Ensuring all students and workers are prepared to work and ready to succeed.
Providing rigorous academic standards in K-12 English language arts and mathematics so that all students graduate ready for college and the workforce.
Raising the performance of U.S. students to ensure that all students read on grade level by the end of the third grade.
Advocating for federal legislation that authorizes programs and policies that improve educational outcomes and help close the skills gap.
Better aligning educational and training goals with business needs ensures a stronger American workforce and a more productive economy.
CTE must be relevant and meaningful for students.
Recipients of Perkins funding must be accountable for results.
The business community must be actively engaged in helping to inform and support CTE at the state, regional and local levels.
The Perkins Act must provide for state / local innovation and reward excellence.
In July 2018, the final version of the bipartisan bill was passed by both houses of Congress and signed by President Trump. This legislation is an excellent example of what can be accomplished when leaders in business, government and education come together to expand education and training pathways for the American people. We must keep pushing forward with initiatives to help prepare America's students and workers for career success in the 21st century.
Working with a coalition of other national organizations from across the business and workforce development communities, Business Roundtable laid out a series of policy principles for the reauthorization of the Higher Education Act (HEA). Other organizations included the Aspen Institute's Skills for America's Future; Business Roundtable; CAEL; Center for Law and Social Policy; Committee for Economic Development; HR Policy Association; Jobs for the Future; National Association of System Heads; National Governors Association; National Skills Coalition; New America Foundation; and the U.S. Chamber of Commerce.
Outcomes are what matter: Student outcomes need to play a stronger role in our quality assurance system and in the rules determining access to public federal higher education funds. As higher education becomes more essential and more expensive, it is imperative that we reduce the financial risk of educational investments for students, employers, and taxpayers. The best way to manage risk is to pay more attention to learning and labor market outcomes. The federal government and its partners in state government and the accrediting community, should ensure that federal higher education funding is linked to the outcomes that matter most to students, employers, and taxpayers: credential attainment, graduation, jobs, earnings, and/or access to more advanced education. Our policies need to reward institutions for helping all students, especially those from historically disadvantaged or underrepresented groups, access and succeed in higher education and transition successfully into the next stage of their personal and professional development. Our policies also need to make it easier for institutions and policymakers to collect and report student outcome data, including employment and earnings.
Federal financial aid policies need to be more flexible: The federal student aid programs authorized under Title IV of HEA do too little to support students who are older, returning to school, or seeking specific skills and credentials for work. While maintaining crucial safeguards against fraud and abuse, eligibility criteria and allowable expenses should be updated to better support students who need to combine work and learning, accelerate their time to degree, and/or attend school part-time and year-round. Creating more opportunities to target grant and loan programs differently, depending on the level of risk or return associated with a particular program, can also provide more safe spaces for innovation in program design.
Higher Education needs to do more to connect learning and work: If students are to graduate ready for the world of work, we need approaches that marry classroom learning with educational opportunities rooted in exposure to real world experience and problem solving. The tools to bridge this divide, such as work-based learning, cooperative education, apprenticeships, and prior learning assessment, are well known but not widely used in postsecondary education. HEA reauthorization should encourage institutions to expand experiential learning opportunities by supporting the use of college work-study and other innovative strategies for fostering stronger linkages between work-based and classroom learning.
Accreditation processes need to be more transparent and rigorous, and protect the interests of students, employers, and taxpayers. Our quality assurance system is fragmented, duplicative, and overly focused on institutional inputs and processes rather than program quality and student outcomes. The lack of transparency and the institutional focus of our accreditation system make it too difficult for stakeholders to identify high-quality programs or mitigate the risks associated with bad programs. The reauthorization of HEA provides an important opportunity to strengthen quality assurance policy and practice.
Quality assurance processes should focus more on programs and credentials: As the labor market grows more competitive and employers look increasingly for specific skills and experience, postsecondary credentials have become more important for securing good jobs. But our quality assurance processes have failed to keep up with the growing demand for and supply of credentials, leaving consumers – both students and employers – with inadequate information about the value, or even composition, of individual credentials. Credentials need to accurately reflect what a student knows and can do. Improved competency and credential validation processes that include employers would go a long way to improve transparency around skill attainment and to ensure that credentials are used for making employment-related decisions. For these reasons and more, greater transparency should be a guiding principle of the reform of accreditation.
Higher education is not an island; HEA shouldn't be either: The reauthorization of HEA creates opportunities to better align the law, particularly the rules surrounding access to the federal student aid programs, with other federal education and training programs. The recent reauthorization of the Workforce Innovation and Opportunity Act is an example of how policymakers can facilitate coordination and alignment among programs offering complementary services. The upcoming reauthorizations of the Higher Education Act, Elementary and Secondary Education Act, and the Carl D. Perkins Career and Technical Education Act create more opportunities for states and institutions to coordinate federally-funded education and training programs through activities like joint planning, data-sharing, cost-sharing, accountability, and dual enrollment.
Policy should encourage innovation and experimentation: Meeting the education challenges of the future will require new ways of designing, delivering, and assessing teaching and learning. Our public policies should support the scaling of evidence-based strategies as well as innovation in higher education to make it more accessible, affordable, and relevant to students, employers, and communities. New models are emerging inside and outside the existing higher education system that are pushing the boundaries of educational design. These include career pathways and stackable credentials, flexible scheduling, concurrent enrollment models with adult education, and competency-based education. HEA should provide safe spaces for experimentation with new outcome-based quality assurance processes, alternative currency for awarding financial aid, and rigorous evaluation of new approaches.
In association with 12 other national organizations, Business Roundtable released a letter to Senators Lamar Alexander and Patty Murray of the Senate Health, Education, Labor, and Pensions (HELP) committee which outlined a set of recommendations for work-study policies.
Funding guarantees for incumbent institutions: When the program was first created it included a relatively small number of colleges and an ad hoc committee of experts determined funding levels. As time passed, policymakers recognized the need to create a funding formula to better target the program's resources toward high need students and ensure fairness. The first major adjustment was made in 1979. But in an effort to protect colleges that might lose funds under the new formula from a financial shock, Congress guaranteed existing colleges 100 percent of their most recent allocation. In subsequent rounds of HEA reauthorization, this "campus base guarantee" has been reset, each time guaranteeing more institutions 100 percent of their previous year's funding, and is currently set at 1999.
While colleges covered by the guarantee are assured, at a minimum, the same base amount of funding each year, funding for the FWS program has not significantly increased. As a result, an ever-growing share of FWS dollars flow to colleges regardless of the financial status of their students, or of the needs of students in other institutions, but based solely on an institution's historic participation in the program. In fact, the campus base guarantee currently accounts for 67% of all FWS dollars allocated (around $600 million), leaving only a third of the program's resources subject to the funding formula designed to target institutions with high need students.
The campus base guarantee has made it almost impossible for the program to respond to the changing demographic and enrollment patterns of American undergraduates. It is the reason why the resources have remained so concentrated in the Northeast – where many institutions began participating in the program in the 1960s – and has failed to gain a foothold in other parts of the country.
A bias toward high-tuition institutions: The annual funds remaining after the campus base guarantee has been met (about $300 million) are distributed via a "fair share formula" that seeks to channel resources to colleges with "high need" students. But the formula does a poor job actually identifying high need students since it is based primarily on the cost of attending a particular institution (i.e. tuition and living expenses), not the economic circumstances of students. Specifically, the formula calculates "institutional need" based on the gap between the cost of attendance and the average expected family contribution (EFC) of students. At institutions with high tuition there is a larger gap between cost of attendance and EFC, so this essentially identifies "high need" students at high tuition schools who are not necessarily low income. The formula is effectively blind to students attending affordable institutions – who are more likely to be low income – and biased toward students at high cost institutions, even if they are not from low- income families.
The fair share formula, combined with the distorting effects of the campus base guarantee, explain why nearly quarter of federal work study participants come from families earning over $90,000 per year, well above the earnings of many low-income students.7 We propose the following reforms as essential steps for better targeting this valuable program toward the students who need it most.
Phase out the "campus-base guarantee" over a 5-year period and replace it with a guarantee of 10% of the total funding per year.
Replace the fair share formula with one that is driven by a combination of Pell enrollment and Pell graduation rates.
Congress should better define what the statute means by FWS positions "must be related to educational goals to the maximum extent practicable" in the reauthorization of HEA.
With this improved guidance, HEA should direct the Department to collect data from colleges on the number of students who are placed into opportunities that relate to their course of study.
Currently, many low-income students face a disadvantage in the labor market because they cannot afford to take unpaid internships like their more affluent counterparts. FWS could play a role in addressing this inequality. Congress should consider lifting the restrictions on students using their FWS funding for jobs with for-profit companies and off campus non-profits.
In June 2018, Business Roundtable launched the Workforce Partnership Initiative (WPI) to tackle current and future skills development challenges and drive economic growth in seven regions around the U.S. CEOs of leading businesses will partner with local colleges and universities to accelerate and scale best-in-class workforce readiness programs and develop a steady talent pipeline that meets the changing needs of each region's growing industries.
The public-private partnerships developed and scaled through the WPI will increase the number and diversity of skilled regional workers by expanding work-based learning programs, creating new industry-recognized credentials and building new and innovative apprenticeship models.
As Congress moves forward with the long-overdue reauthorization of the Workforce Investment Act (WIA), Business Roundtable encourages policymakers to embrace the following principles to ensure a far more effective and efficient workforce investment system in this nation.
Business Driven: To be effective, the federal workforce investment system must provide for a strong and active role for the employer community. This involvement is critical for ensuring training and employment services are relevant to the jobs that are in demand today or will be in demand in the future.
Less Bureaucracy: To be effective, the federal workforce investment system must be streamlined and efficient. Overlapping and duplicative funding streams are unnecessary and the administrative costs diminish funds that could be used for training. Overly complex federal rules and regulations, which micromanage state and local implementation of programs, drive away the willingness of business leaders to participate and create barriers to individuals seeking to obtain necessary services.
Expanded Training for In-Demand Occupations: To be effective, the federal workforce investment system must have "preparing individuals for a career" as a main focus. Currently, too few individuals receive training under the Workforce Investment Act, in part due to the multiple hoops individuals must often go through to access training. This issue, along with better allocation of resources and more efficient delivery of services, must all be addressed.
More Innovation: To be effective, the federal workforce investment system must enable new approaches for the delivery of employment and training services and promote innovations that expand access to training and make training relevant, such as industry-recognized credentials. It must also promote positive outcomes such as through pay-for-performance contracting.
Performance Driven: To be effective, the federal workforce investment system must be driven by real accountability that is focused on outcomes that matter most — such as a job. A key objective is consistency across similar programs and the assurance that grantees and providers of services are accountable for results by measuring placements of those trained, the compensation and quality of the job and success on the job.
Collaboration: To be effective, the federal workforce investment system must be fully engaged with the full array of local and regional partners, particularly community colleges and other postsecondary institutions that share similar missions and urgency for preparing individuals for careers.
Better Information: To be effective, the federal workforce investment system must be driven by high quality data so that: 1) high-quality labor market information drives training opportunities, 2) participants are able to make informed career and training decisions among providers and 3) program outcomes are reliable in order to ensure a robust and fair system of program accountability.
The Workforce Partnership Initiative (WPI) is a CEO-led initiative created by the Business Roundtable to increase the skills and diversity of America's workforce by building strategic partnerships between business, non-profits and educational institutions.
Credential Engine is a non-profit organization working to improve transparency in the credentialing marketplace. By providing a common credentialing language, a centralized registry of credentials, and a credential search engine, Credential Engine will enable job seekers, students, workers, and empower everyone to make more informed decisions about credentials and their value.
With the support of Business Roundtable, the National Network of Business and Industry Associations (National Network) works to bridge the skills gap by better connecting the worlds of learning and work. The National Network helps communicate to educational institutions the skills that employers are looking for, and enables employers to align hiring practices with the skills workers have.


2. Denoting the actions of such an agent (for example, diazepam).
Preventing or reducing anxiety; antianxiety.
An antianxiety medication; a tranquilizer.
2. Denoting the actions of such an agent or medication.
1. Operating to relieve ANXIETY or to treat an anxiety disorder.
2. A drug used to treat anxiety.
A type of medication that helps to relieve anxiety.
adj/n the ability of a substance to alleviate emotional anxiety and stress.
2. Denoting the actions of such an agent (e.g., diazepam).
a mild sedative, such as diazepam, used for relief of anxiety. Called also antianxiety agent and minor tranquilizer.
Thus, it can be suggested that the potential antidepressant and anxiolytic effects of fennel oil observed in this study may be attributed to its antioxidant property (Roby et al.
12] a preclinical study on rats states that OS possess significant antistress activity but the magnitude and efficacy for relieving stress are less when compare to standard anxiolytic agent alprazolam.
Intentional exposure was significantly higher in cases involving painkillers and psychoactive and anxiolytic drugs (p<0.
The excitability of the pyramidal cells of BLA is correlated with anxiety behaviors, where an increase in the excitability is anxiogenic and a decrease is anxiolytic (50,51,52).
The anxiolytic behavior in the cAD rats may be the sum total effect of the impaired function of different neural areas which were affected by colchicine induced neurodegeneration.
In addition, the OF test indicates that EEYP presented anxiolytic activity.
These data indicate that OEP is more potent than a classic anxiolytic drug by the parameters measured in the OF test, such as the amount of time spent in the central area and reduced grooming.
The results suggested that ZEO has strong sedative-hypnotic, anxiolytic and muscle relaxant properties in various animal models.
The present study was designed to investigate in the rat whether or not crocins possess anxiolytic properties.
Several animal models of anxiety have been used in the screening of anxiolytic or anxiogenic properties of compounds, including the elevated plusmaze (EPM), social interaction tests, open fields, etc.
The spa therapy, through its proven anxiolytic activity, is able to favour the substituting of treatment by benzodiapezine, and as a result would make withdrawal easier.
SSRIs increase concentrations of 5-HT in the synapse cleft and thus have anxiolytic effects for almost all subtypes of anxiety disorders, including general anxiety disorder, panic disorder, social anxiety, and obsessive-compulsive disorder.


text:
\section{Introduction}

Two main questions of theoretical physics requiring the knowledge of the structure of spacetime at a fundamental level are the nature of singularities appearing in classical general relativity and the ultraviolet divergences of field theory. It is expected a quantum theory of gravity can provide an answer for such questions as we have learned from simpler quantum physical systems which  improve their behavior as compared to their classical analogues. Not only is compulsory to find how these questions can be answered but in fact to be able to grasp what new concepts, if any, are needed in the theoretical framework that replaces the origin of these issues.

