[Paper Review] High-accuracy emulators for observables in $Λ$CDM, $N_\mathrm{eff}$, $Σm_ν$, and $w$ cosmologies

Boris Bolliet, A. Spurio Mancini+5 more|arXiv (Cornell University)|Mar 2, 2023

Computational Physics and Python Applications4 citations

TL;DR

This paper presents high-accuracy neural network emulators for cosmological observables—CMB power spectra, matter power spectra, distance-redshift relations, and large-scale structure probes—using the CosmoPower framework. Trained on high-precision Einstein-Boltzmann calculations across ΛCDM, $w$CDM, $N_{\mathrm{eff}}$, and $\Sigma m_\nu$ models, the emulators achieve sub-0.5% accuracy up to $\ell=10^4$ (CMB) and $k=50\,\mathrm{Mpc}^{-1}$ (matter power spectrum), enabling Stage-IV CMB and Stage-III LSS data analysis on a laptop via MCMC.

ABSTRACT

We use the emulation framework CosmoPower to construct and publicly release neural network emulators of cosmological observables, including the Cosmic Microwave Background (CMB) temperature and polarization power spectra, matter power spectrum, distance-redshift relation, baryon acoustic oscillation (BAO) and redshift-space distortion (RSD) observables, and derived parameters. We train our emulators on Einstein-Boltzmann calculations obtained with high-precision numerical convergence settings, for a wide range of cosmological models including $Λ$CDM, $w$CDM, $Λ$CDM+$N_\mathrm{eff}$, and $Λ$CDM+$Σm_ν$. Our CMB emulators are accurate to better than 0.5% out to $\ell=10^4$ which is sufficient for Stage-IV data analysis, and our $P(k)$ emulators reach the same accuracy level out to $k=50 \,\, \mathrm{Mpc}^{-1}$, which is sufficient for Stage-III data analysis. We release the emulators via an online repository CosmoPower Organisation, which will be continually updated with additional extended cosmological models. Our emulators accelerate cosmological data analysis by orders of magnitude, enabling cosmological parameter extraction analyses, using current survey data, to be performed on a laptop. We validate our emulators by comparing them to CLASS and CAMB and by reproducing cosmological parameter constraints derived from Planck TT, TE, EE, and CMB lensing data, as well as from the Atacama Cosmology Telescope Data Release 4 CMB data, Dark Energy Survey Year-1 galaxy lensing and clustering data, and Baryon Oscillation Spectroscopic Survey Data Release 12 BAO and RSD data.

Motivation & Objective

• To accelerate cosmological parameter inference by replacing slow Boltzmann code calls with fast neural network emulators.
• To achieve high numerical accuracy (better than 0.5%) in emulating key cosmological observables across diverse models, including $\Lambda$CDM, $w$CDM, $\Lambda$CDM+$N_{\mathrm{eff}}$, and $\Lambda$CDM+$\Sigma m_\nu$.
• To enable high-precision MCMC analyses of current and upcoming CMB and large-scale structure data on standard hardware, such as laptops.
• To release a publicly accessible, continuously updated repository of emulators for the cosmology community.
• To validate the emulators against established codes (class, camb) and real data likelihoods from Planck, ACT DR4, DES Y1, and BOSS DR12.

Proposed method

• The emulators are built using the CosmoPower framework, a TensorFlow-based neural network pipeline for cosmological observables.
• Training data is generated from high-precision Einstein-Boltzmann solver outputs (camb and class) with stringent numerical convergence settings.
• The emulators are trained on a wide range of cosmological parameters, including $\Omega_{\mathrm{b}}$, $\Omega_{\mathrm{c}}$, $n_{\mathrm{s}}$, $A_{\mathrm{s}}$, $H_0$, $N_{\mathrm{eff}}$, $\Sigma m_\nu$, and $w$, across multiple cosmological models.
• The framework supports emulations of CMB temperature (TT), polarization (TE, EE), lensing (PP), matter power spectra (PKL, PKNL), distance-redshift relations (DA, H), $\sigma_8(z)$, and derived parameters (DER).
• Validation is performed by comparing emulator outputs to class and camb results, and by reproducing posterior constraints from real data likelihoods (Planck, ACT DR4, DES Y1, BOSS DR12).
• The emulators are released via an online repository (CosmoPower Organisation) and are designed for integration into standard MCMC pipelines like cobaya and cosmosis.

Experimental results

Research questions

• RQ1 Can neural network emulators achieve sub-0.5% accuracy in CMB and matter power spectrum predictions up to $\ell=10^4$ and $k=50\,\mathrm{Mpc}^{-1}$?
• RQ2 Can these emulators reproduce cosmological parameter constraints from real data (e.g., Planck, ACT DR4, DES Y1, BOSS DR12) with high fidelity?
• RQ3 To what extent can emulator-based MCMC inference replace traditional Boltzmann code calls in terms of speed and accuracy for Stage-III and Stage-IV cosmological surveys?
• RQ4 How robust are the emulators across diverse cosmological models, including extensions like $w$CDM and $\Sigma m_\nu$?
• RQ5 Can the emulators be effectively integrated into standard cosmological analysis frameworks like cobaya and cosmosis?

Key findings

• The CMB emulators achieve sub-0.5% accuracy up to $\ell=10^4$, sufficient for Stage-IV CMB data analysis.
• The matter power spectrum emulators maintain sub-0.5% accuracy up to $k=50\,\mathrm{Mpc}^{-1}$, meeting the requirements for Stage-III large-scale structure surveys.
• The emulators successfully reproduce marginalized posterior distributions from Planck TT, TE, EE, and CMB lensing likelihoods, as well as from ACT DR4, DES Y1, and BOSS DR12 data.
• MCMC parameter inference using the emulators can be performed on a standard laptop, accelerating cosmological analysis by orders of magnitude.
• The emulators are publicly released via the CosmoPower Organisation repository and are designed for ongoing updates with new cosmological models.
• The framework is extensible and will support future emulators for B-mode power spectra and other advanced observables.