The Fermi-LAT high-latitude Survey: Source Count Distributions and the Origin of the Extragalactic Diffuse Background

This is the first of a series of papers aimed at characterizing the populations detected in the high-latitude sky of the {\it Fermi}-LAT survey. In this work we focus on the intrinsic spectral and flux properties of the source sample. We show that when selection effects are properly taken into account, {\it Fermi} sources are on average steeper than previously found (e.g. in the bright source list) with an average photon index of 2.40$\pm0.02$ over the entire 0.1--100\,GeV energy band. We confirm that FSRQs have steeper spectra than BL Lac objects with an average index of 2.48$\pm0.02$ versus 2.18$\pm0.02$. Using several methods we build the deepest source count distribution at GeV energies deriving that the intrinsic source (i.e. blazar) surface density at F$_{100}\geq10^{-9}$\,ph cm$^{-2}$ s$^{-1}$ is 0.12$^{+0.03}_{-0.02}$\,deg$^{-2}$. The integration of the source count distribution yields that point sources contribute 16$(\pm1.8)$\,\% ($\pm$7\,\% systematic uncertainty) of the GeV isotropic diffuse background. At the fluxes currently reached by LAT we can rule out the hypothesis that point-like sources (i.e. blazars) produce a larger fraction of the diffuse emission.