One candidate quantum gravity theory, loop quantum gravity (LQG) \cite{Ashtekar-etal(2004), Rovelli200712,Thiemann200812} which is a non-perturbative, background-independent approach to quantize general relativity is natural to consider in dealing with the nature of spacetime.
In particular, the implementation of the loop quantum gravity program for cosmological models, which is known as loop quantum cosmology (LQC) \cite{Bojowald:2006da,GAMM,Ashtekar:2011ni,CorichiAshtekarPetkov}, has led to the replacement of the big-bang singularity with a quantum bounce for homogeneous and isotropic models (see, for instance, the seminal works \cite{Bojowald:2001xe,Ashtekar:2006uz,Ashtekar:2006wn,Ashtekar:2006es,Vandersloot:2006ws}). Also anisotropic  \cite{Chiou:2007sp,Chiou:2007mg,MartinBenito:2008wx, MartinBenito:2009qu, Ashtekar:2009vc, Ashtekar:2009um,WilsonEwing:2010rh,Singh:2011gp, Fujio:2012zz, Liu:2012xp,Corichi:2012hy, Corichi:2015ala,Singh:2013ava,Modesto:2005zm,Ashtekar:2005qt,Bohmer:2007wi,Campiglia:2007pb,Campiglia:2007pr,Chiou:2008eg,Chiou:2008nm,Modesto:2008im,Gambini:2008dy,Cortez:2012ina,Gambini:2013hna,Gambini:2013ooa,Joe:2014tca,Corichi:2015xia}, as well as inhomogeneous models \cite{Bojowald:2006qu,MenaMarugan:2009dp,Garay:2010sk,MartinBenito:2010up,MartinBenito:2010dz,Olmedo:2011zz,Brizuela:2011ps,MartindeBlas:2012zz,Martin-Benito:2013jqa,Tarrio:2013ija,Fernandez-Mendez:2014raa,Fernandez-Mendez:2014aea,Gomar:2014faa,MartIN-Benito:2015pca} have been studied.

It has been argued in some cases \cite{Ashtekar:2011ni} it is convenient to use effective models capturing their essential quantum aspects mainly when the full quantum dynamics is unknown. The effective approach has been tested by applying the effective dynamics to cases where quantum evolution is fully known, with the astonishing result that the effective dynamics matches quite well, even in the deep quantum regime, with the full quantum dynamics of LQC \cite{Ashtekar:2006wn,Bentivegna:2008bg,Ashtekar:2006es,Diener:2014mia}. Clearly it is crucial to determine whether and when an effective description is pertinent without relying on the full quantum solution.

Motivated by the success of LQC in the study of homogeneous cosmologies, the study of the Schwarzschild black hole interior by using LQC techniques was put forward in  \cite{Modesto:2005zm,Ashtekar:2005qt} exploiting the fact that the interior Schwarzschild geometry is a particular homogneous Kantowski-Sachs model. Their results indicated that quantum Einstein equations were not singular. However, the answer to the question what replaces the classical singularity was not answered. Further developments using an effective approach were done in \cite{Bohmer:2007wi,Campiglia:2007pb,Campiglia:2007pr,Chiou:2008eg,Chiou:2008nm,Modesto:2008im,Gambini:2008dy,Cortez:2012ina,Gambini:2013hna,Gambini:2013ooa,Joe:2014tca,Corichi:2015xia} (For a recent review see \cite{Olmedo:2016ddn}) . Interestingly, \cite{Bohmer:2007wi} argued that there is a connection between the black hole and a Nariai Universe whereas \cite{Corichi:2015xia} found the presence of a white hole instead; the difference between these two works being whether a pair of parameters in the quantization are scale factor dependent or constants, respectively. In particular \cite{Bohmer:2007wi} get quantum corrections both at singularity and horizon, as opposed to \cite{Corichi:2015xia} which corrections are limited to the would be singularity. Actually such difference appears already in cosmological models in regard to the inadequacy of the so called  $\mu_{0}=constant$ prescription \cite{Ashtekar:2011ni,CorichiAshtekarPetkov}; in order to correctly describe the classical regime such parameter should be scale factor dependent.  Yet  \cite{Corichi:2015xia} argue such analysis for cosmological models does not hold for Schwarzschild since it would alter the notion of classical horizon. To us this is an unsettled issue which requires further study and more information, for instance,  the  behavior of effective quantities, like geometric scalars in the effective Raychaudhuri equation, or the effective Kretschmann and curvature scalars (see \cite{Cortez:2012ina,Joe:2014tca} for first steps in this direction).

The present work is aimed at filling the gap between the description using loop quantum model and that using an effective dynamics for the Kantowski-Sachs model representing the Schwarzschild black hole interior. We will derive the effective Hamiltonian constraint via the path integral approach starting from the quantum Hamiltonian in the so called improved dynamics scheme with the quantization parameters depending on the scale factors and consider the transition amplitude between two basis states labelled with different values of a time parameter. After performing the usual partition of the time interval we get the effective action, $S_{{\rm{eff}}}$, as the argument of an imaginary exponential that is to be integrated upon according to Feynman's prescription. It is from $S_{\rm{eff}}$ that the effective Hamiltonian $H_{\rm{eff}}$ will be extracted. Thereafter we will analyze in an analytical manner the effective Hamiltonian theory associated to $H_{\rm{eff}}$ and its impact on the behavior of relevant scalars. More precisely, we will prove that the effective expansion scalar, its time derivative and shear  are bounded. Moreover, it is demonstrated that every scalar polynomial invariant, so, in particular, the Ricci and Kretschmann scalars, are bounded in the effective approach.

 This paper is organized as follows. Section \ref{ClassicalKS} is devoted to the classical setting of the theory. We start by recalling that the Schwarzschild interior geometry can be described by a Kantowski-Sachs model. Thereafter, we recast the model in connection variables and perform a qualitative canonical analysis of the classical dynamics identifying the singular behavior of curvature invariants. Next, in Section \ref{QKS}, within the framework of the improved dynamics prescription, we get the effective Hamiltonian constraint by using the path integral approach. This along the lines of \cite{Liu:2012xp} for Bianchi I. The effective loop quantum black hole interior geometry is analyzed in Section \ref{EffectiveKS}, where it is shown that classically divergent quantities are actually bounded in the effective approach. In Section \ref{sec-disc} we discuss and summarize our main results.

 Throughout this work we will denote by $\mu$ to the so-called improved dynamics for homogeneous models, which has previously been denoted in literature by $\bar{\mu}^{\prime}$ (see, for instance, \cite{Chiou:2008nm}).

\section{Classical Theory}

\label{ClassicalKS}

\subsection{The interior geometry in connection variables}

As it is well known, a Schwarzschild black hole of mass $M$ (i.e., a spherically symmetric vacuum solution to general relativity) is described by the metric
\begin{equation}
ds^{2}=-\left(  1-\frac{2GM}{r}\right)  dT^{2}+{\left(  1-\frac
{2GM}{r}\right)^{-1}  }dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta d\phi^{2}\right), \label{metrica schwarschild}
\end{equation}
in Schwarzschild coordinates $T\in {\mathbb{R}}$, $r\in \mathbb{R}^{+}$, $0\leq \theta \leq \pi$ and $0\leq \phi \leq 2\pi$. At $r=0$ there is a true singularity (the Kretschmann scalar blows up as $r\to 0$) which is wrapped by an event horizon located at the so-called Schwarzschild radius, $r_{s}=2GM$ (where the Schwarzschild coordinates become singular). The exterior (i.e., $r>r_{s}$) spacelike $(\partial / \partial r)^{a}$ and timelike $(\partial / \partial t)^{a}$ vector fields switch into, respectively, timelike and spacelike vector fields at the black hole interior (i.e., $0<r<r_{s}$). For the sake of clarity, let us then rename the interior spatial coordinate $T$ by $x$, and the interior time coordinate $r$ by $t$. Thus, the Schwarzschild interior metric can be written as follows
\begin{equation}
ds^{2}=-\left(  \frac{2GM}{t}-1\right)^{-1}dt^{2}+\left(  \frac{2GM}
{t}-1\right)  dx^{2}+t^{2}\left(d\theta^{2}+\sin^{2}\theta d\phi^{2}\right),\label{schwarzschild interior}
\end{equation}
where $0<t<2GM$ and $x\in \mathbb{R}$. The singularity now corresponds to an {\emph{initial}} singularity at time $t=0$, resembling to a cosmological singularity. In fact, the Schwarzschild interior solution (\ref{schwarzschild interior}) belongs to the class of Kantowski-Sachs cosmological models \cite{Kantowski:1966te}
with homogeneous spatial sections $\Sigma\approx\mathbb{R}\times S^{2}$, i.e., Kantowski-Sachs models with symmetry group $\mathbb{R}\times SO(3)$, which are described by metrics of the form
\begin{equation}
ds^{2}=-N^{2}dt^{2}+X^{2}dx^{2}+Y^{2}\left(d\theta^{2}+\sin^{2}\theta d\phi^{2}\right).\label{KS general metric}
\end{equation}
Here, the metric coefficients $X$, $Y$ and the lapse function $N$, depend on the coordinate time $t$ only.

Since the Schwarzschild interior geometry can be understood as a Kantowski-Sachs spacetime with symmetry group $\mathbb{R}\times SO(3)$, let us consider the Kantowski-Sachs symmetry reduction of canonical general relativity in connection variables to get the connection description for the black hole interior. Following the procedure for the study of homogeneous models \cite{ashbolew} let us introduce an auxiliary metric $\mathring{q}_{ab}$ on the $3$-manifold $\mathbb{R}\times S^{2}$, with compatible triad $\mathring{e}^{a}_{i} $ and co-triad $\mathring{\omega}^{i}_{a}$ that are left-invariant under the action of the Killing fields of $\Sigma$. They carry the symmetry information but ignore the particularities about the minusuperspace model. The usual choice for the auxiliary metric{\footnote{For a different choice see \cite{Corichi:2015xia}.}}, which we will also consider here, is \cite{Ashtekar:2005qt,Modesto:2005zm,Bohmer:2007wi,Chiou:2008eg}:
\begin{equation}
\label{fid-metric}
\mathring{q}_{ab}dy^{a}dy^{b}=dx^{2}+d\theta^{2}+\sin^{2}\theta d\phi^{2}.
\end{equation}
The determinant of the fiducial metric (\ref{fid-metric}) is given by $\mathring{q}=\sin^{2}\theta$, so that the densitized triad $\mathring{E}^{a}_{i}=\sqrt{\mathring{q}}\: \mathring{e}^{a}_{i}$ reads
$\mathring{E}^{a}_{i}=\sin\theta  \mathring{e}^{a}_{i}$. The compatible densitized triad $\mathring{E}^{a}\partial_{a}=\mathring{E}^{a}_{i}\tau^{i}\partial_{a}$, which takes values in the dual of $su(2)$, and its corresponding $su(2)$-valued co-triad $\mathring{\omega}_{a}dy^{a}=\mathring{\omega} ^{i}_{a}\tau_{i}dy^{a}$, are explicitly given by
\begin{equation}
\label{fiducial-cotriad-triad}
\mathring{\omega}_{a}dy^{a}=\tau_{3}dx + \tau_{2} d\theta - \tau_{1}\sin \theta d\phi,\quad
\mathring{E}^{a}\partial_{a}=\tau_{3}\sin\theta\partial_{x}+\tau_{2}\sin\theta\partial_{\theta}-\tau_{1}\partial_{\phi},
\end{equation}
where $\tau_i$ are the standard generators of $SU(2)$, satisfying $[\tau_{i},\tau_{j}]=\epsilon_{ij}^{\:\:\: k}\tau_{k}$.

Note that integrals over $\mathbb{R}\times S^{2}$ involving spatially homogeneous quantities will generally diverge, given the non-compact character of the $x$-direction. To circumvent this feature, which, for instance, is an obstacle to properly calculate the Poisson brackets, one restricts $x$ to an interval of finite length $L$, w.r.t. the fiducial metric, and then perform all integrations over a finite-sized cell ${\cal{V}}_{0}=[0,L]\times S^{2}$ of fiducial volume $V_{0}=4\pi L$.

