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We propose a new method to investigate signatures of a quantum gravity phase in the primordial state of cosmological perturbations. We formulate and study a quantum model of a perturbed Friedmann-Lemaitre-Robertson-Walker universe beyond a tensor-product Born-Oppenheimer-like factorization, that is, without restricting the wave function of the universe to the product of the background and perturbation wave functions. We show that the quantum dynamics generically does not preserve the product form of the universe's wave function, which spontaneously evolves into a more general entangled state. Upon expanding this state in a suitable basis of background wave functions and setting Gaussian initial conditions for the perturbations, we numerically find that each of these wave functions becomes associated with a non-Gaussian state of an inhomogeneous perturbation., Comment: 10 pages, 7 figures

We study the non-unitary relation between quantum gravitational models defined using different internal times. We show that despite the non-unitarity, it is possible to provide a prescription for making unambiguous, though restricted, physical predictions independent of specific clocks. To illustrate this result, we employ a model of quantum gravitational waves in a quantum Friedmann universe., Comment: Corrected typos

We develop a relativistic framework to investigate the evolution of cosmological structures from the initial density perturbations to the highly nonlinear regime. Our approach involves proposing a procedure to match ``best-fit", exact Bianchi IX (BIX) spacetimes to finite regions within the perturbed Friedmann-Lemaitre-Robertson-Walker universe characterized by a positive averaged spatial curvature. This method enables us to approximately track the nonlinear evolution of the initial perturbation using an exact solution. Unlike standard perturbation theory and exact solutions with a high degree of symmetry (such as spherical symmetry), our approach is applicable to generic initial data, with the only requirement being positive spatial curvature. By employing the BIX symmetries, we can systematically incorporate the approximate effects of shear and curvature into the process of collapse. Our approach addresses the limitations of both standard perturbation theory and highly symmetric exact solutions, providing valuable insights into the nonlinear evolution of cosmological structures., Comment: 12 pages, 5 figures

We have discovered a class of dynamically stable coherent states for motion on the half-line. The regularization of the half-line boundary and the consequent quantum motion are expounded within the framework of covariant affine quantization, although alternative approaches are also feasible. The former approach is rooted in affine coherent states and offers a consistent semiclassical representation of quantum motion. However, this method has been known to possess two shortcomings: (a) the dependence of affine coherent states on the choice of a vector, denoted as 'fiducial vector' (which remains unspecified), introduces significant arbitrariness in boundary regularization, and (b) regardless of the choice of 'fiducial vector,' affine coherent states fail to evolve parametrically under the Schr\"odinger equation, thus limiting the accuracy of the semiclassical description. This limitation, in particular, hampers their suitability for approximating the evolution of compound observables. We demonstrate that a distinct and more refined definition of affine coherent states can simultaneously address both of these issues. In other words, these new affine coherent states exhibit parametric evolution only when the 'fiducial vector,' denoted as $|\psi_0>$, possesses a highly specific character, such as being an eigenstate of a well-defined Hamiltonian. Our discovery holds significant relevance in the field of quantum cosmology, particularly in scenarios where the positive variable is the scale factor of the universe, and its regularized motion plays a crucial role in avoiding the big-bang singularity., Comment: 15 pages, 3 figures

We study a semiclassical model of the mixmaster universe. We first derive the quantum model and then introduce its semiclassical approximation. We employ a general integral quantization method that respects the symmetries of the model given by the affine and the Weyl-Heisenberg groups, and can produce a wide class of quantum models. The semiclassical approximation is based on the coherent states. The semiclassical dynamics is complex and cannot be solved by analytical methods. We focus on a key qualitative feature of the dynamics, namely, we investigate whether the primordial anisotropic universe can undergo a spontaneous inflationary phase driven by the anisotropic energy combined with semiclassical corrections. The answer to this question provides a useful perspective on the inflationary paradigm as well as on alternative bouncing models., Comment: 15 pages, 6 figures

