Neutrino Counterparts of Fast Radio Bursts

Fast radio bursts (FRBs) are millisecond-duration pulses of coherent radio emission of extragalactic origin (Lorimer et al. 2007; Thornton et al. 2013; see Cordes & Chatterjee 2019; Petroff et al. 2019 for reviews). Although many models have been proposed for FRBs (Platts et al. 2019), their central engines and emission mechanism remain elusive. The most well-studied class of models, which can for instance accommodate the discovery that many FRBs are observed to recur on timescales from minutes to weeks (e.g., Spitler et al. 2016; CHIME/FRB Collaboration et al. 2019), postulate an origin associated with flaring activity of magnetars (Popov & Postnov 2013; Kulkarni et al. 2014; Lyubarsky 2014; Katz 2016; Murase et al. 2016; Beloborodov 2017; Kumar et al. 2017; Metzger et al. 2017).

Magnetar FRB models received a recent jolt of support following the discovery of a double-peaked radio burst in spatial and temporal coincidence with a double-peaked hard X-ray outburst from the Galactic magnetar SGR 1935+2154 (Bochenek et al. 2020; The CHIME/FRB Collaboration et al. 2020). The energy radiated by the burst (hereafter FRB 200428) in the ∼GHz radio band, ${{ \mathcal E }}_{\mathrm{radio}}\sim {10}^{34}\mbox{--}{10}^{35}$ erg, exceeds by several orders of magnitude that of any radio burst detected prior from a Galactic neutron star (including giant pulses from young pulsars like the Crab) and is within a factor of ≲10 of the least energetic cosmological FRBs of known distance.

The rate of bursts with similar energy and occurrence rate to FRB 200428 is itself insufficient to explain the total extragalactic FRB population, as inferred from the measured all-sky FRB rate. Nevertheless, if the same physical processes can be scaled-up to more energetic bursts, allowing for a more extreme range of magnetar activity than seen in the Milky Way (as needed to explain particularly active recurrent sources like FRB 121102; Spitler et al. 2016), such a unified scenario can account for most or all of the cosmological FRB population (Lu et al. 2020b; Margalit et al. 2020b).

Despite the breakthrough discovery of FRB 200428, several open questions remain, particularly on the theory side. Chief among these is how to distinguish between several proposed variations of the magnetar model, which make qualitatively different predictions for the mechanism and location of the radio emission and the expectation (if any) of each for an accompanying higher-frequency afterglow. Unlike cosmological FRBs, for which no unambiguous multi-wavelength counterpart has yet been detected, the joint detection of X-ray and radio emission from FRB 200428 find a natural explanation in some magnetar models, while placing others in tension (Lu et al. 2020a; Margalit et al. 2020a).

Arguably the most well-developed of these models, in terms of making quantitative predictions for the burst properties motivated by first-principles kinetic plasma simulations, postulate that FRBs are the electromagnetic wave precursors of transient magnetized shocks generated by flare ejecta (e.g., an ultra-relativistically expanding plasmoid) released from the magnetosphere (Lyubarsky 2014; Beloborodov 2017). In these scenarios the radio precursor is generated at the shock front by electrons undergoing the synchrotron maser process (e.g., Gallant et al. 1992; Iwamoto et al. 2019; Plotnikov & Sironi 2019). The "synchrotron maser blast wave model" naturally accounts for the high observed linear polarization of many bursts (e.g., Ravi et al. 2016; Petroff et al. 2017), their complex spectral energy distribution (e.g., Babul & Sironi 2020), and the observed downward drifting of burst sub-pulses (e.g., CHIME/FRB Collaboration et al. 2019; Hessels et al. 2019) due to the decreasing Doppler factor as the blast wave decelerates as it sweeps up material (Metzger et al. 2019; Margalit et al. 2020b).

Even among magnetized shock models, there exist variations depending on the nature of the upstream medium into which the ultra-relativistic flare ejecta from the magnetar collides. Lyubarsky (2014) and Beloborodov (2017) postulated an upstream medium composed of electron/positron plasma, either representing a nebula surrounding the magnetar, or an ultra-relativistic spin-down powered wind, respectively. Metzger et al. (2019) instead considered the upstream medium to be a mildly relativistically expanding, baryon-loaded outflow with an electron-ion composition. Such an ambient environment may be generated by prior flaring activity. The existence of baryonic ejection is supported in at least the most powerful giant Galactic magnetar flares by radio afterglow emission indicating the presence of a trans-relativistic, mass-loaded outflow (Frail et al. 1999; Gelfand et al. 2005).

As shown by Margalit et al. (2020a) and recounted below, the predictions of the baryon-loaded electron-ion upstream model agree quantitatively with the properties of FRB 200428, provided that one interprets the hard X-ray emission from the flare (Mereghetti et al. 2020; Ridnaia et al. 2020; Tavani et al. 2020; Zhang et al. 2020) as incoherent synchrotron radiation from relativistically hot thermal electrons heated by the same shock generating the FRB. Because both the radio and X-ray emission originate from the same physical location, their nearly simultaneous observed arrival time—once accounting for the finite propagation speed of the radio burst through the Galactic interstellar medium (ISM)—is naturally expected (see also Yamasaki et al. 2020; Yuan et al. 2020). Margalit et al. (2020a) summarize several falsifiable predictions of this model, testable by future Galactic or nearby extragalactic FRBs (see also Beniamini & Kumar 2020; Lu et al. 2020a for criticisms of shock models).

An important consequence of an electron-ion upstream medium is the prediction at the shocks of heating, and potential nonthermal acceleration, of ions to relativistic energies. Unlike an electron/positron plasma, relativistic ions can generate neutrino emission via the photohadronic interaction with thermal synchrotron photons, similar to proposed mechanisms of neutrino emission in gamma-ray burst jets (Waxman & Bahcall 1997; Mészáros & Waxman 2001; Dermer & Atoyan 2003; Guetta & Granot 2003).

In this Letter, we estimate the properties of the neutrino bursts predicted to accompany FRBs in the synchrotron maser electron-ion blast wave scenario, and assess their detectability with the IceCube Neutrino Observatory. Tables 1 and 2 summarize several definitions that may be useful to refer to throughout the text.

Table 1. Definitions

Symbol	Description
E	Proton energy
	Photon energy
ε	Neutrino energy
${ \mathcal E }$	Total energy

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Table 2. Key Timescales and Energies

Symbol	Description	Typical Values a
${t}_{\mathrm{FRB}}$	FRB duration	∼0.1–10 ms
${t}_{\mathrm{nth}}$ (Equation (22))	Duration of ν's from thermal protons	$\sim 0.1-1$ ms
${t}_{{\rm{c}}}$ (Equation (15))	Synchrotron emission no longer fast-cooling	$\sim 0.1\mbox{--}{10}^{3}$ s
${t}_{\max }$ (Equation (25))	Duration of ν's from nonthermal protons	$\sim 1\mbox{--}{10}^{3}$ s
${t}_{\mathrm{nr}}$ (Equation (11))	Transition to non-relativistic shock	$\sim {10}^{2}\mbox{--}{10}^{5}$ s
$\bar{E}$	Mean thermal energy of protons	∼10–100 GeV b
${\epsilon }_{\mathrm{pk}}$ (Equation (13))	Synchrotron peak photon energy	$\sim 10\mbox{--}{10}^{3}$ eV
${\epsilon }_{{\rm{c}}}$ (Equation (14))	Synchrotron cooling photon energy	$\sim 10\mbox{--}{10}^{3}$ eV
${\varepsilon }_{{\rm{c}}}$ (Equation (27))	Peak energy of neutrino spectrum ($t\lt {t}_{{\rm{c}}}$)	∼0.1–10 PeV
${\varepsilon }_{\min }$ (Equation (23))	Minimum energy of neutrino spectrum	∼1–10 PeV
${\varepsilon }_{\max }$ (Equation (24))	Maximum energy of neutrino spectrum	∼1–10 PeV

Notes.