Now, by imposing the Kantowski-Sachs symmetry group $\mathbb{R}\times SO(3)$ in the full theory, one gets that the symmetric connection $A=A^{i}_{a}\tau_{i}dy^{a}$ and triad $E=E_{i}^{a}\tau^{i}\partial_{a}$ can be written, after gauge fixing of the Gauss constraint, as follows
\begin{equation}
\label{AE-in-fidu}
A =L^{-1}\,c \left( \mathring{\omega}_{x}dx\right)+b\left( \mathring{\omega}_{\theta}d\theta + \mathring{\omega}_{\phi}d\phi\right)+\Gamma, \quad
E=p_{c}\left(\mathring{E}^{x}\partial_{x}\right)+L^{-1}p_{b}\left(\mathring{E}^{\theta}\partial_{\theta}-\mathring{E}^{\phi}\partial_{\phi}\right).
\end{equation}
Here, $\Gamma=\Gamma^{i}_{a}\tau_{i}dy^{a}=\cos\theta\tau_{3}d\phi$ is the spin-connection compatible with the triad density $E$. Coefficients $b$, $c$, $p_{b}$ and $p_{c}$, which are all only functions of time, capture the non-trivial information about the symmetry reduced model. From Eqs.(\ref{fiducial-cotriad-triad})-(\ref{AE-in-fidu}), it follows that the Kantowsi-Sachs connection and triad are explicitly given by
\begin{equation}
A =
L^{-1}\, c\tau_{3}dx+b\tau_{2}d\theta-b\tau_{1}\sin\theta d\phi+\tau_{3}
\cos\theta d\phi, \label{A reducido}
\end{equation}
\begin{equation}
E= p_{c}\tau_{3}\sin\theta\,\partial_{x}+L^{-1}p_{b}\tau_{2}\sin
\theta\,\partial_{\theta}-L^{-1}p_{b}\tau_{1}\,\partial_{\phi}.
\label{E reducido}
\end{equation}

The phase space resulting from the symmetry reduction and gauge fixing processes is  the symplectic space ${\mathbf{\Gamma}}=[(b,p_{b},c,p_{c}),\Omega]$, with symplectic form \cite{Ashtekar:2005qt,Chiou:2008eg}
 \begin{equation}
 \label{reduced-symps}
 \Omega=\frac{1}{8\pi G\gamma}\int_{{\cal{V}}_0}d^{3}y\: \left(dA^{i}_{a}\wedge dE^{a}_{i}\right)=\frac{1}{2G\gamma}\left(dc\wedge dp_{c}+2db\wedge dp_{b} \right),
 \end{equation}
where $\gamma$ is the so-called Barbero-Immirzi parameter. The only non vanishing Poisson brackets defined by the reduced symplectic form (\ref{reduced-symps}) are
\begin{equation}
\label{poisson-brackets}
\{b,p_{b}\}=G\gamma, \quad
\{c,p_{c}\}=2G\gamma.
\end{equation}

Let us remark that, in fact, we will not consider the whole of the phase space $\bf{\Gamma}$. Indeed, as a part of the gauge-fixing procedure, $p_b$ can be chosen to be a strictly positive function \cite{Chiou:2008nm}, $p_{b}>0$. Besides, since distinct signs of $p_c$ correspond to regions with triads of opposite orientations \cite{Ashtekar:2005qt,Chiou:2008nm}, then $p_{c}$ can be chosen to be strictly positive as well.

Recall that a $3$-metric $q_{ab}$ is related with its compatible densitized triad $E^{a}_{i}$ by $qq^{ab}=E^{a}_{i}E^{b}_{i}$. Thus, from Eqs. (\ref{KS general metric})  and (\ref{E reducido}) it follows that $X^{2}=p_{b}^{2}/(L^{2}p_{c})$ and $Y^{2}=p_{c}$; i.e., in terms of the triad variables, the Kantowski-Sachs metric reads
\begin{equation}
ds^{2}=-N(t)^{2}dt^{2}+\frac{p_{b}^{2}}{L^{2}p_{c}}dx^{2}+p_{c}\left(
d\theta^{2}+\sin^{2}\theta d\phi^{2}\right)  . \label{metrica cinematica}
\end{equation}
Thus, with respect to the metric (\ref{metrica cinematica}), the length of the interval $[0,L]$ in the $x$-direction, the area of $S^{2}$ and the volume of the cell ${\cal{V}}=[0,L]\times S^{2}$, are respectively given by
\begin{equation}
\label{leng-are-vol}
l=p_{b}/\sqrt{p_{c}},\quad A_{S^{2}}=4\pi p_{c} , \quad V=4\pi p_{b}\sqrt{p_{c}}
\end{equation}

Now, in terms of the reduced canonical variables, $(b,p_{b})$ and $(c,p_{c})$, the Hamiltonian constraint takes the form \cite{Ashtekar:2005qt}
\begin{equation}
{\cal{C}}_{\rm{Ham}}=16\pi G\,H_{{\rm{class}}}=-\frac{8\pi N}{\gamma^{2}}\left[  2bc\sqrt{p_{c}}+(b^{2}+\gamma^{2}
)\frac{p_{b}}{\sqrt{p_{c}}}\right] , \label{class-H}
\end{equation}
which defines $H_{{\rm{class}}}$. By choosing the lapse function equal to one, from Eqs.(\ref{poisson-brackets}) and (\ref{class-H}) we obtain that the dynamics is dictated by
\begin{eqnarray}
\dot{b}  &  = & \{b,H_{{\rm{class}}}\} = -\frac{1}{2\gamma\sqrt{p_{c}}}\:\left(b^{2}+\gamma^{2}\right),\label{bdot} \\
\dot{c}  &  =& \{c,H_{{\rm{class}}}\} =\frac{1}{2\, \gamma\,p_{c}^{3/2}}\:\left(b^{2}p_{b}-2bcp_{c}+\gamma^{2}p_{b}\right),\label{cdot}\\
\dot{p}_{b}  &  =& \{p_{b},H_{{\rm{class}}}\} =\frac{1}{\gamma{\sqrt{p_{c}}}}\:\left(bp_{b}+cp_{c}\right),\label{pbdot}\\
\dot{p}_{c}  &  =&\{p_{c},H_{{\rm{class}}}\} =\frac{1}{\gamma}\:\left(2b\sqrt{p_{c}}\right). \label{pcdot}
\end{eqnarray}

A direct calculation shows that the Ricci and Kretschmann scalars of the metric (\ref{metrica cinematica}) are, respectively,
\begin{equation}
R=\frac{2p_{c}\ddot{p}_{b}+p_{b}\left(  2+\ddot{p}_{c}\right)
}{p_{b}p_{c}}, \label{ricciscalarkinematic}
\end{equation}
\begin{align}
K =R_{abcd}R^{abcd} &  =\frac{1}{2\text{$p_{b}$}^{2}\text{$p_{c}$}^{4}}\left[
4\text{$p_{c}$}^{2}\left(  3\dot{p}_{b}^{2}\dot{p}_{c}^{2}-4\text{$p_{c}$}
\dot{p}_{b}\dot{p}_{c}\text{$\ddot{p}_{b}$}+2\text{$p_{c}$}^{2}\text{$\ddot
{p}_{b}$}^{2}\right)  \right. \label{Kretschmannscalarkinematic}\\
&  +4\text{$p_{b}p_{c}$}\left(  \text{$p_{c}\ddot{p}_{b}$}\left(  3\dot{p}
_{c}^{2}-2\text{$p_{c}$}\ddot{p}_{c}\right)  +\dot{p}_{b}\left(  -4\dot{p}
_{c}^{3}+2\text{$p_{c}$}\dot{p}_{c}\ddot{p}_{c}\right)  \right) \nonumber\\
&  \left.  +\text{$p_{b}$}^{2}\left(  7\dot{p}_{c}^{4}+2\text{$p_{c}$}\dot
{p}_{c}^{2}\left(  2-5\ddot{p}_{c}\right)  +\text{$p_{c}$}^{2}\left(
8+6\ddot{p}_{c}^{2}\right)  \right) \right]. \nonumber
\end{align}
It is not difficult to see, by using Eqs.(\ref{pbdot})-(\ref{pcdot}), that $\ddot{p}_{b} =bc/\gamma^{2}$ and $\ddot{p}_{c}    =(b/\gamma)^{2}-1$. Substituting the latter expressions into Eqs.(\ref{ricciscalarkinematic})-(\ref{Kretschmannscalarkinematic}), as well as by imposing the constraint (\ref{class-H}) and employing the dynamical equations (\ref{bdot})-(\ref{pcdot}), we get that on the constraint surface
\begin{equation}
\label{class-RK}
R=0, \qquad K=\frac{12}{\gamma^{4}}\left( \frac{b^{2}+\gamma^{2}}{p_{c}}\right)^{2}.
\end{equation}
Solving the equations (\ref{bdot}) and (\ref{pcdot}), (see Eq. (\ref{pcb}) below) one gets $(b^{2}+\gamma^{2})^{2}=a_{0}/p_{c}$, with $a_{0}$ being a constant depending on initial conditions and $\gamma$. Hence the Kretschmann scalar goes as $1/p_{c}^{3}$. Explicitly,
\begin{equation}
\label{class-RK-2}
K=\frac{12a_{0}}{\gamma^{4}p_{c}^{3}}.
\end{equation}
Thus, the Kretschmann scalar blows up as $p_{c}$ tends to zero, corresponding to the classical singularity.

Let us now examine the solutions to the system (\ref{class-H})-(\ref{pcdot}), and let us inspect the behavior of the expansion scalar and shear.

\subsection{Solutions, expansion scalar and shear}

\label{Classical Qualitative Analysis}

To start, notice that $cp_{c}$ is a constant on the constraint surface. Indeed, from Eqs.(\ref{cdot}) and (\ref{pcdot}) it follows that
\begin{equation}
\{cp_{c},H_{{\rm{class}}}\}=-\frac{\gamma}{16\pi}{\cal{C}}_{\rm{Ham}}.
\end{equation}
Let us denote the constant $cp_{c}$ by $\gamma K_{c}$; i.e.,
\begin{equation}
\label{Kc}
cp_{c}=\gamma K_{c}.
\end{equation}
Since the sign flipping $K_{c}\to -K_{c}$ is associated to the time reversal $t\to -t$ \cite{Chiou:2008nm}, the two regions, $K_{c}>0$ and $K_{c}<0$, are causally disconnected. Let us consider the region $K_{c}>0$, as in \cite{Chiou:2008nm} (for the sake of completeness, we will also discuss the opposite choice, $K_{c}<0$, at the end of the present section). Provided that $p_{c}> 0$, we have that $c$ must be a strictly positive function of time $t$. Since ${\cal{C}}_{\rm{Ham}}=0$ implies that $b$ and $c$ must have opposite signs (see Eq. (\ref{class-H})), we then conclude that $b<0$. On the other hand, viewed as a quadratic equation in $b$, the constraint (\ref{class-H}) has discriminant
$D=\gamma^{2}(K^{2}_{c}-p^{2}_{b}),$
where we have used (\ref{Kc}). Thus, to keep $b$ real, $D$ must be non-negative, which implies that $p_{b}$ is bounded from above
\begin{equation}
p_{b}\leq K_{c}. \label{cotapbclasica}
\end{equation}

Now, note that Eqs.(\ref{bdot}) and (\ref{pcdot}) are actually decoupled equations from the rest of Hamilton's equations. Thus, we have that
\begin{equation}
\frac{dp_{c}}{db}=-\frac{4bp_{c}}{\left(b^{2}+\gamma^{2}\right)}, \label{dpc/db}
\end{equation}
which solution is given by
\begin{equation}
p_{c}=p_{c0}\left(  \frac{b_{0}^{2}+\gamma^{2}}{b^{2}+\gamma^{2}}\right)
^{2}. \label{pcb}
\end{equation}
Here, $p_{c0}$ and $b_{0}$ stand for initial conditions at $t=t_{0}$. Since $\dot{b}<0$ (c.f. Eq.(\ref{bdot})), we have that $b$ is a monotonically decreasing function of time $t$ and, by virtue of Eq.(\ref{pcb}), so is $p_c$. Now, substituting Eq.(\ref{pcb}) in Eq.(\ref{bdot}), we get that
\begin{equation}
\label{b-dot-sol}
\dot{b}=-\alpha_{0}\left(b^{2}+\gamma^{2}\right)^{2}, \qquad \alpha_{0}=\left[2\gamma \sqrt{p_{c0}}\,(b_{0}^{2}+\gamma^{2})\,\right]^{-1}.
\end{equation}
So that,
\begin{equation}
\label{b-sol}
g(b)=-2\gamma^{3}\alpha_{0}(t-t_{0})+g(b_{0}),\qquad g(s)=\frac{\gamma s}{(s^{2}+\gamma^{2})}+\arctan\left(\frac{s}{\gamma}\right).
\end{equation}
Clearly, $g<0$ for all $b<0$. It is easy to see that $g$ decreases monotonically as $b$ evolves in time, which in turn implies that $g$ is a monotonous decreasing function of $t$. Indeed, a straightforward calculation shows that $dg/dt=-2\gamma^{3}\alpha_{0}$; i.e., $g$ is a monotonically decreasing function of time. Note, in addition, that $-\pi /2 < g$. Thus, the relationship (\ref{b-sol}) makes sense (i.e., it is a well-defined relationship providing $b(t)$ at each given $t$ value) only if $2\gamma^{3}\alpha_{0}\Delta t<\pi/2+g(b_{0})$, where $\Delta t =(t-t_{0})$. Hence, as $\Delta t$ approaches to the maximal value $\Delta t_{f}$,
\begin{equation}
\label{time singularity}
\Delta t _{f}=\frac{1}{2\gamma^{3}\alpha_{0}}\left[\frac{\pi}{2}+g(b_{0})\right],
\end{equation}
the solution $b(t)$ will tend to $b\to -\infty$. Then, from Eq.(\ref{pcb}) it follows that  the solution $p_{c}(t)$ will tend to zero as $\Delta t$ approaches $\Delta t _{f}$. By substituting Eq.(\ref{pcb}) into $c=\gamma K_{c}/p_{c}$  [see Eq.(\ref{Kc})], we get that
\begin{equation}
c=\frac{\gamma K_{c}}{p_{c0}}\left(  \frac{b^{2}+\gamma^{2}}{b_{0}^{2}+\gamma^{2}}\right)
^{2}. \label{cb}
\end{equation}
Hence, the solution $c$ must tend to infinity as  $\Delta t \to \Delta t _{f}$. By using Eq.(\ref{Kc}) and Eq.(\ref{pcb}) into the constraint equation ${\cal{C}}_{\rm{Ham}}=0$ [see Eq.(\ref{class-H})], we obtain that
\begin{equation}
\label{pbb}
p_{b}=-2\gamma K_{c}\frac{b}{(b^{2}+\gamma^{2})}.
\end{equation}
Thus, in the limit when $\Delta t$ tends to $\Delta t_{f}$, the solution $p_b$ goes to zero as $1/\vert b \vert$. (Note that $p_{b}/p_{c}$ diverges as $\vert b \vert^{3}$ when $\Delta t \to \Delta t_{f}$).

