We study isotropic and anisotropic (Bianchi I) cosmologies in Palatini $f(R)$ and $f(R,R_{μν}R^{μν})$ theories of gravity and consider the existence of non-singular bouncing solutions in the early universe. We find that all $f(R)$ models with isotropic bouncing solutions develop shear singularities in the anisotropic case. On the contrary, the simple quadratic model $R+a R^2/R_P+R_{μν}R^{μν}/R_P$ exhibits regular bouncing solutions in both isotropic and anisotropic cases for a wide range of equations of state, including dust (for $a<0$) and radiation (for arbitrary $a$). It thus represents a purely gravitational solution to the big bang singularity and anisotropy problems of general relativity without the need for exotic ($w>1$) sources of matter/energy.
We consider the Palatini formalism of the Born-Infeld gravity. In the Palatini formalism, the propagating mode is only graviton, whose situation is different from that in the metric formalism. We discuss about the FRW cosmology by using an effective potential. Especially we consider the condition that the bouncing could occur. We also give some speculations about the black hole formation
We consider the question of deriving initial conditions for scalar fields in driving both an early and late quintessence phase. The dark energy field presents an unresolved uniformity problem. Further difficulties with initial conditions for assisted, kinetic and phantom inflation are presented. We review the use of the canonical measure and find the negative conclusions of Gibbons and Hawking can be allayed by means of a reasonable quantum cosmological input. We remark upon some attempts at incorporating inflationary schemes into cyclic and bouncing models.
It is shown that a quadratic gravitational Lagrangian in the Palatini formulation is able to capture different aspects of quantum gravity phenomenology in a single framework. In particular, in this theory field excitations propagating with different energy-densities perceive different background metrics, a fundamental characteristic of the DSR and Rainbow Gravity approaches. This theory, however, avoids the so-called soccer ball problem. Also, the resulting isotropic and anisotropic cosmologies are free from the big bang singularity. This singularity avoidance occurs non-perturbatively and sha…
The Pre-Big Bang cosmology inspired generations of cosmologists in attempts to cure the initial Big Bang singularity using a fundamental length scale as proposed by String Theory. The existence of a phase of collapse/inflation with increasing curvature followed by a cosmic bounce has been proposed as an alternative to standard inflation in the solution of the horizon and curvature problems. However the generation of a nearly scale-invariant spectrum of perturbations is not an automatic prediction of such scenarios. In this paper I review some general statements about the evolution of perturbat…
We consider a two-component regular cosmology bouncing from contraction to expansion, where, in order to include both scalar fields and perfect fluids as particular cases, the dominant component is allowed to have an intrinsic isocurvature mode. We show that the spectrum of the growing mode of the Bardeen potential in the pre-bounce is never transferred to the dominant mode of the post-bounce. The latter acquires at most a dominant isocurvature component, depending on the relative properties of the two fluids. Our results imply that several claims in the literature need substantial revision.
We show how the large-scale cosmological dynamics can be obtained from the hydrodynamics of isotropic group field theory condensate states in the Gross-Pitaevskii approximation. The correct Friedmann equations are recovered in the semi-classical limit for some choices of the parameters in the action for the group field theory, and quantum gravity corrections arise in the high-curvature regime causing a bounce which generically resolves the big-bang and big-crunch singularities.
We show that bouncing open or flat Friedmann-Robertson-Walker cosmologies are inconsistent with worldsheet string theory to first approximation. Specifically, the Virasoro constraint translates to the null energy condition in spacetime at leading order in the alpha-prime expansion. Thus one must go beyond minimally-coupled Einstein gravity in order to find bounce solutions.
Bouncing cosmologies, suggested by String/M-theory, may provide an alternative to standard inflation to account for the origin of inhomogeneities in our universe. The fundamental question regards the correct way to evolve the scalar perturbations through the bounce. In this work, we determine the evolution of perturbations and the final spectrum for an arbitrary (spatially flat) bouncing cosmology, with the only assumption that the bounce is governed by a single physical scale. In particular, we find that the spectrum of the pre-bounce growing mode of the Bardeen potential (which is scale-inva…
We review the status of bouncing cosmologies as alternatives to cosmological inflation for providing a description of the very early universe, and a source for the cosmological perturbations which are observed today. We focus on the motivation for considering bouncing cosmologies, the origin of fluctuations in these models, and the challenges which various implementations face.
