Implications of a Fast Radio Burst from a Galactic Magnetar

The Galactic soft gamma repeater (SGR) SGR 1935+2154 was first discovered by the Neil Gehrels Swift Observatory's Burst Alert Telescope (BAT) as a gamma-ray burst (GRB) candidate through a series of soft bursts from the Galactic plane (Lien et al. 2014; Stamatikos et al. 2014). The source is associated with supernova remnant (SNR) G57.2+0.8 (Gaensler 2014). Distance estimates are uncertain, ranging from 4.4 to 12.5 kpc (Sun et al. 2011; Pavlović et al. 2013; Kothes et al. 2018; Mereghetti et al. 2020a; Zhou et al. 2020), and throughout this paper, we adopt a distance of $d={d}_{10}\,10\,\mathrm{kpc}$. Subsequent discovery of the coherent X-ray pulsations of SGR 1935+2154 with the Chandra X-ray Observatory established the spin period of the magnetar, $P\,\simeq 3.2\,{\rm{s}}$ (Israel et al. 2014, 2016). Observations of the source with XMM-Newton and NuSTAR during outburst in 2015 provided the magnetar's spin-down rate, $\dot{P}\simeq 1.43\,\times {10}^{-11}\,{\rm{s}}\,{{\rm{s}}}^{-1}$, which implies a surface dipolar magnetic field $B\simeq 2.2\times {10}^{14}\,{\rm{G}}$, spin-down luminosity ${L}_{\mathrm{sd}}\simeq 1.7\times {10}^{34}$ erg s⁻¹, and characteristic spin-down age $P/2\dot{P}\,\simeq 3600$ yr.

We note, however, that the age estimate based on the SNR association, $\gtrsim 16$ kyr, is significantly older (Zhou et al. 2020; see also Kothes et al. 2018). This discrepancy between the dipolar and SNR age estimates is similar to the one observed in the other magnetar associated with an SNR, Swift J1834.9–0846 (Granot et al. 2017; with a spin-down age of 4.9 kyr and an SNR age between 5 and 100 kyr). Since the dipolar age estimate is expected to be inaccurate in case either the surface magnetic field evolves with time (e.g., Colpi et al. 2000; Dall'Osso et al. 2012; Beniamini et al. 2019) or the spin evolution is not dominated by dipolar radiation (Thompson & Blaes 1998; Harding et al. 1999; Beniamini et al. 2020), Granot et al. (2017) concluded that for Swift J1834.9–0846, the SNR age is likely to be the more realistic case. In this case, both Swift J1834.9–0846 and SGR 1935+2154 are significantly (by a factor of ∼4–10) older than the majority of the observed Galactic magnetar population (Beniamini et al. 2019).

Radio observations with the Parkes telescope at 1.5 and $3\,\mathrm{GHz}$ did not reveal any significant radio pulsations from SGR 1935+2154 to a limiting flux of 0.1 and $0.07\,\mathrm{mJy}$, respectively (Burgay et al. 2014). Observations with NCRA/GMRT and ORT at ∼300 and 600 MHz also found no radio pulses down to 0.4 and $0.2\,\mathrm{mJy}$, respectively (Surnis et al. 2014, 2016), followed by additional 14 and 7 μJy limits by Arecibo (at 4.6 and $1.4\,\mathrm{GHz};$ Younes et al. 2017).

The nondetection of periodic radio pulses from the source has not impaired SGR 1935+2154 from being a prolific X-ray burster. This magnetar has been active since its discovery, with at least four outbursts on 2014 July 5, 2015 February 22, 2016 May 14, and 2016 June 18, each with an increasing number of bursts extending to higher energies (Lin et al. 2020a). During its active outburst cycles, SGR 1935+2154 exhibited a remarkably bright burst on 2015 April 12, detected by the four Interplanetary Network (IPN) spacecraft (Kozlova et al. 2016). This burst's long duration of $\sim 1.7\,{\rm{s}}$ and large fluence of $\sim 2.5\times {10}^{-5}$ erg cm⁻² (radiated energy $\sim {10}^{41}$ erg) place it in the rare class of "intermediate" SGR flares (Mazets et al. 1999; Feroci et al. 2003; Mereghetti et al. 2009; Göǧüś et al. 2011).

The trend that the bursts from SGR 1935+2154 have become progressively more energetic in the years since its initial discovery (Younes et al. 2017) persists with the recent outburst, which is the most energetic yet (Mereghetti et al. 2020a; Zhang et al. 2020a). Similar behavior was previously observed for SGR 1806–20, which culminated with its production of the most energetic magnetar giant flare seen to date (Younes et al. 2015).

On 2020 April 10, a short soft X-ray burst was triangulated by the IPN to SGR 1935+2154 (Svinkin et al. 2020). This was followed by a slew of bursts, extending to hard X-rays, detected over the following couple of weeks (Borghese et al. 2020; Cherry et al. 2020; Hurley et al. 2020; Kennea et al. 2020; Lipunov et al. 2020; Marathe et al. 2020; Mereghetti et al. 2020b; Palmer 2020; Ridnaia et al. 2020a, 2020b; Ricciarini et al. 2020; Ridnaia et al. 2020c, 2020d; Veres et al. 2020; Younes et al. 2020).

On April 28, as part of this period of enhanced source activity, a bright millisecond radio burst, the first of its kind, was detected from SGR 1935+2154 (Bochenek et al. 2020a, 2020c; Scholz & CHIME/FRB Collaboration 2020; CHIME/FRB Collaboration et al. 2020b). The radio burst was associated with a short hard X-ray burst (Mereghetti et al. 2020b) peaking at energies ${E}_{\mathrm{peak}}\sim 80\,\mathrm{keV}$ (Li et al. 2020; Mereghetti et al. 2020a; Ridnaia et al. 2020e; Tavani et al. 2020). The detection by the CHIME/FRB back end in the 400–800 MHz band comprises two subburst components. The bursts, each ∼0.5 ms wide and separated by ∼30 ms, had a reported DM of $332.7\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ (CHIME/FRB Collaboration et al. 2020b). This DM value is consistent with the observed $\approx 8.6\,{\rm{s}}$ delay of the radio burst (at 400 MHz) with respect to the peak of the X-ray counterpart flare as being almost entirely due to the cold plasma time delay (Mereghetti et al. 2020a, 2020b; Zhang et al. 2020b). An independent detection of the burst was also reported from the STARE2 radio feeds at the 1.4 GHz band (Bochenek et al. 2020a, 2020c). They reported the burst arrival time and DM value to be consistent with the CHIME detection and constrained the peak fluence to be $\approx 1.5\,\mathrm{MJy}\,\mathrm{ms}$.

The compelling nature of this burst led to a search for track-like muon neutrino events with the IceCube observatory, with no significant neutrino signals detected along its direction (Vandenbroucke 2020). Likewise, Very Large Array follow-up of the source found no persistent or afterglow radio emission down to a flux of ~50 μJy (Ravi et al. 2020a, 2020b).

The millisecond-duration high brightness temperature radio burst of SGR 1935+2154 is unlike any other pulsar/magnetar phenomenology observed to date, with a luminosity exceeding that of even the most luminous Crab giant pulses (e.g., Mickaliger et al. 2012) by several orders of magnitude (Bochenek et al. 2020c; CHIME/FRB Collaboration et al. 2020b). Instead, the burst properties are suggestively similar to cosmological FRBs. As pointed out by Bochenek et al. (2020a), placed at the distance of the nearest localized FRB 180916, $\simeq 149\,\mathrm{Mpc}$ (Marcote et al. 2020), the SGR 1935+2154 outburst would have been potentially detectable as a $\gtrsim 7\,\mathrm{mJy}\,\mathrm{ms}$ burst, coming close to, albeit still lower than, typical FRB fluences. Stated energetically, SGR 1935's (isotropic-equivalent) emitted radio energy is ${E}_{\mathrm{radio}}\,\approx 4\times {10}^{34}{d}_{10}^{2}\,\mathrm{erg}$, where we have normalized to a distance of $10\,\mathrm{kpc}\times {d}_{10}$ and adopted the $1.5\,\mathrm{MJy}\,\mathrm{ms}$ fluence and ${BW}\approx 250\,\mathrm{MHz}$ bandwidth of STARE2 (Bochenek et al. 2020b, 2020c).⁸ The radio burst of SGR 1935+2154 is thus within a factor of ∼10 of the lowest-energy burst observed from any cosmological FRB of known distance to date, $\approx 5\times {10}^{35}\,\mathrm{erg}$ (Marcote et al. 2020). The immediate implication is that magnetar bursts akin to the SGR 1935+2154 April 28 event may be contributing to the extragalactic FRB population (Bochenek et al. 2020c; CHIME/FRB Collaboration et al. 2020b; see also Figure 3 and Section 4).

