Semi-analytical description of clumping factor and CMB free-free distortions from reionization

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.