title: Neutrino Counterparts of Fast Radio Bursts
authors: Brian D. Metzger Ke Fang Ben Margalit
text: 
  
  
  
  Brian D. Metzger
  bdm2129@columbia.edu
  
  
  
  Department of Physics, Columbia University, New York, NY 10027
   
  NHFP Einstein Fellow
  Kavli Institute for Particle Astrophysics and Cosmology (KIPAC), Stanford University, Stanford, CA 94305, USA
  
  
  NHFP Einstein Fellow
  Astronomy Department and Theoretical Astrophysics Center, University of California, Berkeley, Berkeley, CA 94720, USA
  
  
  
  The discovery of a luminous radio burst, ,  with properties similar to those of fast radio bursts (FRB), in coincidence with an X-ray flare from the Galactic magnetar SGR 1935+2154, supports magnetar models for cosmological FRBs.  The burst's X-ray to radio fluence ratio, as well as the X-ray spectral shape and peak energy, are consistent with  being the result of an ultra-relativistic shock (powered, e.g., by an ejected plasmoid) propagating into a magnetized baryon-rich external medium; the shock simultaneously generates X-ray/gamma-rays via thermal synchrotron emission from electrons heated behind the shock, and coherent radio emission via the synchrotron maser mechanism.  Here, we point out that a unique consequence of this baryon-loaded shock scenario is the generation of a coincident burst of high-energy neutrinos, generated by photo-hadronic interaction of relativistic ions-heated or accelerated at the shock-with thermal synchrotron photons.  We estimate the properties of these neutrino burst FRB counterparts and find that a fraction ∼ 10^-8-10^-5 of the flare energy (or ∼ 10^-4-10^-1 of the radio isotropic energy) is channeled into production of neutrinos with typical energies ∼ TeV-PeV.  We conclude by discussing prospects for detecting this signal with IceCube and future high-energy neutrino detectors.
  
  
  
  
  
  
  
  
  
  
  
  § INTRODUCTION
   
  
  
  Fast radio bursts (FRBs) are millisecond duration pulses of coherent radio emission of extragalactic origin (; see ,  for reviews).  Although many models have been proposed for FRBs (), 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. ), postulate an origin associated with flaring activity of magnetars <cit.>.  
  
  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 <cit.>.  The energy radiated by the burst (hereafter FRB 200428) in the ∼ GHz radio band, ℰ_ radio∼ 10^34-10^35 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  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; ), such a unified scenario can account for most or all of the cosmological FRB population <cit.>.  
  
  Despite the breakthrough discovery of , 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  find a natural explanation in some magnetar models, while placing others in tension <cit.>.  
  
  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 <cit.>.  In these scenarios the radio precursor is generated at the shock front by electrons undergoing the synchrotron maser process (e.g. ).  The “synchrotron maser blastwave model" naturally accounts for the high observed linear polarization of many bursts (e.g. ), their complex spectral energy distribution (e.g. ), and the observed downward drifting of burst subpulses (e.g. ) due to the decreasing Doppler factor as the blast wave decelerates as it sweeps up material <cit.>.
  
  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.  <cit.> and <cit.> postulate an upstream medium composed of electron/positron plasma, either representing a nebula surrounding the magnetar, or an ultra-relativistic spin-down powered wind, respectively.  <cit.> instead consider 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 ().  
  
  As shown by <cit.> and recounted below, the predictions of the baryon-loaded electron-ion upstream model agree quantitatively with the properties of , provided one interprets the hard X-ray emission from the flare <cit.> as incoherent synchrotron radiation from relativistically-hot thermal electrons heated by the same shock generating the FRB.  Since 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 ISM-is naturally expected  (see also ).  <cit.> summarize several falsifiable predictions of this model, testable by future Galactic or nearby extragalactic FRBs (see also  for criticisms of shock models). 
  
  An important consequence of an electron-ion upstream medium is the prediction at the shocks of heating, and potential non-thermal 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 <cit.>.  
  
  In this Letter, we estimate the properties of the neutrino bursts predicted to accompany FRBs in the synchrotron maser electron-ion blastwave scenario, and assess their detectability with the IceCube Neutrino Observatory.  Table <ref> summarizes several definitions which may be useful to refer to throughout the text.
  
