Community Living BC fired CEO Rick Mowles Friday after months of controversy surrounding the government appointed organization.
The organization responsible for providing services for disabled adults has been criticized recently over its decision to move clients out of group homes and into private residences.
CLBC Board Chair Denise Turner is downplaying any connection between recent problems in the organization and the firing.
"There's never a good time to let a good person go," Turner said. "We considered it very carefully and made a decision for the direction of the organization. It has nothing to do with what's going on outside."
Moms on the Move, an advocate group for families of individuals with special needs, said the change of leadership won't make a difference. Co-administrator Dawn Steele said the "crisis in community living" is due to a lack of funding, not leadership.
"Unless the new CEO can turn paper into money or cure developmental disabilities, I don't see how that's going to really make a dent in the problem," Steele said.
Advocates for people with disabilities estimate it will take an immediate investment of $70 million to begin addressing the shortfall in services.
In September, B.C.'s former Minister of Social Development Harry Bloy announced an extra $6 million in funding for CLBC, an amount advocates called a slap in the face. Bloy was fired after the public protested the lack of government support.
In light of Mowles being let go, the BC NDP is calling for a full review of the organization.
"The government has found someone to blame for the crisis that they created. The people paying the price are adults with developmental disabilities and their families," BC NDP leader Adrian Dix said.
Mowles will be replaced by CLBC Vice President Doug Wollard.


text:
Die Marxmühle ist eine ehemalige, mit Wasser betriebene Mehrzweckmühle. Sie liegt nordwestlich des Ortsteils Katzenfurt der Gemeinde Ehringshausen im Lahn-Dill-Kreis und steht unter Denkmalschutz.

Geschichte
Die Mühle wurde erstmals 1657 als Schneidmühle erwähnt. Wann sie erbaut wurde, ist nicht bekannt. Sie wurde über ein oberschlächtiges Mühlrad betrieben, dem Wasser aus der Dill mittels eines Mühlgrabens zugeführt wurde und stand im Eigentum der Grafen zu Solms-Greifenstein bzw. später Solms-Braunfels. Die jeweiligen Müller waren Erbpächter. Um 1660 wurde an die Schneidmühle eine Mahlmühle angebaut. Später, um 1697, kam noch eine Papiermühle und 1719 eine Schlagmühle (Ölmühle) hinzu. Die verschiedenen Gewerbezweige wurden teilweise nebeneinander betrieben. Um 1744 ist dann noch von einer Walkmühle die Rede, die für die im benachbarten Ortsteil Daubhausen 1685 angesiedelten französischen Hugenotten arbeitete. Um 1850 übertrug Fürst Ferdinand zu Solms-Braunfels das Eigentum an dem gesamten Mühlenanwesen gegen Zahlung einer Ablösesumme auf den damaligen Erbpächter Johannes Leonard. Leonard verkaufte sie später an Johann Müller weiter, der sie dann 1883 an die Gebrüder Ludwig (Louis) Marx und August Marx veräußerte. Die Gebr. Marx rissen die Schneidmühle ab und bauten eine neue Mahlmühle, die sie von 1884 an betrieben. Der Mahlbetrieb wurde 1910 eingestellt und die Mühle zu einer Presshefe- und Spiritusfabrik umgebaut, die am 1. April 1911 ihren Betrieb aufnahm und bis 1921 bestand. Nach 1921 wurde eine weitere Lagerhalle errichtet und die Wasserkraft des Mühlgrabens von da an zur Stromerzeugung genutzt. Ab 1. Oktober 1925 stellte August Marx auf dem Anwesen Holzwolle her. Während des Zweiten Weltkrieges wurden in Teilen der Mühlengebäude Schweißarbeiten für die Fa. Haas & Sohn in Sinn ausgeführt. Nach dem Krieg betrieb ein Samuel Schuster hier bis Anfang der 1950er Jahre eine mechanische Weberei. Heute beherbergt das Mühlenanwesen einen landwirtschaftlichen Betrieb.

Beschreibung und Ausstattung
Die Gebäude des Mühlenanwesens sind um einen Hof gruppiert. Während der Erbpachtzeit der Mühlenfamilie Dorlas (1756–1850) entstanden die ältesten heute noch vorhandenen Gebäude. Es handelt sich um zwei aneinander gebaute traufständige Wohnhäuser auf ausgebauten Kellern. Die nach der Dillseite ausgerichteten freiliegenden Fachwerke sind datiert: MD 1787 (MD = Martin Dorlas) und HCDI 1815 (HCD = Heinrich Christian Dorlas). Das sich anschließende Speichergebäude wurde 1884 errichtet. Der teilweise verputzte Ziegelbau ist durch eine regelmäßige Anordnung rundbogiger Fenster charakterisiert. Das Speichergebäude ist mit dem benachbarten Lagergebäude baulich verbunden.
Im Jahr 1917 entstand ein Lagerhaus, ein Jahr später ein Brennereigebäude auf dem Gelände. Den baulichen Abschluss der Anlage bildet das 1915 erbaute Wohnhaus mit einem Mansarddach. Hinter dem Wohnhaus befinden sich die Anlagen der Presshefe- und Spiritusfabrik mit einem erhaltenen Schornstein.

Literatur
 Maria Wenzel: Kulturdenkmäler in Hessen. Lahn-Dill-Kreis II (Altkreis Wetzlar), In: Landesamt für Denkmalpflege Hessen (Hrsg.): Denkmaltopographie Bundesrepublik Deutschland, Konrad Theiss Verlag, Stuttgart 2003, ISBN 978-3-8062-1652-3
 Arno W. Fitzler: "Mühlen" in "650 Jahre Hof Edingen 1341 – 1991", Sinn 1991
 Rudolf W. Kopp: Häuser in Katzenfurt und ihre Geschichte, Heft 7

Einzelnachweise

Weblinks
 Kulturdenkmäler in Hessen: Die Marxmühle bei Katzenfurt

Bauwerk in Ehringshausen
Kulturdenkmal in Ehringshausen
Papiermühle
Wassermühle in Hessen
Mühle in Europa
Produzierendes Unternehmen (Lahn-Dill-Kreis)

redpajama_set_name: RedPajamaWikipedia


Trina Braxton is mourning the loss of her ex-husband Gabe Solis after the 43-year-old Braxton Family Values star lost his battle with cancer, TMZ reports.
Solis death comes as a shock to the rest of the world. Trina posted an update on social media earlier today.
With news that Gabe Solis, Trina Braxton's ex-husband passed away, HelloBeautiful put together a gallery of pictures of the former couple before their 2015 divorce and just Gabe living his life to the fullest. R.I.P Gabe.


text:
Frans I (Cognac, 12 september 1494 – Kasteel van Rambouillet, 31 maart 1547) werd in 1515 in Reims tot koning van Frankrijk gekroond en regeerde tot zijn dood in 1547. Frans I was de enige zoon van Karel van Angoulême en Louise van Savoye. Margaretha van Valois was zijn zuster. Hij behoorde tot de linie Valois-Angoulême van het Huis Valois.

Renaissance
Zijn voorgangers, Karel VIII en Lodewijk XII, hadden door de Italiaanse Oorlogen veel tijd doorgebracht in Italië en maakten daar de ontwikkeling van de renaissance mee. Ze hadden geen hoge dunk van deze nieuwe stroming, maar maakten wel de ontwikkeling van de Franse renaissance mogelijk.

Frans I wordt gezien als de eerste Franse vorst die zelf door de renaissance beïnvloed werd. Enkele van de leraren van de jonge Frans, vooral zijn leraren Latijn François Desmoulins de Rochefort en Christophe de Longueil, probeerden hem de nieuwe Italiaanse ideeën bij te brengen. Zijn moeder was geboeid door de kunst van de renaissance en bracht deze passie over op haar zoon. Men kan echter niet zeggen dat Frans een humanistische opleiding heeft gekregen, daarvoor waren te weinig van zijn leraren beïnvloed door de nieuwe ideeën, maar zijn opleiding vertoonde wel enkele humanistische aspecten. Toen Frans I in 1515 op de Franse troon kwam, stond hij op goede voet met vele humanisten. Tijdens zijn bewind kwam de culturele ontwikkeling van Frankrijk in een stroomversnelling. Aan het begin van zijn koningschap gaf hij opdracht tot de bouw van het kasteel van Chambord en betrok Leonardo da Vinci bij het ontwerp. Het ontwerp van de dubbele wenteltrap in dit paleis wordt toegeschreven aan Da Vinci, maar zeker is dat niet.

Frans I nam rond 1530 twee Italiaanse kunstenaars in dienst: Rosso Fiorentino en Francesco Primaticcio (1532). Zij werkten in het Kasteel van Fontainebleau aan de decoratie, de fresco's en de gipspleisterornamenten die de wandtapijten moesten vervangen. Hij bracht er vervolgens een deel van zijn kunstcollectie onder, inclusief de Mona Lisa en de Maagd op de rotsen (van Leonardo da Vinci). Tevergeefs verzocht hij het Vaticaan om een kopie van de Apollo van Belvedère en de Laocoöngroep. Rond 1540 kwamen Sebastiano Serlio en Benvenuto Cellini naar het kasteel en vervaardigden diverse standbeelden voor de tuin en op het dak.

Binnenlands beleid
Tijdens zijn bewind groeide de Franse centrale staat op allerlei vlakken. Krachtens het Edict van Villers-Cotterêts werd in 1539 het Frans de status van officiële bestuurstaal toegekend. Net als zijn voorgangers hield hij vast aan een politiek van interne versterking van zijn rijk. Hij slaagde er onder meer in Auvergne aan het centrale gezag te onderwerpen.

Op het gebied van onderwijs en wetenschappen gaf Frans I in 1539 de aanzet tot het Collège Royal, het latere Collège de France. Zo trad hij op als mecenas van verschillende kunstenaars en wetenschappers, met als bekendste Da Vinci. Hij stimuleerde ook de vertaling van klassieke werken in de Franse taal, onder andere door Guillaume Budé (1467-1540), op wiens suggestie de studie van het Grieks, Latijn en Hebreeuws bevorderd werd.

Zijn eigen macht kon Frans I aanzienlijk versterken dankzij het Concordaat van Bologna. Dit gaf de koning onder andere de macht om zelf kandidaten voor belangrijke functies binnen de katholieke kerk in Frankrijk te benoemen.

Buitenlands beleid
Op buitenlands gebied maakte hij onmiddellijk grote indruk door tijdens de Oorlog van de Liga van Kamerijk in 1515 de Slag bij Marignano voor de poorten van Milaan te winnen en op die manier de controle over Lombardije te verwerven. In 1521 kwam het opnieuw tot spanningen tussen Frankrijk en de Habsburgers om de heerschappij in Noord-Italië. Tijdens de Italiaanse Oorlog van 1521-1526 verloor hij in 1525 de Slag bij Pavia en werd hij gevangengenomen door Karel V, die intussen keizer van het Heilige Roomse Rijk was geworden. Hij was verplicht de Vrede van Madrid (1526) te tekenen.

Op aansturen van zijn moeder Louise de Savoye, regentes tijdens zijn gevangenschap, werd de Liga van Cognac (zijn geboorteplaats) opgericht, een verbond tegen Karel V. Zij ging zo ver dat ze ook een Franco-Ottomaanse alliantie aanging met Süleyman I om Frans vrij te krijgen. Eenmaal vrij viel hij Napels aan. De oorlog werd beëindigd met de Damesvrede van Kamerijk en er kwam voor zeven jaar een einde aan de Italiaanse Oorlogen tussen Frankrijk en de Habsburgers.

Tijdens deze luwe periode stuurde hij Jacques Cartier naar de andere kant van de Atlantische Oceaan om een graantje mee te pikken van de Nieuwe Wereld.

