PySCo-EFT and ECOSMOG-EFT: a tandem of N-body simulation codes for the Effective Field Theory of Dark Energy

Abstract: Modified gravity theories constitute viable alternatives to the standard cosmological model for explaining the observed late-time accelerated expansion of the Universe. The Effective Field Theory of Dark Energy (EFTofDE) is an efficient framework to describe a wide range of such theories with a limited number of parameters. To robustly constrain them by comparison with clustering and weak lensing data from upcoming large-scale structure surveys, high-resolution cosmological N-body simulations are required to obtain accurate predictions for the matter distribution on non-linear scales. We introduce two new N-body simulation codes for EFTofDE cosmologies: PySCo-EFT, a Python-based particle mesh code, and ECOSMOG-EFT, a RAMSES-based code with adaptive mesh refinement. We consider Horndeski models with a luminal gravitational wave speed. We use iterative solvers and multigrid schemes to solve for the additional scalar field equation in both codes, incorporating the non-linear Vainshtein screening mechanism. We present validation and convergence tests of the codes. We obtain a sub-0.5 percent agreement with linear theory on large scales and a similar agreement between the two codes on non-linear scales. The dominant numerical effects on the matter-power-spectrum boost are mass resolution, finite-volume effects, refinement threshold, and starting redshift, but they are limited to below 2% at the largest wavenumbers (k=10 h/Mpc) for the range of tested values. We investigate the impact of the EFTofDE parameters on the matter-power-spectrum ratios between EFTofDE and $Λ$CDM cases. Depending on the EFTofDE parameters, the screening plays a negligible or dominant role compared to the linearised field equations. Our codes provide tools for generating fast and accurate predictions of the impact of the EFTofDE on the clustering of matter, incorporating non-linear screening.

• The paper presents two new N-body simulation codes, PySCo-EFT and ECOSMOG-EFT, that implement the nonlinear Effective Field Theory of Dark Energy.
• It details advanced grid-based methods and adaptive mesh refinement techniques to accurately capture Vainshtein screening and modified gravity effects.
• It validates the codes with sub-percent precision in power spectrum predictions and robust convergence tests across various EFTofDE parameters.

Fast and Robust N-Body Codes for Nonlinear Effective Field Theory of Dark Energy Cosmologies

Introduction and Context

Addressing the nature of cosmic acceleration remains central in modern cosmology. Modified gravity (MG) models, and more generally the Effective Field Theory of Dark Energy (EFTofDE), provide a systematic approach for parameterizing deviations from $\Lambda$CDM at the level of cosmological perturbations. While the EFTofDE framework efficiently encapsulates a variety of MG and dark energy (DE) theories using a handful of time-dependent parameters (notably the $\alpha$-basis), making accurate predictions for the non-linear clustering regime demands simulations that consistently incorporate both the modifications to gravity and nonlinear screening effects, particularly Vainshtein screening.

The paper "PySCo-EFT and ECOSMOG-EFT: a tandem of N-body simulation codes for the Effective Field Theory of Dark Energy" (2604.15434) addresses the computational gap for precision constraints by introducing two new N-body codes targeting the EFTofDE formalism: PySCo-EFT, a Python-based particle-mesh code for rapid explorations, and ECOSMOG-EFT, an extension of the RAMSES-based ECOSMOG-CVG code with adaptive mesh refinement (AMR). Both codes target the cubic subset of luminal Horndeski theories, enabling nonlinear Vainshtein screening and accurate coupling of the scalar field to the metric.

Theoretical Framework and Parameter Choices

The EFTofDE framework enables a model-independent description of a broad class of MG/DE scenarios, structuring cosmological perturbations in terms of the time-dependent $\alpha$-parameters: $\alpha_{\rm M}$, $\alpha_{\rm B}$, and their nonlinear analogs. The focus here is on the cubic Galactic sector with $\alpha_{\rm T} = 0$, respecting the stringent post-GW170817 bounds on gravitational wave speed.

A quasi-static and sub-horizon expansion brings the relevant equations into elliptic, nonlinear PDEs for the metric potentials $\Psi$, $\Phi$, and an additional scalar field $\chi$. The presence of the Vainshtein screening term in the $\chi$ equation forces the use of grid-based solvers rather than Fourier or tree approaches. This includes terms structured as:

$\alpha$0

where the nonlinear term governs screening.

The time evolution of the $\alpha$1-parameters is prescribed as proportional to $\alpha$2, ensuring that deviations from standard cosmology are significant only during DE domination.

Figure 1: The change in the gravitational coupling, $\alpha$3, as a function of the scale factor for various choices of EFT parameters, illustrating the time dependence of modifications to gravity.

Numerical Methods and Code Architecture

Grid Discretization and Solvers

Due to the nonlinearity and structure of the field equations, both codes employ grid-based, iterative numerical strategies:

• PySCo-EFT: Utilizes Jacobi iteration with multigrid V-cycle acceleration in Python/Numba. The scalar field $\alpha$4 is solved in supercomoving coordinates on regular grids, with under-relaxation and explicit stencils for Laplacian and cross derivatives to handle the nonlinearities.
• ECOSMOG-EFT: Extends ECOSMOG-CVG within RAMSES, leveraging AMR and a Gauss-Seidel (red-black) smoother, multigrid FAS scheme, and operator splitting to compute the Laplacian of $\alpha$5. The AMR implementation enables high accuracy in resolving screening in dense environments.

Both codes apply a “force addition” approach: the fifth force from $\alpha$6 supplements the standard Newtonian force, with the Poisson solver itself optionally modified by a time-dependent $\alpha$7.

