Probing Axion Like Particle-Proton Interactions through the Interplay of Dark Matter and Cosmic Rays in Large-Scale Structure

Axion-like particles (ALPs) provide a well-motivated dark matter candidate whose interactions with Standard Model particles may generate observable cosmological signatures. We investigate the diffuse $\gamma$-ray emission produced by the scattering process between ALP dark matter and cosmic-ray (CR) protons inside dark matter halos, and explore its detectability through the cross-correlation between the unresolved $\gamma$-ray background (UGRB) measured by \textit{Fermi} Large Area Telescope (LAT) and the Two Micron All Sky Survey Redshift Survey (2MRS) galaxy catalog in the local Universe ($z\leq0.1$). To model the cosmological $\mathrm{GeV}$-$\mathrm{PeV}$ CR proton population, we develop a phenomenological bottom-up prescription that links CR injection to the star formation rate through the supernova rate, while the steady-state CR density is determined by diffusive escape. Our forecasts indicate that UGRB-galaxy cross-correlation measurements can probe previously unexplored regions of the ALP parameter space over a broad range of ALP masses. More generally, this work presents a general framework for cosmological observables generated by interactions between dark matter and an additional astrophysical field, opening a new avenue for indirect dark matter searches using large-scale structure observations.