c & =& -6 \rho_0 \be {  1-2\be + n(5 \be + 2n -3)  \over  \left({t\over t_0}\right)^{3\be} \left({t \over t_2}\right)^{2n} \left[6\be (2\be-1)\right]^{1-n} } ~~, \\ \nonumber p_c &=& -2 \rho_0 n { 2+4n^2 -\be (2+3\be) +n(8\be -6) \over  \left({t\over t_0}\right)^{3\be} \left({t \over t_2}\right)^{2n} \left[6\be (2\be-1)\right]^{1-n} }~~.   \eeqa
\noindent after defining $t_n \equiv R_n^{-1/2}$ in the weak regime $F_2 \rho \ll \ka$ \cite{accexp}.

Given the Hubble parameter 
\begin{equation}} \def\eeq{\end{equation} H(t) \equiv {\dot{a} \over a} = {\be\over t} ~~, \eeq
\noindent the {\it l.h.s.} of the Friedmann \eq{modFriedmann} falls as $t^{-2}$, so that comparing with the above gives a relation between the exponents $\be$ and $n$:
\begin{equation}} \def\eeq{\end{equation} 3\be + 2n = 2 \rightarrow \be = {2 \over 3}(1-n)~~. \eeq
\noindent Thus, one concludes that any negative exponent $n$ will yield an accelerating Universe for which $\be > 0$ and $q < 0$ at late times. One may also compute the deceleration parameter and EOS parameter for the non-minimally coupled contribution,
\begin{equation}} \def\eeq{\end{equation} q = - 1 + {3 \over 2(1-n)} ~~~~,~~~~w = {n \over 1 - n}~~. \label{eqqz} \eeq
\noindent The latter is the same form found in the previous section. By fitting the numerical solution of \eq{modFriedmann} to the evolution profile of $q(z)$ \cite{gong}, one obtains the best fit values $n = -10$ and $t_{-10} = t_0/2 $, depicted in Fig. \ref{qzgraph}.

This value for the exponent $n$ is very distinct from the $n_{IS} = -1$ and $n_{NFW} = -1/3$ scenarios considered in the previous section. However, these have no cosmological impact on the current scenario, given the broad difference between the relevant timescale $t_0$ and those obtained from $t_{-1} \equiv R_{-1}^{-1/2}$ and $t_{-1/3} \equiv R_{-1/3}^{-1/2}$. The value $n = -10$ may also be regarded as somewhat unnatural, but is highly dependent on the $q(z)$ profile: Fig. \ref{qzgraph} shows that $n = -4$, $t_2 = t_0/4$ also yields a curve within the allowed $3\sigma} \def\vs{\varsigma} \def\ta{\tau} \def\up{\upsilon$ region. Thus, despite this caveat, one concludes that a non-minimal coupling may be used to describe the accelerated expansion of the Universe.
\begin{figure} 

\epsfxsize=\columnwidth \epsffile{qz.eps}
\caption{Evolution of the deceleration parameter $q(z)$ for $n=-4$, $t_2=t_0/4$ (full) and $n=-10$, $t_2=t_0/2$ (dashed); $1\sigma} \def\vs{\varsigma} \def\ta{\tau} \def\up{\upsilon$, $2\sigma} \def\vs{\varsigma} \def\ta{\tau} \def\up{\upsilon$ and $3\sigma} \def\vs{\varsigma} \def\ta{\tau} \def\up{\upsilon$ allowed regions are shaded, white line gives best fit ({\it cf}. Ref. [X]).}
\label{qzgraph}

\end{figure}

\section{Cosmological Perturbations}

In this section one ascertains how the presence of non-trivial functions $f_1(R)$ and $f_2(R)$ affects the evolution of cosmological perturbations --- so that any deviation from the usual GR dynamics can be used to probe the latter \cite{perturbations}. Indeed, it is known that $f(R)$ theories can leave characteristic imprints on the matter power spectrum \cite{pert11,pert12,pert13,pert14,pert15} and modify gravitational lensing \cite{Zhang}.

Cosmological perturbations can be decomposed into scalar, vector and tensorial modes \cite{Mukhanov}; however, since vector modes decay with the cosmological expansion and the tensorial ones are decoupled from the energy density or pressure because the non-diagonal terms of the energy-momentum tensor of matter are zero, one focuses solely on scalar perturbations --- which grow during the expansion of the Universe and lead to the formation of large scale structure (LSS). Thus, in the longitudinal gauge \cite{Mukhanov}, the perturbed FRW metric is

\begin{equation}
ds^{2} = -(1+2\Phi)\, dt^{2}+a^{2}(t)(1-2\Psi)\tilde{g}_{ij}dx^{i}\, dx^{j}~~.
\end{equation}

One obtains the equations ruling the evolution of cosmological perturbations by linearizing the modified field \eq{field0} and non-conservation of the energy-momentum tensor \eq{cov}. In particular, the non-diagonal components of the former yield the relation

\begin{equation}} \def\eeq{\end{equation} \Phi - \Psi = - \de \log \left( F_1 - {F_2 \rho\over \ka} \right)~~, \eeq

\noindent which should be contrasted with the identity $\Phi = \Psi$ obtained in GR.

The relevant scale for the formation of LSS corresponds to modes deep inside the horizon during the matter dominated era, for which $k^{2}/a^{2}H^{2}\gg 1$. Neglecting time derivatives of the potentials and keeping only terms on the energy density fluctuations \cite{Cruz}, one can show from the linearized field \eq{field0} that the generalized Newtonian potential $\Phi$ obeys a modified Poisson equation,

\begin{equation}
\Phi\approx-4\pi \tilde{G}{a^2\over k^2}\delta \rho ~~,
\end{equation}

\noindent with the modified gravitational constant $\tilde{G}$ given by 

\begin{equation}
 {\tilde{G}\over G}\equiv {1+2 {k^{2}\over a^{2}R}\left(2m_{1}-m_{2}\right)\over 1+3 {k^{2}\over a^{2}R}m_{1}}\Sigma~~,
\end{equation}

\noindent introducing the weak lensing parameter $\Sigma$ \cite{weaklensing1,weaklensing2} and quantities $m_1$ and $m_2$,

\begin{equation}\label{defSigma}
\Sigma \equiv { f_2\over F_1- {F_2\rho \over \ka} }~~~~,~~~~ m_{1}\equiv =R{\ka F^\prime_{1}-F^\prime_{2}\rho \over \ka F_{1}-f_2\rho}~~~~,~~~~m_{2} \equiv R{f_2\over f_2}~~.
\end{equation}

\noindent In GR, one has $\Si = 1$, $m_1 =m_2=0$, so that $\tilde{G} = G$ and the usual Poisson equation is recovered.

By the same token, the equation for the second potential has also a Poisson form,

\begin{equation}} \def\eeq{\end{equation} \Psi\approx -4\pi G s{a^{2} \over k^{2}}\delta\rho ~~~~,~~~~ s={1+2{k^{2}\over a^{2}R}\left(m_{1}+m_{2}\right)\over 1+3{k^{2}\over a^{2}R}m_{1}}\Sigma ~~.
\end{equation}

\noindent In GR, one obtains $s=1$, confirming that indeed $\Psi = \Phi$, as expected.

Proceeding along these lines, Ref. \cite{perturbations} shows that the gauge-invariant matter overdensity  $\delta_m\equiv \delta \rho/\rho +3H v$ (where $v$ is the potential velocity, defined through $u_