  The density contrast of the Universe, parametrized in terms of the matter
power spectrum and its variance, can amplify the signal of the free-free
process in the plasma. The damping of fluctuations on scales smaller than the
DM particle free streaming scale corresponds to a suppression of the total
matter power spectrum on large wave numbers k. We derive the time evolution of
the variance of the matter power spectrum for various cosmological models and
parameters by numerically computing the power spectrum with a modified version
of the Boltzmann code CAMB, for different values of the cut-off parameter kmax.
Suitable analytical approximations of the numerical results are presented. We
then characterize the CMB free-free spectral distortion accounting for the
amplification effect coming from clumping factor. Indeed, the clumpiness,
associated to the density contrast of the intergalactic medium, increases at
decreasing redshift. The analysis is carried out for selected astrophysical and
phenomenological cosmological reionization histories, for which we evaluate the
impact of the clumping factor on the free-free distortion and discuss the
wavelength dependence of the predicted signal. Finally, we address a comparison
with other classes of unavoidable CMB spectral distortions and future
observational perspectives. While Comptonization from reionization is dominant
at high frequencies, the free-free signal predicted in the considered models
contributes to the distortion at a level of few (few tens) per cent at
frequencies below about 25 GHz (10 GHz) and represents the main signature below
about 4 GHz. The cosmological signal from the HI 21-cm background is found to
prevail over the free-free distortion in a restricted, model dependent
frequency window between about 0.1 and 0.2 GHz.
