With a well-motivated extension of higher order holonomy corrections, the quantum theory of loop quantum cosmology (LQC) for the $k=0$ Friedmann-Robertson-Walker model (with a free massless scalar) is rigorously formulated. The analytical investigation reveals that, regardless of the order of holonomy corrections and for any arbitrary states, the matter density remains finite, bounded from above by an upper bound, which equals the critical density obtained at the level of heuristic effective dynamics. Particularly, with all orders of corrections included, the dynamical evolution is shown to follow the bouncing scenario in which two Wheeler-DeWitt (WDW) solutions (expanding and contracting) are bridged together through the quantum bounce. These observations provide further evidence that the quantum bounce is essentially a consequence of the intrinsic discreteness of LQC and LQC is fundamentally different from the WDW theory. Meanwhile, the possibility is also explored that the higher order holonomy corrections can be interpreted as a result of admitting generic SU(2) representations for the Hamiltonian constraint operators.
This article addresses the issue of the closure of the algebra of constraints for generic (cosmological) perturbations when taking into account simultaneously the two main corrections of effective loop quantum cosmology, namely the holonomy and the inverse-volume terms. Previous works on either the holonomy or the inverse volume case are reviewed and generalized. In the inverse-volume case, we point out new possibilities. An anomaly-free solution including both corrections is found for perturbations, and the corresponding equations of motion are derived.
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…
Applying the higher order holonomy corrections to the perturbation theory of cosmology, the lattice power law of Loop Quantum Cosmology, $\tildeμ\propto p^β$, is analysed and the range of $β$ is decided to be [-1,0] which is different from the conventional range $-0.1319>β\geq-5/2$ \cite{lqct}. At the same time, we find that there is a anomaly free condition in this theory, and we obtain this condition in the vector and tensor mode. We also find that the nonzero mass of gravitational wave essentially results from the quantum nature of Riemannian geometry of loop quantum gravity.
At the level of heuristic effective dynamics, we investigate the cosmological inflation with holonomy corrections of loop quantum cosmology (LQC) in the $k=0$ Friedmann-Robertson-Walker model with a single inflaton field subject to a simple potential. In the symmetric bouncing scenario of LQC, the condition for occurrence of the quantum bounce naturally and uniquely fixes the initial conditions at the bouncing epoch. Around the quantum bounce, the universe undergoes a short super-inflationary phase, which drives the inflaton field to its potential hill and thus sets the proper initial conditio…
Loop Quantum Cosmology yields two kinds of quantum corrections to the effective equations of motion for cosmological perturbations. Here we focus on the holonomy kind and we study the problem of the closure of the resulting algebra of constraints. Up to now, tensor, vector and scalar perturbations were studied independently, leading to different algebras of constraints. The structures of the related algebras were imposed by the requirement of anomaly freedom. In this article we show that the algebra can be modified by a very simple quantum correction, holding for all types of perturbations. Th…
The detailed formulation of loop quantum cosmology with higher order holonomy corrections has been constructed recently in the homogeneous and isotropic spacetime, yet it is important to extend the higher order holonomy corrections to include the effects of anisotropy which typically grow during the collapsing phase. In this paper we investigate the Bianchi I model in $\barμ'$ scheme which truly captures the regularization of the Hamiltonian constraint. To compare with the earlier works and provide a comparison with the $\barμ'$ scheme, we also investigate the $\barμ$ scheme although it has ma…
In this paper we investigate the scalar mode of first-order metric perturbations over spatially flat FRW spacetime when the holonomy correction is taken into account in the semi-classical framework of loop quantum cosmology. By means of the Hamiltonian derivation, the cosmological perturbation equations is obtained in longitudinal gauge. It turns out that in the presence of metric perturbation the holonomy effects influence both background and perturbations, and contribute the non-trivial terms $S_{h1}$ and $S_{h2}$ in the cosmological perturbation equations.
