We investigate the convergence behaviors of the scalar and the vector perturbations for a big bounce phase in loop quantum cosmology. Two models are discussed: one is the universe filled by a massless scalar field; the other is a toy model which is radiation-dominated in the asymptotic past and future. We find that the behaviors of the Bardeen potential of the scalar mode near both the bounce point and the transition point of the null energy condition are good, moreover, the unlimited growth of the vector perturbation can be avoided in our bounce model. This is different from the bounce models in pure general relativity. And we also find that the maximum of an observable vector mode is inversely proportional to the square of the minimum scalar factor $a_{bounce}$. This conclusion is independent with the bounce model, and we may conclude that the bounce in loop quantum cosmology is reasonable.
We investigate the bounce cosmology induced by inhomogeneous viscous fluids in FRW space-time (non necessarly flat), taking into account the early-time acceleration after the bounce. Different forms for the scale factor and several examples of fluids will be considered. We also analyze the relation between bounce and finite-time singularities and between the corresponding fluids realizing this scenarios. In the last part of the work, the study is extended to the framework of $f(R)$-modified gravity, where the modification of gravity may also be considered as an effective (viscous) fluid produc…
We propose a new five-parameter entropy function that proves to be singular-free during the entire cosmic evolution of the universe, and at the same time, also generalizes the Tsallis, Barrow, Rényi, Sharma-Mittal, Kaniadakis and Loop Quantum Gravity entropies for suitable limits of the parameters. In particular, all the above mentioned known entropies become singular (or diverge) when the Hubble parameter vanishes in course of the universe's evolution (for instance, in bounce cosmology at the instant of bounce), while the newly proposed entropy function seems to be singular-free even at $H = …
For the Bianchi type I space-time (vacuum or with a massless scalar field), the loop quantum cosmology bounce can be viewed as a rapid transition between two classical solutions, with a simple transformation rule relating the Kasner exponents of the two epochs. This transformation rule can be extended to other Bianchi space-times under the assumption that during the loop quantum cosmology bounce the contribution of the spatial curvature to the Hamiltonian constraint is negligible compared to the kinetic terms. For the vacuum Bianchi type IX space-time there are transformation rules for how eac…
We examine the dynamical consequences of homogeneous cosmological magnetic fields in the framework of loop quantum cosmology. We show that a big-bounce occurs in a collapsing magnetized Bianchi I universe, thus extending the known cases of singularity-avoidance. Previous work has shown that perfect fluid Bianchi I universes in loop quantum cosmology avoid the singularity via a bounce. The fluid has zero anisotropic stress, and the shear anisotropy in this case is conserved through the bounce. By contrast, the magnetic field has nonzero anisotropic stress, and shear anisotropy is not conserved …
The detailed formulation for loop quantum cosmology (LQC) in the Bianchi I model with a massless scalar field was recently constructed. In this paper, its effective dynamics with the LQC discreteness corrections is studied and the equations of motion are analytically solved, showing that the big bang singularity is replaced by the big bounces, which take place up to three times, once in each diagonal direction, whenever each of the area scale factors approaches its critical value in the Planck regime measured by the reference of the scalar field momentum.
