Cosmological free-free emission from dark matter halos in the $\Lambda$CDM model

We study the diffuse background free-free emission induced by dark matter halos. Since dark matter halos host ionized thermal plasma, they could be an essential source of cosmological free-free emission. We evaluate the global background intensity and the anisotropy of this free-free emission. We show that the dominant contribution comes from dark matter halos with a mass close to the Jeans mass, $M_{\mathrm{halo}}\sim 10^{10} M_\odot$, around the redshift $z \sim 3$. Therefore, the intensity of the free-free emission is sensitive to the small-scale primordial curvature perturbations that form such small-mass dark matter halos. Although our obtained intensity of the global and anisotropic free-free emission is smaller than the $10\%$ level of the free-free emission observed in the high galactic region, we find that the free-free emission signal is modified by $\sim 20 \%$ even in the parameter set of the spectral index and the running, which is consistent with the recent Planck result. Therefore, the measurement of the cosmological free-free signals has the potential to provide more stringent constraints on the abundance of small-mass dark matter halos and the curvature perturbations including the spectral index and the running, while carefully removing the Galactic free-free emission is required through the multifrequency radio observation or the cross-correlation study with the galaxy surveys or 21-cm intensity map.