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The rising complexity of our terrestrial surrounding is an empirical fact. Details of this process evaded description in terms of physics for long time attracting attention and creating myriad of ideas including non-scientific ones. In this essay we explain the phenomenon of the growth of complexity by combining our up to date understanding of cosmology, non-equilibrium physics and thermodynamics. We argue that the observed increase of complexity is causal in nature, stands in agreement with the second law of thermodynamics and has it's origin in the cosmological expansion. Moreover, we highlight the connection between the leader of complexity growth in localized areas of space with free energy rate density, starting from the largest scales towards the smaller ones. Finally, in the light of recent advances in non-equilibrium statistical mechanics, our belief in the causal structure of modern scientific theories is transferred to biological systems. On relevant scales, adaptation and complexity growth follows a similar pattern, in which free energy rate density is provided by an external electromagnetic radiation. The presented, holistic approach, arms us with predicting power about variety of attributes of complex systems and leads to a chain of successful explanations on all scales of the Universe., Comment: 11 pages, 3 figures. An essay written for the FQXi essay contest: http://fqxi.org/community/forum/topic/2878

In the recent article Phys.\ Lett.\ B {\bf 759} (2016) 424 a new class of field theories called Nonlinear Field Space Theory has been proposed. In this approach, the standard field theories are considered as linear approximations to some more general theories characterized by nonlinear field phase spaces. The case of spherical geometry is especially interesting due to its relation with the spin physics. Here, we explore this possibility showing that classical scalar field theory with such a field space can be viewed as a perturbation of a continuous spin system. In this picture, the spin precession and the scalar field excitations are dual descriptions of the same physics. The duality is studied on the example of the Heisenberg model. It is shown that the Heisenberg model coupled to a magnetic field leads to a non-relativistic scalar field theory, characterized by quadratic dispersion relation. Finally, on the basis of analysis of the relation between the spin phase space and the scalar field theory we propose the \emph{Spin-Field correspondence} between the known types of fields and the corresponding spin systems., Comment: 12 pages, 1 figure. A mistake related to the large spin limit has been corrected. General conclusions remain unchanged

Studies of the effective regime of loop quantum gravity (LQG) revealed that, in the limit of Planckian curvature scales, spacetime may undergo a transition from the Lorentzian to Euclidean signature. This effect is a consequence of quantum modifications of the hypersurface deformation algebra, which in the linearized case is equivalent to a deformed version of the Poincar\'e algebra. In this paper the latter relation is applied to the LQG-inspired hypersurface deformation algebra that is characterized by the above mentioned signature change. While the exact form of the deformed Poincar\'e algebra is not uniquely determined, the algebra under consideration is representative enough to capture a number of qualitative features. In particular, the analysis reveals that the signature change can be associated with two symmetric invariant energy scales, which separate three physically disconnected momentum subspaces. Furthermore, the invariant measure on momentum space is derived, which allows to properly define the average return probability, characterizing a fictitious diffusion process on spacetime. The diffusion is subsequently studied in the momentum representation for all possible variants of the model. Finally, the spectral dimension of spacetime is calculated in each case as a function of the scale parameter. In the most interesting case the deformation is of the asymptotically ultralocal type and the spectral dimension undergoes a reduction to $d_S = 1$ in the UV limit., Comment: 13 pages, 4 figures, v2 some extra results, comments and references added

Shang and Hyndman (2017) proposed a grouped functional time series forecasting approach as a combination of individual forecasts obtained using generalized least squares method. We modify their methodology using generalized exponential smoothing technique for the most disaggregated functional time series in order to obtain more robust predictor. We discuss some properties of our proposals basing on results obtained via simulation studies and analysis of real data related to a prediction of a demand for electricity in Australia in 2016.