The usual Einstein's equations is modified as a one parameter family of equations in the framework of rainbow gravity. In this paper we derive the modified Friedmann-Robertson-Walker (FRW) equations when the cosmological evolution of radiation particles is taken into account. In particular, given some specific dispersion relations, the big bounce solutions to the modified FRW equations can be derived. Notably, to obtain a well defined rainbow metric at the moment of the big bounce, we find it seems necessary to introduce a cosmological constant which depends on the energy of probes as well, im…
The purpose of this short review is to describe cosmological models with a linear inhomogeneous time-dependent equation of state (EoS) for the dark energy, when the dark fluid is coupled with dark matter. This may lead to a bounce cosmology. We consider equivalent descriptions in terms of the EoS parameters for an exponential, a power-law, or a double-exponential law for the scale factor $a$. Stability issues are discussed by considering small perturbations around the critical points for the bounce, in the early as well as in the late, universe. The latter part of the paper is concerned with d…
We investigate how various inflationary and bouncing cosmologies can be realized by imperfect fluids with a generalized equation of state, in the context of both classical and loop quantum cosmology. With regards to the inflationary cosmologies, we study the intermediate inflation scenario, the $R^2$ inflation scenario and two constant-roll inflation scenarios and with regards to the bouncing cosmologies we study the matter bounce scenario, the singular bounce and the super bounce scenario. Within the context of the classical cosmology, we calculate the spectral index of the power spectrum of …
We implement Polymer Quantum Mechanics on the Hamiltonian formulation of the isotropic Universe in both the representations of the standard Ashtekar-Barbero-Immirzi connection and of a new generalized coordinate conjugate to the Universe volume. The resulting dynamics is a bouncing cosmology; when quantizing the volume-like variable the Big Bounce is an intrinsic cut-off on the cosmological dynamics, while when using the standard connection the Bounce density results to be dependent on the initial conditions of the prepared wave packet. Then we compare the nature of the resulting Bounce with w…
We examine the nature of the cosmological big-bounce (BB) transition within the loop geometry underlying Loop Quantum Cosmology (LQC) at classical and quantum levels. Our canonical quantization method is an alternative to the standard LQC. Our method opens the door for analyzes of spectra of physical observables like energy density and volume operator. We find that one cannot determine the energy scale specific to BB by making use of the loop geometry without an extra input from observational cosmology.
Recently the so-called mimetic gravity approach has been used to obtain corrections to Friedmann equation of General Relativity similar to the ones present in loop quantum cosmology. In this paper, we propose an alternative way to derive this modified Friedmann equation via the so-called non-polynomial gravity approach, which consists in adding geometric non-polynomial higher derivative terms to Hilbert-Einstein action, which are nonetheless polynomials and lead to second order differential equation in Friedmann-Lemaître-Robertson-Walker spacetimes. Our explicit action turns out to be a realiz…
In this paper we investigate gravitational waves production during the Big Bounce phase inspired by the Loop Quantum Cosmology. We consider the influence of the holonomy corrections to the equation for tensor modes. We show that they act like additional effective graviton mass, suppressing gravitational waves creation. However, this effects can be treated perturbatively. We investigate the simplified model without these corrections and find its exact analytical solution. For this model we calculate a spectrum of the gravitational waves from the Big Bounce phase. The obtained spectrum decreases…
In this paper, we study a big bounce universe typified by a non-singular big bounce, as opposed to a singular big bang. This cosmological model can describe radiation dominated early universe and matter dominated late universe in FRW model. The connections between thermodynamics and gravity are observed here. In the early stage of both cold and hot universes, we find there is only one geometry containing a 4D de Sitter universe with a general state parameter. We also find the form of the apparent horizon in the early universe strongly depends on the extra dimension, which suggests that the inf…
It has been known for some time that the cosmological Friedmann equation deduced from General Relativity can be also obtained within the Newtonian framework under certain assumptions. We use this result together with quantum corrections to the Newtonian potentials to derive a set a of quantum corrected Friedmann equations. We examine the behavior of the solutions of these modified cosmological equations paying special attention to the sign of the quantum corrections. We find different quantum effects crucially depending on this sign. One such a solution displays a qualitative resemblance to ot…
In this paper we study the flat (k=0) cosmological FRW model with holonomy corrections of Loop Quantum Gravity. The considered universe contains a massless scalar field and the cosmological constant Lambda. We find analytical solutions for this model in different configurations and investigate its dynamical behaviour in the whole phase space. We show the explicit influence of Lambda on the qualitative and quantitative character of solutions. Even in the case of positive Lambda the oscillating solutions without the initial and final singularity appear as a generic case for some quantisation sch…
We provide an analytical solution to the quantum dynamics of a flat Friedmann-Lemaître- Robertson-Walker model with a massless scalar field in the presence of a small and positive cosmological constant, in the context of Loop Quantum Cosmology. We use a perturbative treatment with respect to the model without a cosmological constant, which is exactly solvable. Our solution is approximate, but it is precisely valid at the high curvature regime where quantum gravity corrections are important. We compute explicitly the evolution of the expectation value of the volume. For semiclassical states cha…
We derive the cosmological Big Bounce scenario from the dipole approximation of Loop Quantum Gravity. We show that a non-singular evolution takes place for any matter field and that, by considering a massless scalar field as a relational clock for the dynamics, the semi-classical proprieties of an initial state are preserved on the other side of the bounce. This model thus enhances the relation between Loop Quantum Cosmology and the full theory.
