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Physical Review D

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Nonlinear dynamics in Horndeski gravity: A renormalized approach to effective gravitational coupling

Luca Amendola ^1,*, Carla Bernal ^2,†, and Radouane Gannouji ^2,‡

¹Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
²Instituto de Física, Pontificia Universidad Católica de Valparaíso Avenida Brasil 2950, Valparaíso, Chile

^*Contact author: l.amendola@thphys.uni-heidelberg.de
^†Contact author: carlabernalb@gmail.com
^‡Contact author: radouane.gannouji@pucv.cl

Phys. Rev. D 112, 063533 – Published 17 September, 2025

DOI: https://doi.org/10.1103/64xp-trdr

Abstract

This paper develops a renormalized perturbation theory framework for nonlinear structure formation in a broad class of modified gravity models that exhibit Vainshtein screening, with a focus on a viable subclass of Horndeski theories. We extend earlier perturbative methods, originally applied to the Dvali–Gabadadze–Porrati model, to construct a self-consistent treatment that captures both the linear modifications to gravity at large scales and the nonlinear screening effects at small scales. In the framework, the response of the gravitational potential to matter density fluctuations is characterized by renormalized propagators, leading to the definition of a nonlinear (or renormalized) effective gravitational constant. This paper details several numerical strategies to compute this renormalized gravitational constant. Numerical examples illustrate how the effective gravitational constant evolves with scale and redshift. These results are key to accurately predicting cosmological observables such as the matter power spectrum and bispectrum in modified gravity scenarios.

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