The slow-roll approximation is an analytical approach to study dynamical properties of the inflationary universe. In this article, systematic construction of the slow-roll expansion for effective loop quantum cosmology is presented. The analysis is performed up to the fourth order in both slow-roll parameters and the parameter controlling the strength of deviation from the classical case. The expansion is performed for three types of the slow-roll parameters: Hubble slow-roll parameters, Hubble flow parameters and potential slow-roll parameters. An accuracy of the approximation is verified by comparison with the numerical phase space trajectories for the case with a massive potential term. The results obtained in this article may be helpful in the search for the subtle quantum gravitational effects with use of the cosmological data., Comment: 16 pages, 1 figure

In recent years the idea that not only the configuration space of particles, i.e. spacetime, but also the corresponding momentum space may have nontrivial geometry has attracted significant attention, especially in the context of quantum gravity. The aim of this letter is to extend this concept to the domain of field theories, by introducing field spaces (i.e. phase spaces of field values) that are not affine spaces. After discussing the motivation and general aspects of our approach we present a detailed analysis of the prototype (quantum) Nonlinear Field Space Theory of a scalar field on the Minkowski background. We show that the nonlinear structure of a field space leads to numerous interesting predictions, including: non-locality, generalization of the uncertainty relations, algebra deformations, constraining of the maximal occupation number, shifting of the vacuum energy and renormalization of the charge and speed of propagation of field excitations. Furthermore, a compact field space is a natural way to implement the "Principle of Finiteness" of physical theories, which once motivated the Born-Infeld theory. Thus the presented framework has a variety of potential applications in the theories of fundamental interactions (e.g. quantum gravity), as well as in condensed matter physics (e.g. continuous spin chains), and can shed new light on the issue of divergences in quantum field theories., Comment: 7 pages, v2 additional content and references added, title changed, presentation improved

The four dimensional Causal Dynamical Triangulations (CDT) approach to quantum gravity is already more than ten years old theory with numerous unprecedented predictions such as non-trivial phase structure of gravitational field and dimensional running. Here, we discuss possible empirical consequences of CDT derived based on the two features of the approach mentioned above. A possibility of using both astrophysical and cosmological observations to test CDT is discussed. We show that scenarios which can be ruled out at the empirical level exist., Comment: Presented at the 14th Marcel Grossman Meeting, Rome, July 12-18, 2015

In effective models of loop quantum cosmology, the holonomy corrections are associated with deformations of space-time symmetries. The most evident manifestation of the deformations is the emergence of an Euclidean phase accompanying the non-singular bouncing dynamics of the scale factor. In this article, we compute the power spectrum of scalar perturbations generated in this model, with a massive scalar field as the matter content. Instantaneous and adiabatic vacuum-type initial conditions for scalar perturbations are imposed in the contracting phase. The evolution through the Euclidean region is calculated based on the extrapolation of the time direction pointed by the vectors normal to the Cauchy hypersurface in the Lorentzian domains. The obtained power spectrum is characterized by a suppression in the IR regime and oscillations in the intermediate energy range. Furthermore, the speculative extension of the analysis in the UV reveals a specific rise of the power., Comment: 13 pages, 4 figures

Signature change at high density has been obtained as a possible consequence of deformed space-time structures in models of loop quantum gravity. This article provides a conceptual discussion of implications for cosmological scenarios, based on an application of mathematical results for mixed-type partial differential equations (the Tricomi problem). While the effective equations from which signature change has been derived are shown to be locally regular and therefore reliable, the underlying theory of loop quantum gravity may face several global problems in its semiclassical solutions., Comment: 35 pages, 5 figures

This letter discusses phenomenological aspects of dimensional reduction predicted by the Causal Dynamical Triangulations (CDT) approach to quantum gravity. The deformed form of the dispersion relation for the fields defined on the CDT space-time is reconstructed. Using the Fermi satellite observations of the GRB 090510 source we find that the energy scale of the dimensional reduction is $E_* > 0.7 \sqrt{4-d_{\rm UV}} \cdot 10^{10}$ GeV at (95 $\%$ CL), where $d_{\rm UV}$ is the value of the spectral dimension in the UV limit. By applying the deformed dispersion relation to the cosmological perturbations it is shown that, for a scenario when the primordial perturbations are formed in the UV region, the scalar power spectrum $\mathcal{P}_S \propto k^{n_S-1}$ where $n_S-1\approx \frac{3 r (d_{\rm UV}-2)}{(d_{\rm UV}-1)r-48}$. Here, $r$ is the tensor-to-scalar ratio. We find that within the considered model, the predicted from CDT deviation from the scale-invariance ($n_S=1$) is in contradiction with the up to date Planck and BICEP2 data., Comment: 6 pages, 2 figures, matches version published in EPL

