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An efficient one-loop EFTofLSS framework for Vainshtein-screened Horndeski gravity

This paper introduces an efficient extension of the \texttt{PyBird} framework that employs a direct ODE method to compute one-loop EFTofLSS power spectra for Vainshtein-screened Horndeski gravity, enabling robust constraints on modified gravity models using current cosmological data while demonstrating the importance of exact time-dependent evolution.

Original authors: Stan Verhoeve, Dani de Boe, Gen Ye, Alessandra Silvestri

Published 2026-07-30

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CC BY 4.0

Original authors: Stan Verhoeve, Dani de Boe, Gen Ye, Alessandra Silvestri

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible ocean. For most of its history, this ocean was calm and smooth, but over the last few billion years, gravity has started churning it up, creating massive waves and whirlpools of matter that form galaxies, clusters, and the vast cosmic web we see today. Scientists have long had a set of rules, called "Standard Perturbation Theory," to predict how these waves should behave. It's like having a weather forecast for the cosmos: it works perfectly when the sky is clear, but when the storm gets too wild (when gravity gets too strong and matter clumps too tightly), the old rules start to break down, giving inaccurate predictions.

To fix this, physicists developed a new toolkit called the "Effective Field Theory of Large-Scale Structure" (EFTofLSS). Think of this as a high-tech weather app that doesn't just look at the big clouds but also accounts for the tiny, chaotic raindrops and wind gusts that the old rules missed. This toolkit allows scientists to make precise predictions even in the "stormy" zones of the universe where matter is clumping up. However, there's a catch: some theories suggest that gravity itself might be slightly different from what Einstein predicted, perhaps behaving like a hidden force that only shows up in specific ways. To test these ideas, scientists need to run their new weather app on these "modified gravity" theories, but the math gets incredibly heavy and slow, making it hard to check if the theories match reality.

This paper introduces a clever new way to run that math, making it much faster and more efficient without losing any accuracy. The authors, Stan Verhoeve, Dani de Boe, Gen Ye, and Alessandra Silvestri, have upgraded a piece of software called PyBird. Imagine PyBird as a sophisticated simulator that tries to predict how galaxies should be arranged in the universe based on different rules of gravity. The team replaced the old, clunky method of calculating how these rules change over time with a streamlined, direct approach. Instead of taking a long, winding path through a maze of intermediate steps (which they call the "Green's function" method), their new method takes a straight shot to the answer using a direct equation solver.

The result is a significant speed boost. While the old method had to re-calculate its path every time the scientists wanted to check a different point in time (like checking the weather at 1 PM, then 2 PM, then 3 PM separately), the new method solves the path once and reuses it instantly for all those times. This makes it possible to test complex theories of gravity that were previously too computationally expensive to analyze in detail. They tested this new "fast lane" against massive computer simulations (called N-body simulations) that act as a gold standard for how the universe actually behaves. They found that their new method matches the gold standard almost perfectly, with differences so tiny they are practically invisible.

Using this upgraded toolkit, the team applied their framework to specific theories of "modified gravity" known as Horndeski gravity. These theories suggest that gravity might have a secret "screening" mechanism (called Vainshtein screening) that hides its weird effects on small scales, like inside our solar system, but lets them show up on the huge scales of the universe. They tested two main ways these theories could behave over time: one where the effects grow with the cube of the universe's size, and another where they grow with the amount of dark energy present. By combining their fast calculations with real data from the Planck satellite (which maps the early universe), the BOSS survey (which maps galaxy positions), and the DESI survey (which measures the expansion of the universe), they were able to put strict limits on these theories.

Their findings suggest that while these modified gravity theories are still possible, the data from the full shape of the galaxy distribution (the "full-shape" data) makes the allowed range of these theories much tighter than just looking at the early universe data alone. In fact, the data leans slightly toward the standard theory of gravity (General Relativity), though it doesn't completely rule out the modified versions yet. The authors also checked if using a simplified approximation (called the Einstein-de Sitter approximation) for how gravity evolves over time would change their results. They found that for the most extreme modified gravity scenarios, this simplification could lead to errors as large as four times the expected uncertainty, which is significant. Because their new method is so fast, they decided to skip the simplification and use the exact, complex math instead, ensuring their results are as precise as possible.

In short, this paper doesn't just propose a new theory of gravity; it builds a better, faster engine to test existing theories. By making the calculations efficient, they've opened the door for future, even more precise surveys to either confirm that gravity is exactly as Einstein described or finally catch it doing something unexpected. They showed that with the right tools, we can listen to the universe's whispers in the "stormy" zones of galaxy clusters and understand the true nature of the force that holds it all together.

Technical Summary: An efficient one-loop EFTofLSS framework for Vainshtein-screened Horndeski gravity

Problem Statement
Current and upcoming Stage IV galaxy surveys (e.g., Euclid, DESI) are extending observational reach into the quasi-linear and non-linear regimes of the Universe's large-scale structure (LSS). Fully exploiting the constraining power of these surveys requires theoretical predictions beyond linear perturbation theory, specifically on scales $k \lesssim 0.3 \, h \, \text{Mpc}^{-1}$ . While the Effective Field Theory of Large-Scale Structure (EFTofLSS) provides a robust framework for modeling these scales in $\Lambda$ CDM, extending it to modified gravity (MG) theories presents challenges. Specifically, Horndeski gravity—a general class of scalar-tensor theories encompassing most dynamical dark energy models—requires accurate treatment of non-linear dynamics, Vainshtein screening, and time-dependent background evolution. Existing implementations often rely on the Einstein–de Sitter (EdS) approximation for perturbation kernels or computationally expensive Green's function methods that struggle with the scale-independent growth and complex time evolution characteristic of MG models.