Relations (\ref{pcb}), (\ref{b-sol}), (\ref{cb}) and (\ref{pbb}) provide the solution to the system (\ref{class-H})-(\ref{pcdot}). Once the solution $b(t)$ is obtained from Eq.(\ref{b-sol}), the rest of solution functions, namely $p_{c}(t)$, $c(t)$ and $p_{b}(t)$, are determined by substituting $b(t)$ into equations (\ref{pcb}), (\ref{cb}) and (\ref{pbb}), respectively. The time domain of the solution functions is $t\in [t_{0},t_{0}+\Delta t]$, with $\Delta t \leq \Delta t_{f}$; given an initial data $(b_{0},p_{b0},c_{0},p_{c0})$ at $t=t_0$, with $b_{0}\in \mathbb{R}^{-}$, $p_{b0}\in (0,K_{c})$, and $c_{0},p_{c0}\in \mathbb{R}^{+}$, the solution will tend to the `endpoint'
\begin{equation}
(b\rightarrow-\infty,p_{b}\to 0 ,c\to \infty,p_{c}\to0), \label{endpt}
\end{equation}
as $t$ approaches $t_{f}=t_{0}+\Delta t_{f}$. From (\ref{leng-are-vol}), (\ref{pcb}) and (\ref{pbb}), it follows that the length $l$, the area $A_{S^{2}}$ and the cell volume $V$ will behave as $l\sim \vert b \vert$, $A_{S^{2}}\sim 1/b^{4}$ and $V\sim 1/\vert b \vert^{3}$ as $t\to t_{f}$.

Let us now consider the congruence of timelike geodesics defined by comoving observers in Kantowski-Sachs spacetime (\ref{metrica cinematica}), with $N=1$; that is, the associated vector field to the congruence is $\xi^{a}=(\partial/\partial t)^{a}$. Thus,
the expansion scalar $\theta$ corresponds to $\dot{V}/V$, where $V=4\pi p_{b}\sqrt{p_{c}}$ is the congruence's cross-sectional volume. A simple calculation shows that
\begin{equation}
\label{scalar-factor-r}
\theta=\frac{\dot{p}_{b}}{p_{b}}+\frac{\dot{p}_{c}}{2p_{c}}.
\end{equation}
By using Eqs.(\ref{bdot}), (\ref{pcdot}), (\ref{pcb}) and (\ref{pbb}), as well as calculating $\dot{p}_b$ by employing Eq.(\ref{pbb}), it is not difficult to see that
\begin{equation}
\label{class-exp-scalar}
\theta=\alpha_{0}\left(\frac{b^{2}+\gamma^{2}}{b}\right)\left(3b^{2}-\gamma^{2}\right).
\end{equation}
Clearly, the expansion scalar is a monotonically decreasing function of time (recall that $b<0$ and that it is a monotonically decreasing function of $t$). What is more, irrespective of the initial condition $b_0$, $\theta\to -\infty$ as $t$ tends to the maximal value $t_{f}$ (i.e., the volume shrinks to zero as $t\to t_{f}$, invariably). At finite proper time $\Delta t_{f}$, the congruence of timelike geodesics develop a caustic, and the cell volume becomes zero; in fact, the geodesics of the congruence turn out to be inextendible (i.e., incomplete). Note, however, that depending upon the initial condition $b_0$, there would be a stage where the volume, in fact, will enlarge. Indeed, observe that
$\theta$ is strictly positive for $-\gamma/\sqrt{3}<b<0$, it is zero at $b=-\gamma/\sqrt{3}$, and it is strictly negative for $b<-\gamma/\sqrt{3}$. Thus, if the initial condition $b_{0}$ is in $(-\gamma/\sqrt{3},0)$, we will have that the volume will increase up to a maximum value $V_{{\rm{max}}}$, at $b=-\gamma/\sqrt{3}$, and afterwards it will monotonically decrease up to zero volume, at $t=t_{f}$ (time at which $p_{c}$ vanishes and the Kretschmann scalar blows up).

A direct calculation shows that the time derivative of the expansion scalar (\ref{class-exp-scalar}) is given by
\begin{equation}
\label{class-theta-dot}
\dot{\theta} =-\alpha_{0}^{2}\left(\frac{b^{2}+\gamma^{2}}{b}\right)^{2} \left(9b^{4}+2\gamma^{2}b^{2}+\gamma^{4}\right),
\end{equation}
where we have used (\ref{b-dot-sol}). Since $\dot{\theta}$ is strictly negative, the expansion scalar $\theta$ is a monotonically decreasing function of $t$ (as we have already pointed out).

The shear, which is given by (see for instance \cite{Joe:2014tca})
\begin{equation}
\label{shear-r}
\sigma^{2}=\frac{1}{2}\sigma_{ab}\sigma^{ab}  =\frac{1}{3}\left(  \frac{\dot{p}_{b}}{p_{b}}-\frac{\dot
{p}_{c}}{p_{c}}\right)^2 ,
\end{equation}
reads explicitly as follows
\begin{equation}
\label{class-shear}
\sigma^{2}=\frac{\alpha_{0}^{2}}{3}\left(\frac{b^{2}+\gamma^{2}}{b}\right)^{2}\left(3b^{2}+\gamma^{2}\right)^{2}.
\end{equation}
From Eqs.(\ref{class-exp-scalar}), (\ref{class-theta-dot}) and (\ref{class-shear}), it is a simple exercise to see that
$\dot{\theta}=-(1/3)\theta^{2}-2\sigma^{2}$, which is nothing but Raychaudhuri's equation. (Recall that the congruence is hypersurface orthogonal, so that there is no rotational term. In addition, the term $R_{ab}\xi^{a}\xi^{b}=R_{00}$ is identically zero on shell).

Let us remark that by considering $K_{c}<0$, one gets that $c<0$ (since $p_{c}$ is strictly positive) and that $b>0$ (since $b$ and $c$ must have opposite signs). Exactly as above, it is shown that $p_{b}\leq \vert K_{c} \vert$. The expressions for $p_{c}$, $c$ and $p_{b}$ [respectively, Eqs. (\ref{pcb}), (\ref{cb}) and (\ref{pbb})] will be the same ones, though now with $K_{c}<0$ and $b>0$. Of course, equation (\ref{b-dot-sol}) governing the dynamics of $b$ is the same one, so is its solution (\ref{b-sol}); but now with $g>0$, provided that $b>0$. Since $\dot{b}<0$, then $b$ is a monotonically decreasing function of $t$, which implies that $g$ decreases in time. Rather than technical, the important difference between conventions $K_{c}>0$ and $K_{c}<0$ is conceptual. Recall that associated to the sign of $K_{c}$ is a time reversal, so it is  natural to write the solution to Eq.(\ref{b-dot-sol}) as
\begin{equation}
\label{b-sol-neg-k}
g(b)=2\gamma^{3}\alpha_{0}(t_{0}-t)+g(b_{0}),\qquad g(s)=\frac{\gamma s}{(s^{2}+\gamma^{2})}+\arctan\left(\frac{s}{\gamma}\right),
\end{equation}
with $t_{0}>t$ (for instance, $t_0$ would denote the `present time', whereas $t$ stands for an earlier time). Clearly, $0<g<\pi/2$ for all $b\in \mathbb{R}^{+}$, and it is a monotonically decreasing function of $t$. Equation (\ref{b-sol-neg-k}) implies that $b(t)$ will tend to $b\to \infty$ as $t\to 0$; so that, $p_{c}\to 0$, $c\to -\infty$ and $p_{b}\to 0$ as $t$ approaches zero. In particular, we have that as $t\to 0$, the Kretschmann scalar will diverge as $b^{8}$, whereas the cell volume will collapse to zero as $V\sim 1/b^{3}$ .

The explicit expression for the expansion scalar of a congruence of timelike geodesics constructed from comoving observers (i.e., with associated vector field $\xi^{a}=(\partial /\partial t )^{a}$) is also given by Eq.(\ref{class-exp-scalar}). Note that $\theta>0$ for $0<t<t_{*}$, where $t_{*}$ is `the cosmological time' at which $b(t_{*})=\gamma/\sqrt{3}$, $\theta$ is zero at $t_{*}$, and it is strictly negative for $t>t_{*}$.  The expansion scalar is, in fact, a monotonically decreasing function in time. Note, in addition, that $\theta\to \infty$ at $t=0$. The shear and the time derivative of $\theta$ have, of course, exactly the same expressions as above.

Now, by considering the congruence of `past-directed comoving world lines', which associated vector field is $\xi^{a}=-(\partial /\partial t )^{a}$, one gets that the backward in time (BT) expansion scalar is given by
\begin{equation}
\label{expansion}
\theta_{\rm{BT}}(\tau')=-\alpha_{0}\left(\frac{b^{2}+\gamma^{2}}{b}\right)\left(3b^{2}-\gamma^{2}\right),
\end{equation}
where $b$ is evaluated at $(t_{0}-\tau')$ and the parameter $\tau'$ is from zero to $t_{0}$ (so that $t=0$ corresponds to the limit $\tau'\to t_{0}$). Thus, $\theta_{\rm{BT}}(\tau')\to -\infty$ within a finite `proper time' $t_{0}$; that is to say, the volume shrinks to zero as we approach the `initial singularity'.

\section{Quantum Schwarzschild interior}

\label{QKS}

In this section we implement a path integral quantization of the Schwarzschild black hole interior. To do so we make use of  its Kantowski-Sachs form as well as the similarity of the latter with Bianchi I model, both being anisotropic homogeneous models. In particular, rather than starting from scratch with the LQC techniques (see e.g. \cite{Ashtekar:2011ni, Ashtekar:2009vc,Ashtekar:2005qt}) we perform a sequence of transformations in phase space that ultimately allow us to identify adequate holonomy type variables for the KS model at the hamiltonian level, first, and, second, to introduce its path integral quantization. We follow closely the analyisis for Bianchi I in  \cite{Liu:2012xp}. In this way an effective action, and hence an effective hamiltonian, can be identified from the transition amplitude of the quantum KS model. This effective hamiltonian will be used in the next
section to analyze the effective geometry for Schwarzschild interior.

Let us notice that an effective Hamiltonian was proposed in \cite{Bohmer:2007wi,Chiou:2008eg} motivated by several previous results. Essentially it was defined by  the heuristic replacements  $b\rightarrow \sin(\mu_bb)/\mu_b$ and $c\rightarrow \sin(\mu_cc)/\mu_c$ in (\ref{class-H}), where the use  of $\mu_b,\mu_c$ follows from their appearence  in a form of the hamiltonian constraint in which curvature terms are expressed by holonomies along elementary squares of length related to them  \cite{Ashtekar:2011ni}. Our approach, that was described above, is different from this simple replacement, however, we will regain the effective hamiltonian of \cite{Chiou:2008eg}. Now various criteria turn out to be necessary in the holonomy version of the construction  \cite{Ashtekar:2011ni}. They include (i) the area of the elementary squares used
in the holonomies should not be less than the minimum area gap
$\Delta$ found in the spectrum of the area operator of the full theory, (ii) physical quantities must be independent of a  fiducial
metric introduced along the analysis, as well as  (iii) avoidance of large quantum gravity effects in
classical regimes \cite{Ashtekar:2006uz}.  Such criteria led to propose the following form of the $\mu^{\prime}s$  \cite{Bohmer:2007wi, Chiou:2008eg, Ashtekar:2009vc}
\footnote{The two other possibilities that were explored, do not satisfy all
these criteria, yielding inconsistent physics. See for example
\cite{Joe:2014tca} and references therein.}
\begin{equation}
\mu_{b}:=\sqrt{\frac{\Delta}{p_{c}}}\quad\text{and}\quad\mu_{c}:=\frac
{\sqrt{\Delta p_{c}}}{p_{b}}.\label{mu
\end{equation}

