I assume that the underlying active galactic nuclei (AGN) which power the XRB have a quasar-like spectrum with an energy photon index of one. The spectrum is then constant in a F sense (Fig. 1). The action of photoelectric absorption by increasing amounts of material, characterised by a column density NH, is (Fig. 2) to cut out the lower energy emission from the observed spectrum up to an (approximate) energy E ~ 10 NH8/3 keV, where NH is in units of 1024 cm-2. As the column density exceeds about 1.5 x 1024 cm-2 so the absorber becomes Compton thick and Compton (electron) scattering causes the residual spectrum above this cutoff to decrease in intensity. This means that the intensity observed above about 30 keV is close to the intrinsic unattenuated intensity from Compton-thin sources, and is a lower limit for Compton-thick ones. Therefore the intensity of the XRB at 30 keV equals the normalization of the XRB after correction for absorption by Compton-thin sources. This normalization can be increased by a factor of f-1 if only a fraction f of all the power emerges from sources which are Compton thin. f is at most 3/4 (Maiolino et al 1998) and could be less than one half.
Figure 1. XRB spectrum (solid line) with the assumed unabsorbed spectrum of photon index 2 (dotted line). A typical AGN spectrum with reflection, in which the direct emission is a power law of photon index 2 with an exponential cutoff of 300 keV is shown by the dot-dash line, matching around the XRB peak. If unabsorbed quasars contribute 50 per cent of the XRB at 1 keV, then their contribution lies along the bottom of the figure.
The absorption-corrected XRB spectrum can then be extended into the ultraviolet band assuming the mean quasar spectral energy distribution of Elvis et al (1994). This shows that about 3 per cent of the power from a typical quasar is emitted in the 2-10 keV band. The total absorption-corrected AGN background can now be converted into an energy density AGN and thence, through the use of E = Mc2 or rather (1 + ) = c2 with an accretion efficiency factor and a mean redshift (since photons lose energy in the expansion of the Universe but mass does not), we have the mean density in black holes bh.
Figure 2. Monte-Carlo simulations of of an accretion disc spectrum (a power-law of unit energy index with cold reflection and an exponential cutoff at 360 keV) propagated through a solar abundance spherical cloud of column densities ranging from 1021.25 to 1025.25 cm-2 in steps of a factor 10. Note that the spectra peak around 30 keV, indicated by the vertical line. The lower panel shows the effect of increasing the iron abundance by 5, which causes the peak to shift to 40-50 keV (from Wilman & Fabian 1999).
The resulting value of bh = 6 x 105 M Mpc-3 is about half the value found by Magorrian et al (1998) from a study of ground-based optical data of the cores of nearby galaxies, and in rough agreement with an HST photometric study made by van der Marel (1999). Similar agreement has been obtained by Salucci et al (1999) from a detailed considerations of source counts etc. This close agreement emphasises that most of the mass in black holes has been accreted by a radiatively efficient (but obscured) process, and not by some inefficient process such as an advective flow. The correction required for absorption is extensive and requires that most, about 85%, of the accretion power has been absorbed.