Methodology
The authors present an extension of the public code PyBird, designed to compute one-loop EFTofLSS predictions for biased tracers in redshift space. The methodology involves three primary technical advancements:

Framework Extension: The pipeline is extended to support quasi-static, Vainshtein-screened, luminal Horndeski models (where the speed of gravitational waves $c_T^2 = 1$ ) in both the covariant formalism and the Effective Field Theory of Dark Energy (EFTofDE) language. It also supports the nDGP model. The implementation utilizes the $\alpha$ -basis ( $\alpha_B, \alpha_M, \alpha_K$ ) and interfaces with H–EFTCAMB to supply background evolution and linear perturbation quantities.
Direct ODE Approach: The authors replace the standard Green's function approach for computing time-dependent functions in perturbation kernels with a direct Ordinary Differential Equation (ODE) method. Instead of convolving Green's functions with source terms, the code solves scalar ODEs iteratively to obtain the time-dependent coefficients ( $G, U, V$ ) directly. This approach avoids the need for explicit decomposition into growing and decaying modes, which is difficult in modified gravity scenarios where the growing mode definition is non-trivial.
Exact Time Evolution: Unlike standard EFTofLSS analyses that often employ the EdS approximation for kernels, this framework computes kernels with exact time dependence. The authors validate that this can be achieved at a modest computational cost, removing the need to trade accuracy for efficiency.

The pipeline computes the one-loop galaxy power spectrum multipoles (monopole and quadrupole) including counterterms and stochastic contributions. The authors validate the numerical consistency of their ODE method against the standard Green's function approach for $\Lambda$ CDM, $w$ CDM, and Quintessence, and against independent implementations for nDGP and Cubic Galileon models. Furthermore, the one-loop matter power spectrum is validated against N-body simulations (PySCo-EFT and ECOSMOG-EFT) for the $\alpha_i \propto \Omega_{DE}$ parametrization.

Key Contributions

Computational Efficiency: The direct ODE method significantly improves computational efficiency, particularly when evaluating multiple redshifts. While the Green's function method requires recomputing convolution integrals for each redshift, the ODE method solves the growth equation once, reusing the solution. This results in a speedup of roughly 20–25% for full one-loop spectrum calculations and scales linearly with the number of target redshifts.
Generalized MG Support: The code provides a modular interface for any scale-independent, Vainshtein-screened luminal Horndeski model, bridging the gap between phenomenological parametrizations and covariant theories.
Validation: The framework is rigorously tested against N-body simulations, showing agreement at the sub-percent level for the boost factor $R(k) = P_{\text{EFT}}/P_{\Lambda\text{CDM}}$ in the range $0.1 \lesssim k \lesssim 0.3 \, h \, \text{Mpc}^{-1}$ .
Exact vs. Approximate Kernels: The authors demonstrate that the EdS approximation, while valid for $\Lambda$ CDM, can deviate from exact time evolution by several standard deviations ( $\sim 4\sigma$ ) in extreme modified gravity scenarios at small scales. They show that retaining exact time dependence is computationally feasible and necessary for precision cosmology in MG contexts.

Results
The authors apply the framework to constrain two phenomenological parametrizations of the EFTofDE amplitudes: $\alpha_i \propto a^3$ and $\alpha_i \propto \Omega_{DE}$ , using Planck CMB, BOSS full-shape (FS), and DESI DR2 BAO data.

Constraints: The inclusion of BOSS full-shape data significantly tightens constraints on the modified gravity parameters ( $c_B, c_M$ ) compared to CMB-only analyses. For the $\alpha_i \propto \Omega_{DE}$ case, the combined data yields $c_B = -0.41^{+0.25}_{-0.19}$ and $c_M = 0.32^{+0.61}_{-0.70}$ (68% CL), consistent with General Relativity within the 95% credible interval.
Projection Effects: The analysis finds that projection effects (biases in MAP estimates due to nuisance parameter degeneracies) are not a significant concern when strong CMB priors on $A_s$ and $n_s$ are included.
Model Showcase: The pipeline successfully generates one-loop power spectrum predictions for covariant models like the Cubic Galileon and nDGP, demonstrating its applicability beyond simple amplitude parametrizations.

Significance
The paper establishes a systematic, flexible, and computationally efficient framework for extending EFTofLSS analyses beyond $\Lambda$ CDM. By combining exact time-dependent perturbation theory with a practical implementation in PyBird, the work enables robust one-loop predictions for a broad class of modified gravity theories. The authors argue that the ability to retain exact time dependence at a modest computational cost is crucial for future high-precision surveys (like DESI and Euclid), where the impact of approximations like EdS may become model-dependent and significant. This framework paves the way for precision tests of gravity on cosmological scales using current and upcoming LSS observations.

Technical Summary: An efficient one-loop EFTofLSS framework for Vainshtein-screened Horndeski gravity

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