Extracting the gamma-ray source-count distribution below the Fermi-LAT detection limit with deep learning

We reconstruct the extra-galactic gamma-ray source-count distribution, or $dN/dS$, of resolved and unresolved sources by adopting machine learning techniques. Specifically, we train a convolutional neural network on synthetic 2-dimensional sky-maps, which are built by varying parameters of underlying source-counts models and incorporate the Fermi-LAT instrumental response functions. The trained neural network is then applied to the Fermi-LAT data, from which we estimate the source count distribution down to flux levels a factor of 50 below the Fermi-LAT threshold. We perform our analysis using 14 years of data collected in the $(1,10)$ GeV energy range. The results we obtain show a source count distribution which, in the resolved regime, is in excellent agreement with the one derived from catalogued sources, and then extends as $dN/dS \sim S^{-2}$ in the unresolved regime, down to fluxes of $5 \cdot 10^{-12}$ cm$^{-2}$ s$^{-1}$. The neural network architecture and the devised methodology have the flexibility to enable future analyses to study the energy dependence of the source-count distribution.