Working in Einstein frame we introduce, in order to avoid singularities, holonomy corrections to the $f(R)=R+αR^2$ model. We perform a detailed analytical and numerical study when holonomy corrections are taken into account in both Jordan and Einstein frames obtaining, in Jordan frame, a dynamics which differs qualitatively, at early times, from the one of the original model. More precisely, when holonomy corrections are taken into account the universe is not singular, starting at early times in the contracting phase and bouncing to enter in the expanding one where, as in the original model, i…
We investigate the scalar mode of perturbation of super inflation in the version of loop quantum cosmology in which the gauge invariant holonomy corrections are considered. Given a background solution, we calculate the power spectrum of the perturbation in the classical and LQC conditions. Then we compute the anisotropy originated from the perturbation. It is found that in the presence of the gauge invariant holonomy corrections the power spectrum is exponentially blue and the anisotropy also grows exponentially in the epoch of super inflation.
We study the introduction of holonomy corrections in $f(R)$ gravity. We will show that there are infinitely many ways, as many as canonical transformations, to introduce this kind of corrections, depending on the canonical variables (two coordinates and its conjugate momenta) used to obtain the Hamiltonian. In each case, these corrections lead, at effective level, to different modified holonomy corrected Friedmann equations in $f(R)$ gravity, which are in practice analytically unworkable, i.e. only numerical analysis can be used to understand its dynamics. Finally, we give arguments in favour …
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.…
A well-motivated extension of higher order holonomy corrections in loop quantum cosmology (LQC) for the $k=0$ Friedmann-Robertson-Walker model is investigated at the level of heuristic effective dynamics. It reveals that the quantum bounce is generic, regardless of the order of corrections, and the matter density remains finite, bounded from above by an upper bound in the regime of the Planckian density, even if all orders of corrections are included. This observation provides further evidence that the quantum bounce is essentially a consequence of the loopy nature (i.e. intrinsic discreteness…
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 propagat…
We study the dynamics of the scalar modes of linear perturbations around a flat, homogeneous and isotropic background in loop quantum cosmology. The equations of motion include quantum geometry effects and hold at all curvature scales so long as the wavelengths of the inhomogeneous modes of interest remain larger than the Planck length. These equations are obtained by including holonomy corrections in an effective Hamiltonian and then using the standard variational principle. We show that the effective scalar and diffeomorphism constraints are preserved by the dynamics. We also make some comme…
The holonomy correction is one of the main terms arising when implementing loop quantum gravity ideas at an effective level in cosmology. The recent construction of an anomaly free algebra has shown that the formalism used, up to now, to derive the primordial spectrum of fluctuations was not correct. This article aims at computing the tensor spectrum in a fully consistent way within this deformed and closed algebra.
We study the perturbation of the effective Hamiltonian constraint with holonomy correction from Euclidean loop quantum gravity. The Poisson bracket between the corrected Hamiltonian constraint and the diffeomorphism constraint is derived for vector modes. Some specific form of the holonomy correction function $f^{i}_{cd}$ is found, which satisfies that the constraint algebra is anomaly-free. This result confirms the possibility of non-trivial holonomy corrections from full theory while preserving anomaly-free constraint algebra in the perturbation framework. It also gives valuable hints on the…
Cosmological perturbations in Loop Quantum Cosmology (LQC) are usually studied incorporating either holonomy corrections, where the Ashtekar connection is replaced by a suitable sinus function in order to have a well-defined quantum analogue, or inverse-volume corrections coming from the eigenvalues of the inverse-volume operator. In this paper we will develop an alternative approach to calculate cosmological perturbations in LQC based on the fact that, holonomy corrected LQC in the flat Friedmann-Lemaître-Robertson-Walker (FLRW) geometry could be also obtained as a particular case of telepara…
Using self dual Ashtekar variables, we investigate (at the effective level) the spherically symmetry reduced model of loop quantum gravity, both in vacuum and when coupled to a scalar field. Within the real Ashtekar-Barbero formulation, the system scalar field coupled to spherically symmetric gravity is known to possess a non closed (quantum) algebra of constraints once the holonomy corrections are introduced, which forbids the loop quantization of the model. Moreover, the vacuum case, while not anomalous, introduces modifications which are usually interpreted as an effective signature change …
In this work we investigate which Loop Quantum Cosmology corrected Gauss-Bonnet $F(\mathcal{G})$ gravity can realize two singular cosmological scenarios, the intermediate inflation and the singular bounce scenarios. The intermediate inflation scenario has a Type III sudden singularity at $t=0$, while the singular bounce has a soft Type IV singularity. By using perturbative techniques, we find the holonomy corrected $F(\mathcal{G})$ gravities that generate at leading order the aforementioned cosmologies and we also argue that the effect of the holonomy corrections is minor to the power spectrum…
Most of the phenomenology of loop quantum gravity in the cosmological sector is based on the so-called holonomy correction to the Hamiltonian constraint. It straightforwardly modifies the Friedmann equations. In this work, we investigate the influence of corrections generalizing the one usually used in loop quantum cosmology. We find that a long enough inflation phase can be generated by purely quantum geometrical effects but we also underline the limitations of this scenario. In addition, we study the effects of generalized holonomy corrections on an inflationary phase generated by a massive …
In this article, we study the equations driving the dynamics of a Bianchi-I universe described by holonomy corrected effective loop quantum cosmology. We derive the LQC-modified generalized Friedmann equation, which is used as a guide to find different types of solutions. It turns out that, in this framework, most solutions never reach the classical behavior.