Our results concern quantum chaos of the vacuum Bianchi IX model. We apply the equilateral triangle potential well approximation to the potential of the Bianchi IX model to solve the eigenvalue problem for the physical Hamiltonian. Such approximation is well satisfied in vicinity of the cosmic singularity. Level spacing distribution of the eigenvalues is studied with and without applying the unfolding procedure. In both cases, the obtained distributions are qualitatively described by Brody's distribution with the parameter $\beta\approx0.3$, revealing some sort of the level repulsion. The observed repulsion may reflect chaotic nature of the classical dynamics of the Bianchi IX universe. However, full understanding of this effects will require examination of the Bianchi IX model with the exact potential., Comment: Presented at the conference Random Matrix Theory: Foundations and Applications, 1-6 July 2014, Cracow

Recent calculations in loop quantum cosmology suggest that a transition from a Lorentzian to an Euclidean space-time might take place in the very early Universe. The transition point leads to a state of silence, characterized by a vanishing speed of light. This behavior can be interpreted as a decoupling of different space points, similar to the one characterizing the BKL phase. In this study, we address the issue of imposing initial conditions for the cosmological perturbations at the transition point between the Lorentzian and Euclidean phases. Motivated by the decoupling of space points, initial conditions characterized by a lack of correlations are investigated. We show that the "white noise" gains some support from analysis of the vacuum state in the deep Euclidean regime. Furthermore, the possibility of imposing the silent initial conditions at the trans-Planckian surface, characterized by a vanishing speed for the propagation of modes with wavelengths of the order of the Planck length, is studied. Such initial conditions might result from a loop-deformations of the Poincar\'e algebra. The conversion of the silent initial power spectrum to a scale-invariant one is also examined., Comment: 24 pages, 9 figures. Revised version accepted for publication in IJMPD

This essay addresses the issue of gravitational phase transitions in the early universe. We suggest that a second order phase transition observed in the Causal Dynamical Triangulations approach to quantum gravity may have a cosmological relevance. The phase transition interpolates between a non-geometric crumpled phase of gravity, and an extended phase with classical properties. Transitions of this kind have been postulated earlier in the context of geometrogenesis in Quantum Graphity. We show that critical behavior may also be associated with a signature change event in Loop Quantum Cosmology. In both cases, classical spacetime originates at the critical point associated with a second order phase transition., Comment: Essay written for the Gravity Research Foundation 2014 Awards for Essays on Gravitation

In this paper we study slow-roll inflation with holonomy corrections from loop quantum cosmology. Both tensor and scalar power spectra of primordial perturbations are computed up to the first order in slow-roll parameters and $V/\rho_{c}$, where $V$ is a potential of the scalar field and $\rho_{c}$ is a critical energy density (expected to be of the order of the Planck energy density). Possible normalizations of modes at short scales are discussed. In case the normalization is performed with use of the Wronskian condition applied to adiabatic vacuum, the tensor and scalar spectral indices are not quantum corrected in the leading order. However, by choosing an alternative method of normalization one can obtain quantum corrections in the leading order. Furthermore, we show that the holonomy-corrected equation of motion for tensor modes can be derived from an effective background metric. This allows us to prove that the Wronskian normalization condition for the tensor modes preserves the classical form., Comment: 21 pages

Quantum gravity is sometimes considered as a kind of metaphysical speculation. In this review, we show that, although still extremely difficult to reach, observational signatures can in fact be expected. The early universe is an invaluable laboratory to probe "Planck scale physics". Focusing on Loop Quantum Gravity as one of the best candidate for a non-perturbative and background-independant quantization of gravity, we detail some expected features., Comment: 75 pages, invited topical review for Classical and Quantum Gravity

In this essay we present evidence suggesting that loop quantum gravity leads to deformation of the local Poincar\'e algebra within the limit of high energies. This deformation is a consequence of quantum modification of effective off-shell hypersurface deformation algebra. Surprisingly, the form of deformation suggests that the signature of space-time changes from Lorentzian to Euclidean at large curvatures. We construct particular realization of the loop-deformed Poincar\'e algebra and find that it can be related to curved momentum space, which indicates the relationship with recently introduced notion of relative locality. The presented findings open a new way of testing loop quantum gravity effects., Comment: Essay written for the Gravity Research Foundation 2013 Awards for Essays on Gravitation, 7 pages, 1 figure

