\section{Introduction}
The gravitational collapse for a spherically symmetric stellar object has been analyzed in detail in the standard General Relativity (GR) theory \cite{Weinberg}. By assuming the metric of the space-time to be spherically symmetric and that the collapsing fluid is pressureless, the found metric interior to the object turns to be Robertson-Walker type with a parameter playing the role of spatial curvature and proportional to initial density. The time lapse and the size of the object are given by  a cycloid parametric equation with an angle parameter $\psi$.

In spite of the fact that GR has been one of the most successful theories of the twentieth century, it does not give a satisfactory explanation to some of the latest cosmological and astrophysical observations with usual matter sources. On the one hand, the baryonic matter content has to be supplemented by a dark matter (DM) component. Although there are many possible origins for this additional component \cite{DM}, DM is usually assumed to be in the form of thermal relics that naturally freeze-out with the right abundance in many extensions of the standard model of particles \cite{WIMPs}. Future experiments will be able to discriminate among the large number of candidates and models, such as direct and indirect detection designed explicitly for their search \cite{isearches}, or even at high energy colliders, where they could be produced \cite{Coll}.

On the other hand, a dark energy contribution needs to be considered to provide cosmological acceleration \cite{DE}. Alternatively,
a modified gravitational interaction can potentially generate the present expansion of the Universe without invoking the presence of any exotic dark energy \cite{otras}. In this context, fourth order theories of gravity defined by adding to the action a function of the scalar curvature:
\begin{equation}
S_G=\frac{1}{16\pi G}\int \text{d}^4x \sqrt{\vert g \vert}\left(R+f(R)\right)\,,
\label{Modified action}
\end{equation}
has been extensively studied during the last years \cite{R2a,Rm1,varia,R2b}.