Anisotropic spherically symmetric systems are studied in the connection and densitized triad variables used in loop quantum gravity. The material source is an anisotropic fluid, which is arguably the most commonly used source term in anisotropic studies within general relativity. The gravitational+anisotropic fluid constraints are derived and analyzed and then quantum gravity inspired holonomy replacements are performed. The quantum properties of the fluid are dictated by the modified constraint equations. Particular attention is paid to wormhole throats, as they provide a simplistic model of …
In this paper we derive the uncertainty principle for the Loop Quantum Cosmology homogeneous and isotropic FLWR model with the holonomy-flux algebra. The uncertainty principle is between the variables $c$, with the meaning of connection and $μ$ having the meaning of the physical cell volume to the power $2/3$, i.e $v^{2/3}$ or a plaquette area. Since both $μ$ and $c$ are not operators, but rather the random variables, the Robertson uncertainty principle derivation that works for hermitian operators, can not be used. Instead we use the Wigner-Moyal-Groenewold phase space formalism. The Wigner-M…
We construct a noncommutative extension of the Loop Quantum Cosmology effective scheme for the open FLRW model with a free scalar field via a theta deformation. Firstly, a deformation is implemented in the configuration sector, among the holonomy variable and the matter degree of freedom. We show that this type of noncommutativity retain, to some degree, key features of the Loop Quantum Cosmology paradigm for a free field. Secondly, a deformation is implemented in the momentum sector, among the momentum associated to the holonomy variable and the momentum associated to the matter field. We sho…
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…
Loop quantum gravity introduces two characteristic modifications in the classical constraints of general relativity: the holonomy and inverse-triad corrections. In this paper, a systematic construction of anomaly-free effective constraints encoding such corrections is developed for spherically symmetric spacetimes. The starting point of the analysis is a generic Hamiltonian constraint where free functions of the triad and curvature components as well as non-minimal couplings between geometric and matter degrees of freedom are considered. Then, the requirement of anomaly freedom is imposed in o…
Loop quantum gravity is one of the leading candidate theory to non-perturbatively quantize gravity. In this framework, holonomy corrections to the equation of propagation of gravitons in a FLRW background have been derived. We investigate the consequences of those corrections on the tensor power spectrum in de-Sitter and slow-roll inflations, for n=-1/2. Depending on the value of the Barbero-Immirzi parameter, several observational features could be expected.
