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tokens: fraction of the kinetic energy transferred to CRs by a single SN and SNR is the comoving SN rate. In principle, one should account also for helium nuclei. For simplicity, we assume here that particles can be treated as four protons, and hence be absorbed in the proton spectrum efficiency. In addition to the SFR, to derive SNR we need to know the number of SNe explosions per solar mass of forming stars, which is given by where ( m ) is the initial mass function of Population II/I stars, for which we assume the following form: with x 1.35 and m c 0.35 M ; m lies in the range [0.1, 100] M (Larson 1998 ). We neglect the contribution from Population III stars. Their SF history is very debated and still highly uncertain. In fact, detailed studies, exploiting cosmological hydrodynamical simulations implementing chemical feedback effects, have shown that Population II/I stars dominate the global SFR at any redshift (Tornatore etal. 2007 ; Maio etal. 2010 ). Moreover, the recent surveys hunting for Population III stars in the Milky Way have found no metal-free stars so far, implying that they are rare in the Milky Way even if they exist (Beers Christlieb 2005 ). Combining the above formulae, equation ( 19 ) becomes Particle acceleration in SN explosions is believed to occur through diffusive shock acceleration, which leads to momentum power-law spectra of accelerated particles. With this in mind, we assume that the source function of volume-averaged CR protons injected by SNe (defined as a rate per unit energy and volume) is where E is the proton kinetic energy, E 0 1GeV, m p is the proton mass, v / c is the dimensionless velocity of the particle, 2 is the slope of the differential spectrum of accelerated particles and C ( z ) is a redshift-dependent normalization obtained by imposing that the total kinetic energy rate equals , i.e. In equation ( 24 ), we fix E max 10 6 GeV (Lagage Cesarsky 1983 ), and we verify a posteriori that our conclusions are not strongly dependent on our choice of E min 10keV. From equation ( 23 ), one can easily realize that is the key parameter determining the fraction of the total kinetic energy released that goes into protons with E 1GeV. We keep as the only free parameter of the model. In Fig. 3 , we plot the source function as a function of the kinetic energy for three different values of . We found that protons with kinetic energies below 10MeV represent 0.5percent, 2.7percent and 13percent of the total kinetic energy released for 2,2.2 and 2.5, respectively. Figure 3. Source function of CR protons with respect to their kinetic energy at z 10 for a spectrum slope 2 (blue line), 2.2 (green) and 2.5 (red). 3.2 Energy losses in the IGM CRs can be an efficient heat source especially for a low-density gas. When the CR proton ionizes an atom, it transfers a certain fraction of its kinetic energy to the electron, which is either used for further atomic excitation and ionization or distributed via elastic collisions to other