Abstract: The 70-month Swift-BAT catalogue provides a sensitive view of the extragalactic X-ray sky at hard energies (>10 keV) containing about 800 active galactic nuclei (AGN). We explore its content in heavily obscured, Compton-thick AGN by combining the BAT (14?195 keV) with the lower energy XRT (0.3?10 keV) data. We apply a Bayesian methodology using Markov chains to estimate the exact probability distribution of the column density for each source. We find 53 possible Compton-thick sources (probability range 3?100%) translating to a ~7% fraction of the AGN in our sample. We derive the first parametric luminosity function of Compton-thick AGN. The unabsorbed luminosity function can be represented by a double power law with a break at L? ~ 2 × 1042erg s-1 in the 20?40 keV band. The Compton-thick AGN contribute ~17% of the total AGN emissivity. We derive an accurate Compton-thick number count distribution taking into account the exact probability of a source being Compton-thick and the flux uncertainties. This number count distribution is critical for the calibration of the X-ray background synthesis models, i.e. for constraining the intrinsic fraction of Compton-thick AGN. We find that the number counts distribution in the 14?195 keV band agrees well with our models which adopt a low intrinsic fraction of Compton-thick AGN (~ 12%) among the total AGN population and a reflected emission of ~ 5%. In the extreme case of zero reflection, the number counts can be modelled with a fraction of at most 30% Compton-thick AGN of the total AGN population and no reflection. Moreover, we compare our X-ray background synthesis models with the number counts in the softer 2?10 keV band. This band is more sensitive to the reflected component and thus helps us to break the degeneracy between the fraction of Compton-thick AGN and the reflection emission. The number counts in the 2?10 keV band are well above the models which assume a 30% Compton-thick AGN fraction and zero reflection, while they are in better agreement with models assuming 12% Compton-thick fraction and 5% reflection. The only viable alternative for models invoking a high number of Compton-thick AGN is to assume evolution in their number with redshift. For example, in the zero reflection model the intrinsic fraction of Compton-thick AGN should rise from 30% at redshift z ~ 0 to about 50% at a redshift of z = 1.1.

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