Inferring the Intrinsic Energy Function of FRB 20220912A

Abstract: The statistical analysis of fast radio burst (FRB) samples from repeaters may suffer from a band-limited selection effect, which can bias the observed distribution. We investigated the impact of this selection bias on the energy function through simulations and then applied our analysis to the particular case of FRB 20220912A. Our simulations show that, in the sample of bursts observed by the Five hundred meter Aperture Spherical Telescope (FAST), assuming a unimodal intrinsic energy distribution, the band selection effect alone is insufficient to produce a bimodal energy distribution; only the bimodal central frequency distribution can achieve this. The bursts' energy of FRB 20220912A that primarily fell within the observing band showed no significant correlation with the central frequency. In contrast, bursts with higher central frequency tend to exhibit narrower bandwidth and longer duration. The distribution of the intrinsic energy can be modeled as a log-normal distribution with a characteristic energy of $8.13 \times 10^{37}$ erg, and a power-law function with the index of $1.011 \pm 0.028$. In contrast to the initial energy function reported by \cite{2023ApJ...955..142Z}, the low-energy peak vanishes, and the high-energy decline becomes steeper, which implies the low-energy peak is an observational effect. The bimodality of the energy distribution seems to originate from the intrinsic radiation mechanism.