A quantum cosmological bouncing model may exhibit an ambiguity stemming from the nonclassical nature of the background evolution: two classically equivalent theories can produce two qualitatively different potentials sourcing the perturbations. We derive explicitly the quantum canonical transformation involving the quantum background to show how it leads to inequivalent theories. We identify the relevant quantum parameter describing the difference and expand upon the ambiguity by calculating the expected power spectra produced for initial quantum vacuum fluctuations in the contracting phase of both potentials. We find that one spectral index corresponds to all values of this parameter but one, while the other thus represents a set of measure zero., Comment: 16 pages, 7 figures, follow-up article to gr-qc 2111.02963, matching published version

We develop a complete Hamiltonian approach to the theory of perturbations around any spatially homogeneous spacetime. We employ the Dirac method for constrained systems which is well-suited to cosmological perturbations. We refine the method via the so-called Kucha\vr parametrization of the kinematical phase space. We separate the gauge-invariant dynamics of the three-surfaces from the three-surface deformations induced by linear coordinate transformations. The canonical group of the three-surface deformations and the complete space of gauge-fixing conditions are explicit in our approach. We introduce a frame in the space of gauge-fixing conditions and use it to considerably simplify the prescription for gauge-fixing, partial gauge-fixing and spacetime reconstruction. Finally, we illustrate our approach by considering the perturbed Kasner universe, for which we discuss two kinds of gauges that correspond respectively to the Coulomb-like and the Lorenz-like gauge in electrodynamics., Comment: 27 pages, 2 figures

We consider a simple cosmological model consisting of an empty Bianchi I Universe, whose Hamiltonian we deparametrise to provide a natural clock variable. The model thus effectively describes an isotropic universe with an induced clock given by the shear. Quantising this model, we obtain various different possible bouncing trajectories (semiquantum expectation values on coherent states or obtained by the de Broglie-Bohm formulation) and explicit their clock dependence, specifically emphasising the question of symmetry across the bounce., Comment: 13 pages, 4 figures, for Universe special issue on quantum cosmology

By quantising the background as well as the perturbations in a simple one fluid model, we show that there exists an ambiguity in the choice of relevant variables, potentially leading to incompatible observational physical predictions. In a classical or quantum inflationary background, the exact same canonical transformations lead to unique predictions, so the ambiguity we put forward demands a semiclassical background with a sufficiently strong departure from classical evolution. The latter condition happens to be satisfied in bouncing scenarios, which may thus be having predictability issues. Inflationary models could evade such a problem because of the monotonic behavior of their scale factor; they do, however, initiate from a singular state which bouncing scenarios aim at solving., Comment: 13 pages, 7 figures

We apply the Dirac procedure for constrained systems to the Arnowitt-Deser-Misner formalism linearized around the Bianchi I universe. We discuss and employ basic concepts such as Dirac observables, Dirac brackets, gauge-fixing conditions, reduced phase space, physical Hamiltonian, canonical isomorphism between different gauge-fixing surfaces and spacetime reconstruction. We relate this approach to the gauge-fixing procedure for non-perturbative canonical relativity. We discuss the issue of propagating a basis for the scalar-vector-tensor decomposition as, in an anisotropic universe, the wavefronts of plane waves undergo a nontrivial evolution. We show that the definition of a gravitational wave as a traceless-transverse mode of the metric perturbation needs to be revised. Moreover there exist coordinate systems in which a polarization mode of the gravitational wave is given entirely in terms of a scalar metric perturbation. We first develop the formalism for the universe with a single scalar field and then extend it to the multi-field case. The obtained fully canonical formalism will serve as a starting point for a complete quantization of the cosmological perturbations and the cosmological background., Comment: Added references, corrected typos, clarified notation, improved discussions. Results unchanged