The fact that this Galactic FRB was associated with a contemporaneous X-ray counterpart exhibiting similar subburst temporal structure is an important diagnostic fact that we return to throughout this work. The radiated energy in this X-ray burst is ${E}_{{\rm{X}}}\approx 8\times {10}^{39}{d}_{10}^{2}\,\mathrm{erg}$, where we have adopted the $7\,\times {10}^{-7}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}$ fluence reported by the Hard X-ray Modulation Telescope (HXMT; Li et al. 2020). A similar fluence was also reported by INTEGRAL (Mereghetti et al. 2020a), AGILE (Tavani et al. 2020), and Konus-Wind (Ridnaia et al. 2020e). The ratio of energy radiated in the radio and X-ray bands, or the radio "efficiency," is therefore

$$\begin{eqnarray}&&\eta \equiv \displaystyle \frac{{E}_{\mathrm{radio}}}{{E}_{{\rm{X}}}}\sim {10}^{-5}.\end{eqnarray} \tag{ 1 }$$

This single observation has dramatic implications. In the following section (Section 3), we use this property in discussing whether previous magnetar X-ray bursts may have produced similar FRBs that had gone undetected. In Section 4 we discuss the implications of the radio inefficiency implied by Equation (1) to the population of extragalactic FRBs. Finally, in Section 5 we show that the value of η can serve as a diagnostic differentiating between theoretically proposed FRB models.

Given the 3.6 sr field of view of STARE2 (Bochenek et al. 2020b) and the fact that a single magnetar radio burst has been detected during the ∼300 day operating period of the experiment, we estimate the rate of SGR 1935+2154–like magnetar radio bursts (taking the number of active magnetars in the Galaxy to be N = 29; Olausen & Kaspi 2014) to be ${\lambda }_{\mathrm{mag}}\in [0.36,80]\times {10}^{-2}$ magnetar^–1 yr^–1 at 95% confidence (assuming Poisson statistics; Gehrels 1986).¹¹

The radio burst activity (repetition) rate of SGR 1935 estimated above is plotted in Figure 3 in comparison to cosmological FRBs. We show here the full sample of published localized FRB sources, where the radio-emitted (isotropic-equivalent) energy can be reliably calculated: the repeating FRB 121102 (yellow; Law et al. 2017; Gourdji et al. 2019), the recently localized CHIME repeater FRB 180916 (purple; Marcote et al. 2020; CHIME/FRB Collaboration et al. 2020a), and the apparently nonrepeating FRBs 180924 (green, upper limits denoted by upside-down triangles; Bannister et al. 2019), 190523 (brown; Ravi 2019), and 181112 (blue; Prochaska et al. 2019). We have used quoted rates where possible (Law et al. 2017; Gourdji et al. 2019; CHIME/FRB Collaboration et al. 2020a) and otherwise estimated Poissonian rates based on quoted field exposure times where available. Gray points with error bars show the repetition rate (from Tendulkar et al. 2016) and radio energy implied if some giant magnetar flares produce radio emission with the same efficiency $\eta \sim {10}^{-5}$ between X-ray and radio fluences. This indicates that Galactic magnetars may be energetically capable of powering even the most luminous FRBs, though for one particular flare from SGR 1806–20, such radio emission is ruled out (Tendulkar et al. 2016). Finally, we note that radio pulses observed from M87 by Linscott & Erkes (1980) ∼40 yr ago, though unconfirmed by subsequent follow-up, may have been the earliest recorded FRB detections and correspond to ${E}_{\mathrm{radio}}\sim {10}^{38}\,\mathrm{erg}$ in this figure.

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Figure 3 shows that, at comparable energy, the SGR 1935 radio burst is $\sim {10}^{-5}$ less frequent than the FRB 180916 bursts. Within the hypothesis that magnetars are also the progenitors of such cosmological FRBs, this stark discrepancy in activity rate may be attributable to the magnetar age and internal field strength. Indeed, Margalit et al. (2019) showed that magnetar activity scales strongly with magnetic field, $\dot{E}\propto {B}^{3.2}$ (Dall'Osso et al. 2012; Beloborodov & Li 2016).¹² This is shown schematically with the dotted blue contours in Figure 3, assuming that $\lambda (\gt E)E\sim \dot{E}\propto {B}^{3.2}$, and scaling the B field from $\sim 2\times {10}^{14}\,{\rm{G}}$, the external dipole field of SGR 1935+2154 (note that the internal field is what sets $\dot{E}$, and thus we implicitly assume that the internal field is comparable to the external dipole field for this object). This assumed B field implies an active lifetime of $\sim 70\,\mathrm{kyr}$ for SGR 1935 (e.g., Margalit et al. 2019, their Equation (1)), broadly consistent with the $\gtrsim 16\,\mathrm{kyr}$ source age. Contours of active lifetime (∼magnetar age) are also shown in Figure 3 (gray shaded regions), scaling from SGR 1935's estimated age as ${\tau }_{\mathrm{active}}\propto {B}^{-1.2}$ (Dall'Osso et al. 2012). The 5 orders of magnitude higher repetition rate of FRB 180916 would thus imply ${B}_{180916}\sim {({10}^{5})}^{1/3.2}{B}_{\mathrm{SGR}1935}\sim {10}^{16}\,{\rm{G}}$, in agreement with separate lines of argument, e.g., requirements for FRB 180916 periodicity to be attributable to magnetar precession (Levin et al. 2020). We note that although the $\dot{E}\propto {B}^{3.2}$ scaling may seemingly be in tension with the larger inferred surface fields of SGR 1806–20 and SGR 1900+14 compared to SGR 1935+2154, the relevant field in this scaling is instead the internal B field, which is not directly observable (magnetar activity is also highly stochastic, and the implied $\dot{E}$ should be treated only in a time-averaged sense).

An alternative possibility is that the repetition rate increases significantly with the periodicity (Wadiasingh et al. 2020). In this case, the periodicity of 180916 (and its high activity relative to SGR 1935+2154) may be ascribed to an extremely long period magnetar (Beniamini et al. 2020). Interestingly, this option also requires a large internal magnetic field, $\gt {10}^{16}\,{\rm{G}}$, at birth (see Beniamini et al. 2020 for details).