^a For $\sigma \sim 0.1$ and ${{ \mathcal E }}_{\mathrm{rad}}\sim {10}^{37}\mbox{--}{10}^{43}$ erg in the range of cosmological FRB. ^b For energies we quote values on the timescale $\sim {t}_{\max }$.

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Following Metzger et al. (2019), we consider an ultra-relativistic blast wave generated by flare ejecta from the magnetar (e.g., a magnetically dominated plasmoid; e.g., Yuan et al. 2020) with an isotropic energy ${{ \mathcal E }}_{\mathrm{flare}}$ propagating into an external electron-ion medium with a radial density profile n_ext  ∝ r^−k , where k < 3 (we will soon specialize to the case k = 0). We focus on observer times t greater than the intrinsic width of the flare ejecta δt, as is typically satisfied for FRBs of duration ≳1 ms if the flare duration $\delta t\sim \mathrm{few}\times {R}_{\mathrm{ns}}/c\sim 0.1$ ms is set by the Alfvén crossing time of the inner magnetosphere of a neutron star of radius R_ns ∼ 10 km.

The deceleration of the blast wave follows a self-similar evolution (Blandford & McKee 1976) resulting in the following evolution for the radius, Lorentz factor, and kinetic luminosity of the forward shock,

$$\begin{eqnarray}&&{\rm{\Gamma }}={{\rm{\Gamma }}}_{\mathrm{FRB}}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{\tfrac{(k-3)}{2(4-k)}}\mathop{=}\limits_{k=0}{{\rm{\Gamma }}}_{\mathrm{FRB}}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{-\tfrac{3}{8}},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{r}_{\mathrm{sh}}={r}_{\mathrm{FRB}}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{\tfrac{1}{(4-k)}}\mathop{=}\limits_{k=0}{r}_{\mathrm{FRB}}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{\tfrac{1}{4}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{n}_{\mathrm{ext}}={n}_{\mathrm{FRB}}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{\tfrac{-k}{(4-k)}}\mathop{=}\limits_{k=0}{n}_{\mathrm{FRB}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{L}_{\mathrm{sh}}={L}_{\mathrm{FRB}}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{-1}\simeq \displaystyle \frac{{{ \mathcal E }}_{\mathrm{flare}}}{4{t}_{\mathrm{FRB}}}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{-1},\end{eqnarray} \tag{ 4 }$$

where t_FRB is the observed duration of the FRB (as set by the timescale over which the spectral energy distribution of the maser emission sweeps downward across the observing band; Metzger et al. 2019) and $\{{{\rm{\Gamma }}}_{\mathrm{FRB}},{r}_{\mathrm{FRB}},{n}_{\mathrm{FRB}},{L}_{\mathrm{FRB}}\}$ are the shock properties at t ∼ t_FRB.

The latter quantities are related to observable properties of the FRB, namely its duration ${t}_{\mathrm{FRB}}={t}_{\mathrm{ms}}$ ms, radio frequency ν_obs = ν_GHz GHz, isotropic radio energy ${{ \mathcal E }}_{\mathrm{radio}}={{ \mathcal E }}_{{\rm{r}},40}{10}^{40}$ erg, according to Margalit et al. (2020b)

$$\begin{eqnarray}&&{r}_{\mathrm{FRB}}\approx 5\times {10}^{12}\mathrm{cm}\,{f}_{\xi ,-3}^{-2/15}{\nu }_{\mathrm{GHz}}^{-7/15}{t}_{\mathrm{ms}}^{1/5}{{ \mathcal E }}_{{\rm{r}},40}^{1/3},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{n}_{\mathrm{FRB}}\approx 420\,{\mathrm{cm}}^{-3}\,{f}_{\xi ,-3}^{-4/15}{\nu }_{\mathrm{GHz}}^{31/15}{t}_{\mathrm{ms}}^{2/5}{{ \mathcal E }}_{{\rm{r}},40}^{-1/3},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\mathrm{FRB}}\approx 287\,{f}_{\xi ,-3}^{-1/15}{\nu }_{\mathrm{GHz}}^{-7/30}{t}_{\mathrm{ms}}^{-2/5}{{ \mathcal E }}_{{\rm{r}},40}^{1/6}\end{eqnarray} \tag{ 7 }$$

where ${f}_{\xi }={10}^{-3}{f}_{\xi ,-3}$ is the efficiency of the maser emission, normalized to a characteristic value found by particle-in-cell (PIC) plasma simulations of magnetized shocks (Plotnikov & Sironi 2019; Babul & Sironi 2020) and we have assumed an upstream ratio of electrons to nucleons f_e  = 0.5 (the results can be easily generalized to include an electron–positron component).

The radiative efficiency of the FRB emission is given by Margalit et al. (2020b)

$$\begin{eqnarray}&&\eta \equiv \displaystyle \frac{{{ \mathcal E }}_{\mathrm{radio}}}{{{ \mathcal E }}_{\mathrm{flare}}}\approx 4\times {10}^{-5}{f}_{\xi ,-3}^{4/5}{\nu }_{\mathrm{GHz}}^{-1/5}{t}_{\mathrm{ms}}^{-1/5},\end{eqnarray} \tag{ 8 }$$

such that

$$\begin{eqnarray}&&{{ \mathcal E }}_{\mathrm{flare}}\approx 2.4\times {10}^{44}\mathrm{erg}\,{f}_{\xi ,-3}^{-4/5}{\nu }_{\mathrm{GHz}}^{1/5}{t}_{\mathrm{ms}}^{1/5}{{ \mathcal E }}_{{\rm{r}},40}\end{eqnarray} \tag{ 9 }$$

and the kinetic power of the shock at the epoch of FRB emission is given by

$$\begin{eqnarray}&&{L}_{\mathrm{FRB}}\simeq \displaystyle \frac{{{ \mathcal E }}_{\mathrm{flare}}}{4{t}_{\mathrm{FRB}}}\approx 6\times {10}^{46}\,\mathrm{erg}\,{{\rm{s}}}^{-1}{f}_{\xi ,-3}^{-4/5}{\nu }_{\mathrm{GHz}}^{1/5}{t}_{\mathrm{ms}}^{-4/5}{{ \mathcal E }}_{{\rm{r}},40}\end{eqnarray} \tag{ 10 }$$

The population of cosmological FRBs with known distances exhibit typical ranges ${{ \mathcal E }}_{\mathrm{radio}}\sim {10}^{37}\mbox{--}{10}^{41}$ erg, t_FRB ∼ 0.1−10 ms, such that one infers r_FRB ∼ 10¹²–10¹³ cm, ${{\rm{\Gamma }}}_{\mathrm{FRB}}\sim {10}^{2}\mbox{--}{10}^{3}$, and ${n}_{\mathrm{FRB}}\sim {10}^{2}\mbox{--}{10}^{4}$ cm⁻³ (Margalit et al. 2020b). From the observed rate of downward frequency drifting of the sub-pulses, which in this model could arise from deceleration of the blast wave (Metzger et al. 2019), one infers power-law indices of the external medium spanning a wide-range k ≈ [−2, 2] (Margalit et al. 2020b). In what follows we assume a constant density medium (k = 0) for simplicity, though this too can be easily generalized and our results to follow do not depend qualitatively on this assumption.

