  We consider theories with an arbitrary coupling between matter and gravity
and obtain the perturbation equation of matter on subhorizon scales. Also, we
derive the effective gravitational constant $G_{eff}$ and two parameters
$\Sigma$ and $\eta$, which along with the perturbation equation of the matter
density are useful to constrain the theory from growth factor and weak lensing
observations. Finally, we use a completely solvable toy model which exhibits
nontrivial phenomenology to investigate specific features of the theory. We
obtain the analytic solution of the modified Friedmann equation for the scale
factor $a$ in terms of time $t$ and use the age of the oldest star clusters and
the primordial nucleosynthesis bounds in order to constrain the parameters of
our toy model.
