Observed Steep and Shallow Spectra, Narrow and Broadband Spectra, Multi-frequency Simultaneous Spectra, and Statistical Fringe Spectra in Fast Radio Bursts: Various Faces of Intrinsic Quasi-periodic Spectra?

The spectra of fast radio bursts (FRBs) are crucial for exploring their radiation mechanism(s) (see reviews Cordes & Chatterjee, 2019; Petroff et al., 2019; Xiao et al., 2021; Zhang, 2023, and references therein). Observationally and statistically, FRB spectra exhibit significant diversity, e.g., steep and shallow spectra (Spitler et al., 2016; Macquart et al., 2019; CHIME/FRB Collaboration et al., 2021), narrow and broadband spectra (Kumar et al., 2021; Zhou et al., 2022; Zhang et al., 2023; Kumar et al., 2024; Pleunis et al., 2021), multi-frequency simultaneous spectra (Law et al., 2017; CHIME/FRB Collaboration et al., 2020; Bochenek et al., 2020), and statistical fringe spectra (Lyu et al., 2022; Lyu & Liang, 2023).

The observed spectral shape of FRBs is often approximated by a power-law function $F_{\nu}\propto\nu^{-\alpha}$. Many FRB bursts exhibit shallow spectra with indices $|\alpha|<3$ (Macquart et al., 2019; CHIME/FRB Collaboration et al., 2021; Zhong et al., 2022), while others, both from non-repeating and repeating FRBs, display steep spectra with indices $|\alpha|>3$. For example, FRB 20110523A had $\alpha=7.8\pm 0.4$ (Masui et al., 2015), FRB 20121102A’s bursts spanned a range of $\alpha=-10.4$ to $+13.6$ (Spitler et al., 2016), and many cases exceeded $\alpha>10$ in the first CHIME/FRB FRB catalog (CHIME/FRB Collaboration et al., 2021). These steep spectra may also suggest a narrow bandwidth. This is because, if the intrinsic spectra are described by a broken power-law and the observed spectra capture only the rising or falling segments described by a single power-law, the relationship between spectral index and bandwidth, as presented in the equation (2) of Yang (2023), shows that increasing the absolute value of either the rising index $\alpha_{l}$ or the falling index $\alpha_{h}$ narrows the bandwidth for a given full width $N_{\rm FW}$.

Although broadband spectra are commonly observed in FRBs, extremely narrow spectra have also been reported in some FRBs, such as FRBs 20190711A, 20201124A, 20220912A, 20240114A (Kumar et al., 2021; Zhou et al., 2022; Zhang et al., 2023; Kumar et al., 2024). While these narrow spectra could arise from intrinsic radiation mechanisms or interference processes¹¹1The radiative transfer processes in circumburst medium or emission region (magnetosphere), such as absorption and scattering effects, do not appear to significantly alter the observed spectra or produce narrow/steep spectra (Yang, 2023; Wang et al., 2024)., they seem to be more likely produced by a coherent process, e.g., bunching or maser mechanism (Yang, 2023; Liu et al., 2023; Wang et al., 2024). Interference processes like scintillation, gravitational lensing, or plasma lensing require specific conditions or further observational validation. As discussed and concluded in Yang (2023), for instance, scintillation with $\Delta\nu_{\rm sci}\gtrsim 100$ MHz demands an intermediately dense, turbulent plasma screen at $\sim 10^{15}$ cm from the FRB source. Gravitational lensing cannot explain the observed burst-to-burst variation of the spectra, although it can generate narrow spectra via planet-like objects. While for the plasma lensing, the typical lensing timescale of decades of seconds can cause the spectrum variation during this short time. This requires to be verified by more FRBs in the future.

Multi-frequency simultaneous spectra have been observed only in FRBs 20121102A (Law et al., 2017) and 20200428D²²2It is previous FRB 20200428A, now is referred to as FRB 20200428D, as suggested by Giri et al. (2023), since three earlier bursts were found. In addition to FRBs 20121102A and 20200428D, a burst from FRB 20180916B was also simultaneously observed by the Robert C. Byrd Green Bank Telescope (GBT; 300-400 MHz) and CHIME (400-600 MHz), see Chawla et al. (2020). However, the simultaneous detection in adjacent frequency bands is likely due to downward drifting (Pelliciari et al., 2024). currently (CHIME/FRB Collaboration et al., 2020; Bochenek et al., 2020). One burst in FRB 20121102A was simultaneously detected at $\sim 1.4$ GHz (Arecibo telescope) and $\sim 3$ GHz (VLA telescope), see Law et al. (2017). Since the power-law spectral index derived from observations at $\sim 1.4$ GHz and $\sim 3$ GHz was not in agreement with the spectral index limit from the non-detection at 4.85 GHz conducted simultaneously by the Effelsberg telescope, Law et al. (2017) concluded that the broadband burst spectrum could not be described by a single power-law.