The Immirzi parameter is promoted to be a scalar field and the Hamiltonian analysis of the corresponding dynamical system is performed in the presence of gravity. We identified some SU(2) connections, generalizing Ashtekar-Barbero variables, and we rewrite the constraints in terms of them, setting the classical formulation suitable for loop quantization. Then, we consider the reduced system obtained when restricting to a flat isotropic cosmological model. By mimicking loop quantization via an effective semiclassical treatment, we outline how quantum effects are able to tame the initial singula…
We consider the k=1 Friedman-Robertson-Walker (FRW) model within loop quantum cosmology, paying special attention to the existence of an ambiguity in the quantization process. In spatially non-flat anisotropic models such as Bianchi II and IX, the standard method of defining the curvature through closed holonomies is not admissible. Instead, one has to implement the quantum constraints by approximating the connection via open holonomies. In the case of flat k=0 FRW and Bianchi I models, these two quantization methods coincide, but in the case of the closed $k$=1 FRW model they might yield diff…
Loop quantum cosmology predicts that quantum gravity effects resolve the big-bang singularity and replace it by a cosmic bounce. Furthermore, loop quantum cosmology can also modify the form of primordial cosmological perturbations, for example by reducing power at large scales in inflationary models or by suppressing the tensor-to-scalar ratio in the matter bounce scenario; these two effects are potential observational tests for loop quantum cosmology. In this article, I review these predictions and others, and also briefly discuss three open problems in loop quantum cosmology: its relation to…
We analyse the classical and quantum dynamics of a Bianchi I model in the presence of a small negative cosmological constant characterizing its evolution in term of the dust-time dualism. We demonstrate that in a canonical metric approach, the cosmological singularity is removed in correspondence to a positive defined value of the dust energy density. Furthermore, the quantum Big-Bounce is connected to the Universe turning point via a well-defined semiclassical limit. Then we can reliably infer that the proposed scenario is compatible with a cyclical Universe picture. We also show how, when th…
The detailed formulation for loop quantum cosmology (LQC) in the Bianchi I model with a scalar massless field has been constructed. In this paper, its effective dynamics is studied in two improved strategies for implementing the LQC discreteness corrections. Both schemes show that the big bang is replaced by the big bounces, which take place up to three times, once in each diagonal direction, when the area or volume scale factor approaches the critical values in the Planck regime measured by the reference of the scalar field momentum. These two strategies give different evolutions: In one sche…
Loop quantum cosmology homogeneous models with a massless scalar field show that the big-bang singularity can be replaced by a big quantum bounce. To gain further insight on the nature of this bounce, we study the semi-discrete loop quantum gravity Hamiltonian constraint equation from the point of view of numerical analysis. For illustration purposes, we establish a numerical analogy between the quantum bounces and reflections in finite difference discretizations of wave equations triggered by the use of nonuniform grids or, equivalently, reflections found when solving numerically wave equatio…
The big bounce singularity of a simple 5D cosmological model is studied. Contrary to the standard big bang space-time singularity, this big bounce singularity is found to be an event horizon at which the scale factor and the mass density of the universe are finite, while the pressure undergoes a sudden transition from negative infinity to positive infinity. By using coordinate transformation it is also shown that before the bounce the universe contracts deflationary, and the universe has been existed, according to the proper-time, for an infinitely long time.
