text:
\section{Introduction}\label{sec:intro}

The ubiquitous presence of classical singularities in General Relativity (GR) \cite{Penrose:1969pc,Hawking:1970zqf,Senovilla:2014gza} suggests that it cannot be understood as the ultimate description of the gravitational interaction. Indeed, if gravity is to be quantized, it is well known that, due to its non-renormalizability, GR cannot be seen as a fundamental theory of gravitation but must rather be interpreted as an effective field theory (EFT) with a cutoff at the Planck scale \cite{Donoghue:1994dn,Burgess:2003jk,Donoghue:2012zc}. One can thus try to solve the problem of singularities within the EFT paradigm, by looking for speccific effective theories that do not contain singular solutions for physically sensible initial data. This was the approach followed in \cite{Brandenberger:1993ef}, where higher-order curvature effective theories were found such that all homogeneous and isotropic solutions are non-singular within. The hope is that such kind of theories may shed some light into the properties of the underlying quantum gravity description although, to that end, one has to go beyond the EFT paradigm and understand what possible UV completions these effective theories can have.

The (in principle) most straightforward way to aim for a fundamental theory of gravity is to seek for a renormalizable action for the gravitational interaction that recovers GR in the regimes where it has been tested. Lovelock's theorem ensures that new degrees of freedom will propagate unless we depart from metricity and/or diffeomorphism invariance (namely background independence) if we stick to four space-time dimensions. These new degrees of freedom usually generate ghost instabilities, and standard quantizations of such theories suffer from a breakdown of unitarity \cite{Donoghue:2012zc}, although non-standard quantizations in terms of an anti-linear Hamiltonian (rather than hermitian) have been suggested as a possible solution to such problem \cite{Bender:2007wu,Mannheim:2021xrg,Mannheim:2020ryw,Mannheim:2018ljq}. {{Limitations of this program}} to provide a satisfactory candidate to a fundamental theory of gravity till date suggest that this search requires to go beyond perturbative modifications of the Einstein-Hilbert action.

In the past decades, much theoretical effort has been dedicated to find candidates for quantum theories of gravity that can be interpreted as fundamental, one of the typical requirements for these candidates being the ability to tame the classical singularities. One of such approaches is Loop Quantum Gravity (LQG), a non-perturbative canonical quantization of GR where the canonical phase space coordinates {{of the gravitational sector are the Ashtekar-Barbero variables -- the holonomies of the connection and the fluxes associated with the triads.}} While the dynamics of the full theory remains to be uncovered, much progress has been made in implementing the quantization techniques of LQG to  symmetric scenarios of physical relevance, such as cosmological \cite{Ashtekar:2011ni} and black hole space-times \cite{Ashtekar:2023cod}. The idea behind these approaches is to reduce the dimensionality of the phase space by enforcing the constraints provided by the {{underlying symmetries of the spacetime}}, and then quantize the degrees of freedom of the resulting minisuperspace by using loop quantization techniques. {{As a result,  the classical differentiable geometry of Einsteinian gravity is replaced by a discrete quantum geometry. Extensive investigations in loop quantization of cosmological models show that this underlying quantum geometry results in a bounce which replaces the big bang \cite{Ashtekar:2006rx, Ashtekar:2006wn, Ashtekar:2007em}, with similar results on the resolution of the central singularity in black hole spacetimes (see eg. \cite{Ashtekar:2018lag}). The dynamical resolution of the singularity, which occurs when the energy density reaches the Planckian regime,  manifests itself even at the level of consistent probabilities. Unlike in the Wheeler-DeWitt theory \cite{Craig:2010vf}, the probability for a universe to encounter a singularity is zero, and to have a bounce is unity \cite{Craig:2013mga}.

Interestingly, one can obtain an effective spacetime description in LQC \cite{Taveras:2008ke} which captures the underlying quantum evolution extremely well \cite{Ashtekar:2006wn}. Numerical simulations with high performance computing  confirm this feature \cite{Diener:2014mia}, including in presence of anisotropies \cite{Diener:2017lde, Singh:2018rwa}. These investigations show that for a wide variety of states, bounce in LQC is a robust phenomenon \cite{Diener:2014hba}.  Assuming the validity of the effective dynamics, one can prove that for arbitrary matter there is a generic resolution of singularities in LQC for isotropic as well as anisotropic spacetimes \cite{Singh:2009mz, Singh:2011gp, Singh:2014fsy, Saini:2016vgo, Saini:2017ipg, Saini:2017ggt}. Our manuscript will assume the validity of the effective spacetime description for all regimes and all of the loop cosmology models considered.

As in any approach to quantum gravity, the application of loop quantization techniques also gaces inherent ambiguities in the quantization procedure. One of these ambiguities results from the treatment of the Hamiltonian constraint, which has Euclidean and Lorentzian terms. In the spatially-flat models, using the symmetries of the space-time these terms can be combined before quantization resulting in the standard LQC \cite{Ashtekar:2003hd}. However, such a treatment is not applicable in general and in full LQG one must quantize both  terms independently. As a result,
following Thiemann's regularization of the Hamiltonian constraint \cite{Thiemann:1996aw},  different choices to solve these ambiguities have provided two modified versions of quantum cosmologies \cite{Yang:2009fp,Dapor:2017rwv,Li:2018opr, Li:2018fco,Assanioussi:2019iye, Li:2021mop}, which following Ref. \cite{Li:2018opr} will be referred  to as mLQC-I and mLQC-II, and all
 above models, including standard LQC
 as loop cosmologies. Note that as in standard LQC, one finds that there is a generic resolution of singularities in mLQC-I and mLQC-II \cite{Saini:2018tto}.
 }}

{{
A natural question that permeates the LQG framework is how does one recover the spacetime covariance in the continuum spacetime description. }} From a classical perspective, this question can be reformulated for the different loop cosmologies by asking whether there exist any diff invariant theories that are able to provide an effective classical description that reproduces their background evolution and also the dynamics of their perturbations. At the background level, this question was already answered for {{standard}} LQC in \cite{Olmo:2008nf}, where a diff invariant metric-affine action that reproduced the standard LQC background evolution was found. A natural question that arises after that finding, and the introduction of other loop cosmology models like mLQC-I and mLQC-II, is whether the different choices related to the quantization ambiguities affect this result, or also do yield an effective background evolution that can be described by a diff invariant theory. In the affirmative case, a further direction to explore would be how different are the resulting effective classical theories for each of the models, or whether they can be embedded in a family of diff invariant actions where a choice of parameters leads to the different loop cosmology models.

The exploration of alternative gravity theories in the last two decades offers multiple avenues in order to find diff invariant actions able to capture the dynamics of loop cosmologies. In this sense, an important piece of information that follows from LQC, mLQC-I, and mLQC-II is that the cosmic dynamics satisfies second-order equations {} in which the matter sources act nonlinearly, causing in that way nonperturbative effects (a bounce) without adding new dynamical degrees of freedom. This fact was already used in \cite{Olmo:2008nf} to identify metric-affine theories of the $f(\ensuremath\mathcal{R})$ type as a natural candidate for the effective description of loop cosmologies.

In the metric-affine framework, one assumes that classical gravitation is not described only by a metric, but also by an independent affine or spin connection.\footnote{In principle, given an affine connection there is a canonical spin connection associated to it and viceversa. However, while researchers that view metric-affine gravity theories necessarily as gauge theories of gravity view the spin connection as the fundamental object; and curvature, torsion, and nonmetricity as field-strengths; other researchers see the affine connection as fundamental and regard torsion and nonmetricity just as pieces of this connection with certain geometrical meaning. For a more detailed and up to date account of the details of the metric-affine framework from different perspectives, see e.g. \cite{JimenezCano:2021rlu,Delhom:2021bvq}.}
Due to the independence between metric and  connection, curvature invariants do not contain higher-order derivatives of the metric, and for that reason metric-affine theories have been proposed in the past as a natural framework where  renormalizable ghost-free candidates could be found. It has been recently shown, however, that generic metric-affine theories are also swarmed by ghostly instabilities unless certain restrictive sub-classes or fine-tuned Lagrangians are considered \cite{BeltranJimenez:2019acz,Jimenez:2020dpn,Aoki:2019rvi,Percacci:2019hxn,Delhom:2021bvq,Jimenez-Cano:2022sds}. One of the ghost-free sub-classes are projective-invariant Ricci-Based Gravity (RBG) theories  where the Lagrangian is a generic function of the metric and the symmetrized Ricci tensor of the affine connection (see e.g. \cite{BeltranJimenez:2017doy,Delhom:2021bvq,Olmo:2022rhf}. Indeed, these theories have been shown to be dynamically equivalent to GR coupled to a non-linearly modified matter sector, so that they do not propagate extra degrees of freedom and naturally generate bouncing cosmologies, wormholes, and other exotic compact objects \cite{Afonso:2018bpv,Afonso:2018hyj,Afonso:2018mxn,Delhom:2019zrb,Delhom:2021bvq}. The simplest type of action that we can find within this sub-class is the metric-affine $f(\ensuremath\mathcal{R})$ family of theories, which we already mentioned in relation with the background evolution of LQC \cite{Olmo:2008nf}. Beyond the $f(\ensuremath\mathcal{R})$ family and still ghost-free, one finds the RBG family, which includes quadratic gravity \cite{Olmo:2009xy,Olmo:2012nx} and the EiBI model \cite{BeltranJimenez:2017doy}, among others (see, for instance, \cite{Afonso:2021aho,Boudet:2022wmb,Olmo:2022ops}).