Quantum cosmological models are commonly described by means of semiclassical approximations in which a smooth evolution of the expectation values of elementary geometry operators replaces the classical and singular dynamics. The advantage of such descriptions is that they are relatively simple and display the classical behavior for large universes. However, they may smooth out an important inner structure and to include it a more detailed treatment is needed. The purpose of the present work is to investigate quantum uncertainty in the basic background variables and its influence on primordial gravitational waves. To this end we quantize a model of the Friedmann-Lemaitre-Robertson-Walker universe filled with a linear barotropic cosmological fluid and with gravitational waves. We carefully derive the dynamical equations for the perturbations in quantum spacetime. The quantization yields an equation of motion for the Fourier modes of gravitational radiation, which is a quantum extension to the usual parametric oscillator equation for gravitational waves propagating in an expanding universe. The two quantum effects from the cosmological background that enter the enhanced equation of motion are (i) a repulsive potential resolving the big bang singularity and replacing it with a big bounce and (ii) uncertainties in the numerical values for the background spacetime dynamical variables. First we study the former effect and its consequences for the primordial amplitude spectrum and carefully discuss the relation between the bounce scale and the physical predictions of the model. Next we investigate the latter effect, in particular the extent to which it may affect the primordial amplitude of gravitational waves. Making use of the WKB approximation we find an analytical formula for the amplitude spectrum as a function of the quantum dispersion of the background spacetime., Comment: 16 pages; 14 figures. The new version is accepted by the journal

We discuss the question of time in a Bianchi I quantum cosmology in the framework of singularity avoidance. We show that time parameters fall into two distinct classes, that are such that the time development of the wave function either always leads to the appearance of a singularity (fast-gauge time) or that always prevents it from occurring (slow-gauge time). Furthermore, we find that, in the latter case, there exists an asymptotic regime, independent of the clock choice. This may point to a possible solution of the clock issue in quantum cosmology if there exists a suitable class of clocks all yielding identical relevant physical consequences., Comment: 16 pages, 13 figures, matches version to appear in Phys. Rev. D

The Mixmaster solution to Einstein field equations was examined by C. Misner in an effort to better understand the dynamics of the early universe. We highlight the importance of the quantum version of this model for early universe. This quantum version and its semi-classical portraits are yielded through affine and standard coherent state quantizations and more generally affine and Weyl-Heisenberg covariant integral quantizations. The adiabatic and vibronic approximations widely used in molecular physics can be employed to qualitatively study the dynamics of the model on both quantum and semi-classical levels. Moreover, the semi-classical approach with the exact anisotropy potential can be effective in numerical integration of some solutions. Some promising physical features such as the singularity resolution, smooth bouncing, the excitation of anisotropic oscillations and a substantial amount of post-bounce inflation as the backreaction to the latter are pointed out. Finally, a realistic cosmological scenario based on the quantum mixmaster model, which includes the formation and evolution of local structures is outlined., Comment: 9 figures, Key words and phrases. quantum cosmology; mixmaster; coherent states; quantization; semi-classical portraits; smooth bouncing

We apply the Dirac procedure for constrained systems to the Arnowitt-Deser-Misner formalism linearized around the Friedmann-Lemaitre universe. We explain and employ some basic concepts such as Dirac observables, Dirac brackets, gauge-fixing conditions, reduced phase space, physical Hamiltonian and physical dynamics. In particular, we elaborate on the key concept which is the canonical isomorphism between different gauge-fixing surfaces. We apply our formalism to describe the reduced phase space of cosmological perturbations in some popular in the literature gauges. Our formalism is first developed for the universe with a single fluid and then extended to the multi-fluid case. The obtained results are a starting point for complete quantization of the cosmological perturbations and the cosmological background. Our approach may be used in future to derive the reduced phase space of higher order perturbations and in more generic cosmological spacetimes., Comment: 38 pages, 1 figure, includes a discussion of the relation between the Dirac observables and the gauge-invariant variables such as the Bardeen potentials or the Mukhanov-Sasaki variable

