertson–Walker model coupled to a massless scalar field. We identify the physical quantum Hamiltonian that is positive-definite and generates globally a unitary evolution of the considered quantum system. We examine the properties of expectation values of physical observables in the process of the quantum big bounce transition. The dispersion of evolved observables is studied for the Gaussian state. Calculated relative fluctuations enable an examination of the semi-classicality conditions and possible occurrence of the cosmic forgetfulness. Preliminary estimations based on the cosmological data suggest that there was no cosmic amnesia. Presented results are analytical, and numerical computations are only used for the visualization purposes. Our method may be generalized to sophisticated cosmological models including the Bianchi-type universes.

Observables for a Friedmann-Robertson-Walker model with a cosmological constant in the framework of loop cosmology

We consider a flat cosmological model with a free massless scalar field and the cosmological constant $\Lambda$ in the framework of loop quantum cosmology. The scalar field plays the role of an intrinsic time. We apply the reduced phase space approach. The dynamics of the model is solved analytically. We identify elementary observables and their algebra. The compound physical observables like the volume and the energy density of matter field are analysed. Both compound observables are bounded and oscillate in the $\Lambda<0$ case. The energy density is bounded and oscillates in the $\Lambda>0$ case. However, the volume is unbounded from above, but periodic. The difference between standard and nonstandard loop quantum cosmology is described.

The cosmological perturbation theory in loop cosmology with holonomy corrections

In this paper we investigate the scalar mode of first-order metricperturbations over spatially flat FRW spacetime when the holonomycorrection is taken into account in the semi-classical frameworkof loop quantum cosmology. By means of the Hamiltonian derivation,the cosmological perturbation equations is obtained inlongitudinal gauge. It turns out that in the presence of metricperturbation the holonomy effects influence both background andperturbations, and contribute the non-trivial terms Sh1 and Sh2 in thecosmological perturbation equations.

Turning big bang into big bounce. I. Classical dynamics

The big bounce (BB) transition within a flat Friedmann-Robertson-Walker model is analyzed in the setting of loop geometry underlying the loop cosmology. We solve the constraint of the theory at the classical level to identify physical phase space and find the Lie algebra of the Dirac observables. We express energy density of matter and geometrical functions in terms of the observables. It is the modification of classical theory by the loop geometry that is responsible for BB. The classical energy scale specific to BB depends on a parameter that should be fixed either by cosmological data or determined theoretically at quantum level, otherwise the energy scale stays unknown.

Deformed spaces and loop cosmology

The non-singular bouncing solution of loop quantum cosmology is reproduced by a deformed minisuperspace Heisenberg algebra. This algebra is a realization of the Snyder space, is almost unique and is related to the k-Poincaré one. Since the sign of the deformation parameter it is not fixed, the Friedmann equation of braneworlds theory can also be obtained. Moreover, the sign is the only freedom in the picture and these frameworks are the only ones which can be reproduced by our deformed scheme. A generalized uncertainty principle for loop quantum cosmology is also proposed.

Multi-fluid potential in the loop cosmology

The scalar field can behave like a fluid with equation of state pϕ=wρϕ, where w∈[−1,1]. In this Letter we derive a class of the scalar field potentials for which w=const. Scalar field with such a potential can mimic ordinary matter, radiation, cosmic strings, etc. We perform our calculations in the framework of the loop cosmology with holonomy corrections. We solve the model analytically for the whole parameter space. Subsequently, we perform similar consideration for the model with a phantom field (w<−1). We show that scalar field is monotonic function in both cases. This indicates that it can be treated as a well-defined internal time for these models. Moreover we perform preliminary studies of the scalar field perturbations with this potential. We indicate that non-Gaussian features are present admitting for the possible observational constraints of the model.

Stepping out of homogeneity in loop quantum cosmology

We explore the extension of quantum cosmology outside the homogeneous approximation using the formalism of loop quantum gravity. We introduce a model where some of the inhomogeneous degrees of freedom are present, providing a tool for describing general fluctuations of quantum geometry near the initial singularity. We show that the dynamical structure of the model reduces to that of loop quantum cosmology in the Born–Oppenheimer approximation. This result corroborates the assumptions that ground loop cosmology sheds some light on the physical and mathematical relation between loop cosmology and full loop quantum gravity, and on the nature of the cosmological approximation. Finally, we show that the non-graph-changing Hamiltonian constraint considered in the context of algebraic quantum gravity provides a viable effective dynamics within this approximation.