The maximum distance up to which SGR 1935+2154's radio flare would be detectable by typical FRB search facilities sensitive to ${F}_{\mathrm{lim}}\sim 1\,\mathrm{Jy}\,\mathrm{ms}$ fluence radio pulses is ${D}_{\mathrm{lim}}\,\approx 12\,\mathrm{Mpc}\,{d}_{10}{({F}_{\mathrm{lim}}/1\mathrm{Jy}\mathrm{ms})}^{-1/2}$. The birth rate of ordinary Galactic magnetars in the local universe can be estimated as a fraction, ${f}_{\mathrm{CCSN}}\approx 0.1\mbox{--}1$ (Beniamini et al. 2019), of the CCSN rate, ${{\rm{\Gamma }}}_{\mathrm{CCSN}}\approx (0.71\pm 0.15)\times {10}^{-4}\,{\mathrm{Mpc}}^{-3}\,{\mathrm{yr}}^{-1}$ (Li et al. 2011). Thus, the rate of potentially detectable FRBs produced by SGR 1935+2154–like bursts is

$$\begin{eqnarray}\begin{array}{rcl}{ \mathcal R }(\gt {F}_{\mathrm{lim}}) & \approx & \displaystyle \frac{4\pi }{3}{D}_{\mathrm{lim}}^{3}{f}_{\mathrm{CCSN}}{{\rm{\Gamma }}}_{\mathrm{CCSN}}{\tau }_{\mathrm{active}}{\lambda }_{\mathrm{mag}}\\ & \in & \left[0.004,1.4\right]\,{\mathrm{sky}}^{-1}\,{\mathrm{day}}^{-1}\,\\ & & \times \,{d}_{10}^{3}\left(\displaystyle \frac{{f}_{\mathrm{CCSN}}}{0.1}\right)\left(\displaystyle \frac{{\tau }_{\mathrm{active}}}{{10}^{4}\,\mathrm{yr}}\right){\left(\displaystyle \frac{{F}_{\mathrm{lim}}}{1\mathrm{Jy}\mathrm{ms}}\right)}^{-3/2}.\end{array}\end{eqnarray} \tag{ 3 }$$

This is much lower than the Parkes estimated FRB rate, $\approx {1.7}_{-0.9}^{+1.5}\times {10}^{3}\,{\mathrm{sky}}^{-1}\,{\mathrm{day}}^{-1}$, above a fluence of $2\,\mathrm{Jy}\,\mathrm{ms}$ (Bhandari et al. 2018).

The estimate above shows that magnetars bursting at the same rate and energy as the April 28 SGR 1935+2154 radio burst cannot noticeably contribute to the FRB population. However, magnetar flares clearly span a range of energies (Figure 1), making it natural to ask whether SGR 1935+2154–like magnetars can reproduce the FRB rate if one extrapolates radio production with a similar efficiency to more energetic magnetar flares.

We therefore calculate the all-sky rate of FRBs above a limiting fluence threshold assuming that all magnetars repeat following a luminosity distribution ${\lambda }_{\mathrm{mag}}{(\gt E)\propto (E/{E}_{\min })}^{-\gamma }$ for burst energies between ${E}_{\min }$ and ${E}_{\max }$. We set ${E}_{\max }\,={\eta }_{\mathrm{mag}}{E}_{\mathrm{mag}}\propto {B}^{2}$ dictated by the magnetic energy reservoir of the magnetar (this is consistent with the energies of the three observed Galactic giant flares relative to the magnetic dipole energies of the magnetars producing them; see, e.g., Tanaka et al. 2007) and fix the minimum energy to that observed for the April 28 SGR 1935+2154 flare.¹³ In the following, we assume ${\eta }_{\mathrm{mag}}=\eta \sim {10}^{-5}$ (Equation (1)) is universal for all magnetar FRBs (motivated by the synchrotron maser model discussed in Section 5.2.2). The magnetar birth rate ${{\rm{\Gamma }}}_{\mathrm{birth}}(z)$ is assumed to follow the cosmic star formation rate (Hopkins & Beacom 2006) and is anchored to 10% of the CCSN rate at z = 0 (Li et al. 2011). The integrated rate is thus

$$\begin{eqnarray}\begin{array}{rcl}{ \mathcal R }(\gt {F}_{\mathrm{lim}}) & = & \displaystyle \int {dV}(z){{\rm{\Gamma }}}_{\mathrm{birth}}(z){\tau }_{\mathrm{active}}\\ & & \times \,\displaystyle \int {dE}{\lambda }_{\mathrm{mag}}(E){\rm{\Theta }}\left[E-4\pi {D}_{{\rm{L}}}{\left(z\right)}^{2}{F}_{\mathrm{lim}}\right],\end{array}\end{eqnarray} \tag{ 4 }$$

where dV(z) is the differential comoving volume element, ${D}_{{\rm{L}}}(z)$ is the luminosity distance, and ${\rm{\Theta }}(x)$ is the Heaviside function.

We set ${\tau }_{\mathrm{active}}=24\,\mathrm{kyr}$ and fix $\lambda (\gt {E}_{\min })$ to the estimated rate of SGR 1935 radio bursts. The former is determined from the minimum SNR age estimated by Zhou et al. (2020) and scaled to our adopted distance of $10\,\mathrm{kpc}$. We then integrate over cosmological redshift and calculate the rate as a function of the slope of the luminosity function, γ. Figure 4 shows the resulting all-sky rate as a function of γ (the only free variable in the calculation). For large γ, the rate is dominated by bursts of energy $\sim {E}_{\min }$, and we recover the result of Equation (3), namely, that the rate of Galactic magnetars is too low compared to cosmological FRBs. However, for $\gamma \lt 1.5$, the rate becomes dominated by the high-energy tail of the luminosity function, and the number of detectable magnetar radio bursts increases. In particular, we find that, for $\gamma \lesssim 1$, magnetars of a similar kind to SGR 1935+2154 can account for the entire FRB rate.

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We note that a low value of $\gamma \sim 0.7$ is consistent with estimates of Galactic magnetar X-ray burst fluences (e.g., Göǧüś et al. 1999, 2000; Gavriil et al. 2004) and also potentially with FRB 121102 (Law et al. 2017; although see Gourdji et al. 2019). Separately, a value of $\gamma \sim 0.6$ has been suggested based on the slope count of nonlocalized ASKAP FRBs, with the assumption of a DM–distance relationship (Shannon et al. 2018; Lu & Piro 2019).

Although extending the luminosity function of radio bursts to higher energies can allow (for certain values of γ) ordinary magnetars similar to SGR 1935+2154 to reproduce the number of observed FRBs, this scenario falls short of explaining many other observables. In particular, and as discussed previously, the per-source activity rate for such a population would be far too low to explain prolific repeaters like FRBs 121102 and 180926. For the same reason, most FRB sources would be detected only once, and the relative number of repeaters versus apparently nonrepeaters would be very small. The scenario would also predict average FRB source distances that are much nearer than localized sources or DM-estimated distances.

To further explore the requirements for possible magnetar progenitors of cosmological FRBs, we extend the calculation of the previous section and model a two-component magnetar population as necessitated by the observations: magnetars with low activity levels like SGR 1935+2154 and magnetars that are very active. A natural question this will allow us to address is whether the active population is consistent with an earlier evolutionary stage (a younger version) of the same SGR 1935+2154–like population or whether one requires a distinct population altogether (e.g., a rare subset of magnetars born through unique channels).

We again calculate the all-sky FRB rate using Equation (4), accounting for the two populations contributing to FRB production. As before, the only free parameter describing the ordinary magnetar population is the luminosity function slope γ. The second, "active" population is then scaled from the former population as a function of the internal magnetic field B and relative birth rate f_CCSN.¹⁴

The magnetic field of the magnetar enters in setting ${E}_{\max }\,\propto {B}^{2}$, the active lifetime of sources ${\tau }_{\mathrm{active}}\propto {B}^{-1.2}$ (Dall'Osso et al. 2012; Beloborodov & Li 2016), and the repetition rate ${\lambda }_{\mathrm{mag}}(\gt {E}_{\min })$. The latter is proportional to $\propto {B}^{3.2}$ for γ > 1 and $\propto {B}^{1.2+2\gamma }$ for $\gamma \lt 1$.¹⁵ We normalized ${\tau }_{\mathrm{active}}$ and ${\lambda }_{\mathrm{mag}}({E}_{\min })$ at the magnetic field of SGR 1935+2154 ($\simeq 2.2\times {10}^{14}\,{\rm{G}}$ ¹⁶ ) to the age ($\gtrsim 24{d}_{10}\,\mathrm{kyr}$) and radio repetition rate of SGR 1935. We then integrate Equation (4) and compare the resulting rate, a function of the three free parameters $\{B,\gamma ,{f}_{\mathrm{CCSN}}\}$, to the observed FRB rate (Bhandari et al. 2018). From the point of view of the all-sky rate alone, there exists a degeneracy between the number of FRB sources in the universe ($\propto {f}_{\mathrm{CCSN}}$) and the repetition rate of each source (a function of B). However, this degeneracy can be broken by constraining the observed repetition rate of prolific repeaters.