We develop an approach to quantum dynamics based on quantum phase space trajectories. The latter are built from a unitary irreducible representation of the symmetry group of the respective classical phase space. We use a quantum action functional to derive the basic equations. In principle, our formulation is equivalent to the Hilbert space formulation. However, the former allows for consistent truncations to reduced phase spaces in which approximate quantum dynamics can be derived. We believe that our approach can be very useful in the domain of quantum cosmology and therefore, we use the cosmological phase space example to establish the basic equations of this formalism., Comment: 11 pages, 4 figures, the new version contains improved discussions

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 outcome, the initial singularity is removed and replaced with a quantum bounce. The quantum model depends on a free parameter that is naturally induced from quantisation and determines the scale of the bounce. We study the dynamics of the quantised gravitational waves across the bounce through three different methods ("thin-horizon", analytical and numerical) which give consistent results and we determine the primordial power spectrum for the case of radiation-dominated universe. Next, we use the instantaneous radiation-matter transition transfer function to make approximate predictions for late universe and constrain our model with LIGO and Planck data. We also give an estimate of the quantum uncertainties in the present-day universe., Comment: 38 pages, 8 figures

We employ the framework of affine covariant quantization and associated semiclassical portrait to address two main issues in the domain of quantum gravitational systems: (i) the fate of singularities and (ii) the lack of external time. Our discussion is based on finite-dimensional, symmetry-reduced cosmological models. We show that the affine quantization of the cosmological dynamics removes the classical singularity and univocally establishes a unitary evolution. The semiclassical portrait based on the affine coherent states exhibits a big bounce replacing the big-bang singularity. As a particularly interesting application, we derive and study a unitary quantum dynamics of the spatially homogenous, closed model, the Mixmaster universe. At the classical level it undergoes an infinite number of oscillations before collapsing into a big-crunch singularity. At the quantum level the singularity is shown to be replaced by adiabatic and nonadiabatic bounces. As another application, we consider the problem of time. We derive semiclassical portraits of quantum dynamics of the Friedman universe with respect to various internal degrees of freedom. Next we compare them and discuss the nature of quantum evolution of the gravitational field., Comment: 27 pages, 10 figures, Contribution to the Proceedings of the conference "Coherent States and their Applications: A Contemporary Panorama", November 14-18th, 2016, Centre International de Rencontres Scientifiques, Luminy, Marseille, France

The basic tenet of the present work is the assumption of the lack of external and fixed time in the Universe. This assumption is best embodied by general relativity, which replaces the fixed space-time structure with the gravitational field, which is subject to dynamics. The lack of time does not imply the lack of evolution but rather brings to the forefront the role of internal clocks which are some largely arbitrary internal degrees of freedom with respect to which the evolution of timeless systems can be described. We take this idea seriously and try to understand what it implies for quantum mechanics when the fixed external time is replaced by an arbitrary internal clock. We put the issue in a solid, mathematically rigorous framework. We find that the dynamical interpretation of a quantum state of a timeless system depends on the employed internal clock. In particular, we find that the continuous spectra of well-known dynamical observables like the position of a free particle on the real line may turn discrete if measured in unusual clocks. We discuss the meaning of our result for attempts at quantization of global gravitational degrees of freedom., Comment: 13 pages, 5 figures

We study the spectral properties of the anisotropic part of Hamiltonian entering the quantum dynamics of the Mixmaster universe. We derive the explicit asymptotic expressions for the energy spectrum in the limit of large and small volumes of the universe. Then we study the threshold condition between both regimes. Finally we prove that the spectrum is purely discrete for any volume of the universe. Our results validate and improve the known approximations to the anisotropy potential. They should be useful for any approach to the quantization of the Mixmaster universe., Comment: Version accepted by Phys. Rev. D