Let us begin with the hamiltonian description and consider the classical constraint (\ref{class-H}). It is convenient to define first the following set of canonical variables \cite{Liu:2012xp}
\begin{equation}
\lambda_{b}:=\frac{p_{b}}{\sqrt{G\hbar}},\quad\lambda_{c}:=\sqrt{\frac{{p_{c}
}{{G\hbar}}},\label{muprimvar}\\
\varphi_{b}:=\frac{b}{\sqrt{G\hbar}},\quad\varphi_{c}:=c\sqrt{\frac{{p_{c}
}{{G\hbar}}},
\end{equation}
which Poisson brackets take the form
\begin{equation} \label{PBphilambda}
\hbar\{\varphi_{l},\lambda_{j}\}=\gamma\delta_{l,j}\qquad l,j=b,c.
\end{equation}
Let us observe that the following variables
\begin{equation}
k_{b}:=\frac{\varphi_{b}}{\lambda_{c}}=\frac{b\mu_{b}}{\sqrt{\Delta}},\quad
k_{c}:=\frac{\varphi_{c}}{\lambda_{b}}=\frac{c\mu_{c}}{\sqrt{\Delta
},\label{muprimvar2
\end{equation}
have exponential forms
\begin{eqnarray} \label{U}
U_b := \text{ e}^{\text{i}\sqrt{\Delta}k_{b}} = \text{ e}^{\text{i}b\mu_b}, \quad
U_c :=\text{ e}^{\text{i}\sqrt{\Delta}k_{c}} = \text{ e}^{\text{i}c\mu_c},
\end{eqnarray}
that are amenable for the application of LQC techniques.
In the sequel the following identity will be of much help
\begin{equation}
\frac{\sin(\sqrt{\Delta}k_{l})}{\sqrt{\Delta}}=\frac{U_{l}-(U_{l})^{\ast}
}{2\text{i}\sqrt{\Delta}},  \quad l=b,c.\label{representacion seno
\end{equation}
Using the set of variables $\lambda_b,\lambda_c, k_b,k_c$, given in eqs. (\ref{mu}), (\ref{muprimvar}) and (\ref{muprimvar2}), in
the classical constraint  (\ref{class-H}) yields the following form
\begin{equation} \label{Hmulk}
H_{\mu}=-\frac{\hbar}{2\gamma^{2}}\left[ 2(\lambda_bk_b)(\lambda_ck_c)+\frac{\lambda_{c}}{\lambda_{b}}(\lambda_bk_b)^{2}+\frac{\gamma^{2}}{G\hbar}\frac{\lambda_{b}}{\lambda_{c}}\right].
\end{equation}
Next we consider a small argument approximation for the  $(\lambda_bk_b)$ and $(\lambda_c k_c)$  factors in (\ref{Hmulk}) through
the following relations
\begin{eqnarray}
(\lambda_b k_b) &\approx& \frac{1}{\sqrt{\Delta}} \left[ \lambda_b\sin(\sqrt{\Delta}k_b)  \right] :=\Phi_b, \label{Phib}\\
(\lambda_c k_c) &\approx& \frac{1}{\sqrt{\Delta}} \left[ \lambda_c\sin(\sqrt{\Delta}k_c)  \right] :=\Phi_c, \label{Phic}
\end{eqnarray}
where the $\sin(\sqrt{\Delta}k_b)$ and $\sin(\sqrt{\Delta}k_c)$ will be understood according to (\ref{representacion seno}) so that our
elementary variables will be $\lambda_l,U_l, l=b,c$. Their Poisson brackets can be obtained upon combination of (\ref{PBphilambda}),
(\ref{muprimvar2}) and (\ref{U}). The result is
\begin{eqnarray} \label{PBlambdaU}
\left\{  {\lambda}_b, {U}_b\right\}&=&\frac{\ell_0}{i\hbar}   \frac{{U}_b}{{\lambda}_c}, \nonumber\\
\left\{  {\lambda}_c, {U}_c\right\} &=& \frac{\ell_0}{i\hbar}  \frac{{U}_c}{{\lambda}_b} ,
\end{eqnarray}
with $\ell_0:=\gamma\sqrt{\Delta}$.

Let us now proceed to quantization. Our elementary quantum observables will be $\widehat{\lambda}_{b},\widehat
{\lambda}_{c},\hat{U_b}$ and $\hat{U_c}$.  The
Hilbert space of this system will be $\mathcal{H}^{(2)}_{\mathrm Poly}= \mathcal{H}_{\mathrm Poly}\otimes\mathcal{H}_{\mathrm Poly}$ with
$\mathcal{H}_{\mathrm Poly}=L^{2}(\mathbb{R}
_{\text{Bohr}},\text{d}\mu_{H})$ and  $\mathbb{R}_{\text{Bohr}}$ is the
Bohr compactification of the real line and $\text{d}\mu_{H}$ is its Haar's measure
\cite{Velhinho:2007gg}. We use a basis of eigenkets $|\vec{\lambda}\rangle:=|\lambda_{b},\lambda
_{c}\rangle$ of the operators $\hat{\lambda}_{b}$ and $\hat{\lambda}_{c}$.
These basis satisf
\begin{equation}
\langle\vec{\lambda}^{\prime}|\vec{\lambda}\rangle=\delta_{\vec{\lambda
}^{\prime},\vec{\lambda}},
\end{equation}
where $\delta_{\lambda^{\prime},\lambda}$ is a Kronecker delta. To represent $\hat{U}'s$ we make use of the commutation relations
\begin{eqnarray}
\left[  \hat{\lambda}_b, \hat{U}_b\right] &=& \ell_0  \frac{\hat{U}_b}{\hat{\lambda}_c}, \\
\left[  \hat{\lambda}_c, \hat{U}_c\right] &=& \ell_0  \frac{\hat{U}_c}{\hat{\lambda}_b} ,
\end{eqnarray}
which follow from the application of Dirac's prescription to the Poisson brackets (\ref{PBlambdaU}).
They lead to
\begin{equation}
\hat{U}_{b}|\vec{\lambda}\rangle=|\lambda_{b}+\ell_{0}/\lambda_{c},\lambda
_{c}\rangle,\\
\qquad\hat{U}_{c}|\vec{\lambda}\rangle=|\lambda_{b},\lambda_{c}+\ell
_{0}/\lambda_{b}\rangle.
\end{equation}
Now we proceed to implement the quantum version of (\ref{Phib}) and (\ref{Phic}) as \cite{Ashtekar:2010gz}
\begin{equation}
\hat{\Phi}_{b}:=\frac{1}{\sqrt{\Delta}}\left[  \sqrt{\hat{\lambda}_{b}
\widehat{\sin{\sqrt{\Delta}k_{b}}}\sqrt{\hat{\lambda}_{b}}\right]
\quad\text{and}\quad\hat{\Phi}_{c}:=\frac{1}{\sqrt{\Delta}}\left[  \sqrt
{\hat{\lambda}_{c}}\widehat{\sin{\sqrt{\Delta}k_{c}}}\sqrt{\hat{\lambda}_{c
}\right]  ,\label{phi
\end{equation}
whose action on the basis ar
\begin{align}
\hat{\Phi}_{b}|\vec{\lambda}\rangle &  =-\frac{\text{i}\sqrt{\lambda_{b}
}{2\sqrt{\Delta}}\left[  \sqrt{\lambda_{b}+\ell_{0}/\lambda_{c}}|\lambda
_{b}+\ell_{0}/\lambda_{c},\lambda_{c}\rangle-\sqrt{\lambda_{b}-\ell
_{0}/\lambda_{c}}|\lambda_{b}-\ell_{0}/\lambda_{c},\lambda_{c}\rangle\right]
,\label{phiactua}\\
\hat{\Phi}_{c}|\vec{\lambda}\rangle &  =-\frac{\text{i}\sqrt{\lambda_{c}
}{2\sqrt{\Delta}}\left[  \sqrt{\lambda_{c}+\ell_{0}/\lambda_{b}}|\lambda
_{b},\lambda_{c}+\ell_{0}/\lambda_{b}\rangle-\sqrt{\lambda_{c}-\ell
_{0}/\lambda_{b}}|\lambda_{b},\lambda_{c}-\ell_{0}/\lambda_{b}\rangle\right]
. \label{phiactua2}
\end{align}
Hence the quantum version of the hamiltonian constraint  (\ref{Hmulk}), using (\ref{Phib}) and (\ref{Phic}) first at the classical level and then
 their quantum version (\ref{phi}), becomes
\begin{equation}
\hat{H}_{\mu}=-\frac{\hbar}{2\gamma^{2}}\left[  \hat{\Phi}_{b}\hat{\Phi
_{c}+\hat{\Phi}_{c}\hat{\Phi}_{b}+(\hat{\Phi}_{b})^{2}\frac{\hat{\lambda}_{c
}{2\hat{\lambda}_{b}}+\frac{\hat{\lambda}_{c}}{2\hat{\lambda}_{b}}\hat{(\Phi
}_{b})^{2}+\frac{\gamma^{2}}{G\hbar}\frac{\hat{\lambda}_{b}}{\hat{\lambda_{c
}}\right] , \label{Hmuprima
\end{equation}
for which a symmetric ordering has been introduced. Its building blocks are defined to act upon the eigenbasis
$|\vec\lambda\rangle$, according to (\ref{phiactua}) and (\ref{phiactua2}) while the $\hat{\lambda}$ factors act diagonally.

This hamiltonian will be used now to obtain Feynman's formula for the propagator to go from state $|\vec{\lambda}_{i};\tau_{i}\rangle$ at proper time $\tau_i$ to $|\vec{\lambda}_{f};\tau_{f}\rangle$ at time $\tau_f > \tau_i$. It takes the form
\begin{equation}
\langle
\vec{\lambda}_{f};\tau_{f} |\vec{\lambda}_{i};\tau_{i}\rangle=\langle
\vec{\lambda}_{f}|\text{ e}^{-\text{i}\Delta\tau\hat{H}_{\mu}/\hbar
|\vec{\lambda}_{i}\rangle, \qquad \Delta \tau= \tau_f-\tau_i. \label{aprima1
\end{equation}

To calculate explicitly such propagator we consider as usual a partition of the time interval $\Delta \tau$ and split, accordingly, the time evolution operator as
\begin{equation} \label{Nexp}
\text{ e}^{-\text{i}\Delta\tau\hat{H}_{\mu}/\hbar}=\prod_{n=0}^{N-1}\text{
e}^{-\text{i}\epsilon\hat{H}_{\mu}/\hbar},\quad\text{where}\quad
N\epsilon=\Delta\tau.
\end{equation}
Then using
\begin{equation}
\hat{\mathbb{I}}=\sum_{\vec{\lambda}_{n}}|\vec{\lambda}_{n}\rangle\langle
\vec{\lambda}_{n}|,
\end{equation}
together with  (\ref{Nexp}) allow us to rewrite  (\ref{aprima1}) a
\begin{equation}
\langle \vec{\lambda}_{f};\tau_{f} |\vec{\lambda}_{i};\tau_{i}\rangle=\sum
_{\vec{\lambda}_{N-1},...,\vec{\lambda}_{1}}\prod_{n=0}^{N-1}\langle
\vec{\lambda}_{n+1}|\text{e}^{-\text{i}\epsilon\hat{H}_{\mu}/\hbar
|\vec{\lambda}_{n}\rangle, \label{aprima
\end{equation}
where $\vec{\lambda}_{f}=\vec{\lambda}_{N}$ y $\vec{\lambda}_{i}=\vec{\lambda
}_{0}$. Next we consider that for small $\epsilon
\begin{equation}
\langle\vec{\lambda}_{n+1}|\text{e}^{-\text{i}\epsilon\hat{H}_{\mu}/\hbar
}|\vec{\lambda}_{n}\rangle=\delta_{\vec{\lambda}_{n+1},\vec{\lambda}_{n
}-\text{i}\frac{\epsilon}{\hbar}\langle\vec{\lambda}_{n+1}|\hat{H}_{\mu
|\vec{\lambda}_{n}\rangle+\mathcal{O}(\epsilon^{2}). \label{aprimepartial
\end{equation}
The matrix elements $\langle\vec{\lambda}_{n+1}|\hat{H}_{\mu}|\vec{\lambda
}_{n}\rangle$ can be calculated using (\ref{phiactua}) and (\ref{Hmuprima}):
\begin{align} \label{LHL}
\langle\vec{\lambda}_{n+1}|\hat{H}_{\mu}|\vec{\lambda}_{n}\rangle
=\frac{\hbar}{8\gamma^{2}\Delta}&\left\{\sqrt{\lambda_{b,n+1}\lambda_{b,n}}\sqrt{\lambda_{c,n}\lambda_{c,n+1}}\left(P_n +Q_n\right)+\frac{4\gamma^{2}\Delta}{G\hbar}\frac{\lambda_{b,n}}{\lambda
_{c,n}}\delta_{\vec{\lambda}_{n}\vec{\lambda}_{n+1}}\right. \nonumber\\
&\hspace{1cm} \left.  +\sqrt{\lambda_{b,n}\lambda_{b,n+1}}\frac{\lambda_{b,n}+\lambda_{b,n+1}}{2}\left(  \frac{\lambda_{c,n}}{2\lambda_{b,n}}+\frac{\lambda_{c,n+1}}{2\lambda_{b,n+1}}\right)R_n \right\}.
\end{align}
where
\begin{align} \label{LHL}
P_n&=\left(  \delta_{\lambda_{b,n+1},\lambda_{b,n}+\ell_{0}/\lambda_{c,n+1}}-\delta_{\lambda_{b,n+1},\lambda_{b,n}-\ell_{0}
/\lambda_{c,n+1}}\right) \left(  \delta_{\lambda_{c,n+1},\lambda_{c,n}+\ell_{0}/\lambda_{b,n}}-\delta_{\lambda_{c,n+1},\lambda_{c,n}-\ell
_{0}/\lambda_{b,n}}\right),\nonumber\\
Q_n&=\left(  \delta_{\lambda_{b,n+1},\lambda_{b,n}+\ell_{0}/\lambda_{c,n}}-\delta_{\lambda_{b,n+1},\lambda_{b,n}-\ell_{0}/\lambda_{c,n} }\right)  \left(  \delta_{\lambda_{c,n+1},\lambda_{c,n}+\ell_{0}/\lambda_{b,n+1}}-\delta_{\lambda_{c,n+1},\lambda_{c,n}-\ell_{0
/\lambda_{b,n+1}}\right), \nonumber\\
R_n&=\left(  \delta_{\lambda_{b,n+1},\lambda_{b,n}+2\ell_{0}/\lambda_{c,n+1}}-2\delta_{\lambda_{b,n+1},\lambda_{b,n}}+\delta
_{\lambda_{b,n+1},\lambda_{b,n}-2\ell_{0}/\lambda_{c,n+1}}\right)\delta_{\lambda_{c,n+1},\lambda_{c,n}}.
\end{align}

