A general polynomial emulator for cosmology via moment projection

TL;DR

MomentEmu introduces a moment-projection based polynomial emulator to rapidly map cosmological theory parameters $\boldsymbol{\theta}$ to observables $\boldsymbol{y}$ with symbolic, interpretable expressions. It constructs moment matrices from training data to fit multivariate polynomials, enabling both forward predictions and inverse parameter inference at low cost and with explicit error control. Applied to CMB physics, PolyCAMB-$D_\ell$ and PolyCAMB-peak achieve sub-percent accuracy over large ranges (up to $\ell\le 4050$) and enable millisecond-scale evaluations, delivering Planck-likelihood-based posteriors with substantial speedups over CAMB. The approach offers a lightweight, portable surrogate with transparent functional forms, diagnostic power, and potential extension to broader forward-modeling tasks in cosmology.

Abstract

We present MomentEmu, a general-purpose polynomial emulator for fast and interpretable mappings between theoretical parameters and observational features. The method constructs moment matrices to project simulation data onto polynomial bases, yielding symbolic expressions that approximate the target mapping. Compared to neural-network-based emulators, MomentEmu offers negligible training cost, millisecond-level evaluation, and transparent functional forms. As a proof-of-concept demonstration, we develop two emulators: PolyCAMB-$D_\ell$, which maps six cosmological parameters to the CMB power spectra (TT, EE, BB, TE), and PolyCAMB-peak, which enables a bidirectional mapping between the cosmological parameters and the acoustic peak features of $D_\ell^{\rm TT}$. PolyCAMB-$D_\ell$ achieves sub-percent accuracy over multipoles $\ell \leq 4050$, while PolyCAMB-peak also attains comparable precision and produces symbolic forms consistent with known analytical approximations. The method is well suited for forward modelling, parameter inference, and uncertainty propagation, particularly when the parameter space is moderate in dimensionality and the mapping is smooth. MomentEmu offers a lightweight and portable alternative to regression-based or black-box emulators in cosmological analysis.

Figures (6)

• Figure 1: Diagrams illustrating how MomentEmu operates.
• Figure 2: Emulator training performance (PolyCAMB-$D_\ell^{\textsc{TT}}$) vs polynomial degree.
• Figure 3: Validation of MomentEmu with CMB observables. (a) Comparison of $D_\ell^{\textsc{TT}}$ (top): the CAMB spectrum (dashed line), and the PolyCAMB-$D_\ell$ emulation (thick orange). The five star markers indicate the first five acoustic peaks as predicted by PolyCAMB-peak. The broad feature is an ensemble of emulator outputs (thin blue lines) generated from Gaussian perturbations of the input parameters, which illustrates a typical use case of fast forward modelling for Bayesian inference. (b) Fractional residuals (bottom): fractional differences between PolyCAMB-$D_\ell$ and CAMB, with errors remaining below 0.02% across the full multipole range. The gray dot–dashed lines indicate the Planck 68% confidence interval (upper and lower error bars).
• Figure 4: Corner plot showing the 68% and 95% joint posterior contours for the six $\Lambda$CDM parameters, derived from the Planck "TT/TE/EE+lowE+lowT" likelihoods using the raw CAMB and the PolyCAMB-$D_\ell$ emulator separately, under identical MCMC sampling settings. One-dimensional marginalised posterior distributions are displayed along the diagonal panels, while the off-diagonal panels show the corresponding two-dimensional joint constraints. We employed adaptive MCMC sampling with convergence determined by both sample count ($\geq200,000$ samples) and the Gelman-Rubin diagnostic ($R-1 < 0.02$), with additional confidence level monitoring ($R-1 < 0.2$ at 95% confidence level) to ensure chain mixing and statistical reliability. Both corner plots are based on the last 200,000 samples to ensure comparable statistical robustness. The two contour sets exhibit good agreement, with minor discrepancies attributable to emulation errors, numerical differences, and sampling noise. All the best-fit parameters differs $\leq 0.01\sigma$ between using polyCAMB and CAMB.
• Figure 5: Validation of bidirectional emulation using MomentEmu.