We investigate how the quantum dynamics of gravitational models depends on the internal clock employed in quantisation. Our previous result on the quantum Friedmann-Lemaitre model demonstrates that almost all physical predictions from the quantum cosmological dynamics, e.g. the scale of the quantum bounce or the number of bounces, depend on the choice of internal clock. In the present paper we show that there exists an important prediction concerning the quantum dynamics which holds in all internal clocks. Namely, we prove that any quantum state asymptotically, i.e. away from a quantum interaction regime, admits a unique classical limit described by unique classical solutions irrespectively of the internal clock used for dynamical description of the given state. We demonstrate this property explicitly for the semiclassical Bianchi Type I model, which includes the semiclassical Friedmann-Lemaitre model as a special case. Our methods include the theory of pseudo-canonical transformations and the phase space portraits based on coherent states. As a by-product of the pursuit of our main goal, we elaborate the semiclassical description of anisotropic singularity resolution., Comment: 28 pages, 8 figures, largely improved version

We summarize our investigation of the extent to which the choice of internal clock influences the dynamics in quantum models of gravity. Firstly, at the classical level, we define an extension to the Hamilton-Jacobi theory of contact transformations, which allows for transformations of time coordinates. Secondly, at the quantum level, we employ the extended theory to separate the quantum effects brought by the free choice of internal clock from those originating from inequivalent quantization maps. Next, we show with two examples two kinds of origin of the clock effect in quantum gravitational systems., Comment: 6 pages, Proceedings of the 14th Marcel Grossmann Meeting (Rome, July 12-18, 2015)

A brief summary of the application of coherent states in the examination of quantum dynamics of cosmological models is given. We discuss quantization maps, phase space probability distributions and semiclassical phase spaces. The implementation of coherent states based on the affine group resolves the hardest singularities, renders self-adjoint Hamiltonians without boundary conditions and provides a completely consistent semi-classical description of the involved quantum dynamics. We consider three examples: the closed Friedmann model, the anisotropic Bianchi Type I model and the deep quantum domain of the Bianchi Type IX model., Comment: 6 pages, Proceedings of the 14th Marcel Grossmann Meeting (Rome, July 12-18, 2015)

Following our previous papers concerning the quantization of the vacuum Bianchi-IX model within or beyond the Born-Oppenheimer and adiabatic approximation, we develop a more elaborate analysis of the dynamical properties of the model based the vibronic approach utilized in molecular physics. As in the previous papers, we restrict our approach to the harmonic approximation of the anisotropy potential in order to obtain resoluble analytical expressions., Comment: 29 pages, 12 figures

Following our previous paper, Bergeron et al, Smooth quantum dynamics of the mixmaster universe, Phys. Rev. D 92, 061302(R) (2015), concerning the quantization of the vacuum Bianchi IX model and the Born-Huang-Oppenheimer framework, we present a further analysis of the dynamical properties of the model. Consistently with the deep quantum regime, we implement the harmonic approximation of the anisotropy potential. We thus obtain manageable dynamical equations. We study the quantum anisotropic oscillations during the bouncing phase of the universe. Neglecting the backreaction from transitions between quantum anisotropy states we obtain analytical results. In particular, we identify a parameter which is associated with dynamical properties of the quantum model and describes a sort of phase transition. Once the parameter exceeds its critical value, the Born-Huang-Oppenheimer approximation breaks down. The application of the present result to a simple model of the Universe indicates that the parameter indeed exceeds its critical value and that there takes place a huge production of anisotropy at the bounce. This in turn must lead to a sustained phase of accelerated expansion, an inflationary phase. The quantitative inclusion of backreaction shall be examined in a follow-up paper based on the vibronic approach., Comment: 32 pages, 9 figures

