F(R) gravity and the Hubble parameter

[shadishacker]

Dear all,

Considering Einstein Hilbert lagrangian, by using Einstein field equations one can get the form of Friedman equations and consequently the Hubble parameter.
I know that in f(R) models, Einstein equations get modified. However, what happens to the Friedman equation and the Hubble parameter?
I tried to solve them and get to the form of H, but it seems such a complicated equation.

Using the (00) component, I get
\begin{equation}
H^2=\frac{8\pi G}{3}\rho -\frac{6\alpha}{c^2}(\frac{\ddot{a}^2}{a^2} + H^4)
\end{equation}
What should I do with the
\begin{equation} \ddot{a}^2\end{equation}
in the first equation?!
The (11) component just makes everything more complicated!
I really appreciate any help or idea.
BTW, I am using FRW metric.

 

[shadishacker]

I just realized that the equations consist the forth order of the scale factor (a(t)).
Are these solvable in the matter dominated era? when
\begin{equation}
\rho(z)=\rho_m a^{-3}
\end{equation}

 

[shadishacker]

Has anybody seen any paper on the change of Hubble parameter with f(R)? Is there at least any approximation?