In this paper we study quantum dynamics of the bouncing cosmological model. We focus on the model of the flat Friedman-Robertson-Walker universe with a free scalar field. The bouncing behavior, which replaces classical singularity, appears due to the modification of general relativity along the methods of loop quantum cosmology. We show that there exist a unitary transformation that enables to describe the system as a free particle with Hamiltonian equal to canonical momentum. We examine properties of the various quantum states of the Universe: boxcar state, standard coherent state, and soliton-like state, as well as Schr{\"o}dinger's cat states constructed from these states. Characteristics of the states such as quantum moments and Wigner functions are investigated. We show that each of these states have, for some range of parameters, a proper semiclassical limit fulfilling the correspondence principle. Decoherence of the superposition of two universes is described and possible interpretations in terms of triad orientation and Belinsky-Khalatnikov-Lifshitz conjecture are given. Some interesting features regarding the area of the negative part of the Wigner function have emerged., Comment: 18 pages, 19 figures

The state of asymptotic silence, characterized by causal disconnection of the space points, emerges from various approaches aiming to describe gravitational phenomena in the limit of large curvatures. In particular, such behavior was anticipated by Belinsky, Khalatnikov and Lifshitz (BKL) in their famous conjecture put forward in the early seventies of the last century. While the BKL conjecture is based on purely classical considerations, one can expect that asymptotic silence should have its quantum counterpart at the level of a more fundamental theory of quantum gravity, which is the relevant description of gravitational phenomena in the limit of large energy densities. Here, we summarize some recent results which give support to such a possibility. More precisely, we discuss occurrence of the asymptotic silence due to polymerization of space at the Planck scale, in the framework of loop quantum cosmology. In the discussed model, the state of asymptotic silence is realized at the energy density $\rho = \rho_c/2$, where $\rho_c$ is the maximal allowed energy density, being of the order of the Planck energy density. At energy densities $\rho > \rho_c/2$, the universe becomes 4D Euclidean space without causal structure. Therefore, the asymptotic silence appears to be an intermediate state of space between the Lorentzian and Euclidean phases., Comment: 4 pages, 3 figures, talk presented at the Multiverse and Fundamental Cosmology Conference, 10-14 September, 2012, Szczecin, Poland

The Wick rotation is commonly considered only as an useful computational trick. However, as it was suggested by Hartle and Hawking already in early eighties, Wick rotation may gain physical meaning at the Planck epoch. While such possibility is conceptually interesting, leading to no-boundary proposal, mechanism behind the signature change remains mysterious. We show that the signature change anticipated by Hartle and Hawking naturally appear in loop quantum cosmology. Theory of cosmological perturbations with the effects of quantum holonomies is discussed. It was shown by Cailleteau \textit{et al.} (Class. Quant. Grav. {\bf 29} (2012) 095010) that this theory can be uniquely formulated in the anomaly-free manner. The obtained algebra of effective constraints turns out to be modified such that the metric signature is changing from Lorentzian in low curvature regime to Euclidean in high curvature regime. Implications of this phenomenon on propagation of cosmological perturbations are discussed and corrections to inflationary power spectra of scalar and tensor perturbations are derived. Possible relations with other approaches to quantum gravity are outlined. We also propose an intuitive explanation of the observed signature change using analogy with spontaneous symmetry breaking in "wired" metamaterials., Comment: Proceeding of the conference "Relativity and Gravitation - 100 Years after Einstein in Prague", Prague, June 2012

Holonomy corrections to scalar perturbations are investigated in the loop quantum cosmology framework. Due to the effective approach, modifications of the algebra of constraints generically lead to anomalies. In order to remove those anomalies, counter-terms are introduced. We find a way to explicitly fulfill the conditions for anomaly freedom and we give explicit expressions for the counter-terms. Surprisingly, the "new quantization scheme" naturally arises in this procedure. The gauge invariant variables are found and equations of motion for the anomaly-free scalar perturbations are derived. Finally, some cosmological consequences are discussed qualitatively., Comment: 19 pages, 1 figure, v2, new comments and references added, minor corrections

