15 pages, v2: Extended with a more complete treatment of parity transformations, other minor improvements, v3: Discussion significantly extended, typos corrected, references added, v4: Typos corrected
Using the complex-valued self-dual connection variables, the loop quantum cosmology of a closed Friedmann universe coupled to a massless scalar field is studied. It is shown how the reality conditions can be imposed in the quantum theory by choosing a particular inner product for the kinematical Hilbert space. While holonomies of the self-dual Ashtekar connection are not well-defined in the kinematical Hilbert space, it is possible to introduce a family of generalized holonomy-like operators of which some are well-defined; these operators in turn are used in the definition of the Hamiltonian constraint operator where the scalar field can be used as a relational clock. The resulting quantum theory is closely related, although not identical, to standard loop quantum cosmology constructed from the Ashtekar-Barbero variables with a real Immirzi parameter. Effective Friedmann equations are derived, which provide a good approximation to the full quantum dynamics for sharply-peaked states whose volume remains much larger than the Planck volume, and they show that for these states quantum gravity effects resolve the big-bang and big-crunch singularities and replace them by a non-singular bounce. Finally, the loop quantization in self-dual variables of a flat Friedmann space-time is recovered in the limit of zero spatial curvature and is identical to the standard loop quantization in terms of the real-valued Ashtekar-Barbero variables.
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 us 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.
The dynamical behaviors of interacting dark energy in loop quantum cosmology are discussed in this paper. Based on defining three dimensionless variables, we simplify the equations of the fixed points. The fixed points for interacting dark energy can be determined by the Friedmann equation coupled with the dynamical equations {in Einstein cosmology}. But in loop quantum cosmology, besides the Friedmann equation, the conversation equation also give a constrain on the fixed points. The difference of stability properties for the fixed points in loop quantum cosmology and the ones in Einstein cosm…
Quantum Reduced Loop Gravity is a promising framework for linking Loop Quantum Gravity and the effective semiclassical dynamics of Loop Quantum Cosmology. We review its basic achievements and its main perspectives, outlining how it provides a quantum description of the Universe in terms of a cuboidal graph which constitutes the proper framework for applying loop techniques in a cosmological setting.
The paradigmatic models often used to highlight cosmological features of loop quantum gravity and group field theory are shown to be equivalent, in the sense that they are different realizations of the same model given by harmonic cosmology. The loop version of harmonic cosmology is a canonical realization, while the group-field version is a bosonic realization. The existence of a large number of bosonic realizations suggests generalizations of models in group field cosmology.
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 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 light on the physical and mathematical relation between loop cosmology and f…
In this paper we construct the Hamiltonian constraint operator of loop quantum cosmology using holonomies defined for arbitrary irreducible SU(2) representations labeled by spin J. We show that modifications to the effective semi-classical equations of motion arise both in the gravitational part of the constraint as well as matter terms. The modifications are important for phenomenological investigations of the cosmological imprints of loop quantum cosmology. We discuss the implications for the early universe evolution.
In this letter, we demonstrate that Loop Quantum Cosmology equations, with a near vanishing late time cosmological constant, can be induced on a brane in a Randall-Sundrum II scenario, corresponding to the holographic dual of a strongly coupled string regime in the bulk. Such results can establish a possible connection between Loop Quantum Gravity and string theory.
Here we shall find the green's function of the difference equation of loop quantum cosmology. To illustrate how to use it, we shall obtain an iterative solution for closed model and evaluate its corresponding Bohmian trajectory.
As a first step to generalize the structure of loop quantum cosmology to the theories with the spacetime dimension other than four, the isotropic model of loop quantum cosmology in 2+1 dimension is studied in this paper. We find that the classical big bang singularity is again replaced by a quantum bounce in the model. The similarities and differences between the 2+1 dimensional model and the 3+1 dimensional one are also discussed.