Na de dood van de laatste hertog Francesco II Sforza palmde Karel V het Hertogdom Milaan in, goed voor een nieuwe Italiaanse Oorlog (1535-1538). Deze keer kreeg Frans I steun van het Ottomaanse Rijk, waarbij vooral de inbreng van admiraal Barbarossa Khair ad-Din Pasha doorslaggevend was. Karel V was verplicht de Vrede van Nice te ondertekenen. Karel behield Milaan en Frans kon een groot deel van Piëmont inlijven.

Waren de relaties tussen Frans en Karel nog goed in 1540, toen Karel een bezoek bracht aan Parijs, dan sloegen die om na de moord op de Franse ambassadeur Antonio Rincon in 1541. Die kwam net terug van Constantinopel, waar hij had onderhandeld met het Ottomaanse hof in verband met de nakende oorlog met het Heilige Roomse Rijk, toen hij werd onderschept en vermoord door de keizerlijke troepen. Op 12 juli 1542 verklaarde Frans de oorlog, het begin van alweer een Italiaanse Oorlog (1542-1546). Geldproblemen en een tekort aan manschappen dwongen Frans naar de onderhandelingstafel. De oorlog had uiteindelijk niets opgeleverd.

Dood
Frans I leed sinds 1524 aan syfilis. Eind 1530 had hij last van een fistel, waardoor hij niet meer op een paard kon zitten. De fistel begon te verzweren, en op 31 maart 1547 stierf hij aan een bloedvergiftiging en een nierfalen ten gevolge van die aandoening.

Hij ligt begraven in de Kathedraal van Saint-Denis in Parijs.

Huwelijken en kinderen
Frans trouwde op 18 mei 1514 in Saint-Germain-en-Laye met Claude van Frankrijk (Reine Claude), de oudste dochter van zijn voorganger Lodewijk XII, die al in 1524 overleed. Hij trouwde voor de tweede keer in 1530 met Eleonora van Habsburg (1498-1558), een dochter van Filips de Schone en Johanna van Castilië en daarmee een zuster van keizer Karel.

Met Claude kreeg hij volgende kinderen:
 Louise (1515-1517)
 Charlotte (1516-1524)
 Frans (1518-1536)
 Hendrik II (1519-1559)
 Magdalena (1520-1537), getrouwd met Jacobus V van Schotland
 Karel (1522-1545)
 Margaretha (1523-1574), in 1559 getrouwd met Emanuel Filibert van Savoye.

Voorouders

Koning van Frankrijk
Hertog van Milaan
Huis Valois-Orléans-Angoulême
Persoon in de Italiaanse Oorlogen
Persoon in de 15e eeuw
Frankrijk, F
Hertog van Bretagne
Hertog van Valois


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text:
Buen Suceso
Public Building. Albacete. SPAIN. Built 2006
AWARDS: 1st Prize
The aim was to create an internal space to replace the street as a place of exchange. It was decided to demolish the central bay of the building in which stairs and rooms caused a feeling of utter chaos and made it extremely difficult to reconcile with the idea of space integration.
Replacing the central bay, a large core of communications is built which gives the complex an institutional presence and allows for a healthy coexistence between the different parts of the complex.
The second part of the proposal was to create a sense of unity between the elder and younger people in the building.
38°59'26.5″N 1°50'33.5″W
Recreational Center in Bignasco
Queen´s blocks
Vilnius Concert Hall
© 2023 Arquivio. All Rights Reserved. Muffin group

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}

Two kind of
quantum geometric corrections come  from the discrete nature of space-time assumed in Loop Quantum Cosmology (LQC): inverse volume corrections \cite{bojowald}
and holonomy corrections (see for instance \cite{ashtekar}). Dealing with the flat Friedmann-Lem\^itre-Robertson-Walker (FLRW) geometry, which is the case of our paper,
inverse volume corrections have
problems because of arbitrary re-scalings \cite{as11}, more precisely,
since the scale factor can be arbitrarily re-scaled in a flat metric, these
inverse volume correction could appear an any arbitrary scale loosing its physical meaning. Only in a closed universe they have sense,  leading to a bounce that avoids
the big bang and big crunch singularity \cite{toporensky}.
 On the other hand, holonomy corrections, which are well introduced for compact and no-compact geometries,  provide a big bounce that avoids singularities like the Big Bang and Big Rip
 (see for example \cite{singh}).

On the other side, it is well-known that, in general, f(R) gravity does not avoid singularities, except of particular non-singular cases where $R^2$ term plays an important role as it was demonstrated in
 \cite{bno08}. In order to avoid them, one could
introduce
holonomy corrections in $f(R)$ gravity. The extension of Loop Quantum Gravity (LQG) to $f(R)$ gravity   has  been recently developed in \cite{zm11,zm11a}, where
holonomy corrections  are introduced in Einstein frame (EF), because in that frame
 the gravitational
part of the Hamiltonian is linear in the scalar curvature and the matter part is given by a scalar field. It is important to recall that, the idea to introduce holonomy corrections
via the EF was performed in \cite{Risi} studying the gracefull exit problem in the pre-big bang scenario, i.e., studying the regularization of the singularity that divides the pre
and post big bang branches in pre-big bang models.

This extension simplifies very much
when one consider the flat FLRW geometry. In that case,
in order
to take into account geometric effects, one has to replace the Ashtekar connection by a suitable sinus function (see for instance
\cite{h13a}) obtaining the holonomy corrected Friedmann equation in EF.
Finally,
from the holonomy corrected Friedmann equation in EF and through the relation between the corresponding variables in both frames, one obtains the holonomy corrected
$f(R)$ theory in the Jordan frame (JF).

Our main objective is to apply, for the flat FLRW geometry, holonomy correction to the  $f(R)=R+\alpha R^2$ model (also called $R^2$ gravity)
and study its dynamics. To do this, first of all we perform
 a detailed analysis of $R^2$ gravity without corrections. When
holonomy corrections in the model are taken into account one  obtains a very  complicated dynamical equation in the JF. Fortunately, dynamical equations simplify very much in
EF, (in fact the dynamics is given by  the well-known holonomy corrected Friedmann equation in LQC
plus the Klein-Gordon equation in flat FLRW geometry)
which allows us to perform a very deep analytical an numerical analysis, whose results can be translated to the JF.
Our conclusion is that when holonomy corrections are taken into account the universe starts at the critical point $(H=0,\dot{H}=0)$ (the Hubble parameter and its derivative  vanish)
and makes small oscillations around the critical point before entering the contracting phase, which it leaves bouncing
(see  \cite{brandenberger} for a review of bounce cosmology), and enters the expanding phase where, as in the
classical model, it
reaches an inflationary stage which it leaves
at late times and comes back to  the critical point, once again,  in an oscillating way.

The paper is organized as follows:

In Section II, we review $f(R)$ gravity in Jordan and Einstein frames.
In Section III, we introduce holonomy corrections to $f(R)$ gravity. The idea is very simple: working in EF,  $f(R)$ gravity is formulated as Einstein gravity plus and scalar field.
Then, the idea, as in standard LQC for the flat FLRW geometry, is to replace the Ashtekar connection by a suitable sinus function.
Section IV is devoted to the study of $R^2$ gravity without holonomy corrections. After performing the change of variable $p^2=H$ where $H$ is
the Hubble parameter,
the obtained dynamical equation  can be understood as the dynamics of a particle under the action of a quadratic potential with dissipation. This system is very simple and the phase portrait
can be drawn with all the details.
In Section V, we analyze the model with holonomy corrections. We start working in EF due to the simplicity of equations and, once we have studied the dynamics in EF,
we  obtain the dynamics in JF
form the formulae that relate both frames.
Moreover, we obtain in EF the corrected expressions of the slow-roll parameters and the values of the spectral index for scalar perturbations and the ratio of tensor to scalar
perturbations, showing that holonomy corrections help to match correctly the theoretical results obtained from $R^2$ gravity with current observations.
Section VI is devoted to discuss a possible unification of inflation and current cosmic acceleration in  the framework of Loop Quantum $f(R)$ theories. We will show that
 when one consider the current suggested models for such unification
this extension and/or its analytical study is, in general, unworkable. The only model we have been able to deal with is $R^2$ plus an small cosmological constant. For such a model, we have
performed a detailed analytical
study and the results are shown at the end of the work.
\section{Classical dynamical equations in different frames}

In this Section we review the relations between Jordan and Einstein frames in $f(R)$ gravity for the flat FLRW geometry.

 The  Lagrangian in JF for the flat FLRW geometry is given by ${\mathcal L}_{JF}=\frac{a^3}{2}f(R)$, where the scalar curvature is
 $R=6\dot{H}+12H^2$ being $H=\frac{\dot{a}}{a}$  the Hubble parameter, and
the corresponding modified Friedmann equation in $f(R)$ gravity can be obtained form Ostrogradskii's construction  \cite{h13a}  giving as a result
\begin{eqnarray}\label{1}
 6f_{RR}(R)\dot{R}H+(6H^2-R)f_R(R)+f(R)=0,
\end{eqnarray}
 where $f_R(R)\equiv \frac{\partial f(R)}{\partial R}$.
Taking the derivative of equation (\ref{1}) with respect to time and using the relation $R=6(\dot{H}+2H^2)$ one obtains the equivalent equation
 \begin{eqnarray}\label{1a}
 f_{RR}(R)(\ddot{R}-\dot{R}H)+f_{RRR}(R)\dot{R}^2+2f_R(R)\left(\frac{R}{2}-2H^2\right)=0.
\end{eqnarray}

To work in the Einstein frame (EF), one has to perform the change of variables \cite{dsy}
\begin{eqnarray}\label{2}
\tilde{a}=\sqrt{f_R(R)}a; \quad d\tilde{t}=\sqrt{f_R(R)}dt.
\end{eqnarray}

Then, in that frame the Lagrangian density, for flat FLRW geometries, is
\begin{eqnarray}\label{3}
{\mathcal L}_{EF}=\tilde{a}^3\left(\frac{1}{2}\tilde{R}+\frac{1}{2}(\tilde{\phi}')^2-V(\tilde{\phi})\right)
\Longleftrightarrow {\mathcal L}_{EF}=-3(\tilde{a}')^2\tilde{a}+\tilde{a}^3\left(\frac{1}{2}(\tilde{\phi}')^2-V(\tilde{\phi})\right),
\end{eqnarray}
where $'$ means the derivative with respect the time $\tilde{t}$. Here,
$\tilde{a}$ and $\tilde{\phi}$ have to be considered as independent variables, and of course, $\tilde{R}=6{\tilde{H}}'+12\tilde{H}^2$.

The relation between both frames is given through the relations
\begin{eqnarray}\label{4}
 \tilde{\phi}=\sqrt{\frac{3}{2}}\ln(f_R(R));\quad V(\tilde{\phi})=\frac{Rf_R(R)-f(R)}{2f_R^2(R)},
\end{eqnarray}
and
a simple calculation shows that the Friedmann equation in the EF, i.e. $\tilde{H}^2=\frac{1}{3}\tilde{\rho}$,
  obtained from the Hamiltonian constrain
\begin{eqnarray}\label{ham}{\mathcal H}_{EF}\equiv\tilde{a}'\frac{\partial {\mathcal L}_{EF}}{\partial\tilde{a}'}+
\tilde{\phi}'\frac{\partial {\mathcal L}_{EF}}{\partial\tilde{\phi}'}-{\mathcal L}_{EF}=
-3(\tilde{a}')^2\tilde{a}+\tilde{a}^3\left(\frac{1}{2}(\tilde{\phi}')^2+V(\tilde{\phi})\right)=0,
\end{eqnarray} where
$\tilde{\rho}\equiv
\frac{1}{2}(\tilde{\phi}')^2+V(\tilde{\phi})$,
is equivalent to equation (\ref{1}).
However, the Friedmann equation in EF, $\tilde{H}^2=\frac{1}{3}\tilde{\rho}$, is a constrain instead of a dynamical equation. The dynamics  is given by  the conservation
equation $\tilde{\rho}'=-3\tilde{H}(\tilde{\phi}')^2$ or the Raychauduri one $\tilde{H}'=-\frac{1}{2}(\tilde{\phi}')^2$ which are equivalent  to equation (\ref{1a}).

