Unimodular gravity is based on a modification of the usual Einstein-Hilbert action that allows one to recover general relativity with a dynamical cosmological constant. It also has the interesting property of providing, as the momentum conjugate to the cosmological constant, an emergent clock variable. In this paper we investigate the cosmological reduction of unimodular gravity, and its quantization within the framework of flat homogeneous and isotropic loop quantum cosmology. It is shown that the unimodular clock can be used to construct the physical state space, and that the fundamental features of the previous models featuring scalar field clocks are reproduced. In particular, the classical singularity is replaced by a quantum bounce, which takes place in the same condition as obtained previously. We also find that requirement of semi-classicality demands the expectation value of the cosmological constant to be small (in Planck units). The relation to spin foam models is also studied, and we show that the use of the unimodular time variable leads to a unique vertex expansion.
We consider quantization of the positive curvature Friedmann cosmology in the unimodular modification of Einstein's theory, in which the spacetime four-volume appears as an explicit time variable. The Hamiltonian admits self-adjoint extensions that give unitary evolution in the Hilbert space associated with the Schrödinger equation. The semiclassical estimate to the no-boundary wave function of Hartle and Hawking is found. If this estimate is accurate, there is a continuous flux of probability into the configuration space from vanishing three-volume, and the no-boundary wave function evolves n…
We combine the unimodular gravity and mimetic gravity theories into a unified theoretical framework, which is proposed to provide a suggestive proposal for a framework that may assist in the discussion and solution search of the cosmological constant problem and of the dark matter issue. After providing the formulation of the unimodular mimetic gravity and investigating all the new features that the vacuum unimodular gravity implies, by using the underlying reconstruction method, we realize some well known cosmological evolutions, with some of these being exotic for the ordinary Einstein-Hilbe…
Some models are presented in which the strength of the gravitational coupling of the potential energy relative to the same coupling for the kinetic energy is, in a precise sense, adjustable. The gauge symmetry of these models consists of those coordinate changes with unit jacobian.
It is well known that classical Unimodular Gravity (UG) is equivalent to General Relativity, but is not clear the equivalence at the quantum level. Here we exhibit, in the UG framework, the Quantum Cosmology (QC) for a Friedmann-Lemaitre-Robertson-Walker (FLRW) universe, considering an inflationary scenario and also introducing a deformation on the commutation relations for the minisuperspace variables (NCC).
We investigate the inflationary realization in the context of unimodular $F(T)$ gravity, which is based on the $F(T)$ modification of teleparallel gravity, in which one imposes the unimodular condition through the use of Lagrange multipliers. We develop the general reconstruction procedure of the $F(T)$ form that can give rise to a given scale-factor evolution, and then we apply it in the inflationary regime. We extract the Hubble slow-roll parameters that allow us to calculate various inflation-related observables, such as the scalar spectral index and its running, the tensor-to-scalar ratio,…
The unimodular theory of gravity admits a canonical quantization of minisuperspace models without the problem of time. We derive instead a kind of Schrödinger equation. We have found unitarily evolving wave packet solutions for the special case of a massless scalar field and a spatially flat Friedmann universe. We show that the longterm behaviour of the expectation values of the canonical quantities corresponds to the evolution of the classical variables. The solutions provided in an explicit example can be continued beyond the singularity at t=0, passing a finite minimal extension of the univ…
Unimodular quantum cosmology admits wavepacket solutions that evolve according to a kind of Schrödinger equation. Though this theory is equivalent to general relativity on the classical level, its canonical structure is different and the problem of time does not occur. We present an Ehrenfest theorem for the long term evolution of the expectation value of the scale factor for a spatially flat Friedmann universe with a scalar field. We find that the classical and the quantum behaviour in the asymptotic future coincide for the special case of a massless scalar field. We examine the general behav…
We work out the one-loop contribution to the lepton anomalous magnetic moment coming from Unimodular Gravity. We use Dimensional Regularization and Dimensional Reduction to carry out the computations. In either case, we find that Unimodular Gravity gives rise to the same one-loop correction as that of General Relativity.
The question of the averaging of inhomogeneous spacetimes in cosmology is important for the correct interpretation of cosmological data. In this paper we suggest a conceptually simpler approach to averaging in cosmology based on the averaging of scalars within unimodular gravity. As an illustration, we consider the example of an exact spherically symmetric dust model, and show that within this approach averaging introduces correlations (corrections) to the effective dynamical evolution equation in the form of a spatial curvature term.
We compute the one-loop effective action in unimodular gravity, starting from two different classical formulations of the theory. We find that the effective action is the same in both cases, and agrees with the one of General Relativity.
