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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 $v^i_π$ 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 $v^i_π$ with a single coordinate transformation, and thus the standard $Λ$CDM consistency relations for large-scale structure are violated by terms proportional to the relative velocity $v^i - v^i_π$. 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 $Λ$CDM with exact time dependence.