The Asymptotically Safe Gravity provides a framework for the description of gravity from the trans-Planckian regime to cosmological scales. According to this scenario, the cosmological constant and Newton's coupling are functions of the energy scale whose evolution is dictated by the renormalization group equations. The formulation of the renormalization group equations on foliated spacetimes, based on the Arnowitt-Deser-Misner (ADM) formalism, furnishes a natural way to construct the RG energy scale from the spectrum of the laplacian operator on the spatial slices. Combining this idea with a …
We consider the early time cosmology of f(R) theories in Palatini formalism and study the conditions that guarantee the existence of homogeneous and isotropic models that avoid the Big Bang singularity. We show that for such models the Big Bang singularity can be replaced by a cosmic bounce without violating any energy condition. In fact, the bounce is possible even for pressureless dust. We give a characterization of such models and discuss their dynamics in the region near the bounce. We also find that power-law lagrangians with a finite number of terms may lead to non-singular universes, wh…
Nonsingular cosmologies are investigated in the framework of f(R) gravity within the first order formalism. General conditions for bounces in isotropic and homogeneous cosmology are presented. It is shown that only a quadratic curvature correction is needed to predict a bounce in a flat or to describe cyclic evolution in a curved dust-filled universe. Formalism for perturbations in these models is set up. In the simplest cases, the perturbations diverge at the turnover. Conditions to obtain smooth evolution are derived.
We confront various nonsingular bouncing cosmologies with the recently released BICEP2 data and investigate the observational constraints on their parameter space. In particular, within the context of the effective field approach, we analyze the constraints on the matter bounce curvaton scenario with a light scalar field, and the new matter bounce cosmology model in which the universe successively experiences a period of matter contraction and an ekpyrotic phase. Additionally, we consider three nonsingular bouncing cosmologies obtained in the framework of modified gravity theories, namely the …
We investigate the post-bounce background dynamics in a certain class of single bounce scenarios studied in the literature, in which the cosmic bounce is driven by a scalar field with negative exponential potential such as the ekpyrotic potential. We show that those models can actually lead to cyclic evolutions with repeated bounces. These cyclic evolutions, however, do not account for the currently observed late-time accelerated expansion and hence are not cosmologically viable. In this respect we consider a new kind of cyclic model proposed recently and derive some cosmological constraints o…
In this note, we construct a family of non-singular time-dependent solutions of a six-dimensional gravitational theory that are warped products of a four dimensional bouncing cosmological solution and a two dimensional internal manifold. The warp factor is time-dependent and breaks translation invariance along one of the internal directions. When the warp factor is periodic in time, the non-compact part of the geometry bounces periodically. The six dimensional geometry is supported by matter that does not violate the null energy condition. We show that this 6D geometry does not admit a closed …
It is argued that the Universe reheating in bouncing cosmologies could be explained via gravitational particle production, as due to a sudden phase transition in the contracting regime. To this end, it is shown that gravitational production of massive particles conformally coupled with gravity in a matter-ekpyrotic bouncing Universe, where the sudden phase transition occurs in the contracting regime, yields a reheating temperature which is in good agreement with cosmological observations.