The appearance of Hamiltonian constraint in the canonical formalism for general relativity reflects the lack of a fixed external time. The dynamics of general relativistic systems can be expressed with respect to an arbitrarily chosen internal degree of freedom, the so called internal clock. We investigate the way in which the choice of internal clock determines the quantum dynamics and how much different quantum dynamics induced by different clocks are. We develop our method of comparison by extending the Hamilton-Jacobi theory of contact transformations to include a new type of transformations which transform both the canonical variables and the internal clock. We employ our method to study the quantum dynamics of the Friedmann-Lemaitre model and obtain semiclassical corrections to the classical dynamics, which depend on the choice of internal clock. For a unique quantisation map we find the abundance of inequivalent semiclassical corrections induced by quantum dynamics taking place in different internal clocks. It follows that the concepts like minimal volume, maximal curvature and the number of quantum bounces, often used to describe quantum effects in cosmological models, depend on the choice of internal clock., Comment: 14 pages, 6 figures

We present a quantum model of the vacuum Bianchi-IX dynamics. It is based on four main elements. First, we use a compound quantization procedure: an affine coherent state quantization for isotropic variables and a Weyl quantization for anisotropic ones. Second, inspired by standard approaches in molecular physics, we make an adiabatic approximation (Born-Oppenheimer-like approximation). Third, we expand the anisotropy potential about its minimum in order to deal with its harmonic approximation. Fourth, we develop an analytical treatment on the semiclassical level. The resolution of the classical singularity occurs due to a repulsive potential generated by the affine quantization. This procedure shows that during contraction the quantum energy of anisotropic degrees of freedom grows much slower than the classical one. Furthermore, far from the quantum bounce, the classical recollapse is reproduced. Our treatment is put in the general context of methods of molecular physics, which can include both adiabatic and nonadiabatic approximations., Comment: 26 pages, 6 figures

We present the affine coherent state quantization of the Bianchi I model. As in our previous paper on quantum theory of Friedmann models, we employ a variable associated with a perfect fluid to play a role of clock. Then we deparameterize the model. A distinctive feature, absent in isotropic models, is an extra non-holonomic constraint, which survives the deparameterization and constrains the range of physical variables. The appearance of the constraint reflects the `amplification' of singularity due to anisotropy. The quantization smoothes the extra constraint and allows quantum contracting trajectories to be smoothly transformed into expanding ones. Making use of affine coherent state we develop a semiclassical description. Figures are included to illustrate our result., Comment: 21 pages, 4 figures

We present a quantum version of the vacuum Bianchi IX model by implementing affine coherent state quantization combined with a Born-Oppenheimer-like adiabatic approximation. The analytical treatment is carried out on both quantum and semiclassical levels. The resolution of the classical singularity occurs by means of a repulsive potential generated by our quantization procedure. The quantization of the oscillatory degrees of freedom produces a radiation energy density term in the semiclassical constraint equation. The Friedmann-like lowest energy eigenstates of the system are found to be dynamically stable., Comment: 2 figures

It is often conjectured that a choice of time function merely sets up a frame for the quantum evolution of gravitational field, meaning that all choices should be in some sense compatible. In order to explore this conjecture (and the meaning of compatibility), we develop suitable tools for determining the relation between quantum theories based on different time functions. First, we discuss how a time function fixes a canonical structure on the constraint surface. The presentation includes both the kinematical and the reduced perspective, and the relation between them. Second, we formulate twin theorems about the existence of two inequivalent maps between any two deparameterizations, a {\it formal canonical} and a {\it coordinate} one. They are used to separate the effect of choice of clock from other effects. We show, in an example, how the spectra of quantum observables are transformed under the change of clock and prove, via a general argument, the existence of choice-of-time-induced semiclassical effects. Finally, we study an example, in which we find that the semiclassical discrepancies can in fact be arbitrarily large for dynamical observables. We conclude that the values of critical energy density or critical volume in the bouncing scenarios of quantum cosmology cannot in general be at the Planck scale and always need to be given with reference to a specific time function., Comment: 22 pages, 7 figures

