[Paper Review] Sesame: A power spectrum emulator pipeline for beyond-$Λ$CDM models

Renate Mauland, Hans A. Winther+1 more|arXiv (Cornell University)|Sep 23, 2023

Computational Physics and Python ApplicationsComputer Science3 references3 citations

TL;DR

This paper introduces Sesame, a computational pipeline that enables efficient emulation of the non-linear matter power spectrum boost in beyond-ΛCDM models using fast approximate N-body simulations (COLA) and neural networks. By computing the ratio $ B(k,z) = P_{\text{beyond-}\Lambda\rm CDM}(k,z)/P_{\Lambda\rm CDM}(k,z) $, it achieves ~1–2% accuracy up to $ k \sim 3-5\,h\,\mathrm{Mpc}^{-1} $ with only ~3,000 CPU hours, significantly reducing resource needs compared to traditional emulators.

ABSTRACT

The mysterious nature of the dark sector of the $Λ$CDM model is one of the main motivators behind the study of alternative cosmological models. A central quantity of interest for these models is the matter power spectrum, which quantifies structure formation on various scales and can be cross-validated through theory, simulations, and observations. We present a tool that can be used to create emulators for the non-linear matter power spectrum, and similar global clustering statistics, for models beyond $Λ$CDM with very little computation effort and without the need for supercomputers. We use fast approximate N-body simulations to emulate the boost, $B(k,z)=P_{ ext{beyond}-Λ m CDM}(k,z) / P_{Λ m CDM}(k, z)$, and then rely on existing high-quality emulators made for $Λ$CDM to isolate $P_{ ext{beyond}-Λ m CDM}(k,z)$. Since both the $Λ$CDM and beyond-$Λ$CDM models are simulated in our approach, some of the lack of power on small scales is factored out, allowing us to extend the emulator to $k \sim 3-5\,h\,\mathrm{Mpc}^{-1}$ and still maintain good accuracy. In addition, errors from the simulation and emulation process can be estimated and factored into the covariance when using the emulator on data. As an example of using the pipeline, we create an emulator for the well-studied $f(R)$ model with massive neutrinos, using approximately 3000 CPU hours of computation time. Provided with the paper is a fully functioning pipeline that generates parameter samples, runs a Boltzmann solver to produce initial conditions, runs the simulations, and then gathers all the data and runs it through a machine learning module to develop the emulator. This tool, named Sesame, can be used by anyone to generate a power spectrum emulator for the cosmological model of their choice.

Motivation & Objective

• To develop a scalable, low-resource pipeline for creating emulators of the matter power spectrum in beyond-ΛCDM cosmological models.
• To overcome the high computational cost of full N-body simulations and supercomputer-dependent emulator training.
• To enable accurate emulation of the power spectrum boost $ B(k,z) = P_{\text{beyond-}\Lambda\rm CDM}/P_{\Lambda\rm CDM} $ up to non-linear scales ($ k \sim 3-5\,h\,\mathrm{Mpc}^{-1} $) using approximate simulations.
• To integrate error estimation from both simulation resolution and emulation uncertainty into the final covariance for data analysis.
• To provide a fully reproducible, open-source pipeline (Sesame) for researchers to generate custom emulators for any beyond-ΛCDM model.

Proposed method

• Use the comoving Lagrangian acceleration (COLA) method to perform fast, approximate N-body simulations for both ΛCDM and beyond-ΛCDM models.
• Compute the power spectrum boost $ B(k,z) = P_{\text{beyond-}\Lambda\rm CDM}(k,z)/P_{\Lambda\rm CDM}(k,z) $ to isolate deviations from ΛCDM.
• Leverage existing high-accuracy ΛCDM emulators to reconstruct the full $ P_{\text{beyond-}\Lambda\rm CDM}(k,z) $ from the boost and ΛCDM power spectra.
• Train a neural network on the simulated boost data using the PyTorch Lightning framework to create a differentiable, interpolative emulator.
• Perform convergence tests on simulation setup and use Latin hypercube sampling to efficiently explore cosmological parameter space.
• Estimate total error by combining simulation resolution effects (from high-res N-body comparisons) and emulation uncertainty from training data.

Experimental results

Research questions

• RQ1 Can a low-cost, approximate simulation method like COLA be used to accurately emulate the non-linear matter power spectrum boost in beyond-ΛCDM models?
• RQ2 To what extent can the cancellation of resolution-related artifacts in the boost ratio extend the dynamic range of the emulator to $ k \sim 3-5\,h\,\mathrm{Mpc}^{-1} $?
• RQ3 How accurately can a neural network emulator trained on COLA-simulated boosts reproduce the true power spectrum for models like $ f(R) $ with massive neutrinos?
• RQ4 Can the error budget of the emulator be reliably estimated and propagated into cosmological data analysis?
• RQ5 Is it feasible to build a fully automated, reusable pipeline for custom beyond-ΛCDM model emulation with minimal supercomputing requirements?

Key findings

• The Sesame pipeline achieves ~1–2% accuracy in the non-linear boost for $ f(R) $ gravity with massive neutrinos at redshift $ z = 0 $, extending to $ k \sim 3-5\,h\,\mathrm{Mpc}^{-1} $.
• The fully linear boost emulator for $ f(R) $ gravity has below-percent accuracy compared to simulations, demonstrating high fidelity in the weakly non-linear regime.
• The method reduces required computation time to approximately 3,000 CPU hours for a full emulator, compared to millions of CPU hours for traditional ΛCDM emulator pipelines.
• By computing the ratio $ B(k,z) $, the pipeline mitigates resolution-related errors from approximate simulations, enabling reliable emulation at higher $ k $-values.
• The pipeline provides a complete, open-source workflow including parameter sampling, CLASS-based initial conditions, COLA simulations, data gathering, and neural network training with error estimation.
• The error from both COLA simulations and the emulator is quantified and can be folded into the covariance matrix during data fitting, ensuring robust statistical inference.