At this point we can see from (\ref{phiactua})-(\ref{phiactua2}) and hence in the matrix elements of $\hat{H}_{\mu}$  given by Eq. (\ref{Hmuprima}), that states supported on a regular  (equally spaced) $\vec{\lambda}-$lattice do not fit into our quantum KS model.
This is a difficulty that also appears in the Bianchi I models and thus, to proceed further,  we can use the approximation proposed in \cite{Liu:2012xp} for that case. It
consists of exploiting the fact that we are looking for a continuous yet quantum effective approximation \cite{Varadarajan:1999it,Ashtekar:2002vh}. Hence, effectively, one can  replace at leading order the Kronecker deltas by Dirac's in (\ref{LHL}). This implies
that one is approximating at leading order a description from ${\cal H}_{\mathrm Poly}^{(2)}$ to ${\cal H}_{\mathrm Sch}^{(2)}={\cal H}_{\mathrm Sch}\otimes {\cal H}_{\mathrm Sch}, {\cal H}_{\mathrm Sch}=L^2(\mathbb{R},dx)$, so that $\vec{\lambda}$ is now a
continuous variable. Within this approximation it is useful to adopt the following integral form of Dirac's delta
\begin{equation}
\delta(\lambda_{n+1}-\lambda_{n})=\frac{1}{2\pi\gamma}\int_{\mathbb{R
}\text{d}\varphi_{n+1}\text{ e}^{-\text{i}\varphi_{n+1}(\lambda_{n+1
-\lambda_{n})/\gamma}.
\end{equation}
Then, eq. (\ref{aprimepartial}) can be expressed as
\begin{align}
\langle\vec{\lambda}_{n+1}|\text{e}^{-\text{i}\epsilon\hat{H}_{\mu}
|\vec{\lambda}_{n}\rangle=&\left(  \frac{1}{2\pi\gamma}\right)  ^{2}\int
d\vec{\varphi}_{n+1}\text{ e}^{-\text{i}\vec{\varphi}_{n+1}(\vec{\lambda}_{n+1
-\vec{\lambda}_{n})/\gamma}\nonumber\\
&\hspace{1.5cm}\times\left\{  1+\text{i}\frac{\epsilon}{2\gamma
^{2}\Delta}\left[M_n+N_n +  L_n+\frac{\gamma^{2}\Delta}{G\hbar
\frac{\lambda_{b,n}}{\lambda_{c,n}}\right]  \right\}  +\mathcal{O
(\epsilon^{2}),
\end{align}
where
\begin{align}
M_n&=\sqrt{\lambda_{b,n+1}\lambda_{b,n}}\sqrt{\lambda_{c,n}\lambda_{c,n+1}}\sin(\sqrt{\Delta}\varphi_{b,n+1}/\lambda_{c,n+1})\sin(\sqrt{\Delta}\varphi_{c,n+1}/\lambda_{b,n}),\\
N_n&=\sqrt{\lambda_{b,n+1}\lambda_{b,n}}\sqrt{\lambda_{c,n}\lambda_{c,n+1}}\sin(\sqrt{\Delta}\varphi_{b,n+1}/\lambda_{c,n})\sin(\sqrt{\Delta}\varphi_{c,n+1}/\lambda_{b,n+1}),\\
L_n&=\sqrt{\lambda_{b,n}\lambda_{b,n+1}}\frac{\lambda_{b,n}+\lambda_{b,n+1}}{2}\left(  \frac{\lambda_{c,n}}{2\lambda_{b,n}}+\frac{\lambda_{c,n+1}}{2\lambda_{b,n+1}}\right)\sin(\sqrt{\Delta}\varphi_{b,n+1}/\lambda_{c,n+1})^{2}.
\end{align}
here $\vec{\varphi}=(\varphi_{b},\varphi_{c})$. This last expression allow us
to rewrite the propagator in the for
\begin{equation} \label{TA}
\langle \vec{\lambda}_{f};\tau_{f} |\vec{\lambda}_{i};\tau_{i}\rangle =\left(
\frac{1}{2\pi\gamma}\right)  ^{2N} \int d{\vec{\lambda}_{N-1}... d\vec{\lambda
}_{1}}\int\text{d}\vec{\varphi}_{N}...\text{d}\vec{\varphi}_{1}\text{ e}^{\text{i}/\hbar
S_{\mu}^{N}}+\mathcal{O}(\epsilon^{2}),
\end{equation}
wher
\begin{align}\label{SNprime}
S_{\mu}^{N}  &  =\epsilon\sum_{n=0}^{N-1}-\frac{\hbar}{\gamma}\vec{\varphi
}_{n+1}\frac{\vec{\lambda}_{n+1}-\vec{\lambda}_{n}}{\epsilon}+\frac{\hbar
}{2\gamma^{2}\Delta}\left[  \sqrt{\lambda_{b,n+1}\lambda_{b,n}}\sqrt
{\lambda_{c,n}\lambda_{c,n+1}}\right. \nonumber \\
&  \times\left(  \sin(\sqrt{\Delta}\varphi_{b,n+1}/\lambda_{c,n+1})\sin
(\sqrt{\Delta}\varphi_{c,n+1}/\lambda_{b,n})+\sin(\sqrt{\Delta}\varphi
_{b,n+1}/\lambda_{c,n})\sin(\sqrt{\Delta}\varphi_{c,n+1}/\lambda
_{b,n+1})\right) \nonumber\\
&  \left.  +\sqrt{\lambda_{b,n}\lambda_{b,n+1}}\frac{\lambda_{b,n
+\lambda_{b,n+1}}{2}\left(  \frac{\lambda_{c,n}}{2\lambda_{b,n}}+\frac
{\lambda_{c,n+1}}{2\lambda_{b,n+1}}\right)  \sin(\sqrt{\Delta}\varphi
_{b,n+1}/\lambda_{c,n+1})^{2}+\frac{\gamma^{2}\Delta}{G\hbar}\frac
{\lambda_{b,n}}{\lambda_{c,n}}\right]  .
\end{align}
Now we take the limit $N\rightarrow\infty$ and  Eq.  (\ref{SNprime}) takes the for
\begin{align}
S_{\mu}=  &  \lim_{N\rightarrow\infty}S_{\mu}^{N}= \int_{\tau_{i}}^{\tau_{f
}\text{d}\tau \left\{ -\frac{\hbar}{\gamma}\vec{\varphi}\cdot\dot{\vec{\lambda
}\nonumber\label{Sprime} \right.\\
&  \left. +\frac{\hbar}{2\gamma^{2}\Delta}\left[  \lambda_{b}\lambda_{c}\sin
(\sqrt{\Delta}\varphi_{b}/\lambda_{c})\left(  2\sin(\sqrt{\Delta}\varphi
_{c}/\lambda_{b})+\sin(\sqrt{\Delta}\varphi_{b}/\lambda_{c})\right)
+\frac{\gamma^{2}\Delta}{G\hbar}\frac{\lambda_{b}}{\lambda_{c}}\right]  \right\}.
\end{align}
Therefore we can see that the effective hamiltonian is
\begin{equation}
H_{\mu}^{\text{eff}}=-\frac{\hbar}{2\gamma^{2}\Delta}\left[  \lambda
_{b}\lambda_{c}\sin(\sqrt{\Delta}\varphi_{b}/\lambda_{c})\left(  2\sin
(\sqrt{\Delta}\varphi_{c}/\lambda_{b})+\sin(\sqrt{\Delta}\varphi_{b
/\lambda_{c})\right)  +\frac{\gamma^{2}\Delta}{G\hbar}\frac{\lambda^{b
}{\lambda_{c}}\right]  .
\end{equation}
Using (\ref{muprimvar}) to return to the original variables $(b,c,p_{b
,p_{c})$ we get finally
\begin{equation}
H_{\mu}^{\text{eff}}=-\frac{1}{2G\gamma^{2}}\left[  2\sqrt{p_{c}}\frac{\sin
\mu_{b}b}{\mu_{b}}\frac{\sin\mu_{c}c}{\mu_{c}}+\frac{p_{b}}{\sqrt{p_{c}
}\left(  \frac{\sin\mu_{b}b}{\mu_{b}}\right)  ^{2}+\gamma^{2}\frac{p_{b
}{\sqrt{p_{c}}}\right]  . \label{eff ham
\end{equation}

Let us emphasize that the classical hamiltonian (\ref{class-H}) is recovered by taking the small argument limit $\vert \mu_{l}l\vert <<1$ ($l=b,c$) in the effective hamiltonian (\ref{eff ham}); i.e., the classical model, namely, the classical hamiltonian and equations of motion, are recovered from the effective one in the regime $\vert \mu_{l}l\vert <<1$.

The hamiltonian (\ref{eff ham}) is the key piece defining and governing the effective quantum geometry. Effective states $(b,c,p_{b},p_{c})$ lie in the constraint surface $H_{\mu}^{\text{eff}}=0$, ``evolving" along the gauge integral curves of the hamiltonian vector field generated by $H_{\mu}^{\text{eff}}$. In the next section we will focus on analyze how geometrical quantities behave in the effective quantum scenario provided
by $H_{\mu}^{\text{eff}}$.

\section{Effective loop quantum dynamics}

\label{EffectiveKS}

To investigate the effective geometry, let us begin by considering the Hamilton equations associated to the hamiltonian (\ref{eff ham}), $\dot{\zeta}=\{\zeta,H_{\mu}^{\text{eff}}\}$, with the Poisson brackets  (\ref{poisson-brackets}), and the effective scalar constraint $H_{\mu}^{\text{eff}}=0$,
\begin{equation}
\dot{b}=\frac{-\gamma^{2}\mu_{b}^{2}-\sin\left(b\mu_{b}\right)
\left[\sin\left(b\mu_{b}\right) -2c\mu_{c}\cos\left(  c\mu_{c}\right)   +2\sin\left(  c\mu_{c}\right)  \right]  }{2\gamma
\sqrt{\Delta}\mu_{b}}, \label{beffdot
\end{equation}

\begin{equation}
\dot{c}=\frac{\gamma^{2}\mu_{b}^{2}+2b\mu_{b}\cos\left(  b\mu
_{b}\right)  \left[  \sin\left(  b\mu_{b}\right)  +\sin\left(
c\mu_{c}\right)  \right]  -\sin\left(  b\mu_{b}\right)  \left[
\sin\left(  b\mu_{b}\right)+2c\mu_{c}\cos\left(  c\mu_{c}\right)   +2\sin\left(  c\mu_{c}\right)  \right]  }{2\gamma
\sqrt{\Delta}\mu_{c}}, \label{ceffdot
\end{equation}

\begin{equation}
\dot{p}_{b}=\frac{\sqrt{\Delta}\cos\left(  b\mu_{b}\right)  \left[
\sin\left(  b\mu_{b}\right)  +\sin\left(  c\mu_{c}\right)
\right]  }{\gamma\mu_{b}\mu_{c}}, \label{pbeffdot
\end{equation}

\begin{equation}
\dot{p}_{c}=\frac{2\sqrt{\Delta}\cos\left(  c\mu_{c}\right)
\sin\left(  b\mu_{b}\right)  }{\gamma\mu_{b}^{2}}. \label{pceffdot
\end{equation}

At this point two remarks are in order. First, provided that both $p_{b}$ and $p_{c}$ are strictly positive quantities, the first term in (\ref{eff ham}) must be strictly negative in order to satisfy the constraint. Second, since $\gamma \mu_{b}\mu_{c}=\gamma \Delta /p_{b}$ and $\gamma \mu_{b}^{2}=\gamma \Delta /p_{c}$ [cf. Eq.  (\ref{mu})], Eq.(\ref{pbeffdot}) and Eq.(\ref{pceffdot}) can be written in the form $d(\ln p_{i})/dt=f_{i}(b\mu_{b},c\mu_{c})$, $i=a,b$.

A solution to the effective model is a sufficiently smooth{\footnote{Let $f=(l,p_{l})$, $l=a,b$, be a solution to Eqs. (\ref{beffdot})-(\ref{pceffdot}), and let us suppose that $f$ is, at least, of class $C^{1}$. Thus, it follows from Eqs. (\ref{beffdot})-(\ref{pceffdot}) that $f$ is, in fact, a $C^{\infty}$ function.}}, real  solution to Eqs. (\ref{beffdot})-(\ref{pceffdot}) which, in addition, satisfies the scalar constraint (\ref{eff ham}). Let us refer to solutions of the effective model as effective solutions. Since the dynamics is pure gauge, each point in the constraint surface is an appropriate initial condition for effective solutions. Now, to fix notation, let $\chi_{0}$ be the initial condition $(b_{0},c_{0},p_{b0},p_{c0})$ at $t=t_0$ (the reference initial time) to the effective solution $\chi=(b,c,p_{b},p_{c})$.