Figure 5 shows, for values of the magnetic field $B(\gamma ,{f}_{\mathrm{CCSN}})$ required to fit the observed FRB rate (and its uncertainties), the region in $\{\gamma ,{f}_{\mathrm{CCSN}}\}$ parameter space where the average repetition rate of the active magnetar population equals the observed repetition rate of FRBs 121102 and 180916. For this, we take a fiducial value ${\lambda }_{\mathrm{mag}}(\gt {10}^{38}\,\mathrm{erg})=1\,{\mathrm{hr}}^{-1}$ (see Figure 3); however, the uncertainties encompass a few orders of magnitude of leeway in this assumed value. On the basis of Figure 5, one can conclude that, regardless of the luminosity function slope γ, fitting both the all-sky FRB rate and the activity level of repeating FRBs requires a rare class of progenitors, f_CCSN ≪ 1. This is in line with many previous studies (e.g., Nicholl et al. 2017), and here we have extended these by adding the possibly inevitable (though likely subdominant) contribution of a second population of "normal magnetars" and utilizing scaling relations anchored to the new observations of SGR 1935+2154.

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The fact that f_CCSN ≪ 1 is required of the active population implies that these sources cannot be interpreted as younger incarnations of the SGR 1935+2154–like magnetar population as a whole, since this would require the same birth rate for both populations, i.e., ${f}_{\mathrm{CCSN}}\sim 0.1$. In the above, we have assumed isotropic radio emission, as we expect beaming to be modest at most (Section 4). If radio emission is significantly beamed, then the number of FRB-emitting magnetars in the universe would be larger.

To further compare the resulting population against observational constraints, we calculate the expected number of repeating and apparently nonrepeating FRB sources detected by a mock survey of this population. We assume a limiting fluence of ${F}_{\mathrm{lim}}=4\,\mathrm{Jy}\,\mathrm{ms}$ and average repeat-field exposure time of ${T}_{\exp }=40\,{\rm{h}}{\rm{r}}$, parameters motivated by the CHIME FRB survey. We then calculate the (Poissonian) number of sources for which only a single burst would be detected,

$$\begin{eqnarray}&&{N}_{{\rm{n}}{\rm{o}}{\rm{n}}{\rm{r}}{\rm{e}}{\rm{p}}}=\int dV(z){f}_{{\rm{m}}{\rm{a}}{\rm{g}}}{{\rm{\Gamma }}}_{{\rm{b}}{\rm{i}}{\rm{r}}{\rm{t}}{\rm{h}}}(z){\tau }_{{\rm{a}}{\rm{c}}{\rm{t}}{\rm{i}}{\rm{v}}{\rm{e}}}\mu {e}^{-\mu },\end{eqnarray} \tag{ 5 }$$

where

$$\begin{eqnarray}&&\mu (z)\equiv {\lambda }_{\mathrm{mag}}\left(\gt 4\pi {D}_{{\rm{L}}}{\left(z\right)}^{2}{F}_{\mathrm{lim}}\right){T}_{\exp },\end{eqnarray} \tag{ 6 }$$

and summing the contribution from both active and SGR 1935+2154–like populations. The number of sources classified as repeaters by the same survey is similarly calculated as

$$\begin{eqnarray}&&{N}_{\mathrm{rep}}=\int {dV}(z){f}_{\mathrm{mag}}{{\rm{\Gamma }}}_{\mathrm{birth}}(z){\tau }_{\mathrm{active}}\left[1-\displaystyle \frac{1}{2}{\rm{\Gamma }}\left(2,\mu \right)\right],\end{eqnarray} \tag{ 7 }$$

where ${\rm{\Gamma }}(2,\mu )$ is the incomplete gamma function.

Figure 6 shows N_rep and N_nonrep as a function of γ and for a representative value of ${f}_{\mathrm{CCSN}}={10}^{-4}$ that is consistent with the constraints on FRB 121102–like activity and the all-sky FRB rate (Figure 5). The number of detected repeating and nonrepeating sources can be compared to values from the CHIME FRB survey, shown as horizontal curves (V. Kaspi 2020, private communication). The figure shows that, for values 1 ≲ γ ≲ 1.5, the absolute number and relative ratio of repeating and nonrepeating FRBs can be reproduced simultaneously with the all-sky rate and per-source activity rate. For these values of γ, the rates of both repeaters and apparent nonrepeaters are dominated by the active population. At low values of γ, the observed FRBs are dominated by the less active ordinary magnetar population. This results in a significant reduction in the relative number of repeating versus nonrepeating sources that is inconsistent with observations. This substantiates our previous claim that a single population of (or, equivalently, a population dominated by) SGR 1935+2154–like magnetars cannot account for the number of known repeaters.

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Finally, using the cumulative distribution of detected events implied by Equations (5) and (7), we calculate the characteristic distances at which repeating and nonrepeating FRBs would be detected by this mock survey. The right panel of Figure 6 shows the median distance of detected repeating and apparently nonrepeating FRB sources as a function of γ. Confirmed repeaters are detected, on average, at a lower distance, broadly consistent with the $149\,\mathrm{Mpc}$ distance of the first localized CHIME repeater (and note that FRB 121102, at a much larger distance of $972\,\mathrm{Mpc}$, is detected only once by CHIME, consistent with the median distance of apparently nonrepeating sources detected by the mock survey).

As the model shows, many potential rare magnetar formation channels could, in principle, be consistent with the all-sky FRB and repeater rate constraints (Figure 5). One way to further break this degeneracy is via host galaxy demographics. A high-B (and hence potentially particularly slowly rotating; Beniamini et al. 2020) tail of the magnetar population should be formed in otherwise ordinary CCSNe and hence track star-forming galaxies almost exclusively. Superluminous supernovae (SLSNe) and long-duration GRBs (LGRBs) should originate predominantly (Fruchter et al. 2006; Lunnan et al. 2015; Blanchard et al. 2016), though not exclusively (e.g., Perley et al. 2017), in dwarf star-forming galaxies. By comparison, NS mergers, white dwarf–NS mergers, and accretion-induced collapse (AIC) should originate from a range of star-forming and non-star-forming galaxies (Margalit et al. 2019), weighted more toward massive galaxies than SLSNe/LGRBs. Attempts to perform an analysis along these lines are already underway (e.g., Margalit et al. 2019; Li & Zhang 2020), though it should be cautioned that without at least arcsecond localization, it is usually challenging to uniquely identify the host galaxy (Eftekhari et al. 2018), much less the local environment within the host galaxy, as it becomes available with VLBI localization (Bassa et al. 2017; Marcote et al. 2020).

In the low-twist models (Wadiasingh & Timokhin 2019; Wadiasingh et al. 2020), magnetic field dislocations and oscillations in the NS surface can lead to pair cascades, assuming that the background charge density is sufficiently low. The latter then result in coherent radio emission. Since it is the same dislocation that supposedly results in short X-ray bursts and (in some cases) FRBs, a prediction of this model is that all FRBs should be associated with short magnetar bursts (but not vice versa). This is consistent with the observations of SGR 1935. Another attractive feature of this model is that FRBs will typically be associated with older magnetars, consistent with the putative age of SGR 1935. However, the association with older ages in the model is due to the requirement of longer periods, while the period of SGR 1935+2154 is very typical as compared to other Galactic magnetars. Similarly, the model favors strong dipole field strengths, while the dipole field of SGR 1935+2154 does not appear to be exceptional relative to other magnetars.