From Equations (4) and (7), the blast wave transitions from ultra-relativistic to non-relativistic expansion (Γ ≃ 1) after a time

$$\begin{eqnarray}&&{t}_{\mathrm{nr}}\simeq {t}_{\mathrm{FRB}}{{\rm{\Gamma }}}_{\mathrm{FRB}}^{8/3}\simeq 3.5\times {10}^{3}{\rm{s}}\,{f}_{\xi ,-3}^{-8/45}{\nu }_{\mathrm{GHz}}^{-28/45}{t}_{\mathrm{ms}}^{-1/15}{{ \mathcal E }}_{{\rm{r}},40}^{4/9},\end{eqnarray} \tag{ 11 }$$

which is minutes to hours for typical parameters.

In addition to generating the coherent FRB emission, the shock heats the plasma to ultra-relativistic temperatures. Again guided by the results of PIC simulations of magnetized shocks, we assume that the post-shock energy is shared roughly equally with the electrons and ions, heating each to a Maxwellian distribution of energies (e.g., Sironi & Spitkovsky 2011). Given the strong magnetic field behind the shock, the electrons radiate (incoherent) synchrotron emission. The latter peaks at a photon energy (Metzger et al. 2019)

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{pk}}=\displaystyle \frac{{\hslash }{eB}^{\prime} }{{m}_{e}c}{\bar{\gamma }}^{2}{\rm{\Gamma }},\end{eqnarray} \tag{ 12 }$$

where $\bar{\gamma }=({m}_{p}/{m}_{e}){\rm{\Gamma }}/2$ is the mean thermal Lorentz factor, $B^{\prime} =\sqrt{64\pi \sigma {{\rm{\Gamma }}}^{2}{m}_{p}{c}^{2}{n}_{\mathrm{ext}}}$ is the post-shock magnetic field, and σ is the magnetization of the upstream medium. Thus we have

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{pk}}\approx 235\,\mathrm{MeV}{\sigma }_{-1}^{1/2}{f}_{\xi ,-3}^{-2/5}{\nu }_{\mathrm{GHz}}^{1/10}{{ \mathcal E }}_{{\rm{r}},40}^{1/2}{t}_{\mathrm{ms}}^{-7/5}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{-3/2},\end{eqnarray} \tag{ 13 }$$

in the hard X-ray/γ-ray range, where σ₋₁ ≡ σ/(0.1). ⁵ The cooling frequency is given by

$$\begin{eqnarray}&&\begin{array}{l}{\epsilon }_{{\rm{c}}}=\displaystyle \frac{{\hslash }{eB}^{\prime} }{{m}_{e}c}{\gamma }_{{\rm{c}}}^{2}{\rm{\Gamma }}\\ \approx \ 0.023\,\mathrm{MeV}\,{\sigma }_{-1}^{-3/2}{f}_{\xi ,-3}^{2/3}{\nu }_{\mathrm{GHz}}^{-65/30}{t}_{\mathrm{ms}}^{-1}{{ \mathcal E }}_{{\rm{r}},40}^{-1/6}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{-1/2},\end{array}\end{eqnarray} \tag{ 14 }$$

where ${\gamma }_{{\rm{c}}}=(6\pi {m}_{e}c/{\sigma }_{{\rm{T}}}{\rm{\Gamma }}{B}^{{\prime} 2}t)$ is the Lorentz factor of the electrons that cool on the expansion time. The post-shock electrons thus remain fast-cooling (${\epsilon }_{{\rm{c}}}\lesssim {\epsilon }_{\mathrm{pk}}$) for a timescale

$$\begin{eqnarray}&&{t}_{{\rm{c}}}\approx 10\,{\rm{s}}\,{\sigma }_{-1}^{2}{f}_{\xi ,-3}^{-16/15}{\nu }_{\mathrm{GHz}}^{34/15}{t}_{\mathrm{ms}}^{3/5}{{ \mathcal E }}_{{\rm{r}},40}^{2/3},\end{eqnarray} \tag{ 15 }$$

much longer than the FRB duration; thus the radiated X-ray/gamma-ray energy ${E}_{{\rm{X}}/\gamma }\approx {{ \mathcal E }}_{\mathrm{flare}}/2$ is equal to that given to the electrons.

Considering the application to FRB 200428 (Margalit et al. 2020b), the value $\eta \sim {10}^{-4}\mbox{--}{10}^{-5}$ predicted by Equation (8) agrees with the measured ratio of the radio and X-ray fluence (${{ \mathcal E }}_{\mathrm{radio}}\sim {10}^{34}\mbox{--}{10}^{35}$ erg; Bochenek et al. 2020; The CHIME/FRB Collaboration et al. 2020; ${E}_{{\rm{X}}/\gamma }\sim {10}^{39}\mbox{--}{10}^{40}$ erg; Mereghetti et al. 2020; Zhang et al. 2020) for ${f}_{\xi }\sim {10}^{-3}$. For the same parameters (e.g. ${{ \mathcal E }}_{{\rm{r}},40}\sim {10}^{-5}$), Equation (13) predicts a peak frequency (given the observed duration t ∼ 3 ms of the X-ray spikes of FRB 200428) of ${\epsilon }_{\mathrm{pk}}\sim 300{\sigma }_{-1}^{1/2}$ keV, which is also consistent with the spectral peak of the observed power-law X-ray/gamma-ray emission. Finally, the energy spectrum of the X-ray/gamma-ray emission is well fit by a power law + cut-off (e.g., Mereghetti et al. 2020), which is similar to predictions for a fast-cooling thermal synchrotron spectrum (Giannios & Spitkovsky 2009). Taken together, these observations support a magnetized shock propagating into a baryon-loaded medium as the origin of FRB 200428 and its associated X-ray burst (Margalit et al. 2020a).