We present results concerning propagation of the Gaussian state across the cosmological quantum bounce. The reduced phase space quantization of loop quantum cosmology is applied to the Friedman-Robertson-Walker universe with a free massless scalar field. Evolution of quantum moments of the canonical variables is investigated. The covariance turns out to be a monotonic function so it may be used as an evolution parameter having quantum origin. We show that for the Gaussian state the Universe is least quantum at the bounce. We propose explanation of this counter-intuitive feature using the entropy of squeezing. The obtained time dependence of entropy is in agreement with qualitative predictions based on von Neumann entropy for mixed states. We show that, for the considered Gaussian state, semiclassicality is preserved across the bounce, so there is no cosmic forgetfulness., Comment: 7 pages, 7 figures; matches version published in Phys. Rev. D

We present the method of describing an evolution in quantum cosmology in the framework of the reduced phase space quantization of loop cosmology. We apply our method to the flat Friedman-Robertson-Walker model coupled to a massless scalar field. We identify the physical quantum Hamiltonian that is positive-definite and generates globally an unitary evolution of considered quantum system. We examine properties of expectation values of physical observables in the process of the quantum big bounce transition. The dispersion of evolved observables are studied for the Gaussian state. Calculated relative fluctuations enable an examination of the semi-classicality conditions and possible occurrence of the cosmic forgetfulness. Preliminary estimations based on the cosmological data suggest that there was no cosmic amnesia. Presented results are analytical, and numerical computations are only used for the visualization purposes. Our method may be generalized to sophisticated cosmological models including the Bianchi type universes., Comment: 28 pages, 7 figures. Matches version published in Class. Quantum Grav

We investigate vector perturbations with holonomy corrections in the framework of loop quantum cosmology. Conditions to achieve anomaly freedom for these perturbations are found at all orders. This requires the introduction of counter-terms in the hamiltonian constraint. We also show that anomaly freedom requires the diffeomorphism constraint to hold its classical form when scalar matter is added although the issue of a vector matter source, required for full consistency, remains to be investigated. The gauge-invariant variable and the corresponding equation of motion are derived. The propagation of vector modes through the bounce is finally discussed., Comment: 16 pages, 1 figure. Matches version published in Class. Quantum Grav

We apply the nonstandard loop quantum cosmology method to quantize a flat Friedmann-Robertson-Walker cosmological model with a free scalar field and the cosmological constant $\Lambda>0$. Modification of the Hamiltonian in terms of loop geometry parametrized by a length $\lambda$ introduces a scale dependence of the model. The spectrum of the volume operator is discrete and depends on $\Lambda$. Relating quantum of the volume with an elementary lattice cell leads to an explicit dependence of $\Lambda$ on $\lambda$. Based on this assumption, we investigate the possibility of interpreting $\Lambda$ as a running constant., Comment: 6 pages, 3 figures, version accepted for publication in Phys. Rev. D

Cosmological models where the standard big bang is replaced by a bounce have been studied for decades. The situation has, however, dramatically changed in the past years for two reasons: first, because new ways to probe the early Universe have emerged, in particular, thanks to the cosmic microwave background, and second, because some well grounded theories -especially loop quantum cosmology- unambiguously predict a bounce, at least for homogeneous models. In this article, we investigate into the details the phenomenological parameters that could be constrained or measured by next-generation B-mode cosmic micorwave background experiments. We point out that an important observational window could be opened. We then show that those constraints can be converted into very meaningful limits on the fundamental loop quantum cosmology parameters. This establishes the early Universe as an invaluable quantum gravity laboratory., Comment: 12 pages, 5 figures; typos corrected Published in Physical Review D

In the recent paper by Mielczarek \emph{et al.} (JCAP {\bf 1007} (2010) 004) an idea of the method which can be used to put some constraint for the reheating phase was proposed. Another method of constraining the reheating temperature has been recently studied by Martin and Ringeval (Phys.\ Rev.\ D {\bf 82} (2010) 023511). Both methods are based on observations of the cosmic microwave background (CMB) radiation. In this paper, we develop the idea introduced in this first article to put constraint on the reheating after the slow-roll inflation. We restrict our considerations to the case of a massive inflaton field. The method can be, however, easily extended to the different inflationary scenarios. As a main result, we derive an expression on the reheating temperature $T_{\text{RH}}$. Surprisingly, the obtained equation is independent on the unknown number of relativistic degrees of freedom $g_*$ produced during the reheating. Based on this equation and the WMAP 7 observations, we find $T_{\text{RH}}=3.5\cdot 10^6$ GeV, which is consistent with the current constraints. The relative uncertainty of the result is, however, very high and equal to $\sigma(T_{\text{RH}})/T_{\text{RH}} \approx 53$. As we show, this uncertainty will be significantly reduced with future CMB experiments., Comment: 6 pages, 3 figures. Matches version published in Phys. Rev. D