We compute the complete post-Newtonian limit of the Palatini form of f(R) gravities using a scalar-tensor representation. By comparing the predictions of these theories with laboratory and solar system experiments, we find a set of inequalities that any lagrangian f(R) must satisfy. The constraints imposed by those inequalities allow us to find explicit functions that bound from above and from below the possible nonlinear terms of the lagrangian. We conclude that the lagrangian f(R) must be almost linear in R and that nonlinear corrections that grow at low curvatures are incompatible with obse…
The observed temperature fluctuations in the cosmic microwave background can be traced back to primordial curvature modes that are sourced by adiabatic and/or entropic matter perturbations. In this paper, we explore the entropic mechanism in the context of non-singular bouncing cosmologies. We show that curvature modes are naturally generated during `graceful exit,' i.e., when the smoothing slow contraction phase ends and the universe enters the bounce stage. Here, the key role is played by the kinetic energy components that come to dominate the energy density and drive the evolution towards t…
The so called $f(X)$ hybrid metric-Palatini gravity presents a unique viable generalisation of the $f(R)$ theories within the metric-affine formalism. Here the cosmology of the $f(X)$ theories is studied using the dynamical system approach. The method consists of formulating the propagation equation in terms of suitable (expansion-normalised) variables as an autonomous system. The fixed points of the system then represent exact cosmological solutions described by power-law or de Sitter expansion. The formalism is applied to two classes of $f(X)$ models, revealing both standard cosmological fix…
We present a full investigation of scalar perturbations in a rather generic model for a regular bouncing universe, where the bounce is triggered by an effective perfect fluid with negative energy density. Long before and after the bounce the universe is dominated by a source with positive energy density, which may be a perfect fluid, a scalar field, or any other source with an intrinsic isocurvature perturbation. Within this framework, we present an analytical method to accurately estimate the spectrum of large-scale scalar perturbations until their reentry, long after the bounce. We also prop…
Cyclic universes with bouncing solutions are candidates for solving the big bang initial singularity problem. Here we seek bouncing solutions in a modified Gauss-Bonnet gravity theory, of the type $R+f(G)$, where $R$ is the Ricci scalar, $G$ is the Gauss-Bonnet term, and $f$ some function of it. In finding such a bouncing solution we resort to a technique that reduces the order of the differential equations of the $R+f(G)$ theory to second order equations. As general relativity is a theory whose equations are of second order, this order reduction technique enables one to find solutions which a…
We review matter bounce scenarios where the matter content is dark matter and dark energy. These cosmologies predict a nearly scale-invariant power spectrum with a slightly red tilt for scalar perturbations and a small tensor-to-scalar ratio. Importantly, these models predict a positive running of the scalar index, contrary to the predictions of the simplest inflationary and ekpyrotic models, and hence could potentially be falsified by future observations. We also review how bouncing cosmological space-times can arise in theories where either the Einstein equations are modified or where matter…
We calculate the running of the scalar index in the ekpyrotic and matter bounce cosmological scenarios, and find that it is typically negative for ekpyrotic models, while it is typically positive for realizations of the matter bounce where multiple fields are present. This can be compared to inflation, where the observationally preferred models typically predict a negative running. The magnitude of the running is expected to be between $10^{-4}$ and up to $10^{-2},$ leading in some cases to interesting expectations for near-future observations.
We review the general features of nonsingular universes ({\em i.e.} those that go from an era of accelerated collapse to an expanding era without displaying a singularity) as well as cyclic universes. We discuss the mechanisms behind the bounce, and analyze examples of solutions that implement these mechanisms. Observational consequences of such regular cosmologies are also considered, with emphasis in the behavior of the perturbations.
In the framework of massive gravity with a de Sitter reference metric, we study homogeneous and isotropic solutions with positive spatial curvature. Remarkably, we find that bounces can occur when cosmological matter satisfies the strong energy condition, in contrast to what happens in classical general relativity. This is due to the presence in the Friedmann equations of additional terms, which depend on the scale factor and its derivatives and can be interpreted as an effective fluid. We present a detailed study of the system using a phase space analysis. After having identified the fixed po…
We elaborate further on the evolution properties of cosmological fluctuations through a bounce. We show this evolution to be describable either by ``transmission'' and ``reflection'' coefficients or by an effective unitary S-matrix. We also show that they behave in a time reversal invariant way. Therefore, earlier results are now interpreted in a different perspective and put on a firmer basis.
