Violation of the consistency relations for large-scale structure with
  dark energy
  We study infrared effects in perturbation theory for large-scale structure
coupled to the effective field theory of dark energy, focusing on, in
particular, Degenerate Higher-Order Scalar-Tensor (DHOST) theories. In the
subhorizon, Newtonian limit, DHOST theories introduce an extra large-scale
velocity $v^i_\pi$ which is in general different from the matter velocity
$v^i$. Contrary to the case in Horndeski theories, the presence of this extra
large-scale velocity means that one cannot eliminate the long-wavelength
effects of both $v^i$ and $v^i_\pi$ with a single coordinate transformation,
and thus the standard $\Lambda$CDM consistency relations for large-scale
structure are violated by terms proportional to the relative velocity $v^i -
v^i_\pi$. We show, however, that in non-linear quantities this violation is
determined by the linear equations and the symmetries of the fluid system. We
find that the size of the baryon acoustic oscillations in the squeezed limit of
the bispectrum is modified, that the bias expansion contains extra terms which
contribute to the squeezed limit of the galaxy bispectrum, that infrared modes
in the one-loop power spectrum no longer cancel, and that the equal-time double
soft limit of the tree-level trispectrum is non-vanishing. In addition, we give
explicit expressions for how these violations depend on the relative velocity.
Many of our computations are also relevant for perturbation theory in
$\Lambda$CDM with exact time dependence.
