Constraint on the early-formed dark matter halos using the free-free
  emission in the Planck foreground analysis
  We provide a new constraint on the small-scale density fluctuations,
evaluating the diffuse background free-free emission from dark matter halos in
the dark ages. If there exists a large amplitude of the matter density
fluctuations on small scales, the excess enhances the early formation of dark
matter halos. When the virial temperature is sufficiently high, the gas in a
halo is heated up and ionized by thermal collision. The heated ionized gas
emits photons by the free-free process. We would observe the sum of these
photons as the diffuse background free-free emission. Assuming the analytical
dark matter halo model including the gas density and temperature profile, we
calculate the intensity of the diffuse background free-free emission from
early-formed dark matter halos in the microwave frequency range. Comparing with
the recent foreground analysis on cosmic microwave background, we obtain the
constraint on the excess of the density fluctuations on small scales. Our
constraint corresponds to $P_\zeta \lesssim 10^{-7}$ for $k \simeq
1-100~\mathrm{Mpc}^{-1}$ with assuming the delta-function-type curvature power
spectrum. Therefore, our constraint is the most stringent constraint on the
perturbations below $1~\rm Mpc$ scales.