We study oscillatory universes within the context of Loop Quantum Cosmology. We make a comparative study of flat and positively curved universes sourced by scalar fields with either positive or negative potentials. We investigate how oscillating universes can set the initial conditions for successful slow-roll inflation, while ensuring that the semi-classical bounds are satisfied. We observe rich oscillatory dynamics with negative potentials, although it is difficult to respect the semi-classical bounds in models of this type.
An action functional for the loop quantum cosmology difference equation is presented. It is shown that by guessing the general form of the solution and optimizing the action functional with respect to the parameters in the guessed solution one can obtain approximate solutions which are reasonably good.
This brief article sums up results obtained in arXiv:0911.2653, which develops a constrained SU(2) lattice gauge theory in the "dipole" approximation. This is a further step toward the issue of a (inhomogeneous) loop quantum cosmology and its merging into loop quantum gravity.
We study the problem of gauge-invariance and gauge-dependence in one-loop quantum cosmology. We formulate some requirements which should be satisfied by boundary conditions in order to give gauge-independent path integral. The case of QED is studied in some detail. We outline difficulties in gauge-invariant quantization of gravitational field in a bounded region.
For many years, the most active area of quantum cosmology has been the issue of choosing boundary conditions for the wave function of a universe. Recently, loop quantum cosmology, which is obtained from loop quantum gravity, has shed new light on this question. In this case, boundary conditions are not chosen by hand with some particular physical intuition in mind, but they are part of the dynamical law. It is then natural to ask if there are any relations between these boundary conditions and the ones provided before. After discussing the technical foundation of loop quantum cosmology which l…
In this paper we present a model of Riemannian loop quantum cosmology with a self-adjoint quantum scalar constraint. The physical Hilbert space is constructed using refined algebraic quantization. When matter is included in the form of a cosmological constant, the model is exactly solvable and we show explicitly that the physical Hilbert space is separable consisting of a single physical state. We extend the model to the Lorentzian sector and discuss important implications for standard loop quantum cosmology.
A temporally discrete Schroedinger time evolution equation is proposed for isotropic quantum cosmology coupled to a massless scalar source. The approach employs dynamically determined intrinsic time and produces the correct semiclassical limit.
Loop Quantum Gravity is a background independent, nonperturbative approach to the quantization of General Relativity. Its application to models of interest in cosmology and astrophysics, known as Loop Quantum Cosmology, has led to new and exciting views of the gravitational phenomena that took place in the early universe, or that occur in spacetime regions where Einstein's theory predicts singularities. We provide a brief introduction to the bases of Loop Quantum Cosmology and summarize the most important results obtained in homogeneous scenarios. These results include a mechanism to avoid the…
Loop quantum cosmology(LQC) is the symmetric model of loop quantum gravity. In this paper, we generalize the structure of loop quantum cosmology to the theories with arbitrary spacetime dimensions. The isotropic and homogenous cosmological model in n+1 dimensions is quantized by the loop quantization method. Interestingly, we find that the underlying quantum theories are divided into two qualitatively different sectors according to spacetime dimensions. The effective Hamiltonian and modified dynamical equations of n+1 dimensional LQC are obtained. Moreover, our results indicate that the classi…
We shall show that it is possible to make a causal interpretation of loop quantum cosmology using the momentum as the dynamical variable. We shall show that one can derive Bohmian trajectories. For a sample cosmological solution with cosmological constant, the trajectory is plotted.
Loop quantum cosmology (LQC) is very powerful to deal with the behavior of early universe. And the effective loop quantum cosmology gives a successful description of the universe in the semiclassical region. We consider the apparent horizon of the Friedmann-Robertson-Walker universe as a thermodynamical system and investigate the thermodynamics of LQC in the semiclassical region. The effective density and effective pressure in the modified Friedmann equation from LQC not only determine the evolution of the universe in LQC scenario but are actually also found to be the thermodynamic quantities.…
The continuum and semiclassical limits of isotropic, spatially flat loop quantum cosmology are discussed, with an emphasis on the role played by the Barbero-Immirzi parameter γin controlling space-time discreteness. In this way, standard quantum cosmology is shown to be the simultaneous limit γ\to 0, j \to \infty of loop quantum cosmology. Here, j is a label of the volume eigenvalues, and the simultaneous limit is technically the same as the classical limit \hbar \to 0, l \to \infty of angular momentum in quantum mechanics. Possible lessons for semiclassical states at the dynamical level in th…
Open FRW model in Loop Quantum Cosmology is under consideration. 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 quantum scalar constraint originally introduced by Vandersloot. The quantum scalar constraint operator with negative cosmological constant is proved to be essentially self-adjoint.