In this work, we will assess the question of whether the effective continuum description of the background evolution provided by mLQC-I and mLQC-II can also be derived from a covariant effective action within the $f(\ensuremath\mathcal{R})$ class. We will show that there exists a common family of metric-affine $f(\ensuremath\mathcal{R})$ actions that faithfully describe the background evolution predicted by LQC and mLQC-II, as well as the post-bounce phase of mLQC-I, and which reduce to the one found in \cite{Olmo:2008nf} for a particular choice of parameters. Interestingly, we find that the non-perturvative features of the evolution of the energy density with respect to the curvature are universally captured by a generic logarithmic correction $\ln(\ensuremath\mathcal{R}/\ensuremath\mathcal{R}_\mathrm{bounce})$ in the effective Lagrangian. Furthermore, once the general form of the family of effective $f(\ensuremath\mathcal{R})$ theories is specified, we will be able to write the values of the parameters leading to each of the considered loop cosmologies in terms of fundamental constants of the underlying LQC theory, {concretely the Barbero-Immirzi parameter $\gamma$ and the critical density at which the bounce occurs. In standard LQC this bounce density is given by $ \rho_\mathrm{b} = \rho_\textrm{c}= 3/8 \pi G \gamma^2 \Delta$ where $\Delta = 4 \sqrt{3} \pi \gamma \ell_{\mathrm{Pl}}^2$ is the minimum area gap in quantum geometry. In mLQC-I and mLQC-II the bounce density is related to $\rho_c$ as discussed later.}Let us emphasize that, as explained above, this family of theories does not propagate extra degrees of freedom, is free from ghostly instabilities, and its dynamics can be mapped into GR coupled to a nonlinear matter sector.

Other attempts to derive the effective dynamics of loop cosmologies already exist in the literature \cite{Ribeiro:2021gds,Miranda:2021oig}. However, they relay on an {\it order reduction} method that basically boils down to a perturbative approach designed to ignore the extra dynamical degrees of freedom of the considered theories. Such approaches, which somehow try to capture intrinsically nonperturbative phenomena within a perturbative approach, do not guarantee energy conservation (they break the Bianchi identities associated to diffeomorphism invariance), and are not robust under small perturbations, which excite the artificially frozen degrees of freedom. On the contrary, metric-affine $f(R)$ theories (and, more generally, RBGs) satisfy by construction the Bianchi identities, have no new dynamical degrees of freedom which may be excited by small perturbations, and generically exhibit nonperturbative effects \cite{Olmo:2022rhf}.

The structure of the paper is the following. In section \ref{sec:BackgroundEvolution}, we will briefly review the main features of the cosmological background evolution for the different loop cosmology models and for metric-affine $f(\ensuremath\mathcal{R})$ theories, showing how the conditions for which both are equivalent at the physical level lead to a non-linear ODE for the metric-affine $f(\ensuremath\mathcal{R})$ Lagrangian. {{We will provide this description in cosmic time.}} In section \ref{sec:NumSol}, we will discuss the sensitivity of the solutions to small changes of at low curvatures, and physically meaningful numerical solutions for this ODE will be obtained. In section \ref{sec:Analytics}, we will find a family of metric-affine $f(\ensuremath\mathcal{R})$ Lagrangians that approximate the numerical solutions to the three ODEs corresponding to LQC, mLQC-I and mLQC-II to a high degree of accuracy, reducing to the known one for LQC found in \cite{Olmo:2008nf}. We will finish with a discussion on the results and future directions to be pursued.

\section{Background evolution equivalence}\label{sec:BackgroundEvolution}
In this section we will establish the conditions under which the background evolution dictated by the effective theory is physically equivalent to the one described by the corresponding loop cosmological model by focusing in the relevant physical observables. To that end, we will first describe background evolution in both, the loop cosmology and the explicitly covariant pictures, and then discuss the conditions for their equivalence.

\subsection{Background evolution in loop cosmologies}\label{sec:BckEvolLQC}

The background evolution of loop cosmology models can be expressed in terms of a modified Friedmann equation which modifies the GR evolution at energy densities close to the bounce, while quickly recovering GR away from the bounce scale. This modified Friedmann dynamics is obtained by quantizing the only gravitational degree of freedom that is left after a symmetry reduction that enforces homogeneity and isotropy in the phase space, namely the scale factor. This quantisation is carried out {\it \`a la Loop}, namely by rewriting GR in terms of Ashtekar-Barbero variables and using Dirac's method to quantise constrained theories. In isotropic and homogeneous backgrounds, of all the constraints in GR, only the Hamiltonian one is relevant, and it can be written as
\begin{equation}
\mathcal{H}_\mathrm{grav}=\mathcal{H}^{(E)}_\mathrm{grav}-(1+\gamma^2)\mathcal{H}^{(L)}_\mathrm{grav}\,,
\end{equation}
where $\mathcal{H}^{(E)}_\mathrm{grav}$ and $\mathcal{H}^{(L)}_\mathrm{grav}$ are the Euclidean and Lorentzian parts respectively  \cite{Yang:2009fp}, and $\gamma$ is the Barbero-Immirzi parameter, whose value can be fixed by black hole entropy computations in LQG to be $\gamma\approx0.2375$. {{As already mentioned, the quantization process has some ambiguities related to several possible treatments that can be given to the Lorentzian part of the constraint before quantising. These ambiguities lead to different effective Hamiltonians to describe cosmological evolution, leading to different Loop Cosmology models described by their respective modified Friedmann equations. In this paper, we consider two models that stem from these quantization ambiguities concerning how one treats the Euclidean and Lorentzian terms, which results in  mLQC-I and mLQC-II. Detailed investigations of their properties show that though the mLQC-II background evolution is consistent with a symmetric quantum bounce as in standard LQC, this is not the case for mLQC-I, which has an asymmetric bounce for a consistent evolution \cite{Li:2018opr, Li:2021mop}. A striking difference between LQC, mLQC-II and mLQC-I is that the latter unlike the other two quantizations does severely forbids a cyclic evolution in cosmic time \cite{Li:2021fmu}. }}

While the Friedmann equation for LQC with a scalar field matter sector reads
\begin{equation}
3H_{\rm LQC}^2=\kappa\rho\lr{1-\frac{\rho}{\rho_\textrm{b}}},
\label{eq:LQCFriedmann}
\end{equation}
where $\rho_\textrm{b} = \rho_\textrm{c}$ is the value of the energy density at the bounce, the modified loop cosmology models are described respectively by the following modified Friedmann equations
\begin{equation}
\resizebox{.9\columnwidth}{!}{$3H^2_{\rm II}=\kappa \rho\left(1+\gamma^{2} \frac{\rho}{\rho_{c}}\right)\left(1-\frac{\left(\gamma^{2}+1\right) \rho / \rho_{c}}{\left(1+\sqrt{\gamma^{2} \rho / \rho_{c}+1}\right)^{2}}\right)$}
\label{eq:mLQC-IIFriedmann}
\end{equation}
for mLQC-II evolution \cite{Li:2018fco}, and
{{
\begin{equation}
\resizebox{.9\columnwidth}{!}{$3H^2_{\rm I+}=\kappa \alpha \rho_{\Lambda}\left(1-\frac{\rho}{\rho_\textrm{b}^{\rm{I}}}\right)\left[1+\frac{1-2\gamma^2+\sqrt{1-\rho/\rho_\textrm{b}^{\rm{I}}}}{4\gamma^2(1+\sqrt {1-\rho/\rho_\textrm{b}^{\rm{I}}})}\frac{\rho}{\rho_\textrm{b}^{\rm{I}}}\right],$}
\label{eq:mLQC-IpreFriedmann}
\end{equation}

\begin{equation}
\resizebox{.9\columnwidth}{!}{$3H^2_{\rm I-}=\kappa{{\rho}}\left(1-\frac{\rho}{\rho_\textrm{b}^{\rm I}}\right)\left[1+\frac{\gamma^2}{1+\gamma^2}\left(\frac{\sqrt{\rho/\rho_\textrm{b}^{\rm I}}}{1+\sqrt{1-\rho/\rho_\textrm{b}^{\rm I}}}\right)^2\right],$}
\label{eq:mLQC-IFriedmann}
\end{equation}
}}
for pre-bounce and post-bounce phases of mLQC-I evolution respectively. Here $\rho_\textrm{b}^\mathrm{I}=\rho_\textrm{c}/(4(1+\gamma^2))$ is the mLQC-I bounce density, and we can define analogously $\rho_\textrm{b}^\mathrm{II}=4(1+\gamma^2)\rho_\textrm{c}$ which gives the mLQC-II bounce density.

