A unified picture of Galactic and cosmological fast radio bursts

On April 28, 2020, the Canadian Hydrogen Intensity Mapping Experiment (CHIME, 400-800 MHz) and the Survey for Transient Astronomical Radio Emission 2 (STARE2, 1.3-1.5 GHz) independently detected a fast radio burst (hereafter FRB 200428), which is spatially coincident with the well known Galactic Soft Gamma-ray Repeater (SGR) 1935+2154 (The CHIME/FRB Collaboration et al., 2020b; Bochenek et al., 2020a). The arrival-time difference between these two frequency bands is consistent with dispersive delay due to plasma along the line of sight with dispersion measure $\mathrm{DM}=332.8\pm 0.1\rm\,pc\,cm^{-3}$. The burst had two $\sim 1\rm\,ms$ components separated by about 30 ms as measured by CHIME, the first at lower frequencies (400-550 MHz) and the second at higher frequencies (550-800 MHz). The FRB occurred in a side lobe of CHIME, so its inferred fluence of a few hundred kJy ms may suffer large uncertainty (The CHIME/FRB Collaboration et al., 2020b). However, STARE2 provided a more accurate fluence measurement of $1.5\rm\,MJy\,ms$ (Bochenek et al., 2020a).

The SGR 1935+2154 was first detected by Swift with a burst of $\gamma$-rays (Stamatikos et al., 2014). Subsequent X-ray follow-up observations identified this source as a magnetar with rotational period $P=3.24\,\rm s$ and characteristic surface dipolar magnetic field $B\simeq 2.2\times 10^{14}\rm\,G$ (Israel et al., 2016). This magnetar has had multiple episodes of outbursts since the initial discovery (Lin et al., 2020a). SGR 1935+2154 is spatially associated with the supernova remnant G57.2+0.8 (Sieber & Seiradakis, 1984; Gaensler, 2014; Surnis et al., 2016), which is at a distance between 6.7 and 12.5 kpc from us (Kothes et al., 2018). We adopt $d\simeq 10\rm\,kpc$ but our results are unaffected by the distance uncertainty.

A hard X-ray burst was detected from SGR 1935+2154 by several instruments including INTEGRAL (Mereghetti et al., 2020), Insight-HXMT (Li et al., 2020), AGILE (Tavani et al., 2020), Konus-Wind (Ridnaia et al., 2020), and the arrival time is in agreement with that of the FRB after de-dispersion. The X-ray burst had fluence of $7\times 10^{-7}\rm\,erg\,cm^{-2}$ in the 1-150 keV range, and the lightcurve in the hardest band (27-250 keV) of HXMT showed two distinct peaks separated by about 30 ms (Li et al., 2020), further confirming the association with the FRB 200428. The isotropic energy ($\nu E_{\nu}$) ratio between the radio and the hard X-ray bands is $\sim 3\times 10^{-5}$.

Understanding the origin of FRBs — mysterious bright millisecond-duration radio flashes first discovered about a decade ago (Lorimer et al., 2007) — has been a major scientific goal of many current or future telescopes, such as Parkes (Thornton et al., 2013; Bhandari et al., 2018), Arecibo (Spitler et al., 2016), UTMOST (Caleb et al., 2017), ASKAP (Shannon et al., 2018), CHIME (CHIME/FRB Collaboration et al., 2019), FAST (Li et al., 2013), and DSA (Ravi et al., 2019). Before the discovery of FRB 200428, all localized FRBs were from cosmological distances (Chatterjee et al., 2017; Bannister et al., 2019; Prochaska et al., 2019; Ravi et al., 2019; Marcote et al., 2020). Even with precise localizations of these events in their host galaxies, it is so far inconclusive what the progenitors of FRBs are and by what process the powerful radio emission is generated. Many ideas have been proposed (see Katz, 2018; Petroff et al., 2019; Cordes & Chatterjee, 2019, for recent reviews). They fall into two general categories: (1) emission within the magnetosphere of a neutron star (NS), and (2) emission from a relativistic outflow which interacts with the surrounding medium at large distances from the NS or black hole.