It has recently been shown that f(R) theories formulated in the Palatini variational formalism are able to avoid the big bang singularity yielding instead a bouncing solution. The mechanism responsible for this behavior is similar to that observed in the effective dynamics of loop quantum cosmology and an f(R) theory exactly reproducing that dynamics has been found. I will show here that considering more general actions, with quadratic contributions of the Ricci tensor, results in a much richer phenomenology that yields bouncing solutions even in anisotropic (Bianchi I) scenarios. Some implica…
Background boucing cosmologies in the framework of General Relativity, driven by a single scalar field filling the Universe, and with a quasi-matter domination period, i.e., depicting the so-called Matter Bounce Scenario, are reconstructed for geometries with positive spatial curvature. These cosmologies lead to a nearly flat power spectrum of the curvature fluctuations in co-moving coordinates for modes that leave the Hubble radius during the quasi-matter domination period, and whose spectral index and its running, which are related with the effective Equation of State parameter given by the …
For bouncing cosmologies such as the ekpyrotic/cyclic scenarios we show that it is possible to make predictions for density perturbations which are independent of the details of the bouncing phase. This can be achieved, as in inflationary cosmology, thanks to the existence of a dynamical attractor, which makes local observables equal to the unperturbed solution up to exponentially small terms. Assuming that the physics of the bounce is not extremely sensitive to these corrections, perturbations can be evolved even at non-linear level. The resulting spectrum is not scale invariant and thus inco…
In this work, we explore cosmological sudden singularities arising in the dynamically equivalent scalar-tensor representation of generalized hybrid metric-Palatini gravity. Using a FLRW background, we show that the structure of the field equations prevents sudden singularities from arising at time derivatives of the scale factor of orders lower than four, but that they are allowed to appear for time derivatives of higher orders. Imposing an ansatz for the scale factor, we provide an explicit solution where these sudden singularities appear in the fourth-order time derivative of the scale facto…
We consider the field equations for a flat FRW cosmological model, in a generic $f(R)$ gravity model and cast them into a, completely normalized-dimensionless, system of O.D.Es for the scale factor and the function $f(R)$, with respect to the scalar curvature $R$. It is shown that under reasonable assumptions one can produce simple analytical and numerical solutions describing bouncing cosmological models where in addition there are late-time accelerating. Possibility of extending these results is briefly discussed.
These notes are a summary of lectures given at the Instituto de Astronomia of the Universidad Nacional Autonoma de Mexico (UNAM), the Dipartimento di Fisica of the Universita degli studi di Salerno (Italy), and the Instituto de Fisica Corpuscular of the Universitat de Valencia (Spain) during the year 2012. Basic mathematical aspects of Palatini theories of gravity, which are constructed assuming that metric and connection are independent geometrical entities, are briefly introduced and discussed, and followed by a detailed derivation of the field equations of some rather general Lagrangian the…
Given the proliferation of bouncing models in recent years, we gather and critically assess these proposals in a comprehensive review. The Planck data shows an unmistakably red, quasi scale-invariant, purely adiabatic primordial power spectrum and no primary non-Gaussianities. While these observations are consistent with inflationary predictions, bouncing cosmologies aspire to provide an alternative framework to explain them. Such models face many problems, both of the purely theoretical kind, such as the necessity of violating the NEC and instabilities, and at the cosmological application lev…
For contracting/expanding bouncing cosmologies, the formulation of junction conditions at a bouncing was recently revisited by the author in collaboration with B. Le Floch and G. Veneziano. The regime of interest here is the so-called quiescent regime, in which a monotone behavior of the metric is observed and asymptotic expansions can be derived. Here, we overview our new methodology based on the notion of {\sl singularity scattering maps} and {\sl cyclic spacetimes}, and we present our main conclusions. In particular, we provide a classification of all allowed bouncing junction conditions, i…
For bouncing cosmologies, a fine set of parameters is introduced in order to describe the nearly matter dominated phase, and which play the same role that the usual slow-roll parameters play in inflationary cosmology. It is shown that, as in the inflation case, the spectral index and the running parameter for scalar perturbations in bouncing cosmologies can be best expressed in terms of these small parameters. Further, they explicitly exhibit the duality which exists between a nearly matter dominated Universe in its contracting phase and the quasi de Sitter regime in the expanding one. The res…
We study the generalized version of energy-momentum squared gravity (EMSG) in the Palatini formalism. This theory allows the existence of a scalar constructed with energy-momentum tensor as $T_{αβ}T^{αβ}$ in the generic action of the theory. We study the most general form of this theory in the Palatini framework and present the underlying field equations. The equations of motion of a massive test particle have been derived. The weak field limit of the theory is explored and the generalized version of the Poisson equation is obtained. Moreover, we explore the cosmological behavior of the theory…
We exhibit a situation in which cosmological perturbations of astrophysical relevance propagating through a bounce are affected in a scale-dependent way. Involving only the evolution of a scalar field in a closed universe described by general relativity, the model is consistent with causality. Such a specific counter-example leads to the conclusion that imposing causality is not sufficient to determine the spectrum of perturbations after a bounce provided it is known before. We discuss consequences of this result for string motivated scenarios.