\subsection{Background evolution in $f(\ensuremath\mathcal{R})$}

In the Metric-affine formalism, the fundamental fields that describe the gravitational interaction are the metric and the affine connection, which is {\it a priori} independent from the metric and has dynamics of its own dictated by the action of the theory. Metric-affine (or Palatini) $f(\ensuremath\mathcal{R})$ theories are defined by a Lagrangian of the form
\begin{equation}
\mathcal{S}=\frac{1}{2\kappa}\int d^4x \sqrt{-g}f(\ensuremath\mathcal{R})+S_m
\end{equation}
where $\ensuremath\mathcal{R}$ is the Ricci scalar associated to the Ricci tensor of the independent affine connection $\Gamma$ contracted with the metric $g^{\mu\nu}$, and $S_m$ is the matter action. As is well known, metric-affine $f(\ensuremath\mathcal{R})$ theories do not propagate extra degrees of freedom and are free of ghosts \cite{Olmo:2011uz}, which can now be easily understood by noting that they lay inside the projective-invariant subclass of Ricci-Based gravity theories \cite{Olmo:2022rhf}, whose projective symmetry protects them from propagating pathological degrees of freedom \cite{BeltranJimenez:2019acz,Jimenez:2020dpn}. For a matter sector with an equation of state with $\omega=1$ (e.g. a massless scalar field), this family of theories predicts a modified Friedmann equation of the form
\begin{equation}
3H_{ f(\ensuremath\mathcal{R})}^{2}=\frac{f_{\ensuremath\mathcal{R}}\left[2 \kappa^{2} \rho+\mathcal{R} f_{\ensuremath\mathcal{R}}-f\right]}{2\left(f_{\ensuremath\mathcal{R}}+\frac{f_{\ensuremath\mathcal{R} \ensuremath\mathcal{R}}}{2} \frac{\dot{\ensuremath\mathcal{R}}}{H}\right)^{2}}\,,\label{eq:fRFriedmann}
\end{equation}
where the following relations can be derived from the field equations of the theory (see {\it e.g. } \cite{Olmo:2011uz})
\begin{align}
&\rho=\frac{\ensuremath\mathcal{R} f_\ensuremath\mathcal{R}-2f}{2\kappa},\label{eq:DensityOfR}\\
&\frac{\dot{\ensuremath\mathcal{R}}}{H}=-12\kappa \rho/(\ensuremath\mathcal{R} f_{\ensuremath\mathcal{R}\cR}-f_{\ensuremath\mathcal{R}}) \ .
\end{align}
This allows to rewrite the right-hand side of \eqref{eq:fRFriedmann} as a function of $f(\ensuremath\mathcal{R})$ and its two derivatives $f_\ensuremath\mathcal{R}$ and $f_{\ensuremath\mathcal{R}\cR}$ as
\begin{equation}
3H_{ f(\ensuremath\mathcal{R})}^{2}=\frac{f_{\ensuremath\mathcal{R}}\lr{2\mathcal{R} f_{\ensuremath\mathcal{R}}-3f}}{2\left(f_{\ensuremath\mathcal{R}}-3f_{\ensuremath\mathcal{R} \ensuremath\mathcal{R}}\lr{\frac{\ensuremath\mathcal{R} f_\ensuremath\mathcal{R}-2f}{\ensuremath\mathcal{R} f_{\ensuremath\mathcal{R}\cR}-f_{\ensuremath\mathcal{R}}}}\right)^{2}}\,.\label{eq:fRFriedmannODE}
\end{equation}
To find bouncing cosmologies within the $f(\ensuremath\mathcal{R})$ subfamily of metric-affine theories we need to have either $f_\ensuremath\mathcal{R}=0$ or $2\ensuremath\mathcal{R} f_\ensuremath\mathcal{R}-3f=0$ at some value of the affine curvature $\ensuremath\mathcal{R}_\mathrm{bounce}$ and the quotient of \eqref{eq:fRFriedmannODE} being regular as $\ensuremath\mathcal{R}$ approaches $\ensuremath\mathcal{R}_\mathrm{bounce}$. We will proceed with the bounce given by the condition $f_\ensuremath\mathcal{R}=0$.

\subsection{Physical conditions for equivalence}\label{sec:PhysCondEquiv}

\begin{table*}
  \begin{center}
    \begin{tabular}{|c|c|c|c|}
    \hline
       & A & B & C\\
      \hline
      \textbf{LQC} & $\sqrt{2\left(\mathcal{R} f_{\ensuremath\mathcal{R}}-2 f\right)\left(2{R}_{\rm c}-\left(\mathcal{R} f_{\ensuremath\mathcal{R}}-2 f\right)\right)}$ & $2\sqrt{{R}_{\rm c} f_{\ensuremath\mathcal{R}}\left(2 \mathcal{R} f_{\ensuremath\mathcal{R}}-3 f\right)}$ & $1$\\
     \hline
     \textbf{mLQC-I$^-$}  & $\sqrt{2(\ensuremath\mathcal{R} f_{\ensuremath\mathcal{R}}-2f)\left(\frac{\mathcal{R}_\textrm{c}}{2(1+\gamma^2)}-\ensuremath\mathcal{R} f_{\ensuremath\mathcal{R}}+2f\right)}$ & $\sqrt{\frac{{R}_{\rm c}}{1+\gamma^2} f_{\ensuremath\mathcal{R}}\left(2 \mathcal{R} f_{\ensuremath\mathcal{R}}-3 f\right)}$ & $\sqrt{\frac{\mathcal{R}_\textrm{c}-(1+\gamma^2)(\ensuremath\mathcal{R} f_\ensuremath\mathcal{R}-2f)+\sqrt{\mathcal{R}_\textrm{c}^2-2(1+\gamma^2)\mathcal{R}_\textrm{c}(\ensuremath\mathcal{R} f_{\ensuremath\mathcal{R}}-2f)}}{(1+\gamma^2)\left(\mathcal{R}_\textrm{c}+2f-\ensuremath\mathcal{R} f_\ensuremath\mathcal{R}+\sqrt{\mathcal{R}_\textrm{c}^2-2(1+\gamma^2)\mathcal{R}_\textrm{c}(\ensuremath\mathcal{R} f_{\ensuremath\mathcal{R}}-2f)}\right)}}$\\
     \hline
     \textbf{mLQC-I$^+$}  & $\sqrt{\frac{4 \sqrt{3} \left(\frac{\mathcal{R}_\textrm{c}}{2 \left(\gamma ^2+1\right)}-R f_{\ensuremath\mathcal{R}}+2 f\right)}{\left(\gamma ^2+1\right) \gamma   \kappa ^2 \mathcal{R}_\textrm{c}}}$ & $\sqrt{\frac{{R}_{\rm c}}{1+\gamma^2} f_{\ensuremath\mathcal{R}}\left(2 \mathcal{R} f_{\ensuremath\mathcal{R}}-3 f\right)}$ & $\sqrt{\frac{\gamma^2\left(\sqrt{\frac{\mathcal{R}_\textrm{c}}{2(1+\gamma^2)}}+\sqrt{\frac{\mathcal{R}_\textrm{c}}{2(1+\gamma^2)}-\ensuremath\mathcal{R} f_{\ensuremath\mathcal{R}}+2f}
\right)}{2\gamma^2+1-\sqrt{\frac{\mathcal{R}_\textrm{c}}{2(1+\gamma^2)}-\ensuremath\mathcal{R} f_{\ensuremath\mathcal{R}}+2f}
}}$\\
     \hline
     \textbf{mLQC-II}  & $\sqrt{2\left(\mathcal{R} f_{\ensuremath\mathcal{R}}-2 f\right)\left(2{R}_{\rm c}+\gamma^2\left(\mathcal{R} f_{\ensuremath\mathcal{R}}-2 f\right)\right)}$ & $2\sqrt{{R}_{\rm c} f_{\ensuremath\mathcal{R}}\left(2 \mathcal{R} f_{\ensuremath\mathcal{R}}-3 f\right)}$ & $\sqrt{\frac{\gamma^2\left(1+\sqrt{\gamma^2\frac{\ensuremath\mathcal{R} f_\ensuremath\mathcal{R}-2f}{2\mathcal{R}_\textrm{c}}+1}\right)}{2\gamma^2+1-\sqrt{\gamma^2\frac{\ensuremath\mathcal{R} f_\ensuremath\mathcal{R}-2f}{2\mathcal{R}_\textrm{c}}+1}}}$\\
     \hline
    \end{tabular}
    \caption{\label{tab:EqCoef}
This table contains the values of the different factors that define the particular form of the general ODE which enforces equivalence between the background evolution of metric-affine $f(\ensuremath\mathcal{R})$ theories and the different loop cosmology models that we will deal with.}
  \end{center}
\end{table*}