From Eq.  (\ref{eff ham}), it follows that effective solutions $\chi$ must satisfy, in particular, that
\begin{equation}
\sin(\mu_{b}b)=-\sin(\mu_{c}c)\pm \sqrt{\sin^{2}(\mu_{c}c)-\frac{\Delta \gamma^2}{p_{c}}}.
\end{equation}
Since effective solutions $\chi$ are real ones, the discriminant must necessarily be nonnegative, so that $\sin^{2}(\mu_{c}c)\geq \Delta \gamma^2/p_{c}$. Thus, in particular, we have that $p_{c}$ is bounded from below\footnote{This bound is consistent with that found in
Ref. \cite{Chiou:2008nm}, where $p_{c}\geq\Delta\gamma^{2}/3.$} by $\Delta\gamma^{2}$; i.e.,
\begin{equation}
p_{c}\geq\Delta\gamma^{2}. \label{cota pc
\end{equation}
This expression implies that the area of $S^{2}$ [cf. Eq. (\ref{leng-are-vol})] cannot be less than $4\pi\Delta\gamma^{2}$ in the effective geometry.
 By using inequality (\ref{cota pc}) into the relations defining $\mu_{b}$ and $\mu_{c}$ [cf. Eq. (\ref{mu})], we get that
\begin{equation}
\label{mub-muc-fbounds}
 \frac{\Delta \gamma}{p_{c}}\leq \mu_{b}\leq \frac{1}{\gamma},\qquad
\frac{\Delta \gamma}{p_{b}}\leq \mu_{c}\leq \frac{p_{c}}{\gamma p_{b}}.
\end{equation}
Thus, in particular,
\begin{equation}
\label{cota mub}
\gamma\mu_{b}\leq1.
\end{equation}

Using again that the first term in (\ref{eff ham}) must be strictly negative it must be the case that $\sin(\mu_{b}b)$ and $\sin(\mu_{c}c)$ must have opposite constant signs,
\begin{equation}
\sin(\mu_{l}l)>0,\quad \sin(\mu_{l'}l')<0, \label{sinmubsinmuc
\end{equation}
with $l$ being equal to $b$ or $c$, and $l'$ being the complementary of $l$; that is, for $l=b$ ($l=c$), $l'=c$ ($l'=b$). Strict inequalities (\ref{sinmubsinmuc}), and the continuity of the functions $\mu_{b}b$ and $\mu_{c}c$, imply that $2n_{0}\pi<\mu_{l}l <(2n_{0}+1)\pi$ and $(2m_{0}-1)\pi<\mu_{l'}l' <2m_{0}\pi$, for some $n_{0},m_{0}\in \mathbb{Z}$ fixed and determined by the effective solution $\chi$; in fact, by its corresponding initial condition $\chi_0$. Indeed, given an initial condition $\chi_{0}$, we will have that $\sin(\mu_{l}l)_{0}>0$ and that $\sin(\mu_{l'}l')_{0}<0$, so that $n_{0}$ is the greatest integer $n$ satisfying that $n<(\mu_{l}l)_{0}/2\pi$, whereas $m_{0}$ is the least integer $m$ satisfying that $(\mu_{l'}l')_{0}/2\pi<m$. By continuity, $\mu_{l}l$ and $\mu_{l'}l'$ must remain, respectively, in $\big{(}2n_{0}\pi,(2n_{0}+1)\pi\big{)}$ and in $\big{(}(2m_{0}-1)\pi,2m_{0}\pi\big{)}$; otherwise, the constraint will be violated. We then have disjoint sectors, and they are as many as the distinct pairs $(n_{0},m_{0})$ that the initial conditions define. Although we will perform our analysis by considering a generic sector, it is worth remarking that it is only within the $(0,0)$-sector that the regime $\mu_{d} \vert d\vert <<1$ can be consistently treated.

Let us introduce a more symmetric notation through
$$ N_{l}=
\begin{cases}
  (2n_{0}+1)  ,& \text{if } n_{0}\geq 0\\
    2\vert n_{0}\vert ,              & \text{if }  n_{0}<0
\end{cases}, \qquad
N_{l'}=
\begin{cases}
  2m_{0}  ,& \text{if } m_{0}\geq 1\\
    (2\vert m_{0}\vert +1) ,              & \text{if }  m_{0}\leq 0
\end{cases} $$
In terms of $N_{d}$ (with $d$ being $b$ or $c$) we have that $\mu_{d}\vert d \vert$ is confined to be in $\big( \pi(N_{d}-1),\pi N_{d}\big)$. Explicitly,  given an effective solution $\chi$, the quantities $\mu_{b}\vert b \vert$ and $\mu_{c}\vert c \vert$ are bounded by $\pi(N_{b}-1)<\mu_{b}\vert b \vert <\pi N_{b}$ and by $ \pi(N_{c}-1)<\mu_{c}\vert c \vert <\pi N_{c}$, where $N_{b}$ and $N_{c}$ are (strictly) positive fixed integers determined by the initial condition $\chi_{0}$.

Now, inequalities $\pi(N_{d}-1)<\mu_{d}\vert d \vert$ and $\mu_{d}\vert d \vert <\pi N_{d}$ imply that $0<\vert\sin(\mu_{d}d)\vert \leq 1$ and that $0\leq \vert\cos(\mu_{d}d)\vert < 1$. Thus, for any given phase-space function $g$, we will have the strict inequality
\begin{equation}
\label{cos-stbound}
 \vert g\cos(\mu_{d}d)\vert< \vert g \vert.
\end{equation}
In particular, we have that the strict inequality $\vert\sin(\mu_{d}d)\cos(\mu_{d'}d')\vert<1$ must be satisfied.

Now, let us consider $\vert \dot{p}_{b}\vert$ and $\vert \dot{p}_{c}\vert$. From Eqs. (\ref{pbeffdot})-(\ref{pceffdot}) -written in terms of the explicit expressions for $\gamma \mu_{b}\mu_{c}$ and $\gamma \mu_{b}^{2}$- it follows by using the triangle inequality, the boundedness of the sine function and Eq.(\ref{cos-stbound}) that
\begin{equation}
\label{dpb-dpc-bounds}
\left\vert \dot{p}_{b}\right\vert < \left(\frac{3}{2\gamma \sqrt{\Delta}}\right)p_{b}, \qquad \left\vert \dot{p}_{c}\right\vert < \left(\frac{2}{\gamma \sqrt{\Delta}}\right)p_{c}.
\end{equation}
To get the first inequality we have used, in addition, the relationship $\sin(2b\mu_{b})=2\cos(b\mu_{b})\sin(b\mu_{b})$ in Eq. (\ref{pbeffdot}). Similar calculations employing relations (\ref{cota pc})-(\ref{mub-muc-fbounds}) in Eqs. (\ref{beffdot}) and (\ref{ceffdot})  shows that  $\vert \dot{b}\vert$ and $\vert \dot{c}\vert$ are bounded from above by
\begin{equation}
\vert \dot{b}\vert < \frac{1}{2\sqrt{\Delta}}+\left(\frac{3+2\mu_{c}\vert c \vert}{2\gamma^{2}\Delta^{3/2}}\right)p_{c}\leq \left(\frac{2+\mu_{c}\vert c\vert}{\gamma^{2}\Delta^{3/2}}\right)p_{c},\quad \vert \dot{c}\vert <\left(\frac{4+2\mu_{c}\vert c \vert+3\mu_{b}\vert b \vert}{2\gamma^{2}\Delta^{3/2}}\right) p_{b}.
\end{equation}
Since $\mu_{d}\vert d\vert< \pi N_{d}$, we obtain that
\begin{equation}
\label{db-dc-bounds}
\vert \dot{b}\vert <  \left(\frac{2+\pi N_{c}}{\gamma^{2}\Delta^{3/2}}\right)p_{c}
,\quad \vert \dot{c}\vert < \left(\frac{4+ 2\pi N_{c}+3\pi N_{b}} {2\gamma^{2}\Delta^{3/2}}\right) p_{b}.
\end{equation}

 Inequalities (\ref{dpb-dpc-bounds}) and (\ref{db-dc-bounds}) imply that $\vert \dot{\chi} \vert$ is bounded from above by $F_{(N_{b},N_{c})}\vert \chi \vert$, where
$$F_{(N_{b},N_{c})}^{2}=\max\left\{\frac{9}{4\gamma^{2}\Delta}+\frac{(4+2\pi N_{c}+3\pi N_{b})^{2}}{4\gamma^{4}\Delta^{3}}\, , \,\frac{4}{\gamma^{2}\Delta}+\frac{(2+\pi N_{c})^{2}}{\gamma^{4}\Delta^{3}}\right\}.$$
All effective solutions  in the sector labelled by $(N_{b},N_{c})$  turn out  defined for $t\in \mathbb{R}$.
In addition, let us remark that effective solutions are bounded by the exponential function. Indeed, recall that Eq.(\ref{pbeffdot}) and Eq.(\ref{pceffdot}) can be written in the form $d(\ln p_{d})/dt=f_{d}(b\mu_{b},c\mu_{c})$. Thus, combining Eqs. (\ref{pceffdot}) and (\ref{cota pc}), employing the boundedness of the sine and Eq. (\ref{cos-stbound}), it is not difficult to see that
\begin{equation}
\label{pc-bounds}
\Delta\gamma^{2}\, \leq \, p_{c} \, <\, p_{c0}\:e^{ 2\vert t-t_{0} \vert/ \gamma{\sqrt{\Delta}}}.
\end{equation}
Similarly, from Eq.(\ref{pbeffdot}) it follows that
\begin{equation}
\label{pb-bounds}
p_{b0}\:e^{-3\vert t-t_{0}\vert/\gamma\sqrt{4\Delta}}< p_{b}<p_{b0}\:e^{3\vert t-t_{0}\vert/\gamma\sqrt{4\Delta}}.
\end{equation}
Since $\pi(N_{d}-1)<\mu_{d}\vert d \vert <\pi N_{d}$, using inequalities (\ref{cota pc}) and (\ref{pc-bounds})-(\ref{pb-bounds}), we get that
\begin{equation}
\label{b-mod-bound}
\gamma \pi (N_{b}-1)<\vert b \vert < \frac{\pi N_{b}}{(\mu_{b})_{0}}e^{\vert t-t_{0} \vert/ \gamma{\sqrt{\Delta}}},
\end{equation}
\begin{equation}
\label{c-mod-bound}
\frac{\pi (N_{c}-1)}{(\mu_{c})_{0}}e^{-5\vert t-t_{0}\vert/\gamma\sqrt{4\Delta}}<\vert c \vert < \frac{\pi N_{c}\,p_{b0}}{\Delta \gamma}e^{3\vert t-t_{0}\vert/\gamma\sqrt{4\Delta}}
\end{equation}
Thus, in contrast to the classical model, $b$ and $c$ are finite quantities at every time: there is not a finite proper time limit, $t_f$, at which $b$ and $c$ will become infinite. Besides, {\emph{for all}} $t\in \mathbb{R}$, $p_{c}$ is bounded from below by a positive number, namely $\Delta \gamma^{2}$, and $p_{b}$ is a strictly positive quantity as well. Note, in addition, that Eqs. (\ref{pc-bounds})-(\ref{pb-bounds}) prevent the metric (\ref{metrica cinematica}) to have a coordinate singularity in the effective approach.

Let us now focus on the behavior of geometrical and invariant quantities. From Eqs. (\ref{pbeffdot})-(\ref{pceffdot}), using the explicit expressions for $\gamma \mu_{b}\mu_{c}$ and $\gamma \mu_{b}^{2}$, it immediately follows that
\begin{equation}
\label{dtheta-b}
\left\vert \theta_{\rm{eff}}\right\vert =\left\vert\frac{\dot{p}_{b}}{p_{b}}+\frac{\dot{p}_{c}}{2p_{c}}\right\vert = \frac{1}{\gamma\sqrt{\Delta}} \left\vert \sin(b\mu_{b}+c\mu_{c})+\frac{1}{2}\sin(2b\mu_{b})\right\vert \leq  \frac{3}{\gamma\sqrt{4\Delta}},
\end{equation}
\begin{equation}
\label{dshear-b}
\left\vert \sigma_{\rm{eff}}\right\vert  = \frac{1}{\sqrt{3}}\left\vert\frac{\dot{p}_{b}}{p_{b}}-\frac{\dot{p}_{c}}{p_{c}}\right\vert
=\frac{1}{\gamma\sqrt{3\Delta}} \left\vert  \sin(c\mu_{c}-b\mu_{b})-\cos(c\mu_{c})\sin(b\mu_{b})+\frac{1}{2}\sin(2b\mu_{b})\ \right\vert
<  \frac{5}{\gamma\sqrt{12\Delta}},
\end{equation}
where we have used Eq.(\ref{cos-stbound}) to get the strict inequality in the last term of Eq. (\ref{dshear-b}). The boundedness of the expansion scalar ensures, in particular, that the volume of a cell will remain different from zero at any finite proper time. Indeed, let $V_{r}=(4\pi p_{b}\sqrt{p_{c}})\,\vert_{t_{r}}$ be the volume of the cell ${\cal{V}}=[0,L]\times S^{2}$ at an arbitrary reference proper time $t_{r}$ of comoving observers in the effective Kantowski-Sachs geometry (with $N=1$), and let $t$ be any other finite proper time. It is a simple matter to see that $\vert \theta_{\rm{eff}}\vert\leq 3/\gamma\sqrt{4\Delta}$ implies that
\begin{equation}
\label{two-bounds}
V_{r}e^{-3\vert t-t_{r}\vert/\gamma\sqrt{4\Delta}}\leq V\leq V_{r}e^{3\vert t-t_{r}\vert/\gamma\sqrt{4\Delta}},
\end{equation}
where $V$ is the volume of the cell $\cal{V}$ at time $t$. The volume $V$ is a well-defined, strictly positive quantity at any finite proper time $t\in \mathbb{R}$ and, consequently, the congruence of timelike geodesics defined by comoving observers [i.e., the integral curves of the vector field $\xi^{a}=(\partial/\partial t)^{a}$] will not develop a caustic (at finite proper times).

