Mercury (Hg): a relativistic $N$-body simulation for Horndeski gravity

Abstract

We present Mercury, a relativistic $N$-body code for cosmological simulations in the $\alpha$-basis formulation of Horndeski gravity. Mercury extends a particle-mesh scheme by incorporating the dynamics of the additional scalar degree of freedom and the corresponding modified Einstein equations, expressed in terms of the time-dependent functions ${\alpha }_{\mathrm{K}},{\alpha }_{\mathrm{B}},{\alpha }_{\mathrm{M}},{\alpha }_{\mathrm{T}}$ and the effective Planck mass $M_{\ast }^2$. This generalises our previous implementation of Horndeski subclass models, namely $k$-essence and kinetic gravity braiding (KGB), to the general linear Horndeski $\alpha$-basis. The code evolves the additional scalar degree of freedom and its effective stress--energy tensor alongside the nonlinear matter and metric fields, so that nonlinear structure formation sources the scalar-field evolution and feeds back on the gravitational potentials. Through its interface with hi_class, Mercury supports both phenomenological $\alpha$-function parametrisations and covariant Horndeski theories; we demonstrate the latter using nKGB, the Cubic Galileon, and Brans--Dicke models. We validate the implementation against hi_class in the linear regime and recover the expected nonlinear $k$-essence and KGB limits. For the luminal Horndeski models considered here, we show how a running effective Planck mass generates gravitational slip through the scalar anisotropic stress and modifies the friction terms governing the vector and tensor metric perturbations. The evolution of $M_{\ast }^2$ also modifies the normalisation and scale dependence of the effective dark-energy density contrast, leaving a corresponding imprint on its power spectrum. The present framework provides an intermediate step towards nonlinear Horndeski simulations, in which nonlinear scalar self-interactions and screening effects can be incorporated consistently within the same relativistic framework.

arXiv comment44 pages, 15 figures, 4 tables