  
  
  § MAGNETIZED RELATIVISTIC SHOCKS
  
  
  Following <cit.>, we consider an ultra-relativistic blast wave generated by flare ejecta from the magnetar (e.g. a magnetically-dominated plasmoid; e.g. ) with an isotropic energy ℰ_ 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 δ t, as is typically satisfied for FRBs of duration ≳ 1 ms if the flare duration δ t ∼ few × R_ ns/c ∼ 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 blastwave follows a self-similar evolution <cit.> resulting in the following evolution for the shock, Lorentz factor, and kinetic luminosity of the forward shock,
  
      Γ   =   Γ_ FRB(t/t_ FRB)^(k-3)/2(4-k)k=0= Γ_ FRB(t/t_ FRB)^-3/8  ,
      
      r_ sh   =    r_ FRB(t/t_ FRB)^1/(4-k)k=0=r_ FRB(t/t_ FRB)^1/4 , 
      
      n_ ext   =    n_ FRB(t/t_ FRB)^-k/(4-k)k=0=n_ FRB,
      
      
      L_ sh   =    L_ FRB(t/t_ FRB)^-1≃ℰ_ flare/4 t_ FRB(t/t_ FRB)^-1 ,
  
  where t_ FRB is the observed duration of the FRB (time for the spectral energy distribution of the maser emission to sweep across the observing band; ) and {Γ_ FRB, r_ FRB, n_ FRB,L_ FRB} are the shock properties at t ∼ t_ FRB.  
  
  The latter quantities are related to observable properties of the FRB, namely its duration t_ FRB = t_ ms ms, radio frequency ν_ obs = ν_ GHz GHz, isotropic radio energy ℰ_ radio = ℰ_ r,40 10^40 erg, according to <cit.>
  
      r_ FRB≈ 5× 10^12 cm  f_ξ,-3^-2/15ν_ GHz^-7/15t_ ms^1/5ℰ_ r,40^1/3, 
      
      n_ FRB≈ 420  cm^-3 f_ξ,-3^-4/15ν_ GHz^31/15t_ ms^2/5ℰ_ r,40^-1/3 , 
      Γ_ FRB≈ 287  f_ξ,-3^-1/15ν_ GHz^-7/30t_ ms^-2/5ℰ_ r,40^1/6
  
  where f_ξ = 10^-3f_ξ,-3 is the efficiency of the maser emission, normalized to a characteristic value found by particle-in-cell (PIC) plasma simulations of magnetized shocks <cit.> and we have assumed a 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
  
      η≡ℰ_ radio/ℰ_ flare≈ 4× 10^-5f_ξ,-3^4/5ν_ GHz^-1/5t_ ms^-1/5,
  
  such that 
  
      ℰ_ flare≈ 2.4× 10^44 erg f_ξ,-3^-4/5ν_ GHz^1/5t_ ms^1/5ℰ_ r,40
  
  and the kinetic power of the shock at the epoch of FRB emission is given by
  
      L_ FRB≃ℰ_ flare/4 t_ FRB≈ 6× 10^46  erg s^-1f_ξ,-3^-4/5ν_ GHz^1/5t_ ms^-4/5ℰ_ r,40
  
  The population of cosmological FRBs with known distances exhibit typical ranges ℰ_ radio∼ 10^37-10^41 erg, t_ FRB∼ 0.1-10 ms, such that one infers r_ FRB∼ 10^12-10^13 cm, Γ_ FRB∼ 10^2-10^3, and n_ FRB∼ 10^2-10^4 cm^-3 <cit.>.  From the observed rate of downward frequency drifting of the sub-pulses, which in this model could arise from deceleration of the blast wave <cit.>, one infers power-law indices of the external medium spanning a wide-range k≈ [-2,2] <cit.>.  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 (<ref>, <ref>), the blastwave transitions from ulta-relativistic to non-relativistic expansion (Γ≃ 1) after a time
  
      t_ nr≃ t_ FRBΓ_ FRB^8/3≃ 3.5× 10^3 s  f_ξ,-3^-8/45ν_ GHz^-28/45t_ ms^-1/15ℰ_ r,40^4/9,
  
  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. ).  Given the strong magnetic field behind the shock, the electrons radiate (incoherent) synchrotron emission.  The latter peaks at a photon energy <cit.>
  
      ϵ_ pk = ħ eB'/m_e cγ̅^2Γ,
  
  where γ̅ = (m_p/m_e)Γ/2 is the mean thermal Lorentz factor, B' = √(64πσΓ^2m_p c^2n_ ext) is the post-shock magnetic field, and σ is the magnetization of the upstream medium.  Thus we have,
  