The isotropic specific energy of FRB 200428, $E_{\nu}\sim 2\times 10^{26}\rm\,erg\,Hz^{-1}$ (for a distance of 10 kpc), is about a factor of $\sim$30 lower than the faintest burst detected from FRB 180916 at cosmological distances (Marcote et al., 2020) but exceeds that of the brightest known giant radio pulses from NSs by four orders of magnitude. Apart from this energetic argument, we provide further evidence based on volumetric rate (in §2) that FRB 200428 belongs to the faint end of the cosmological FRB population. Therefore, the detection of FRB 200428 in the Milky Way provides an extraordinary opportunity to understand the FRB phenomenon, in the following three major aspects: (1) strongly magnetized NSs or magnetars can make FRBs (as advocated by many authors, e.g., Popov & Postnov, 2010; Kulkarni et al., 2014; Katz, 2016; Kumar et al., 2017; Lyubarsky, 2014; Beloborodov, 2017; Metzger et al., 2019; Wadiasingh & Timokhin, 2019), (2) the associated X-ray emission (and future identifications of other counterparts) provides valuable clue for the emission mechanism, (3) the close proximity may allow us to disentangle many of the propagation effects from the intrinsic emission properties.

This paper aims to explore the implications of FRB 200428. In §2, we compare the rate of FRB 200428-like events with that of the cosmological FRB population. In §3, we compare SGR 1935+2154 with the sources of other actively repeating FRBs and discuss how they may be understood in a general framework of the magnetar progenitors from different formation channels. In §4, we compare the rates of magnetar X-ray bursts and FRBs, and discuss the physical link between them. Finally, we closely examine the possible emission mechanisms for FRB 200428 and for other FRBs in §5. A brief summary is provided in §6. We use the widely adopted, convenient, subscript notation of $X_{n}\equiv X/10^{n}$ in the CGS units.

Based on a single detection in about one year of STARE2 operation (Bochenek et al., 2020b), we roughly estimate the Galactic FRB rate to be $\sim 10\rm\,yr^{-1}$ above specific energy of $E_{\nu}\sim 5\times 10^{25}\rm\,erg\,Hz^{-1}$ (the detection threshold energy for a distance of 10 kpc). This leads to a volumetric rate of $\sim 10^{8}\rm\,Gpc^{-3}\,yr^{-1}$ (see Bochenek et al., 2020a, for a more detailed calculation). This should be compared with the bright end of rate density distribution $R\sim 10^{2.6}\rm\,Gpc^{-3}\,yr^{-1}$ above $E_{\nu}=10^{32}\rm\,erg\,Hz^{-1}$ as measured by ASKAP also at 1.4 GHz (Shannon et al., 2018; Lu & Piro, 2019). We find the slope for the cumulative rate distribution to be $\beta\simeq\Delta\mathrm{log}R/\Delta\mathrm{log}E_{\nu}\simeq 0.8$, which is insensitive to Poisson error of a factor of a few. This agrees with the slope of the rate distribution found within the (small) ASKAP sample $0.3\lesssim\beta\lesssim 0.9$ (Lu & Piro, 2019) as well as the joint analysis of the Parkes and ASKAP samples $0.5\lesssim\beta\lesssim 1.1$ (Luo et al., 2020). This agreement suggests that FRB 200428 contributes a significant fraction of the FRB rate density at the faint end near $E_{\nu}\sim 10^{26}\rm\,erg\,Hz^{-1}$, as illustrated by Fig. 1. Combining this with the fact that the specific energy of FRB 200428 is only a factor of $\sim$30 below the faintest known cosmological FRB (Marcote et al., 2020), we conclude that the magnetar nature of the progenitor and emission mechanism of FRB 200428 is likely representative of the whole FRB population.