We describe the quantum evolution of the vacuum Bianchi II universe in terms of the transition amplitude between two asymptotic quantum Kasner-like states. For large values of the momentum variable the classical and quantum calculations give similar results. The difference occurs for small values of this variable due to the Heisenberg uncertainty principle. Our results can be used, to some extent, as a building block of the quantum evolution of the vacuum Bianchi IX universe., Comment: 12 pages, 3 figures, version accepted by PRD

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 dilation. The main issue of our approach is the appearance of a quantum centrifugal potential allowing for regularization of the singularity, essential self-adjointness of the Hamiltonian, and unambiguous quantum dynamical evolution., Comment: 22+9 pages, 3 figures

We apply the reduced phase space quantization to the Kasner universe. We construct the kinematical phase space, find solutions to the Hamilton equations of motion, identify Dirac observables and arrive at physical solutions in terms of Dirac observables and an internal clock. We obtain the physical Hilbert space, which is the carrier space of the self-adjoint representation of the Dirac observables. Then we discuss the problem of time. We demonstrate that the inclusion of evolution in a gravitational system, at classical level as well as at quantum level, leads respectively to canonically and unitarily inequivalent theories. The example of Hubble operator in two different clock variables and with two distinct spectra is given., Comment: 24 pages

We analyze the quantum Bianchi I model in the setting of the nonstandard loop quantum cosmology. Elementary observables are used to quantize the volume operator. The spectrum of the volume operator is bounded from below and discrete. The discreteness may imply a foamy structure of spacetime at semiclassical level. The results are described in terms of a free parameter specifying loop geometry to be determined in astro-cosmo observations. An evolution of the quantum model is generated by the so-called true Hamiltonian, which enables an introduction of a time parameter valued in the set of all real numbers., Comment: 18 pages, version accepted for publication by Class. Quant. Grav

This paper is devoted to investigation of the quantum Friedman-Robertson-Walker universe with matter satisfying the equation of state $p=w\rho$, where $w$ is an almost arbitrary constant. The procedure starts with a reduced Lagrangian, which describes the system in a gauge fixed, so that the evolution parameter corresponds to the cosmological time. Then we construct the phase space, which is believed to correspond to the reduced phase space consisting of Dirac's observables. The physically relevant quantities are mapped into operators. We show that the operators have self-adjoint realizations and that there exist quantum states for which the evolution across singularity is well-defined., Comment: 15 pages, 2 figures, improved version, submitted for publication

Recent developments in observational cosmology call for understanding the nature of the cosmological singularity (CS). Our work proposes modelling the vicinity of CS by a time dependent orbifold (TDO). Our model makes sense if quantum elementary objects (particle, string, membrane) can go across the singularity of TDO, and our work addresses this issue. We find quantum states of elementary objects, that can propagate in TDO. Our results open door for more detailed examination., Comment: This is my PhD Thesis, about 70 pages; defended at the Institute for Nuclear Studies, Warsaw, Poland, November 2009

In this paper we argue that the compactified Milne space is a promising model of the cosmological singularity. It is shown that extended objects like strings propagate in a well-defined manner across the singularity of the embedding space. Then a proposal for quantization of extended objects in the case of a membrane is given., Comment: 4 pages, prepared for the proceedings of the Grassmannian Conference in Fundamental Cosmology (Grasscosmofun'09), 14-19 September 2009, Szczecin, Poland

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 in some way to specify the big bounce transition., Comment: 14 pages, no figures, version accepted for publication in Class. Quant. Grav

We examine spectrum of the physical volume operator within the non-standard loop quantum cosmology. The spectrum is discrete with equally distant levels defining a quantum of the volume. The discreteness may imply a foamy structure of spacetime at semi-classical level which may be detected in astro-cosmo observations., Comment: 5 pages, no figures, improved version