3.1. Proton Heating and Acceleration

A fraction ∼1/2 of the shock's energy goes into thermally heating ions, which hereafter we take to be protons for simplicity. The protons achieve a relativistic Maxwellian distribution with an average particle energy of $\bar{E}^{\prime} \approx {\rm{\Gamma }}{m}_{p}{c}^{2}/2$ in the co-moving frame of the shocked gas (hereafter denoted by a prime'). Though less certain, particularly for a strongly magnetized upstream (σ ≫ 1), the shocks may accelerate a small fraction of the protons to higher energies $\gg \bar{E}^{\prime}$ (e.g., via diffusive acceleration; Blandford & Ostriker 1978), into a power-law nonthermal spectrum of the form

$$\begin{eqnarray}&&\displaystyle \frac{{{dN}}_{p}^{{\prime} }}{{dE}^{\prime} }\propto {\left(\displaystyle \frac{E^{\prime} }{\bar{E}^{\prime} }\right)}^{-q},\,\bar{E}^{\prime} \lt E^{\prime} \lt {E}_{\max }^{{\prime} },\end{eqnarray} \tag{ 16 }$$

where ${E}_{\max }^{{\prime} }$ is a maximum cut-off energy, to be estimated below. Taking q ≃ 2, the fraction of the total shock kinetic luminosity placed into protons of energy $\sim E^{\prime} \geqslant \bar{E}^{\prime}$ can be roughly written as

$$\begin{eqnarray}{f}_{E^{\prime} }\approx \left\{\begin{array}{ll}1/2 & E^{\prime} \sim \bar{E}^{\prime} \\ {\epsilon }_{\mathrm{rel}}/{\rm{\Lambda }} & E^{\prime} \gg \bar{E}^{\prime} ,\end{array}\right.\end{eqnarray} \tag{ 17 }$$

where _rel ≲ 0.1 is an acceleration efficiency and ${\rm{\Lambda }}\equiv \mathrm{ln}({E}_{\max }^{{\prime} }/\bar{E}^{\prime} )\sim 10$ (see below).

In order to estimate the value of ${E}_{\max }^{{\prime} }$ we equate the proton acceleration time (estimated by the Larmor gyration time) ${t}_{\mathrm{acc}}\sim 2\pi E^{\prime} /{eB}^{\prime} c$ to various loss timescales. The most stringent constraint ⁶ comes from equating ${t}_{\mathrm{acc}}$ to the dynamical time, ${t}_{\mathrm{dyn}}\simeq {r}_{\mathrm{sh}}/({\rm{\Gamma }}c)$, over which adiabatic losses occur. This gives

$$\begin{eqnarray}&&\begin{array}{l}{E}_{\max }^{{\prime} }\simeq \displaystyle \frac{{eB}^{\prime} {r}_{\mathrm{sh}}}{2\pi {\rm{\Gamma }}}={{er}}_{\mathrm{sh}}c\sqrt{(16/\pi )\sigma {m}_{p}{n}_{\mathrm{ext}}}\\ \approx \,850\,\mathrm{TeV}{\sigma }_{-1}^{1/2}{f}_{\xi ,-3}^{-4/15}{\nu }_{\mathrm{GHz}}^{17/30}{t}_{\mathrm{ms}}^{2/5}{{ \mathcal E }}_{{\rm{r}},40}^{1/6}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{1/4}.\end{array}\end{eqnarray} \tag{ 18 }$$

Note that ${E}_{\max }^{{\prime} }\sim 1\,\mathrm{PeV}\sim {10}^{3}\mbox{--}{10}^{4}\bar{E}^{\prime}$ on the timescale of the FRB $t\sim {t}_{\mathrm{FRB}}$, increasing by $t\sim {t}_{\mathrm{nr}}$ to ≳10 PeV.

3.2. Pion Production and Neutrino Emission

Although the electrons generally cool faster than the dynamical timescale at early times (t ≪ t_c; Equation (15)), this is not true of the protons due to their greater mass. The protons retain their energy, enabling them to interact with the thermal synchrotron photons via the photomeson process $p+\gamma \to p+{\pi }^{\pm ,0}$ (Berezinskii et al. 1990), generating high-energy gamma-rays and neutrinos via the decay of π⁰ and π^±, respectively. The rate of hadronuclear interactions (e.g. $p+p\to p+p+{\pi }^{\pm ,0}$) are low in comparison because of the high photon-to-baryon ratio behind the shock.

The energy distribution of the thermal synchrotron emission from the electrons peaks at a photon energy

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{pk}}^{{\prime} }\approx \displaystyle \frac{{\epsilon }_{\mathrm{pk}}}{{\rm{\Gamma }}}\approx 0.8\,\mathrm{MeV}{\sigma }_{-1}^{1/2}{f}_{\xi ,-3}^{-1/3}{\nu }_{\mathrm{GHz}}^{1/3}{t}_{\mathrm{ms}}^{-1}{{ \mathcal E }}_{{\rm{r}},40}^{1/3}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{-9/8},\end{eqnarray} \tag{ 19 }$$

where we have used Equations (7) and (13). Immersed in this radiation field, pion production peaks at the Δ-resonance at a proton energy (e.g., Waxman & Bahcall 1997)

$$\begin{eqnarray}\begin{array}{rcl}{E}_{\mathrm{thr}}^{{\prime} } & \approx & \displaystyle \frac{{E}_{{\rm{\Delta }}}^{{\prime} }}{{\epsilon }_{\mathrm{pk}}^{{\prime} }}\displaystyle \frac{{m}_{p}{c}^{2}}{2}\\ & \approx & 170\,\mathrm{GeV}{\sigma }_{-1}^{-1/2}{f}_{\xi ,-3}^{1/3}{\nu }_{\mathrm{GHz}}^{-1/3}{t}_{\mathrm{ms}}{{ \mathcal E }}_{{\rm{r}},40}^{-1/3}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{9/8},\end{array}\end{eqnarray} \tag{ 20 }$$

where ${E}_{{\rm{\Delta }}}^{{\prime} }\approx 0.3\,\mathrm{GeV}$. Comparing this threshold to the proton thermal energy,

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{E}_{\mathrm{thr}}^{{\prime} }}{\bar{E}^{\prime} }\approx \displaystyle \frac{2{E}_{\mathrm{thr}}^{{\prime} }}{{\rm{\Gamma }}{m}_{p}{c}^{2}}\approx \displaystyle \frac{{E}_{{\rm{\Delta }}}^{{\prime} }}{{\epsilon }_{\mathrm{pk}}}\\ \approx 1.3\,{\sigma }_{-1}^{-1/2}{f}_{\xi ,-3}^{2/5}{\nu }_{\mathrm{GHz}}^{-1/10}{t}_{\mathrm{ms}}^{7/5}{{ \mathcal E }}_{{\rm{r}},40}^{-1/2}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{3/2}.\end{array}\end{eqnarray} \tag{ 21 }$$

Thus, on timescales of the FRB emission (t ∼ t_FRB), protons which are either thermal ($E^{\prime} \sim \bar{E}^{\prime}$), or only moderately into the nonthermal tail ($E^{\prime} \gtrsim \bar{E}^{\prime} ;$ Equation (16)), are capable of pion-producing on synchrotron photons. However, at times $t\gg {t}_{\mathrm{FRB}}$, only the nonthermal protons with $E^{\prime} \gg \bar{E}^{\prime}$ have enough energy to produce pions. This transition ($\bar{E}^{\prime} ={E}_{\mathrm{thr}}^{{\prime} }$) from thermal to nonthermal-only production occurs at a time

$$\begin{eqnarray}&&{t}_{\mathrm{nth}}\approx 8.5\times {10}^{-4}{\rm{s}}\,{\sigma }_{-1}^{1/3}{f}_{\xi ,-3}^{-4/15}{\nu }_{\mathrm{GHz}}^{1/15}{t}_{\mathrm{ms}}^{1/15}{{ \mathcal E }}_{{\rm{r}},40}^{1/3}.\end{eqnarray} \tag{ 22 }$$