The first condition for background evolution equivalence can be formulated as an equivalence of the Hubble rates of both pictures. On the explicitly covariant side, the Hubble rate is a function of the metric-affine $f(\ensuremath\mathcal{R})$ Lagrangian, and its first two derivatives, as seen in \eqref{eq:fRFriedmann}. On the other hand, in the loop cosmology side, the Hubble rate can be written in terms of the energy density for each of the models as in Eqs. \eqref{eq:LQCFriedmann} to \eqref{eq:mLQC-IFriedmann}. Using now the relations that exist in the $f(\ensuremath\mathcal{R})$ side between energy density, the Lagrangian, and its first derivative \eqref{eq:DensityOfR}; we can enforce the equivalence of Hubble rates by requiring that the $f(\ensuremath\mathcal{R})$ Lagrangian solves a 2nd-order nonlinear ODE. This equation will be different for each of the loop cosmology models here considered, but they can all be written in the general form
{
\begin{equation}\label{eq:GeneralODE}
f_{\ensuremath\mathcal{R}\cR}=f_\ensuremath\mathcal{R} \lr{\frac{B C-f_{\ensuremath\mathcal{R}}A}{B C \ensuremath\mathcal{R}+2A(\ensuremath\mathcal{R} f_{\ensuremath\mathcal{R}}-3f)}}
\end{equation}
where $A$, $B$ and $C$ are functions of $\ensuremath\mathcal{R}$, $f(\ensuremath\mathcal{R})$ and $f_\ensuremath\mathcal{R}$ which depend on the loop cosmology models under consideration. }We show the particular values of these coefficients for LQC, mLQC-II and the two asymmetric branches of mLQC-I in Table I.

Given a pair of boundary conditions for $f(\ensuremath\mathcal{R})$ and $f_\ensuremath\mathcal{R}$ at some value of $\ensuremath\mathcal{R}$, this equation yields a unique solution in the interval $[\ensuremath\mathcal{R}_\mathrm{bounce},0]$ provided that the right hand side of \eqref{eq:GeneralODE} is regular within the whole interval. The required boundary conditions can be found analytically by enforcing physical equivalence at the bounce in the values of the energy density and the acceleration of the scale factor for the two descriptions. { Under the constraint $f_\ensuremath\mathcal{R}=0$, enforced in order to have a bounce in the $f(\ensuremath\mathcal{R})$ description\footnote{There are other bouncing solutions with $f_\ensuremath\mathcal{R}\neq 0$, but these end up determining the values of $f_{\ensuremath\mathcal{R}\cR}$ and even higher derivatives at the bounce, so that they are not suited to yield initial conditions for the ODE in \eqref{eq:GeneralODE}.}, these conditions lead to a unique set of allowed boundary conditions: imposing equality of the Hubble function, the bounce density yields the value of the Lagrangian at the bounce, while enforcing equality in the acceleration of the scale factor yields the value of the affine curvature at the bounce, which has the value of $-12 \dot{H}_\textrm{bounce}$ for the three models}. The specific values for each of the models are given in  Table \ref{tab:BounceCond}. With these initial conditions we can, in principle, obtain a solution to the corresponding ODE, which would provide a particular Lagrangian of the $f(\ensuremath\mathcal{R})$ type for each of the loop cosmology models. Note that, according to the definitions on Table \ref{tab:EqCoef}, there is no obvious reason to expect that these Lagrangians should bear any relation among them, beyond the fact of laying within the metric-affine $f(\ensuremath\mathcal{R})$ subclass. However, as we will see below, their nonperturbative behavior can be parametrised by a subfamily of $f(\ensuremath\mathcal{R})$ functions where nonperturbative deviations from the Einstein-Hilbert Lagrangian arise via logarithmic corrections of the form $\ln(\ensuremath\mathcal{R}/\ensuremath\mathcal{R}_\mathrm{bounce})$.

\begin{table}
  \begin{center}
    \begin{tabular}{|c|c|c|c|}
    \hline
      &   \textbf{LQC} & \textbf{mLQC-I} & \textbf{mLQC-II}\\
      \hline
     $\mathcal{R}_\mathrm{b}$ & $-12\mathcal{R}_\textrm{c}$ & $-\frac{3(1+2\gamma^2)}{(1+\gamma^2)^2}\mathcal{R}_\textrm{c}$ & $-48(1+	\gamma^2)(1+2\gamma^2)\mathcal{R}_\textrm{c}$\\
     \hline
     $f(\mathcal{R}_\mathrm{b})$  &$-\mathcal{R}_\textrm{c}$ & $-\frac{1}{4(1+\gamma^2)}\mathcal{R}_\textrm{c}$ & $-4(1+\gamma^2)\mathcal{R}_\textrm{c}$\\
     \hline
    \end{tabular}
    \caption{
This table contains the values of the affine curvature and the effective Lagrangian at the Bounce conditions for the equivalence between the metric-affine $f(\ensuremath\mathcal{R})$ description and the effective dynamics of the respective loop cosmology models. They are found by enforcing equivalence of the bounce energy density and the acceleration of the scale factor.}
\label{tab:BounceCond}
  \end{center}
\end{table}

\section{Numerical solutions}\label{sec:NumSol}

Though an analytical solution of the ODEs that encode background evolution equivalence for the different models would be ideal, to the best of our knowledge, it is not possible to obtain it. Thus, we need to find a numerical solution that we can later approximate with an analytical ansatz. Even obtaining physically meaningful numerical solutions of these ODEs is not generally a straightforward task, due to the fact that the boundary conditions at the bounce, obtained in section \ref{sec:PhysCondEquiv} from requiring physical consistency of both descriptions at the bounce, are not appropriate for numerical analysis, the reason being that the right hand side of \eqref{eq:GeneralODE} is indeterminate for these initial conditions. Hence, we have to look for boundary conditions elsewhere within the interval $[\mathcal{R}_\textrm{bounce},0]$ and find the appropriate numerical solution by requiring that the bounce that it describes satisfies the physically consistent bounce conditions up to a certain degree of precision.

Given that, in order for the effective $f(\ensuremath\mathcal{R})$ Lagrangians to be physically meaningful, they must approximate GR at low curvatures, we can try to fix boundary conditions near $\ensuremath\mathcal{R}=0$, where we know that $f_\ensuremath\mathcal{R}$ should be close to unity. Given a value of $\ensuremath\mathcal{R}$ at which we can fix boundary conditions, $\mathcal{R}_\textrm{bounce}$, the values of $f(\ensuremath\mathcal{R})$ and $f_\ensuremath\mathcal{R}$ at $\mathcal{R}_\textrm{bounce}$ can be tuned so that the numerical solution reproduces the right conditions at the bounce up to the desired accuracy. In that sense, we note that, for any choice of $\mathcal{R}_\textrm{bounce}$ within the interval $[\mathcal{R}_\textrm{bounce},0]$, there is an upper limit to the boundary value of $f_\ensuremath\mathcal{R}$ above which the numerical solution diverges before reaching the bounce. Below this critical value, the ODEs are well behaved and the conditions for existence and uniqueness of the solution are satisfied across the relevant interval $[\mathcal{R}_\textrm{bounce},0]$. More interestingly, it appears that the physically consistent bounce conditions are best approximated for the numerical solution with a boundary value for $f_\ensuremath\mathcal{R}$ equal to this critical value. The robustness of this result has ben verified by giving boundary values close to the critical value, both above and below, and running an iterative Newton-Raphson method where the goal is to achieve $f_\ensuremath\mathcal{R}=0$ at the bounce. The iterations show how the boundary value of the derivative converges to the critical value as the number of iterations increase in both LQC and mLQC-II, as well as in mLQC-I$^-$. However, the solutions are extremely sensitive to a small change in the value of the derivative. A deviation in $f_\ensuremath\mathcal{R}$ on the seventh significant digit at $\mathcal{R}_\textrm{bounce}=10^{-25}$ yields variations in the energy density at the bounce of around the 10\% of its value. However, note that it is remarkable that there are solutions which satisfy the appropriate boundary conditions at the bounce to a desired degree of precision while at the same time reducing to GR at low curvatures. This could signal that the Einstein-Hilbert Lagrangian might be an attractor at $\ensuremath\mathcal{R}=0$ in solution space as, otherwise, solutions satisfying the boundary bounce conditions would not be, in general, close to GR near $\ensuremath\mathcal{R}=0$.

\begin{figure*}[htb!]
\includegraphics[width=0.44\textwidth]{lagI}\hspace{1cm}
\includegraphics[width=0.423\textwidth]{lagII}
\caption{Plots of the the effective Lagrangian given by the numerical solutions to \eqref{eq:GeneralODE} for mLQC-I and mLQC-II and its analytical approximations given by \eqref{GenericModel} with the parameters in Table \ref{tab:ParamVal} and eqs. \eqref{eq:betaLQC} and \eqref{eq:Relationbetas}. The values of the affine curvature and energy density are given in units of $\mathcal{R}_\textrm{c}$ and the bounce curvature for each model. We can see how the pre-bounce branch is described by an almost constant Lagrangian as corresponds to a quasi-de-Sitter phase.}
\label{fig:mLQCI}
\end{figure*}