The big bounce (BB) transition within a flat Friedmann-Robertson-Walker model is analyzed in the setting of loop geometry underlying the loop cosmology. We solve the constraint of the theory at the classical level to identify physical phase space and find the Lie algebra of the Dirac observables. We express energy density of matter and geometrical functions in terms of the observables. It is the modification of classical theory by the loop geometry that is responsible for BB. The classical energy scale specific to BB depends on a parameter that should be fixed either by cosmological data or determined theoretically at quantum level, otherwise the energy scale stays unknown., Comment: 14 pages, 1 figure, version accepted for publication in Physical Review D

We present quantum theory of a membrane propagating in the vicinity of a time dependent orbifold singularity. The dynamics of a membrane, with the parameters space topology of a torus, winding uniformly around compact dimension of the embedding spacetime is mathematically equivalent to the dynamics of a closed string in a flat FRW spacetime. The construction of the physical Hilbert space of a membrane makes use of the kernel space of self-adjoint constraint operators. It is a subspace of the representation space of the constraints algebra. There exist non-trivial quantum states of a membrane evolving across the singularity., Comment: 16 pages, no figures, version accepted for publication in Journal of High Energy Physics

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., Comment: 5 pages, no figures, version published by Phys Rev D80, 063506 (2009)

We analyze classical theory of a membrane propagating in a singular background spacetime. The algebra of the first-class constraints of the system defines the membrane dynamics. A membrane winding uniformly around compact dimension of embedding spacetime is described by two constraints, which are interpreted in terms of world-sheet diffeomorhisms. The system is equivalent to a closed bosonic string propagating in a curved spacetime. Our results may be used for finding a quantum theory of a membrane in the compactified Milne space., Comment: 10 pages, no figures, revtex4; version submitted for publication

The appearance of the big bounce (BB) in the evolution of the universe is analyzed in the setting of loop quantum cosmology (LQC). Making use of an idea of a minimum length turns classical Big Bang into BB. We argue why the spectrum of the kinematical area operator of loop quantum gravity cannot be used for the determination of this length. We find that the fundamental length, at the present stage of development of LQC, is a free parameter of this model., Comment: 9 pages, no figures, revtex4; version published by Acta Physica Polonica

We present analytical results on the propagation of a classical string in non-zero modes through the singularity of the compactified Milne space. We restrict our analysis to a string winding around the compact dimension of spacetime. The compact dimension undergoes contraction to a point followed by re-expansion. We demonstrate that the classical dynamics of the string in excited states is non-singular in the entire spacetime., Comment: Version accepted for publication in Class. Quantum Grav

Our results concern the transition of a quantum string through the singularity of the compactified Milne (CM) space. We restrict our analysis to the string winding around the compact dimension (CD) of spacetime. The CD undergoes contraction to a point followed by re-expansion. We demonstrate that both classical and quantum dynamics of considered string are well defined. Most of presently available calculations strongly suggest that the singularity of a time dependent orbifold is useless as a model of the cosmological singularity. We believe that our results bring, to some extent, this claim into question., Comment: 9 pages, 2 figures, revtex4; version accepted for publication in Class. Quantum Grav

We examine the transition of a particle across the singularity of the compactified Milne (CM) space. Quantization of the phase space of a particle and testing the quantum stability of its dynamics are consistent to one another. One type of transition of a quantum particle is described by a quantum state that is continuous at the singularity. It indicates the existence of a deterministic link between the propagation of a particle before and after crossing the singularity. Regularization of the CM space leads to the dynamics similar to the dynamics in the de Sitter space. The CM space is a promising model to describe the cosmological singularity deserving further investigation by making use of strings and membranes., Comment: 19 pages, 7 figures, revtex4, added references, version accepted for publication in Class. Quantum Grav

We present classical and quantum dynamics of a test particle in the compactified Milne space. Background spacetime includes one compact space dimension undergoing contraction to a point followed by expansion. Quantization consists in finding a self-adjoint representation of the algebra of particle observables. Our model offers some insight into the nature of the cosmic singularity., Comment: 17 pages, no figures, RevTeX4, accepted for publication in Class. Quantum Grav