Quantum Cosmological Model Research Articles

Dynamics of an interacting dark energy model in Einstein and loop quantum cosmology

We investigate the background dynamics when dark energy is coupled to dark matter in the Universe described by Einstein cosmology and loop quantum cosmology. We introduce a new general form of dark sector coupling, which presents a more complicated dynamical phase space. Differences in the phase space in obtaining the accelerated scaling attractor in Einstein cosmology and loop quantum cosmology are disclosed.

Wave function of the Universe in the early stage of its evolution

In quantum cosmological models, constructed in the framework of Friedmann–Robertson–Walker metrics, a nucleation of the Universe with its further expansion is described as a tunneling transition through an effective barrier between regions with small and large values of the scale factor a at non-zero (or zero) energy. The approach for describing this tunneling consists of constructing a wave function satisfying an appropriate boundary condition. There are various ways for defining the boundary condition that lead to different estimates of the barrier penetrability and the tunneling time.In order to describe the escape from the tunneling region as accurately as possible and to construct the total wave function on the basis of its two partial solutions unambiguously, we use the tunneling boundary condition that the total wave function must represent only the outgoing wave at the point of escape from the barrier, where the following definition for the wave is introduced: the wave is represented by the wave function whose modulus changes minimally under a variation of the scale factor a. We construct a new method for a direct non-semiclassical calculation of the total stationary wave function of the Universe, analyze the behavior of this wave function in the tunneling region, near the escape point and in the asymptotic region, and estimate the barrier penetrability. We observe oscillations of the modulus of the wave function in the external region starting from the turning point which decrease with increasing of a and which are not shown in semiclassical calculations. The period of such an oscillation decreases uniformly with increasing a and can be used as a fully quantum dynamical characteristic of the expansion of the Universe.

Effects of the quantization ambiguities on the Big Bounce dynamics

In this paper we investigate dynamics of the modified loop quantum cosmology models using dynamical systems methods. Modifications considered come from the choice of the different field strength operator $\hat{F}$ and result in different forms of the effective Hamiltonian. Such an ambiguity of the choice of this expression from some class of functions is allowed in the framework of loop quantisation. Our main goal is to show how such modifications can influence the bouncing universe scenario in the loop quantum cosmology. In effective models considered we classify all evolutional paths for all admissible initial conditions. The dynamics is reduced to the form of a dynamical system of the Newtonian type on a 2-dimensional phase plane. These models are equivalent dynamically to the FRW models with the decaying effective cosmological term parametrised by the canonical variable $p$ (or by the scale factor $a$). We find that for the positive cosmological constant there is a class of oscillating models without the initial and final singularities. The new phenomenon is the appearance of curvature singularities for the finite values of the scale factor, but we find that for the positive cosmological constant these singularities can be avoided. For the positive cosmological constant the evolution begins at the asymptotic state in the past represented by the deSitter contracting (deS$_{-}$) spacetime or the static Einstein universe H=0 or $H=-\infty$ state and reaches the deSitter expanding state (deS$_{+}$), the state H=0 or $H=+\infty$ state. In the case of the negative cosmological constant we obtain the past and future asymptotic states as the Einstein static universes.

Fermions in loop quantum cosmology and the role of parity

Fermions play a special role in homogeneous models of quantum cosmology because the exclusion principle prevents them from forming sizable matter contributions. They can thus describe the matter ingredients only truly microscopically and it is not possible to avoid strong quantum regimes by positing a large matter content. Moreover, possible parity-violating effects are important especially in loop quantum cosmology whose basic object is a difference equation for the wavefunction of the universe defined on a discrete space of triads. The two orientations of a triad are interchanged by a parity transformation, which leaves the difference equation invariant for ordinary matter. Here, we revisit and extend loop quantum cosmology by introducing fermions and the gravitational torsion they imply, which renders the parity issue non-trivial. A treatable locally rotationally symmetric Bianchi model is introduced which clearly shows the role of parity. General wavefunctions cannot be parity-even or odd, and parity-violating effects in matter influence the microscopic big bang transition which replaces the classical singularity in loop quantum cosmology.

Numerical techniques for solving the quantum constraint equation of generic lattice-refined models in loop quantum cosmology

To avoid instabilities in the continuum semiclassical limit of loop quantum cosmology models, refinement of the underlying lattice is necessary. This lattice refinement leads to new dynamical difference equations which, in general, do not have a uniform step size, implying complications in their analysis and solutions. We propose a numerical method based on Taylor expansions, which can give us the necessary information to calculate the wave function at any given lattice point. The method we proposed can be applied in any lattice-refined model, while in addition the accuracy of the method can be estimated. Moreover, we confirm numerically the stability criterion which was earlier found following a von Neumann analysis. Finally, the ''motion'' of the wave function due to the underlying discreteness of the space-time is investigated, for both a constant lattice, as well as lattice refinement models.