The pions generated decay into neutrinos of lab-frame energy $\varepsilon \approx {\rm{\Gamma }}E^{\prime} /20$ (Kelner & Aharonian 2008). The minimum neutrino energy at time t is therefore

$$\begin{eqnarray}&&\begin{array}{l}{\varepsilon }_{\min }\approx \max [{\rm{\Gamma }}\bar{E}^{\prime} /20,{\rm{\Gamma }}{E}_{\mathrm{thr}}^{{\prime} }/20]\\ \approx \ \max \left\{\begin{array}{l}1.9\,\mathrm{TeV}{f}_{\xi ,-3}^{-2/15}{\nu }_{\mathrm{GHz}}^{-7/15}{t}_{\mathrm{ms}}^{-4/5}{{ \mathcal E }}_{{\rm{r}},40}^{1/3}{\left(\tfrac{t}{{t}_{\mathrm{FRB}}}\right)}^{-3/4},t\lesssim {t}_{\mathrm{nth}}\\ 2.5\,\mathrm{TeV}{\sigma }_{-1}^{-1/2}{f}_{\xi ,-3}^{4/15}{\nu }_{\mathrm{GHz}}^{-17/30}{t}_{\mathrm{ms}}^{3/5}{{ \mathcal E }}_{{\rm{r}},40}^{-1/6}{\left(\tfrac{t}{{t}_{\mathrm{FRB}}}\right)}^{3/4},t\gtrsim {t}_{\mathrm{nth}}\end{array}\right.\end{array}\end{eqnarray} \tag{ 23 }$$

From Equation (18), one can also define an absolute maximum neutrino energy, as set by the accelerator,

$$\begin{eqnarray}&&{\varepsilon }_{\max }\approx \displaystyle \frac{{\rm{\Gamma }}{E}_{\max }^{{\prime} }}{20}\approx 12\,\mathrm{PeV}\,{\sigma }_{-1}^{1/2}{f}_{\xi ,-3}^{-1/3}{\nu }_{\mathrm{GHz}}^{1/3}{{ \mathcal E }}_{{\rm{r}},40}^{1/3}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{-1/8}\end{eqnarray} \tag{ 24 }$$

Equating ${\varepsilon }_{\min }(t\gt {t}_{\mathrm{nth}})$ (Equation (23)) to ${\varepsilon }_{\max }$, we obtain

$$\begin{eqnarray}&&{t}_{\max }\approx 17\,{\rm{s}}\,{\sigma }_{-1}^{8/7}{f}_{\xi ,-3}^{-24/35}{\nu }_{\mathrm{GHz}}^{36/35}{t}_{\mathrm{ms}}^{11/35}{{ \mathcal E }}_{{\rm{r}},40}^{4/7}.\end{eqnarray} \tag{ 25 }$$

This is the maximal timescale for significant neutrino emissions because at t > t_max the proton energy required to interact with the most energetic synchrotron photons (of energy ∼_pk) exceeds E_max.

Before t_max, higher-energy neutrinos with ${\varepsilon }_{\min }\lt \varepsilon \lt {\varepsilon }_{\max }$ can be created by the interaction of protons with energies $E^{\prime} \gg {E}_{\mathrm{thr}}^{{\prime} },\bar{E}^{\prime}$ with photons of lower energies $\epsilon ^{\prime} \ll {\epsilon }_{\mathrm{pk}}^{{\prime} }$ below the synchrotron peak. In fact, while the synchrotron emission is fast-cooling ($t\lt {t}_{{\rm{c}}};$ Equation (15)) such interactions dominate neutrino production (by nonthermal protons) because the photon number spectrum in the fast-cooling regime,

$$\begin{eqnarray}&&\epsilon ({{dN}}_{\gamma }/d\epsilon )\propto {\epsilon }^{-1/2},{\epsilon }_{{\rm{c}}}\lesssim \epsilon \lesssim {\epsilon }_{\mathrm{pk}},\end{eqnarray} \tag{ 26 }$$

peaks at _c < _pk. ⁷

In the fast-cooling regime the energy distribution of neutrinos from nonthermal protons will thus peak at an energy,

$$\begin{eqnarray}&&\begin{array}{l}{\varepsilon }_{{\rm{c}}}\approx \displaystyle \frac{{\rm{\Gamma }}{E}_{{\rm{\Delta }}}^{{\prime} }}{20{\epsilon }_{{\rm{c}}}^{{\prime} }}\displaystyle \frac{{m}_{p}{c}^{2}}{2}\\ \approx \ 26\,\mathrm{PeV}\,{\sigma }_{-1}^{3/2}{f}_{\xi ,-3}^{-4/5}{\nu }_{\mathrm{GHz}}^{17/10}{t}_{\mathrm{ms}}^{1/5}{{ \mathcal E }}_{{\rm{r}},40}^{1/2}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{-1/4},\end{array}\end{eqnarray} \tag{ 27 }$$

where ${\epsilon }_{{\rm{c}}}^{{\prime} }={\epsilon }_{{\rm{c}}}/{\rm{\Gamma }}$. The value of ${\varepsilon }_{{\rm{c}}}$ decreases in time, matching ε_min at t = t_c.

In principle, pions and muons could suffer synchrotron losses before decaying into neutrinos. This happens above a energy critical energy

$$\begin{eqnarray}&&\begin{array}{l}{\varepsilon }_{\pi ,\mu }^{\mathrm{syn}}={\left(\displaystyle \frac{6\pi {m}_{\pi ,\mu }c}{{\sigma }_{T}{B}^{{\prime} 2}{\tau }_{\pi ,\mu }}\right)}^{1/2}\left(\displaystyle \frac{{m}_{\pi ,\mu }}{{m}_{e}}\right){m}_{\pi ,\mu }\,{c}^{2}{\rm{\Gamma }}\\ =\,(30,1.7)\times {10}^{18}\,\mathrm{eV}\,{\sigma }_{-1}^{-1/2}{f}_{\xi ,-3}^{2/15}{\nu }_{\mathrm{GHz}}^{-31/30}{t}_{\mathrm{ms}}^{-1/5}{{ \mathcal E }}_{{\rm{r}},40}^{1/6},\end{array}\end{eqnarray} \tag{ 28 }$$

where m_π , μ and τ_π , μ are the masses and rest-frame lifetimes of pions and muons, respectively. Because ${\varepsilon }_{\pi }^{\mathrm{syn}}\gg 4\,{\varepsilon }_{\max }$ and ${\varepsilon }_{\mu }^{\mathrm{syn}}\gg 3\,{\varepsilon }_{\max }$ (where the prefactors account for the rough energy partition between pions/muons/neutrinos in the decays), we conclude that synchrotron cooling of the secondaries is negligible.

