New measures to test modified gravity cosmologies

The observed accelerated expansion of the Universe may be explained by dark energy or the breakdown of general relativity (GR) on cosmological scales. When the latter case, a modified gravity scenario, is considered, it is often assumed that the background evolution is the same as the ΛCDM model but the density perturbation evolves differently. In this paper, we investigate more general classes of modified gravity, where both the background and perturbation evolutions are deviated from those in the ΛCDM model. We introduce two phase diagrams, α−fσ8 and H−fσ8 diagrams; H is the expansion rate, fσ8 is a combination of the growth rate of the Universe and the normalization of the density fluctuation which is directly constrained by redshift-space distortions, and α is a parameter which characterizes the deviation of gravity from GR and can be probed by gravitational lensing. We consider several specific examples of Horndeski's theory, which is a general scalar-tensor theory, and demonstrate how deviations from the ΛCDM model appears in the α−fσ8 and H−fσ8 diagrams. The predicted deviations will be useful for future large-scale structure observations to exclude some of the modified gravity models.