Note that combining, in EF, the conservation and Friedmann equation one obtains
\begin{eqnarray}
 (\tilde{\rho}')^2=3\tilde{\rho}(\tilde{\phi}')^2,
\end{eqnarray}
and coming back to the JF this equation is a second order differential equation in $R$ (it only contains $R$, $\dot{R}$ and $\ddot{R}$) which is equivalent to equations
(\ref{1}) and (\ref{1a}).

Finally, we show
the following relations between both frames, which will be important when we extend LQC to $R^2$ gravity:
\begin{eqnarray}\label{rel}
 {H}=\sqrt{f_R(R)}\left(\tilde{H}-\frac{1}{\sqrt{6}}\tilde{\phi}' \right);\quad R=f_R(R)\left(\tilde{R}+(\tilde{\phi}')^2+
\sqrt{6}\frac{\partial V(\tilde{\phi})}{\partial\tilde{\phi}} \right).
\end{eqnarray}

\section{$f(R)$ Loop Quantum Cosmology}
The idea to extend Loop Quantum Cosmology to $f(R)$ theories  ($f(R)$ LQC) has been recently developed in \cite{zm11,zm11a}.  For a flat FLRW geometry the idea is very simple and
 goes as follows: Working in EF, in the same way as in standard LQC,
we can see that the classical variable $\tilde{\beta}\equiv\gamma\tilde{H}$, where $\gamma$ is the Barbero-Immirzi parameter,  and the
volume $\tilde{V}\equiv\tilde{a}^3$ are canonically conjugated variables with Poisson bracket $\{\tilde{\beta},\tilde{V}\}=\frac{\gamma}{2}$ \cite{singh09}.
Then, in order to take into account the discrete nature of the space, one has to choose  a Hilbert space
where quantum states was represented by  almost
periodic functions. However, in this space the variable $\tilde\beta$ does not correspond to a well-defined quantum operator in this space, and since it
appears in the Hamiltonian (\ref{ham}), because it could be written as
\begin{eqnarray}\label{ham1}{\mathcal H}_{EF}=
-3\frac{\tilde{\beta}^2}{\gamma^2}\tilde{V}+\tilde{V}\left(\frac{1}{2}(\tilde{\phi}')^2+V(\tilde{\phi})\right)=0,
\end{eqnarray}
in order to have a well-defined quantum theory one needs to use the general holonomy corrected Hamiltonian in Loop Quantum Gravity
(see for instance \cite{aps06, abl03}).

At effective level, this is equivalent
 to introduce the square root of the minimum eigenvalue of the area operator in LQG, namely
 $\lambda=\sqrt{\frac{\sqrt{3}}{2}\gamma}$,  and
 make the replacement (see \cite{he10,dmw09,bo08} for a detailed discussion about the justification of this replacement)
\begin{eqnarray}\label{rep}
 \tilde{\beta}\rightarrow \frac{\sin(\lambda\tilde{\beta})}{\lambda},
\end{eqnarray}
in the Hamiltonian (\ref{ham1}), while keeping on the Poisson bracket $\{\tilde{\beta},\tilde{V}\}=\frac{\gamma}{2}$.
Here is important to notice that, after the introduction of holonomy corrections, $\tilde{\beta}$ stops to be equal to $\gamma \tilde{H}$. This can be showed from
the Hamilton equation $\tilde{V}'=\{\tilde{V},{\mathcal H}_{EF,LQC}\}$
 (being ${\mathcal H}_{EF,LQC}$ the new Hamiltonian obtained from (\ref{ham1}) after the replacement (\ref{rep})), because this equation can be written as
 $\tilde{V}'=-\frac{\gamma}{2}\frac{\partial{\mathcal H}_{EF,LQC} }{\partial \tilde{\beta}}=3\frac{\sin \lambda\tilde{\beta}\cos \lambda\tilde{\beta}}{\lambda\gamma}$
which is equivalent to
\begin{eqnarray}
 \tilde{H}=\frac{\sin 2\lambda\tilde{\beta}}{2\lambda\gamma}\Longleftrightarrow \tilde{\beta}=\frac{1}{2\lambda}\arcsin 2\lambda\gamma \tilde{H}.
\end{eqnarray}

\begin{remark}
It is stated in \cite{corichi} that there are many different inequivalent loop quantization, i.e., different pairs of canonically
conjugated variables leading to inequivalent quantum realizations. Here we have used the so-called
new quantization of LQC (also known $\bar{\mu}$ quantization) \cite{aps06} based in the use of variables $(\tilde{\beta},\tilde{V})$,
which is the unique choice consistent with the physical requirements proposed in \cite{corichi}.
\end{remark}

\begin{remark}
 It is important to stress that the replacement (\ref{rep}) is only valid for spatially flat models which is our case. When the spatial curvature does not vanish holonomy corrections
 has to be introduced in a non-trivial way (see for instance \cite{Gupt})
\end{remark}

Finally, from the Hamilton equation $\tilde{V}'=\{\tilde{V},{\mathcal H}_{EF,LQC}\}$ and the Hamiltonian constrain
${\mathcal H}_{EF,LQC}=0$, one obtains the
corresponding holonomy corrected version of the classical Friedmann equation \cite{singh09}, that is,
\begin{eqnarray}\label{5}
\tilde{H}^2=\frac{1}{3}\tilde{\rho}\left(1-\frac{\tilde{\rho}}{\tilde{\rho}_c}\right),
\end{eqnarray}
where $\tilde{\rho}_c\equiv \frac{3}{\lambda^2\gamma^2}$ is the so-called {\it critical density} in the EF.

As has been discussed in detail in \cite{aho} this equation depicts an ellipse in the plane $(\tilde{H},\tilde{\rho})$, and the
dynamics along this curve is very simple: For a non-phantom field the universe moves  clockwise from the contracting to the expanding phase
starting and ending at the critical point  $(0,0)$ and
bouncing
only once at $(0,\tilde{\rho}_c)$.

Finally,
note that
in the JF, the holonomy corrected Friedmann equation acquires the complicated form
\begin{eqnarray}\label{fri}
  6f_{RR}(R)\dot{R}H+(6H^2-R)f_R(R)+f(R)=

-\frac{\left(\frac{3}{2}f_{RR}^2(R)\dot{R}^2+(Rf_R(R)-f(R))f_R(R)\right)^2}{2f_R^4(R)\tilde{\rho}_c}.
\end{eqnarray}

\section{$R^2$ gravity}
In this Section we study with all the details the classical  model $f(R)=R+\alpha R^2$, with $\alpha>0$.
This model contains a quadratic correction to the scalar curvature and is a modified version of the  Starobinsky model \cite{s80}, where the author considered quantum vacuum
effects due to  massless fields conformally coupled with gravity. Note that such (eternal) trace-anomaly driven inflation was proposed earlier in ref \cite{dc}.

For this model, the classical equation (\ref{1}) becomes
\begin{eqnarray}\label{sta}
 12\alpha H\dot{R}+6H^2+12\alpha RH^2-\alpha R^2=0\Longleftrightarrow H^2=-12\alpha\left(3\dot{H}H^2+H\ddot{H}-\frac{1}{2}\dot{H}^2\right),
\end{eqnarray}
which coincides, when the parameter $\beta$ vanishes, with the dynamical  equation studied in \cite{s80}
\begin{eqnarray}\label{sta1}
 H^2=-12\alpha\left(3\dot{H}H^2+H\ddot{H}-\frac{1}{2}\dot{H}^2\right)+\beta H^4, \quad \mbox{where}\quad \beta>0.
\end{eqnarray}

It is very simple to show that equation (\ref{sta}) leads to an inflationary epoch \cite{mms, no11}. Effectively, when the slow-roll initial condition
$|\dot{H}|\ll H^2$ is fulfilled, equation (\ref{sta}) becomes $\ddot{H}=-3H\dot{H}-\frac{H}{12\alpha}$, which has  the following particular solution
in the expanding phase ($H>0$)
\begin{eqnarray}
\dot{H}(t)=-\frac{1}{36\alpha}\Longrightarrow
 H(t)=\frac{t_1-t}{36\alpha}\Longrightarrow a(t)=a(t_1)e^{-18\alpha H^2(t)}\quad \mbox{for}\quad t<t_1.
\end{eqnarray}

If  $t_i$ and  $t_f$ are the beginning and the end of inflation ($t_i<t_f<t_1$), then one will have
\begin{eqnarray}
 a(t_f)=a(t_i)e^{18\alpha (H^2(t_i)-H^2(t_f))}\cong a(t_i)e^{18\alpha H^2(t_i)},
\end{eqnarray}
and  the $60$ e-folds needed to solve the flatness and horizon problems  will be obtained when $\alpha H^2(t_i)$ is approximately $3.3$.

Unfortunately, $R^2$  gravity contains singularities at early times, that is, all solutions have divergent scalar curvature at early times. To show that, one has to perform the change
of variables $p^2(t)=H(t)>0$ \cite{aw} (in this model the universe doesn't bounce), then equation (\ref{sta}), which is not well-defined at singular value $H=0$, becomes
the following well-defined equation
\begin{eqnarray}\label{sis}
 \frac{d}{dt}\left(\frac{\dot{p}^2}{2}+W(p) \right)=-3p^2\dot{p}^2,
\end{eqnarray}
where $W(p)=\frac{p^2}{48\alpha}$.

We can see that the system (\ref{sis}) is dissipative. To understand its dynamics, we can imagine a "particle" rolling down along the parabola $W(p)$ losing energy
and oscillating, at late times, around $p=0$. As a
consequence, when time goes back the "particle" gains energy and finally $|p|\rightarrow \infty$ ($H\rightarrow \infty$), i.e., all the solutions
are singular at early times.
One also can check this fact as follows: We write equation (\ref{sis}) as
\begin{eqnarray}\label{sis1}
 \ddot{p}+\frac{p}{24\alpha}=-3p^2\dot{p},
\end{eqnarray}
and look for, at early times, solutions of the form $p(t)=\frac{C}{(t-\bar{t})^r}$, where $C$ and $r$ are parameters. Inserting this expression in
(\ref{sis1}) and retaining the leading terms when $t\gtrsim \bar{t}$, one obtains the equation:
\begin{eqnarray}
 \frac{r(r+1)C}{(t-\bar{t})^{r+2}}= \frac{3rC^3}{(t-\bar{t})^{3r+1}},
\end{eqnarray}
which has singular solutions at $t=\bar{t}$ of the form $p(t)=\sqrt{\frac{1}{2(t-\bar{t})}}$.

\begin{remark}
 In the contracting phase we can perform the change of variable $p^2(t)=-H(t)>0$, obtaining the system
 \begin{eqnarray}\label{sis0}
 \frac{d}{dt}\left(\frac{\dot{p}^2}{2}+W(p) \right)=3p^2\dot{p}^2,
\end{eqnarray}
where $W(p)=\frac{p^2}{48\alpha}$. We can see that the in the contracting phase the system is anti-dissipative (the universe gains energy), in this case the universe starts oscillating
around the bottom of the potential leaving it gradually and becomes singular at late times.
\end{remark}

Equation (\ref{sis1}) is also useful to obtain the inflationary period and the dynamics at late times. Effectively,
 when initially one has $\ddot{p}\cong 0$ equation (\ref{sis1}) becomes $p\dot{p}=-\frac{1}{72\alpha}$, whose inflationary solution is once again
\begin{eqnarray}\label{infla}
 H(t)=p^2(t)=\frac{t_1-t}{36\alpha}.
\end{eqnarray}

