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

This paper introduces and validates two new N-body simulation codes, PySCo-EFT and ECOSMOG-EFT, designed to accurately model non-linear structure formation and Vainshtein screening in Effective Field Theory of Dark Energy cosmologies, providing essential tools for constraining modified gravity theories with upcoming large-scale structure surveys.

The Gist

Imagine the universe as a giant, invisible ocean. For decades, scientists have been trying to understand why this ocean is not just sitting still, but actually expanding faster and faster. The standard explanation is a mysterious force called "Dark Energy," but we don't really know what it is.

Some scientists think the standard explanation is wrong. They propose that gravity itself might work differently on huge cosmic scales than it does here on Earth. This is called "Modified Gravity."

The problem is that these theories are incredibly complex math puzzles. To see if they are true, scientists need to run massive computer simulations of the universe, watching how galaxies form and move over billions of years. But these simulations are so hard to build that, until now, no one had a good toolkit to test these specific "Modified Gravity" ideas.

Enter the "Tandem": PySCo-EFT and ECOSMOG-EFT

This paper introduces two new computer programs (codes) designed to solve this puzzle. Think of them as a dynamic duo, like a fast sports car and a rugged off-road truck, working together to explore the same terrain.

1. The Fast Sports Car: PySCo-EFT

• What it is: A Python-based code.
• The Analogy: Imagine you want to test a new recipe. You don't want to bake a whole wedding cake just to see if the sugar amount is right. You want a quick, small test batch.
• How it works: PySCo-EFT is the "quick test batch." It's fast and flexible. It allows scientists to run hundreds of simulations in a short time to get a rough idea of how different theories behave. It's great for exploring the "what-ifs" quickly.

2. The Rugged Off-Road Truck: ECOSMOG-EFT

• What it is: A high-precision code based on a powerful engine called RAMSES.
• The Analogy: Now that you know the recipe works, you want to bake the massive wedding cake for the actual event. You need a truck that can handle heavy loads and navigate rough, bumpy roads (like the dense clusters of galaxies).
• How it works: ECOSMOG-EFT is the heavy lifter. It uses "Adaptive Mesh Refinement," which is like having a camera that automatically zooms in super-close on the interesting parts (dense galaxies) while keeping the empty space blurry. It gives you the ultra-high-definition, accurate results needed to compare with real telescope data.

The Secret Sauce: The "Vainshtein Screening"

Here is the tricky part. If Modified Gravity is real, it should change how galaxies move. But we know gravity works perfectly here in our solar system. So, how can gravity be different on big scales but normal on small scales?

Nature uses a "shield" called the Vainshtein screening mechanism.

• The Metaphor: Imagine a loudspeaker playing music. Far away, the music is loud and clear (Modified Gravity is active). But if you stand right next to the speaker, the sound is so intense it distorts and effectively "hides" the true nature of the music, making it sound like normal silence (Standard Gravity).
• In the paper: The new codes are the first to successfully simulate this "shield." They can calculate how the universe behaves when the "shield" is active in dense areas (like galaxy clusters) and inactive in empty space.

What Did They Find?

The authors tested their two codes against each other and against standard math theories.

1. They agree: Even though the two codes use different math engines (one is like a sports car, one is a truck), they produced almost identical results. This proves the math is solid.
2. The "Shield" matters: They found that if you ignore the "shield" (the screening mechanism), your predictions are wildly wrong. You might think gravity is 15% stronger than it actually is in certain places. The shield is crucial for getting the right answer.
3. The Future: These tools are ready for the next generation of telescopes (like Euclid and DESI). When these telescopes take pictures of millions of galaxies, scientists will use PySCo and ECOSMOG to say, "Okay, does the universe look like the standard model, or does it look like one of these Modified Gravity theories?"

In a Nutshell

This paper is about building the best possible simulation tools to test if our understanding of gravity is broken. They built a fast tool for quick ideas and a super-accurate tool for final answers, and they proved that both work perfectly together to handle the complex "shielding" effects of the universe. This sets the stage for solving one of the biggest mysteries in physics: What is Dark Energy?

