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 to zero for the low energy modes. Based on this observation we indicate that this effect can lead to the low CMB multipoles suppression and gives a potential way to test Loop Quantum Cosmology models. We also consider a scenario with a post-bounce inflationary phase. The obtained power spectrum gives qualitative explanation of the CMB spectra, including low multipoles suppression. This result is a challenge to construct a consistent bounce+inflation model in the Loop Quantum Cosmology.
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…
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…
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.
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…
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 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…
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 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.
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…
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 …
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…
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…
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.
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 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 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,…
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…
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…
The four-fermion gravitational interaction is induced by torsion, and gets dominating on the Planck scale. The regular, axial-axial part of this interaction by itself does not stop the gravitational compression. However, the anomalous, vector-vector interaction results in a natural way both in big bounce and in inflation.
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…
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…
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 Einstein-Cartan-Sciama-Kibble theory of gravity naturally extends general relativity to account for the intrinsic spin of matter. Spacetime torsion, generated by spin of Dirac fields, induces gravitational repulsion in fermionic matter at extremely high densities and prevents the formation of singularities. Accordingly, the big bang is replaced by a bounce that occurred when the energy density $ε\propto gT^4$ was on the order of $n^2/m_\textrm{Pl}^2$ (in natural units), where $n\propto gT^3$ is the fermion number density and $g$ is the number of thermal degrees of freedom. If the early Uni…
The properties of an extension of the New Massive 3D Gravity by scalar matter with Higgs-like self-interaction are investigated. Its perturbative unitarity consistency is verified for a family of cosmological Bounce solutions found by the superpotential method. They correspond to the lower bound ${λ= -1}$ of the BHT unitarity window and describe eternally accelerated 3D Universe between two initial/final stable $dS_3$ vacua states.
We present a rich class of exact solutions which contains radiation-dominated and matter-dominated models for the early and late universe. They include a variable cosmological ``constant'' which is derived from a higher dimension and manifests itself in spacetime as an energy density for the vacuum. This is in agreement with observational data and is compatible with extensions of general relativity to string and membrane theory. Our solutions are also typified by a non-singular ``big bounce'' (as opposed to a singular big bang), where matter is created as in inflationary cosmology.
We analyze the dynamics of a homogeneous and isotropic universe in the Einstein$--$Cartan theory of gravity. The coupling between the spin and torsion prevents gravitational singularities and replaces the Big Bang with a nonsingular big bounce, at which the universe transitions from contraction to expansion. We show that a closed universe exists only when the product of the scale factor and temperature is higher than a particular threshold, contrary to a flat universe and an open universe, which are not restricted. During inflation, this product must increase to another threshold, so that the …
We examine the nature of the cosmological Big Bounce transition within the loop geometry underlying loop quantum cosmology at classical and quantum levels. Our canonical quantization method is an alternative to the standard loop quantum cosmology. An evolution parameter we use has a clear interpretation. Our method opens the door for analyses of spectra of physical observables like the energy density and the volume operator. We find that one cannot determine the energy scale specific to the Big Bounce by making use of the loop geometry without an extra input from observational cosmology.
We analyze the big bounce transition of the quantum FRW model in the setting of the nonstandard loop quantum cosmology (LQC). Elementary observables are used to quantize composite observables. The spectrum of the energy density operator is bounded and continuous. The spectrum of the volume operator is bounded from below and discrete. It has equally distant levels defining a quantum of the volume. The discreteness may imply a foamy structure of spacetime at semiclassical level which may be detected in astro-cosmo observations. The nonstandard LQC method has a free parameter that should be fixed…
We report on the possibility to use dark matter particle's mass and its cross section as a smoking gun signal of the existence of a Big Bounce at the early stage in the evolution of our currently observed universe. A model independent study of dark matter production in the pre-bounce contraction and the post-bounce expansion epochs of the bounce universe reveals a new venue for achieving the observed relic abundance of our present universe, in which a significantly smaller amount of dark matter with a smaller cross section -- as compared to the prediction of Standard Cosmology -- is produced a…
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.
The Einstein-Cartan-Sciama-Kibble theory of gravity removes the constraint of general relativity that the affine connection be symmetric by regarding its antisymmetric part, the torsion tensor, as a dynamical variable. The minimal coupling between the torsion tensor and Dirac spinors generates a spin-spin interaction which is significant in fermionic matter at extremely high densities. We show that such an interaction averts the unphysical big-bang singularity, replacing it with a cusp-like bounce at a finite minimum scale factor, before which the Universe was contracting. This scenario also e…
An alternative to the Big Bang cosmologies is obtained by the Big Bounce cosmologies. In this paper, we study a bounce cosmology with a Type IV singularity occurring at the bouncing point, in the context of $F(R)$ modified gravity. We investigate the evolution of the Hubble radius and we examine the issue of primordial cosmological perturbations in detail. As we demonstrate, for the singular bounce, the primordial perturbations originating from the cosmological era near the bounce, do not produce a scale invariant spectrum and also the short wavelength modes, after these exit the horizon, do n…
Exact global solutions of five-dimensional cosmological models compactified on a $S_1/Z_2$ orbifold with two 3-branes are presented, and evolution of a simple model is studied. It is found that on all 4D spacetime hypersurfaces, except a singular one, the expanding universe was started not from a big bang but from a big bounce, and before the bounce the universe was in a deflationary contracting phase. It is also found that whether the energy density on the second brane is positive, zero, or negative depends on the size of the fifth dimension.