In the low-twist model, both the radio emission and the X-rays arrive from the magnetosphere. As such, this model predicts a radius-to-frequency mapping leading to frequency selection, similar to the case in pulsars and potentially a polarization angle that is varying with time (as the magnetic field orientation changes relative to the observer). The X-ray burst associated with an FRB should exhibit a standard blackbody (or double blackbody) spectrum as seen in other short X-ray bursts.

In synchrotron maser models, a version of which was first proposed by Lyubarsky (2014), FRBs are created via the coherent synchrotron maser process that is naturally produced in magnetized relativistic shocks (Gallant et al. 1992; Plotnikov & Sironi 2019). Such shocks are expected to arise from relativistic flares that may be ejected during magnetar outbursts. A number of variants on the synchrotron maser model exist that differ regarding the nature of the upstream medium and the required shock properties; however, in all cases, the bursts are powered by tapping into a small fraction of the kinetic energy of the outflow and predict corresponding (though differing) high-frequency counterparts to FRBs.

5.2.1. Magnetar Wind Nebula (Lyubarsky 2014)

Lyubarsky (2014) proposed that FRB production occurs as the ultrarelativistic flare ejecta collides with the pulsar wind nebula. The radius of the termination shock is estimated to be

$$\begin{eqnarray}&&{r}_{{\rm{s}}}=\sqrt{\tfrac{{L}_{\mathrm{sd}}}{4\pi {pc}}},\end{eqnarray} \tag{ 8 }$$

where p is the pressure inside the nebula. Taking ${L}_{\mathrm{sd}}\simeq 1.7\,\times {10}^{34}$ erg s⁻¹ and $p\sim {10}^{-9}$ erg cm⁻³ (estimated from the energy of 10⁵¹ erg of a typical supernova and the observed $\sim 20\,\mathrm{pc}$ size of the SNR surrounding SGR 1935+2154; Kothes et al. 2018), we find ${r}_{{\rm{s}}}\approx 6\times {10}^{15}$ cm. The light-crossing timescale to this radius is ${r}_{{\rm{s}}}/c\sim 2$ days. However, because the flare ejecta and the resulting shock are also moving close to the speed of light, radio photons from the shocked nebula could, in principle, still arise nearly simultaneously with the X-rays, which in this model presumably must be generated from the inner magnetosphere.¹⁷ If the X-ray photons are emitted within the magnetosphere concurrently with the relativistic flare ejection, the X-rays would be expected to arrive a short time, $\sim {r}_{{\rm{s}}}/2{{\rm{\Gamma }}}^{2}c\sim \mathrm{ms}$, before the radio, which is potentially in tension with the timing estimated by Mereghetti et al. (2020a). However, we note that various uncertainties in this timing may affect this conclusion (see Section 5.2.2 for further details).

Looking more closely at the predictions for the radio emission requires rescaling the results of Lyubarsky (2014), who considered a giant flare that carries away a significant fraction of the magnetic energy of the star. The recent flare from SGR 1935+2154 was less energetic by a factor of $\sim {10}^{6}$, corresponding to a strength of the magnetic field in the pulse smaller by a factor of $\sim {10}^{3}$, i.e., dimensionless constant $b\sim {10}^{-3}$ in the notation of Lyubarsky (2014). Following Equation (8) of Lyubarsky (2014) for these parameters, the predicted peak frequency of the maser emission from the forward shock is estimated to be ${\nu }_{\mathrm{pk}}\sim 10\,\mathrm{MHz}$. Given the drop-off in the $\nu {L}_{\nu }$ spectrum of the maser from ${\nu }_{\mathrm{pk}}$ to $\sim 100{\nu }_{\mathrm{pk}}$ by a factor of $\sim {10}^{-4}$ (Plotnikov & Sironi 2019), the fraction of the total radio emission in the 1.4 GHz band of STARE2 would be only $\sim {10}^{-4}$. Given an intrinsic efficiency of the synchrotron maser emission of ${f}_{\xi }\sim {10}^{-2}\mbox{--}{10}^{-3}$ (see next section), the resulting net efficiency of the radio emission of $\sim {10}^{-6}\mbox{--}{10}^{-7}$ is at best marginally consistent with the observations if the energy of the flare ejecta were comparable to the released X-ray fluence. The fact that the second radio burst detected by CHIME appears to have a rising spectrum is also in tension with this model (since ${\nu }_{\mathrm{pk}}\sim 10\,\mathrm{MHz}$, as discussed above), although scintillation may affect the observed spectrum (Simard & Ravi 2020).

5.2.2. Baryonic Shell (Metzger et al. 2019; Margalit et al. 2020)

In this version of the synchrotron maser model, first proposed by Beloborodov (2017), the ultrarelativistic head of the magnetar flare collides not with the magnetar wind nebula but rather with matter ejected from a recent, earlier flare. Motivated by the inference from the radio afterglow of the 2004 giant flare from SGR 1806–20 of a slow ejecta shell generated by the burst (Gelfand et al. 2005; Granot et al. 2006), Metzger et al. (2019) considered the upstream medium to be a subrelativistic baryon-loaded shell with an electron–ion composition.

The low efficiency of radio emission implied that the X-ray and radio observations of SGR 1935+2154 (Equation (1)) are consistent with the predictions of this model. The radio inefficiency is attributable to a combination of the intrinsic synchrotron maser efficiency ${f}_{\xi }\sim {10}^{-2}\mbox{--}{10}^{-3}$ (for moderate magnetization; Plotnikov & Sironi 2019) and a further reduction by a factor of $\sim {10}^{-2}$ due to the effects of induced Compton scattering suppressing the low-frequency portion of the maser's intrinsic SED (see Metzger et al. 2019, their Section 3.2).¹⁸

Following the methodology of Margalit et al. (2020), we can use the energy, frequency, and duration of the observed radio burst to derive the intrinsic parameters of the flare demanded by the synchrotron maser model. Adopting the observed quantities from Section 2, we find that the energy of the relativistic flare, ${E}_{\mathrm{flare}};$ the Lorentz factor Γ of the shocked gas at the time the observed radio flux is emitted; the radius of the shock from the central magnetar at this time, ${r}_{\mathrm{sh}};$ and the external density of the upstream baryonic shell at this location, ${n}_{\mathrm{ext}}$, are given by

$$\begin{eqnarray}&&{E}_{\mathrm{flare}}\approx 7\times {10}^{39}\,\mathrm{erg}\,{f}_{\xi ,-3}^{-4/5}{d}_{10}^{2},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{\rm{\Gamma }}\approx 56\,{f}_{\xi ,-3}^{-1/15}{d}_{10}^{1/3},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{n}_{\mathrm{ext}}\approx 4\times {10}^{4}\,{\mathrm{cm}}^{-3}\,{f}_{\xi ,-3}^{-4/15}{d}_{10}^{-2/3},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{r}_{\mathrm{sh}}\approx 1.1\times {10}^{11}\,\mathrm{cm}\,{f}_{\xi ,-3}^{-2/15}{d}_{10}^{2/3}.\end{eqnarray} \tag{ 13 }$$

In the above, we adopt the STARE2 fluence and burst duration (Bochenek et al. 2020b) and express the results as a function of the maser efficiency ${f}_{\xi }={10}^{-3}{f}_{\xi ,-3}$.

From our inferred parameters, the (very) local DM contributed by the immediate upstream medium ahead of the shock is $\mathrm{DM}\gtrsim {n}_{\mathrm{ext}}{r}_{\mathrm{sh}}\approx {10}^{-3}\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ and can exceed this value significantly if the upstream medium extends to larger radii. In the context of the shock model, it may be expected that DM variations could exist between radio bursts on this order of magnitude or larger. Note, however, that nonlinear wave interaction with the upstream plasma, as well as strong upstream magnetic fields, may inhibit the DM of this local environment (e.g., Lu & Phinney 2020).

