CRRfast: An emulator for the Cosmological Recombination Radiation with effects from inhomogeneous recombination

The Cosmological Recombination Radiation (CRR) is one of the guaranteed $\Lambda$CDM Spectral Distortion (SD) signals. Even if very small in amplitude, it provides a direct probe of the three recombination eras, opening the path for testing one of the key pillars in our cosmological interpretation of the measured CMB anisotropies. Here we develop a new emulator, CRRfast, to quickly and accurately represent the CRR for a wide range of cosmologies, using the state-of-the-art CosmoSpec code as a reference. CRRfast has been made publicly available both as stand-alone code and as part of CLASS, thereby completing the set of $\Lambda$CDM sources of SDs that can be modeled with CLASS. With this newly-developed pipeline we investigate the full constraining power of SDs within $\Lambda$CDM and highlight possible future applications to experimental design optimization. Furthermore, we show that the inhomogeneous evolution of the recombination process imprints second-order contributions to the CRR spectrum, leading to a broadening and shifting of the CRR features. These second-order terms are naturally captured by the emulator and allow us to evaluate the $\Lambda$CDM contributions to the average CRR as well as to illustrate the effect of perturbed recombination due to Primordial Magnetic Fields (PMFs). As it turns out, while the $\Lambda$CDM variance effects can be neglected, they could be significantly enhanced in the beyond-$\Lambda$CDM models. In particular in the case of PMFs we demonstrate that through these non-linear terms the parameter space relevant to the Hubble tension could be tested with future CMB spectrometers.