Emerging singularities in the bouncing loop cosmology

In this paper we calculate $\mathcal{O}({\ensuremath{\mu}}^{4})$ corrections from holonomies in the loop quantum gravity, usually not taken into account. Allowance of the corrections of this kind is equivalent with the choice of the new quatization scheme. Quantization ambiguities in the loop quantum cosmology allow for this additional freedom and presented corrections are consistent with the standard approach. We apply these corrections to the flat Friedmann-Robertson-Walker cosmological model and calculate the modified Friedmann equation. We show that the bounce appears in the models with the standard $\mathcal{O}({\ensuremath{\mu}}^{2})$ quantization scheme and is shifted to the higher energies ${\ensuremath{\rho}}_{\mathrm{bounce}}=3{\ensuremath{\rho}}_{\mathrm{c}}$. Also, a pole in the Hubble parameter appears for ${\ensuremath{\rho}}_{\mathrm{pole}}=\frac{3}{2}{\ensuremath{\rho}}_{\mathrm{c}}$ corresponding to hyperinflation/deflation phases. This pole represents a curvature singularity at which the scale factor is finite. In this scenario the singularity and bounce coexist. Moreover, we find that an ordinary bouncing solution appears only when quantum corrections in the lowest order are considered. Higher order corrections can lead to nonperturbative effects.

Thermal fluctuations in loop cosmology

Quantum gravitational effects in loop quantum cosmology lead to a resolution of the initial singularity and have the potential to solve the horizon problem and generate a quasi scale-invariant spectrum of density fluctuations. We consider loop modifications to the behavior of the inverse scale factor below a critical scale in closed models and assume a purely thermal origin for the fluctuations. We show that the no-go results for scale invariance in classical thermal models can be evaded even if we just consider modifications to the background (zeroth order) gravitational dynamics. Since a complete and systematic treatment of the perturbed Einstein equations in loop cosmology is still lacking, we simply parametrize their expected modifications. These change quantitatively, but not qualitatively, our conclusions. We thus urge the community to more fully work out this complex aspect of loop cosmology, since the full picture would not only fix the free parameters of the theory, but also provide a model for a noninflationary, thermal origin for the structures of the Universe.

Formation and Evolution of Structure in Loop Cosmology

Inhomogeneous cosmological perturbation equations are derived in loop quantum gravity, taking into account corrections, in particular, in gravitational parts. This provides a framework for calculating the evolution of modes in structure formation scenarios related to inflationary or bouncing models. Applications here are corrections to the Newton potential and to the evolution of large scale modes which imply nonconservation of curvature perturbations possibly noticeable in a running spectral index. These effects are sensitive to quantization procedures and test the characteristic behavior of correction terms derived from quantum gravity.

Loop cosmological implications of a nonminimally coupled scalar field

Nonminimal actions with matter represented by a scalar field coupled to gravity are considered in the context of a homogeneous and isotropic universe. The coupling is of the form -(1/2){xi}{phi}{sup 2}R. The possibility of successful inflation is investigated taking into account features of loop cosmology. For that end a conformal transformation is performed. That brings the theory into the standard minimally coupled form (Einstein frame) with some effective field and its potential. Both analytical and numerical estimates show that a negative coupling constant is preferable for successful inflation. Moreover, provided fixed initial conditions, larger {xi} leads to a greater number of e-folds. The latter is obtained for a reasonable range of initial conditions and the coupling parameter and indicates a possibility for successful inflation.

Coordinate time dependence in quantum gravity

The intuitive classical space-time picture breaks down in quantum gravity, which makes a comparison and the development of semiclassical techniques quite complicated. Using ingredients of the group averaging method to solve constraints one can nevertheless introduce a classical coordinate time into the quantum theory, and use it to investigate the way a semiclassical continuous description emerges from discrete quantum evolution. Applying this technique to test effective classical equations of loop cosmology and their implications for inflation and bounces, we show that the effective semiclassical theory is in good agreement with the quantum description even at short scales.