Technical Summary

1. Problem Statement

The standard $\Lambda$CDM cosmological model faces challenges in explaining the late-time accelerated expansion of the Universe, leading to the exploration of Modified Gravity (MG) theories. The Effective Field Theory of Dark Energy (EFTofDE) provides a unified, model-independent framework to describe a wide range of MG scenarios (specifically Horndeski theories) using a minimal set of time-dependent parameters ($\alpha$-basis).

However, constraining these theories against upcoming high-precision Stage-IV galaxy surveys (e.g., Euclid, DESI, LSST) requires accurate predictions of matter distribution on non-linear scales. Existing tools are limited:

• Einstein-Boltzmann solvers (e.g., hi_class, EFTCAMB) are restricted to the linear regime.
• Perturbation theory and halo models offer approximations but lack the precision of full numerical simulations.
• Current N-body codes for MG often lack implementations of the full EFTofDE formalism, specifically the non-linear Vainshtein screening mechanism required to satisfy local gravity constraints, or they are limited to specific sub-classes of models (like Galileons) rather than the general $\alpha$-basis.

There is a critical need for robust, high-resolution N-body simulation codes that can solve the non-linear scalar field equations inherent to EFTofDE to generate precise predictions for clustering and weak lensing.

2. Methodology

The authors developed two complementary N-body simulation codes, PySCo-EFT and ECOSMOG-EFT, designed to solve the modified Poisson equations and the non-linear scalar field equation within the EFTofDE framework.

Theoretical Framework

• Model: The authors focus on a subset of Horndeski theories with a luminal gravitational wave speed ($\alpha_T = 0$). This restricts the non-linear operators to a cubic screening model involving two time-dependent parameters: the braiding parameter ($\alpha_B$) and the running of the Planck mass ($\alpha_M$).
• Equations: The system involves three fields: the gravitational potentials $\Psi$ and $\Phi$, and the scalar field $\chi$. The scalar field equation is a non-linear elliptical partial differential equation (PDE) containing cross-derivative terms (Vainshtein screening).
  • The screening term is proportional to $C_4 \equiv -4\alpha_B + 2\alpha_M$.
  • The equations are solved in the quasi-static approximation on sub-horizon scales.

Numerical Implementation

The non-linearity of the scalar field equation precludes the use of Fourier-based solvers or tree methods. The authors implemented grid-based iterative solvers coupled with multigrid schemes (Full Approximation Storage - FAS) to handle the elliptical nature and accelerate convergence.

• PySCo-EFT:

  • Type: Fast, Python-based Particle-Mesh (PM) code.
  • Solver: Uses Jacobi iterative method with under-relaxation.
  • Approach: Solves directly for the scalar field value $\chi_{i,j,k}$ at each grid point using a quadratic formula derived from the discretized equation.
  • Use Case: Rapid parameter space exploration and preliminary assessments.

• ECOSMOG-EFT:

  • Type: High-resolution code based on RAMSES with Adaptive Mesh Refinement (AMR).
  • Solver: Uses Gauss-Seidel iterations with red-black ordering.
  • Approach: Employs Operator Splitting. Instead of solving for $\chi$, it solves for the Laplacian $[\nabla^2\chi]$ first, then updates $\chi$ via a Newton-Gauss-Seidel step. This method improves convergence in high-density regions.
  • Use Case: Accurate simulations of structure formation in highly dense environments (halos) where AMR is crucial.

• Force Calculation: The standard gravitational force is augmented by a "fifth force" derived from the scalar field gradient: $\vec{F}_{total} = \frac{G_{eff}}{G}\vec{F}_{Newton} + (\alpha_B - \alpha_M)\nabla\chi$.

