This paper investigates what constraints can be placed on the fraction of Compton-thick (CT) AGN in the Universe from the modeling of the spectrum of the diffuse X-ray background (XRB). We present a model for the synthesis of the XRB that uses as input a library of AGN X-ray spectra generated by the Monte Carlo simulations described by Brightman & Nandra. This is essential to account for the Compton scattering of X-ray photons in a dense medium and the impact of that process on the spectra of obscured AGN. We identify a small number of input parameters to the XRB synthesis code which encapsulate the minimum level of uncertainty in reconstructing the XRB spectrum. These are the power-law index and high energy cutoff of the intrinsic X-ray spectra of AGN, the level of the reflection component in AGN spectra and the fraction of CT AGN in the Universe. We then map the volume of the space allowed to these parameters by current observations of the XRB spectrum in the range 3-100 keV. One of the least constrained parameters is the fraction of CT AGN. Statistically acceptable fits to the XRB spectrum at the 68% confidence level can be obtained for CT fractions in the range 5-50%. This is because of degeneracies among input parameters to the XRB synthesis code and uncertainties in the modeling of AGN spectra (e.g. reflection). The most promising route for constraining the fraction of CT AGN in the Universe is via the direct detection of those sources in high energy (>10keV) surveys. It is shown that the observed fraction of CT sources identified in the SWIFT/BAT survey, limits the intrinsic fraction of CT AGN, at least at low redshift, to 10-20% (68% confidence level). We also make predictions on the number density of CT sources that current and future X-ray missions are expected to discover. Testing those predictions will constrain the intrinsic fraction of CT AGN as a function of redshift.
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