Bouncing cosmologies are often proposed as alternatives to standard inflation for the explanation of the homogeneity and flatness of the universe. In such scenarios, the present cosmological expansion is preceded by a contraction phase. However, during the contraction, in general the anisotropy of the universe grows and eventually leads to a chaotic mixmaster behavior. This would either be hard to reconcile with observations or even lead to a singularity instead of the bounce. In order to preserve a smooth and isotropic bounce, the source for the contraction must have a super-stiff equation of…
We calculate the gravitational waves (GW) spectrum produced in various Early Universe scenarios from gauge field sources, thus generalizing earlier inflationary calculations to bouncing cosmologies. We consider generic couplings between the gauge fields and the scalar field dominating the energy density of the Universe. We analyze the requirements needed to avoid a backreaction that will spoil the background evolution. When the scalar is coupled only to $F \tilde F$ term, the sourced GW spectrum is exponentially enhanced and parametrically the square of the vacuum fluctuations spectrum, ${\cal…
We study the dynamics of a homogeneous and isotropic Friedmann-Robertson-Walker universe in the context of the Eddington-inspired Born-Infeld theory of gravity. We generalize earlier results, obtained in the context of a radiation dominated universe, to account for the evolution of a universe permeated by a perfect fluid with an arbitrary equation of state parameter $w$. We show that a bounce may occur for $κ>0$, if $w$ is time-dependent, and we demonstrate that it is free from tensor singularities. We argue that Eddington-inspired Born-Infeld cosmologies may be a viable alternative to the inf…
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…
In a quantum theory of gravity spacetime behaves classically when quantum probabilities are high for histories of geometry and field that are correlated in time by the Einstein equation. Probabilities follow from the quantum state. This quantum perspective on classicality has important implications: (a) Classical histories are generally available only in limited patches of the configuration space on which the state lives. (b) In a given patch states generally predict relative probabilities for an ensemble of possible classical histories. (c) In between patches classical predictability breaks d…
In this paper I examine the phase space dynamics in the framework of Non-Projectable Hořava-Lifshitz bouncing cosmologies. By considering a closed Friedmann-Lemaître-Robertson-Walker (FLRW) geometry, the first integral contains a correction term that leads to nonsingular metastable bounces in the early evolution of the universe. The matter content of the model is a massive conformally coupled scalar field, dust and radiation. A nonvanishing cosmological constant connected to a de Sitter attractor in the phase space is also assumed. In narrow windows of the parameter space, labeled by an intege…
The string effective action at tree level contains, in its bosonic sector, the Einstein-Hilbert term, the dilaton, and the axion, besides scalar and gauge fields coming from the Ramond-Ramond sector. The reduction to four dimensions brings to scene moduli fields. We generalize this effective action by introducing two arbitrary parameters, $ω$ and $m$, connected with the dilaton and axion couplings. In this way, more general frameworks can be analyzed. Regular solutions with a bounce can be obtained for a range of (negative) values of the parameter $ω$ which, however, exclude the pure string co…
We study the conditions under which thermal fluctuations generated in the contracting phase of a non-singular bouncing cosmology can lead to a scale-invariant spectrum of cosmological fluctuations at late times in the expanding phase. We consider point particle gases, holographic gases and string gases. In the models thus identified, we also study the thermal non-Gaussianities of the resulting distribution of inhomogeneities. For regular point particle radiation, we find that the background must have an equation of state $w = 7/3$ in order to obtain a scale-invariant spectrum, and that the non…
The simplest possible classical model leading to a cosmological bounce is examined in the light of the non-Gaussianities it can generate. Concentrating solely on the transition between contraction and expansion, and assuming initially purely Gaussian perturbations at the end of the contracting phase, we find that the bounce acts as a source such that the resulting value for the post-bounce $f_{\mathrm{NL}}$ may largely exceed all current limits, to the point of potentially casting doubts on the validity of the perturbative expansion. We conjecture that if one can assume that the non-Gaussianit…