On the other hand, to obtain the dynamics at late times we follow the same method used in chaotic inflation for a quadratic potential (see page $240$ of \cite{mukh}).  Performing the
change of variable
\begin{eqnarray}\label{newvar}
\dot{p}(t)=\sqrt{2}f(t)\cos(\theta(t)),\quad {p}(t)=\sqrt{48\alpha}f(t)\sin(\theta(t)),
\end{eqnarray}
and inserting these expressions in equations (\ref{sis}) and (\ref{sis1}) one gets the system
\begin{eqnarray}\label{sis2}\left\{\begin{array}{ccc}
 \dot{f}&=&-18\alpha f^3(1-\cos(4\theta))\\
  \dot{\theta}&=& \frac{1}{\sqrt{24\alpha}}+144\alpha f^2\sin^3(\theta)\cos(\theta).              \end{array}\right.
\end{eqnarray}

Since $p$ goes to zero at late times, we can disregard the second term in the right hand side in the second equation of (\ref{sis2}), obtaining
$\dot{\theta}=\frac{1}{\sqrt{24\alpha}}$, whose solution is $\theta(t)=\frac{t}{\sqrt{24\alpha}}+\omega$, $\omega$  being a constant of integration. Inserting this approximate solution in the
first equation of (\ref{sis2}), we obtain a solvable equation whose solution is given by
\begin{eqnarray}
 f(t)=\sqrt{\frac{1}{36\alpha t\left(1-\frac{\sin\left(\frac{2t}{\sqrt{6\alpha}}+4\omega\right)}{\frac{2t}{\sqrt{6\alpha}}}\right)}}
 \cong
 \sqrt{\frac{1}{36\alpha t}\left(1+\frac{\sin\left(\frac{2t}{\sqrt{6\alpha}}+4\omega\right)}{\frac{2t}{\sqrt{6\alpha}}}\right)},
\end{eqnarray}
and thus the Hubble parameter reads
\begin{eqnarray}
 H(t)\cong\frac{4}{3t}
 \left(1+\frac{\sin\left(\frac{2t}{\sqrt{6\alpha}}+4\omega\right)}{\frac{2t}{\sqrt{6\alpha}}
 }\right)\sin^2\left(\frac{t}{\sqrt{24\alpha}}+\omega\right).
\end{eqnarray}

Now,
choosing $\omega=\pi/2$ one obtains the well-known result \cite{mms,w,wa}
\begin{eqnarray}
 H(t)\cong\frac{4}{3t}
 \left(1+\frac{\sin\left(\frac{2t}{\sqrt{6\alpha}}\right)}{\frac{2t}{\sqrt{6\alpha}}
 }\right)\cos^2\left(\frac{t}{\sqrt{24\alpha}}\right),
\end{eqnarray}
and after integrating by parts  one gets as Starobinsky in \cite{s80}
\begin{eqnarray}\label{H3}
 a(t)\cong t^{2/3}\left(1+\frac{2}{3}\frac{\sin\left(\frac{t}{\sqrt{6\alpha}}\right)}{\frac{t}{\sqrt{6\alpha}}}\right)\cong t^{2/3}.
\end{eqnarray}

These analytic results are supported numerically in figure $1$:

\vspace{2cm}

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.4]{H_Hpunt_p_starobinsky.png}
\includegraphics[scale=0.4]{H_Hpunt_p_starobinsky_alfa.png}
\end{center}
\caption{{\protect\small Phase portrait for $\alpha=0.1$. The universe comes from a  singularity at early times, when time goes forward it enters in the attractor
inflationary phase, leaving it at early times
when the universe
starts to oscillate around $(0,0)$ without bouncing. In the first figure we have taken values of $H$ up to $6$ to show clearly the inflationary stage, and the second one
the values of $H$ are up to $1$ to show, in more detail, the oscillatory phase. It's clear from the pictures that orbits are unbounded, coming form $\infty$ at early times.}}
\end{figure}

An important remark is in order:
Note that in the  model of \cite{s80} one obtains the same equation (\ref{sis}) but with the potential  $W(p)=\frac{p^2}{48\alpha}-\frac{\beta p^6}{144\alpha}$.
In this case the potential has an stable minimum at $p=0$ and two unstable  maximums at $p=\pm\beta^{-1/4}$, which corresponds to the unstable de Sitter solution
$H=\beta^{-1/2}$. From the shape of this potential one deduces that
there are only two unstable non-singular solutions (the ones that start at the de Sitter points and end at the bottom of the potential), and two that only are
singular at late time (the ones that start at the de Sitter points and ends at $|p|=\infty$),
all the other solutions are singular at early times.  At late times, there are two kinds of solutions: the ones that
have enough energy to overpass  the wedge of the potential and become singular at late times, and others with less energy that fall down against the wedge of the potential without clearing
it  due to the dissipation and, approaching to $p=0$ with the same oscillatory behavior as in the $R+\alpha R^2$  model (see figure 2 for the shape of potentials, and figure 3
for the phase portrait of the Starobinsky model).

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.3]{Wquadratic.png}
\includegraphics[scale=0.3]{Wperturbat.png}
\end{center}
\caption{{\protect\small Shape of the potentials $W(p)$ for $\alpha=0.01$ and $\beta=1$.
The first picture corresponds to the potential given by  $R^2$ gravity and the second one to the potential given by the  model suggested in \cite{s80}.
The dynamics  is very easy: one can imagine a "particle" moving under the action of the potential $W$
and losing energy.  For the first potential particles   come at early times from $|p|=\infty$ and ends at late time at  $p=0$ in an oscillating way. For the second potential there are
two unstable de Sitter solutions at $p=\pm(1/\beta)^{-1/4}$, so the "particle" could start at early times at these points and fall down
into the wedge of the potential  ending,  at late time, at  $p=0$ in
an oscillatory way. These are the only non-singular solutions. All the other orbits are singular at early and/or late times.
}}
\end{figure}

\vspace{1cm}

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.35]{retrat_fase_Starobinsky.png}
\end{center}
\caption{{\protect\small Phase portrait for $\alpha=0.01$ and $\beta=1$ of the Starobinsky model. The unique non-singular solutions
are the de Sitter one which correspond to the saddle point $(1,0)$, which is the unique critical point of the system (painted brown), and the black curve that starts at the critical  point
and ends oscillating at $(0,0)$.
}}
\end{figure}

Note that what is really important in the Starobinsky model at late times,
 is the oscillatory behavior of the scale factor rather than
its amplitude, because at late times the period of oscillation of the scale factor is much shorter than the Hubble time, meaning that for a few oscillations the amplitude of the scale
factor can be considered constant.
This behavior can be thought of as oscillations of a decaying field called {\it scalaron} \cite{s80} that
creates light conformally coupled particles, which finally
thermalize yielding a hot Friedmann universe that matches with the Standard Model.

\section{Loop Quantum $R^2$ gravity}

We start this section showing that there exists a wide range of values of $\alpha$ and $\tilde{\rho}_c$ for which the $R^2$ LQC model does not have any singularity.
First at all, from the holonomy corrected Friedmann equation (\ref{fri}) we deduce that
\begin{eqnarray}
0\leq \tilde{\rho}\leq \tilde{\rho}_c \quad \mbox{and}\quad
-\sqrt{\frac{\tilde{\rho}_c}{12}}\leq \tilde{H}\leq \sqrt{\frac{\tilde{\rho}_c}{12}}.
\end{eqnarray}

On the other hand,
 equation (\ref{4}), applied to $R^2$ gravity,  leads to the positive potential
\begin{eqnarray}\label{pot}
 V(\tilde{\phi})=\frac{1}{8\alpha}\left(1-e^{-\sqrt{\frac{2}{3}}\tilde{\phi}} \right)^2,
\end{eqnarray}
 and thus, one also has
\begin{eqnarray}
0\leq (\tilde{\phi}')^2\leq 2\tilde{\rho}_c \quad \mbox{and}\quad
0\leq V(\tilde{\rho})\leq \tilde{\rho}_c.
\end{eqnarray}

Using the Raychaudhuri equation in LQC, $\tilde{H}'=-\frac{1}{2}(\tilde{\phi}')^2\left(1-\frac{2\tilde{\rho}}{\tilde{\rho}_c} \right)$, one deduces
that
\begin{eqnarray}
 |\tilde{H}'|\leq \frac{1}{2}(\tilde{\phi}')^2\leq \tilde{\rho}_c\Longrightarrow |\tilde{R}|\leq 7\tilde{\rho}_c.
\end{eqnarray}

Moreover,
the potential (\ref{pot})
 satisfies
\begin{eqnarray}
 \frac{\partial V(\tilde{\phi})}{\partial \tilde{\phi}}=\frac{1}{f_R(R)}\sqrt{\frac{V(\tilde{\phi})}{3\alpha}},
\end{eqnarray}
which means (see the second equation of (\ref{rel}))
\begin{eqnarray}
 R=f_R(R)\left(\tilde{R}+(\tilde{\phi}')^2\right)+\sqrt{\frac{2V(\tilde{\phi})}{\alpha}},
\end{eqnarray}
and thus,
\begin{eqnarray}
 R=\frac{1}{1-2\alpha(\tilde{R}+(\tilde{\phi}')^2)}\left(\tilde{R}+(\tilde{\phi}')^2+ \sqrt{\frac{2V(\tilde{\phi})}{\alpha}}\right).
\end{eqnarray}

From the bound $1-2\alpha(\tilde{R}+(\tilde{\phi}')^2)\geq 1-18\alpha\tilde{\rho}_c$ one easily deduces
\begin{eqnarray}
 |R|\leq \frac{1}{1-18\alpha\tilde{\rho}_c}\left(18\tilde{\rho}_c+ \sqrt{\frac{2\tilde{\rho}_c}{\alpha}}\right),
\end{eqnarray}
which is always bounded provided we choose $\alpha<\frac{1}{18\tilde{\rho}_c}$.

Finally, since $|R|$ is bounded, from the first equation of (\ref{rel}) one deduces that $|H|$ is bounded, and consequently
$|\dot{H}|=\frac{1}{6}\left|R-12H^2\right|$ is bounded, meaning that $R^2$ gravity  in LQC has no singularities.

In fact, as we will see, in any case there are singularities when one takes into account holonomy corrections. However, when
$8\alpha\tilde{\rho}_c>1$ the scalar curvature $R$ can achieve very large values. To show that, we have to perform a detailed
analysis in EF.

\subsection{$R^2$ LQC  in Einstein frame}
To perform a deeper analysis of the model we will work in EF, where the dynamical equations are simpler than in the JF one.
In fact,  when $f(R)=R+\alpha R^2$ equation
(\ref{fri}) becomes
\begin{eqnarray}\label{equacio}
 12\alpha\dot{R}H+6H^2(1+2\alpha R)-\alpha R^2=-\frac{\alpha^2[2\alpha(R^3+3\dot{R}^2)+R^2 ]^2}{2(1+2\alpha R)^4\tilde{\rho}_c}.
\end{eqnarray}

On the other hand,
in EF, the field $\tilde{\phi}$ satisfies the equation
\begin{eqnarray}\label{field}
 \tilde{\phi}''+3\tilde{H}\tilde{\phi}'+\frac{\partial V(\tilde{\phi})}{\partial \tilde{\phi}}=0,
\end{eqnarray}
where the potential $\tilde{\phi}$ is given by (\ref{pot}).

As we have already explained, due to the holonomy effects in EF the universe starts in the contracting phase  with zero energy, and the energy density increases as far as it catches up with the
critical value $\tilde{\rho}_c$, where the universe bounces and enters in the expanding phase.