We explore the possibility of an Ekpyrotic contraction phase harbouring a mechanism for Baryogenesis. A Chern-Simons coupling between the fast-rolling Ekpyrotic scalar and the Standard Model Hypercharge gauge field enables the generation of a non-zero helicity during the contraction phase. The baryon number subsequently produced at the Electroweak Phase Transition is consistent with observation for a range of couplings and bounce scales. Simultaneously, the gauge field production during the contraction provides the seeds for galactic magnetic fields and sources gravitational waves, which may p…
In this work, we calculate the deflection angle of light in a spacetime that interpolates between regular black holes and traversable wormholes, depending on the free parameter of the metric. Afterwards, this angular deflection is substituted into the lens equations which allows to obtain physically measurable results, such as the position of the relativistic images and the magnifications.
Torsion is a geometrical object, required by quantum mechanics in curved spacetime, which may naturally solve fundamental problems of general theory of relativity and cosmology. The black-hole cosmology, resulting from torsion, could be a scenario uniting the ideas of the big bounce and inflation, which were the subject of a recent debate of renowned cosmologists.
The big bang singularity of the expanding-universe Friedmann solution of the Einstein gravitational field equation can be regularized by the introduction of a degenerate metric and a nonzero length scale $b$. The result is a nonsingular bounce of the cosmic scale factor with a contracting prebounce phase and an expanding postbounce phase. The corresponding maximum values of the curvature and the energy density occur at the moment of the bounce and are proportional to powers of $1/b$. This article presents a detailed calculation of the dynamics of such a nonsingular bounce. In addition, a compa…
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 de…
We analyze the big bounce transition of the quantum FRW model in the setting of the nonstandard loop quantum cosmology (LQC). Elementary observables are used to quantize compound observables. The spectrum of the energy density operator is bounded and continuous. The spectrum of the volume operator is bounded from below and discrete. It has equally distant levels defining a quantum of the volume. The discreteness may imply a foamy structure of spacetime at semiclassical level which may be detected in astro-cosmo observations. The nonstandard LQC method has a free parameter that should be fixed …
The dynamics of an inhomogeneous universe is studied with the methods of Loop Quantum Cosmology as an example of the quantization of vacuum cosmological spacetimes containing gravitational waves (Gowdy spacetimes). The analysis performed at the effective level shows that: (i) The initial Big Bang singularity is replaced (as in the case of homogeneous cosmological models) by a Big Bounce, joining deterministically two large universes, (ii) the universe size at the bounce is at least of the same order of magnitude as that of the background homogeneous universe, (iii) for each gravitational wave …
A gauge-invariant quantum theory of the Friedmann-Robertson-Walker (FRW) universe with dust is studied in terms of the Ashtekar variables. We use the reduced phase space quantization which has following advantages: (i) fundamental variables are all gauge invariant, (ii) there exists a physical time evolution of gauge-invariant quantities, so that the problem of time is absent and (iii) the reduced phase space can be quantized in the same manner as in ordinary quantum mechanics. In the FRW model, the dynamical components of the Ashtekar variables are given by a single quantity $p$ and its conju…
Recent observations of Type Ia supernovae provide evidence for the acceleration of our universe, which leads to the possibility that the universe is entering an inflationary epoch. We simulate it under a ``big bounce'' model, which contains a time variable cosmological ``constant'' that is derived from a higher dimension and manifests itself in 4D spacetime as dark energy. By properly choosing the two arbitrary functions contained in the model, we obtain a simple exact solution in which the evolution of the universe is divided into several stages. Before the big bounce, the universe contracts …
It is stated that holonomy corrections in loop quantum cosmology introduce a modification in Friedmann's equation which prevent the big rip singularity. Recently in \cite{h12} it has been proved that this modified Friedmann equation is obtained in an inconsistent way, what means that the results deduced from it, in particular the big rip singularity avoidance, are not justified. The problem is that holonomy corrections modify the gravitational part of the Hamiltonian of the system leading, after Legendre's transformation, to a non covariant Lagrangian which is in contradiction with one of the …
We examine the cosmological sector of a gauge theory of gravity based on the SO(4,2) conformal group of Minkowski space. We allow for conventional matter coupled to the spacetime metric as well as matter coupled to the field that gauges special conformal transformations. An effective cosmological constant is generated dynamically via solution of the equations of motion, and this allows us to recover the late time acceleration of the universe. Furthermore, gravitational fields sourced by ordinary cosmological matter (i.e. dust and radiation) are significantly weakened in the very early universe…
We consider evolutions of linear fluctuations as the background Friedmann world model goes from contracting to expanding phases through smooth and non-singular bouncing phases. As long as the gravity dominates over the pressure gradient in the perturbation equation the growing-mode in the expanding phase is characterized by a conserved amplitude, we call it a C-mode. In the spherical geometry with a pressureless medium, we show that there exists a special gauge-invariant combination Φwhich stays constant throughout the evolution from the big-bang to the big-crunch with the same value even afte…