The number density of galactic magnetars, SGR 1935+2154 being one of them, is of the order $3\times 10^{8}\rm\,Gpc^{-3}$. The progenitors of highly active repeaters like FRB 180916 (CHIME/FRB Collaboration et al., 2019; Marcote et al., 2020) are much rarer in the Universe with a number density of $\sim 7$–$700\rm\,Gpc^{-3}$, which is estimated by the expectation number between 0.05 and 4.7 (90% C.L., Gehrels, 1986) of such repeaters in about half of the sphere (visible by CHIME) within $150\rm\,Mpc$. If these active repeaters are also powered by magnetars, they must belong to a type of “active magnetars” not seen in the Milky Way. If one assumes that all active magnetars will evolve to normal magnetars over time by reducing the bursting rate, the volume density of these active magnetars’ “descendants” would be at most a factor of $\sim 10^{4}/30\sim 300$ times greater than the volume density of active magnetars, where $10^{4}$ yr is the typical age of SGR 1935+2154-like galactic magnetars and $\sim$$30\,\rm yr$ is a conservative estimate of the characteristic age of active magnetars. This gives a volume density of $\sim 2\times 10^{3}$–$2\times 10^{5}\rm\,Gpc^{-3}$ of these descendants, still 3–4 orders of magnitude smaller than the galactic magnetar volume density. The discrepancy is even larger if the characteristic age of active magnetars is longer than 30 yr. This deficit cannot be fully reconciled by reasonable beaming correction¹¹1Here, beaming correction is given by the total beaming factor, which is defined as the fraction of the whole $4\pi$ sky occupied by the union of the solid angles spanned by all bursts from a given source. Due to the star’s rotation, the total beaming factor is typically substantially larger than that for individual bursts., because we would not have seen FRB 200428 from one of the $\sim$30 magnetars in our galaxy if the average beaming fraction is $\ll 1/30$. We can then draw the conclusion that “active magnetars” and SGR 1935+2154-like galactic magnetars must be two distinct populations (as also suggested by Margalit et al., 2020b).

One possibility is that the progenitors of FRB 180916 (or FRB 121102, Spitler et al., 2016) may be produced from rare, extreme explosions such as long gamma-ray bursts (LGRBs), superluminous supernovae (SLSNe, e.g., Metzger et al., 2017), or NS mergers (e.g., Margalit et al., 2019; Wang et al., 2020b), so that they have relatively short (e.g. millisecond) periods at births (Usov, 1992; Zhang & Mészáros, 2001; Metzger et al., 2011). These magnetars likely stored more toroidal magnetic energy inside the star which provides a larger energy reservoir to power bursting activities (e.g., Thompson & Duncan, 1993). In contrast, Galactic magnetars were likely born with a more moderate initial spin, as evidenced by the limited energy in their surrounding supernova remnants (e.g., Vink & Kuiper, 2006). These magnetars may store less toroidal magnetic energy inside the star and are relatively less active compared with their active cousins. The possible dichotomy of FRB magnetar progenitor is consistent with the host galaxy data of the localized FRBs (Li & Zhang, 2020): whereas FRB 121102 has a host galaxy similar to that of LGRBs or SLSNe (Tendulkar et al., 2017; Metzger et al., 2017; Nicholl et al., 2017), other four hosts resemble the Milky Way galaxy that hosts regular magnetars (Bannister et al., 2019; Ravi et al., 2019; Prochaska et al., 2019; Marcote et al., 2020).

The hard X-ray burst associated with FRB 200428 was one of the numerous X-ray bursts that SGRs generate during their active periods. The ratio of the energy release in the radio and X-ray bands is $f_{\rm r}\sim 3\times 10^{-5}$. In the following, we discuss the implications on the physical link between emission in these two bands and possible beaming of the radio emission.

The Galactic SGR X-ray burst rate is of order $0.1\rm\,yr^{-1}$ (volumetric rate $\sim 2\times 10^{6}\rm\,Gpc^{-3}\rm\,yr^{-1}$) above $E_{\rm x}=10^{44}\rm\,erg$, and the energy dependence has a similar power-law form as that for FRBs (e.g., Ofek, 2007; Kulkarni et al., 2014; Beniamini et al., 2019). For a fiducial value of the radio-to-X-ray flux ratio $f_{\rm r}=10^{-4}f_{\rm r,-4}$ to connect the rates of X-ray bursts to FRBs (Chen et al., 2020), $E_{\rm x}=10^{44}\rm\,erg$ corresponds to FRB specific energy of $E_{\nu}\simeq 10^{31}f_{\rm r,-4}\rm\,erg\,Hz^{-1}$ (for $1\rm\,GHz$ bandwidth), above which the volumetric rate is $\sim 3\times 10^{3}f_{\rm r,-4}^{-0.8}\rm\,Gpc^{-3}\,yr^{-1}$ (Lu & Piro, 2019; Luo et al., 2020). We see that only a small fraction ($10^{-3}$ to $10^{-2}$) X-ray bursts may be associated with FRBs. This also agrees with the fact that 29 of the X-ray bursts from SGR 1935+2154 had concurrent observations by FAST but no radio signal was detected down to fluence limit of $\sim$10 mJy ms (Lin et al., 2020b).

