What $f(R)$ models pass most of the known constraints? In most papers and talks about $f(R)$ gravity authors repeatedly state that the model proposed by Starobinsky 2007
$$
f(R)=R+\lambda\,R_{0} \bigg[\bigg(1+\frac{R^{2}}{R_{0}^{2}}\bigg)^{-n}-1\bigg]
$$
passes all or most of the known constraints.
My question is: Are there any other models which also pass all known constraints?
I am mainly concerned with models with modifications that become important for small $R$.
 A: I think in general there is a wide class of models that should pass cosmological and astrophysical constraints.  The Living Review on this is a good place to look.  In section 14 they explain the basic constraints on allowed types of $f(R)$ models.  It seems like they amount to

*

*$f_{,R} > 0$ and $f_{,RR}>0$ when $R>R_0$, where $R_0$ is the scalar curvature of the present universe (something like the Hubble rate squared).  This is to avoid ghosts or tachyons.

*$f(R)$ approaches the usual GR Lagrangian for $R>R_0$, to pass local gravity constraints like solar system tests.

There is another restriction on the derivatives of $f$ to ensure the late time de Sitter phase exists and is stable.
Section 4.2 of that article also describes some viable $f(R)$ models that have noticeable infrared (or low curvature) effects.  But basically it amounts to choosing a function that satisfies the above constraints (of which there are many), and then simply checking to see if it gives the desired behavior on cosmological scales.
