Consistency relations for large-scale structure in modified gravity and the matter bispectrum

Abstract

We study perturbation theory for large-scale structure in the most general scalar-tensor theories propagating a single scalar degree of freedom, which include Horndeski theories and beyond. We model the parameter space using the effective field theory of dark energy. For Horndeski theories, the gravitational field and fluid equations are invariant under a combination of time-dependent transformations of the coordinates and fields. This symmetry fixes the perturbation-theory kernels in the squeezed limit and ensures that the well-known consistency relations for large-scale structure, originally derived in general relativity, hold in modified gravity as well. For theories beyond Horndeski, instead, the gravitational field and fluid equations are invariant under separate transformations. In the absence of a common symmetry, the perturbation-theory kernels are modified in the squeezed limit and the consistency relations for large-scale structure do not hold. We show, however, that the modification of the squeezed limit depends only on the linear theory. We investigate the observational consequences of this violation by computing the matter bispectrum. In the squeezed limit, the largest effect is expected when considering the cross-correlation between different tracers. Moreover, the individual contributions to the 1-loop matter power spectrum do not cancel in the infrared limit of the momentum integral, modifying the power spectrum on non-linear scales.

UV sensitive one-loop matter power spectrum in degenerate higher-order scalar-tensor theories

“…The first-and second-order solutions for the matter density perturbations in DHOST theories have already been obtained in the literature [45,46], but here for completeness we replicate the previous discussion. The third-order solution, which is obtained for the first time in this paper, is presented in the next section.…”

“…Substituting Eqs. ( 4) and (5) to the action (1) and expanding it in terms of the perturbations, one obtains the action for the perturbations [21,23,24,44,45,53]. In doing so we take the quasi-static approximation and neglect time derivatives compared to spatial derivatives.…”

“…It should be noted that δ(n) appears in Eq. ( 23) when one considers theories more general than Horndeski [18,23,24,44,45,53,54]. Given the concrete form of O (n) , it is straightforward to write the explicit form of τ…”

“…Refs. [45,46] have investigated the bi and tri-spectra of the matter density perturbations and also the one-loop matter power spectrum in the context of DHOST theories. They have, in particular, focused on the behaviors at the infrared (IR) limit: the squeezed limit for the bispectrum, the double soft limit for the trispectrum, and the IR contributions in the loop integrals for the one-loop power spectrum, and studied a consistency relation for large scale structure and its violation.…”

“…So far, in DHOST theories, the third-order solution for the matter density perturbations has been obtained only at the IR limit as done in Refs. [45,46], and the complete form of the one-loop matter power spectrum has not been derived yet. The goal of this paper is therefore to derive the third-order solution for the matter density perturbations and to investigate the one-loop matter power spectrum in its complete form.…”