This small fraction of association may be explained by (a combination of) the following two possible reasons. The first is that most X-ray bursts are accompanied by an FRB but the radio emission is highly beamed, with a beaming fraction of $\Omega_{\rm frb}/4\pi\sim 10^{-3}$–$10^{-2}$. This may be realized if FRBs are only generated along magnetic field lines near the poles. The second explanation is that only a small fraction of X-ray bursts may be physically associated with FRB but in each association the FRB beaming fraction is order unity.

In the next section, we discuss the implications of the association between FRBs and magnetar X-ray bursts on the coherent emission mechanism.

Models for the generation of FRB coherent radio emission can be divided into two broad classes based on the distance from the NS where they operate. The first class consists of the “far-away” models where relativistic ejecta from a neutron star (or black hole) dissipates its energy at large distances by interacting with the circum-stellar medium (CSM) and the radio emission is generated by a maser process (Lyubarsky, 2014; Waxman, 2017; Beloborodov, 2017, 2019; Metzger et al., 2019; Margalit et al., 2020a). The second class are the “close-in” models which describe that the coherent processes occur within the magnetosphere of a neutron star (Pen & Connor, 2015; Cordes & Wasserman, 2016; Lyutikov et al., 2016; Kumar et al., 2017; Zhang, 2017; Lu & Kumar, 2018; Yang & Zhang, 2018; Kumar & Bošnjak, 2020; Lyubarsky, 2020; Wang et al., 2020a). These two general classes of models have very different predictions regarding the FRB temporal and spectral properties, and multiwavelength counterparts. In the Appendix, we present a detailed analysis of the “far-away” models and show that they face a number of difficulties explaining the available radio and X-ray data for FRB 200428.

In this section, we focus on the generation of FRB radiation in the magnetosphere of a magnetar. An additional motivation for our consideration of this model is that at least some FRBs show very rapid variability time as short as tens of micro-seconds (Farah et al., 2018; Prochaska et al., 2019; Cho et al., 2020), which corresponds to the light-crossing time of a few km and suggests that the radiation might be produced close to a neutron star²²2The transverse size of the source and the distance from the compact object is larger when the radiation is produced in a relativistic outflow moving toward the observer with a Lorentz factor $\gamma$ by a factor $\gamma$ and $\gamma^{2}$ respectively (e.g., Katz, 2019)..

The basic scenario we suggest is that a disturbance emanating from the NS surface spreads through the magnetosphere. The dissipation of the energy near the surface in the closed field line region produces X-ray emission. The disturbance propagating to distances much larger than the NS radius, above the magnetic poles, is converted into coherent radio waves (Fig. 2).

Let us first consider that the energy in the outburst near the surface of the NS is carried by a beam of $e^{\pm}$ pairs of isotropic equivalent luminosity $L_{\rm b}$ and Lorentz factor $\gamma_{\rm b}$. The $e^{\pm}$ number density at distance $R=10^{8}R_{8}\rm\,cm$ from the NS in the beam comoving frame is given by $n_{\rm b}\sim 3\times 10^{16}\mathrm{\,cm^{-3}}\,R_{8}^{-2}\gamma_{\rm b}^{-2}$ for $L_{\rm b}=4\pi R^{2}\gamma_{\rm b}^{2}n_{\rm b}m_{\rm e}c^{3}\sim 10^{38}\rm\,erg\,s^{-1}$, which is the minimum particle beam luminosity so as to generate the observed radio flux of FRB 200428. This corresponds to a plasma frequency of $\nu_{\rm p}\sim 10^{13}R_{8}^{-1}\mathrm{\,Hz}$ in the observer frame. Moreover, the cyclotron frequency is $\nu_{\rm B}\sim 3\times 10^{15}R_{8}^{-3}\mathrm{\,Hz}$ for surface dipolar magnetic field strength of $B_{\rm ns}=10^{15}\,\rm G$. Most maser processes resulting from an interaction between highly relativistic beam of particles and mildly or sub-relativistic plasma produce radiation near the plasma frequency or the appropriately Doppler shifted cyclotron frequency. The estimates for these frequencies show the difficulty for the particle beam based class of maser models to produce GHz radiation with the observed luminosity of FRB 200428.

