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Emulating the nonlinear effects of modified gravity on the matter power spectrum for reconstruction

This paper presents a neural-network emulator trained on nearly one million samples to efficiently approximate the nonlinear matter power spectrum corrections in modified gravity models, achieving high accuracy and enabling robust, cost-effective model-independent reconstruction and forecasting for future cosmological surveys.

Original authors: Lanyang Yi, Kazuya Koyama, Qi Xiong, Gong-Bo Zhao

Published 2026-08-03

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CC BY 4.0

Original authors: Lanyang Yi, Kazuya Koyama, Qi Xiong, Gong-Bo Zhao

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 decades, scientists have been trying to figure out what's making this ocean expand faster and faster. We know there's a mysterious force pushing it apart, often called "dark energy," but we also wonder if our understanding of gravity itself might be slightly off. Gravity is the invisible glue that holds stars and galaxies together, but on the largest scales, it might behave differently than Isaac Newton or Albert Einstein predicted. To test these ideas, astronomers look at the "fingerprint" of the universe: the way matter is clumped together in a cosmic web. This pattern, called the "matter power spectrum," changes over time. In the early, calm universe, these changes were simple and predictable. But as the universe got older, gravity started pulling matter into messy, chaotic clumps, creating a "nonlinear" mess that is incredibly hard to calculate. If we want to know if gravity is truly modified, we need to understand this messy part perfectly, but doing the math for every possible theory takes so long that it would make a supercomputer fall asleep.

This paper is about building a "reference" for that messy math. The authors, Lanyang Yi and colleagues, created a smart computer program (a neural network) that acts like a super-fast emulator. Instead of solving the difficult, time-consuming equations for every single theory of modified gravity, this emulator learns the pattern of how the "messy" part of the universe changes compared to the "clean" part. They trained this AI on about 900,000 different cosmic scenarios. The results are impressive: for most situations, the emulator predicts the cosmic fingerprint with an accuracy of within 1.5%. Even in the most extreme cases where gravity behaves very strangely, it stays accurate up to a certain scale. The team tested this tool with fake data that mimics what future telescopes like DESI and CSST will see, and it successfully recovered the correct answers without getting confused. Essentially, they've built a high-speed shortcut that allows scientists to explore thousands of weird gravity theories quickly, helping us figure out if the universe is expanding because of a mysterious energy or because our laws of gravity need a little tune-up.

Technical Summary: Emulating the Nonlinear Effects of Modified Gravity on the Matter Power Spectrum for Reconstruction

Problem Statement
Model-independent reconstruction of Modified Gravity (MG) functions, specifically the phenomenological parameters $\mu(a)$ and $\Sigma(a)$ , offers a powerful way to test General Relativity (GR) without assuming a specific MG theory. However, incorporating nonlinear information from small scales is essential to tighten constraints on these models, as MG theories can significantly alter structure formation even when linear predictions are degenerate with $\Lambda$ CDM. The primary bottleneck in this process is computational cost: calculating the nonlinear matter power spectrum ( $P_{NL}$ ) for MG models using the standard pipeline (MGCAMB combined with the ReACT halo model reaction) is prohibitively expensive for repeated likelihood evaluations required in high-dimensional parameter space exploration (e.g., MCMC). This expense limits the ability to include nonlinear data in model-independent reconstruction pipelines.

Methodology
To address this computational bottleneck, the authors construct a neural-network emulator designed to predict the nonlinear correction factor, defined as the ratio $R_{MG}(k, z) = P_{NL}^{MG}(k, z) / P_{L}^{MG}(k, z)$ , rather than the full power spectrum. This approach isolates the model-dependent nonlinear modifications, reducing the dynamic range the network must learn while preserving the accurate linear predictions from the Boltzmann code.