      ϵ_ pk≈ 
      235  MeVσ_-1^1/2f_ξ,-3^-2/5ν_ GHz^1/10ℰ_ r,40^1/2t_ ms^-7/5(t/t_ FRB)^-3/2,
  
  in the hard X-ray/γ-ray range, where σ_-1≡σ/(0.1).[The value of σ in the baryon-loaded upstream is uncertain.  A minimum value σ≳ 10^-3 is required for operation of the synchrotron maser, while very large values σ≫ 1 are also disfavored due to the declining radiative efficiency with increasing σ <cit.>.]  The cooling frequency is given by
  
      ϵ_ c   =   ħ eB'/m_e cγ_ c^2Γ≈
            0.023  MeV  σ_-1^-3/2f_ξ,-3^2/3ν_ GHz^-65/30t_ ms^-1ℰ_ r,40^-1/6(t/t_ FRB)^-1/2 ,
  
  where γ_ c = (6π m_e c/σ_ TΓ B'^2t) is the Lorentz factor of the electrons that cool on the expansion time.  The post-shock electrons thus remain fast-cooling (ϵ_ c≲ϵ_ pk) for a timescale
  
      t_ c≈ 10  s σ_-1^2f_ξ,-3^-16/15ν_ GHz^34/15t_ ms^3/5ℰ_ r,40^2/3,
  
  much longer than the FRB duration; thus the radiated X-ray/gamma-ray energy E_ X/γ≈ℰ_ flare/2 is equal to that given to the electrons.
  
  Considering the application to  <cit.>, the value η∼ 10^-4-10^-5 predicted by equation <ref> agrees with the measured ratio of the radio and X-ray fluence (ℰ_ radio∼ 10^34-10^35 erg; ; E_ X/γ∼ 10^39-10^40 erg; ) for f_ξ∼ 10^-3.  For the same parameters, equation (<ref>) predicts a peak frequency (given the observed duration t ∼ 3 ms of the X-ray spikes of ) of ϵ_ pk∼ 300σ_-1^1/2 keV, 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. ), similar to predictions for a fast-cooling thermal synchrotron spectrum <cit.>.  
  Taken together, these observations support a magnetized shock propagating into a baryon-loaded medium as the origin of  and its associated X-ray burst <cit.>.  
  
  
  
  
  § HIGH-ENERGY NEUTRINO EMISSION
  
  
  
  
   §.§ Proton Heating and Acceleration
  
  
  A fraction ∼ 1/2 of the shock's energy goes into thermally heating ions, which we hereafter take to be protons for simplicity.  The protons achieve a relativistic Maxwellian distribution with an average particle energy of E̅' ≈Γ 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 ≫E̅' (e.g. via diffusive acceleration; ), into a power-law non-thermal spectrum of the form 
      dN'_p/dE'∝(E'/E̅')^-q,    E̅' < E'  < E_max',
  
  where E_max' 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 ∼ E' ≥E̅' can be roughly written as,
  
      f_E'≈
      1/2     E ∼E̅
      ϵ_ rel/Λ    E ≫E̅ ,
  
  where ϵ_ rel≲ 0.1 is an acceleration efficiency and Λ≡ ln(E_max'/E̅') ∼ 10 (see below).
  
  In order to estimate the value of E_max' we equate the proton acceleration time (estimated by the Larmor gyration time) t_ acc∼ 2π E'/eB'c to various loss timescales.   The most stringent constraint[Other proton loss timescales, such as due to synchrotron cooling, are much longer-and hence unconstraining on E_max- than the dynamical timescale for shock parameters of interest.] comes from equating t_ acc to the dynamical time, t_ dyn≃ r_ sh/(Γ c), over which adiabatic losses occur.  This gives
  
      E_max'    ≃   eB'r_ sh/2πΓ = er_ shc√((16/π)σ m_p n_ ext)
         ≈    850   TeVσ_-1^1/2f_ξ,-3^-4/15ν_ GHz^17/30t_ ms^2/5ℰ_ r,40^1/6(t/t_ FRB)^1/4.
  
  Note that E_max' ∼ 1  PeV∼ 10^3-10^4E̅' on the timescale of the FRB t ∼ t_ FRB, increasing by t ∼ t_ nr to ≳ 10 PeV.
  