The derived shock properties are shown in Figure 7 in comparison to those derived for cosmological FRBs within the same model for an assumed efficiency ${f}_{\xi }={10}^{-3}$. One important thing to note is that the flare energy ${E}_{\mathrm{flare}}$ that the model demands (based on the radio observation of SGR 1935+2154 alone) agrees remarkably well with the independently observed X-ray energy, ${E}_{{\rm{X}}}\approx 8\times {10}^{39}\,\mathrm{erg}$. As we discuss below, such an agreement is naturally expected if electrons heated at the shock generate the X-rays via synchrotron radiation in the fast-cooling regime.

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Stated more directly, the model predicts a ratio (Margalit et al. 2019),

$$\begin{eqnarray}\begin{array}{rcl}{\eta }_{\mathrm{theory}} & \equiv & \displaystyle \frac{{E}_{\mathrm{radio}}}{{E}_{\mathrm{flare}}}={\left(\displaystyle \frac{1215}{512{\pi }^{2}}\displaystyle \frac{{m}_{{\rm{e}}}}{{m}_{{\rm{p}}}}\right)}^{1/5}{f}_{{\rm{e}}}^{1/5}{f}_{\xi }^{4/5}{\left({\nu }_{\mathrm{obs}}\cdot {t}_{\mathrm{FRB}}\right)}^{-1/5}\\ & \sim & 3\times {10}^{-5}{\left(\displaystyle \frac{{f}_{{\rm{e}}}}{0.5}\right)}^{1/5}{f}_{\xi ,-3}^{4/5}{\left(\displaystyle \frac{{\nu }_{\mathrm{obs}}\cdot {t}_{\mathrm{FRB}}}{1\mathrm{GHz}\cdot \mathrm{ms}}\right)}^{-1/5},\end{array}\end{eqnarray} \tag{ 14 }$$

that matches the observed ratio, $\eta \equiv {E}_{\mathrm{radio}}/{E}_{{\rm{X}}}\sim {10}^{-5}$ (Equation (1)), for the expected values of the maser efficiency ${f}_{\xi }\sim {10}^{-3}\mbox{--}{10}^{-2}$ (Plotnikov & Sironi 2019), provided that ${E}_{{\rm{X}}}\sim {E}_{\mathrm{flare}}$. In the above, ${f}_{{\rm{e}}}$ is the ratio of electron to ion number densities in the upstream medium, and ${f}_{{\rm{e}}}\sim 1$ for an electron–ion plasma we consider.

The same outwardly propagating shock that generates the coherent precursor radio emission (the FRB) in this scenario also generates incoherent synchrotron radiation from relativistically hot electrons heated by the shock (Lyubarsky 2014; Metzger et al. 2019), somewhat akin to a GRB afterglow. Using the shock parameters implied by the radio observations (Figure 7), we now assess the predicted properties of the high-frequency counterpart, showing it to be in accord with the X-ray emission from SGR 1935+2154.

The peak frequency of the "afterglow" is set by the characteristic synchrotron frequency, which, for an ultrarelativistic blast wave decelerating into an effectively stationary upstream medium of magnetization $\sigma ={10}^{-1}{\sigma }_{-1}$, is given by (Metzger et al. 2019; their Equations (56) and (57))

$$\begin{eqnarray}\begin{array}{rcl}{E}_{\mathrm{peak}} & \sim & h{\nu }_{\mathrm{syn}}\approx 110\,\mathrm{keV}\ {\left(\displaystyle \frac{{E}_{\mathrm{flare}}}{{10}^{39}\mathrm{erg}}\right)}^{1/2}{\sigma }_{-1}^{1/2}{\left(\displaystyle \frac{t}{3\mathrm{ms}}\right)}^{-3/2}\\ & \approx & 290\,\mathrm{keV}\,{\sigma }_{-1}^{1/2}\ {\left(\displaystyle \frac{t}{3\mathrm{ms}}\right)}^{-3/2}{f}_{\xi ,-3}^{-2/5}{d}_{10},\end{array}\end{eqnarray} \tag{ 15 }$$

where t is the time since the peak of the flare, and we have assumed that half of the kinetic power dissipated at the shock goes into heating electrons into a relativistic Maxwellian distribution (supported, e.g., by particle-in-cell simulations of magnetized shocks; Sironi & Spitkovsky 2011). In the second line of Equation (15), we have substituted the flare energy from Equation (13) needed to reproduce the radio burst properties from SGR 1935+2154, and we take the X-ray burst duration to be $\sim 3\,\mathrm{ms}$, consistent with observations (Li et al. 2020). Although the value of σ in the flare ejecta of a magnetar flare is uncertain theoretically, there is only a small dynamical range it can attain if this model is to explain FRB radio observations: a minimum magnetization $\sigma \gtrsim {10}^{-3}$ is required for the synchrotron maser to operate in the first place, while the declining efficiency of the maser emission with increasing σ places an upper limit of σ ≲ 1.

For the same parameters, the cooling frequency is given by

$$\begin{eqnarray}\begin{array}{rcl}h{\nu }_{{\rm{c}}} & \approx & 1.4\,\mathrm{keV}\,{\sigma }_{-1}^{-3/2}\ {\left(\displaystyle \frac{{n}_{\mathrm{ext}}}{{10}^{4}{\mathrm{cm}}^{-3}}\right)}^{-3/2}{\left(\displaystyle \frac{{\rm{\Gamma }}}{100}\right)}^{-4}{\left(\displaystyle \frac{t}{3\mathrm{ms}}\right)}^{-2}\\ & \approx & 22\,\mathrm{keV}\,{\sigma }_{-1}^{-3/2}\ {\left(\displaystyle \frac{t}{3\mathrm{ms}}\right)}^{-1/2}{f}_{\xi ,-3}^{2/3}{d}_{10}^{-1/3},\end{array}\end{eqnarray} \tag{ 16 }$$

where the particular temporal scaling $\propto {t}^{-1/2}$ is derived assuming a radially constant density profile (${n}_{\mathrm{ext}}\propto {r}^{0}$).

The fact that ${\nu }_{{\rm{c}}}\lesssim {\nu }_{\mathrm{syn}}$ on the timescales $t\sim 3\,\mathrm{ms}$ of interest shows that the postshock electrons are fast-cooling, and hence a large fraction of the flare energy, ${E}_{\mathrm{flare}}$, is emitted as hard X-rays of energy ${E}_{\mathrm{peak}}\sim 10-100\,\mathrm{keV}$. The predicted X-ray spectrum is thus fast-cooling synchrotron emission from relativistically hot electrons with a thermal Maxwellian energy distribution (see Giannios & Spitkovsky 2009, their Figure 3), resulting in an ordinary fast-cooling spectrum $\nu {L}_{\nu }\propto {\nu }^{1/2}$ between ν_c and ν_syn and an exponential cutoff at an energy $\sim {\nu }_{\mathrm{syn}}$. This is broadly consistent with the observed photon index of ∼1.5 inferred by HXMT for the SGR 1935+2154 X-ray counterpart (Li et al. 2020). Indeed, modeling of other bright magnetar flares suggests that they can be well fit by a cutoff power-law spectrum in the X-rays (van der Horst et al. 2012).

Extending the same model to the shock properties derived for the observed populations of cosmological FRBs predicts that the afterglow emission for these more energetic bursts will occur at much higher energies, ${E}_{\mathrm{peak}}\ \gtrsim$ MeV–GeV, in the gamma-ray band (Figure 8). Unfortunately, gamma-ray satellites like Swift and Fermi are generally not sensitive enough to detect this emission to the cosmological distances of most FRB sources (Metzger et al. 2019; Chen et al. 2020; Margalit et al. 2020). We furthermore emphasize that this predicted short (∼milliseconds duration) gamma-ray signal from the shocks is distinct from the longer-lasting and typically softer gamma-ray emission observed from giant Galactic magnetar flares (e.g., Hurley et al. 2005; Palmer et al. 2005), which is instead well explained as a pair fireball generated by dissipation very close to the NS surface. The latter being relatively isotropic compared to the relativistically beamed radio emission from the ultrarelativistic shocks (hypothesized to accompany the beginning of the flare; Lyubarsky 2014; Beloborodov 2017) might explain the nondetection of FRB-like emission from the 2004 giant flare of SGR 1806–20 (Tendulkar et al. 2016).