The method of placing boundary conditions at low curvatures is appropriate for the post-bounce phases where, according to our experience, the gravity Lagrangian must resemble the Einstein-Hilbert action at low curvatures up to a high degree of precision. For LQC and mLQC-II, this procedure completely determines a covariant Lagrangian that describes the full cosmological evolution at the background level, due to the fact that the pre- and post-bounce branches are symmetric within these models. However, this is not true for mLQC-I, for which the pre-bounce branch differs from the post-bounce one in cosmic time.   Thus, though we have experimental data about how the low-curvature regime of the mLQC-I post-bounce evolution looks like, this is not the case for the pre-bounce branch, for which we do not have any available data constraining the behavior of the Lagrangian at low curvatures, even though this phase has been constrained by CMB observations \cite{Agullo:2018wbf,Li:2019qzr,Li:2020mfi,Gomar:2020orw}. Indeed, the pre-bounce branch is asymptotically de Sitter with an asymptotic curvature that is not far from the Planck scale.\footnote{ {{Such an asymmetric evolution is a direct manifestation of the form of the Hamiltonian constraint in mLQC-I, with a similar evidence seen also for a loop quantization of Kantowski-Sachs spacetime \cite{Dadhich:2015ora}. }}}
This makes it impossible to assess the low curvature behavior of the corresponding effective $f(\ensuremath\mathcal{R})$ Lagrangian for the pre-bounce phase, as we know that, once the curvature reaches the value of the effective cosmological constant, the curvature-dependent terms of the Lagrangian become irrelevant as compared to the effective cosmological constant, thus leading to a high degree of arbitrariness. Thus, we can only hope for unveiling the form of the Lagrangian from scales close to the effective cosmological constant to the bounce curvature.

Even so, we have been able to find numerical solutions for the pre-bounce phase of mLQC-I which match the value of the energy density, Lagrangian, and first derivative of the post-bounce phase. Both pre-bounce and post-bounce numerical solutions are plotted in figure \ref{fig:mLQCI}. We see that the solutions are almost constant, both for the Lagrangian and the energy density. However, the values of these numerical solutions do not correspond to the correct values of the asymptotically de Sitter phase, which are $f_{\mathrm{asymp.}}(\ensuremath\mathcal{R})= -0.297$ and $\rho_\Lambda=0.314\rho_\textrm{b}^\mathrm{I}$. Our investigations show that it is difficult to approach more precise values of the asymptotic energy density by changing the boundary conditions at the bounce, due to the sensitivity of the ODE to small changes in the boundary conditions.

\section{Analytical approximations}\label{sec:Analytics}

Following \cite{Olmo:2008nf}, we now approximate the numerical solutions that describe LQC, mLQC-II and the post-bounce phase of mLQC-I. Our aim is to see whether the quantum gravitational  non-perturbative behavior of the different models can be fitted into the same family of metric-affine Lagrangians. Hence, we proceed by generalizing the analytical approximation to LQC introduced in \cite{Olmo:2008nf} to a three parameter family of metric-affine $f(\ensuremath\mathcal{R})$ Lagrangians of the form
\begin{equation}\label{GenericModel}
f(\ensuremath\mathcal{R})=\xi \ensuremath\mathcal{R}\left[1+\frac{(\ensuremath\mathcal{R}-\alpha \mathcal{R}_\textrm{c})^2}{\beta \alpha^2 \mathcal{R}_\textrm{c}^2}-\frac{1}{2}\ln\left(\frac{\ensuremath\mathcal{R}^2}{\alpha^2 \mathcal{R}_\textrm{c}^2}\right)\right] .
\end{equation}
Here the nonperturbative effects near the bounce can be mostly accounted for by a logarithmic correction. The parameter  $\alpha$ determines the value of the affine curvature at the bounce in units of $\mathcal{R}_\textrm{c}$, and $\xi$ determines the value of the Lagrangian at the bounce. $\beta$ is a free parameter controlling the perturbative information of the effective action (namely its polynomial piece) that can be used to fit the numerical solution. In order for $f_\ensuremath\mathcal{R}$ to have only one root on the interval $[0,\mathcal{R}_\textrm{bounce}]$, it is necessary that $\beta\geq2$. By requiring that the bounce curvature and value for the Lagrangian are consistent with the respective models, the values of $\alpha$ and $\xi$ are completely determined for each case (see table \ref{tab:ParamVal}), leaving only $\beta$ to fit the numerical results. As we will see later, there is an unexpected relation between the values of $\beta$ obtained by doing independent fits for each of the models in terms of the Barbero-Immirzi parameter given by
\begin{equation}\label{eq:Relationbetas}
\beta_{\textrm{I}}=(1+\gamma)\beta_{\textrm{LQC}}\qquad\text{and}\qquad \beta_{\textrm{II}}=\frac{\beta_{\textrm{LQC}}}{(1+\gamma)}.
\end{equation}

\begin{figure*}[htb!]
\centering
\includegraphics[width=0.45\textwidth]{densLQC}\\
\includegraphics[width=0.45\textwidth]{densI}\hspace{1cm}
\includegraphics[width=0.45\textwidth]{densII}

\caption{Plot of the energy density in units of the corresponding bounce densities for the numerical solution to \eqref{eq:GeneralODE} and the analytical approximation of each model given by \eqref{GenericModel} with the parameters in Table \ref{tab:ParamVal} and eqs. \eqref{eq:betaLQC} and \eqref{eq:Relationbetas}. The top plot is LQC, the bottom left is mLQC-I and the bottom right is mLQC-II. order of the models is standard LQC, post-bounce branch of mLQC-I, and mLQC-II. Note that for LQC we have used a the value for $\beta$ in terms of fundamental LQC parameters  \eqref{eq:betaLQC} $(\approx3.85)$, which coincides with our best-fit value, instead of the one used in \cite{Olmo:2008nf} $(\approx 3.76)$, although both results are practically indistinguishable at the observable level. The mLQC-I analytic approximation captures the non-perturbative behavior of the energy density at high curvatures very accurately and, although some deviation can be found at intermediate scales, this can be overcome by adding fine tuned polynomial corrections at the $1\%$ level if desired. The pre-bounce branch is almost constant as corresponding to an asymptotically de-Sitter phase, although its value does not correspond to the expected one $\rho_\Lambda\approx 0.3\rho_\textrm{b}^\textrm{I}$. We can seee that the behavior of the energy density is accurately descirbed by the analytic approximation at all scales, particularly near the bounce, for the post-bounce phase of mLQC-I as well as for all regimes of LQC and mLQC-II.}
\label{fig:DensityPlot}
\end{figure*}

\begin{table}
  \begin{center}
    \begin{tabular}{|c|c|c|}
    \hline
      &  $\alpha$ & $\xi$ \\
       \hline
       \textbf{LQC} & $-12$ & $-\frac{1}{\alpha_\textrm{LQC}}$\\
        \hline
       \textbf{mLQC-I$^-$}  & $-\frac{3 (1+2\gamma^2)}{(1+\gamma^2)^2}$ & -$\frac{1}{4(1+\gamma^2)\alpha_{\textrm{I}}}$\\
       \hline
       \textbf{mLQC-II}  & $-48(1+2\gamma^2)(1+\gamma^2)$ & $-\frac{4(1+\gamma^2)}{\alpha_{\textrm{II}}}$\\
     \hline
    \end{tabular}
    \caption{This table contains the values of $\alpha$ and $\xi$ that have to be chosen in the general Lagrangian \eqref{GenericModel} so that it yields the correct values of the affine curvature and the effective Lagrangian at the bounce for each of the loop cosmology models. These conditions determine two of the three parameters of the family. The third parameter can be found by fiting the numerical solutions for each of the models, and related to fundamental constants from the underlying quantum dynamics by \eqref{eq:betaLQC} and \eqref{eq:Relationbetas}.}
    \label{tab:ParamVal}

 \end{center}
\end{table}

Also, note that the value of $\beta$ is (inversely) correlated to the bounce density of each model. The best fit values for $\beta$, obtained by minimising the area between the numerical solution and the analytical approximation, are given by
\begin{align}\label{betafits}
&\beta_\textrm{LQC}=3.84 \hspace{1cm} \beta_{\textrm{I}}^{-}=4.77 \hspace{1cm} \beta_{\textrm{II}}=3.10.
\end{align}
where $\beta_{\textrm{I}}^{-}$ refers to the post-bounce phase of mLQC-I. This values satisfy the relation \eqref{eq:Relationbetas} to the reported precision. It is also interesting to note that the above value for $\beta_{\textrm{LQC}}$ is also approximated by
\begin{equation}\label{eq:betaLQC}
\beta_{\textrm{LQC}}=3\pi \frac{\rho_\textrm{c}}{\rho_{\textrm{Pl}}}.
\end{equation}