We analyze the cosmological implementation of Palatini $f(R)$ theories, constructed with a Nieh-Yan term and solved with respect to the torsion. We consider the relevant case of the quadratic correction to the Hilbert-Palatini action in the Ricci scalar, mimicking the Starobinsky model of the metric formulation. We point out the emergence of peculiar cosmological scenarios, depending on the sign of such correction, able to reproduce bouncing settings and to restore the standard Universe dynamics in the late asymptotic limit. Furthermore, we outline the settling of Little-Rip dynamics, which ca…
A recently derived inequality on volume fluctuations of a bouncing cosmology, valid for states which are semiclassical long after the bounce, does not restrict pre-bounce fluctuations sufficiently strongly to conclude that the pre-bounce state was semiclassical except in a very weak sense.
We will perform a detailed study of the matter-ekpyrotic bouncing scenario in Loop Quantum Cosmology using the methods of the dynamical systems theory. We will show that when the background is driven by a single scalar field, at very late times, in the contracting phase, all orbits depict a matter dominated Universe, which evolves to an ekpyrotic phase. After the bounce the Universe enters in the expanding phase, where the orbits leave the ekpyrotic regime going to a kination (also named deflationary) regime. Moreover, this scenario supports the production of heavy massive particles conformall…
We investigate the classical and quantum behavior of a Bianchi I model in the presence of a stiff matter contribution when the Vilenkin interpretation of the wave function of the Universe is taken into account. We study its evolution in the so-called polymer representation of quantum mechanics, in order to characterize the modifications that a discrete nature in the isotropic variable of the Universe induces on the morphology of the cosmological singularity. We demonstrate that in such a model the Big-Bang singularity is removed at a semiclassical level in favor of a Big-Bounce when a lattice …
Recently, a de-Sitter epoch has been found in the new model of loop quantum cosmology which is governed by the scalar constraint with both of Euclidean and Lorentz terms. The singularity free bounce in the new LQC model and the emergent cosmology constant strongly suggest that the effective stress energy tensor induced by quantum corrections must violate the standard energy conditions. In this paper, we do an explicit calculation to analyze the behaviours of specific representative energy conditions, i.e., average null, strong and dominant energy conditions. It turns out that the averaging nul…
This brief article sums up the possible imprints of loop quantum gravity effects on the cosmological microwave background. We focus on semi-classical terms and show that "Big Bounce" corrections, together with the "pre Big Bounce" state, could modify the observed spectrum.
We consider an isotropic and homogeneous universe in loop quantum cosmology. 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 t…
We consider the matter bounce scenario in Loop Quantum Cosmology (LQC) for physical potentials that at early times provide a nearly matter dominated Universe in the contracting phase, having a reheating mechanism in the expanding phase, i.e., being able to release the energy of the scalar field creating particles that thermalize in order to match with the hot Friedmann Universe, and finally at late times leading to the current cosmic acceleration. For these models, numerically solving the dynamical equations we have seen that the teleparallel version of LQC leads to theoretical results that fi…
Within the context of the Fermi-bounce curvaton mechanism, we analyze the one-loop radiative corrections to the four fermion interaction, generated by the non-dynamical torsion field in the Einstein-Cartan-Holst-Sciama-Kibble theory. We show that contributions that arise from the one-loop radiative corrections modify the energy-momentum tensor, {\it mimicking} an effective Ekpyrotic fluid contribution. For these reasons, we call this effect {\it quantum Ekpyrotic} mechanism. This leads to the dynamical washing out of anisotropic contributions to the energy-momentum tensor, without introducing …
We show that there exist modified theories of gravity in which the metric satisfies second-order equations and in which the Big Bang singularity is replaced by a cosmic bounce without violating any energy condition. In fact, the bounce is possible even for presureless dust. We give a characterization of such theories, which are formulated in the Palatini formalism, and discuss their dynamics in the region near the bounce. We consider spatially flat and non-flat homogeneous and isotropic universes.
