a fraction $f_{\rm esc}$ of the UV photons in the intrinsic spectrum of the galaxies escape from the source into the IGM. The SED of the stellar component does not have any significant X-ray photons that are required for heating the neutral / partially ionized IGM. We call this model, consisting of only stars, the [**Galaxy model**]{}.

In the [**mini-QSO model**]{}, we assume the model source SED has, in addition to the stellar component, a component powered by intermediate mass black holes of mass $10^3-10^6\, {{\rm M}_{\odot}}$. We assume that the X-rays emitted from these mini-QSOs follow a power law with a spectral index $\alpha$ [@vanden01; @vignali03]. We also introduce a parameter $f_X$ which is the ratio of the X-ray to UV luminosity, where we assume that the UV and X-ray bands span from 10.2 to 100 eV and 100 eV to 10 KeV respectively. We assume that our fiducial source model has stellar mass $M_{\star} =  10^7~ {{\rm M}_{\odot}}$, age $t_{\rm age} = 20$ Myr. We choose $f_{\rm esc}=0.1$, $\alpha=1.5$ [@laor97; @vanden01; @vignali03] and $f_X=0.05$ as the fiducial values.[^9] We set the fiducial value of the parameter as $f_X = 0.05$. This corresponds to an accreting BH to galaxy 