In summary, although all of the timescales increase with the flare energy, the hierarchy ${t}_{\mathrm{nth}}\lt {t}_{{\rm{c}}}\lt {t}_{\max }\lt {t}_{\mathrm{nr}}$ is generally preserved for $\sigma \lesssim 1$ and moderate variations around fiducial parameters (Table 2). At early times ($t\lesssim {t}_{\mathrm{nth}}$) we expect ∼1–10 TeV neutrinos generated by the interaction of the thermal protons with synchrotron photons of energy ∼_pk. This prediction is relatively robust (within the baryon shock model) because proton heating at the shocks is inevitable. Furthermore, because t_nth ∼ t_FRB the thermal neutrino emission occurs simultaneously with the FRB emission, and hence we do not need to assume a continuation of the external medium outside the region directly probed by radio observations.

At later times (t ≳ t_nth) the thermal protons are soon no longer sufficiently energetic to produce pions, but nevertheless neutrinos may still be produced by the power-law nonthermal protons. Insofar as the proton spectrum is flat (q ≃ 2), the dominant interaction while the synchrotron emission is fast-cooling (t ≪ t_c) will be between photons of energy ∼_c  ≪ _pk on high-energy protons ∼20ε_c generating neutrinos of energy ε_c ∼ 1–10 PeV. However, at times t ≫ t_c the synchrotron spectrum becomes slow-cooling and the interaction will be dominated by photons of energy ∼_pk generating neutrinos of energy ε_min(t ≳ t_c)  ≳  10 PeV.

3.3. Neutrino Radiative Efficiency

What fraction of the available proton luminosity $\sim {L}_{\mathrm{sh}}/2$ will be radiated as neutrinos? One can define an "optical depth" for neutrino production over each decade in observer time ∼t,

$$\begin{eqnarray}&&{\tau }_{p\gamma }\approx {n}_{\gamma }^{{\prime} }{\sigma }_{p\gamma }{\kappa }_{p\gamma }r^{\prime} \end{eqnarray} \tag{ 29 }$$

where ${n}_{\gamma }^{{\prime} }$ is the target photon number density given below, $r^{\prime} \approx {r}_{\mathrm{sh}}/{\rm{\Gamma }}$ is the characteristic radius of the post-shock region in the co-moving frame (through which relativistic protons accelerated at the shock traverse to escape without losses), ${\sigma }_{p\gamma }$ is the cross section of photopion production, which is $\sim 5\times {10}^{-28}$ cm² (Particle Data Group et al. 2004) corresponding to its value near the Δ-resonance (and $\sim 1.6\times {10}^{-28}$ cm² above the resonance). The factor ${\kappa }_{p\gamma }\sim 0.15$ is the average fraction of energy lost from a proton per collision ("elasticity"). Consider first proton interactions with photons of energy  = _pk that generate neutrinos of energy ${\varepsilon }_{\min }$ (Equation (23)). Their number density is given by ${n}_{\gamma }^{{\prime} }\approx {U}_{\gamma }^{{\prime} }/{\epsilon }_{\mathrm{pk}}^{{\prime} }$, where

$$\begin{eqnarray}{U}_{\gamma }^{{\prime} }\approx \displaystyle \frac{{L}_{\mathrm{sh}}/2}{4\pi {r}_{\mathrm{sh}}^{2}c{{\rm{\Gamma }}}^{2}}\left\{\begin{array}{cc}1 & t\leqslant {t}_{{\rm{c}}}\\ {\left(t/{t}_{{\rm{c}}}\right)}^{-1/2} & t\geqslant {t}_{{\rm{c}}},\end{array}\right.\end{eqnarray} \tag{ 30 }$$

where the factor of 1/2 in the numerator arises from the half of the shock power placed into electrons in the fast-cooling regime. The factor ${(t/{t}_{{\rm{c}}})}^{-1/2}$ factor accounts for the reduction in the radiated power when the electrons become slow-cooling at $t\gtrsim {t}_{{\rm{c}}}$ (Equation (15)).

Combining with the above

$$\begin{eqnarray}\begin{array}{l}{\tau }_{p\gamma }[{\varepsilon }_{\nu ,\min }]\approx \displaystyle \frac{{L}_{\mathrm{sh}}{\sigma }_{p\gamma }{\kappa }_{p\gamma }}{8\pi {r}_{\mathrm{sh}}c{{\rm{\Gamma }}}^{2}{\epsilon }_{\mathrm{pk}}}\left\{\begin{array}{cc}1 & t\leqslant {t}_{{\rm{c}}}\\ {\left(t/{t}_{{\rm{c}}}\right)}^{-1/2} & t\geqslant {t}_{{\rm{c}}},\end{array}\right.\\ \approx 3.9\,\times \,{10}^{-8}{\sigma }_{-1}^{-1/2}{f}_{\xi ,-3}^{-2/15}{\nu }_{\mathrm{GHz}}^{31/30}{t}_{\mathrm{ms}}^{6/5}{{ \mathcal E }}_{{\rm{r}},40}^{-1/6}\\ \times \left\{\begin{array}{cc}\left(\tfrac{t}{{t}_{\mathrm{FRB}}}\right) & t\leqslant {t}_{{\rm{c}}}\\ \left(\tfrac{{t}_{{\rm{c}}}}{{t}_{\mathrm{FRB}}}\right){\left(\tfrac{t}{{t}_{{\rm{c}}}}\right)}^{1/2} & t\geqslant {t}_{{\rm{c}}}\end{array}\right.\end{array}\end{eqnarray} \tag{ 31 }$$

At time $t\lesssim {t}_{{\rm{c}}}$, the photon density ${n}_{\gamma }$ at $\epsilon ={\epsilon }_{{\rm{c}}}$ is larger by a factor ${(t/{t}_{{\rm{c}}})}^{-1/2}$ than at ${\epsilon }_{\mathrm{pk}}$ (Equation (26)); thus, the optical depth to generating ${\varepsilon }_{\nu ,c}$ neutrinos is larger by the same factor:

$$\begin{eqnarray}&&\begin{array}{l}{\tau }_{p\gamma }[{\varepsilon }_{\nu ,c}]\approx {\tau }_{p\gamma }[{\varepsilon }_{\nu ,\min }]{\left(t/{t}_{{\rm{c}}}\right)}^{-1/2}\\ \approx \,3.9\,\times \,{10}^{-6}{\sigma }_{-1}^{1/2}{f}_{\xi ,-3}^{-2/3}{\nu }_{\mathrm{GHz}}^{65/30}{t}_{\mathrm{ms}}{{ \mathcal E }}_{{\rm{r}},40}^{1/6}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{1/2},t\leqslant {t}_{{\rm{c}}}\end{array}\end{eqnarray} \tag{ 32 }$$

Interestingly, ${\tau }_{{\rm{p}},\gamma }$ at the peak of the neutrino energy spectrum increases $\propto {t}^{1/2}$ in both fast- and slow-cooling regimes (both before and after ${t}_{{\rm{c}}}$). The fluence emitted in neutrinos over a timescale $\sim t$ is thus given by

$$\begin{eqnarray}&&\begin{array}{l}{{ \mathcal E }}_{\nu }\approx \displaystyle \frac{3}{8}{\displaystyle \int }^{t}{\tau }_{p\gamma }{f}_{E^{\prime} }{L}_{\mathrm{sh}}{dt}\sim \displaystyle \frac{3}{8}{\tau }_{p\gamma }{f}_{E^{\prime} }{L}_{\mathrm{sh}}t\\ \approx \ 8.9\times {10}^{37}\mathrm{erg}\,{f}_{E^{\prime} }{\sigma }_{-1}^{1/2}{f}_{\xi ,-3}^{-22/15}{\nu }_{\mathrm{GHz}}^{71/30}{t}_{\mathrm{ms}}^{6/5}{{ \mathcal E }}_{{\rm{r}},40}^{7/6}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{1/2}\end{array}\end{eqnarray} \tag{ 33 }$$

where the factor 3/8 accounts for the fraction of the proton energy that goes into neutrino products (Waxman & Bahcall 1999) and ${f}_{E^{\prime} }\sim 0.01-0.5$ (Equation (17)) is the fraction of the shock power placed into protons obeying the threshold condition.