Voids and Field Breakdown

The discriminant of the quadratic equation for $\alpha$8 can become negative in deep voids, reflecting the quasi-static breakdown or actual pathologies in certain MG parameter regimes. To ensure numerical stability and maintain power spectrum fidelity, the codes follow an established fix: the discriminant is set to a small positive value in such cells, as their contribution to the nonlinear regime is subdominant.

Validation, Convergence, and Robustness

Spherical Solutions

Direct tests against analytical solutions for static, spherically symmetric configurations confirm the numerical accuracy of the solvers at the $\alpha$90.2% level for the scalar field and $\alpha$02.5% for the gravitational potential outside the innermost cells.

Figure 2: Solutions for $\alpha$1 on a fixed grid with 32$\alpha$2 cells, $\alpha$3 Mpc, demonstrating the match between numerical and analytical spherically symmetric solutions.

Figure 3: Top: The absolute value of the gravitational potential $\alpha$4 for a $\alpha$5 cell box (circles), analytical (solid/dashed), and GR (dotted) solutions; Bottom: Ratio of numerical to analytical results.

Power Spectrum Comparison: Nonlinear and Linear EFTofDE

Both codes are cross-validated in matched PM-only mode, with matter power spectra and their boosts compared for a variety of EFT parameter choices and screening/non-screening cases. The relative differences in the power spectrum boost $\alpha$6 between both codes are uniformly below 1% for $\alpha$7.

Figure 4: EFT boost $\alpha$8 comparing ECOSMOG-EFT (solid), PySCo-EFT (dashed), and linear theory for nonlinear (thick) and linearised (thin) simulations.

Figure 5: The relative error in $\alpha$9 between PySCo-EFT and ECOSMOG-EFT for nonlinear and linearised runs, staying below the $\alpha_{\rm M}$0 level up to $\alpha_{\rm M}$1.

Comprehensive convergence testing addresses mass resolution, box size, multigrid cycles, AMR refinement threshold, starting redshift, and solver accuracy, demonstrating percent-level robustness in the $\alpha_{\rm M}$2 for all reasonable parameter choices. Notably, convergence to sub-percent differences in $\alpha_{\rm M}$3 is achieved for $\alpha_{\rm M}$4 and still within $\alpha_{\rm M}$5 at $\alpha_{\rm M}$6.

Impact of EFTofDE Parameters and Physical Insights

Varying the braiding ($\alpha_{\rm M}$7) and Planck mass run rate ($\alpha_{\rm M}$8) in full AMR simulations reveals the following:

• Large $\alpha_{\rm M}$9: Increases the large-scale boost $\alpha_{\rm B}$0, meaning more power in both linear and quasi-linear regimes due to enhanced coupling between the scalar field and the metric.
• Screening: For substantial $\alpha_{\rm B}$1, nonlinear screening (Vainshtein) becomes crucial. Linearised approaches without screening systematically overpredict the small-scale boost, in some cases diverging by $\alpha_{\rm B}$2 or more for large $\alpha_{\rm B}$3. Screening suppresses this, restoring GR on the smallest scales.
• Large $\alpha_{\rm B}$4: Alters the time dependence of $\alpha_{\rm B}$5, yielding scale-dependent modifications in the power spectrum at $\alpha_{\rm B}$6. Negative $\alpha_{\rm B}$7 values enhance small-scale clustering.

  Figure 6: EFT boost $\alpha_{\rm B}$8 for varying $\alpha_{\rm B}$9, showing the effect of braiding on the clustering amplitude, and agreement with linear theory on the largest scales.

  

  Figure 7: EFT boost $\alpha_{\rm T} = 0$0 as a function of $\alpha_{\rm T} = 0$1, highlighting the differential impact of the Planck mass running.

A key numerical result is sub-0.5% agreement between simulation boost $\alpha_{\rm T} = 0$2 and the analytical linear theory prediction at large scales, validating the approach. The codes demonstrate that linearised EFTofDE, while computationally cheaper and viable for small $\alpha_{\rm T} = 0$3, cannot be trusted for models with strong screening effects.

Implications and Future Directions

These simulation tools—PySCo-EFT for fast parameter-space surveys and ECOSMOG-EFT for precision, high-resolution, AMR-enabled runs—cover the use cases of both exploration and production of percent-accuracy theoretical templates. This is essential in the context of upcoming and ongoing Stage-IV galaxy surveys (Euclid, DES, LSST, etc.), which demand such theoretical precision in modeling nonlinear clustering and gravitational lensing.

The demonstrated sub-percent consistency between codes, analytical predictions, and robust convergence properties imply that these tools can enable next-generation constraints on EFTofDE parameters from large-scale structure, including scale-dependent and environment-dependent effects driven by screening. The codes can also be extended to support more complex Horndeski scenarios (e.g. arbitrary $\alpha_{\rm T} = 0$4 time evolution or different Lagrangian sectors), or to accept alternative background expansions.

A further significant aspect is the detailed assessment of where linearised field equations fail and when full nonlinear screening must be explicitly simulated, clarifying the requirements for theoretical systematics budgets in cosmological parameter inference.

Conclusion

This paper provides rigorously validated, robust, and efficient N-body simulation codes for EFTofDE cosmologies, covering both rapid exploration and high-resolution predictions with full screening. The new capability to generate accurate nonlinear observables for modified gravity models, including for the first time in both Python and RAMSES/AMR environments, will be central to analyses of forthcoming LSS and lensing data. The codes set a technical benchmark for future theoretical and observational studies of cosmic acceleration beyond $\alpha_{\rm T} = 0$5CDM (2604.15434).