Decoding FRB energetics and frequency features hidden by observational incompleteness

Abstract

Fast radio bursts (FRBs) are millisecond-duration radio flashes of extragalactic origin, with magnetars implicated as viable central engines. Yet their triggering and radiation mechanisms remain unknown. Radio telescopes inevitably record bursts incompletely, as limited sensitivity and finite bandwidth lead to observational truncation. Here we establish a general analytical framework to reconstruct intrinsic population-level frequency characteristics and energetic parameters directly from observationally truncated FRB data. Applying this method to 2,223 bursts of FRB 20121102A observed by three different telescopes, we show that the narrow spectra of repeating FRBs are predominantly an observational selection effect. Only intrinsically high-energy bursts are genuinely narrowband. We further quantify, for the first time, the number and energy of completely undetected bursts, and reveal intrinsic long-term frequency evolution of the source. Our methodology transforms incomplete archival observations into physically meaningful probes, bridging instrumental readouts and intrinsic FRB physics.

Figures (15)

• Figure 1: A schematic illustration of the observational cutoff of FRBs on the $F_\nu$-$\nu$ plane. The solid black curve represents the intrinsic spectrum of the burst. The dashed gray line represents the fluence threshold $F_{\nu,{\rm{thre}}}$ of the telescope. The orange shaded region shows the operating band of the telescope, with lower and upper limits denoted as $f_{\rm{\ell}}$ and $f_{\rm{h}}$, respectively. The spectrum of the burst intersects with $F_\nu=F_{\nu, {\rm{thre}}}$ at two frequencies, $\nu_{\rm{\ell}}$ and $\nu_{\rm{h}}$, which define the intrinsic bandwidth of the burst. Under the influence of both the sensitivity cutoff and operating-band cutoff, the observed bandwidth is determined by $\nu_1$ and $\nu_2$. Note that this figure only shows one of the six possible cases, determined by the relationship of $\nu_{\rm{\ell}}, \nu_{\rm{h}}, f_{\rm{\ell}}$ and $f_{\rm{h}}$.
• Figure 2: Comparison between the observed parameters $\left(F_{\nu, {\rm{obs}}}^{\rm{equiv}}, \delta\nu_{\rm{obs}}, \Delta{t}_{\rm{obs}}, W_{\rm{obs}}^{\rm{equiv}} \right)$ and the derived spectral parameters $\left(F,\nu_{\rm{p}}, \sigma_\nu\right)$ for the band-unlimited bursts of FRB 20121102A observed by Arecibo. (A) The pairwise comparison between $\left(F_{\nu, {\rm{obs}}}^{\rm{equiv}}, \delta\nu_{\rm{obs}}, \Delta{t}_{\rm{obs}}, W_{\rm{obs}}^{\rm{equiv}}\right)$ and $\left(F, \nu_{\rm{p}}, \sigma_\nu\right)$. Each main plot illustrates the 2D distribution of the band-unlimited bursts. The contour lines correspond to the kernel density estimation (KDE) of the distribution. 1D histograms are also plotted along the axes. The correlation coefficient ($\tau_{\rm{b}}$) is calculated for each pair of parameters by using the Kendall tau method. The corresponding coincidence probability ($p$) is also calculated and marked. A trendline (dash-dotted line) is plotted when the correlation is statistically significant. (B) A pairwise plot for the three parameters of $\left(F, \nu_{\rm{p}}, \sigma_\nu\right)$.
• Figure 3: Effects of sensitivity cutoff and the characteristics of reconstructed band-unlimited bursts. (A) The distribution of band-unlimited bursts on the $F$-$\sigma_\nu$ plane. The data points are color-coded based on $\frac{\delta\nu_{\rm{obs}}}{\sigma_\nu}$ and their sizes are proportional to $\lg\left[F_{\nu,{\rm{obs}}}^{\rm{equiv}} \left(\rm{Jy\ ms}\right)\right]$. The inset plots the data points on the $F$-$\frac{\delta\nu_{\rm{obs}}}{\sigma_\nu}$ plane. (B) The distribution of band-unlimited bursts on the $\nu_{\rm{h}}$-$\nu_{\rm{\ell}}$ plane, with data points color-coded based on $\frac{\delta\nu_{\rm{obs}}}{\sigma_\nu}$ and scaled in size proportional to $\lg\left[\sigma_\nu \left(\rm{MHz}\right)\right]$. (C) Observed energy distribution as compared with the intrinsic energy distribution for band-unlimited bursts.
• Figure 4: Operating-band cutoff effects on burst distribution and the reconstructed intrinsic profile (Arecibo sample). (A) The distribution of observed bursts on the $\nu_{\rm{h}}$-$\nu_{\rm{\ell}}$ plane. The yellow-orange-red contour lines represent the KDE, while the blue contour lines show their intrinsic distribution. 1D histograms of the bursts are shown along the axes. The dashed orange lines stand for the operating band limits of Arecibo, corresponding to $\nu_{\rm{\ell}}=f_{\rm{\ell}}$, $\nu_{\rm{\ell}}=f_{\rm{h}}$, $\nu_{\rm{h}}=f_{\rm{\ell}}$ and $\nu_{\rm{h}}=f_{\rm{h}}$. The region below the red dashed line is non-physical, while the pink region is undetectable. (B) Constraints on the parameters of the reconstructed intrinsic distribution function $N\left(\nu_{\rm{\ell}}, \nu_{\rm{h}}; \mu_{\nu_{\rm{\ell}}}, \mu_{\nu_{\rm{h}}}, \sigma_{\nu_{\rm{\ell}}}, \sigma_{\nu_{\rm{h}}}, \rho\right)$. (C) The observed distribution of bursts on the $\delta\nu$-$\nu_{\rm{p}}$ plane. The line styles are similar to that of (A). Comparing with (A), we see that the accumulation caused by distortion in the 1D histograms of $\nu_{\rm{p}}$ and $\delta\nu$ is not restricted to the edge bin of the observed histogram, but can occur in every bin. Such histograms can be effectively explained by the "squeezing" effect. The point $\left(f_{\rm{c,Arecibo}},\delta f_{\rm{Arecibo}}\right)$ represents the central frequency and width of Arecibo operating band.
• Figure 5: Effects of operating-band cutoff on FRB energetics (Arecibo sample). (A) The distribution of band-limited bursts on the $E$-$E_{\rm{obs}}$ plane. Here, $E$ denotes the estimated intrinsic energy and $E_{\rm{obs}}$ the observed energy. The data points are color-coded based on $\nu_{\rm{p,obs}}$ and scaled in size proportional to $\lg\left(\delta\nu_{\rm{obs}}\right)$. (B) Observed energy distribution as compared with the recovered intrinsic energy distribution for detected bursts, including band-unlimited bursts and band-limited bursts. A shifted log-normal function is used to fit the distribution. (C) Constraints on the parameters of the standard log-normal distribution for the proper energy.