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

Ahmad Nouri-Zonoz, Farbod Hassani, Julian Adamek, Emilio Bellini, Martin Kunz

Nov 2, 2026arXiv:2611.03637v1

astro-ph.COgr-qcphysics.comp-ph

Scorecard· 16/16

5.5/10 impact

Useful, well-documented public code generalising a line of relativistic DE simulations to the full linear α-basis, but physics remains linear in the scalar sector and validation is modest.

Abstract

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.

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Mercury (Hg): a relativistic N-body simulation for Horndeski gravity — impact assessment

Core Contribution

Mercury generalises the authors' earlier k-evolution and KGB-evolution codes, which are built on gevolution, to the general linear Horndeski α-basis (α_K, α_B, α_M, α_T plus a running effective Planck mass M*²). The scalar degree of freedom is evolved on the lattice with its linear equation of motion, but it is sourced by nonlinear matter and metric fields. Its effective stress-energy, including the anisotropic stress, feeds back into a modified Hamiltonian constraint and trace-free solver. The solver also changes the vector and tensor friction terms through α_M and the tensor speed through α_T.

The main practical novelty is the interface to hi_class. Any model in hi_class, whether specified through α-parametrisations or covariant Horndeski actions, can be run in a relativistic N-body setting. The paper demonstrates six models: three EFT limits (kess, KGB, luminal Horndeski) and three covariant ones (nKGB, Cubic Galileon, Brans–Dicke).

The key conceptual limitation is that the scalar dynamics stay linear. There are no nonlinear scalar self-interactions and no screening, so this is an "intermediate step" and not a full nonlinear Horndeski simulation. Much of the novelty is therefore engineering and generalisation, not a new physical regime.

Methodological Rigor

The formulation is coherent. The Jordan-frame, minimally coupled matter treatment keeps the gevolution particle equations unchanged. The leapfrog scheme for ζ and π is explicitly derived, and the Hamiltonian-constraint and trace-free modifications are clearly localised in the source terms and the damping coefficients.

Validation is mostly qualitative. Agreement with hi_class on large scales and recovery of the k-essence and KGB limits are shown, but the paper has no quantitative error metrics in the linear regime. It has no independent code comparison, for example against the EFT N-body codes it cites (PySCo-EFT, ECOSMOG-EFT). It also has no tests with analytic nonlinear solutions.

The consistency of treating a linear scalar equation inside a nonlinear weak-field metric is not tested quantitatively. Examples would be checks on momentum-constraint violation or on the size of neglected terms. The paper admits that α functions do not fix nonlinear dynamics. Its closing claim that the small δ_DE "supports the linear treatment" is self-referential and not demonstrated for models with α_B of order unity, such as nKGB and the Cubic Galileon.

The physical interpretation of results is largely qualitative. The explanation of EFT-LH having less Φ power is stated as "consistent with" and not isolated by controlled experiments. The vector and tensor results use the quasistatic elliptic treatment for h_ij, so the α_M friction-term modification for tensors is derived but not actually exercised. α_T = 0 throughout, so the modified tensor speed is untested.

Convergence tests (Appendix A) are reasonable. They cover three resolutions, and the departure from convergence at about 0.12–0.16 of Nyquist is characterised. Individual spectra are good to roughly 2% only for k ≲ 0.3 h/Mpc, while ratios are more robust. The convergence estimates are extrapolated and not measured at the production resolution.

The runs use a single realisation per model, with no cosmic-variance or seed tests. Box-scale spectra are shown without error bars.

The comparison to the "linear hi_class" result for χ in models with α_M = α_T = 0 is physically sensible: nonlinear matter anisotropic stress dominates the slip. This is a useful sanity check.

Potential Impact

It provides a public CPU/GPU code with a clean pipeline from a theory specification (hi_class) to relativistic simulations, giving a reusable tool for the relativistic-simulation community. It lowers the barrier to producing nonlinear predictions for broad classes of Horndeski/EFT-of-DE models, in a framework that retains the metric potentials, vector and tensor modes, and light-cone observables (lensing, ISW-RS, redshift effects).

Its scientific payoff is bounded. Without screening or nonlinear scalar terms, the most interesting nonlinear Horndeski phenomenology (Vainshtein screening, scalar instabilities) is not captured. For the luminal models studied, α-basis linear scalar dynamics may already be adequate on quasi-linear scales, and the observable differences from linear or semi-analytic treatments remain modest. The paper includes no forecast or data-facing demonstration, for example an observable that changes by a detectable amount.

It is directly relevant to the Euclid/DESI/LSST era, and the EFT-LH parameters come from recent data fits (CMB+DESI+ISWL+Pantheon+), which is a good motivational link.

Competing codes (ECOSMOG-EFT, PySCo-EFT, Galileon N-body codes) handle nonlinear screening in Newtonian frameworks. Mercury's distinctive advantage is relativistic completeness, which matters mainly for horizon-scale and relativistic observables.

Timeliness & Relevance

Interest is high in dynamical dark energy (DESI hints at w0wa, with phantom crossing) and in scalar–tensor extensions. Simulation tools that go beyond ΛCDM are a recognised bottleneck. The paper is timely, but it is an incremental technical step in a series (k-evolution, KGB-evolution) from the same group.

Strengths

General α-basis coverage with a hi_class interface, supporting both the EFT and covariant routes, in a public code with GPU support.

Clear exposition of the implementation, including lattice discretisation, workflow stages, and the shooting procedure.

Good handling of gauge transformations and initial conditions, and a transparent statement of approximations.

A useful demonstration that α_M generates linear slip and modifies the damping of the vector and tensor sectors.

Honest discussion of limitations and the roadmap toward nonlinear scalar dynamics.

Limitations

The scalar sector is linear, so there is no screening and no genuinely new nonlinear Horndeski physics.

Validation is mostly visual and qualitative, with no quantitative code comparison and no tests of the unexercised α_T or dynamical-tensor sectors.

There is no performance or scaling data; it is deferred to future work.

There is no observational or forecasting application, and the results largely re-demonstrate qualitative behaviour already seen in KGB-evolution.

The "nonlinear dark energy" snapshots arise only from sourcing by nonlinear matter and are not a test of the theory's own nonlinearity.

Overall

This is a solid, useful code paper that will be cited by those using or extending gevolution-based codes and by people interested in relativistic modified-gravity simulations. It is unlikely to change the field's direction until a nonlinear scalar sector is added.

Rating:5.5/ 10

Significance 5.5Rigor 5.5Novelty 4.5Clarity 7

Generated Nov 3, 2026

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