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 $Ω_{\text{gw}}$ 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.
Bouncing cosmologies are obtained by adding to the Einstein-Hilbert action a term of the form $\sqrt{-g}f(χ)$, with $χ$ a scalar depending on the Hubble parameter only, not on its derivatives, and which is here shown to arise from the divergence of the unitary time-like eigenvector of the stress tensor. At background level, the dynamical equations for a given $f$-theory are calculated, showing that the simplest bouncing cosmology resulting leads to exactly the same equations as those for holonomy corrected Loop Quantum Cosmology (LQC). When dealing with perturbations, the equation for tensor o…
Guided by recent developments towards the implementation of the deformation quantization program within the Loop Quantum Cosmology (LQC) formalism, in this paper we address the introduction of both the integral and differential representation of the star product for LQC. To this end, we consider the Weyl quantization map for cylindrical functions defined on the Bohr compactification of the reals. The integral representation contains all of the common properties that characterize a star product which, in the case under study here, stands for a deformation of the usual pointwise product of cylin…
A Cuscuton Representation of the Loop Quantum Cosmology Bounce
University of Waterloo（滑铁卢大学）；Perimeter Institute for Theoretical Physics（珀蒂默理论物理研究所）；Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）
Considering as usual that the underlying geometry of our universe is well described by the spatially flat Friedmann-Lemaitre-Robertson-Walker line element, we review how the background of holonomy corrected Loop Quantum Cosmology (LQC) could be obtained as a simple modified version of the mimetic gravity. We also analyze the scalar and tensor perturbations of this modified mimetic model, from which we find that at the level of tensor perturbations it is indistinguishable from General Relativity while at the level of scalar perturbations, the modified mimetic model, which has the same backgroun…
We show that the standard representation of homogeneous isotropic loop quantum cosmology (LQC) is the GNS-representation that corresponds to the unique state on the reduced quantum holonomy-flux $^*$-algebra that is invariant under residual diffeomorphisms $-$ both when the standard algebra is used as well as when one uses the extended algebra proposed by Fleischhack. More precisely, we find that in both situations the GNS-Hilbert spaces coincide, and that in the Fleischhack case the additional algebra elements are just mapped to zero operators. In order for the residual diffeomorphisms to hav…
We do not observe quantum effects on cosmological scales. Thus, if loop quantum cosmology (LQC) is to provide an accurate depiction of the real world, it must allow for quantum states of spacetime geometry which are semi-classical in two respects: they must be sharply peaked around a single, classical geometry, and they must have small quantum fluctuations. It is generally assumed that Gaussian states exhibit both of these properties. After all, they do in ordinary quantum mechanics. In this paper, we derive exact closed-form expressions for the fluctuations of Gaussian states in LQC and their…
In this work we use the Loop Quantum Cosmology modified scalar-tensor reconstruction techniques in order to investigate how bouncing and inflationary cosmologies can be realized. With regard to the inflationary cosmologies, we shall be interested in realizing the intermediate inflation and the Type IV singular inflation, while with regard to bouncing cosmologies, we shall realize the superbounce and the symmetric bounce. In all the cases, we shall find the kinetic term of the LQC holonomy corrected scalar-tensor theory and the corresponding scalar potential. In addition, we shall include a stu…
Warm inflation is analyzed in the context of Loop Quantum Cosmology (LQC). The bounce in LQC provides a mean through which a Liouville measure can be defined, which has been used previously to characterize the a priori probability for inflation in LQC. Here we take advantage of the tools provided by LQC to study instead the a priori probability for warm inflation dynamics in the context of a monomial quartic inflaton potential. We study not only the question of how a general warm inflation dynamics can be realized in LQC with an appropriate number of e-folds, but also how such dynamics is cons…
Matter and quasi-matter bounce scenarios are studied for an F(R) gravity model with holonomy corrections and a Lagrange multiplier, with a scale factor $a(t) = \left(a_0t^2 + 1 \right)^n$, where the Hubble parameter squared has a linear and a quadratic dependence on the effective energy density. Provided $n < 1/2$, it is shown that the primordial curvature perturbations are generated deeply into the contracting era, at large negative time, which makes the low-curvature limit a good approximation for calculating the perturbation power spectrum. Moreover, it is shown that, for $n$ within this ra…
In the spatially flat case of loop quantum cosmology, the connection $\bar{k}$ is usually replaced by the holonomy $\frac{\sin(\barμk)}{\barμ}$ in the effective theory. In this paper, instead of the $\barμ$ scheme, we use a generalised, undertermined function $g(\bar{k},\bar{p})$ to represent the holonomy and by using the approach of anomaly free constraint algebra we fix all the counter terms in the constraints and find the restriction on the form of $g(\bar{k},\bar{p})$, then we derive the gauge invariant equations of motion of the scalar, tensor and vector perturbations and study the inflat…
The effective dynamics of scalar-tensor theory (STT) in the Jordan frame is studied in the context of loop quantum cosmology with holonomy corrections. After deriving the effective Hamiltonian from the connection dynamics formulation, we obtain the holonomy-corrected evolution equations of STT on spatially flat Friedmann-Robterson-Walker background, which exhibit some interesting features unique to the Jordan frame of STT. In particular, the linear term of the cosine function appearing in the equations could lead to dynamics much different from the classical theory in the low-energy limit. In …
We introduce a new family of coherent states for loop quantum gravity, inspired by the twisted geometry parametrization. We compute their peakedness properties and compare them with the heat-kernel coherent states. They show similar features for the area and the holonomy operators, but improved peakedness in the direction of the flux. At the gauge-invariant level, the new family is built from tensor products of coherent intertwiners. To study the peakedness of the holonomy operator, we introduce a new shift operator based on the harmonic oscillator representation associated with the twisted ge…
We investigate the consequences of different regularizations and ambiguities in loop cosmological models on the predictions in the scalar and tensor primordial spectrum of the cosmic microwave background using the dressed metric approach. Three models, standard loop quantum cosmology (LQC), and two modified loop quantum cosmologies (mLQC-I and mLQC-II) arising from different regularizations of the Lorentzian term in the classical Hamiltonian constraint are explored for chaotic inflation in spatially-flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe. In each model, two different treatmen…
Non-minimally coupled loop quantum inflation with inverse-volume corrections
Universidad de Tarapacá（塔拉帕卡大学）；Pontificia Universidad Católica de Valparaíso（瓦帕莱索天主教 Pontificia 大学）；Università degli Studi di Napoli “Federico II”（那不勒斯费德里科二世大学）；Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）
Quantum corrections to cosmic perturbations for a bouncing background
Universidad Autonoma de Chihuahua（奇瓦瓦自治大学）；Universidad Autonoma Metropolitana Cuajimalpa（墨西哥城自治大学夸希马尔帕校区）；Universidad Autonoma Metropolitana Unidad Iztapalapa（墨西哥城自治大学伊斯塔帕拉帕校区）
In previous work, we showed that in loop quantum cosmology of scalar-tensor theory (STT) with the holonomy correction the background equations of motion in the Jordan frame have two branches, i.e., the $b_{+}$ branch and the $b_{-}$ branch. In the low-energy limit, the $b_{+}$ branch of the equations of motion reproduce the equations of motion of classical STT while the $b_{-}$ branch of equations of motion do not reproduce the classical equations. In this paper, we investigate cosmological dynamics in an expanding universe whose background is described by the the $b_{-}$ branch of equations o…