Statistical spectral fringe patterns in FRBs 20121102A and 20190520B have emerged based on peak frequency distributions³³3In the absence of peak frequency data for bursts, Lyu et al. (2022) and Lyu & Liang (2023) approximate the central frequencies as the peak frequencies. We adopt the same approach in this work. (Lyu et al., 2022; Lyu & Liang, 2023), though it is still unclear whether such fringe patterns result from statistical and/or observational selection effects, such as the signal-to-noise ratio threshold, detection algorithm, frequency range, burst bandwidth, and/or statistical fluctuation, as elaborated in Lyu et al. (2022).

These observed spectral characteristics motivate us to think that the intrinsic FRB spectra may be quasi-periodic. If such quasi-periodic spectra stem from a coherent process, such as coherent curvature radiation by quasi-periodic structured bunches moving along the same trajectory proposed by Yang (2023), the observed steep and shallow spectra, narrow and broadband spectra, multi-frequency simultaneous spectra, and statistical fringe spectra could all be manifestations of such intrinsic quasi-periodic spectra. This paper aims to explore this.

The structure of the paper is organized as follows. Section 2 explores how intrinsic quasi-periodic FRB spectra can result in observed steep, shallow, narrow, broadband spectra, multi-frequency simultaneous spectra, and statistical fringe spectra. Section 3 provides a summary and discussion.

For coherent curvature radiation emitted by strictly periodic multiple bunches moving along the same trajectory⁴⁴4Since coherent radiation is still radiated into multiples of $2\pi\omega_{m}$ with narrow bandwidths, even for periodically distributed bunches with relative random phase (Yang, 2023), we use strictly periodically distributed bunches in this work for convenience., the intrinsic spectrum is quasi-periodic and can be expressed as (Yang, 2023)

where $N_{\rm b}$ is the number of bunches, and $P_{m}=1/\omega_{m}$ represents the period of the bunch distribution. The coherent peak frequencies appear at $\omega/\omega_{m}=2n\pi$, where $n\in\mathbb{Z}^{+}$. The spectrum of a single bunch is given by

which assumes an electron–positron pair bunch, where each pair bunch consists of one electron clump and one positron clump separated by a distance $\Delta$ (Yang et al., 2020). Here, $N_{\rm p}$ is the number of coherent pairs in one bunch, i.e., the number of electron/positron in each clump. The reason for using this type of bunch will be discussed in Section 2.2. $\left|E_{1}(\omega)\right|^{2}$, the spectrum of curvature radiation by a single charge, is described by (Yang & Zhang, 2018)

where $\omega_{c}=3c\gamma^{3}/(2\rho)$ is the characteristic frequency of curvature radiation, with $\rho$ as the curvature radius, $\gamma$ the Lorentz factor, and $c$ the speed of light.

It is important to note that the term $\sin^{2}\left(\frac{N_{\rm b}\omega}{2\omega_{m}}\right)\sin^{-2}\left(\frac{\omega}{2\omega_{m}}\right)$, which is related to the coherence properties of multiple bunches, implies a higher level coherence. Not only are the separated electron-positron pairs within each bunch coherent, but also the bunches themselves form coherent “clusters” we call. As a result, the total power spectrum from coherent curvature radiation by these bunch-forming clusters is

where $N_{\rm c}$ is the number of clusters, and $N=N_{\rm c}N_{\rm p}^{2}$ is set. Note that in this case $N_{\rm b}$ would represent the number of bunches in one cluster, rather than the total number of bunches. However, if $N_{\rm b}=1$, which corresponds to the case of uniformly distributed bunches, $N_{\rm c}$ would revert to being the total number of bunches.

Such that the observed flux $F_{\nu}$, representing the energy received per unit time per unit frequency per unit area, is written by (Yang & Zhang, 2018)

where $D$ is the distance between the source and observer, and $T$ is the time interval between adjacent clusters. For simplicity, we assume $T\sim 1/\nu_{c}\sim 1{\rm ns}$ for $\nu_{c}\sim{\rm GHz}$.

