  We study structure formation in phenomenological models in which the
Friedmann equation receives a correction of the form
$H^{\alpha}/r_c^{2-\alpha}$, which realize an accelerated expansion without
dark energy. In order to address structure formation in these model, we
construct simple covariant gravitational equations which give the modified
Friedmann equation with $\alpha=2/n$ where $n$ is an integer. For $n=2$, the
underlying theory is known as a 5D braneworld model (the DGP model). Thus the
models interpolate between the DGP model ($n=2, \alpha=1$) and the LCDM model
in general relativity ($n \to \infty, \alpha \to 0$). Using the covariant
equations, cosmological perturbations are analyzed. It is shown that in order
to satisfy the Bianchi identity at a perturbative level, we need to introduce a
correction term $E_{\mu \nu}$ in the effective equations. In the DGP model,
$E_{\mu \nu}$ comes from 5D gravitational fields and correct conditions on
$E_{\mu \nu}$ can be derived by solving the 5D perturbations. In the general
case $n>2$, we have to assume the structure of a modified theory of gravity to
determine $E_{\mu \nu}$. We show that structure formation is different from a
dark energy model in general relativity with identical expansion history and
that quantitative features of the difference crucially depend on the conditions
on $E_{\mu \nu}$, that is, the structure of the underlying theory of modified
gravity. This implies that it is essential to identify underlying theories in
order to test these phenomenological models against observational data and,
once we identify a consistent theory, structure formation tests become
essential to distinguish modified gravity models from dark energy models in
general relativity.