Indeed, the best fits obtained with the $\beta$ for each model as a free parameter and those obtained by the above value for $\beta_{\textrm{LQC}}$ and  the relations \eqref{eq:Relationbetas} to determine all the $\beta$ parameters in $\beta_{\textrm{II}}$ in terms of fundamental constants of LQC lead to analytic approximations that are virtually indistinguishable from one another, see figure  \ref{fig:DensityPlot}. As we can see, the non-perturbative behavior is matched by the analytic approximations to a high degree of accuracy, finding deviations only at the percent level in mLQC-I$^-$ and -II. However, if one looks close to the mLQC-I bounce ($\ensuremath\mathcal{R}=0.999 \ensuremath\mathcal{R}_{\textrm{b}}$), it is possible to see that the energy density does not approach the bounce with the correct asymptotic behavior. This can be understood by noticing that while the non-perturbative behavior of the Lagrangian at the bounce is perfectly fit in LQC and mLQC-II, the second derivative of the effective Lagragian has a divergence at the bounce for the numerical solution in the mLQC-I case, which is not accounted for by the logarithmic corrections. By studying the asymptotic behavior of the numerical solution, it can be seen that this divergence corresponds to a square-root-like asymptotic behavior of the first derivative, so that the second derivative diverges when approaching the bounce as $(\alpha\mathcal{R}_\textrm{c}-R)^{-1/2}$. This can be corrected by modifying the above family with a square root term, so that the mLQC-I$^-$ branch is approximated by
\begin{equation}\label{GenericModelSquareRoot}
\begin{split}
f(\ensuremath\mathcal{R})=&\xi \ensuremath\mathcal{R}\left[1+\frac{(\ensuremath\mathcal{R}-\alpha \mathcal{R}_\textrm{c})^2}{\beta \alpha^2 \mathcal{R}_\textrm{c}^2}-\frac{1}{2}\ln\left(\frac{\ensuremath\mathcal{R}^2}{\alpha^2 \mathcal{R}_\textrm{c}^2}\right)\right]\\
&+\frac{\gamma^2}{4\pi^2}\sqrt{\frac{(\alpha \mathcal{R}_\textrm{c}-\ensuremath\mathcal{R})^3}{\alpha \mathcal{R}_\textrm{c}}}
\end{split}
\end{equation}
which perfectly matches the behavior of the numerical solution, including the divergenve at the bounce of its second derivative (see fig. \ref{fig:2der}). Although this correction seems to drive the effective metric-affine description of mLQC-I away from the same family of theories as LQC or mLQC-II, we note that this is not the case if one focuses only on the correspondence of the relevant physical observables near the bounce. In fact, the contribution of the square-root correction to the energy density of the effective metric-affine description is given by
\begin{equation}
\frac{{\gamma^2}}{16 \pi^2  \kappa} (\ensuremath\mathcal{R}-4 \alpha  {\mathcal{R}_\textrm{c}}) \sqrt{\frac{\alpha \mathcal{R}_\textrm{c}-\ensuremath\mathcal{R}}{\alpha\mathcal{R}_\textrm{c}}},
\end{equation}
which describes with slightly more accuracy the asymptotic behavior of $\rho$ as we approach the bounce but which vanishes at the bounce, although it introduces a negative contribution to the energy density at vanishing curvature given by $-\gamma^2\alpha\mathcal{R}_\textrm{c}/ 4\pi^2\kappa$. By taking the appropriate limits, one can also check that the contribution of the correction to the Hubble factor vanishes both, at the bounce, and at low curvatures, while the global fit is as good as the one without the square root correction. This shows that a logarithmic correction is universal in all three loop cosmology models in accounting for the non-perturbative behavior of the energy density as the bounce is approached.

\begin{figure}[h!]
\hspace{-0.5cm}\includegraphics[width=0.45\textwidth]{lag2derI.pdf}
\caption{Second derivative of the efective Lagrangian given by the numerical solutions to \eqref{eq:GeneralODE} for mLQC-I  and the analytical approximation given by \eqref{GenericModelSquareRoot} with the parameters in Table \ref{tab:ParamVal} and eqs. \eqref{eq:betaLQC} and \eqref{eq:Relationbetas}. The square-root term in \eqref{GenericModelSquareRoot} is necessary to account for the non-perturvative behavior of the second derivative in the Lagrangian, and it does not contribute to the energy at the bounce, so that the behavior of the energy density is still well described by the universal Logarithmic correction in \eqref{eq:GeneralODE}. The square-root term in \eqref{GenericModelSquareRoot} adds a constant term to the Lagrangian that causes the small deviations at lower curvatures seen in fig. \ref{fig:DensityPlot}. These deviations can be solved by adding a small polynomial correction to \eqref{GenericModelSquareRoot}.}
\label{fig:2der}
\end{figure}

We conclude that the family of theories given in \eqref{GenericModel} provides an accurate description of the full background evolution of LQC and mLQC-II, as well as the post-bounce phase of mLQC-I.  Let us, however, elaborate on the need of introducing the square-root correction in \eqref{GenericModelSquareRoot} to fit the behaviour of the second derivative of the Lagrangian with respect to the curvature for mLQC-I. A possible explanation for the physical need of this correction would be that, due to the fact that mLQC-I has an asymmetric bounce, then both branches have to be described by different effective $f(\ensuremath\mathcal{R})$ models, which opens the door to have a non-smooth Lagrangian at the bounce, even if this does not permeate all physical observables. Although this seems natural from the loop cosmology side, the reader should notice that the ODE for the mLQC-I post-bounce branch does not have any encoded information about the pre-bounce branch. Therefore, it is rather surprising that, precisely for this model but not for the symmetric ones, the smoothness of the Lagrangian is lost at the bounce. On the other hand, the fact that this lack of smoothness does not affect the relevant physical observables, in principle, allows for a physically well behaved transition from one branch to the other.

\section{Outlook}

In this work we have established a correspondence between the background evolution of different loop cosmology models and that of metric-affine $f(\ensuremath\mathcal{R})$ theories, expanding previous results and proving that the background dynamics of different loop cosmologies appears to be generically indistinguishable from that of some diffeomorphism invariant theory. Furthermore, we have shown that the evolution of the energy density with the curvature can be accurately described for the three models of loop cosmologies analyzed here by a single family of $f(\ensuremath\mathcal{R})$ theories.  This is so despite the fact that one of the loop cosmology models has an asymmetric bounce, while the other two have symmetric ones.

Structural differences in the effective Lagrangian of the symmetric and asymmetric bounce models only become relevant in the second derivative of the Lagrangian with respect to the affine curvature, which for the symmetric bounce models is well captured by the family \eqref{GenericModel}, but requires an extra term given in \eqref{GenericModelSquareRoot} for  mLQC-I in order to faithfully track the divergent behavior of $f_{\ensuremath\mathcal{R}\cR}$ as the bounce is approached. We ignore if  this divergence of $f_{\ensuremath\mathcal{R}\cR}$ in mLQC-I is just an accident or a generic signature of asymmetric models.
In any case, the lack of analyticity in $f_{\ensuremath\mathcal{R}\cR}$
has no effect on physical aspects such as the energy density or the acceleration of the expansion factor at the bounce, which are the key observables used to construct our numerical solutions starting with initial data close to GR at low curvatures. In fact, the square root correction of mLQC-I in \eqref{GenericModelSquareRoot} does not contribute to the energy density or the Hubble rate at the bounce, so that the non-perturbative behavior of the relevant physical observables is not affected by it. Furthermore, the near-bounce behavior of the energy density is universally well accounted for by a logarithmic correction as in \eqref{GenericModel}, as can be seen in fig. \ref{fig:DensityPlot}. A limitation of our analysis is that for the pre-bounce branch of mLQC-I we have been unable to recover the asymptotic value of the energy density in the emergent Planckian phase.

Finally, let us comment on the fact that the effective dynamics of the three loop cosmology models can be embedded into the same family of metric-affine $f(\ensuremath\mathcal{R})$ Lagrangians. This is an a priori unexpected result. In fact, the proposed family of metric-affine $f(\ensuremath\mathcal{R})$ theories does not encode any information about the common origin of the three models and the ambiguities that generate them so that, in principle, there was no guarantee that they could fit within a one-parameter family of Lagrangian theories. It is also remarkable the fact that the only free parameter of the family that can be used to fit the behavior of the energy density satisfies a simple relation between its best fit values for the three models given in \eqref{eq:Relationbetas}, and even more remarkable that the values of this parameter and these relations can be written in terms of fundamental constants that originate in the underlying quantum description. These numerical coincidences call for a closer scrutiny in order to understand whether they are mere chance or if there exists some underlying reason for this to occur.