We propose a covariant ghost-free unimodular $F(R)$ gravity theory, which contains a three-form field and study its structure using the analogy of the proposed theory with a quantum system which describes a charged particle in uniform magnetic field. Newton's law in non-covariant unimodular $F(R)$ gravity as well as in unimodular Einstein gravity is derived and it is shown to be just the same as in General Relativity. The derivation of Newton's law in covariant unimodular $F(R)$ gravity shows that it is modified precisely in the same way as in the ordinary $F(R)$ theory. We also demonstrate th…
This study considers the operator $\hat{T}$ corresponding to the classical spacetime four-volume $\tilde{T}$ (on-shell) of a finite patch of spacetime in the context of Unimodular Loop Quantum Cosmology for the homogeneous and isotropic model with flat spatial sections and without matter sources. Since the spacetime four-volume is canonically conjugate to the cosmological "constant", the operator $\hat{T}$ is constructed by solving its canonical commutation relation with $\hatΛ$ - the operator corresponding to the classical cosmological constant on-shell $\tildeΛ$. This conjugacy, along with t…
In teleparallel gravity, we apply Lorenz type gauge fixing to cope with redundant degrees of freedom in the vierbein field. This condition is mainly to restore the Lorentz symmetry of the teleparallel torsion scalar. In cosmological application, this technique provides standard cosmology, turnaround, bounce or $Λ$CDM as separate scenarios. We reconstruct the $f(T)$ gravity which generates these models. We study the stability of the solutions by analyzing the corresponding phase portraits. Also, we investigate Lorenz gauge in the unimodular coordinates, it leads to unify a nonsingular bounce an…
Unimodular Gravity is normally assumed to be equivalent to General Relativity for all matters but the character of the Cosmological Constant. Here we discuss this equivalence in the presence of a non-minimally coupled scalar field. We show that when we consider gravitation to be dynamical in a QFT sense, quantum corrections can distinguish both theories if the non-minimal coupling is non-vanishing. In order to show this, we construct a path integral formulation of Unimodular Gravity, fixing the complicated gauge invariance of the theory and computing all one-loop divergences. We find a combina…
We suggest a Lorentz non-invariant generalization of the unimodular gravity theory, which is classically equivalent to general relativity with a locally inert (devoid of local degrees of freedom) perfect fluid having an equation of state with a constant parameter $w$. For the range of $w$ near $-1$ this dark fluid can play the role of dark energy, while for $w=0$ this dark dust admits spatial inhomogeneities and can be interpreted as dark matter. We discuss possible implications of this model in the cosmological initial conditions problem. In particular, this is the extension of known microcan…
We compute the lowest order gravitational UV divergent radiative corrections to the S matrix element of the $fermion + fermion\rightarrow fermion + fermion$ scattering process in the massive Yukawa theory, coupled either to Unimodular Gravity or to General Relativity. We show that both Unimodular Gravity and General Relativity give rise to the same UV divergent contribution in Dimensional Regularization. This is a nontrivial result, since in the classical action of Unimodular Gravity coupled to the Yukawa theory, the graviton field does not couple neither to the mass operator nor to the Yukawa…
We derive the fixed-$Λ$ and unimodular propagators using the path integral formalism as applied to the Einstein-Cartan action. The simplicity of the action (which is linear in the lapse function) allows for an exact integration starting from the lapse function and the enforcement of the Hamiltonian constraint, leading to a product of Chern-Simons states if the connection is fixed at the endpoints. No saddle point approximation is needed. Should the metric be fixed at the endpoints, then, depending on the contour chosen for the connection, Hartle-Hawking or Vilenkin propagators are obtained. Th…
It is shown that the one-loop effective action of unimodular gravity is the same as that of ordinary gravity, restricted to unimodular metrics. The only difference is in the treatment of the global scale degree of freedom and of the cosmological term. A constant vacuum energy does not gravitate, addressing one aspect of the cosmological constant problem.
