ECOSMOG-EFT: Modified Gravity Simulations

• ECOSMOG-EFT is a simulation code that implements the Effective Field Theory framework for dark energy and cubic Horndeski models using adaptive mesh refinement.
• It employs operator-splitting and multigrid methods to solve the non-linear scalar field equations, achieving sub-percent accuracy in reproducing large-scale structure formation.
• Validated against analytic and code benchmarks, the code delivers precise predictions for observables critical to next-generation cosmological surveys.

ECOSMOG-EFT is a RAMSES-based adaptive-mesh-refinement (AMR) N-body simulation code for models in the Effective Field Theory of Dark Energy (EFTofDE) framework, supporting the non-linear, cubic Horndeski class of modified gravity models with a luminal gravitational wave speed. ECOSMOG-EFT numerically evolves both the standard cosmological N-body equations and an additional scalar field governed by the non-linear Vainshtein screening mechanism, accurately capturing the impact of modified gravity on structure formation from linear to deeply non-linear scales. The code has been validated against both analytic and code-to-code benchmarks, and provides sub-percent-level accuracy in reproducing large-scale structure observables relevant for upcoming cosmological surveys (Ganjoo et al., 16 Apr 2026).

1. Theoretical Framework

ECOSMOG-EFT is constructed to simulate cosmologies governed by the cubic sector of Horndeski's theory in the effective-field-theory (EFT) formalism, specifically restricting to models where the gravitational wave speed $c_{\rm GW}$ equals the speed of light (i.e., $\alpha_T=0$). The starting point is the EFT action for perturbations about a flat FLRW background, expressed in the Bellini & Sawicki $\alpha$-parametrization: $$S = \int d^4x\sqrt{-g} \left[ \frac{M^2(t)}{2}R - \Lambda(t) - c(t)\,g^{00} + \frac{1}{2}M^2(t)\alpha_K(t)(\delta g^{00})^2 - \frac{1}{2}M^2(t)\alpha_B(t)\delta g^{00}\delta K + \dots \right],$$ where the EFT functions $\{\alpha_K(a),\alpha_B(a),\alpha_M(a)\}$ parameterize kinetic, braiding, and Planck-mass running effects. Enforcing $\alpha_T=0$ (from GW170817 constraints) removes all non-trivial tensor speed contributions.

At the nonlinear level and in the quasi-static, subhorizon regime, the action up to third-order perturbations (Eqs. 3–5) involves only the cubic Horndeski operator, with the key dynamical fields $\{\Psi,\Phi,\chi\}$ (Newtonian potentials and scalar fluctuation). The governing action reads: $S = S_m + S_g^{(2)} + S_g^{(3)},$ with explicit forms for $A_{ab}$ and $B_{abc}$ given in Eq. 6 and Eq. 9, enforcing all time-dependence of EFT functions and screening coefficients ($\alpha_T=0$0).

The resulting field equations (Eqs. 11–12) are: $\alpha_T=0$1

$\alpha_T=0$2

$\alpha_T=0$3

The last equation encodes the non-linear Vainshtein screening, and $\alpha_T=0$4 parameterizes its strength. In supercomoving code units, the code implements these as Eqs. 17-18.

2. Numerical Algorithms and AMR Implementation

ECOSMOG-EFT extends the RAMSES and ECOSMOG-CVG AMR framework, with grid refinement triggered when the number of particles per cell exceeds $\alpha_T=0$5 (default), supporting up to six levels of refinement above the base $\alpha_T=0$6.

The solver for the non-linear $\alpha_T=0$7 equation uses operator-splitting and Full Approximation Storage (FAS) multigrid techniques. By recasting the $\alpha_T=0$8 equation as a local quadratic for $\alpha_T=0$9 (Section 2.3.2), operator splitting (with $\alpha$0) is applied to make the discrete update for $\alpha$1 in each cell depend only on neighbouring cells, improving convergence properties on adaptive grids. The FAS V-cycle incorporates red-black Gauss-Seidel smoothing, restriction of residuals to coarser grids, and prolongation of corrections back to finer levels, ensuring high efficiency in the AMR environment.