In this sense, to understand the significance of these results concerning the universality of the three effective descriptions, one should assess the generality of the logarithmic term in nonsingular metric-affine cosmologies. It is important to clarify if it is a general feature of models with an upper bound on the energy density or if it is somehow characteristic of the quantum effects of loop cosmologies.   Another important task consists on exploring other scenarios with different symmetry-reduction patterns in phase space, such as black hole like space-times, where LQG-related quantum corrections can also lead to singularity resolution and upper bounds on the space-time curvature \cite{Ashtekar:2018lag}.

\section*{Aknowledgements}

We would like to thank Beatriz Elizaga-Navascu\'es for useful comments on the manuscript. AD is supported by NSF grants PHY-1903799, PHY-2206557 and Hearne Institute. PS is supported by NSF grant PHY-2110207. This work is also supported by the Spanish Grant PID2020-116567GB- C21 funded by MCIN/AEI/10.13039/501100011033, the project PROMETEO/2020/079 (Generalitat Valenciana), and by the European Union's Horizon 2020 research and innovation program under the H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740.

redpajama_set_name: RedPajamaArXiv


Five SquashSmarts team members held their heads high this week as they took center state at their Middle School graduation.
Middle School: McMichael - Class Salutatorian!
Fashion, with a Passion, for Kids!
The latest fashion craze emerged at the 2004 United States Junior Open: the SquashSmarts star bracelet.
Designed, developed and marketed by 16-year old Sandra Mumanachit of Narberth, PA, the hip, red, round bracelets are etched with the name of Philadelphia's first-and-only urban youth squash program, SquashSmarts..
The idea for the bracelets stemmed from the yellow silicon 'LiveStrong' Lance Armstrong version which has become a cult collectors' item in recent months. And like the Armstrong original, all proceeds from the SquashSmarts bracelets benefited a worthy cause – the kids from SquashSmarts!
Of the 400 bracelets, available for $3.00/piece, all sold out within a matter of hours. Sales were diligently handles by Sandra, her sister, Sarah, and the four SquashSmarts team members (Tempest Bowden, Tyreek Davis, Rachel Edmonds and Kareem Price) who qualified for the US Open. All made great effort to spread the newest fad.
For more information on the one-of-a-kind SquashSmarts star bracelet (available in small and large) please contact 215-895-4963 or email info@squashsmarts.org memo: star bracelets.
The key to doubles…partner selection! On December 9th, SquashSmarts paired with Cozen O'Connor, The Racquet Club of Philadelphia and Freeman's Auction House for the 2nd annual Gary Waite Doubles Hardball Squash Exhibition and Live Auction.
International Squash Doubles Association (ISDA) founder and #1 Gary Waite (Toronto, CAN) paired with local hero and RCOP pro Rob Whitehouse (East Falls, PA). The talented two-some ultimately defeated their challengers which featured 8-time US National Champion Morris Clothier (New York, NY) and #5 ISDA professional Josh McDonald (New York, NY) by a margin of three to one.
Between breaks in play, Jonathan Freeman of Freeman's Auction House (America's oldest operating auction house) called a shot-gun style live auction featuring fine items from the Philadelphia sporting and fashion worlds.
Before the exhibition, Mr. Waite took time out to teach and share his skills, knowledge and passion for doubles squash with the eager children from SquashSmarts.
Learn more about the SquashSmarts program, our model and our vision. Click here.
SquashSmarts inaugural Embrace Camp, America's first-and-only youth camp to combine intensive squash instruction with communityservice, was a huge success. For information or a brochure, please contact info@squashsmarts.org.
SquashSmarts is pleased to welcome David Kanthor as its new Squash Director. David grew up in Rochester, NY and attendedHaverford College where he played four years on its Men's Varsity Squash team. He then was awarded a Masters Degree in community Planning from the University of Baltimore, with his course of study focused on community advocacy. David will coordinate the community service programs to complement the Squash Director duties he will fill.
SquashSmarts is pleased to announce that Stephen Gregg has joined the organization as its new Executive Director and Julie Williams as its new Academic Director.
Gregg served the past four years as the Director of Junior Development for the United States Squash Racquets Association. "We are thrilled and fortunate to recruit a talent like Steve" remarked Vincent McGuinness, President of the SquashSmarts Board of Directors and a Senior Member at Cozen O'Connor law offices in Philadelphia.
SquashSmarts, currently based at Drexel University, is working with other local groups with plans to build an eight court facility in theHunting Park section of Philadelphia. Joining Gregg will be Williams, who was appointed the new Academic Director of the program.


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Flickreel recently had the pleasure of speaking with Alexandra Shipp, who stars as a young Storm in X-Men: Apocalypse. The 24-year-old actress told us all about playing an iconic superhero, why Storm is a force to be reckoned with, her life growing up in Arizona, and what the future holds for her…
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Watch along live from the BFI IMAX blue carpet at the X-Men: Apocalypse Global Fan Screening, as the cast are interviewed:
John Boyega on Joining the Star Wars Legacy
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Posted on December 9, 2021 December 14, 2021 by Ben Slowey
Natural Satellite are a chamber pop duo consisting of Karli and Jasen Reisdorf. Based in Menasha, their music blends elements of pop, folk, and jazz with Karli on cello and vocals while Jasen is on saxophone. Their most recent single "Best You Can" came out in October.
ME: How did you guys get involved with Wisconsin Music Ventures?
KARLI: Actually, we met Allison virtually during the shutdown. She reached out to us, and I think we were the first band to do her virtual brunch concert that she had going on. So it was unique circumstances to meet someone through Zoom and make a connection that way, but we did get to meet her in person over the summer and it was a very wonderful thing to have happen for us during the shutdown, connecting with her.
ME: Same; I also connected with her during the shutdown and it's great that the scene has someone like her to make things work in a sustainable and safe way.
ME: So you guys are based in Menasha, right?
KARLI: Correct.
ME: Tell me, what is there to do in Menasha? What's it known for?
KARLI: The paper mill (laughs). I honestly love Menasha and I think it's a hidden gem in the valley. The downtown is wonderful; I really think it's becoming something.
JASEN: Yeah, ever since Faith Technologies built a facility that's like their new main headquarters in downtown Menasha, I think that's helped bring some money in there. It's starting to grow and thrive; there's a Midwest Sun Splash music festival that gets put on right downtown. It was kind of dumpy when we first moved here but it's gotten better.
KARLI: It's been fun to see it grow and I think the people here are wonderful. I love when we're playing a show and we say that we're from Menasha and there's the one person that's like "yeah! Menasha!" (laughs).
ME: Right on, that's awesome. I've never been so I'm always curious what each little town has to offer. There's always something.
ME: So, winter is upon us. What is your favorite snow activity to do?
KARLI: We don't do anything (laughs).
JASEN: Drinking.
ME: Well, we do live in the beer-soaked tundra.
KARLI: That's true! I don't know; we really don't have any snow activities but now that we have a daughter, I'm hoping to do more of that this winter, like the sledding and snowman-making.
JASEN: And watching football.
KARLI: We're kind of lame (laughs).
ME: That's okay; that sounds like how most people spend their winter. Do you guys ever throw snowballs at each other?
JASEN: No, I'd throw my arm out.
KARLI: You better not throw a snowball at me!
ME: Yeah, that shit can hurt, like if it has ice in it. That's why you'd get in trouble at recess for that.
ME: Tell me about your most recent song "Best You Can."
KARLI: We decided that we were going to start recording ourselves because we need more music out there. We can play as good as we can but it doesn't matter if we don't have anything online for people to listen to, so that was the motivation. We had a show that didn't go very well and I was kind of upset about it, and then I was like "you know what? we tried our best and that's all you can do" so that's where those words came from. We recorded it here at home and it was exciting; it was like the first project we got to do here in our house together, and we're working on more music because of it now.
ME: Very nice. That was exactly what my last question was: what are you guys working on now? What are your goals for 2022?
JASEN: Well, hopefully we just continue to stack on each success that we have so that it can continue to blossom into more opportunities as we've seen things happen for us this year. I'm working with a group here in Appleton with some of my old college friends, and we're doing some original jazz stuff, so that's something I want to work on a little more too. And for us, producing more too.
KARLI: Yeah, it's just going to be a year of creating and jumping onto new opportunities. We're hoping to record a full-length album this year.
Natural Satellite (naturalsatellitemusic.com)
Posted in Spotlight of the WeekTagged Menasha music, Natural Satellite, Wisconsin pop, WMV Green Room member


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. Theological Review - Near East School Of Theology. [Beirut, Lebanon: Near East School Of Theology. . Print.
Theological review - Near East School of Theology.
[Beirut, Lebanon] Near East School of Theology.