In this paper we study possible observational consequences of the bouncing cosmology. We consider a model where a phase of inflation is preceded by a cosmic bounce. While we consider in this paper only that the bounce is due to loop quantum gravity, most of the results presented here can be applied for different bouncing cosmologies. We concentrate on the scenario where the scalar field, as the result of contraction of the universe, is driven from the bottom of the potential well. The field is amplified, and finally the phase of the standard slow-roll inflation is realized. Such an evolution modifies the standard inflationary spectrum of perturbations by the additional oscillations and damping on the large scales. We extract the parameters of the model from the observations of the cosmic microwave background radiation. In particular, the value of inflaton mass is equal to $m=(2.6 \pm 0.6) \cdot 10^{13}$ GeV. In our considerations we base on the seven years of observations made by the WMAP satellite. We propose the new observational consistency check for the phase of slow-roll inflation. We investigate the conditions which have to be fulfilled to make the observations of the Big Bounce effects possible. We translate them to the requirements on the parameters of the model and then put the observational constraints on the model. Based on assumption usually made in loop quantum cosmology, the Barbero-Immirzi parameter was shown to be constrained by $\gamma<1100$ from the cosmological observations. We have compared the Big Bounce model with the standard Big Bang scenario and showed that the present observational data is not informative enough to distinguish these models., Comment: 25 pages, 8 figures, JHEP3.cls

Loop quantum cosmology provides an efficient framework to study the evolution of the Universe beyond the classical Big Bang paradigm. Because of holonomy corrections, the singularity is replaced by a "bounce". The dynamics of the background is investigated into the details, as a function of the parameters of the model. In particular, the conditions required for inflation to occur are carefully considered and are shown to be generically met. The propagation of gravitational waves is then investigated in this framework. By both numerical and analytical approaches, the primordial tensor power spectrum is computed for a wide range of parameters. Several interesting features could be observationally probed., Comment: 11 pages, 14 figures. Matches version published in Phys. Rev. D

We consider a flat cosmological model with a free massless scalar field and the cosmological constant $\Lambda$ in the framework of loop quantum cosmology. The scalar field plays the role of an intrinsic time. We apply the reduced phase space approach. The dynamics of the model is solved analytically. We identify elementary observables and their algebra. The compound physical observables like the volume and the energy density of matter field are analysed. Both compound observables are bounded and oscillate in the $\Lambda<0$ case. The energy density is bounded and oscillates in the $\Lambda>0$ case. However, the volume is unbounded from above, but periodic. The difference between standard and nonstandard loop quantum cosmology is described., Comment: 10 pages, 4 figures, version accepted for publication in PRD

The dynamical systems methods are used to study evolution of the polymerised scalar field cosmologies with the cosmological constant. We have found all evolutional paths admissible for all initial conditions on the two-dimensional phase space. We have shown that the cyclic solutions are generic. The exact solution for polymerised cosmology is also obtained. Two basic cases are investigated, the polymerised scalar field and the polymerised gravitational and scalar field part. In the former the division on the cyclic and non-cyclic behaviour is established following the sign of the cosmological constant. The value of the cosmological constant is upper bounded purely from the dynamical setting., Comment: 10 pages, 4 figs, JHEP3.cls

In this paper we investigate power spectrum of a smoothed scalar field. The smoothing leads to the regularisation of the UV divergences and can be related with the internal structure of the considered field or the space itself. We apply procedure of smoothing to the quantum fluctuations generated during the phase of cosmic inflation. We study whether this effect can be probed observationally and conclude that the modifications of the power spectrum due to the smoothing on the Planck scale are negligible and far beyond the observational abilities. Subsequently we investigate whether smoothing in any other form can be probed observationally. We introduce phenomenological smoothing factor $e^{-k^2\sigma^2}$ to the inflationary spectrum and investigate its effects on the spectrum of CMB anisotropies and polarisation. We show that smoothing can lead to suppression of high multipoles in the spectrum of the CMB. Based on five years observations of WMAP satellite we indicate that the present scale of high multipoles suppression is constrained by $\sigma < 2.86$ Mpc (95% CL). This corresponds to the constraint $\sigma < 100 \mu$m at the end of inflation. Despite this value is far above the Planck scale, the other processes of smoothing can be possibly studied with this constraint, e.g. diffusion or decoherence of primordial perturbations., Comment: 7 pages, 3 figures