The article gives an account of several aspects of the space known as the Bohr compactification of the line, featuring as the quantum configuration space in loop quantum cosmology, as well as of the corresponding configuration space realization of the so-called polymer representation. Analogies with loop quantum gravity are explored, providing an introduction to (part of) the mathematical structure of loop quantum gravity, in a technically simpler context.
The loop quantum cosmology of the Bianchi type II string cosmological model in the presence of a homogeneous magnetic field is studied. We present the effective equations which provide modifications to the classical equations of motion due to quantum effects. The numerical simulations confirm that the big bang singularity is resolved by quantum gravity effects.
We follow the Feynman procedure to obtain a path integral formulation of loop quantum cosmology starting from the Hilbert space framework. Quantum geometry effects modify the weight associated with each path so that the effective measure on the space of paths is different from that used in the Wheeler-DeWitt theory. These differences introduce some conceptual subtleties in arriving at the WKB approximation. But the approximation is well defined and provides intuition for the differences between loop quantum cosmology and the Wheeler-DeWitt theory from a path integral perspective.
We introduce a new framework for loop quantum gravity: mimicking the spinfoam quantization procedure we propose to study the symmetric sectors of the theory imposing the reduction weakly on the full kinematical Hilbert space of the canonical theory. As a first application of Quantum-Reduced Loop Gravity we study the inhomogeneous Bianchi I model. The emerging quantum cosmological model represents a simplified arena on which the complete canonical quantization program can be tested. The achievements of this analysis could elucidate the relationship between Loop Quantum Cosmology and the full th…
In this article, we study the equations driving the dynamics of a Bianchi-I universe described by holonomy corrected effective loop quantum cosmology. We derive the LQC-modified generalized Friedmann equation, which is used as a guide to find different types of solutions. It turns out that, in this framework, most solutions never reach the classical behavior.
In loop quantum cosmology, the values of volume fluctuations and correlations determine whether the dynamics of an evolving state exhibits a bounce. Of particular interest are states that are supported only on either the positive or the negative part of the spectrum of the Hamiltonian that generates this evolution. It is shown here that the restricted support on the spectrum does not significantly limit the possible values of volume fluctuations.
The dressed-metric approach is shown to violate general covariance by demonstrating that it cannot have an off-shell completion in which the correct infinitesimal relations of space-time hypersurface deformations are realized. The main underlying reason -- a separation of background degrees of freedom and modes of inhomogeneity that is incompatible with covariance -- is shared with other approaches such as hybrid loop quantum cosmology.
To inherit more features of full loop quantum Brans-Dicke theory, the Euclidean and Lorentzian terms of the Hamiltonian constraint are quantized independently in loop quantum Brans-Dicke cosmology. An alternative Hamiltonian constraint operator and its effective expression are obtained in the cosmological model. A residual quantum correction term is found in the effective Hamiltonian constraint, which has no analog in the effective Hamiltonian of the loop quantum cosmology from general relativity. The dynamics driven by this effective Hamiltonian constraint is analyzed in detail. For the physi…
One possible description of the very early stages of the evolution of the universe is provided by Chaotic Inflationary Cosmology. For that model the role of the inflaton field is played by quantum gravitational effects. We study if such a picture may arise within the framework of Loop Quantum gravity by studying a simple model. While we were unable to reach definitive conclusions we believe the general approach proposed in this paper may prove fruitful in the future.