Let us now demonstrate that the effective Ricci and Kretschmann scalars, $R_{\rm{eff}}$ and $K_{\rm{eff}}$, are in fact well-behaved, finite quantities. Provided that $R_{\rm{eff}}$ and $K_{\rm{eff}}$ have second order terms in the time derivatives of $p_{b}$ and $p_{c}$, we shall first calculate $\{\dot{p}_{d},H_{\mu}^{\rm{eff}}\}$. By using Eqs. (\ref{pbeffdot})-(\ref{pceffdot}), as well as Eqs. (\ref{beffdot})-(\ref{ceffdot}), a straightforward calculation shows that
\begin{eqnarray}
\label{pbeffdotdot}
\ddot{p}_{b}= \frac{p_{b}}{2\gamma^{2}\Delta}\Bigg{(}
&\gamma^{2}&\mu_{b}^{2}\sin(2b\mu_{b})\left[\sin(b\mu_{b})\cos(c\mu_{c})+\frac{1}{4}\sin(2c\mu_{c})\right]\nonumber \\
\, &+&\sin(2c\mu_{c})\left[\sin(c\mu_{c})\cos(b\mu_{b})+\frac{1}{4}\sin(2b\mu_{b})\sin^{2}(b\mu_{b})\right]\nonumber \\
\, &+& \cos(c\mu_{c})\left[\sin(2c\mu_{c})\cos(b\mu_{b})+\cos(c\mu_{c})\sin^{2}(b\mu_{b})\sin(2b\mu_{b})\right](\mu_{b}b-\mu_{c}c)\Bigg{)},
\end{eqnarray}

\begin{eqnarray}
\label{pceffdotdot}
\ddot{p}_{c}=-\cos(b\mu_{b}&-&c\mu_{c})+\frac{p_{c}}{\gamma^{2}\Delta}\Bigg{(}2\sin^{2}(b\mu_{b})\left[1+\cos^{2}(c\mu_{c})\right]-\frac{1}{2}\sin(2b\mu_{b})\sin(2c\mu_{c})\nonumber \\ \, &-& \sin^{2}(b\mu_{b})\cos(b\mu_{b}+c\mu_{c})
+\sin(2b\mu_{b})\left[1+\sin(b\mu_{b})\sin(c\mu_{c})\right](\mu_{c}c-\mu_{b}b)\Bigg{)}.
\end{eqnarray}

Employing the triangle inequality, condition (\ref{cos-stbound}), and the boundedness of $\mu_{b}$ [cf. (\ref{cota mub})] as well as of the sine function, we get that
\begin{equation}
\left\vert\ddot{p}_{b}\right\vert< \frac{1}{\gamma^{2}\Delta}\left(\frac{5}{4}+\big{[}\mu_{b}\vert b\vert +\mu_{c}\vert c\vert\big{]}\right)p_{b},\quad \left\vert\ddot{p}_{c}\right\vert < 1+\frac{1}{\gamma^{2}\Delta}\left(\frac{11}{2}+2\big{[}\mu_{b}\vert b\vert +\mu_{c}\vert c\vert\big{]}\right)p_{c}.
\end{equation}
Using that $\mu_{d}\vert d \vert < \pi N_{d}$, we arrive to
\begin{equation}
\label{pb-pb-ddot-bounds}
\left\vert\ddot{p}_{b}\right\vert<  \left(\frac{5+{4\pi [N_{b}+ N_{c}]}}{4\gamma^{2}\Delta}\right)p_{b},
\quad \left\vert\ddot{p}_{c}\right\vert<
1+\left(\frac{11+{4\pi [N_{b}+ N_{c}]}}{2\gamma^{2}\Delta}\right)p_{c}\leq \left(\frac{13+{4\pi [N_{b}+ N_{c}]}}{2\gamma^{2}\Delta}\right)p_{c},
\end{equation}
where the last inequality in the second expression follows from $1\leq p_{c}/(\gamma^{2}\Delta)$.

In order to simplify notation, let us introduce the quotients $x:=\dot{p_{b}}/p_{b}$, $y:=\dot{p_{c}}/p_{c}$, $v:=\ddot{p_{b}}/p_{b}$ and $w:=\ddot{p_{c}}/p_{c}$. So, inequalities (\ref{dpb-dpc-bounds}) and (\ref{pb-pb-ddot-bounds}) read as follows
\begin{equation}
\label{quotients-bounds}
\vert x \vert < \frac{3}{2\gamma\sqrt{\Delta}},\quad \vert y \vert < \frac{2}{\gamma\sqrt{\Delta}}, \quad \vert v \vert <
 \left(\frac{5+{4\pi [N_{b}+ N_{c}]}}{4\gamma^{2}\Delta}\right)
,\quad \vert w \vert < \left(\frac{13+{4\pi [N_{b}+ N_{c}]}}{2\gamma^{2}\Delta}\right).
\end{equation}

The Ricci scalar, which is given by $R_{\rm{eff}}=2v+w+(2/p_{c})$ [cf. Eq. (\ref{ricciscalarkinematic})], is thus bounded by
\begin{equation}
\vert R_{\rm{eff}}\vert \leq 2 \vert v \vert + \vert w \vert+\frac{2}{p_{c}}.
\end{equation}
By using Eq. (\ref{cota pc}) and Eq. (\ref{quotients-bounds}) we have that the Ricci scalar [in the sector labelled by $(N_{b},N_{c})$] is bounded by
\begin{equation} \label{Rbound}
\vert R_{\rm{eff}}\vert <\left(\frac{11+{4\pi [N_{b}+ N_{c}]}}{\gamma^{2}\Delta}\right).
\end{equation}

Let us now focus on the Kretschmann scalar, $K_{\rm{eff}}$. From Eq. (\ref{Kretschmannscalarkinematic}), it is easy to see that in terms of the quotients $x$, $y$, $v$ and $w$, $K_{\rm{eff}}$ is given by
\begin{eqnarray}
K_{\rm{eff}}=&4&v^{2}+3w^{2}-4vw-8vxy+4wyx+6vy^2\nonumber \\
\,&-&5wy^{2} +6x^{2}y^{2}-8xy^{3}+\frac{7}{2}y^{4}+\frac{2}{p_{c}}y^{2}+\frac{4}{p_{c}^{2}}.
\end{eqnarray}
Clearly, the effective Kretschmann scalar turns out to be a bounded quantity. The explicit bound is obtained by using the inequalities (\ref{cota pc}) and (\ref{quotients-bounds}), as well as the triangle inequality. A straightforward calculation shows that
\begin{equation} \label{Kbound}
\vert K_{\rm{eff}} \vert < \frac{\xi}{\gamma^{4}\Delta^{2}}, \qquad \xi=4\left(6[N_{b}+N_{c}]^{2}\pi^{2}+59[N_{b}+N_{c}] \pi+160\right)+\frac{23}{2}.\end{equation}

In addition, since $\dot{\theta}_{\rm{eff}}=v-x^{2}+(w-y^{2})/2$, we get for effective solutions that
\begin{equation}
\left \vert \dot{\theta}_{{\rm{eff}}}\right\vert\leq \vert v \vert+x^{2}+\frac{1}{2}\vert w \vert +\frac{1}{2}y^{2}<\frac{1}{\gamma^{2}\Delta}\left(\frac{35}{4}+ 2(N_{b}+N_{c})\pi \right)
\end{equation}
This, together with Eqs. (\ref{dtheta-b})-(\ref{dshear-b}), proves that $(R_{00})_{\rm{eff}}$ is a bounded quantity as well.

In general, we have that any quantity of the form
\begin{equation}
\label{g-form}
\Lambda:=\sum_{j=1}^{N}C_{j}\left(\ddot{p}_{b}\right)^{n_{j}}\left(\ddot{p}_{c}\right)^{m_{j}}\left(\dot{p}_{b}\right)^{r_{j}}\left(\dot{p}_{c}\right)^{s_{j}}\left(p_{b}\right)^{\alpha_{j}}\left(p_{c}\right)^{\beta_{j}},
\end{equation}
where $n_{j}$, $m_{j}$, $r_{j}$ and $s_{j}$ are nonnegative integers, and $\alpha_{j}$ and $\beta_{j}$ are any two real numbers, is a bounded quantity on shell. Indeed, from inequalities (\ref{dpb-dpc-bounds}) and (\ref{pb-pb-ddot-bounds}) it follows that
\begin{equation}
\label{general-b}
\vert \Lambda_{\rm{eff}} \vert < \sum_{j=1}^{N}\vert C_{j}\vert\left(\frac{A_{bc}}{4\gamma^{2}\Delta}\right)^{n_{j}}\left(\frac{B_{bc}}{2\gamma^{2}\Delta}\right)^{m_{j}}
\left(\frac{3}{2\gamma\sqrt{\Delta}}\right)^{r_{j}} \left(\frac{2}{\gamma\sqrt{\Delta}}\right)^{s_{j}}
\left(p_{b}\right)^{n_{j}+r_{j}+\alpha_{j}}\left(p_{c}\right)^{m_{j}+s_{j}+\beta_{j}},
\end{equation}
where $A_{bc}:=5+{4\pi [N_{b}+ N_{c}]}$ and  $B_{bc}:=13+{4\pi [N_{b}+ N_{c}]}$. Then, we have that in the effective approach of the KS model, any effective quantity $\Lambda_{\rm{eff}}$ of the form (\ref{g-form}) will be bounded by (\ref{general-b}). Provided that any scalar polynomial invariant $P$ associated to the metric (\ref{metrica cinematica}) -with the lapse function being set to the unit constant function- will take the form (\ref{g-form}), as it is actually the case for the Ricci and Kretschmann scalars, we can assert that in the effective geometry of the KS model $P_{\rm{eff}}$ will be bounded everywhere, even though its classical counterpart is not (i.e., even if $P_{\rm{class}}$ diverges at some regime).

\section{Discussion}
\label{sec-disc}
The quest for the fundamental nature of spacetime may shed light on long standing problems as the singularities appearing in classical general relativity and the ultraviolet divergences of field theories. Hence quantum gravity theories that endow with quantum character to spacetime acquire particular interest. Loop quantum gravity, in particular, has yielded homogenous cosmological models in which the classical singularity is replaced by a quantum bounce and thus Schwarzschild interior which classically amounts to a homogeneous, Kantowski-Sachs, model is amenable for a similar treatment. Indeed the loop quantization of Schwarschild interior showed that the would be classical singularity is actually traversable and later on some heuristic effective models confirmed the same result but also added possible replacements for the singularities like another black hole, a Nariai universe or a white hole. However, connecting the quantum treatment with the effective model was left open. In this paper we have advanced a proposal that links the loop quantum description of the Schwarzschild interior with an effective model that is based on a path integral scheme. Specifically we have built a transition amplitude between two loop quantum states of the Kantowski-Sacks model as a path integral, Eq. (\ref{aprima}), consisting of an imaginary exponential of an action in phase space from which the effective heuristic hamiltonian constraint descends, Eqs. (\ref{TA}), (\ref{Sprime}) and (\ref{eff ham}). Although this strategy was originally used for homogeneous isotropic, as well as some anisotropic, models the particular case of Kantowski-Sachs had not been dealt with before.

Armed with the effective constraint we embarked in the study of the ensuing dynamics that happened to lead to rather simple analytic bounds for the basic phase space variables, Eqs. (\ref{pc-bounds})-(\ref{c-mod-bound}) and their time derivatives. In particular expansion and shear turn out to be bounded too as it is the volume, Eqs. (\ref{dtheta-b})-(\ref{two-bounds}). Similarly, by considering the second order time derivatives of the basic phase space variables, according to the effective dynamics, we get that both the effective Ricci  and Kretschmann scalars, Eqs. (\ref{Rbound}) and (\ref{Kbound}), are bounded. This bounded character actually holds for any product  of the form (\ref{g-form}) containing second order and first order time derivatives as well as powers of the variables $p_b,p_c$. It is a remarkable fact that analytic results were obtained from the effective dynamics which, although simple in appearance, could only be treated numerically in previous works.

There are several interesting points which can be further explored along the lines we have followed in the present work. One of them concerns
our analysis performed considering a generic sector labeled by $(N_l,N_{l'})$ indicated by the detailed form of the effective hamiltonian constraint. Since it is in the regime $\mu_{d} \vert d\vert <<1$, from which the classical behavior can be recovered through a semiclassical approximation this selects only the $(0,0)$-sector. Thus, it is natural to ask about the physical relevance of the other sectors. Another thing we have not done in the present work is an analysis of the would be classical horizon. Since we have adopted the improved quantization for the Kantowski-Sachs model it is expected that it will differ from the recent results of \cite{Corichi:2015xia} that adopts an effective dynamics preserving the classical horizon definition. Further work is required to clarify other possible physical differences.
Indeed, for example, recent phenomenological results on black hole evaporation \cite{Barrau:2016qri,Barrau:2015ana} require connecting interior effective descriptions like the one studied presently with that corresponding to the exterior. It would be interesting to combine our path integral analysis of the Schwarszchild interior including a coupling to a scalar field along the lines of \cite{Hartle:1976tp} and extend it to the exterior region in order to investigate further quantum gravity corrections to the black hole emission (See e.g. \cite{Tecotl:2015cya} which applies polymer path integral to a mechanical model to study the problematics of the black hole semiclassical approximation.)
Finally, important consequences of the features we have found here may play a role in the geodesic analysis in regard to completeness and perhaps complement recent results for the cosmological case \cite{Joe:2014tca,Saini:2016vgo}.

\section*{Acknowledgements}
This work was partially supported by CONACyT Grant No. 237351 ``Implicaciones f\'isicas de la estructura del espacio tiempo" and DGAPA-UNAM Grant No. IN113115 ``Teor\'ia de campos en fondos curvos, gravedad cu\'antica y holograf\'ia".

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