  
  
  
  
  
  
  
   §.§ Pion Production and Neutrino Emission
  
  
  
  Although the electrons generally cool faster than the dynamical timescale at early times (t ≪ t_ c; eq. <ref>), 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 +γ→ p + π^±,0 <cit.>, generating high-energy gamma-rays and neutrinos via the decay of π^0 and π^±, respectively.  The rate of hadronuclear interactions (e.g. p + p →  p + p + π^±,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
  
      ϵ_ pk' ≈ϵ_ pk/Γ≈
       0.8  MeVσ_-1^1/2f_ξ,-3^-1/3ν_ GHz^1/3t_ ms^-1ℰ_ r,40^1/3(t/t_ FRB)^-9/8 ,
  
  where we have used equations (<ref>) and (<ref>).  Immersed in this radiation field, pion production peaks at the Δ-resonance at a proton energy (e.g. )
  
      E_ thr' ≈E_Δ'/ϵ_ pk'm_p c^2/2≈
            170  GeVσ_-1^-1/2f_ξ,-3^1/3ν_ GHz^-1/3t_ msℰ_ r,40^-1/3(t/t_ FRB)^9/8,
  
  where E_Δ' ≈ 0.3 GeV.  Comparing this threshold to the proton thermal energy,
  
      E_ thr'/E̅'≈2 E_ thr'/Γ m_p c^2≈ E_Δ'/ϵ_ pk
            ≈ 1.3  σ_-1^-1/2f_ξ,-3^2/5ν_ GHz^-1/10t_ ms^7/5ℰ_ r,40^-1/2(t/t_ FRB)^3/2.
  
  Thus, on timescales of the FRB emission (t ∼ t_ FRB), protons which are either thermal (E' ∼E̅'), or only moderately into the non-thermal tail (E' ≳E̅'; eq. <ref>), are capable of pion-producing on synchrotron photons.  However, at times t ≫ t_ FRB, only the non-thermal protons with E' ≫E̅' have enough energy to produce pions.
  This transition (E̅' = E_ thr') from thermal to non-thermal-only production occurs at a time
  
      t_ nth≈ 8.5× 10^-4 s  σ_-1^1/3f_ξ,-3^-4/15ν_ GHz^1/15t_ ms^1/15ℰ_ r,40^1/3 .
  
  
  The pions generated decay into neutrinos of lab-frame energy ε≈Γ E'/20  ().  The minimum neutrino energy at time t is therefore   
  
      ε_ min≈ max[ΓE̅'/20,Γ E_ thr'/20] ≈
             max 
      1.9  TeV f_ξ,-3^-2/15ν_ GHz^-7/15t_ ms^-4/5ℰ_ r,40^1/3(t/t_ FRB)^-3/4, t ≲ t_ nth
      
      2.5  TeVσ_-1^-1/2f_ξ,-3^4/15ν_ GHz^-17/30t_ ms^3/5ℰ_ r,40^-1/6(t/t_ FRB)^3/4, t ≳ t_ nth
  
  
  From equation (<ref>), one can also define an absolute maximum neutrino energy, as set by the accelerator,
  
      ε_ max≈Γ E_ max'/20≈ 12  PeV  σ_-1^1/2f_ξ,-3^-1/3ν_ GHz^1/3ℰ_ r,40^1/3(t/t_ FRB)^-1/8
  
  Equating ε_ min(t > t_ nth) (eq. <ref>) to ε_ max, we obtain
  
      t_ max≈ 17  s  σ_-1^8/7f_ξ,-3^-24/35ν_ GHz^36/35t_ ms^11/35ℰ_ r,40^4/7.
  
  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 ∼ϵ_ pk) exceeds E_ max.  
  
  
  cc
  Definitions
  700pt
  
  
  Symbol     Description 
   
  E     Proton energy 
  
  ϵ     Photon energy 
  
  ε     Neutrino energy 
  
  ℰ     Total energy 
  
  
  
  
  ccc
  Key Timescales
  700pt
  
  
  Symbol     Description     
  Typical Value^† 
   
  t_ FRB     FRB duration     ∼ 0.1-10 ms 
  
  t_ nth (eq. <ref>)     Duration of ν's from thermal protons     ∼ 0.1-1 ms 
  
  t_ c (eq. <ref>)     Synchrotron emission no longer fast-cooling      ∼ 0.1-10^3 s 
   t_ max (eq. <ref>)     Duration of ν's from non-thermal protons     ∼ 1-10^3 s 
   
  t_ nr (eq. <ref>)     Transition to non-relativistic shock     ∼ 10^2-10^5 s 
  
  ^†For σ∼ 0.1 and ℰ_ rad∼ 10^37-10^43 erg in the range of cosmological FRB.
  