Stringent convergence criteria are enforced: each V-cycle must reduce the residual by at least a factor of 2.5, with final tolerances of $\alpha$2 on the base grid and $\alpha$3 on refined levels.

Boundary conditions are periodic; refinement and mass resolution are tunable, with the standard configuration using $\alpha$4 in a box of $\alpha$5 Mpc. All variables are stored in supercomoving code units.

3. Validation and Numerical Performance

ECOSMOG-EFT underwent a suite of verification tests:

• Static Spherical Test (Sec 4.1): For a truncated, isothermal sphere ($\alpha$6, $\alpha$7), the $\alpha$8 solver matches analytic $\alpha$9 to better than 0.2% for both $$S = \int d^4x\sqrt{-g} \left[ \frac{M^2(t)}{2}R - \Lambda(t) - c(t)\,g^{00} + \frac{1}{2}M^2(t)\alpha_K(t)(\delta g^{00})^2 - \frac{1}{2}M^2(t)\alpha_B(t)\delta g^{00}\delta K + \dots \right],$$0 and $$S = \int d^4x\sqrt{-g} \left[ \frac{M^2(t)}{2}R - \Lambda(t) - c(t)\,g^{00} + \frac{1}{2}M^2(t)\alpha_K(t)(\delta g^{00})^2 - \frac{1}{2}M^2(t)\alpha_B(t)\delta g^{00}\delta K + \dots \right],$$1 grids. The computed $$S = \int d^4x\sqrt{-g} \left[ \frac{M^2(t)}{2}R - \Lambda(t) - c(t)\,g^{00} + \frac{1}{2}M^2(t)\alpha_K(t)(\delta g^{00})^2 - \frac{1}{2}M^2(t)\alpha_B(t)\delta g^{00}\delta K + \dots \right],$$2 (including Vainshtein screening) agrees to within 2.5% down to a four-cell core radius.
• Linear Regime (Sec 4.2): ECOSMOG-EFT and the companion PySCo-EFT (particle-mesh) code both reproduce the EFTofDE-to-$$S = \int d^4x\sqrt{-g} \left[ \frac{M^2(t)}{2}R - \Lambda(t) - c(t)\,g^{00} + \frac{1}{2}M^2(t)\alpha_K(t)(\delta g^{00})^2 - \frac{1}{2}M^2(t)\alpha_B(t)\delta g^{00}\delta K + \dots \right],$$3CDM power spectrum boost $$S = \int d^4x\sqrt{-g} \left[ \frac{M^2(t)}{2}R - \Lambda(t) - c(t)\,g^{00} + \frac{1}{2}M^2(t)\alpha_K(t)(\delta g^{00})^2 - \frac{1}{2}M^2(t)\alpha_B(t)\delta g^{00}\delta K + \dots \right],$$4 to 0.4% agreement with linear theory at $$S = \int d^4x\sqrt{-g} \left[ \frac{M^2(t)}{2}R - \Lambda(t) - c(t)\,g^{00} + \frac{1}{2}M^2(t)\alpha_K(t)(\delta g^{00})^2 - \frac{1}{2}M^2(t)\alpha_B(t)\delta g^{00}\delta K + \dots \right],$$5, and to within 1% code-to-code up to $$S = \int d^4x\sqrt{-g} \left[ \frac{M^2(t)}{2}R - \Lambda(t) - c(t)\,g^{00} + \frac{1}{2}M^2(t)\alpha_K(t)(\delta g^{00})^2 - \frac{1}{2}M^2(t)\alpha_B(t)\delta g^{00}\delta K + \dots \right],$$6.
• Nonlinear and Parameter Sensitivity (Appendix A/B): Across mass resolutions ($$S = \int d^4x\sqrt{-g} \left[ \frac{M^2(t)}{2}R - \Lambda(t) - c(t)\,g^{00} + \frac{1}{2}M^2(t)\alpha_K(t)(\delta g^{00})^2 - \frac{1}{2}M^2(t)\alpha_B(t)\delta g^{00}\delta K + \dots \right],$$7 to $$S = \int d^4x\sqrt{-g} \left[ \frac{M^2(t)}{2}R - \Lambda(t) - c(t)\,g^{00} + \frac{1}{2}M^2(t)\alpha_K(t)(\delta g^{00})^2 - \frac{1}{2}M^2(t)\alpha_B(t)\delta g^{00}\delta K + \dots \right],$$8), box sizes, refinement thresholds, solver parameters, and starting redshifts, $$S = \int d^4x\sqrt{-g} \left[ \frac{M^2(t)}{2}R - \Lambda(t) - c(t)\,g^{00} + \frac{1}{2}M^2(t)\alpha_K(t)(\delta g^{00})^2 - \frac{1}{2}M^2(t)\alpha_B(t)\delta g^{00}\delta K + \dots \right],$$9 varies by less than 1% at $\{\alpha_K(a),\alpha_B(a),\alpha_M(a)\}$0 and below 2% even at $\{\alpha_K(a),\alpha_B(a),\alpha_M(a)\}$1. Full $\{\alpha_K(a),\alpha_B(a),\alpha_M(a)\}$2-solver runs confirm that Vainshtein screening suppresses $\{\alpha_K(a),\alpha_B(a),\alpha_M(a)\}$3 for $\{\alpha_K(a),\alpha_B(a),\alpha_M(a)\}$4 whenever $\{\alpha_K(a),\alpha_B(a),\alpha_M(a)\}$5. For negligible $\{\alpha_K(a),\alpha_B(a),\alpha_M(a)\}$6, the linearized solver suffices; when $\{\alpha_K(a),\alpha_B(a),\alpha_M(a)\}$7, errors in the non-screened approach can exceed 10–30% by $\{\alpha_K(a),\alpha_B(a),\alpha_M(a)\}$8.
• Science Runs (Sec 4.3): Varying $\{\alpha_K(a),\alpha_B(a),\alpha_M(a)\}$9 and $\alpha_T=0$0 demonstrates the boost $\alpha_T=0$1 rises with $\alpha_T=0$2 and the sign of $\alpha_T=0$3. Screening reduces $\alpha_T=0$4 to unity at small scales in non-linear runs, where linearized results significantly diverge.