We study unimodular gravity in the context of cosmology, particularly some interesting consequences that might be able to describe the background cosmology and the late cosmic acceleration. We focus our attention on the hypothesis of \textit{non conservation of the energy momentum tensor}. This characteristic has an interesting outcome: we can obtain a modified Friedmann equation along with the acceleration equation and also new fluid equations related to a third order derivative of the scale factor, known in cosmography as the jerk parameter. As a consequence of this theory, it seems that rad…
We work out the one-loop and order $κ^2 m_ϕ^2$ UV divergent contributions, coming from Unimodular Gravity and General Relativity, to the S matrix element of the scattering process $ϕ+ ϕ\rightarrow ϕ+ ϕ$ in a $λϕ^4$ theory with mass $m_ϕ$. We show that both Unimodular Gravity and General Relativity give rise to the same UV divergent contributions in Dimension Regularization. This seems to be at odds with the known result that in a multiplicative MS dimensional regularization scheme the General Relativity corrections, in the de Donder gauge, to the beta function $β_λ$ of the $λ$ coupling do not …
In this paper we study a modified version of unimodular general relativity in the context of $f(G)$, $G$ denoting the Gauss-Bonnet invariant. We attach attention to Bianchi-type I and Friendmann-Robertson-Walker universes and search for unimodular $f(G)$ models according to the de Sitter and power-law solutions. Assuming unimodular $f(G)$ gravity as perfect fluid and making use of the slow-roll parameters, inflationary model has been reconstructed in concordance with the Planck observational data. Moreover, we investigate the realization of the bounce and loop quantum cosmological ekpyrotic pa…
We study the finite time singularity correspondence between the Jordan and Einstein frames for various $F(R)$ gravity theories. Particularly we investigate the ordinary pure $F(R)$ gravity case and the unimodular $F(R)$ gravity cases, in the absence of any matter fluids. In the ordinary $F(R)$ gravity cases, by using specific illustrative examples, we show that it is possible to have various correspondences of finite time singularities, and in some cases it is possible a singular cosmology in one frame might be non-singular in the other frame. In the unimodular $F(R)$ gravity case, the unimodu…
It has been suggested that the homogeneous black hole interior spacetime, when quantized following the techniques of loop quantum cosmology, has a resolved singularity replaced by a black-to-white hole transition. This result has however been derived so far only using effective classical evolution equations, and depends on details of the so-called polymerization scheme for the Hamiltonian constraint. Here we propose to use the unimodular formulation of general relativity to study the full quantum dynamics of this mini-superspace model. When applied to such cosmological models, unimodular gravi…
The Hamiltonian formalism of the generalized unimodular gravity theory, which was recently suggested as a model of dark energy, is shown to be a complicated example of constrained dynamical system. The set of its canonical constraints has a bifurcation -- splitting of the theory into two branches differing by the number and type of these constraints, one of the branches effectively describing a gravitating perfect fluid with the time-dependent equation of state, which can potentially play the role of dark energy in cosmology. The first class constraints in this branch generate local gauge symm…
Using general features of recent quantizations of the Hamiltonian constraint in loop quantum gravity and loop quantum cosmology, a dynamical interpretation of the constraint equation as evolution equation is presented. This involves a transformation from the connection to a dreibein representation and the selection of an internal time variable. Due to the discrete nature of geometrical quantities in loop quantum gravity also time turns out to be discrete leading to a difference rather than differential evolution equation. Furthermore, evolving observables are discussed in this framework which …
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…
In this paper, we discuss the properties of one-parameter sequences that arise when solving the Hamiltonian constraint in Bianchi I loop quantum cosmology using a separation of variables method. In particular, we focus on finding an expression for the sequence for all real values of the parameter, and discuss the pre-classicality of this function. We find that the behavior of these preclassical sequences imply time asymmetry on either side of the classical singularity in Bianchi I cosmology.
We present a construction of the Hubble operator for the spatially flat isotropic loop quantum cosmology. This operator is a Dirac observable on a subspace of the space of physical solutions. This subspace gets selected dynamically, requiring that its action be invariant on the physical solution space. As a simple illustrative application of the expectation value of the operator, we do find a generic phase of (super)inflation, a feature shown by Bojowald from the analysis of effective Friedmann equation of loop quantum cosmology.
Recently, Chamseddine and Mukhanov introduced a higher-derivative scalar-tensor theory which leads to a modified Friedmann equation allowing for bouncing solutions. As we note in the present work, this Friedmann equation turns out to reproduce exactly the loop quantum cosmology effective dynamics for a flat isotropic and homogeneous space-time. We generalize this result to obtain a class of scalar-tensor theories, belonging to the family of mimetic gravity, which all reproduce the loop quantum cosmology effective dynamics for flat, closed and open isotropic and homogeneous space-times.
Loop quantum cosmology is a symmetry reduced quantization of cosmological spacetimes based on loop quantum gravity. While it has been successful in resolution of various cosmological singularities and connecting Planck scale physics to phenomenology, its connection with loop quantum gravity has remained elusive. It is therefore important to integrate more and more features of the full theory into this framework and understand the reliability of physical predictions. In particular, if one wishes to connect the effective Hamiltonian in loop quantum cosmology to an expectation value of the scalar…
Loop quantum cosmology is considered in inflation era. A slow roll scalar field solution with power law potential is presented in the neighborhood of transition time, i.e. when the universe enters inflation phase from super-inflation. The compatibility of the model with Planck 2013 data is discussed. The domain of validity of second and the generalized second laws of thermodynamics for this solution and some other examples is studied.
