tokens: a complete exploration of the parameter space, we first show a comparison between the model cumulative halo distributions and the observed number density of galaxies in the illustrative case of a sterile neutrino with (the candidate origin of the potential 3.5 keV line) in the limit of small Higgs portal coupling . In such a freeze-in limit (see Section 3 ), the scalar mass m S and the Yukawa coupling y will be related for any given value of , thus reducing the parameter space to be explored. In Figure 3 (left panel) we show the cumulative halo distributions in such a regime for a scalar mass of and for different values of the Yukawa coupling y , and we compare them to the observed number density of high-redshift HFF galaxies. The strong dependence of the maximum predicted number density on y is highlighted in the right-hand panel of Figure 3 , which shows that present data allow us to set a constraint at the 2- σ confidence level. Figure 3. Left panel: the cumulative mass functions computed at z = 6 for SD sterile neutrino models with and , for different values of the Yukawa coupling y (shown by the labels on the right) in the freeze-in limit. The shaded areas correspond to the observed number density of galaxies with within within the 1- σ , 2- σ , and 3- σ confidence levels. Right panel: the maximum value (including the theoretical uncertainties) of the predicted number density of DM halos at z = 6 for the case with as a function of the Yukawa coupling y . The upper shaded areas represent the observed number density of galaxies with within the 1- σ , 2- σ , and 3- σ confidence levels. We then extend our exploration to cover the whole parameter space of SD production models for sterile neutrinos. With this aim, we consider a grid of λ and y values for six different values of the scalar mass . The latter are chosen to cover the three different regimes where different production channels are dominant for the production of scalars from their coupling to the SM particles (see König et al. 2016 , and Section 3 ): the light scalar case , the intermediate case , and the heavy scalar case (in terms of the Higgs mass m h = 125 GeV). These different regimes have a rather strong impact on the distribution function of the scalar which directly translates into that of the sterile neutrino and, hence, on the resulting power spectrum. For each value of m S , we compute the power spectrum corresponding to each point in the plane as described in Section 3 . Then we represent in Figure 4 , left panel, the regions of the parameter space consistent with the galaxy number densities measured in the HFF (). These regions clearly split into a freeze-out (for ) and freeze-in (for ) family. For the freeze-out family, decreasing the scalar mass m S leads to a tighter bound on y , while yielding an approximate lower bound of for the Higgs portal coupling. For the freeze-in family, decreasing the scalar mass m S pushes the admitted values of λ to progressively
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