4. Input Parameters, Compilation, and Usage

ECOSMOG-EFT adopts the standard ECOSMOG/RAMSES parameter file interface, with two additional EFTofDE-specific entries:

• $\alpha_T=0$5
• $\alpha_T=0$6

The time dependence is hard-coded as: $\alpha_T=0$7 pivoted at $\alpha_T=0$8. Including $\alpha_T=0$9 or arbitrary $\{\Psi,\Phi,\chi\}$0 requires minor code modifications.

Repository is available at https://github.com/hganjoo/ecosmogeft.git Compilation requires the RAMSES library, MPI, and HDF5, following ECOSMOG-CVG build instructions. The computational cost per EFTofDE run (with $\{\Psi,\Phi,\chi\}$1 particles and 6 AMR levels) is approximately 10 times that of $\{\Psi,\Phi,\chi\}$2CDM RAMSES.

5. Output Products and Analysis

ECOSMOG-EFT outputs standard RAMSES-format snapshots containing particle positions, velocities, and optionally the scalar field $\{\Psi,\Phi,\chi\}$3 on the AMR grid. Analysis pipelines (PKLibrary, Pylians) are compatible, supporting fast computation of power spectra via CIC-mesh density assignment and FFT, as well as boosted observables $\{\Psi,\Phi,\chi\}$4, bispectra, and halo catalogs via standard post-processing. Lightcone and weak lensing pipelines can be attached via the existing ECOSMOG modules.

6. Scientific Impact and Scope

ECOSMOG-EFT enables the generation of accurate predictions for the non-linear matter distribution in a broad range of modified gravity and dark energy models, as encoded in the cubic Horndeski/EFTofDE framework subject to luminal gravitational wave constraints. The code attains better than 1% numerical control at $\{\Psi,\Phi,\chi\}$5 and maintains sub-2% accuracy to $\{\Psi,\Phi,\chi\}$6 across varied initialization and refinement parameters. It directly supports the theoretical requirements of next-generation large-scale structure and weak lensing surveys, providing robust tools to explore parameterized departures from $\{\Psi,\Phi,\chi\}$7CDM including Vainshtein screening, and allows the field to constrain or falsify broad classes of modified gravity via direct simulation (Ganjoo et al., 16 Apr 2026).