Training Data: The emulator is trained on approximately $9 \times 10^5$ samples generated using MGCAMB and ReACT. The dataset combines samples from previous reconstruction chains (to ensure physical motivation in the MG parameter space) and random samples generated within $3\sigma$ posterior ranges of standard cosmological parameters.
Input Parameters: The input vector includes standard cosmological parameters ( $\log A, n_s, h, \Omega_b h^2, \Omega_c h^2$ ), the reconstructed MG function nodes ( $\mu_{1-11}, \Omega_{X,1-10}$ ), redshift ( $z$ ), and the nonlinear screening parameter ( $p_1$ ). The $\Sigma$ nodes are excluded as they do not directly affect the matter power spectrum.
Architecture: The emulator utilizes the CosmoPower framework with a fully connected neural network featuring four hidden layers of 512 nodes each. It predicts the ratio $R_{MG}$ across 420 $k$ -modes in the range $10^{-5} \le k < 10 \text{ Mpc}^{-1}$ .
Validation Strategy: The emulator undergoes a three-tiered validation process:
1. Representative Spectra: Comparison against MGCAMB+ReACT for $\Lambda$ CDM and specific MG models where linear spectra are nearly degenerate with $\Lambda$ CDM.
2. Independent Validation Set: Statistical analysis of 2000 unseen points across the parameter space.
3. Synthetic Data MCMC: End-to-end testing using synthetic BAO+RSD (DESI-like) and Weak Lensing (CSST-like) data generated from a known $\Lambda$ CDM fiducial model to verify that the emulator does not introduce biases in parameter recovery.

Key Results

Accuracy: For $\Lambda$ CDM and moderate MG nonlinear corrections ( $p_1 = 0, 1$ ), the emulator reproduces the reference nonlinear power spectra to within 1.5% across the full scale range. In extreme nonlinear cases ( $p_1 = 2$ ), this 1.5% accuracy is maintained for $k < 0.8 \text{ Mpc}^{-1}$ , with deviations increasing at smaller scales due to stronger nonlinear effects.
Statistical Performance: Over the independent validation set, the mean residual is close to zero. The $2\sigma$ scatter remains below 1% for $k < 0.5 \text{ Mpc}^{-1}$ and increases to approximately 2% on smaller scales.
MCMC Recovery: In synthetic data analyses, the emulator successfully recovers the input $\Lambda$ CDM cosmology and the GR limits ( $\mu=\Sigma=1$ ) of the reconstructed MG functions within posterior uncertainties. Deviations observed in the reconstructed functions at low redshift are attributed to projection effects and degeneracies with galaxy bias parameters rather than emulator errors.
Computational Efficiency: The emulator reduces the time required to compute nonlinear power spectra from approximately 80 seconds (MGCAMB+ReACT) to roughly 20 seconds, with the majority of the time still consumed by the linear spectrum calculation.

Applications Demonstrated
The paper illustrates three specific applications of the emulator:

Degeneracy Breaking: The emulator can distinguish between MG models that have nearly degenerate linear power spectra but different nonlinear corrections, serving as a fast diagnostic tool.
Current Survey Likelihoods: In a test using DES-Y3 3 $\times$ 2pt likelihoods, the emulator's systematic errors (even in a worst-case scenario) were found to be subdominant to current measurement uncertainties (maximum deviation $\sim 0.1\sigma$ ), validating its use for current photometric surveys.
Future Forecasts: Principal Component Analysis (PCA) of DESI+CSST forecasts indicates that the emulator enables the recovery of MG functions with significantly improved precision. The DESI-V+CSST-F combination is projected to reduce errors by approximately 46% for $\Sigma$ modes and 19% for $\mu$ modes compared to earlier survey configurations.

Significance and Limitations
The authors claim that this work provides a reliable, fast, and accurate tool for model-independent MG reconstruction pipelines, enabling the inclusion of nonlinear information from Stage-IV surveys (like DESI and CSST) without the computational prohibitions of full halo-model calculations. The emulator allows for the exploration of high-dimensional parameter spaces that were previously inaccessible.

The paper notes several limitations:

The emulator is valid only within the training $k$ -range; extrapolation is not controlled.
The reconstruction assumes scale independence, fixing all ReACT nonlinear parameters except $p_1$ .
Accuracy degrades in extreme nonlinear regimes (high $p_1$ , high $k$ ).
The ReACT calibration is limited to $z \le 2.5$ , requiring approximations for higher-redshift bins in surveys like CSST.
Current validation relies on synthetic data; application to real observed data is proposed for future work.

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