Thermal history of the low-density neutral inter galactic medium (IGM) due to heating by accreting stellar black holes in high mass X-ray binaries (BH-HMXBs). This figure shows the gas temperature of the IGM as a function of redshift z for three possible values of f_X as defined in Eq. (6): f_X = 0.1 (blue dashed line), f_X = 1.0 (black dotted line), f_X = 10.0 (red solid line)). As discussed in the text, most likely f_X > 1.