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If the X-rays from magnetar flares are attributable to thermal synchrotron shock emission, this may be imprinted in correlations between X-ray observables. Since the electrons behind the shock are fast-cooling, the X-ray fluence ${F}_{{\rm{X}}}$ should scale linearly with the flare energy ${E}_{\mathrm{flare}}$, from which one predicts from Equation (15) a correlation (for fixed σ)

$$\begin{eqnarray}&&{E}_{\mathrm{peak}}\propto {F}_{{\rm{X}}}^{1/2}{t}_{{\rm{X}}}^{-3/2}\end{eqnarray} \tag{ 17 }$$

between the spectral energy peak, burst fluence, and some measure of the burst duration ${t}_{{\rm{X}}}$.

For X-ray bursts from the Galactic magnetar SGR J1550–5418, van der Horst et al. (2012) reported a correlation between the fluence and "emission" time, ${\tau }_{90}\propto {F}_{{\rm{X}}}^{0.47}$. Taking ${t}_{{\rm{X}}}\propto {\tau }_{90}$, Equation (17) predicts ${E}_{\mathrm{peak}}\propto {F}_{X}^{-0.2}$, which is close to but slightly shallower than the correlation ${E}_{\mathrm{peak}}\propto {F}_{X}^{-0.44}$ found by van der Horst et al. (2012) using the entire burst sample. However, note that the most energetic bursts studied by van der Horst et al. (2012) exhibit a flatter or even positive correlation of ${E}_{\mathrm{peak}}$ with fluence. A correlation very close to ${E}_{\mathrm{peak}}\propto {F}_{X}^{-0.44}$ is also predicted from Equation (17) using the empirical relationship ${F}_{{\rm{X}}}\propto {t}_{{\rm{X}}}^{1.54}$ found between the bursts from different magnetars (Gavriil et al. 2004; see Figure 1). Thus, we advance the hypothesis that hard X-ray emission in even ordinary magnetar flares is generated by internal shocks in baryon-loaded outflows.

Scaling the shock model to the lower values of ${r}_{\mathrm{sh}},{\rm{\Gamma }}$ than derived for SGR 1935+2154 above would also have potential consequences for the X-ray spectrum. In particular, the synchrotron spectrum could become opaque to pair creation in some X-ray counterparts of Galactic magnetar flares. Following Lithwick & Sari (2001) and Beniamini & Giannios (2017), we calculate the pair creation opacity corresponding to the model parameters described above and find the external shock to be optically thin to pair creation. Pair creation could, however, become important for somewhat lower values of ${r}_{\mathrm{sh}},{\rm{\Gamma }}$, which could be realized if the dissipation is due to internal shocks at a radius that is much lower than ${r}_{\mathrm{sh}}$. For example, keeping ${E}_{\mathrm{flare}}$ fixed and reducing the Lorentz factor to ${\rm{\Gamma }}\sim 4$ will lead to a pair creation cutoff at the internal shock site at ∼40 keV. We speculate that this might explain why the high temperature of X-ray bursts described by a double blackbody spectrum typically peaks at energies $\lesssim 40$ keV.

Finally, the explanation provided here for the X-ray spectra of Galactic magnetar short bursts might apply most robustly only to the smaller subset of bright SGR bursts that are luminous enough to drive radiation-driven outflows (i.e., large values of ${L}_{{\rm{X}}}/{L}_{\min }$; see Figure 2). In this regard, it is interesting that the brightest bursts analyzed by van der Horst et al. (2012) exhibit a different correlation between their peak energy and flux to the rest of the bursts, consistent with the idea that the physical mechanism driving these bursts is different.

Returning to the radio burst emission, Metzger et al. (2019) predicted a downward drift in frequency¹⁹ as the shock decelerates and the (Lorentz boosted) synchrotron maser emission sweeps from high to low frequencies. The presence (or lack) of this feature in the SGR 1935+2154 burst would therefore provide a helpful diagnostic of FRB models and a probe of the density profile in this specific burst (see Margalit et al. 2020 for application to CHIME repeaters). In addition to the in-band drift that may potentially be detectable by CHIME (although the nontrivial frequency response of the CHIME side lobes may hinder this), the same process could manifest as a small arrival time delay between the STARE2 detection at $1.4\,\mathrm{GHz}$ and the CHIME detection at lower frequencies.

The relative delay between the observed radio emission (corrected for DM) and its hard radiation counterpart observed at peak (Figure 8) should be, at most, comparable to the radio burst duration ($\lesssim \mathrm{ms}$ in the case of SGR 1935+2154's April 28 burst). This results from the fact that the high-frequency photons are optically thin at the shock deceleration radius (when the emission peaks), while the radio is optically thick to induced scattering at this time and only escapes at a time $t\sim$ burst width later. Updated timing by Mereghetti et al. (2020a) suggests that the radio photons may in fact precede the X-ray peaks by a few milliseconds, which would be at odds with the above prediction. This is an important observation; however, we note that ∼millisecond-resolution timing is currently very challenging based on limited X-ray photon counts and possible systematics associated with precision-level dedispersion of the radio bursts. Such dedispersion would require $\sim 0.03\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ fidelity (with respect to the $\approx 332\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ measured DM) to achieve millisecond-level timing; however, the difference between the DMs quoted by CHIME and STARE2 is already comparable to this (and much larger than the quoted statistical uncertainties in either value). Furthermore, the value of the DM is known to depend on the dedispersion method used and can change by as much as $\sim 2\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ between signal-to-noise–maximizing and "structure-maximizing" dedispersion methods (Hessels et al. 2019). We thus urge caution when interpreting such timing.

As shown in Figure 9, the shock model predicts that the X-ray emission should be accompanied by longer-lived synchrotron emission at lower frequencies as the shock propagates to larger radii, again similar to a GRB afterglow (Sari et al. 1998). For frequencies below ν_syn and ν_c, the peak flux will rise as ${F}_{\nu }\propto {\nu }^{1/2}$ as ${\nu }_{\mathrm{syn}}\propto {t}^{-\alpha }$ decreases in time until reaching the observing frequency, where $\alpha =3/2$ during the relativistic phase and $\alpha =3$ once the blast wave transitions to become nonrelativistic. Thus, if ν_syn is passing through the X-ray band (${\nu }_{{\rm{X}}}\sim 10$ keV) on a timescale of ${t}_{{\rm{X}}}\sim 0.1\,{\rm{s}}$, it will reach the optical band on a timescale of seconds and the radio band (∼10 GHz) on a timescale of less than 1 hr. For an external medium with a radially constant density, the flux density ${F}_{\nu }$ at peak (${\nu }_{\mathrm{obs}}\sim {\nu }_{\mathrm{syn}}$) is constant during the relativistic phase and thus approximately equal to that initially achieved at the higher frequencies. Using the X-ray fluence of ${F}_{X}\sim 7\times {10}^{-7}$ erg cm^–2 of the FRB-generating flare from SGR 1935+2154 as a proxy for the latter, we predict a peak radio afterglow flux density ${F}_{\mathrm{radio}}\sim {F}_{X}/({\nu }_{{\rm{X}}}\cdot {t}_{{\rm{X}}})\sim 0.3$ Jy (although self-absorption severely mitigates this estimate; see Figure 9). After peak, the predicted flux will decay exponentially as the synchrotron emission occurs from electrons on the Wien tail of the thermal Maxwellian. Thus, such an afterglow is consistent with upper limits on radio afterglow emission of 0.05 mJy taken on a timescale of a few days (Ravi et al. 2020a, 2020b). Also note that the radius sampled by the shock at these late times is a factor of $\gtrsim 30$ times larger than at the time of the earlier radio emission, and there is no guarantee that the dense medium extends that far from the magnetar.

