GokuNEmu: A Neural Network Emulator Based on the Goku Simulation Suite

GokuNEmu is a neural network (NN) emulator for the nonlinear matter power spectrum, trained on simulations from the Goku suite using the T2N-MusE emulation technique.

The nonlinear matter power spectrum describes how matter clusters across cosmic scales and is a key link between cosmological models and observations of large-scale structure and gravitational lensing. GokuNEmu provides fast predictions across a broad cosmological parameter space—including models with dynamical dark energy and massive neutrinos—so users can explore and constrain cosmology without running a new, computationally expensive $N$-body simulation for every parameter choice. GokuNEmu is the fastest matter power spectrum emulator, taking approximately 1 millisecond per single-cosmology evaluation; see the speed benchmark.

Post-publication update: The current package no longer uses the models trained on the full combined data set. It now builds both GokuNEmu-N and GokuNEmu-W internally from the corresponding N_L* and W_L* models. Inputs within the Goku-N parameter box are evaluated with GokuNEmu-N; all other inputs are evaluated with GokuNEmu-W. The L1-L2 transition uses sigmoid sharpness 6 for GokuNEmu-N and 4 for GokuNEmu-W. The new models were trained using updated, more robust T2N-MusE methods introduced before commit 49acdc1. The training data were also updated: the power spectra were regenerated with pmesh on higher-resolution grids, improving their accuracy at the smallest scales (highest $k$) by reducing aliasing effects. In addition, inference was accelerated to approximately 1 millisecond per single-cosmology evaluation. This routing is automatic and does not change the user interface described below.

Updated leave-one-out validation

Leave-one-out cross-validation fractional mean absolute errors for the updated GokuNEmu-W and GokuNEmu-N models, shown across scale and redshift in the same format as the paper. The gray band marks the 1% error region.

Cosmological parameter space

GokuNEmu takes ten cosmological parameters. The bounds of the broad Goku-W parameter space are:

• Total matter density, $\Omega_m$ (Om): $[0.22, 0.40]$
• Baryon density, $\Omega_b$ (Ob): $[0.040, 0.055]$
• Reduced Hubble parameter, $h$ (hubble): $[0.60, 0.76]$
• Primordial scalar amplitude, $A_s$ (As): $[1.0 \times 10^{-9}, 3.0 \times 10^{-9}]$
• Scalar spectral index, $n_s$ (ns): $[0.80, 1.10]$
• Dark-energy equation-of-state parameter, $w_0$ (w0): $[-1.30, 0.25]$
• Dark-energy evolution parameter, $w_a$ (wa): $[-3.0, 0.5]$
• Sum of neutrino masses, $\sum m_\nu$ (mnu): $[0.0, 0.6],\mathrm{eV}$
• Effective number of relativistic species, $N_\mathrm{eff}$ (Neff): $[2.2, 4.5]$
• Running of the scalar spectral index, $\alpha_s$ (alphas): $[-0.05, 0.05]$

Installation

We recommend installing GokuNEmu via pip:

pip install gokunemu

Note for Intel Mac users:
PyTorch 2.2 was the final release with macOS x86_64 binaries. Use Python 3.10 or 3.11 and install an Intel-compatible PyTorch stack before GokuNEmu:

python -m pip install "numpy<2" "torch==2.2.2"
python -m pip install gokunemu

Usage

Example notebooks are provided in the examples/ directory:

• example.ipynb: Demonstrates how to use GokuNEmu for predicting the nonlinear matter power spectrum.
• speed_benchmark.ipynb: Benchmarks the runtime performance.

Training data

The data used as the training set for the emulator are available at https://github.com/astro-YYH/T2N-MusE.

Citation

If you use GokuNEmu, please cite:

• The main GokuNEmu paper: https://arxiv.org/abs/2507.07177
• The Goku simulations (optional): https://arxiv.org/abs/2501.06296
• The T2N-MusE emulation method (optional): https://arxiv.org/abs/2507.07184

License

This project is licensed under the MIT License.