Structure formation in modified gravity models

We study structure formation in phenomenological models in which the Friedmann equation receives a correction of the form H (alpha)/r(c)(2-alpha), to realize an accelerated expansion without dark energy. In order to address structure formation in these models, 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 five-dimensional (5D) braneworld model (the DGP model). Thus the models interpolate between the DGP model (n = 2, alpha = 1) and the lambda cold dark matter (LCDM) model in general relativity (n -> infinity , alpha -> 0). Using the covariant equations, cosmological perturbations are analysed. 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 that of 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.