We consider another possibility, that the energy released near the polar region of the NS is carried by magnetic disturbances – Alfvén waves – which damp far away from the surface, but well inside the light cylinder, and produce radio waves (Kumar & Bošnjak, 2020). Let us consider that the amplitude of the Alfvén wave at the NS surface is $\delta B$ and its transverse wavenumber is $k_{\perp}=2\pi/\lambda_{\perp}$, where $\lambda_{\perp}$ is the wavelength perpendicular to the NS magnetic field. Both $\delta B$ and $k_{\perp}$ decrease with radius as $R^{-3/2}$ as the wave packet follows the curved magnetic field lines and fans out such that its transverse size increases as $R^{3/2}$. The wave becomes charge starved at a radius $R$ where the plasma density is below the critical density

where $q$ is electron charge, $\delta B_{10}=\delta B/10^{10}\rm\,G$ and $\lambda_{\perp 4}=\lambda_{\perp}/10^{4}{\rm\,cm}$ are measured at the NS surface.

When the wave arrives at the charge starvation radius $R$, a strong electric field develops along the background magnetic field and accelerates clumps of particles that were formed due to two-stream instability associated with the Alfvén wave current density. These particle clumps move along curved field lines and produce coherent curvature radiation. The clumps that form due to two-stream instability have a broad spectrum of longitudinal sizes $\ell_{\parallel}\lesssim c/\nu_{\rm p}$ ($c$ being the speed of light), and radio emission is generated by those ones with $\ell_{\parallel}\simeq\lambda_{\rm frb}/2=15\nu_{9}^{-1}\rm\,cm$. The number of particles that can radiate coherently is $N_{\rm coh}\simeq\pi n_{\rm c}\ell_{\parallel}\ell_{\perp}^{2}$, where the transverse size is given by $\ell_{\perp}\simeq\sqrt{R\lambda_{\rm frb}}$ such that the photon arrival time does not differ by more than half an FRB wave period. This choice of $\ell_{\perp}$ is because the other two relevant length scales — the Alfvén transverse wavelength $\lambda_{\perp}$ and the causal length $R/\gamma$ — are typically much longer than $\sqrt{R\lambda_{\rm frb}}$. The clump Lorentz factor $\gamma$ is related to the characteristic frequency of curvature emission $\nu=3\gamma^{3}c/(4\pi R_{\rm B})$ and the curvature radius of magnetic field lines $R_{\rm B}$,

The total luminosity is $N_{\rm coh}^{2}$ times the curvature luminosity $L_{\rm curv}\simeq 16\gamma^{8}q^{2}c/3R_{\rm B}^{2}$ from an individual particle, provided that the observer is located within the relativistic beaming cone (of angular size $\sim\gamma^{-1}$), so we obtain

where we have denoted the magnetic colatitude of the field line on the NS surface as $\theta=10^{-1.5}\theta_{-1.5}\rm\,rad$ and the corresponding curvature radius for a dipolar geometry is $R_{\rm B}\simeq 0.8(R_{\rm ns}/\theta)(R/R_{\rm ns})^{1/2}$.

The luminosity is mainly set by the charge starvation radius $R$, and the initial amplitude $\delta B$ as well as the transverse wavelength $\lambda_{\perp}$ of the Alfvén waves. Our poor understanding of the charge density profile of the magnetosphere does not allow us to directly determine $R$. Generally, Alfvén waves launched near the magnetic poles where field lines extend to large distances are much more likely to become charge starved and produce coherent radiation. Here, we can use observed luminosity of FRB 200428, $L_{\rm frb}\sim 3\times 10^{38}\rm\,erg\,s^{-1}$ (assuming a distance of $\sim$$10\rm\,kpc$ and frequency bandwidth of $\Delta\nu\sim 1.4\rm\,GHz$), to constrain $R/R_{\rm ns}\sim 20\,(\delta B_{10}/\lambda_{\perp 4})^{6/11}$.

The spectrum of the emergent radio waves depends on the size distribution of particle clumps and their Lorentz factors. The emergent power at frequency $\nu$ depends on the Fourier transform of particle number density $\tilde{n}(k)$ at wave number $k\sim 2\pi\nu/c$, and the distribution of particle Lorentz factor ($\gamma$) on this scale. We note that the transverse size of the coherent patch $\ell_{\perp}$ is typically much smaller than the causal length $R/\gamma$, which means that Doppler effect only slightly broadens the spectrum by $\Delta\nu/\nu\simeq(\gamma\ell_{\perp}/R)^{2}\sim 0.1$, and the spectrum can have large intrinsic variations over a narrow band as radiation arrives from different clumps at different observer time.