One can define a dimensionless neutrino radiative efficiency,

$$\begin{eqnarray}&&\begin{array}{l}{\eta }_{\nu }\equiv \displaystyle \frac{{{ \mathcal E }}_{\nu }}{{{ \mathcal E }}_{\mathrm{flare}}}\approx \displaystyle \frac{3}{32}{\tau }_{p\gamma }[{\varepsilon }_{\nu ,c}]{f}_{E^{\prime} }\\ \approx \ 3.7\,\times \,{10}^{-7}{f}_{E^{\prime} }{\sigma }_{-1}^{1/2}{f}_{\xi ,-3}^{-2/3}{\nu }_{\mathrm{GHz}}^{65/30}{t}_{\mathrm{ms}}{{ \mathcal E }}_{{\rm{r}},40}^{1/6}{\left(\displaystyle \frac{t}{{t}_{\mathrm{FRB}}}\right)}^{1/2}\end{array}\end{eqnarray} \tag{ 34 }$$

which is valid at all times ${t}_{\mathrm{FRB}}\lesssim t\lesssim {t}_{\max }$.

During the early epoch $t\lesssim {t}_{\mathrm{nth}}$, when neutrinos are produced by thermal protons, we have ${f}_{E^{\prime} }\sim 0.5$ (Equation (17)) and hence ${\eta }_{\nu }\sim {10}^{-7}\mbox{--}{10}^{-6}$ for fiducial parameters. At times $t\gg {t}_{\mathrm{nth}}\sim {t}_{\mathrm{FRB}}$ when only nonthermal protons can produce neutrinos, we have ${f}_{E^{\prime} }\sim {10}^{-2}$ if the shocks place a fraction ${\epsilon }_{\mathrm{rel}}\sim 0.1$ of their energy into the power-law tail. Although the efficiency starts low in this case, it grows in time ${\eta }_{\nu }\propto {t}^{1/2}$ until reaching a peak value once $t\sim {t}_{\max }$ of

$$\begin{eqnarray}&&\begin{array}{l}{\eta }_{\nu }[{t}_{\max }]\approx 4.8\times {10}^{-7}\left(\displaystyle \frac{{f}_{E^{\prime} }}{{10}^{-2}}\right)\\ \times \ {\sigma }_{-1}^{\tfrac{15}{14}}{f}_{\xi ,-3}^{-\tfrac{106}{105}}{\nu }_{\mathrm{GHz}}^{\tfrac{563}{210}}{t}_{\mathrm{ms}}^{\tfrac{23}{35}}{{ \mathcal E }}_{{\rm{r}},40}^{\tfrac{19}{42}},\end{array}\end{eqnarray} \tag{ 35 }$$

corresponding to a total radiated neutrino energy

$$\begin{eqnarray}&&\begin{array}{l}{{ \mathcal E }}_{\nu }={\eta }_{\nu }[{t}_{\max }]{{ \mathcal E }}_{\mathrm{flare}}\\ \approx \ 1.1\,\times \,{10}^{38}\mathrm{erg}\left(\displaystyle \frac{{f}_{E^{\prime} }}{{10}^{-2}}\right){\sigma }_{-1}^{15/14}{f}_{\xi ,-3}^{-38/21}{\nu }_{\mathrm{GHz}}^{121/42}{t}_{\mathrm{ms}}^{6/7}{{ \mathcal E }}_{{\rm{r}},40}^{61/42}.\end{array}\end{eqnarray} \tag{ 36 }$$

The total neutrino emission from the shocks peaks at an energy,

$$\begin{eqnarray}&&{\varepsilon }_{\max }[{t}_{\max }]\approx 3.6\,\mathrm{PeV}\,{\sigma }_{-1}^{5/14}{f}_{\xi ,-3}^{-26/105}{\nu }_{\mathrm{GHz}}^{43/210}{{ \mathcal E }}_{{\rm{r}},40}^{11/42}{t}_{\mathrm{ms}}^{3/35}\end{eqnarray} \tag{ 37 }$$

which scales weakly with FRB properties and model parameters.

Figure 1 shows an example calculation of the characteristic energies versus time, while Figure 2 shows the build-up of the neutrino spectrum with time as the shock propagates outwards (bottom panel). Figure 3 shows several key properties related to neutrino emission—such as the duration, spectral peak, total radiated energy, and fluence—for a sample of FRBs with measured properties (${t}_{\mathrm{FRB}}$, ${{ \mathcal E }}_{\mathrm{radio}}$, ${\nu }_{\mathrm{obs}}$) and known distances. Approximate IceCube sensitivity curves ⁸ are shown for comparison.

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The upshot of our work is the prediction of an efficiency of up to ${\eta }_{\nu }\sim {10}^{-8}\mbox{--}{10}^{-5}$ (Equation (35); Figure 3) for converting magnetar flare energy into a neutrino burst of energy ∼1–10 PeV, near the peak sensitivity range of IceCube. The energy radiated in neutrinos is typically ∼10–100 times smaller than the radio isotropic energy, making the signal extremely challenging to detect.

Consider first the recent FRB 200428 from the Galactic magnetar SGR 1935+2154. Taking ${{ \mathcal E }}_{\mathrm{flare}}\sim 2{E}_{{\rm{X}}/\gamma }\sim {10}^{40}$ erg, we predict a neutrino burst of energy ${{ \mathcal E }}_{\nu }\lesssim {10}^{32}$ erg, corresponding to a fluence (given an assumed source distance of 10 kpc) of ${F}_{\nu }\sim {10}^{-12}$ GeV cm⁻² (Figure 3). This is ∼11 orders of magnitude smaller than the IceCube upper limit of $5\times {10}^{-2}$ GeV cm⁻² (Vandenbroucke 2020). Thus, the neutrino non-detection of FRB 200428 is not constraining on the baryon shock model.