Performing the change of variable $\sqrt{\frac{2}{3}}\tilde{\phi}=\ln\tilde{\psi}$, i.e. $\tilde{\psi}=f_R(R)=1+2\alpha R$ (essentially $\tilde{\psi}$ is like $R$), one gets
\begin{eqnarray}\label{field1}
 \tilde{\psi}''\tilde{\psi}-(\tilde{\psi}')^2+3\tilde{H}\tilde{\psi}'\tilde{\psi}+\frac{1}{6\alpha}\left(\tilde{\psi}-1\right)=0.
\end{eqnarray}

From equation (\ref{field1}) one can show that the orbits in the plane $(\tilde{\psi},\tilde{\psi}')$, are  symmetric with respect the axis $\tilde{\psi}'=0$
in the expanding and contracting phase, because equation
(\ref{field1})  remains invariant after performing the replacement $\tilde{t}\rightarrow-\tilde{t}$ and $\tilde{H}\rightarrow-\tilde{H}$. To be more precise, consider in the plane
$(\tilde{\psi},\tilde{\psi}')$,  a trajectory (a solution of (\ref{field1})) $\sigma_1(t)=(\tilde{\psi}(t),\tilde{\psi}'(t))$ in the contracting $\tilde{H}<0$ (resp. expanding $\tilde{H}>0$) phase. Then,
$\sigma_2(t)=(\tilde{\psi}(-t),-\tilde{\psi}'(-t))$ is a trajectory in the expanding $\tilde{H}>0$ (resp. contracting $\tilde{H}<0$) phase.

The energy density, using the new variables, is  given by
\begin{eqnarray}
 \tilde{\rho}=\frac{3}{4\tilde{\psi}^2}\left((\tilde{\psi}')^2 +\frac{1}{6\alpha}(\tilde{\psi}-1)^2 \right),
\end{eqnarray}
which means that $\tilde{H}$ vanishes at the point $(\tilde{\psi},\tilde{\psi}')=(1,0)$ and over the curve $\tilde{\rho}=\tilde{\rho}_c$, with equation
\begin{eqnarray}\label{curva}
\frac{(\tilde{\psi}')^2}{\frac{4\tilde{\rho}_c}{3(1-8\alpha\tilde{\rho}_c)}}+
\frac{(\tilde{\psi}-\frac{1}{1-8\alpha\tilde{\rho}_c})^2}{\frac{8\alpha\tilde{\rho}_c}{(1-8\alpha\tilde{\rho}_c)^2}}=1,
\end{eqnarray}
which produces an ellipse for $1-8\alpha\tilde{\rho}_c>0$, an hyperbola for $1-8\alpha\tilde{\rho}_c<0$ and a parabola for $1-8\alpha\tilde{\rho}_c=0$.
Note also that $(1,0)$ is the unique critical point
 corresponding to $\tilde{\rho}=0$, which means that all the orbits start and end at this point
 (the universe starts and ends at this point), and in the curve (\ref{curva}) the universe in EF bounces, because it corresponds to
$\tilde{\rho}=\tilde{\rho}_c$.

From the previous  analysis we can conclude that
the dynamics, working in EF, goes as follows: the universe
starts in the contracting phase $\tilde{H}<0$  oscillating around  the unique critical point $(1,0)$ and increasing the amplitude of oscillations, then it reaches
 the curve $\tilde{\rho}=\tilde{\rho}_c$ where it bounces and enters in the expanding phase
 $\tilde{H}>0$ coming back once again to $(1,0)$ in an oscillatory way (our analytical study is supported numerically in figure $4$).

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.4]{psi_psipunt_starobinsky_lqc_ef.png}
\includegraphics[scale=0.4]{psi_psipunt_starobinsky_lqc_ef_hiperbola.png}
\end{center}

\caption{{\protect\small In the first picture we have the phase space portrait of an orbit in EF for the case $1-8\alpha\tilde{\rho}_c>0$ ($\alpha=0.1$ and $\tilde{\rho}_c=1$).
The universe starts, in the contracting phase $\tilde{H}<0$, oscillating arround $(1,0)$ (red curve)
and  arriving to the ellipse defined by equation (\ref{curva}) (blue curve), where the universe bounces entering in the
expanding phase $\tilde{H}>0$ and coming back to  $(1,0)$ oscillating (black curve). In second picture we draw an orbit in EF for the case
$1-8\alpha\tilde{\rho}_c<0$ ($\alpha=0.1$ and $\tilde{\rho}_c=15$). The dynamics is similar, the only difference is that now the blue curve is an hyperbola. At the top of the picture
we have inserted and increased in size the oscillatory behavior around the critical point $(1,0)$.}}
\end{figure}

Two important remarks are in order:
\begin{enumerate}\item
 Strictly speaking, the phase portrait in the plane $(\tilde{\psi},\tilde{\psi}')$ shows the dynamics of two dynamical systems,
 because equation (\ref{field1}) defines two different differential equations, one with $\tilde{H}>0$ and the other one with $\tilde{H}<0$. Then, since we have two different
  autonomous  dynamical systems, at each point of the plane $(\tilde{\psi},\tilde{\psi}')$ two different orbits, one with   $\tilde{H}>0$ and the other one with $\tilde{H}<0$,
 cross.
\item
It is important to realize  that
the system does not contain singularities because all the orbits start and end at the critical point $(1,0)$.
In the case $1-8\alpha\tilde{\rho}_c>0$, the variables $\tilde{\psi}$ and  $\tilde{\psi}'$
move inside an ellipse (a compact domain) meaning that, in this case, all the quantities are bounded. Effectively,
inside the ellipse the quantities $\tilde{H}$, $\tilde{R}$, $\tilde{\phi}$, $\tilde{\phi}'$
and $\tilde{\psi}=1+2\alpha R$ are bounded. Consequently, it follows from (\ref{rel}) that $H$ is bounded. On the other hand, in the case $1-8\alpha\tilde{\rho}_c<0$
the variables $\tilde{\psi}$ and  $\tilde{\psi}'$ move inside an unbounded region delimited  by an hyperbola, meaning that there are
orbits where $\tilde{\psi}$, and
consequently the scalar curvature $R$,  achieve very large values, which never happens in the other case.

\end{enumerate}

 \subsubsection{Inflation in Einstein frame}

 The slow-roll parameters in EF are given by (see for example \cite{sl})
 \begin{eqnarray}\label{sr}
  \tilde{\epsilon}\equiv-\frac{\tilde{H}'}{\tilde{H}^2}\quad \mbox{and}\quad \tilde{\eta}\equiv \tilde{\epsilon}-\tilde{\delta}=2\tilde{\epsilon}-
  \frac{\tilde{\epsilon}'}{2\tilde{H}\tilde{\epsilon}},
 \end{eqnarray}
where $\tilde{\delta}=\frac{\tilde{\phi}''}{\tilde{H}\tilde{\phi}'}$.

Slow-roll dynamics requires $(\tilde{\phi}')^2\ll V(\tilde{\phi})$ and $\tilde{\phi}''\ll \tilde{H}\tilde{\phi}'$. Then, in the slow-roll phase the dynamical equations read
\begin{eqnarray}\label{slequation}
 \tilde{H}^2=\frac{V(\tilde{\phi})}{3}\left(1-\frac{V(\tilde{\phi})}{\tilde{\rho}_c}\right)\quad \mbox{and}\quad 3\tilde{H}\tilde{\phi}'+
 \frac{\partial V(\tilde{\phi}) }{\partial \tilde{\phi}}=0,
\end{eqnarray}
and thus, in this phase, the slow-roll parameters are approximately
\begin{eqnarray}\label{sr1}
  \tilde{\epsilon}\cong\frac{1}{2}
  \left( \frac{1}{V(\tilde{\phi})} \frac{\partial V(\tilde{\phi}) }{\partial \tilde{\phi}}\right)^2
  \frac{\left(1-\frac{2V(\tilde{\phi})}{\tilde{\rho}_c} \right)}{\left(1-\frac{V(\tilde{\phi})}{\tilde{\rho}_c} \right)^2}\quad
  \mbox{and}\quad \tilde{\eta}\cong
  \frac{1}{V(\tilde{\phi})} \frac{\partial^2 V(\tilde{\phi}) }{\partial \tilde{\phi}^2}\frac{1}{\left(1-\frac{V(\tilde{\phi})}{\tilde{\rho}_c} \right)}.
 \end{eqnarray}

For the potential given by $R^2$ gravity, i.e. for (\ref{pot}), slow-roll conditions ($|\tilde{\epsilon}|\ll 1$
and $|\tilde{\eta}\ll 1$) are only satisfied for large positive values of the field. In that case,  equation (\ref{sr1}) becomes
\begin{eqnarray}\label{sr2}
  \tilde{\epsilon}\cong\frac{4}{3}
  \frac{e^{-\sqrt{\frac{8}{3}}\tilde{\phi}}}{(1-e^{-\sqrt{\frac{2}{3}}\tilde{\phi}})^4}
  \frac{\left(1-\frac{(1-e^{-\sqrt{\frac{2}{3}}\tilde{\phi}})^2}{4\alpha\tilde{\rho}_c} \right)}{\left(1-\frac{(1-e^{-\sqrt{\frac{2}{3}}\tilde{\phi}})^2}{8\alpha\tilde{\rho}_c} \right)^2}
 \end{eqnarray}
  and
  \begin{eqnarray}
  \tilde{\eta}\cong
  \frac{4}{{3}}
  \frac{2e^{-\sqrt{\frac{8}{3}}\tilde{\phi}}-e^{-\sqrt{\frac{2}{3}}\tilde{\phi}}}{(1-e^{-\sqrt{\frac{2}{3}}\tilde{\phi}})^2}
  \frac{1}{\left(1-\frac{(1-e^{-\sqrt{\frac{2}{3}}\tilde{\phi}})^2}{8\alpha\tilde{\rho}_c} \right)}.
 \end{eqnarray}

 To calculate
 inflation ends, the values of the slow-roll parameters must be of the order $1$, which happens, for positive values of the field  $\tilde{\phi}$, when it satisfies the equation
 \begin{eqnarray}
  \frac{e^{-\sqrt{\frac{2}{3}}\tilde{\phi}}}{(1-e^{-\sqrt{\frac{2}{3}}\tilde{\phi}})^2}\cong\frac{\sqrt{3}}{2},
 \end{eqnarray}
whose solution is
 \begin{eqnarray}
  \tilde{\phi}_{end}=-\sqrt{\frac{3}{2}}\ln\left(\frac{1+\sqrt{3}-\sqrt{2\sqrt{3}+1}}{\sqrt{3}} \right)>0.
 \end{eqnarray}

And to calculate
 the number of e-folds that the scale factor increases during the period of inflation
 \begin{eqnarray}\tilde{N}\equiv \int_{\tilde{t}_{in}}^{\tilde{t}_{end}}\tilde{H}d\tilde{t}=
 \int_{\tilde{\phi}_{in}}^{\tilde{\phi}_{end}}\frac{\tilde{H}}{\tilde{\phi}'}d\tilde{\phi},
 \end{eqnarray}
we have to use the slow roll equations (\ref{slequation}) obtaining
 \begin{eqnarray}\tilde{N}\cong
 \int_{\tilde{\phi}_{end}}^{\tilde{\phi}_{in}}
 \frac{V(\tilde{\phi})}{\frac{\partial V(\tilde{\phi}) }{\partial \tilde{\phi}}}
 \left(1-\frac{V(\tilde{\phi})}{\tilde{\rho}_c}\right)
 d\tilde{\phi}

 \end{eqnarray}

 In the case of our potential (\ref{pot}), the final number of e-folds is approximately
 \begin{eqnarray}
  \tilde{N}\cong \frac{3}{4}e^{\sqrt{\frac{2}{3}}\tilde{\phi}_{in}}.
 \end{eqnarray}

 On the other hand, for a given value of $\tilde{N}$ the slow-roll parameters  are:
 \begin{eqnarray}
  \tilde{\epsilon}\cong \frac{3}{4\tilde{N}^2}\frac{\left(1-\frac{1}{4\alpha\tilde{\rho}_c} \right)}{\left(1-\frac{1}{8\alpha\tilde{\rho}_c} \right)^2}\quad \mbox{and}\quad
  \tilde{\eta}\cong -\frac{1}{\tilde{N}}\frac{1}{\left(1-\frac{1}{8\alpha\tilde{\rho}_c} \right)}.
 \end{eqnarray}

 With these values,  the spectral index of scalar perturbations, namely $\tilde{n}_s$, and the ratio of tensor to scalar perturbations, namely $\tilde{r}$, are approximately
 \begin{eqnarray}
  \tilde{n}_s\cong 1-6\tilde{\eta}+2\tilde{\eta}\cong 1 -\frac{2}{\tilde{N}}\frac{1}{\left(1-\frac{1}{8\alpha\tilde{\rho}_c} \right)}, \quad
  \tilde{r}\cong 16\tilde{\epsilon}\cong \frac{12}{\tilde{N}^2}\frac{\left(1-\frac{1}{4\alpha\tilde{\rho}_c} \right)}{\left(1-\frac{1}{8\alpha\tilde{\rho}_c} \right)^2},
 \end{eqnarray}
which coincide, when  holonomy corrections are disregarded, i.e. when $\tilde{\rho}_c\rightarrow \infty$, with the values obtained in \cite{bno}.