An affine transport equation was used recently to study properties of angular momentum and gravitational-wave memory effects in general relativity. In this paper, we investigate local properties of this transport equation in greater detail. Associated with this transport equation is a map between the tangent spaces at two points on a curve. This map consists of a homogeneous (linear) part given by the parallel transport map along the curve plus an inhomogeneous part, which is related to the development of a curve in a manifold into an affine tangent space. For closed curves, the affine transpo…
We consider the matter bounce scenario in Loop Quantum Cosmology (LQC) for physical potentials that at early times provide a nearly matter dominated Universe in the contracting phase, having a reheating mechanism in the expanding phase, i.e., being able to release the energy of the scalar field creating particles that thermalize in order to match with the hot Friedmann Universe, and finally at late times leading to the current cosmic acceleration. For these models, numerically solving the dynamical equations we have seen that the teleparallel version of LQC leads to theoretical results that fi…
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 $V/ρ_{c}$, where $V$ is a potential of the scalar field and $ρ_{c}$ 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 qu…
Anomaly-free perturbations of loop quantum cosmology with holonomy corrections reveal an $Ω$ -deformed spacetime structure, $Ω:=1-2ρ/ρ_c$, where $Ω<0$ indicates a Euclidean-like space and $Ω>0$ indicates a Lorentz-like space. It would be reasonable to give the initial value at the spacetime transition point, $ρ=ρ_c/2$, but we find that it is impossible to define a Minkowski-like vacuum even for large $k$ modes at that time. However, if we loosen the condition and give the initial value slightly after $Ω=0$, e.g., $Ω\simeq 0.2$, the vacuum state can be well defined and, furthermore the slow rol…
We study the scalar modes of linear perturbations in loop quantum cosmology. This is done on a lattice where each cell is taken to be homogeneous and isotropic and can be quantized via standard homogeneous loop quantum cosmology techniques. The appropriate interactions between nearby cells are included in the Hamiltonian in order to obtain the correct physics. It is shown that the quantum theory is anomaly-free: the scalar and diffeomorphism constraint operators weakly commute with the Hamiltonian. Finally, the effective theory encoding the leading order quantum gravity corrections is derived …
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 gravitationa…
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 sch…
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…
Considering an arbitrary, varying equation of the state parameter, the thermodynamic properties of the dark energy fluid in a semiclassical loop quantum cosmology scenario, which we consider the inverse volume modification, is studied. The equation of the state parameters are corrected as a semiclassical one during considering the effective behavior. Assuming that the apparent horizon has Hawking temperature, the modified entropy-area relation is obtained, we find that this relation is different from the one which is obtained by considering the holonomy correction. Considering the dark energy …
We discuss a new scenario for early cosmology when the inflationary de Sitter phase emerges dynamically. This genuine quantum effect occurs as a result of dynamics of the topologically nontrivial sectors in a strongly coupled QCD- like gauge theory in an expanding universe. We test these ideas by explicit computations in hyperbolic space $ \mathbb{H}^3_κ\times \mathbb{S}^1_{κ^{-1}}$. We argue that the key element for this idea to work is the presence of nontrivial holonomy computed along $\mathbb{S}^1_{κ^{-1}}$. The effect is non-local in nature, non-analytical in coupling constant and can not…
It is stated that holonomy corrections in loop quantum cosmology introduce a modification in Friedmann's equation which prevent the big rip singularity. Recently in \cite{h12} it has been proved that this modified Friedmann equation is obtained in an inconsistent way, what means that the results deduced from it, in particular the big rip singularity avoidance, are not justified. The problem is that holonomy corrections modify the gravitational part of the Hamiltonian of the system leading, after Legendre's transformation, to a non covariant Lagrangian which is in contradiction with one of the …
We study the tensor modes of linear metric perturbations within an effective framework of loop quantum cosmology. After a review of inverse-volume and holonomy corrections in the background equations of motion, we solve the linearized tensor modes equations and extract their spectrum. Ignoring holonomy corrections, the tensor spectrum is blue tilted in the near-Planckian superinflationary regime and may be observationally disfavoured. However, in this case background dynamics is highly nonperturbative, hence the use of standard perturbative techniques may not be very reliable. On the other han…
We discuss scalar-tensor cosmology with an extra $R^{-1}$ correction by the Noether Symmetry Approach. The existence of such a symmetry selects the forms of the coupling $ω(ϕ)$, of the potential $V(ϕ)$ and allows to obtain physically interesting exact cosmological solutions.