Validation Strategy

• Static Test: Solved for a static, spherically symmetric mass distribution to verify the solver against analytical solutions.
• Code Comparison: Ran identical cosmological initial conditions (2LPT) in both codes to compare the matter power spectrum boost ($R(k) = P_{EFT}/P_{\Lambda CDM}$).
• Convergence Tests: Systematically varied numerical parameters (mass resolution, box size, refinement thresholds, starting redshift, smoothing cycles) to ensure robustness.

3. Key Contributions

1. First N-body Implementations: This work presents the first N-body simulation tools capable of exploring the EFTofDE parameter space in the $\alpha$-basis, including the full non-linear Vainshtein screening mechanism.
2. Dual-Code Approach: The introduction of a fast Python-based code (PySCo-EFT) and a high-precision AMR code (ECOSMOG-EFT) allows for both rapid screening of parameters and high-fidelity modeling of non-linear structures.
3. Handling Void Pathologies: The authors address a numerical instability where the scalar field discriminant becomes negative in underdense regions (voids). They implement an ad-hoc workaround (setting the discriminant to a near-zero value) that allows simulations to proceed without significantly affecting the matter power spectrum, which is dominated by overdensities.
4. Linearized vs. Non-linear Comparison: Both codes include a "linearized" mode (ignoring screening terms) to quantify the specific impact of non-linear screening on observables.

4. Results

• Solver Accuracy:
  • In static tests, the numerical solutions for the scalar field $\chi$ matched analytical solutions within 0.2%.
  • The gravitational potential $\Psi$ matched analytical solutions within 2.5% down to scales of 4 grid cells.
• Code Agreement:
  • PySCo-EFT and ECOSMOG-EFT showed excellent agreement in the matter power spectrum boost, with relative differences < 1% across all probed scales ($k \lesssim 3 h \text{ Mpc}^{-1}$), despite using different solvers (Jacobi vs. Gauss-Seidel) and different variables ($\chi$ vs. $\nabla^2\chi$).
• Convergence:
  • Numerical effects (finite mass resolution, volume, refinement threshold) were found to be limited to < 2% even at the highest wavenumbers ($k = 10 h \text{ Mpc}^{-1}$).
  • Agreement with linear theory on large scales ($k < 0.05 h \text{ Mpc}^{-1}$) was within 0.5%.
• Physical Impact of Parameters:
  • $\alpha_B$ (Braiding): Increasing the magnitude of $\alpha_B$ increases the coupling between the scalar field and the metric, leading to a higher power spectrum boost on large scales.
  • $\alpha_M$ (Planck Mass Run): A negative $\alpha_M$ strengthens gravity over time, enhancing small-scale structure formation.
  • Screening Effect: In non-linear simulations, the Vainshtein screening mechanism successfully suppresses the fifth force in high-density regions, restoring General Relativity (boost $\to$ 1) at small scales.
  • Linear Approximation Validity: The linearized approximation (ignoring screening) is valid only when the screening coefficient $|C_4|$ is small. For large $|C_4|$, the linearized approximation induces significant errors (up to 35% at $k=2 h \text{ Mpc}^{-1}$), incorrectly predicting a continued rise in the power spectrum boost at small scales.

5. Significance

This work provides essential infrastructure for the next generation of cosmological surveys. By enabling accurate, non-linear predictions for EFTofDE models, these codes allow researchers to:

• Constrain MG Theories: Directly compare simulation outputs with clustering and weak lensing data from Euclid, DESI, and LSST to constrain the $\alpha$-parameters.
• Quantify Screening: Accurately model the transition from modified gravity on large scales to General Relativity on small scales, a critical feature for interpreting galaxy-scale observations.
• Optimize Resources: Utilize the fast PySCo-EFT for broad parameter scans and the high-resolution ECOSMOG-EFT for detailed studies of specific viable models.

The authors conclude that these tools are the first of their kind to combine the EFTofDE formalism, the $\alpha$-basis, and non-linear screening in both Python and AMR frameworks, paving the way for precise tests of gravity on cosmological scales.