  
  
  Before t_ max, higher-energy neutrinos with ε_ min < ε < ε_ max can be created by the interaction of protons with energies E' ≫ E_ thr', E̅' with photons of lower energies ϵ' ≪ϵ_ pk' below the synchrotron peak.  In fact, at times t < t_ c (eq. <ref>) such interactions dominate neutrino production (by non-thermal protons) because the photon number spectrum in the fast-cooling regime,
  
      ϵ(dN_γ/dϵ) ∝ϵ^-1/2, ϵ_ c≲ϵ≲ϵ_ pk,
  
  peaks at ϵ_ c < ϵ_ pk.[At times t ≳ t_ c the spectrum below ϵ_ pk steepens to the standard slow-cooling spectrum ϵ(dN_γ/dϵ) ∝ϵ^1/3 for which photons of energy ϵ_ pk dominate by both energy and number.] 
  
  Before t_ c the energy distribution of neutrinos from non-thermal protons will thus peak at an energy,
  
      ε_ c   ≈   Γ E_Δ'/20 ϵ_ c'm_p c^2/2≈
         26  PeV    σ_-1^3/2f_ξ,-3^-4/5ν_ GHz^17/10t_ ms^1/5ℰ_ r,40^1/2(t/t_ FRB)^-1/4,
  
  where ϵ_ c' = ϵ_ c/Γ.  The value of ε_ 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 
  
      ε_π,μ^ syn   =   (6π m_π,μ c/σ_T B'^2 τ_π,μ)^1/2(m_π,μ/m_e) m_π,μ c^2Γ
         =    (30,   1.7)× 10^18  eV σ_-1^-1/2f_ξ,-3^2/15ν_ GHz^-31/30t_ ms^-1/5ℰ_ r,40^1/6,
  
  where m_π,μ and τ_π,μ are the masses and rest-frame lifetimes of pions and muons, respectively. Because ε_π^ syn≫ 4 ε_ max and ε_μ^ syn≫ 3 ε_ 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. 
  
  
  The timescales discussed in this section are summarized in Table <ref>.  Although all of the timescales increase with the flare energy, the hierarchy t_ nth < t_ c < t_ max < t_ nr is generally preserved for σ≲ 1 and moderate variations around fiducial parameters.
  
  In summary, at early times (t ≲ t_  nth) we expect ∼ 1-10 TeV neutrinos generated by the interaction of the thermal protons with synchrotron photons of energy ∼ϵ_ pk.  This prediction is relatively robust (within the baryon shock model) because proton heating at the shocks is inevitable and since t_ nth∼ t_ FRB does not need to assume a continuation of the external medium outside the region probed by the observed FRB.
  
  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 non-thermal protons.  Insofar as the proton spectrum is flat (q ≃ 2) the dominant interaction will initially (t ≪ t_ c) be between photons of energy ∼ϵ_ c≪ϵ_ 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 ∼ϵ_ pk generating neutrinos of energy ε_ min(t ≳ t_ c) ≳ 10 PeV.   
  
  
  
   §.§ Neutrino Radiative Efficiency
  
  
  
  
  
  
  What fraction of the available proton luminosity ∼ L_ sh/2 will be radiated as neutrinos?  One can define an “optical depth" for neutrino production over each decade in observer time ∼ t,
  
      τ_pγ   ≈    n_γ'σ_p γκ_pγ r'
  
  where n_γ' is the target photon number density given below, r' ≈ r_ sh/Γ is the characteristic radius of the post-shock region in the comoving frame (through which relativistic protons accelerated at the shock traverse to escape without losses), σ_pγ is the cross section of photopion production, which is ∼ 5× 10^-28 cm^2 <cit.> corresponding to its value near the Δ-resonance (and ∼ 1.6× 10^-28 cm^2 above the resonance).  The factor κ_pγ∼ 0.15 is the average fraction of energy lost from a proton per collision (“elasticity").  
  Consider first proton interactions with photons of energy ϵ = ϵ_ pk which generate neutrinos of energy ε_ min (eq. <ref>).  Their number density is given by n_γ' ≈ U_γ'/ϵ_ pk', where
  
      U_γ' ≈L_ sh/2/4π r_ sh^2 c Γ^2
      1     t ≤ t_ c
      (t/t_ c)^-1/2    t ≥ t_ c ,
  
  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_ c)^-1/2 factor accounts for the reduction in the radiated power when the electrons become slow-cooling at t ≳ t_ c (eq. <ref>).
  
