Signatures of Horndeski gravity on the Dark Matter Bispectrum
  We present a detailed study of second-order matter perturbations for the
general Horn- deski class of models. Being the most general scalar-tensor
theory having second-order equations of motion, it includes many known gravity
and dark energy theories and General Relativity with a cosmological constant as
a specific case. This enables us to estimate the leading order dark matter
bispectrum generated at late-times by gravitational instability. We parametrize
the evolution of the first and second-order equations of motion as proposed by
Bellini and Sawicki (2014), where the free functions of the theory are assumed
to be proportional to the dark energy density. We show that it is unnatural to
have large 10% ( 1%) deviations of the bispectrum introducing even larger ~ 30%
(~ 5%) deviations in the linear growth rate. Considering that measurements of
the linear growth rate have much higher signal-to-noise than bispectrum
measurements, this indicates that for Horndeski models which reproduce the
expansion history and the linear growth rate as predicted by GR the dark matter
bispectrum kernel can be effectively modelled as the standard GR one. On the
other hand, an observation of a large bispectrum deviation that can not be
explained in terms of bias would imply either that the evolution of
perturbations is strongly different than the evolution predicted by GR or that
the theory of gravity is exotic (e.g., breaks the weak equivalence principle)
and/or fine-tuned.
