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

Aurelio Amerio, Alessandro Cuoco, Nicolao Fornengo

TL;DR

The paper addresses reconstructing the extragalactic gamma-ray source-count distribution $dN/dS$ below the Fermi-LAT detection threshold. It introduces a convolutional neural network trained on synthetic, PSF- and exposure-aware sky maps and applies it to 14 years of Fermi-LAT data in the (1-10) GeV band to recover $dN/dS$ down to fluxes of about $5\times10^{-12}$ cm$^{-2}$ s$^{-1}$. The baseline result shows agreement with the resolved catalog in the bright regime and a power-law $dN/dS\propto S^{-2}$ in the unresolved regime, with robustness checks across foreground models, latitude cuts, and sphere-based CNN implementations. The approach provides a data-driven, computationally efficient alternative to likelihood-based methods and paves the way for studying energy dependence and potential exotic components.

Abstract

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.

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

TL;DR

The paper addresses reconstructing the extragalactic gamma-ray source-count distribution below the Fermi-LAT detection threshold. It introduces a convolutional neural network trained on synthetic, PSF- and exposure-aware sky maps and applies it to 14 years of Fermi-LAT data in the (1-10) GeV band to recover down to fluxes of about cm s. The baseline result shows agreement with the resolved catalog in the bright regime and a power-law in the unresolved regime, with robustness checks across foreground models, latitude cuts, and sphere-based CNN implementations. The approach provides a data-driven, computationally efficient alternative to likelihood-based methods and paves the way for studying energy dependence and potential exotic components.

Abstract

We reconstruct the extra-galactic gamma-ray source-count distribution, or , 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 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 in the unresolved regime, down to fluxes of cm s. 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.
Paper Structure (21 sections, 5 equations, 28 figures, 3 tables)

This paper contains 21 sections, 5 equations, 28 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Data selection
  3. Synthetic map generation
  4. Differential source-count distribution
  5. Galactic foreground and isotropic background
  6. Count maps
  7. Neural network architecture and training
  8. Spherical neural networks
  9. Data pre-processing
  10. Network architecture
  11. Bayesian error estimation and cost function
  12. Training and validation
  13. Frequentist error estimation and Bias
  14. Results
  15. Conclusions
  16. ...and 6 more sections

Figures (28)

  • Figure 1: Fermi-LAT photon-counts map in units of counts per pixel in the 1-10 GeV energy range
  • Figure 2: Fermi-LAT mean exposure map in the 1-10 GeV energy range for the 14-years data set, in units of cm$^2$ s, at the Healpix resolution $N_{\rm side} = 128$ ($n=7$).
  • Figure 3: Average Fermi-LAT point spread function (PSF) for energies in $(1, 10)$ GeV.
  • Figure 4: Left: The 1-10 GeV $dN/dS$ for resolved Fermi-LAT sources, obtained from the 4FGL-DR3 Fermi-LAT:2022byn catalog. Right: An example of $dN/dS$ with 3 breaks $S_j$ ($j=1,3$) and four slopes, where $n_i$ ($i=1,4$) is the inclination of the slope.
  • Figure 5: Simulated flux map of point sources extracted from a specific source-count distribution, before (upper panel) and after (lower panel) the PSF is applied, in units of photons/(cm$^2$ s sr). The maps are shown at Healpix resolution $N_{\rm side} = 128$ ($n=7$).
  • ...and 23 more figures