Effects of inverse triad corrections and (point) holonomy corrections, occuring in loop quantum gravity, are considered on the properties of Reissner-Nordström black holes. The version of inverse triad corrections with unmodified constraint algebra reveals the possibility of occurrence of three horizons (over a finite range of mass) and also shows a mass threshold beyond which the inner horizon disappears. For the version with modified constraint algebra, coordinate transformations are no longer a good symmetry. The covariance property of spacetime is regained by using a \emph{quantum} notion …
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…
We present a covariant model of a spherically symmetric black hole with corrections motivated by loop quantum gravity. The effective modifications, parametrized by a positive constant $λ$, are implemented through a canonical transformation and a linear combination of the constraints of general relativity, in such a way that the theory remains free of anomalies and general relativity is recovered for $λ=0$. In addition, the corresponding metric is constructed in a fully covariant way to ensure that gauge transformations on phase space correspond to coordinate changes. The solution for each gaug…
We show that the effective dynamics of the recently proposed isotropic condensate state of group field theory with Laplacian kinetic operator can be equivalent to that of homogeneous and isotropic loop quantum cosmology in the improved dynamics quantization scheme, where the area of elementary holonomy plaquettes is constant. This constitutes a somewhat surprising example of a cosmological model of quantum gravity where the operations of minisuperspace symmetry reduction and quantization can actually commute.
The first-order loop quantum gravity correction of the simplest, classical general-relativistic Friedmann Hamiltonian constraint, emerging from a holomorphic spinfoam cosmological model peaked on homogeneous, isotropic geometries, is studied. The quantum Hamiltonian constraint, satisfied by the EPRL transition amplitude between the boundary cosmological coherent states, includes a contribution of the order of the Planck constant $\hbar$ that also appears in the corresponding semiclassical symplectic model. The analysis of this term gives a quantum-gravitational correction to the classical Frie…
The scalar field can behave like a fluid with equation of state $p_ϕ=wρ_ϕ$, 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 indic…
The influence of non-minimal coupling of a scalar field and the Gauss-Bonnet term on the inflationary stage of evolution of the universe is investigated in this paper. The main cosmological effects of such a coupling were considered. The deviations between Einstein-Gauss-Bonnet inflation and standard one based on Einstein gravity were determined. The corrections of a weak GB coupling preserving the type of the scalar field potential to standard inflationary models is considered as well.
Gauge invariance of scalar perturbations is studied together with the associated equations of motion. Extending methods developed in the framework of hamiltonian General Relativity, the Hamilton-Jacobi equation is investigated into the details in Loop Quantum Cosmology. The gauge-invariant observables are built and their equations of motions are reviewed both in Hamiltonian and Lagrangian approaches. This method is applied to scalar perturbations with either holonomy or inverse-volume corrections.
In the context of the Loop Quantum Cosmology we have analysed the holonomy correction to the classical evolution of the simplified Bianchi I model in the presence of vector fields. For the Universe dominated by a massive vector field or by a combination of a scalar field and a vector field a smooth transition between Kasner-like and Kasner-unlike solutions for a Bianchi I model has been demonstrated. In this case a lack of initial curvature singularity and a finite maximal energy density appear already at the level of General Relativity, which simulates a classical Big Bounce.
Scalar cosmological perturbations in loop quantum cosmology (LQC) is revisited in a covariant manner, using self dual Ashtekar variables. For real-valued Ashtekar-Barbero variables, this `deformed algebra' approach has been shown to implement holonomy corrections from loop quantum gravity (LQG) in a consistent manner, albeit deforming the algebra of modified constraints in the process. This deformation has serious conceptual ramifications, not the least of them being an effective `signature-change' in the deep quantum regime. In this paper, we show that working with self dual variables lead to…
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…
We consider the ADM formalism as a tool to build bouncing cosmologies. In this approach, the foliation of the spacetime has to be fixed in order to go beyond General Relativity modifying the gravitational sector. Once a preferred slicing, which we choose based on the matter content of the universe following the spirit of Weyl's postulate, has been fixed, $f$ theories depending on the extrinsic and intrinsic curvature of the slicing are covariant for all the reference frames preserving the foliation, i.e., the constraint and dynamical equations have the same form for all these observers. Moreov…
Quantum fluctuations or other moments of a state contribute to energy expectation values and can imply interesting physical effects. In quantum cosmology, they turn out to be important for a discussion of density bounds and instabilities of initial-value problems in the presence of signature change in loop-quantized models. This article provides an effective description of these issues, accompanied by a comparison with existing numerical results and an extension to squeezed states. The comparison confirms that canonical effective methods are well-suited for computations of properties of physic…