A Bianchi IX Mixmaster spacetime is the most general spatially homogeneous solution of Einstein's equations and it can represent the space-averaged Universe. We introduce two novel mechanisms resulting in a Mixmaster Universe with non-singular bounces which are quasi-isotropic. A fluid with a non-linear equation of state allows non-singular bounces. Using negative anisotropic stresses successfully isotropises this Universe and mitigates the well known Mixmaster chaotic behaviour. Thus the Universe can be an eternal Mixmaster, going through an infinite series of different cycles separated by bo…
In recent years, numerical simulations with Gaussian initial states have demonstrated the existence of a quantum bounce in loop quantum cosmology in various models. A key issue pertaining to the robustness of the bounce and the associated physics is to understand the quantum evolution for more general initial states which may depart significantly from Gaussianity and may have no well defined peakedness properties. The analysis of such states, including squeezed and highly non-Gaussian states, has been computationally challenging until now. In this manuscript, we overcome these challenges by us…
We investigate whether successful Gravitational Leptogenesis can take place during an Ekpyrotic contraction phase. Two possible paths by which this can occur are coupling the Ekpyrotic scalar to a gravitational Chern-Simons term, or to a $ U(1) $ gauge field Chern-Simons term. These couplings lead to the production of chiral gravitational waves, which generate a lepton number asymmetry through the gravitational-lepton number anomaly. This lepton asymmetry is subsequently reprocessed by equilibrium sphaleron processes to produce a baryon asymmetry. We find successful Gravitational Leptogenesis …
We consider the most general quadratic in curvature stringy motivated non-local action for the modified Einstein's gravity. We present exact analytic cosmological solutions including the bouncing ones and develop the relevant techniques for the further study of this type of models. We also elaborate on the perturbation formalism and argue that the found bouncing solution is stable during the bounce phase.
It is known that in loop quantum cosmology (LQC) the universe avoids the singularity by a bounce when the matter density approaches the critical density $ρ_c$ (the order of Planck density). After incorporating the inverse volume modifications both in the gravitational and matter part in the improved framework of LQC, we find that the inverse volume modification can decrease the bouncing energy scale, and the presence of nonsingular bounce is generic. For the backward evolution in the expanding branch, in terms of different initial states the evolution trajectories classify into two classes. On…
We analyze the dynamics of a homogeneous and isotropic universe in the Einstein--Cartan theory of gravity. The spin of fermions produces spacetime torsion that prevents gravitational singularities and replaces the big bang with a nonsingular big bounce. We show that a closed universe exists only when a particular function of its scale factor and temperature is higher than some threshold value, whereas an open and a flat universes do not have such a restriction. We also show that a bounce of the scale factor is double: as the temperature increases and then decreases, the scale factor decreases,…
In this paper, we analyze the modified $f(\mathcal{R})$ gravity models in Friedmann--Lema\^ıtre--Robertson--Walker (FLRW) background. The actions of bouncing cosmology are studied under consideration of different viable models in $f(\mathcal{R})$ gravity theory that can resolve the difficulty of singularity in standard Big-Bang cosmology. Under different viable models in $f(\mathcal{R})$ gravity theory, the cosmological constraints are plotted in provisions of cosmic-time, then investigated the bounce circumstance. In addition, the red-shift parameter is used to reconstruct the modified gravit…
We quantise and solve the dynamics of gravitational waves in a quantum Friedmann-Lemaitre-Robertson-Walker spacetime filled with perfect fluid. The classical model is formulated canonically. The Hamiltonian constraint is de-parametrised by setting a fluid variable as the internal clock. The obtained reduced (i.e. physical) phase space is then quantised. Our quantisation procedure is implemented in accordance with two different phase space symmetries, namely, the Weyl-Heisenberg symmetry for the perturbation variables, and the affine symmetry for the background variables. As an appealing outcom…
We numerically investigate the solutions to the effective equations of the Bianchi II model within the "improved" Loop Quantum Cosmology (LQC) 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 point-like 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 an…