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Finally, the ejection of the baryonic outflow is predicted to result in a small decrease in the spin frequency of the magnetar due to the temporary opening up of field lines by the mass-loaded wind (Thompson & Blaes 1998; Harding et al. 1999; Beniamini et al. 2020). The magnitude of the spin frequency change increases with the luminosity of the baryonic wind and its duration, both of which are highly uncertain. However if the X-ray burst concurrent with the FRB produces such an outflow, then there are at least two other bursts within the latest outburst, with even larger luminosities (Ridnaia et al. 2020b; Veres et al. 2020), that may have also produced an outflow. Taking, for example, the brightest of these bursts, which occurred on April 22, we can calculate a lower limit on the spin frequency decrease of ${\rm{\Delta }}{\rm{\Omega }}=2\times {10}^{-8}{L}_{41}^{1/2}{t}^{1/2}{{\rm{s}}}^{-1}$ (Beniamini et al. 2020).

The amount of baryonic matter necessary to explain the observed radio burst within this version of the synchrotron maser model is very small, ${M}_{{\rm{b}}}\sim 4\pi {r}_{\mathrm{sh}}^{3}{n}_{\mathrm{ext}}{m}_{p}/3\sim 4\times {10}^{14}\,{\rm{g}}$, although the baryon shell may extend to radii ${r}_{{\rm{b}}}\gg {r}_{\mathrm{sh}}$, in which case the mass estimate would increase by $\sim {({r}_{{\rm{b}}}/{r}_{\mathrm{sh}})}^{3}$. The energy associated with this shell is only $\sim {M}_{{\rm{b}}}{v}_{{\rm{b}}}^{2}/2\,\sim 4\times {10}^{34}\,\mathrm{erg}$, where we have assumed a characteristic expansion velocity for the baryonic shell of ${v}_{{\rm{b}}}=0.5c$, which also implies that the shell must have been ejected $\gtrsim {r}_{\mathrm{sh}}/{v}_{{\rm{b}}}\sim 10\,{\rm{s}}$ prior to the radio burst. Although the luminosity of the X-ray bursts from SGR 1935+2154 does not typically surpass the magnetic Eddington limit (Figure 1), the physics of baryon ejection from active magnetars is still uncertain, and there may be alternative ejection mechanisms that can provide the necessary environment. The modest amount of material required in this baryon shell implies that, despite the uncertainty in the details, the ejection process is energetically unprohibitive for active magnetars.

5.2.3. Spin-down-powered Wind (Beloborodov 2017, 2020)

Beloborodov (2017, 2020) argued that the upstream medium into which the relativistic flare collides is an electron/positron plasma from a spin-down-powered component to the pulsar wind. Given the low spin-down power of SGR 1935+2154, however, this scenario might be somewhat disfavored.

The observed radio fluence at $\nu =1.4\,\mathrm{GHz}$ in this model can be estimated from Equation (91) of Beloborodov (2020) and reexpressed as a function of the spin-down power ${L}_{\mathrm{sd}}$ and pair multiplicity ${ \mathcal M }$,

$$\begin{eqnarray}&&{E}_{\mathrm{radio}}\sim 3\times {10}^{31}\,\mathrm{erg}\,{L}_{\mathrm{sd},34}^{1/12}{{ \mathcal M }}_{3}^{2/3}.\end{eqnarray} \tag{ 18 }$$

In the above, we have assumed that the flare energy is ${10}^{40}\,\mathrm{erg}$, motivated by the X-ray counterpart of SGR 1935+2154, and we have omitted scaling with nuisance parameters for clarity. For the spin-down power of SGR 1935+2154 and standard pair multiplicity of 10³, this fluence is ∼3 orders of magnitude lower than the observed energy of SGR 1935+2154's radio burst.

This tension may be alleviated by an enhancement of the magnetar wind (because of, e.g., opening of field lines by the magnetar flare) or an increased pair multiplicity shortly preceding the flare (Beloborodov 2020). From Equation (18), we find that a pair multiplicity of $\sim 5\times {10}^{7}$ would be required to fit the observed fluence. This is much higher than pulsar pair multiplicities, though it has been suggested that much larger values $\gg {10}^{3}$ may be attainable for magnetars immediately preceding a flare (Beloborodov 2020).

The high pair loading in this version of the synchrotron maser model, as in the Lyubarsky (2014) model, pushes the high-energy thermal synchrotron counterpart to lower frequencies (${\nu }_{\mathrm{syn}}\propto {f}_{{\rm{e}}}^{-2}$), leading to an afterglow peaking optical/UV band (Beloborodov 2020) instead of the X-ray/gamma-ray band predicted in the baryonic model (Figure 8). In this scenario, as in the magnetar wind nebula case (Section 5.2.1), the X-rays from the flare must be created by a process that is not directly related to the FRB mechanism.

In curvature models, the FRB is produced by curvature radiation from bunched electrons streaming along the magnetic field lines of the magnetar (e.g., Kumar et al. 2017; Lu & Kumar 2018). These models predict fluence ratios $\eta \sim 1$ in all bands (Chen et al. 2020) and thus cannot explain the properties of the observed X-ray burst of SGR 1935+2154 (whose fluence ratio is $\eta \sim {10}^{-5}\ll 1$, but also in terms of X-ray spectrum) as a bona fide FRB counterpart. This shortcoming can potentially be dismissed by arguing that the same magnetar activity leading to particle bunching, acceleration, and FRB emission also produces "normal" short X-ray bursts. However, this scenario makes no prediction as to the quantitative relationship between the radio and X-ray burst properties, and the observed fluence ratio of $\eta \sim {10}^{-5}$ is ad hoc within this framework. These considerations also apply to models where FRBs are produced by reconnection in the outer magnetosphere (Lyubarsky 2020) or, indeed, to any magnetospheric models. If true, a testable prediction of the scenario (and one that would be in tension with synchrotron maser models) may be large variations in the value of η for future events.

Another class of FRB models discussed in the literature power FRBs by the kinetic energy of an outflow interacting with the magnetosphere of an NS (Zhang 2017, 2018; Ioka & Zhang 2020). The outflow in these "cosmic comb" models has been suggested to range from AGN-driven winds to GRB jets or winds from a binary companion to the NS. None of the above are likely to be applicable in the context of SGR 1935+2154's radio burst, in tension with predictions of this model (although see Wang et al. 2020).

A final class of NS-related models we discuss are the "spin-down models," in which the radio bursts are powered by rotational energy of the NS (Connor et al. 2016; Cordes & Wasserman 2016; Muñoz et al. 2020). Though in principle applicable to magnetars, these models typically envision normal pulsars as progenitors. Indeed, considering the very low spin-down power of SGR 1935+2154 in comparison to pulsars, and the fact that the latter are $\gtrsim 100$ times more common than Galactic magnetars (in terms of their effective active lifetime), it seems unnatural that a spin-down-powered FRB would be first detected for SGR 1935+2154. A prediction of this model would be a change in NS spin period due to the released burst energy. Accounting for the radiated energy of SGR 1935+2154's burst, including its associated X-ray counterpart, we find that a change ${\rm{\Delta }}{\rm{\Omega }}\approx -4\times {10}^{-6}\,{{\rm{s}}}^{-1}{d}_{10}^{2}$ in the angular velocity of SGR 1935+2154 needs to have occurred. A reduction in the spin frequency is also expected in the Metzger et al. (2019) version of the synchrotron maser model due to the opening of magnetic field lines by the baryonic outflow; however, this effect is much smaller (Section 5.2.2). The Beloborodov (2020) synchrotron maser model (Section 5.2.3) also necessitates some level of period change due to a significant enhancement of the pulsar wind required immediately preceding the FRB; however, this too would be expected to be smaller than the ${\rm{\Delta }}{\rm{\Omega }}$ estimated above. Future timing of SGR 1935+2154 may help test this prediction of the spin-down model.