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

We present Mercury, a relativistic $N$-body code for cosmological simulations in the $α$-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 $α_{\rm K}, α_{\rm B}, α_{\rm M}, α_{\rm T}$ and the effective Planck mass $M_*^2$. This generalises our previous implementation of Horndeski subclass models, namely $k$-essence and kinetic gravity braiding (KGB), to the general linear Horndeski $α$-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 $α$-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 $M_*^2$ 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.