\section{Gravitational collapse}
In order to study the gravitational collapse inside this modified gravity theories, so-called $f(R)$ theories, we can make use of the spherically symmetric metric. The collapsing object is considered to be approximated pressureless and homogeneous. Therefore, we can search a separable solution for this metric as follows \cite{Weinberg}:
\begin{eqnarray}
&&\text{d}s^2 =\text{d}t^2-U(r,t)\text{d}r^2-V(r,t)(\text{d}\theta^2+\text{sin}^2\theta \text{d}\phi^2),\\
&&U(r,t)=A^2(t)h(r),\quad V(r,t)=A^2(t)r^2.
\label{Intervalo colapso, U y V}
\end{eqnarray}
Components $tt$, $rr$ and $\theta\theta$ for the modified tensorial equations may be written respectively in terms of the functions $A(t)$ and $h(r)$. Once we have calculated $h(r)$, the resulting metric is formally the same as the one obtained in GR. We consider $\rho\,(t)=\rho\,(t=0)/A(t)^3$ (given by the energy motion equation for dust matter). Furthermore, provided that the fluid is assumed to be at rest for $t=0$, initial conditions $\dot{A}(t=0)=0$ and $A(t=0)=1$ hold. We define $R(t=0)\equiv R_0$ and $\rho\,(t=0)\equiv \rho_0$ to simplify notation. Taking into account the previous conditions, the evolution equation for $A$ is found to be \cite{Cembranos:2012fd}:
\begin{eqnarray}
\dot{A}^2&=&-\frac{1}{6(1-f_R^2(R_0))}\big[\vphantom{\ddot{f}_R(R_0)}\,8\pi G\rho_0\left(2-f_R(R_0)\right)
- f(R_0)(1+f_R(R_0))
\\
&&
-3\ddot{f}_R(R_0)f_R(R_0)\,\big]
+\frac{1}{(1-f_R^2)}\,\frac{8\pi G}{3}\rho_0\,A^{-1}
-\frac{1}{6(1-f_R^2)}\big[8\pi G\rho_0\,A^{-1}f_R
\nonumber\\
&&
+3A^2\ddot{f}_{R}f_R-3A\dot{A}\dot{f}_{R}(1-f_R)+A^2f(R)(1+f_R)\big]\,.\nonumber
\label{Simplified Eq for dotA}
\end{eqnarray}
Let us remind at this stage that the zeroth order solution of GR is given by the parametric equations of a cycloid \cite{Weinberg}:
\begin{equation}
t=\frac{\psi+\text{sin}\,\psi}{2\sqrt{k}},\;\;\; A_G=\frac{1}{2}(1+\text{cos}\,\psi).
\label{Eqs cicloide}
\end{equation}
Expression \eqref{Eqs cicloide} clearly implies that a sphere with initial density $\rho_0$ and negligible pressure will collapse from rest to a state of infinite proper energy density in a finite time that we will denote $T_G$. This time is obtained for the first value of $\psi$ such as $A_G=0$, i.e. for $\psi=\pi$. In order to study the modification to the gravitational collapse in $f(R)$ theories, we have expanded $A$ around $A_G$ and $f(R)$ around the scalar curvature in GR ($R=R_G$) \cite{Cembranos:2012fd},i.e. $A=A_G+g(\psi)$, and $f(R)\simeq f(R_G)+f'(R_G)(R-R_G)$.
The presence of a function $f(R)$ in the gravitational Lagrangian represents a correction of first order with respect to the usual Einstein-Hilbert action. By substituting the above series expansions in expression \eqref{Simplified Eq for dotA} until first order in $\varepsilon$,  we find that equation (\ref{Simplified Eq for dotA}) becomes \cite{Cembranos:2012fd}:
\begin{eqnarray}
\text{tg}\left(\frac{\psi}{2}\right)g'\,&=&\,-\frac{1}{2}\text{cos}^{-2}\left(\frac{\psi}{2}\right) g
+\frac{1}{12k}\text{cos}^{2}\left(\frac{\psi}{2}\right)\left(f(R_{G0})+3 k f_{R}(R_{G0})\right)
\\
&-&\frac{1}{4}f_R(R_G)-\frac{1}{12k}\text{cos}^{6}\left(\frac{\psi}{2}\right) f(R_G)+\frac{1}{4\sqrt{k}}\,\text{sin}^{}\left(\frac{\psi}{2}\right)\text{cos}^{3}\left(\frac{\psi}{2}\right)\dot{f}_{R}(R_G)
\,. \nonumber
\label{Eq g' with generic Rs}
\end{eqnarray}
By studying this equation for particular $f(R)$ models, it is possible to constraint an important region of the parameter space of
this gravitational theories (See Fig. \ref{validity}). This validity tests are interesting since the higher contraction that these models generally present in the perturvative regime, can alleviate some tension of the standard $\Lambda$CDM model with observations of structures at high redshift \cite{clusters}.
\begin{figure}[tp]
	\centering
		\includegraphics[width=0.48\textwidth]{Validityregimemodel1.eps}\;\;\;\;
		\includegraphics[width=0.48\textwidth]{Validityregimemodel2.eps}
		\caption{\footnotesize{
Validity of the perturbative regime for $f(R)= \epsilon R^2$ \cite{R2a,R2b} (left panel) and $f(R)= \epsilon R^{-1}$ \cite{Rm1} (right panel). As it is indicated in the figures, in the first case, it is the high density region, the one that exhibits a non linear regime; whereas it is the low density region for the second model (on the right). The right area on the left panel is excluded by the condition $\varepsilon\leq 2.3\times10^{22}\,\text{GeV}^{-2}$ \cite{R2b}. Finally, the density marking the beginning of structure formation (SF) $\rho_{\text{SF}}\simeq1.5 \times 10^{-38}$ GeV$^4$, and the dark energy (DE) density $\rho_{\text{DE}}\simeq2.8 \times 10^{-47}$ GeV$^4$, have also been plotted for reference \cite{Cembranos:2012fd}.}}
\label{validity}
\end{figure}


\begin{theacknowledgments}
This work has been supported by MICINN (Spain) project numbers FIS 2008-01323, FIS2011-23000, FPA 2008-00592, FPA2011-27853-01, Consolider-Ingenio MULTIDARK CSD2009-00064 (Spain) and URC (South Africa).
\end{theacknowledgments}

\bibliographystyle{aipproc}  


