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 demonstration, we develop two emulators: PolyCAMB-$D_\ell$, which maps six cosmological parameters to the CMB temperature power spectrum, and PolyCAMB-peak, which enables bidirectional mapping between parameters and acoustic peak features. PolyCAMB-$D_\ell$ achieves an accuracy of $0.03\%$over $\ell \leq 2510$, while PolyCAMB-peak also reaches sub-percent accuracy 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.