Shingles, also known as herpes zoster, is an infection. It is the same varicella-zoster virus (VZV) that causes chickenpox . Inactive copies of the virus live in your nerves. If these viruses become reactivated, then you get shingles.
What are the risk factors for shingles?What are the symptoms of shingles?How is shingles diagnosed?What are the treatments for shingles?Are there screening tests for shingles?How can I reduce my risk of developing shingles?What questions should I ask my doctor?What is it like to live with shingles?Where can I get more information about shingles?
NINDS shingles information page. National Institute of Neurological Disorders and Stroke website. Available at: http://www.ninds.nih.gov/disorders/shingles/shingles.htm. Updated June 27, 2018. Accessed July 16, 2018.
Shingles (herpes zoster). Centers for Disease Control and Prevention website. Available at: http://www.cdc.gov/shingles/about/overview.html. Updated June 15, 2018. Accessed July 16, 2018.


Dance PORT Derida is a one-year educational and training program in the field of contemporary dance. It's developed by Derida Dance Company and financed by America for Bulgaria Foundation. Derida Dance Center is the place in Sofia, which offers free daily training to maintain good physical condition. How is that possible? In 2012, after a successful completion of the "Probiotic for Contemporary dance in Bulgaria" - a program aimed at professional dancers, Derida Dance Company and America for Bulgaria Foundation provided a new opportunity for free participation in the program of Dance PORT Derida. This initiative is aimed at dancers, without experience, but with creative potential. For one year they will have the opportunity to build a solid dance foundation and skills for future career in the field of contemporary dance and physical theater.
- To prepare psychophysics of participants for a future career in the field of contemporary dance and physical theater.
- To develop and enrich the culture of movement and athleticism of the participants, while keeping their creative individuality.
For participation in Module 1 of the program Dance PORT Derida can apply everybody who is willing to engage actively with contemporary dance. Module 1 does not require previous experience and education related to dance. There will be a pre-selection for the participants.
- Training: contemporary dance, classical ballet, yoga, pilates, stretching, etc.
- Interaction with video composition and stage lighting.
- Participation in open for public presentations.
- Composition of the movement.
- History of contemporary dance and physical theater.
- Talk-lectures with guest choreographers and dancers.
How will determine the level of participants during the training?
Throughout the period are organized presentations in which the results of the training are represented. Participants who do not show progress in the relevant stage of training will lose their right to attend the trainings/program.
What's next after Dance PORT Derida?
One of the main objectives of the program after successfully completed the training is to have a rich culture of the movement, creative personality and knowledge, which are a prerequisite for active creative engagement and future careers.
In the last stage of the program Dance PORT Derida all participants are included in a show, produced by the company. Successful graduates from Dance PORT Derida can join projects scheduled for next year.
If participants are not present to 75% of the classes, they lose the right to participate in the program.
What is the duration and timing?
Training program lasts 1 year.
The training program includes over 550 hours and is held Monday through Friday in the afternoon. Periodically there will be organized workshops and presentations with varying intensity and duration.
Systematic approach will be implemented. The goal is to develop the potential of movement and self-expression of each participant.
The classes intend to increase the strength, the endurance and the flexibility.
- To provide an opportunity to maintain a good physical condition.
- To encourage new contacts and future cooperation between program participants and guest choreographers.
- To improve the professional quality and competitiveness of the artists in the field.
Module 2 of Dance Port Derida is aimed at dancers who have completed the specialized schools, and artists made no less than five projects in the field of contemporary dance.
Training program lasts 1 year. All exercises and workshops will be held on weekdays, with varying intensity.


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Connected cars draw crowds in Barcelona, but the industry is focused on 5G V2X
Tesla X trades the S's sleek form for size and impact.
Posted by Jeremy CowanMarch 1, 2019
As one wag put it at last year's Mobile World Congress, "There sure are a load of mobile phones at this car show." While drivers focused on the cars in Barcelona this year, the industry has been diving into the details of vehicle autonomy. And, as with so much at MWC19, says Jeremy Cowan, 5th Generation (5G) mobile communications is at the heart of it.
During #MWC19 the 5G Automotive Association (5GAA) gathered experts from the automotive, technology and telecom industries to explain how the deployment of 'Cellular Vehicle-to-Everything' (C-V2X) communication technology on the path to 5G is shaping future mobility. LTE-V2X, the initial version of C-V2X, is now close to commercial launch. It will allow vehicles to communicate with each other and their surroundings and, together with 5G enhancements, will facilitate broad improvements in road safety.
LTE-V2X
C-V2X communication is the new, high-speed cellular communications platform that enables vehicles to communicate with one another, with roadside infrastructure, and other road users (such as pedestrians, cyclists, and motorcyclists). It uses either direct short-range communications or cellular networks. While C-V2X network-based solutions are already widely deployed, direct communication solutions will be commercially available this year. The C-V2X platform is intended to deliver safety, mobility, traffic efficiency, and environmental benefits. C-V2X is designed with an evolutionary path to 5G.
"By bringing together key actors of the mobility ecosystem – across the automotive, information and communications technology industries – 5GAA continuously strives to develop forward-looking solutions for intelligent transportation services. Both for today and tomorrow," says Thierry Klein, 5GAA vice chair and head of the Disruptive Innovation Program at Nokia Bell Labs. "These end-to-end integrated solutions bring enhanced safety, sustainability, and convenience to all road users. 5GAA is very excited to be pioneering the revolution towards a smarter and more connected mobility world," he adds.
Tesla speeds away from horrors of 2018
Model X is a step up from the S in interior quality.
Tesla may wish to forget much about 2018, with troubles ranging from production bottlenecks that some Wall Street analysts predicted would bring the company down, through dwindling cash reserves and ill-advised tweets, to a joint smoked too publicly by CEO, Elon Musk. But, with assistance from some astute car-making hires in the second half of the year, the company is putting 2018 behind it and now claims to be producing around 1,000 of the well-received Model 3s a week. It hopes to manage 7,000 a week before 2019 is out.
Tesla showed its new-to-Europe electric sports utility vehicle (SUV), the Model X on its stand in Barcelona. Forget what you know about Tesla's sizing from the Model S, this is a big vehicle with seating for seven. It also feels a step up in production quality. Yes, there's still a disappointing 'ting' when you close the door (particularly with the frameless windows down) that you won't find in comparable vehicles from other premium car makers like Audi, BMW and Mercedes, but the interior ambience from switchgear, dashboard, seating and the huge central screen are a notable improvement.
The company is not short of ambition. It is putting the finishing touches to its launch plans for Model Y (Tesla's Models are S, 3, X, Y – see what they did there?), a smaller SUV based on the same platform and many of the same parts as the 3. It also has plans for a truck (a semi, for US readers), a pick-up and a new roadster.
Mercedes' new electric crossover vehicle, the EQC, was also popular in Barcelona.
Mobile operators collaborate on solutions for car makers
Meanwhile, mobile network operators are getting their act together in readiness for 5G. AT&T and Vodafone Business are working together to accelerate Internet of Things (IoT) connectivity and innovation in the automotive industry. The Tier-1 operators will bring together their expertise to develop connected car solutions and experiences for customers that they say will be "superior and consistent" across their combined footprints in North America, Europe and Africa.
Vehicle makers face numerous challenges when deploying connected car solutions across countries. Each market has different requirements and regulations, and it can be complicated for automakers to work across multiple network operators and vendors. So, the goal is to simplify the deployment process, improve operations, deliver innovative solutions, and make the network certification process easier.
"This alliance with Vodafone Business is a natural extension of our existing relationship," says Chris Penrose, president, Internet of Things Solutions, AT&T. "We each have rich experience in connected vehicle technology. By working together, we can innovate faster and help our global customers bring connectivity, entertainment and telematics to more vehicles across our respective footprints."
This is echoed by Stefano Gastaut, IoT director, Vodafone Business, who says, "Our work with AT&T will benefit automotive manufacturers and their customers around the world as we simplify processes and provide a consistent experience to accelerate IoT adoption in this fast-moving market. As technology complexity increases, this is the right time to make technology adoption easier for the automotive industry to help them achieve their business outcomes."
The companies say they will prioritise projects to enhance safety, security and entertainment capabilities. The key areas of focus will be:
5G and autonomous vehicle technology
V2X capabilities (vehicle-to-everything)
In-vehicle entertainment
Connected car applications and services
Global service quality models
Connected car/ smart cities intersection.
The author is Jeremy Cowan, editorial director of IoT Now and VanillaPlus.
Comment on this article below or via Twitter: @IoTNow_OR @jcIoTnow
Automotive, Blogs, Blogs, Connected vehicles, Connectivity, Internet of Things/Cloud Services, IoT News, Telematics, Transport, Transport & Telematics
5G Automotive Association, 5GAA, AT&T, Audi, BMW, Elon Musk, Mercedes, Nokia Bell Labs, Tesla, Vodafone
Jeremy Cowan
5GAAAT&TAutomotiveConnected vehicles
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Audi and Huawei sign MoU for strategic cooperation on intelligent connected vehicles
5G is now, Huawei invite ICT stakeholders in Spain to explore 5G together
5GAA, Audi, Ford and Qualcomm demo C-V2X direct communications interoperability to improve road safety
Infineon joins the 5G Automotive Association to contribute to connected automated driving
Ficosa is first Spanish company to join 5G Automotive Association, aiming to boost connected mobility
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