We have demonstrated that the observed steep and shallow spectra, narrow and broadband spectra, multi-frequency simultaneous spectra, and statistical fringe spectra in FRBs may be various manifestations of intrinsic quasi-periodic spectra, arising from coherent curvature radiation by structured bunches composed of separated electron and positron clumps. Our analysis, modelings, and MC simulations lead to the following main results:

• •

  For a given bunch distribution period $\omega_{m}$, the slenderness of the spectrum at each peak frequency is determined by the number of bunches per cluster, $N_{\rm b}$. In general, the observed spectrum above the telescope sensitivity becomes narrower and steeper as $N_{\rm b}$ increases. Moreover, the spectral index is inversely correlated with the bandwidth. The appearance of broadband spectra in most FRB bursts indicates that the number of bunches per cluster is typical small, perhaps $N_{\rm b}\lesssim 5$ for most bursts, although this is also dependent on the telescope sensitivity. Furthermore, if the telescope’s band envelope only covers the rising or falling half of the spectrum, the observed spectrum may exhibit a single power-law shape rather than a full Gaussian-like or broken power-law profile. In addition, a case study for the most fractionally narrow-banded burst in FRB 20190711A was conducted. Its extremely narrow spectrum can be well fit by a quasi-periodic spectrum relevant to a relatively large $N_{\rm b}=25$.

• •

  The observed multi-frequency simultaneous spectra for FRB 20200428D and a burst in FRB 20121102A can be effectively modeled by intrinsic quasi-periodic spectra within typical parameter ranges, as illustrated in Figure 2. From the results in Table 1, it is evident that the number of bunches per cluster, $N_{\rm b}$, is likewise relatively small for both FRBs 20121102A and 20200428D.

• •

  The statistical fringe patterns in peak frequency distributions for FRBs 20121102A and 20190520B can be reproduced if their bursts display intrinsic quasi-periodic spectra with highly universal $\omega_{m}$ values. Additionally, we discuss potential formation mechanisms of structured bunches. If these bunches originate from pair cascades in charge-starvation regions, the period of the bunch distribution reflects the period of pair cascades, which could be used to constrain the properties of the FRB source, such as its magnetic field. Alternatively, if the bunches are formed via two-stream instability involving Langmuir wave, the period of the bunch distribution could help estimate the wavelength of the Langmuir wave at the point where the linear regime breaks down.

As to whether the quasi-periodic bunch model can explain burst-to-burst variation in the spectra for an FRB, we illustrate as follows. FRB 20121102A shows significant spectral variation from burst to burst. For example, although one burst was detected simultaneously by both the Arecibo and VLA, several bursts in Law et al. (2017) were detected only by the VLA, with no detections by the Arecibo. If the distribution properties of quasi-periodic structured bunches, specifically, the $\omega_{m}$ values are not centered around a single universal $\omega_{m,c}$ value, then the quasi-periodic bunch model not only can explain the multi-frequency simultaneous spectra for the FRB 20121102A burst (as shown by the black solid line in the upper panel of Figure 2, under the parameter values in Table 1), but also possibly explain the non-detections from the Arecibo despite being detected at the VLA for several bursts (shown by the gray solid line in the same panel of Figure 2, under the parameter values $\omega_{m,c}=1.2\times 10^{9}~{}{\rm rad~{}s^{-1}}$, $N_{\rm b}=6$, $\gamma=214$, $\log_{10}(\rho/{\rm cm})=6.5$, $\log_{10}N=45.2$, and $\Delta=3$). Although this requires two distinct $\omega_{m,c}$ values, such a requirement is also noted in the fit in the upper left panel of Figure 1 of Lyu et al. (2022). This may be another reason why a relatively large $p_{\rm KS}$ value cannot be achieved when simulating with a single universal $\omega_{m,c}$ value for the fringe pattern of peak frequency distribution in FRB 20121102A bursts, in addition to the incompleteness of the observed sample and other observational effects mentioned in Section 2.3.1.

From a prospective viewpoint, we think that the intrinsic quasi-periodic spectrum may also account for the spectral zebra pattern observed in the high-frequency interpulse of the Crab pulsar’s radio emission (Hankins & Eilek, 2007; Hankins et al., 2016). While several models have been suggested to explain this phenomenon (see review, Eilek & Hankins, 2016), including the recent magnetosphere diffraction screen model (Medvedev, 2024), the quasi-periodic spectrum could provide a compelling alternative explanation. This will be explored elsewhere.