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 $\Lambda$CDM 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 $\Lambda$CDM model. We
introduce two phase diagrams, $\alpha{\rm-}f\sigma _8$ and $H{\rm-}f\sigma _8$
diagrams; $H$ is the expansion rate, $f\sigma_8$ 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 $\alpha$ 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 $\Lambda$CDM model appears in the $\alpha{\rm-}f\sigma _8$ and
$H{\rm-}f\sigma _8$ diagrams. The predicted deviations will be useful for
future large-scale structure observations to exclude some of the modified
gravity models.