In this paper we consider realistic model of inflation embedded in the framework of loop quantum cosmology. Phase of inflation is preceded here by the phase of a quantum bounce. We show how parameters of inflation depend on the initial conditions established in the contracting, pre-bounce phase. Our investigations indicate that phase of the bounce easily sets proper initial conditions for the inflation. Subsequently we study observational effects that might arise due to the quantum gravitational modifications. We perform preliminary observational constraints for the Barbero-Immirzi parameter $\gamma$, critical density $\rho_{\text{c}}$ and parameter $\lambda$. In the next step we study effects on power spectrum of perturbations. We calculate spectrum of perturbations from the bounce and from the joined bounce+inflation phase. Based on these studies we indicate possible way to relate quantum cosmological models with the astronomical observations. Using the Sachs-Wolfe approximation we calculate spectrum of the super-horizontal CMB anisotropies. We show that quantum cosmological effects can, in the natural way, explain suppression of the low CMB multipoles. We show that fine-tuning is not required here and model is consistent with observations. We also analyse other possible probes of the quantum cosmologies and discuss perspectives of their implementation., Comment: 11 pages, 7 figures

In this paper we study dynamics of the closed FRW model with holonomy corrections coming from loop quantum cosmology. We consider models with a scalar field and cosmological constant. In case of the models with cosmological constant and free scalar field, dynamics reduce to 2D system and analysis of solutions simplify. If only free scalar field is included then universe undergoes non-singular oscillations. For the model with cosmological constant, different behaviours are obtained depending on the value of $\Lambda$. If the value of $\Lambda$ is sufficiently small, bouncing solutions with asymptotic de Sitter stages are obtained. However if the value of $\Lambda$ exceeds critical value $\Lambda_{\text{c}} =\frac{\sqrt{3}m^2_{\text{Pl}}}{2\pi\gamma^3} \simeq 21 m^2_{\text{Pl}}$ then solutions become oscillatory. Subsequently we study models with a massive scalar field. We find that this model possess generic inflationary attractors. In particular field, initially situated in the bottom of the potential, is driven up during the phase of quantum bounce. This subsequently leads to the phase of inflation. Finally we find that, comparing with the flat case, effects of curvature do not change qualitatively dynamics close to the phase of bounce. Possible effects of inverse volume corrections are also briefly discussed., Comment: 18 pages, 11 figures

In this paper we consider tensor perturbations produced at a bounce phase in presence of the holonomy corrections. Here bounce phase and holonomy corrections originate from Loop Quantum Cosmology. We re-derive formulas for the corrections for the model with a scalar field content. Background dynamics with a free scalar field and multi-fluid potential are considered. Since the considerations are semi-classical effects of quantum fluctuations of the background dynamics are not taken into account. Quantum and classical back-reaction effects are also neglected. To find spectrum of the gravitational waves both analytical approximations as well as numerical investigations were performed. We have found analytical solutions on super-horizontal and sub-horizontal regimes and derived corresponding tensor power spectra. Also occupation number $n_{\bf k}$ and parameter $\Omega_{\text{gw}}$ were derived in sub-horizontal limit, leading to its extremely low present value. Final results are numerical power spectra of the gravitational waves produced in presence of quantum holonomy corrections. The obtained spectrum has two UV and IR branches where $\mathcal{P}_T \propto k^2$, however with the different prefactors. Spectrum connecting these regions is in the form of oscillations. We have found good agreement between numerical spectrum and this obtained from the analytical model. Obtained spectrum can be directly applied as initial conditions for the inflationary modes. We mention possible resulting observational features of the CMB in particular B-type polarization., Comment: 13 pages, 12 figures. Numerical computations improved. Major revision