A very important remark is in order: The latest Planck data gives for the spectral index the approximate value $\tilde{n}_s=0.9603\pm 0.0073$. If one disregards the loop corrections, to achieve
the value $0.96$ one has to take  $\tilde{N}=50$ e-folds, which does not give enough inflation to solve the flatness and horizon problems. However, if one takes into account
holonomy corrections, for the values $8\alpha \tilde{\rho}_c\cong 6$ and $\tilde{N}= 60$ (the minimum number of e-folds required to solve the horizon and flatness problems)
one obtains  the desired result. Moreover, for these same values one obtains $\tilde{r}= 0.0031$, which satisfies the current bound $\tilde{r}< 0.11$.

To be more precise, if one disregards loop corrections, $60$ e-folds are only achieved when $0.9666\leq\tilde{n}_s\leq \tilde{n}_{s,max}=0.9676$,
in fact for $\tilde{n}_s=0.9676$ one obtains $61.72$ e-folds, which means that, in this model without corrections, it is impossible for the universe to inflate more that $61.72$ e-folds. However,
including
loop quantum effects one easily achives a greater number of e-folds; for example, for $\tilde{n}_s=0.9676$ one obtains $70$ e-folds choosing $8\alpha\tilde{\rho}_c\cong 8.46$.
To sum up, we have shown that
loop quantum corrections
could be essential to match correctly $R^2$ inflation with the current observational data.

\subsection{$R^2$ LQC in Jordan frame}
To study the dynamics in the JF from the results obtained in the EF,
 we look for the points in the space $(\tilde{\psi},\tilde{\psi}')$ where the universe could  bounce in the JF, i.e., we look for the points  where $H=0$. Since,
${H}=\sqrt{f_R(R)}\left(\tilde{H}-\frac{1}{\sqrt{6}}\tilde{\phi}' \right)$, one has to solve the equation
$\tilde{H}^2=\frac{1}{{4}}\frac{(\tilde{\psi}')^2}{\tilde{\psi}^2}$, which gives, for  $\tilde{\psi}>1$ the following curve
\begin{eqnarray}\label{curva1}
\frac{(\tilde{\psi}')^2}{\frac{\tilde{\rho}_c}{12(1-\sqrt{8\alpha\tilde{\rho}_c})}}+
\frac{\left(\tilde{\psi}-\frac{1-\sqrt{2\alpha\tilde{\rho}_c}}{1-\sqrt{8\alpha\tilde{\rho}_c}}\right)^2}{\frac{2\alpha\tilde{\rho}_c}
{\left(1-\sqrt{8\alpha\tilde{\rho}_c}\right)^2}}=1,
\end{eqnarray}
which, as in EF, produces an ellipse for $1-8\alpha\tilde{\rho}_c>0$, an hyperbola for $1-8\alpha\tilde{\rho}_c<0$ and a parabola for $1-8\alpha\tilde{\rho}_c=0$. And,
for $0<\tilde{\psi}<1$ the curve is
\begin{eqnarray}\label{curva2}
\frac{(\tilde{\psi}')^2}{\frac{\tilde{\rho}_c}{12(1+\sqrt{8\alpha\tilde{\rho}_c})}}+
\frac{\left(\tilde{\psi}-\frac{1+\sqrt{2\alpha\tilde{\rho}_c}}{1+\sqrt{8\alpha\tilde{\rho}_c}}\right)^2}{\frac{2\alpha\tilde{\rho}_c}
{\left(1+\sqrt{8\alpha\tilde{\rho}_c}\right)^2}}=1,
\end{eqnarray}
which is always an ellipse. Then, when in EF the orbits in the plane $(\tilde{\psi},\tilde{\psi}')$ reach those curves, the universe in the JF could bounce. To assure that it bounces the equation $\tilde{H}=\frac{\tilde{\psi}'}{2\tilde{\psi}}$ must be satisfied.

Now we are ready to explain the dynamics in JF from the results already obtained in EF: In EF the dynamics starts in the contracting phase
and ends in the expanding one at the critical point $(\tilde{\psi},\tilde{\psi}')=(1,0)$. From the relation  between both frames
\begin{eqnarray}\label{rel1}
H=\sqrt{\tilde{\psi}}\left(\tilde{H}-\frac{\tilde{\psi}'}{2\tilde{\psi}}\right),\quad \dot{H}=\frac{\tilde{\psi}'}{2}\left(\tilde{H}-\frac{\tilde{\psi}'}{2\tilde{\psi}}\right)+
\tilde{\psi}\left(\tilde{H}'-
\frac{1}{2}\left(\frac{\tilde{\psi}'}{\tilde{\psi}}\right)'\right),
\end{eqnarray}
which is obtained from the first equation of (\ref{rel}) and its derivative, one deduces that, in JF, the universe starts and ends at $(H=0,\dot{H}=0)$. Note that to calculate
explicitly $\dot{H}$ one has to use the Raychaudhuri equation $\tilde{H}'=-\frac{3}{4}\left(1-\frac{2\tilde{\rho}}{\tilde{\rho}_c}\right)\left(\frac{\tilde{\psi}'}{\tilde{\psi}}\right)^2$
and the field equation (\ref{field1}).
Moreover, since in EF the orbits of the system at early and late times oscillate  around the point $(\tilde{\psi},\tilde{\psi}')=(1,0)$, crossing many times
the curves (\ref{curva1}) and (\ref{curva2}) one can conclude that
in JF the orbits of the system at early times oscillate around the point $(H,\dot{H})=(0,0)$ meaning that the universe makes small bounces many times, and
when it leaves this  oscillatory regime  it enters in the contracting
phase and bounces (in EF when the orbit reach the curve (\ref{curva})) to enter in the expanding phase, where the universe inflates and finally, at late times, it goes asymptotically
to the critical point $(0,0)$ in an oscillating way, that is, bouncing again many times.

Note that this behavior is completely different from the one obtained disregarding holonomy corrections where, in JF, as we have already seen in section IV, the universe never bounces
and is singular at early times. Moreover,
it is important to remark that the holonomy corrected  equation (\ref{equacio}) is not singular at $H=0$, and thus the orbits can cross the axis $H=0$, which allows the
universe to bounce. Of course, that does not happen in  classical $R^2$ gravity  where the corresponding dynamical equation (eq. (\ref{sta})) is not defined at $H=0$.

Numerically, the dynamics in the plane $(H,\dot{H})$ is  easily derived
via (\ref{rel1})
from the one in EF, which is very simple as we have already shown. In figure 5, we have depicted in the plane $(H,\dot{H})$ the orbits depicted  in figure 4.

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.45]{H_Hpunt_starobinsky_lqc_ef.png}
\includegraphics[scale=0.45]{H_Hpunt_starobinsky_lqc_ef_hiperbola.png}
\end{center}

\caption{{\protect\small In the first picture we have the phase space portrait of an orbit in JF for the case $1-8\alpha\tilde{\rho}_c>0$ ($\alpha=0.1$ and $\tilde{\rho}_c=1$).
The universe starts oscillating around $(0,0)$
then enters in  the contracting phase $(H<0)$ and  bounces entering in the
expanding phase ${H}>0$  coming back to  $(0,0)$ oscillating. In the second picture we draw an orbit in JF for the case
$1-8\alpha\tilde{\rho}_c<0$ ($\alpha=0.1$ and $\tilde{\rho}_c=15$). The dynamics is similar to that described in the other picture, but there is enough inflation here in the
expanding phase.}}
\end{figure}

Note also that  equation $(\ref{field1})$ defines two different dynamical systems, which means that in  the plane $(H,\dot{H})$, two different
orbits, one with  $\tilde{H}>0$ and the other one with $\tilde{H}<0$,  cross at each point. Moreover, the invariance of the equation
(\ref{field1}) with respect to the the replacement $\tilde{t}\rightarrow-\tilde{t}$ and $\tilde{H}\rightarrow-\tilde{H}$, means that the phase portrait in the plane
$(H,\dot{H})$ has a symmetry with respect the axis $H=0$. More precisely, given a piece of an orbit with $\tilde{H}>0$ (resp. $\tilde{H}<0$) in EF, there is  a symmetric
piece with respect to the axis $H=0$,
 of an orbit with $\tilde{H}<0$ (resp. $\tilde{H}>0$) in EF.

A final remark is in order: In JF, the dynamics of our extension of LQC to $R^2$ gravity in the vaccum (we do not have considered any scalar field),  is given by equation
(\ref{equacio}) which is a second order differential equation on $H$, mathematically meaning that is a first order differential system in variables $(H,\dot{H})$.
Working in EF the dynamics is depicted by
equation (\ref{field}) which is also a second order  in $\tilde{\phi})$ (note that from the holonomy corrected Friedmann equation $\tilde{H}$ is merely a function of
$\tilde{\phi}$ and $\tilde{\phi}'$, in the same way as in standard LQC), meaning that the dynamics is given by a first order differential system in variables
($\tilde{\phi},\tilde{\phi}'$).

The same happens in $f(R)$ gravity ``\`a la Palatini'' where the connection is a free variable (see for instance \cite{olmo}) and in teleparallel $f(T)$ gravity \cite{aho},
when the stress-tensor is depicted by an
scalar field $\phi$, because in both cases the corresponding modifed Friedmann equation relates the Hubble parameter with the energy density, i.e., the Hubble parameter ia a
function of ${\phi}$ and $\dot{\phi}$, meaning that the dinamics is given by the conservation equation
\begin{eqnarray}
 \ddot{\phi}+3H\dot{\phi}+\frac{\partial V}{\partial \phi}=0,
\end{eqnarray}
which is a second order differential equation in $\phi$.

However, if one considers  standard $f(R)$ gravity or $f(R)$ LQC, i.e., if the connection is fixed to be the Levi-Civit\`a one,  coupled with a scalar field $\phi$, the number of degrees
of freedom will increasse because appart from the modified
Friemann equation in $f(R)$ gravity, which is second order in $H$, one has to consider the conservation equation, which is second order in $\phi$, meaning that one will
have a first order differential system in the plane $(H,\dot{H},\phi,\dot{\phi})$.

 \section{Inflation and Dark Energy in $R^2$ LQC}
 Some time ago the unification of the early time inflation with late time Dark Energy (DE) in frames of modified gravity was proposed (\cite{no03}). Later,
 several improved models containing Dark Energy (DE) have been suggested to unify inflation with the current acceleration of the universe.  In this work,  the idea is to add to $R^2$ gravity
 a correction $g(R)$ given a model of the form $f(R)=R+\alpha R^2+g(R)$ that takes into account the accelerated expansion of the universe and passes the Solar system tests. Two of the
 best regarded examples
 of these
 corrections are:
 $g(R)=\lambda(e^{-bR}-1)$ being $\lambda$ and $b$ positive constants \cite{cenosz}, and
 $g(R)=-m^2\frac{c_1\left(R/m^2 \right)^n}{c_2\left(R/m^2 \right)^n+1}$, where $n>0$ and $c_1, c_2$ are dimensionless parameters \cite{hs}.