What about a more energetic Galactic magnetar flare? Although no FRB was detected in association with the 2004 giant flare from SGR 1806-20 (Tendulkar et al. 2016), this could be explained if the shock-generated plasmoid ejection was directed away from our line of sight (in which case the FRB would be dimmer due to relativistic beaming effects). Assuming a more fortuitous geometry of a future flare and scaling ${{ \mathcal E }}_{\mathrm{flare}}$ up by a factor of $\sim {10}^{6}$ to the $\sim {10}^{46}$ erg energy scale of the gamma-ray emission from giant flares, the predicted flux for the nearest magnetar (1E 1048.1–5937 at $\approx 2.7\,\mathrm{kpc};$ Gaensler et al. 2005) could reach $\sim {10}^{-2}$ GeV cm⁻². In this case the detection of a neutrino from a giant flare becomes potentially feasible (see also Gelfand et al. 2005), particularly given proposed future upgrades to neutrino detectors (e.g., The IceCube-Gen2 Collaboration et al. 2020). Note that an FRB energy $\gtrsim {10}^{12}$ Jy ms is thus required to produce detectable neutrino emission in this case.

Individual extragalactic FRBs discovered to date are unlikely to be detectable due to their much greater distances (bottom panel of Figure 3). However, thanks to new surveys like the Australian Square Kilometre Array Pathfinder (ASKAP; Bannister et al. 2019) and the Canadian Hydrogen Intensity Mapping Experiment (CHIME)/FRB (CHIME/FRB Collaboration et al. 2018), the rate of FRB discoveries is rapidly growing and may reach thousands or more over the coming years.

The distribution of FRB energies ${dN}/d{{ \mathcal E }}_{\mathrm{radio}}\propto {{ \mathcal E }}_{\mathrm{radio}}^{-\gamma }$ (where $\gamma \lesssim 2;$ Law et al. 2017; Lu & Piro 2019; Cruces et al. 2020; though see e.g., Gourdji et al. 2019) is such that the total radiated energy is dominated by the rarest, most energetic bursts. Given the neutrino radiative efficiency of the shocks ${\eta }_{\nu }\propto {{ \mathcal E }}_{\mathrm{radio}}^{19/42}$ (Equation (35)), the total radiated neutrino energy is also dominated by the high-${{ \mathcal E }}_{\nu }$ bursts.

A key question then becomes the upper cut-off energy to the FRB energy distribution, ${{ \mathcal E }}_{{\rm{r}},\max }$. Modeling the ASKAP FRB sample, Lu & Piro (2019) found a range of allowed values ${{ \mathcal E }}_{{\rm{r}},\max }\approx 3\times {10}^{42}-2\times {10}^{44}$ erg for a burst frequency width $\nu \sim 1.4\,\mathrm{GHz}$. In what follows, we optimistically assume ⁹ ${{ \mathcal E }}_{{\rm{r}},\max }\approx {10}^{44}$ erg, bursts for which the local volumetric rate is estimated as ${ \mathcal R }(z=0)\sim 6$ Gpc⁻³ yr⁻¹ (adopting a $\gamma =1.6$ power-law luminosity function; Lu & Piro 2019; Luo et al. 2020). At this rate, the closest FRB of this energy per year occurs at a distance ${D}_{\min }\sim$ 0.34 Gpc. Given its predicted neutrino energy ${{ \mathcal E }}_{\nu }\sim 5\times {10}^{43}$ erg (Equation (36)), its neutrino flux would be ${F}_{\nu }\approx {{ \mathcal E }}_{\nu }/(4\pi {D}_{\min }^{2})\sim {10}^{-9}$ GeV cm⁻², i.e., still ∼7 orders of magnitude below the IceCube detection threshold.

The sensitivity could be improved by a stacked joint analysis of a large FRB sample (e.g., Aartsen et al. 2018; Kheirandish et al. 2019). Unfortunately, given the relatively long duration ${t}_{\max }\sim 3000\,{\rm{s}}$ for the most energetic ${{ \mathcal E }}_{\mathrm{radio}}\approx {10}^{44}$ erg bursts described above, the search becomes background-dominated after only a few bursts, after which the detector sensitivity quickly saturates. Although less-energetic FRBs are more common and produce shorter-lived neutrino bursts ${t}_{\max }\propto {{ \mathcal E }}_{\mathrm{radio}}^{4/7}$, their sharply lower radiated energies ${{ \mathcal E }}_{\nu }\propto {{ \mathcal E }}_{\mathrm{radio}}^{61/42}$ largely cancel out these benefits.

In addition to a burst of neutrinos (from ${\pi }^{\pm }$ decay), we predict a burst of ∼TeV−PeV gamma-rays from neutral pion decay. As shown in Figure 4, the TeV thermal neutrinos fall in the sensitivity range of ground-based water Cherenkov detectors, including the Altitude Water Cherenkov Observatory (HAWC; Martinez-Castellanos 2019), and upcoming and future detectors such as the Large High Altitude Air Shower Observatory (LHAASO; di Sciascio & Lhaaso Collaboration 2016) and the Southern Wide-field Gamma-ray Observatory (SWGO; Huentemeyer et al. 2019). However, again for fiducial parameters we require a nearby source $\lesssim 0.1\,\mathrm{kpc}$ for a detection. Furthermore, the pion decay signal could be overwhelmed by nonthermal leptonic emission, depending on the electron acceleration efficiency of the shocks.

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With the discovery of FRB 200428, flaring magnetars are the leading FRB model. However, several distinct mechanisms and environments around a magnetar for generating the radio burst—and any accompanying higher-frequency afterglow—have been proposed, which have proven challenging to distinguish observationally. A unique prediction of baryon-loaded shock-powered FRB models (Metzger et al. 2019; Margalit et al. 2020b) are neutrinos generated by ions heated or accelerated at the shock interact with thermal synchrotron shock photons via the photohadronic process.

We predict a burst of ∼TeV−PeV neutrinos of total energy ${{ \mathcal E }}_{\nu }\approx {10}^{35}\mbox{--}{10}^{44}$ erg (depending most sensitively on the FRB isotropic energy; Equation (36)) which lasts for a timescale ${t}_{\max }\sim 0.1\mbox{--}1000\,{\rm{s}}$ (Equation (25)) following the radio burst (once accounting for the time delay due to the finite propagation speed of the radio waves as inferred from the dispersion measure). Our calculations make several optimistic assumptions, including (1) nonthermal particle acceleration at magnetized shocks with an efficiency of 10%; (2) the presence of a baryon-loaded medium which extends at least a distance ${r}_{\max }\equiv r[{t}_{\max }]\sim (3\mbox{--}30){r}_{\mathrm{FRB}}$ beyond the radius ${r}_{\mathrm{FRB}}$ probed by the FRB emission itself. ¹⁰

Although the detection prospects with present neutrino observatories are extremely challenging (to put it lightly), the detection of even a single neutrino from an FRB would be a smoking gun for this model. The most promising potentiality is a giant flare from a nearby Galactic magnetar with the fortuitous geometry of the shock-generating plasmoid being directed along our line of sight.

We thank Ali Kheirandish for helpful discussion about IceCube sensitivities to FRBs. We thank Israel Martinez for helpful discussion about HAWC sensitivities to bursts. B.D.M. acknowledges support from the Simons Foundation (grant 606260). Support for K.F. was provided by NASA through the NASA Hubble Fellowship grant #HST-HF2-51407 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. B.M. is supported by NASA through the NASA Hubble Fellowship grant #HST-HF2-51412.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555.