We present a new cosmological model derived from Loop Quantum Gravity. The formulation is based on a projection of the kinematical Hilbert space of the full theory down to a subspace representing the proper arena for an inhomogeneous Bianchi I model. This procedure gives a direct link between the full theory and its cosmological sector. The emerging quantum cosmological model represents a simplified arena on which the complete canonical quantization program can be tested. The achievements of this analysis could also shed light on Loop Quantum Cosmology and its relation with the full theory.
Loop quantum cosmological methods are extended to homogeneous models in diagonalized form. It is shown that the diagonalization leads to a simplification of the volume operator such that its spectrum can be determined explicitly. This allows the calculation of composite operators, most importantly the Hamiltonian constraint. As an application the dynamics of the Bianchi I model is studied and it is shown that its loop quantization is free of singularities.
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.
We develop a Gauss-Bonnet extension of Loop Quantum Cosmology, by introducing holonomy corrections in modified $F(\mathcal{G})$ theories of gravity. Within the context of our formalism, we provide a perturbative expansion in the critical density, a parameter characteristic of Loop Quantum Gravity theories, and we result in having leading order corrections to the classical $F(\mathcal{G})$ theories of gravity. After extensively discussing the formalism, we present a reconstruction method that makes possible to find the Loop Quantum Cosmology corrected $F(\mathcal{G})$ theory that can realize va…
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.
In the framework of loop quantum cosmology anomaly free quantizations of the Hamiltonian constraint for Bianchi class A, locally rotationally symmetric and isotropic models are given. Basic ideas of the construction in (non-symmetric) loop quantum gravity can be used, but there are also further inputs because the special structure of symmetric models has to be respected by operators. In particular, the basic building blocks of the homogeneous models are point holonomies rather than holonomies necessitating a new regularization procedure. In this respect, our construction is applicable also for…
We compute the one-loop divergences in a theory of gravity with Lagrangian of the general form $f(R,R_{μν}R^{μν})$, on an Einstein background. We also establish that the one-loop effective action is invariant under a duality that consists of changing certain parameters in the relation between the metric and the quantum fluctuation field. Finally, we discuss the unimodular version of such a theory and establish its equivalence at one-loop order with the general case.
We find a number of complex solutions of the Einstein equations in the so-called unimodular version of general relativity, and we interpret them as saddle points yielding estimates of a gravitational path integral over a space of almost everywhere Lorentzian metrics on a spacetime manifold with topology of the "no-boundary" type. In this setting, the compatibility of the no-boundary initial condition with the definability of the quantum measure reduces reduces to the normalizability and unitary evolution of the no-boundary wave function ψ. We consider the spacetime topologies R^4 and RP^4 # R^…
Isotropic models in loop quantum cosmology allow explicit calculations, thanks largely to a completely known volume spectrum, which is exploited in order to write down the evolution equation in a discrete internal time. Because of genuinely quantum geometrical effects the classical singularity is absent in those models in the sense that the evolution does not break down there, contrary to the classical situation where space-time is inextendible. This effect is generic and does not depend on matter violating energy conditions, but it does depend on the factor ordering of the Hamiltonian constra…
Volume operators measuring the total volume of space in a loop quantum theory of cosmological models are constructed. In the case of models with rotational symmetry an investigation of the Higgs constraint imposed on the reduced connection variables is necessary, a complete solution of which is given for isotropic models; in this case the volume spectrum can be calculated explicitly. It is observed that the stronger the symmetry conditions are the smaller is the volume spectrum, which can be interpreted as level splitting due to broken symmetries. Some implications for quantum cosmology are pr…
We study the quantum cosmology of a flat Friedmann-Lemaître-Robertson-Walker universe filled with a (free) massless scalar field and a perfect fluid that represents radiation or a cosmological constant whose value is not fixed by the action, as in unimodular gravity. We study two versions of the quantum theory: the first is based on a time coordinate conjugate to the radiation/dark energy matter component, i.e., conformal time (for radiation) or unimodular time. As shown by Gryb and Thébault, this quantum theory achieves a type of singularity resolution; we illustrate this and other properties…
This is an introduction to loop quantum cosmology (LQC) reviewing mini- and midisuperspace models as well as homogeneous and inhomogeneous effective dynamics.
A concise description is presented of the basic features of the formalism of non-canonical spacetime volume-forms and its application in modified gravity theories and cosmology. The well known unimodular gravity theory appears as a very special case. Concerning the hot issues facing cosmology now, we specifically briefly outline the construction of: (a) unified description of dark energy and dark matter as manifestations of a single material entity -- a special scalar field "darkon"; (b) quintessential models of universe evolution with a gravity-"inflaton"-assisted dynamical Higgs mechanism --…