 The problem with this kind of  models is that they lead to  very complicated potentials in EF,  complicating considerably their extension to LQC.
 Moreover, it is nearly impossible to perform a
 detailed analytical study and it is not evident how to perform numerical computations. For this reason in order to deal with DE we will consider the simplest model:
  we will add an small cosmological constant to our model, i.e., we will consider the $f(R)=R+\alpha R^2-2\Lambda$ model.

  When one does not take into
 account holonomy corrections, the system after the change $p=H^2$ has the same form as (\ref{sis}) but with the potential $W(p)=\frac{p^2}{48\alpha}+\frac{\Lambda}{144\alpha p^2}$.
 This potential satisfies $V(0)=V(\infty)=\infty$ meaning that the dynamics can be restricted to positive values of $p$. The potential only has a minimum at the point
 $p=\left(\frac{\Lambda}{3}\right)^{1/4}$ (de Sitter solution), and thus at late times all the solutions go asymptotically to this point oscillating around it. Moreover,
 the inflationary solution given in (\ref{infla}) is also an attractor when  the cosmological constant is taken into account. Finally, it is easy to show that the solutions are
 singular at early times. When a cosmological constant is considered, there are two kind of solutions:
 the ones that, as in $R^2$ gravity without cosmological constant, are given by $p(t)=\sqrt{\frac{1}{2(t-\bar{t})}}$, and the other ones given by
$p(t)=\left(\frac{\Lambda}{36\alpha}(t-\bar{t})\right)^{1/6}$, which vanish at $t=\bar{t}$ but have divergent scalar curvature.

Incorporating the cosmological constant to the EF model we have obtained the following potential $V(\tilde{\phi})=\frac{1}{8\alpha}\left(1-e^{-\sqrt{\frac{2}{3}}\tilde{\phi}} \right)^2+
 \Lambda e^{-\sqrt{\frac{8}{3}}\tilde{\phi}}$, which has a minimum at $\tilde{\phi}_{min}=\sqrt{\frac{3}{2}}\ln (1+8\alpha\Lambda)$. That means that, at late times in
 the plane $(\tilde{\phi},\tilde{\phi}')$ of EF, all the solutions oscillate around $\tilde{Q}_{min}\equiv (\tilde{\phi}_{min},0)$. When we introduce Loop Quantum effects
 in EF, the orbits will oscillate initially around $\tilde{Q}_{min}$ in the contracting phase, i.e.,  $\tilde{H}<0$. In fact, $\tilde{Q}_{min}$  in the contracting
 phase corresponds to the anti de Sitter solution $\tilde{H}_-=-\sqrt{\frac{V(\tilde{\phi}_{min})}{3}\left(1-\frac{V(\tilde{\phi}_{min})}{\tilde{\rho}_c} \right)}$,
 where $V(\tilde{\phi}_{min})=\frac{\Lambda}{8\alpha\Lambda+1}$ is the minimum value of the potential. After leaving the anti de Sitter phase  the orbits move into the contracting
 phase before bouncing and entering in the expanding one where the universe inflates, and finally oscillate asymptotically to the de Sitter solution
 $\tilde{H}_+\equiv -\tilde{H}_-$.

In JF, the dynamics is very similar: the universe starts oscillating around  the anti de Sitter solution $H_-=\sqrt{8\alpha\Lambda+1}\tilde{H}_-$, after leaving the
anti de Sitter phase moves in the contracting phase $H<0$, which it leaves bouncing, enters the expanding phase where it inflates and finally, at late times, it oscillates around
 the de Sitter
solution $H_+=\sqrt{8\alpha\Lambda+1}\tilde{H}_+$. This oscillatory behavior at late  times is essential, because it excites the light fields coupled with gravity
that will re-heat the universe \cite{mms,w,s80}, yielding a hot universe that matches with the $\Lambda$CDM model.

\vspace{1cm}

\section{Conclusions}
We have introduced holonomy corrections to $R^2$ gravity in order to avoid early time singularities that appear in this model.
We have performed a detailed analytical and numerical analysis which shows that the new model is not singular due to
 the quantum geometric corrections (holonomy corrections) coming from the discrete nature of space-time assumed in LQC. The new model
 is more involved than the original one. For this reason, in order to understand the dynamics in  JF, a previous analysis must
 be performed in EF, where the dynamical equations greatly simplify. This allows us to perform a detailed study of its dynamics, what is essential in order
 to have a global idea  of the system in JF. From this analysis we conclude that, when quantum geometric corrections are taken into account, the universe evolves from the
 contracting phase to the expanding one through a big bounce, and when it enters in the expanding phase, as in the classical model, it inflates in such a way that,
 these holonomy corrections lead to theoretical predictions that match  correctly with current observational data.
Finally, to remark that
it would be interesting to study different versions of F(R) gravity, for instance, with several power-law type terms in order to understand how such theories which normally
do not support the inflation behave in LQC approach.

\vspace{1cm}
{\bf Acknowledgments:}
This investigation has been
supported in part by MINECO (Spain), project MTM2011-27739-C04-01, MTM2012-38122-C03-01,
 by AGAUR (Generalitat de Ca\-ta\-lu\-nya),
contracts 2009SGR 345 and 994, and by
project TSPU-139 of Russ. Min. of Education and Science (SDO).

redpajama_set_name: RedPajamaArXiv


text:
The Pandemic and All-Hazards Preparedness and Advancing Innovation Act (PAHPAI) is legislation introduced and passed by the U.S. Congress in 2019 that aims to improve the nation's preparation and response to public health threats, including both natural threats and deliberate man-made threats.

A previous bill (with a near-identical name), the Pandemic and All-Hazards Preparedness Act (PAHPA), was signed into law in 2006 and reauthorized in 2013 in order to create a system that prepares for, and responds to, public health threats that could turn into emergencies.

The 2019 bill (PAHPAI) was introduced by U.S. Senators Richard Burr (R-NC), Bob Casey (D-PA), Lamar Alexander (R-TN), and Patty Murray (D-WA). Congress passed the bill and sent it to President Trump for his signature in June 2019. (The bill number is S. 1379).

Background
After the September 11, 2001 terrorist attacks and the anthrax attacks that followed, a critical need arose: to develop and stockpile medical countermeasures (certain drugs and vaccines) needed to protect people from chemical, biological, radiological, nuclear (CBRN) and pandemic threats.

However, medical countermeasures that address these types of threats, such as anthrax or the plague, have little or no commercial market. Therefore, the government needed to incentivize the private market to develop them. To do so, Congress passed the Project BioShield Act in 2004. That law created a federal program called the Special Reserve Fund (SRF) to incentivize development of these types of medical countermeasures.

Since 9/11, the United States has faced a number of additional sizable public health emergencies. The anthrax attacks of 2001, the destruction caused by Hurricane Katrina in 2005, and the Ebola outbreak in 2014 all raised alarm and concern about how the government should prepare for, and respond to, such events. Because of these events, Congress has attempted to vastly improve the federal government's ability to address these threats.

In 2006, passage of the Pandemic and All-Hazards Preparedness Act (PAHPA), among other things, created the Biomedical Advanced Research and Development Authority (BARDA) to provide companies with money and technical assistance in the advanced research and development of medical countermeasures.

Several key existing laws give authorities, budgets and policies and procedures to several government agencies tasked with responding to public health emergencies.

 The Public Health Service Act (PHS) created a streamlined federal response to public health emergencies by giving the United States Public Health Service the job of preventing the spread of infectious diseases from foreign countries into the United States.
 The Public Health Security and Bioterrorism Preparedness Act of 2002 gave funding to hospitals and health systems to augment the work of PHS.
 In 2004, the Project BioShield Act allowed the government to give incentives to the private sector to create drugs that could protect people from biological weapons and naturally-occurring biological threats.

When disasters such as wildfires, hurricanes, and disease outbreaks strike, many sectors (public health, healthcare, and the private sector) play a role in saving lives. According to the American Hospital Association, PAHPA is "the backbone of our nation's health security" by giving the U.S. Department of Health and Human Services (HHS) the authority to respond to disasters.

After implementing the 2013 PAHPA reauthorization, Congress recognized certain federal programs, policies and procedures that needed improvement. The purpose of the 2019 PAHPAI bill is to implement these improvements.

Bioterrorism and antimicrobial resistance
PAHPAI focuses on a variety of threats, but pays special attention to two specific threats: bioterrorism and "antimicrobial resistance."

Bioterrorism occurs when a person or group of people intentionally release germs that can cause serious illness or death. Depending on the type of germ (also known as a "biological agent"), sometimes bioterrorism attacks can go almost completely undetected by technology because the biological agents are so microscopic in size.

Another emerging public health threat that is causing concern is the increase in "microbial resistance" among infectious diseases. Microbial resistance occurs when a person who has an infectious disease is given a countermeasure, such as an antibiotic medicine, and it does nothing to help the person get better.

Therefore, one of the goals of the PAHPAI law is to develop new therapies that are effective at treating infectious diseases.

Provisions of the legislation

An "authorization" bill in Congress is legislation that creates, or continues, the existence of federal agencies and programs. Authorization bills also define the terms and conditions under which the agencies and programs operate, authorize Congress to appropriate funds to those programs, and spell out how the programs must use their funds. Therefore, a "reauthorization" bill is legislation that redefines these specifics for federal agencies and programs.

PAHPAI reauthorizes federal government programs that were established in the past by the Public Health Service Act and the Federal Food, Drug, and Cosmetic Act. The law makes significant changes and improvements to the agencies and programs under those two laws, in order to make the federal government more effective at preventing and responding to public health security threats such as bioterrorism attacks or disease outbreaks.

PAHPAI focuses on making changes in the law in order to improve the government's response to a number of public health threats, including:

 Biological attacks (such as anthrax)
 Multidrug-resistant pathogens
 Natural disasters (such as wildfires, hurricanes and flooding)
 Other deliberate attacks (such as nuclear weapons)
 Pandemics (such as outbreaks of infectious diseases such as Zika, Ebola, measles, and flu)

The legislation is divided among seven titles which are described below.

Title I - Strengthening the National Health Security Strategy
Title I changes the National Health Security Strategy (NHSS) so that it includes a detailed description of all public health threats, as well as the processes to respond to each type of threat and emergency. It also adds zoonotic disease and disease outbreaks related to food and agriculture into the National Health Strategy.

The NHSS is a vision, set forth by HHS, to responding and managing public health emergencies. According to HHS:The 2019-2022 National Health Security Strategy (NHSS) provides a vision to strengthen our nation's ability to prevent, detect, assess, prepare for, mitigate, respond to, and recover from disasters and emergencies. It describes strategies to improve readiness and adapt operational capabilities to address new and evolving threats. By coordinating a whole-of-government approach that engages external partners and supports public health authorities and health care stakeholders, we can better safeguard the health and well-being of people across the country.The NHSS is created every four years by HHS and the Office of the Assistant Secretary for Preparedness and Response (ASPR).