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 $κ$-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.
Loop quantum cosmology allows for arbitrary superpositions of the triad variable. We show here how these superpositions can become indistinguishable from a classical mixture by the interaction with fermions. We calculate the reduced density matrix for a locally rotationally symmetric Bianchi I model and show that the purity factor for the triads decreases by decoherence. In this way, the universe assumes a definite orientation.
In this work, we extend the formalism of hybrid loop quantum cosmology for primordial perturbations around a flat, homogeneous, and isotropic universe to the new treatment of Friedmann-Lemaître-Robertson-Walker geometries proposed recently by Dapor and Liegener, based on an alternative regularization of the Hamiltonian constraint. In fact, our discussion is applicable also to other possible regularization schemes for loop quantum cosmology, although we specialize our analysis to the Dapor-Liegener proposal and construct explicitly all involved quantum operators for that case.
We calculate the full spectrum, as observed today, of the cosmological gravitational waves generated within a model based on loop quantum cosmology. It is assumed that the universe, after the transition to the classical regime, undergoes a period of inflation driven by a scalar field with a chaotic-type potential. Our analysis shows that, for certain conditions, loop quantum effects leave a clear signature on the spectrum, namely, an over-production of low-frequency gravitational waves. One of the aims of our work is to show that loop quantum cosmology models can be tested and that, more gener…
We generalize the perturbations theory of loop quantum cosmology to a hydrodynamical form and define an effective curvature perturbation on an uniform density hypersurfaces $ζ_e$. As in the classical cosmology, $ζ_e$ should be gauge-invariant and conservation on the large scales. The evolutions of both the adiabatic and the non-adiabatic perturbations for a multi-fluids model are investigated in the framework of the effective hydrodynamical theory of loop quantum cosmology with the inverse triad correction. We find that, different from the classical cosmology, the evolution of the large-scales…
In this paper we discuss the inflationary universe in the context of a Chaplygin gas equation of state within the framework of the effective theory of loop quantum cosmology. Under the slow-roll approximation, we calculate the primordial perturbations for this model. We give the general expressions of the scalar spectral index, its running, and the tensor-to-scalar ratio, etc. For the chaotic inflation with a quadratic potential, using the WMAP 5-year results, we determine the parameters of the Chaplygin inflation model in loop quantum cosmology. The results are consistent with the WMAP observ…
The loop quantum cosmology "improved dynamics" of the Bianchi type IX model are studied. The action of the Hamiltonian constraint operator is obtained via techniques developed for the Bianchi type I and type II models, no new input is required. It is shown that the big bang and big crunch singularities are resolved by quantum gravity effects. We also present the effective equations which provide modifications to the classical equations of motion due to quantum geometry effects.
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 …
A free massless scalar field is coupled to homogeneous and isotropic loop quantum cosmology. The coupled model is investigated in the vicinity of the classical singularity, where discreteness is essential and where the quantum model is non-singular, as well as in the regime of large volumes, where it displays the expected semiclassical features. The particular matter content (massless, free scalar) is chosen to illustrate how the discrete structure regulates pathological behavior caused by kinetic terms of matter Hamiltonians (which in standard quantum cosmology lead to wave functions with an …
We develop an action principle for those models arising from isotropic loop quantum cosmology, and show that there is a natural conserved quantity $Q$ for the discrete difference equation arising from the Hamiltonian constraint. This quantity $Q$ relates the semi-classical limit of the wavefunction at large values of the spatial volume, but opposite triad orientations. Moreover, there is a similar quantity for generic difference equations of one parameter arising from a self-adjoint operator.
