One-loop kernels in scale-dependent Horndeski theory

TL;DR

This paper develops a framework to compute nonlinear cosmological perturbation kernels in scale-dependent Horndeski gravity, enabling accurate predictions of galaxy clustering and redshift-space distortions for testing gravity theories.

Contribution

It derives explicit second- and third-order kernels in Horndeski theories, incorporating scale dependence, bias, and RSD, facilitating improved analysis of large-scale structure data.

Findings

Kernels depend solely on the linear growth mode.

Explicit formulas for kernels including bias and RSD.

Framework suitable for galaxy power spectrum analysis in modified gravity.

Abstract

We investigate the nonlinear evolution of cosmological perturbations in theories with scale-dependent perturbation growth, first in general and then focusing on Horndeski gravity. Within the framework of standard perturbation theory, we derive the second- and third-order kernels and show that they are fully determined by two effective functions, \( h_1 \) and \( h_c \), which parametrize deviations from general relativity. Using the Wronskian method, we obtain solutions for the nonlinear growth functions and present explicit expressions for the resulting kernels, including bias and redshift space distortions, valid in the limit in which the $k$-dependent part is subdominant. We show that the kernels are entirely dependent on the linear growing mode: once this is calculated, the kernels are analytic up to a time integral. We also include redshift-space distortions (RSD) and scale-dependent bias. Our approach provides a physically motivated framework for evaluating the one-loop galaxy power spectrum in scale-dependent theories, suitable for the forecasts and actual data analysis.