Title II - Improving Preparedness and Response
Title II is further divided into 10 sections:

 Section 201: Improving Benchmarks and Standards for Preparedness and Response
 Requires the government to re-evaluate several important metrics of two programs: the Public Health Emergency Preparedness (PHEP) cooperative agreement and the Hospital Preparedness Program (HPP)
 Section 202: Amendments to Preparedness and Response Programs
 Reauthorizes the PHEP cooperative agreement and HPP through 2023
 Makes some changes to the two programs
 Section 203: Regional Health Care Emergency Preparedness and Response Systems
 Requires the federal government to give specific instructions to local hospitals and health systems to be able to respond to a bioterror attack, an emerging infectious disease outbreak, or a pandemic
 Was written with lessons learned from the Ebola virus outbreak in 2014 in mind
 Sets up a number of processes to improve coordination between the government and health responders
 Section 204: Military and Civilian Partnership for Trauma Readiness
 Brings together military trauma teams with civilian trauma centers (such as hospitals)
 Sets up a system for the two to cooperate in case of a national emergency
 Section 205: Public Health and Health Care Situational Awareness and Biosurveillance Capabilities
 Modernizes and adds more resources to the government's biosurveillance programs
 Expands coordination across government agencies to improve the "situational awareness" that is necessary to monitor, identify, and respond to CBRN threats
 Creates a budget for the government's biosurveillance network
 Reauthorizes biosurveillance and situational awareness programs through 2023
 Section 206: Strengthening and Supporting the Public Health Emergency Fund
 Clarifies specific incidents and how the Public Health Emergency Fund dollars can be spent
 Section 207: Improving All-Hazards Preparedness and Response by Public Health Emergency Volunteers
 Encourages states to let doctors and other health professionals to cross state lines to do their work during a public health emergency
 Helps states find ways to get more volunteers during public health emergencies
 Section 208: Clarifying State Liability Law for Volunteer Health Care Professionals
 Clarifies that in whichever state the disaster occurs, the liability laws for health care professionals from that state are applied
 Section 209: Report on Adequate National Blood Supply
 Requires a report with ways to get more blood donors and increase the national blood supply
 Section 210: Report on the Emergency Preparedness and Response Capabilities and Capacities of Hospitals, Long-term Care Facilities, and Other Health Care Facilities
 Requires a report that shows how prepared and ready to respond health care facilities are

Title III - Reaching All Communities

Title III is further divided into six sections:

 Section 301: Strengthening and Assessing the Emergency Response Workforce
 Recent emergencies have put strain on the emergency response workforce and its capabilities, so this section closes the gaps needed to address these strains
 Reauthorizes the National Disaster Medical System and Medical Reserve Corps through 2023
 Section 302: Health System Infrastructure to Improve Preparedness and Response
 This section gives federal authorities more flexibility in getting emergency medical supplies and restocking the Strategic National Stockpile (SNS). (The SNS is a large supply of medicine and medical supplies to be used in public health emergencies. An example is the anthrax vaccine.)
 Section 303: Considerations for At-Risk Individuals
 Focuses federal efforts on making sure that at-risk individuals (such as pregnant and postpartum women and infants) are taken into consideration when doing emergency planning
 Section 304: Improving Emergency Preparedness and Response Considerations for Children
 Similar to Section 303, except focused on children
 Section 305: National Advisory Committees on Disasters
 Reauthorizes the National Advisory Committee on Children and Disasters through 2023
 Creates new advisory committees focused on senior citizens and people with disabilities
 Section 306: Guidance for Participation in Exercises and Drills
 Requires the government to finish writing a "guidance" on using public health emergency workers in drills and practice exercises

Title IV - Prioritizing a Threat-Based Approach

Title IV is further divided into five sections:

 Section 401: Assistant Secretary for Preparedness and Response
 Encourages the HHS Assistant Secretary for Preparedness and Response (ASPR) to work with the intelligence agencies, Department of Defense, and public health agencies in carrying out his/her mission
 Section 402: Public Health Emergency Medical Countermeasure Enterprise (PHEMCE)
 Makes the existence of the PHEMCE permanent in federal law (The PHEMCE is a council of the heads of several agencies and is in charge of coordinating the federal government in gathering medical countermeasures to prepare for chemical, biological, radiological and nuclear threats (CBRN) and emerging infectious diseases)
 Section 403: Strategic National Stockpile
 Reauthorizes the SNS through 2023
 Focuses the efforts to replenish and supply the SNS based on a "threat-based approach"
 Section 404: Preparing for Pandemic Influenza, Antimicrobial Resistance, and Other Significant Threats
 Gives more authority to the head of the Biomedical Advanced Research and Development Authority (BARDA) to create initiatives to combat threats that are a significant risk to U.S. national security, in order to "accelerate and support advanced research, development, and procurement of countermeasures" to combat such threats
 (BARDA is a federal agency that helps companies inventing countermeasures such as vaccines and drugs move from the research phase all the way to FDA approval and inclusion of the vaccines and drugs into the SNS. It is focused on medical countermeasures that combat chemical, biological, radiological, and nuclear (CBRN) threats, as well as threats from pandemic influenza (PI) and emerging infectious diseases (EID). To-date, BARDA has supported 42 FDA approvals for products addressing CBRN, PI, and EID threats.)
 Section 405: Reporting on the Federal Select Agent Program
 Requires more reporting on the Select Agent Program (a federal program that "oversees the possession, use and transfer of biological select agents and toxins, which have the potential to pose a severe threat to public, animal or plant health or to animal or plant products")

Title V - Increasing Communication in Medical Countermeasure Advanced Research and Development
Title V is further divided into five sections:

 Section 501: Medical Countermeasure Budget Plan
 Expands the Countermeasure Budget Plan to coordinate more detail and communication regarding medical countermeasures
 Section 502: Material Threat and Medical Countermeasure Notifications
 Requires the government to better notify specific oversight committees in Congress "of current material threat determinations on an annual basis, and promptly notify Congress each time there is a change to such determinations."
 Section 503: Availability of Regulatory Management Plans
 Requires the FDA to better help new medical countermeasure development companies apply to the government by posting the process online
 Section 504: BARDA and the Special Reserve Fund
 Reauthorizes both BARDA and BioShield Special Reserve Fund through 2023 (Project BioShield is the federal program to develop, acquire, stockpile, and distribute medical countermeasures to protect the U.S. against weapons of mass destruction)
 Section 505: Additional Strategies for Combating Antibiotic Resistance
 Officially writes into U.S. federal law the findings of the Presidential Advisory Council on Combating Antibiotic-Resistant Bacteria

Title VI - Advancing Technologies for Medical Countermeasures

Title VI is further divided into seven sections:

 Section 601: Administration of Countermeasures
 Makes clear that BARDA can use existing resources to help get new technology related to medical countermeasures
 Section 602: Updating Definitions of Other Transactions
 Makes some technical changes to procedures
 Section 603: Medical Countermeasure Master Files
 Uses data to help companies developing new medical countermeasures submit their applications
 Requires the FDA to take certain steps to help Master File maintenance
 Section 604: Animal Rule Report
 Requires a report about the use of the "animal rule" in the development of medical countermeasures
 (In 2002, the FDA created a rule that allows the results of animal testing to be used for approval of products that will treat people in the event of a serious public health emergency, such as biological, chemical, radiological, or nuclear attacks)
 Section 605: Review of the Benefits of Genomic Engineering Technologies and Their Potential Role in National Security
 Creates a high-level meeting of people and organizations that help the federal government, along with private companies, to talk about potential ways that "genomic engineering technologies" could help improve national security
 Requires the ASPR to create a public report about the results of the meeting
 Section 606: Report on the Development of Vaccines to Prevent Future Epidemics
 Requires that a report be written about how the government worked with other countries in the past during public health emergencies; specifically how they did advanced R&D on products to help pandemics and epidemics
 Section 607: Strengthening Mosquito Abatement for Safety and Health (SMASH)
 Allows the government to continue using tools that help states and local governments control and watch mosquito populations, especially those that carry the Zika virus
 Continues a program that gives grants to epidemiology and laboratory efforts that are trying to control the spread of infectious diseases

Title VII - Miscellaneous Provisions
Title VII is further divided into seven sections:

 Section 701: Reauthorizations and Extensions
 Extends a number of federal programs and laws, such as funding for VA hospitals, flu vaccine tracking, and the distribution of flu vaccines during a pandemic
 Section 702: Location of Materials in the Stockpile
 Limits what the government can disclose to the public about the Strategic National Stockpile, in cases where doing so could affect U.S. national security
 Section 703: Cybersecurity
 Requires the government to create a strategy to protect the public's health in case of a cyber attack that threatens U.S. national security
 Section 704: Report
 Requires that the government publish a report about what the ASPR has done to help migrant children and their families reunite, as the result of federal immigration policies
 Section 705: Technical Changes
 Makes some technical changes to various laws

Legislative history

Original law

In 2006, during the 109th Congress (2005-2006), Senator Richard Burr introduced S. 3678, the Pandemic and All-Hazards Preparedness Act. It passed Congress and was signed into law by President George W. Bush (Public Law 109-417).

Senators Richard Burr and the late Edward Kennedy (D-MA) spearheaded the original legislation.

First reauthorization
In 2013, the original Pandemic and All-Hazards Preparedness Act of 2006 needed to be reauthorized. Senator Richard Burr introduced S. 242, the Pandemic and All-Hazards Preparedness Reauthorization Act. The bill, however, never made it to a full vote in the Senate. In the House, Rep. Mike Rogers introduced the same legislation, H.R. 307. That bill passed Congress and was signed into law by President Barack Obama (Public Law 113-5).

Second reauthorization
In 2018, during the 115th Congress (2017-2018) the law once again needed to be updated and reauthorized. Senator Burr introduced S. 2852, the Pandemic and All-Hazards Preparedness and Advancing Innovation Act of 2018. The bill made it through committee consideration but never reached a full vote in the Senate. In the House, Rep. Susan Brooks introduced companion legislation, H.R. 6378. Brooks' bill passed the House, but was never considered in the Senate. Once the 115th Congress ended, both bills were effectively dead.

The following year, in 2019 during the first year of the 116th Congress (2019-2020), Senator Burr re-introduced the All-Hazards Preparedness and Advancing Innovation Act as S. 1379. The bill was passed by Congress and on June 24, 2019 it was signed by President Trump on June 24, 2019 (PL 116-22).

Table summary of legislative versions

S. 1379 procedural history
The Pandemic and All-Hazards Preparedness and Advancing Innovation Act of 2019 (S. 1379) was introduced in the Senate by Richard Burr on May 8, 2019. On May 15, the Senate Committee on Health, Education, Labor, and Pensions passed the bill by unanimous consent. That same day, the full Senate passed the bill by voice vote.

On June 4, the House passed the bill by voice vote. The legislation was sent to President Trump on June 13, who signed it into law on June 24, 2019 as Public Law 116-22.

See also

 Assistant Secretary for Preparedness and Response (ASPR)
 Bioterrorism
 BioWatch
 Blue Ribbon Study Panel on Biodefense
 Centers for Disease Control and Prevention
 Government Accountability Office (GAO)
 Medical Reserve Corps
National Biosurveillance Strategy
National Disaster Medical System
National Intelligence Assessments on Infectious Diseases
National Response Framework
Select agent
Strategic National Stockpile

References

External links
Antibiotic / Antimicrobial Resistance (AR / AMR) - CDC
 Blue Ribbon Study Panel on Biodefense
 Federal Select Agent Program
 Hospital Preparedness Program (HPP)
 National Biodefense Strategy summary
 National Biodefense Strategy (full text)
National Health Security Strategy 2019-2022 (full text)
National Strategy for Chemical, Biological, Radiological, Nuclear, and Explosives (CBRNE) Standards
Public Health and Social Services Emergency Fund
Public Health Emergency Preparedness (PHEP) Cooperative Agreement
Statement from the President on the National Biodefense Strategy and National Security Presidential Memorandum (9/18/18)

United States Department of Health and Human Services
United States federal health legislation
United States federal emergency management legislation
Acts of the 116th United States Congress


Ryan Wilson on the Roshell.
As a fishermen-owned cooperative, our relentless commitment to quality begins with the careful handling of our catch the moment it comes on board and continues all the way to when our customers receive our product and beyond, as so many of our customers have been loyal to us for years for that extra personal touch that we provide.
At Seafood Producers Cooperative, our people are the difference. At the SPC plant in Sitka, skippers, deck hands, and dock workers take great pride in the quality product that they help bring to SPC customers. Below is a gallery of pictures that show the people that make up our cooperative.
Deckhand on the F/V Seaboy, a Southeast Alaska troller.
Guillermo Gutierrez with a beautiful Alaskan salmon.
Eric Jordan waving from the I Gotta.
Matt Konoske waiting for a bite on the Infinity.
Gary McMaster of the Brat.
Norm Pillen has been a longtime member both as a fisherman and tender operator.
Michelle Istoc processing frozen seafood.
Photos by Brian Adams Photography, Kendall Whitney, Dan Lamont and Amy Grondin.