Signature change from Schutz’s canonical quantum cosmology and its classical analogue

We study the signature change in a perfect fluid Friedmann-Robertson-Walker quantum cosmological model. In this work the Schutz's variational formalism is applied to recover the notion of time. This gives rise to a Schrodinger-Wheeler-DeWitt equation with arbitrary ordering for the scale factor. We use the eigenfunctions in order to construct wave packets and evaluate the time-dependent expectation value of the scale factor which coincides with the ontological interpretation. We show that these solutions exhibit signature transitions from a finite Euclidean to a Lorentzian domain. Moreover, such models are equivalent to a classical system where, besides the perfect fluid, a repulsive fluid is present.

Cosmological evolution of the interacting phantom (quintessence) model in loop quantumgravity

The dynamics of the interacting dark energy model in loop quantum cosmology(LQC) is studied in this paper. The dark energy has a constant equation of statewx and interacts with dark matter through a form3cH(ρx+ρm). We find that for thequintessence model (wx>−1) the cosmological evolution in LQC is the same as that inclassical Einstein cosmology, whereas for phantom dark energy(wx<−1), although there are the same critical points in LQC and classical Einsteincosmology, the loop quantum effect significantly reduces the parameter spacetime(c,wx) required bystability. If parameters c and wx satisfy the conditions that the critical points are existing and stable, the universewill enter an era dominated by dark energy and dark matter with a constantenergy ratio between them, and accelerate forever; otherwise it will enteran oscillatory regime. Comparing our results with the observations we find at1σ confidence level that the universe will accelerate forever.

Numerical solutions to lattice-refined models in loop quantum cosmology

In this paper, we develop an intuitive and efficient numerical technique to solve the quantum evolution equation of generic lattice-refined models in loop quantum cosmology. As an application of this method, we extensively study the solutions of the recently introduced, lattice-refined anisotropic model of the Schwarzschild interior geometry. Our calculations suggest that the results obtained from the approach are accurate, robust and in complete agreement with the expectations from the von Neumann stability analysis of the model.

Scaling Law for the Cosmological Constant from Quantum Cosmology with Seven Extra Dimensions

According to a model of quantum cosmology the maximum number of degrees of freedom allowed in our three dimensions was determined by the size of seven extra dimensions in an initial excited state before inflation. The size of the extra dimensions can be inferred from a simple scheme for unifying the strong force and gravity. Coupled with the Bekenstein-Hawking entropy bound, these considerations lead to a scaling law for the cosmological constant that has been proposed independently by several authors.

The status of quantum geometry in the dynamical sector of loop quantum cosmology

This paper is motivated by the recent papers by Dittrich and Thiemann and, respectively, Rovelli discussing the status of quantum geometry in the dynamical sector of quantum geometry. Since the papers consider model examples, we also study the issue in the case of an example, namely on the loop quantum cosmology model of a space-isotropic universe. We derive the Rovelli–Thiemann–Dittrich partial observables corresponding to the quantum geometry operators of LQC in both Hilbert spaces: the kinematical one and the physical Hilbert space of solutions to the quantum constraints. We find that quantum geometry can be used to characterize the physical solutions, and the operators of quantum geometry preserve many of their kinematical properties.

Perfect fluid quantum Universe in the presence of negative cosmological constant

We present perfect fluid Friedmann–Robertson–Walker quantum cosmological models in the presence of negative cosmological constant. In this work the Schutz’s variational formalism is applied for radiation, dust, cosmic string, and domain wall dominated Universes with positive, negative, and zero constant spatial curvature. In this approach the notion of time can be recovered. These give rise to Wheeler–DeWitt equations for the scale factor. We find their eigenvalues and eigenfunctions by using Spectral Method. After that, we use the eigenfunctions in order to construct wave packets for each case and evaluate the time-dependent expectation value of the scale factors, which are found to oscillate between finite maximum and minimum values. Since the expectation values of the scale factors never tends to the singular point, we have an initial indication that these models may not have singularities at the quantum level.

An open FRW model in loop quantum cosmology

An open FRW model in loop quantum cosmology is considered. The left and right invariant vector fields and holonomies along them are studied. It is shown that in the hyperbolic geometry of k = −1 it is possible to construct a suitable loop which provides us with the quantum scalar constraint originally introduced by Vandersloot [19]. The quantum scalar constraint operator with negative cosmological constant is proved to be essentially self-adjoint.

Quantum FRW cosmological solutions in the presence of Chaplygin gas and perfect fluid

We present a Friedmann–Robertson–Walker quantum cosmological model in the presence of Chaplygin gas and perfect fluid for early and late time epochs. In this work, we consider perfect fluid as an effective potential and apply Schutz's variational formalism to the Chaplygin gas which recovers the notion of time. These give rise to Schrödinger–Wheeler–DeWitt equation for the scale factor. We use the eigenfunctions in order to construct wave packets and study the time dependent behavior of the expectation value of the scale factor using the many-worlds interpretation of quantum mechanics. We show that contrary to the classical case, the expectation value of the scale factor avoids singularity at quantum level. Moreover, this model predicts that the expansion of Universe is accelerating for the late times.