The FRB emission is produced at a radius $R\sim 20R_{\rm ns}$, as described above, with an uncertainty by a factor of a few. Thus, Alfvén waves should be launched within the polar angle $\theta\lesssim 0.1\rm\,rad$ in order to ensure that these waves are able to propagate out to $\sim 10^{2}R_{\rm ns}$ and pass through charge starvation point. Furthermore, $\theta$ cannot be much smaller than $0.02\rm\,rad$ because otherwise the beaming cone of field lines at $\sim 20R_{\rm ns}$ would rotate outside observer line of sight in 30 ms³³3It might be tempting to consider the possibility that the two radio pulses separated by 30 ms were in fact due to one continuous event that produced a hollow cone of radio emission, and the two pulses corresponded to the sweep of the cone across the line of sight as the NS rotated. However, two hard X-ray pulses also separated by $\sim 30$ ms cast doubt on this possibility, since it requires that the hard X-ray emission is also beamed into the same hollow cone as the radio emission whereas the softer X-rays were presumably not beamed. and the second radio pulse seen from FRB 200428 would have been missed. These constraints on the magnetic colatitude motivates the choice of $\theta=0.03\rm\,rad$ as our fiducial value in eq. (3). All things being similar for different X-ray bursts from SGR 1935+2154, we expect to see one FRB for $\sim 10^{2}$ X-ray bursts (this seems consistent with the available data for this object, Lin et al., 2020b). If the Alfvén wave packet has an azimuthal angular span of $\delta\phi\sim 1\rm\,rad$, then the solid angle of FRB emission at the charge starvation radius is $\Omega_{\rm frb}\sim\delta\phi\,\theta^{2}(R/R_{\rm ns})\sim 10^{-2}\rm\,sr$. The beaming fraction of $\Omega_{\rm frb}/4\pi\sim 10^{-3}$ is consistent with that inferred from the volumetric rate of X-ray bursts and FRBs in §4.

In Fig. 3, we show the solutions for different FRBs with a wide range of luminosities, along with a number of physical constraints on the charge starvation radius and the initial amplitude of Alfvén disturbance. For simplicity, we fix $B_{\rm ns}=3\times 10^{14}\rm\,G$, $P=3\rm\,s$, $\theta=10^{-1.5}\rm\,rad$, and $\nu_{\rm frb}=1\rm\,GHz$. The biggest uncertainty lies on $\lambda_{\perp}$, the transverse wavelength of the Alfvén waves on the NS surface, which depends on how the initial disturbance is launched. For $\lambda_{\perp}=10^{4}\rm\,cm$, our model predicts FRB luminosities in the range $10^{36}$ to $10^{48}\rm\,erg\,s^{-1}$ and hence provides a viable explanation for faint bursts like FRB 200428 as well as bright events like FRB 190523 (Ravi et al., 2019). The maximum luminosity is due to the wave electric field, parallel to the magnetic field, at the charge starvation radius exceeding the Schwinger limit (Lu & Kumar, 2019). We also predict that the FRB luminosity function must have a (so-far unobserved) flattening at the lower end, although the exact minimum luminosity $L_{\rm min}$ depends on the unknown $\lambda_{\perp}$. This is because, for very small initial Alfvén amplitude, charge starvation occurs far away from the NS surface where the plasma frequency is below the GHz band, and in this case all charge clumps have longitudinal sizes $\ell_{\parallel}>\lambda_{\rm frb}$ and hence the coherent emission at GHz frequencies is strongly suppressed. When the line of sight is outside the beaming cone of angular size $\sim\gamma^{-1}$, the observed luminosity is heavily suppressed by relativistic effects and hence may be below $L_{\rm min}$, but the chance of detection is very small.