We study the scalar modes of linear perturbations in loop quantum cosmology. This is done on a lattice where each cell is taken to be homogeneous and isotropic and can be quantized via standard homogeneous loop quantum cosmology techniques. The appropriate interactions between nearby cells are included in the Hamiltonian in order to obtain the correct physics. It is shown that the quantum theory is anomaly-free: the scalar and diffeomorphism constraint operators weakly commute with the Hamiltonian. Finally, the effective theory encoding the leading order quantum gravity corrections is derived …
We present some results concerning the large volume limit of loop quantum cosmology in the flat homogeneous and isotropic case. We derive the Wheeler-De Witt equation in this limit. Looking for the action from which this equation can also be obtained, we then address the problem of the modifications to be brought to the Friedman's equation of motion and to the equation of motion of the scalar field, in the classical limit.
The use of techniques from loop quantum gravity for cosmological models may solve some difficult problems in quantum cosmology. The solutions under a number of circumstances have been well studied. We will analyse the behaviour of solutions in the closed model, focusing on the behaviour of a universe containing a massless scalar field. The asymptotic behaviour of the solutions is examined, and is used to determine requirements of the initial conditions.
The covariant entropy bound conjecture is an important hint for the quantum gravity, with several versions available in the literature. For cosmology, Ashtekar and Wilson-Ewing ever show the consistence between the loop gravity theory and one version of this conjecture. Recently, S. He and H. Zhang proposed a version for the dynamical horizon of the universe, which validates the entropy bound conjecture for the cosmology filled with perfect fluid in the classical scenario when the universe is far away from the big bang singularity. However, their conjecture breaks down near big bang region. We…
In this article, 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 are in complete agreement with the expectations from the von Neumann stability analysis of the model.
Since the past Iagrg meeting in December 2004, new developments in loop quantum cosmology have taken place, especially with regards to the resolution of the Big Bang singularity in the isotropic models. The singularity resolution issue has been discussed in terms of physical quantities (expectation values of Dirac observables) and there is also an ``improved'' quantization of the Hamiltonian constraint. These developments are briefly discussed. This is an expanded version of the review talk given at the 24$^{\mathrm{th}}$ IAGRG meeting in February 2007.
We compute the predictions for the power spectrum of scalar perturbations from a recent new proposal for the effective Hamiltonian of loop quantum cosmology. The model provides an attractive picture of the early cosmos, in which our classical Friedmann-Lemaître-Robertson-Walker universe emerges from a quantum phase where the spacetime curvature remains constant and of Planckian size. We compare the predictions for the cosmic microwave background with previous results obtained within loop quantum cosmology, and discuss the differences and similarities. The analysis provides an example of the wa…
In this paper we introduce a numerical approximation technique to obtain pre-classical solutions to models of loop quantum gravity. In particular, we apply the technique to vacuum Bianchi I cosmological models and recover known solutions. We also present a pre-classical solution to the Bianchi I LRS model with cosmological constant, which has not appeared elsewhere.
A path integral formulation of the Bianchi I models containing a massless scalar field in loop quantum cosmology is constructed. Following the strategy used in the homogenous and isotropic case, the calculation is extended to the simplest non-isotropic models according to the $\barμ$ and $\barμ^{\prime}$ scheme. It is proved from the path integral angle that the quantum dynamic lacks the full invariance with respect to fiducial cell scaling in the $\barμ$ scheme, but it does not in the $\barμ^{\prime}$ scheme. The investigation affirms the equivalence of the canonical approach and the path int…
A Bianchi type-I cosmological model in the presence of a magnetic flux along a cosmological string is considered. The first objective of this study is to investigate Einstein equations using a tractable assumption usually accepted in the literature. Quantum effects of the present cosmological model are examined in the framework of loop quantum cosmology. Finally we draw a parallel between the classical and quantum approaches.
We study the anisotropic Bianchi I loop quantum cosmology in 2+1 dimensions. Both the $\mubar$ and $\mubar'$ schemes are considered in the present paper and the following expected results are established: (i) the massless scalar field again play the role of emergent time variables and serves as an internal clock; (ii) By imposing the fundamental discreteness of length operator, the total Hamiltonian constraint is obtained and gives rise the evolution as a difference equation; and (iii) the exact solutions of Friedmann equation are constructed rigorously for both classical and effective level. …