Quantum Stephani exact cosmological solutions and the selection of time variable

We study a perfect fluid Stephani quantum cosmological model. In the present work, the Schutz's variational formalism which recovers the notion of time is applied. This gives rise to a Wheeler–DeWitt equation for the scale factor. We use the eigenfunctions in order to construct wave packets for each case. We study the time-dependent behavior of the expectation value of the scale factor, using many-worlds and de Broglie–Bohm interpretations of quantum mechanics.

Inflationary nonsingular quantum cosmological model

A stiff matter-dominated universe modeled by a free massless scalar field minimally coupled to gravity in a Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) geometry is quantized. Generalized complex-width Gaussian superpositions of the solutions of the Wheeler-DeWitt equation are constructed and the Bohm-de Broglie interpretation of quantum cosmology is applied. A planar dynamical system is found in which a diversity of quantum Bohmian trajectories are obtained and discussed. One class of solutions represents nonsingular inflationary models starting at infinity past from flat space-time with Planckian size spacelike hypersurfaces, which inflates without inflaton but due to a quantum cosmological effect, until it makes an analytical graceful exit from this inflationary epoch to a decelerated classical stiff matter expansion phase.

Behavior of nonlinear anisotropies in bouncing Bianchi I models of loop quantum cosmology

In homogeneous and isotropic loop quantum cosmology, gravity can behave repulsively at Planckian energy densities leading to the replacement of the big bang singularity with a big bounce. Yet in any bouncing scenario it is important to include nonlinear effects from anisotropies which typically grow during the collapsing phase. We investigate the dynamics of a Bianchi I anisotropic model within the framework of loop quantum cosmology. Using effective semiclassical equations of motion to study the dynamics, we show that the big bounce is still predicted with only differences in detail arising from the inclusion of anisotropies. We show that the anisotropic shear term grows during the collapsing phase, but remains finite through the bounce. Immediately following the bounce, the anisotropies decay and with the inclusion of matter with equation of state $w&lt;+1$, the universe isotropizes in the expanding phase.

Stephani–Schutz quantum cosmology

We study the Stephani quantum cosmological model in the presence of a cosmological constant in radiation dominated Universe. In the present work the Schutz's variational formalism which recovers the notion of time is applied. This gives rise to Wheeler–DeWitt equations which can be cast in the form of Schrödinger equations for the scale factor. We find their eigenvalues and eigenfunctions by using the spectral method. Then we use the eigenfunctions in order to construct wave packets and evaluate the time-dependent expectation value of the scale factor, which is found to oscillate between non-zero finite maximum and minimum values. Since the expectation value of the scale factor never tends to the singular point, we have an initial indication that this model may not have singularities at the quantum level.

Schrödinger–Wheeler–DeWitt Equation in Chaplygin Gas FRW Cosmological Model

We present a Chaplygin gas Friedmann–Robertson–Walker quantum cosmological model. In this work the Schutz’s variational formalism is applied with positive, negative, and zero constant spatial curvature. In this approach the notion of time can be recovered. These give rise to Schrödinger–Wheeler–DeWitt equation for the scale factor. We use the eigenfunctions in order to construct wave packets for each case. We study the time dependent behavior of the expectation value of the scale factor, using the many-worlds interpretations of quantum mechanics.

Closed Friedmann–Robertson–Walker model in loop quantum cosmology

The basic idea of loop quantum cosmology (LQC) applies to every spatially homogeneous cosmological model; however only the spatially flat (so-called k = 0) case has been understood in detail in the literature thus far. In the closed (so-called k = 1) case certain technical difficulties have been the obstacle to development. In this work the difficulties are overcome, and a new LQC model of the spatially closed, homogeneous, isotropic universe is constructed. The topology of the spacelike section of the universe is assumed to be that of SU(2) or SO(3). Surprisingly, according to the new results achieved in this paper, the two cases can be distinguished from each other just by the local properties of the quantum geometry of the universe! The quantum Hamiltonian operator of the gravitational field takes the form of a difference operator, where the elementary step is the quantum of the 3-volume derived in the flat case by Ashtekar, Pawlowski and Singh. The mathematical properties of the operator are studied: it is essentially self-adjoint, bounded from above by 0, the 0 itself is not an eigenvalue, the eigenvectors form a basis. An estimate on the dimension of the spectral projection on any finite interval is provided.

Large scale effective theory for cosmological bounces

An exactly solvable bounce model in loop quantum cosmology is identified which serves as a perturbative basis for realistic bounce scenarios. Its bouncing solutions are derived analytically, demonstrating why recent numerical simulations robustly led to smooth bounces under the assumption of semiclassicality. Several effects, easily included in a perturbative analysis, can however change this smoothness. The effective theory is not only applicable to such situations where numerical techniques become highly involved but also allows one to discuss conceptual issues. For instance, consequences of the notoriously difficult physical inner product can be implemented at the effective level. This indicates that even physical predictions from full quantum gravity can be obtained from perturbative effective equations.