We discuss epistemological and methodological aspects of the Bayesian approach in astrophysics and cosmology. The introduction to the Bayesian framework is given for a further discussion concerning the Bayesian inference in physics. The interplay between the modern cosmology, Bayesian statistics, and philosophy of science is presented. We consider paradoxes of confirmation, like Goodman's paradox, appearing in the Bayesian theory of confirmation. As in Goodman's paradox the Bayesian inference is susceptible to some epistemic limitations in the logic of induction. However Goodman's paradox applied to cosmological hypotheses seems to be resolved due to the evolutionary character of cosmology and accumulation new empirical evidences. We argue that the Bayesian framework is useful in the context of falsificability of quantum cosmological models, as well as contemporary dark energy and dark matter problem., Comment: RevTeX4, 14 pages, 6 figures; v2 new title, improvements and corrections, more on Bayesian inference with example in cosmology

The scalar field can behave like a fluid with equation of state $p_{\phi}=w\rho_{\phi}$, where $w \in [-1,1]$. In this Letter we derive a class of the scalar field potentials for which $w=$ const. Scalar field with such a potential can mimic ordinary matter, radiation, cosmic strings, etc. We perform our calculations in the framework of the loop cosmology with holonomy corrections. We solve the model analytically for the whole parameter space. Subsequently, we perform similar consideration for the model with a phantom field ($w<-1$). We show that scalar field is monotonic function in both cases. This indicates that it can be treated as a well-defined internal time for these models. Moreover we perform preliminary studies of the scalar field perturbations with this potential. We indicate that non-Gaussian features are present admitting for the possible observational constraints of the model., Comment: 6 pages, 6 figures, new section on perturbations added. Version to appear in PLB

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., Comment: 19 pages, 9 figures

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., Comment: RevTeX4, 28 pages, 11 figs; rev.2 new section on exact solutions; (v3) published version

In this paper we study the description of the Universe based on the low energy superstring theory modified by the Loop Quantum Gravity effects.This approach was proposed by De Risi et al. in the Phys. Rev. D {\bf 76} (2007) 103531. We show that in the contrast with the string motivated pre-Big Bang scenario, the cosmological realisation of the $t$-duality transformation is not necessary to avoid an initial singularity. In the model considered the universe starts its evolution in the vacuum phase at time $t\to - \infty$. In this phase the scale factor $a\to 0$, energy density $\rho \to 0$ and coupling of the interactions $g^2_s \to 0$. After this stage the universe evolves to the non-singular hot Big Bang phase $\rho \to \rho_{\text{max}} < \infty$. Then the standard classical universe emerges. During the whole evolution the scale factor increases monotonically. We solve this model analytically. We also propose and solve numerically the model with an additional dilaton potential in which the universe starts the evolution from the asymptotically free vacuum phase $g^2_s \to 0$ and then evolves non-singularly to the emerging dark energy dominated phase with the saturated coupling constant $g^2_s \to \text{const}$., Comment: JHEP3 LaTeX class, 19 pages, 9 figures, v2: added some comments and references, v3: new numerical result added, new figures

In this paper we calculate $\mathcal{O}(\mu^4)$ corrections from holonomies in the Loop Quantum Gravity, usually not taken into account. Allowance of the corrections of this kind is equivalent with the choice of the new quatization scheme. Quantization ambiguities in the Loop Quantum Cosmology allow for this additional freedom and presented corrections are consistent with the standard approach. We apply these corrections to the flat FRW cosmological model and calculate the modified Friedmann equation. We show that the bounce appears in the models with the standard $\mathcal{O}(\mu^2)$ quantization scheme is shifted to the higher energies $\rho_{\text{bounce}} = 3 \rho_{\text{c}}$. Also a pole in the Hubble parameter appears for $\rho_{\text{pole}} = {3/2} \rho_{\text{c}}$ corresponding to \emph{hyper-inflation/deflation} phases. This pole represents a curvature singularity at which the scale factor is finite. In this scenario the singularity and bounce co-exist. Moreover we find that an ordinary bouncing solution appears only when quantum corrections in the lowest order are considered. Higher order corrections can lead to the nonperturbative effects., Comment: RevTeX4, 8 pages, 4 figures; v2 change of title, more discussion on co-existence of singularity and bounce

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 schemes., Comment: 12 pages, 12 figures, added references and changed introduction and summary