arXiv2025
Despite hundreds of detected fast radio bursts (FRBs), the faint-end slope ($γ$) of their energy distribution remains poorly constrained, hindering understanding of whether bright, cosmological FRBs and faint, Galactic magnetar SGR 1935+2154-like bursts share a common origin. In this study, we constrain this faint-end slope, modeled with a Schechter-like distribution, by searching for potential associations between bursts from the CHIME/FRB Catalog-1 and galaxies in the local volume. We cross-matched Catalog-1 FRBs with 495 local volume galaxies within 21 Mpc, identified from the HECATE catalog, and found no associations. Assuming the FRB energy function extends to $\sim 3 \times 10^{34}$ erg-the energy of the Galactic magnetar burst from SGR 1935+2154-this null result constrains $γ$ to be $<$ 2.3 (95% confidence upper limit), representing the first empirical estimate for extragalactic FRBs at such low energies. This finding supports the hypothesis that the FRB population is dominated by bright, likely cosmological bursts with a relatively flat energy distribution ($γ< 2.5$). However, the constraint weakens if higher energy thresholds are assumed. A flatter energy function is consistent with the observed anti-correlation between FRB dispersion measure and fluence, as seen across various observational bands. While the contribution of low-energy bursts, such as those from the Galactic magnetar SGR 1935+2154, appears minimal, our results suggest that normal magnetars like SGR 1935+2154 could dominate the FRB population if their burst rates and energies scale with age and magnetic field. Upcoming CHIME/FRB Catalog-2 data and targeted nearby galaxy surveys will further refine these constraints, offering critical insight into whether FRBs arise from a single population or diverse origins.