What fraction of energy in this event reached near the magnetic poles and contributed to FRB emission? Suppose initially the outburst started far away from the magnetic pole, since most free energy in the tangled magnetic fields is near the equator (Thompson et al., 2002; Gourgouliatos et al., 2016). Crustal deformations during the flare excite seismic oscillations, preferentially toroidal shear modes which preserve the shape of the star (Duncan, 1998; Piro, 2005), and the disturbance propagates along the crust to other parts of the star. Due to the small thickness of the crust $h\sim 0.5$–$1\rm\,km$, the wave undergoes many reflections off the surface before reaching the polar region. The distance traveled by the wave between two consecutive reflections is $\ell\sim\sqrt{hR_{\rm ns}}$, and the minimum number of reflections between the trigger to the magnetic pole is $\pi R_{ns}/2\ell\sim 4$. The FRB duration is given by propagation delay between different paths $t_{\rm frb}\sim\ell/v_{\rm s}\sim 1\rm\,ms$ for wave speed $v_{\rm s}\sim 0.01c$. Each time the waves reach the surface, high-frequency ($\gg 10^{4}\rm\,Hz$) Fourier components is largely transmitted into magnetospheric Alfvén modes (Blaes et al., 1989). The Alfvén waves launched at $\theta\gtrsim 0.1\rm\,rad$ are trapped in closed field lines, cascade to smaller scales, and create an $e^{\pm}$-photon plasma that radiates most of the energy as X-rays. For low-frequency seismic components $\lesssim 10^{4}\rm\,Hz$, since the corresponding Alfvén wavelength is $\gtrsim 3R_{\rm ns}$, their transmission to the magnetosphere preferentially occurs near the poles where the magnetic field lines are sufficiently extended (Thompson & Duncan, 1995). The energy per unit surface area transmitted into the magnetospheric Alfvén waves in the polar region can be estimated to be $F_{\rm A}/F\sim T(1-T)^{N_{\rm r}}h/R_{\rm ns}$, where the fluence normalization $F=E/4\pi R_{\rm ns}^{2}$ is from uniformly distributing the total energy over the NS surface, $T$ is the transmission coefficient from crustal shear waves to Alfvén waves, and $N_{\rm r}\sim 5$ is the typical number of reflections. The frequency spectrum of seismic oscillations and their propagation properties are still highly uncertain. For $0.03\lesssim T\lesssim 0.5$ (Blaes et al., 1989; Bransgrove et al., 2020), we roughly estimate $F_{\rm A}/F$ to be of order $10^{-3}$. Following the field lines from the NS surface to the charge starvation radius, the energy per solid angle drops by another factor of $R_{\rm ns}/R\sim 0.1$. Assuming a fraction of order unity of the Alfvén luminosity is converted into coherent radio emission, we expect the FRB to X-ray luminosity ratio to be of order $10^{-4}$, which is in rough agreement with observed fluence ratio between these two bands.

The first Galactic FRB from a magnetar, with its associated X-ray counterpart, provides an extraordinary opportunity to understand the FRB phenomenon as a whole. We explore the implications of FRB 200428 in various aspects. We find that FRB 200428-like events likely contribute a significant fraction of the cosmological FRB rate function at the faint end near specific energy $E_{\nu}\sim 10^{26}\rm\,erg\,Hz^{-1}$. We compared SGR 1935+2154 with the sources of other active repeaters (e.g., FRB 121102) and discuss how they may be understood in a general framework of the magnetar progenitors from different formation channels. Then, we compare the rates of SGR X-ray bursts and FRBs and find that only a small fraction (of order $10^{-3}$–$10^{-2}$) of X-ray bursts may be accompanied by FRBs.

We consider two broad classes of FRB emission mechanisms. First, the “far-away” models describe that a relativistic outflow drives a shock into the surrounding medium at large distances and generates radio emission by a plasma maser process. We carried out a detailed analysis of these models and found a number of difficulties explaining the radio and X-ray data from FRB 200428. The second class are the “close-in” models where radio emission is generated by a coherent process within the NS magnetosphere. We propose a scenario that magnetic disturbance near the stellar surface propagates to larger radii in the form of Alfvén waves which then damp and produce radio emission. FRB 200428 was associated with an X-ray burst, and the hard X-ray lightcurve had two prominent spikes that occurred at nearly the same time as the two FRB pulses. The coincidence of hard X-ray and radio peaks and their relative fluxes can be understood in this scenario. This model provides a unified picture for faint bursts like FRB 200428 as well as the bright events seen at cosmological distances.

The data underlying this article will be shared on reasonable request to the corresponding author.

WL thank Shri Kulkarni for his encouragement throughout this project. This work has been funded in part by an NSF grant AST-2009619. WL was supported by the David and Ellen Lee Fellowship at Caltech.