The spatially flat Friedman-Robertson-Walker (FRW) cosmological model with a massless scalar field in loop quantum cosmology admits a description in terms of a completely solvable model. This has been used to prove that: i) the quantum bounce that replaces the big bang singularity is generic; ii) there is an upper bound on the energy density for all states and iii) semiclassical states at late times had to be semiclassical before the bounce. Here we consider a family of exact solutions to the theory, corresponding to generalized coherent Gaussian and squeezed states. We analyze the behavior of…
I present a streamlined review of how the separate universe approach to cosmological perturbation theory can be used to study the dynamics of long-wavelength scalar perturbations in loop quantum cosmology, and then use it to calculate how long-wavelength curvature perturbations evolve across the loop quantum cosmology bounce assuming a constant equation of state. A similar calculation is possible for tensor modes using results from a complementary approach to cosmological perturbation theory in loop quantum cosmology based on an effective Hamiltonian constraint. An interesting result is that t…
We examine the possibility of dealing with gravitational singularities on a quantum level through the use of coherent state or wavelet quantization instead of canonical quantization. We consider the Robertson-Walker metric coupled to a perfect fluid. It is the simplest model of a gravitational collapse and the results obtained here may serve as a useful starting point for more complex investigations in future. We follow a quantization procedure based on affine coherent states or wavelets built from the unitary irreducible representation of the affine group of the real line with positive dilati…
The absence of isotropic singularity in loop quantum cosmology can be understood in an effective classical description as the universe exhibiting a Big Bounce. We show that with scalar matter field, the big bounce is generic in the sense that it is independent of quantization ambiguities and details of scalar field dynamics. The volume of the universe at the bounce point is parametrized by a single parameter. It provides a minimum length scale which serves as a cut-off for computations of density perturbations thereby influencing their amplitudes.
We show that the Nieh-Yan topological invariant breaks projective symmetry and loses its topological character in presence of non vanishing nonmetricity. The notion of the Nieh-Yan topological invariant is then extended to the generic metric-affine case, defining a generalized Nieh-Yan term, which allows to recover topologicity and projective invariance, independently. As a concrete example a class of modified theories of gravity is considered and its dynamical properties are investigated in a cosmological setting. In particular, bouncing cosmological solutions in Bianchi I models are derived.…
We extend the notion of the Nieh-Yan invariant to generic metric-affine geometries, where both torsion and nonmetricity are taken into account. Notably, we show that the properties of projective invariance and topologicity can be independently accommodated by a suitable choice of the parameters featuring this new Nieh-Yan term. We then consider a special class of modified theories of gravity able to promote the Immirzi parameter to a dynamical scalar field coupled to the Nieh-Yan form, and we discuss in more detail the dynamics of the effective scalar tensor theory stemming from such a revised…
Recent work in quantum loop cosmology suggests the universe undergoes a bounce when evolving from a previous collapsing phase. However, with non-inflationary matter sources, the scenario appears to strongly violate the holography bound S\leq A/4 during the bounce, where A now represents the cross-sectional area of the bounce. We also give a simple argument why any inflationary phase after the bounce is unlikely due to prior dissipation of a scalar field kinetic energy phase into regular matter components.
Horava-Lifshitz gravity, a recent proposal for a UV-complete renormalizable gravity theory, may lead to a bouncing cosmology. In this note we argue that Horava-Lifshitz cosmology may yield a concrete realization of the matter bounce scenario, and thus give rise to an alternative to inflation for producing a scale-invariant spectrum of cosmological perturbations. In this scenario, quantum vacuum fluctuations exit the Hubble radius in the pre-bounce phase and the spectrum is transformed into a scale-invariant one on super-Hubble scales before the bounce because the long wavelength modes undergo …
The propagation of a scalar field in an open FLRW bounce-type quantum spacetime is examined, which arises within the framework of the IKKT matrix theory. In the first part of the paper, we employ general-relativity tools to study null and timelike geodesics at the classical level. This analysis reveals that massless and massive non-interacting particles can travel across the big bounce. We then exploit quantum-field-theory techniques to evaluate the scalar field propagator. In the late-time regime, we find that it resembles the standard Feynman propagator of flat Minkowski space, whereas for e…
