A general polynomial emulator for cosmology via moment projection

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.