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 .