We consider linear perturbation equations for long-wavelength scalar metric perturbations in generalised gravity, applicable to non-singular cosmological models including a bounce from collapse to expansion in the very early universe. We present the general form for the perturbation equations which follows from requiring that the inhomogeneous universe on large scales obeys the same local equations as the homogeneous Friedmann-Robertson-Walker background cosmology (the separate universes approach). In a pseudo-longitudinal gauge this becomes a homogeneous second-order differential equation for…
We address the issue of fermionic particle creation in cosmological bouncing models governed by General Relativity, but where the bounce itself takes place due to quantum effects. If the energy scale of the bounce is not very close to the Planck energy, the Wheeler-DeWitt approach can be used to furnish sensible singularity-free background models with a contracting phase preceding an expanding phase in which the fermions evolve. The fermionic fields considered are massive, neutral and minimally coupled to gravity. We are particularly interested in neutrinos, neutrons and heavy neutrinos motiva…
The results from Planck2015, when combined with earlier observations from WMAP, ACT, SPT and other experiments, were the first observations to disfavor the "classic" inflationary paradigm. To satisfy the observational constraints, inflationary theorists have been forced to consider plateau-like inflaton potentials that introduce more parameters and more fine-tuning, problematic initial conditions, multiverse-unpredictability issues, and a new 'unlikeliness problem.' Some propose turning instead to a "postmodern" inflationary paradigm in which the cosmological properties in our observable unive…
In this letter we point out the existence of solutions to General Relativity with a negative cosmological constant in four dimensions, which contain solitons as well as traversable wormholes. The latter connect two asymptotically locally AdS$_{4}$ spacetimes. At every constant value of the radial coordinate the spacetime is a spacelike warped AdS$_{3}$. We compute the dual energy momentum tensor at each boundary showing that it yields different results. We also show that these vacuum wormholes can have more than one throat and that they are indeed traversable by computing the time it takes for…
We present nonsingular, homogeneous and isotropic bouncing solutions of the conformal Galileon model. We show that such solutions necessarily begin with a radiation-dominated contracting phase. This is followed by a quintom scenario in which the background equation of state crosses the cosmological constant boundary allowing for a nonsingular bounce which in turn is followed by Galilean Genesis. We analyze the spectrum of cosmological perturbations in this background. Our results show that the fluctuations evolve smoothly and without any pathology, but the adiabatic modes form a blue tilted sp…
In this paper, I construct a class of everywhere regular geometric sigma models that possess bouncing solutions. Precisely, I show that every bouncing metric can be made a solution of such a model. My previous attempt to do so by employing one scalar field has failed due to the appearance of harmful singularities near the bounce. In this work, I use four scalar fields to construct a class of geometric sigma models which are free of singularities. The models within the class are parametrized by their background geometries. I prove that, whatever background is chosen, the dynamics of its small p…
Primordial Black Hole Formation in Dust-Radiation Bouncing Cosmologies
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；University of Science and Technology of China（中国科学技术大学）；Universidade Estadual de Londrina（朗德里纳州立大学）；Université de Savoie, CNRS, IN2P3, LAPP（萨瓦省大学、法国国家科学研究中心、法国国家核子和粒子物理研究所、阿讷西粒子物理实验室）；Chengdu University of Technology（成都理工大学）；The University of Electronic Science and Technology of China（电子科技大学）；COSMO– Centro Brasileiro de Pesquisas Físicas（巴西物理研究中心）
We consider alternative inflationary cosmologies in massive gravity with degenerate reference metrics and study the feasibilities of the emergent universe scenario, bouncing and cyclic universes. We focus on the construction of the Einstein static universe, classes of exact solutions of bouncing and cyclic universes in degenerate massive gravity. We further study the stabilities of the Einstein static universe against both homogeneous and inhomogeneous scalar perturbations and give the parameters region for a stable Einstein static universe.
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…
