We present the first fast and detailed computation of the cosmological recombination radiation released during the hydrogen (redshift z ~ 1300) and helium (z ~ 2500 and z ~ 6000) recombination epochs, introducing the code CosmoSpec. Our computations include important radiative transfer effects, 500-shell bound-bound and free-bound emission for all three species, the effects of electron scattering and free-free absorption as well as interspecies (HeII --> HeI --> HI) photon feedback. The latter effect modifies the shape and amplitude of the recombination radiation and CosmoSpec improves significantly over previous treatments of it. Utilizing effective multilevel atom and conductance approaches, one calculation takes only ~ 15 seconds on a standard laptop as opposed to days for previous computations. This is an important step towards detailed forecasts and feasibility studies considering the detection of the cosmological recombination lines and what one may hope to learn from the ~ 6.1 photons emitted per hydrogen atom in the three recombination eras. We briefly illustrate some of the parameter dependencies and discuss remaining uncertainties in particular related to collisional processes and the neutral helium atom model.
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