  Modified gravity theories have richer observational consequences for
large-scale structure than conventional dark energy models, in that different
observables are not described by a single growth factor even in the linear
regime. We examine the relationships between perturbations in the metric
potentials, density and velocity fields, and discuss strategies for measuring
them using gravitational lensing, galaxy cluster abundances, galaxy
clustering/dynamics and the ISW effect. We show how a broad class of gravity
theories can be tested by combining these probes. A robust way to interpret
observations is by constraining two key functions: the ratio of the two metric
potentials, and the ratio of the Gravitational ``constant'' in the Poisson
equation to Newton's constant. We also discuss quasilinear effects that carry
signatures of gravity, such as through induced three-point correlations.
  Clustering of dark energy can mimic features of modified gravity theories and
thus confuse the search for distinct signatures of such theories. It can
produce pressure perturbations and anisotropic stresses, which breaks the
equality between the two metric potentials even in general relativity. With
these two extra degrees of freedom, can a clustered dark energy model mimic
modified gravity models in all observational tests? We show with specific
examples that observational constraints on both the metric potentials and
density perturbations can in principle distinguish modifications of gravity
from dark energy models. We compare our result with other recent studies that
have slightly different assumptions (and apparently contradictory conclusions).
