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Thermal and Resonant Emission of Dark Ages Halos in the Rotational Lines of HeH$^{+}$

We analyse the thermal emission and resonant scattering of CMB radiation from the dark ages halos in the rotational lines of the helium hydride ion (HeH$^{+}$), one of the first chemical compounds in the Universe. Evaluating the optical depth, thermal and resonant brightness temperatures and spectral fluxes of dark ages halos is based on computing the cross-sections and rate coefficients of excitation/de-excitation of the lowest five rotational states of HeH$^{+}$ by inelastic collisions with atomic hydrogen. It was shown that in the Dark Ages the collisional excitation/de-excitation by atoms of neutral hydrogen and electrons are competitive, whereas in the denser regions, e.g., in the halos, the contribution of collisions with atomic hydrogen is dominant. We demonstrate the peak-like time-dependence of halo luminosities in HeH$^{+}$ lines.