  Combining with the above
  
      τ_pγ[ε_ν,min]    ≈   L_ shσ_pγκ_pγ/8π r_ sh c Γ^2ϵ_ pk
      1     t ≤ t_ c
      
      (t/t_ c)^-1/2    t ≥ t_ c ,
      
         ≈    3.9× 10^-8σ_-1^-1/2f_ξ,-3^-2/15ν_ GHz^31/30t_ ms^6/5ℰ_ r,40^-1/6×
            (t/t_ FRB)     t ≤ t_ c
      (t_ c/t_ FRB)(t/t_ c)^1/2    t ≥ t_ c
  
  At time t ≲ t_ c, the photon density n_γ at ϵ = ϵ_ c is larger by a factor (t/t_ c)^-1/2 than at ϵ_ pk (eq. <ref>); thus, the optical depth to generating ε_ν,c neutrinos is larger by the same factor:
  
      τ_pγ[ε_ν,c] ≈τ_pγ[ε_ν,min](t/t_ c)^-1/2
            ≈ 3.9× 10^-6σ_-1^1/2f_ξ,-3^-2/3ν_ GHz^65/30t_ msℰ_ r,40^1/6(t/t_ FRB)^1/2, t ≤ t_ c
  
  Interestingly, τ_ p,γ at the peak of the neutrino energy spectrum increases ∝ t^1/2 both before and after t_ c.  The fluence emitted in neutrinos over a timescale ∼ t is thus given by
  
      ℰ_ν   ≈   3/8∫^tτ_pγf_ E'L_ sh dt ∼3/8τ_pγf_ E'L_ sht 
         ≈    8.9× 10^37 erg  f_ E'σ_-1^1/2f_ξ,-3^-22/15ν_ GHz^71/30t_ ms^6/5ℰ_ r,40^7/6(t/t_ FRB)^1/2
  
  where the factor 3/8 accounts for the fraction of the proton energy that goes into neutrino products <cit.> and f_ E'∼ 0.01-0.5 (eq. <ref>) is the fraction of the shock power placed into protons obeying the threshold condition.
  
  One can define a neutrino radiative efficiency, 
  
      η_ν   ≡   ℰ_ν/ℰ_ flare≈3/32τ_pγ[ε_ν,c] f_ E'
         ≈    
      3.7× 10^-7f_ E'σ_-1^1/2f_ξ,-3^-2/3ν_ GHz^65/30t_ msℰ_ r,40^1/6(t/t_ FRB)^1/2
   
  which is valid at all times t_ FRB≲ t ≲ t_ max. 
  
  During the early epoch t ≲ t_ nth, when neutrinos are produced by thermal protons, we have f_ E'∼ 0.5 (eq. <ref>) and hence η_ν∼ 10^-7-10^-6 for fiducial parameters.  At times t ≫ t_ nth∼ t_ FRB when only non-thermal protons can produce neutrinos, we have f_ E'∼ 10^-2 if the shocks place a fraction ϵ_ rel∼ 0.1 of their energy into the power-law tail.  Although the efficiency starts low in this case, it grows in time η_ν∝ t^1/2 until reaching a peak value once t ∼ t_ max of 
  
      η_ν[t_ max] ≈ 4.8× 10^-7(f_ E'/10^-2)
      σ_-1^15/14f_ξ,-3^-106/105ν_ GHz^563/210t_ ms^23/35ℰ_ r,40^19/42,
  
  corresponding to a total radiated neutrino energy 
  
      E_ν = η_ν[t_ max] ℰ_ flare≈
             1.1× 10^38 erg(f_ E'/10^-2)σ_-1^15/14f_ξ,-3^-38/21ν_ GHz^121/42t_ ms^6/7ℰ_ r,40^61/42
  
  The total neutrino emission from the shocks peaks at an energy,
  
      ε_ max[t_ max] ≈ 
      3.6  PeV  σ_-1^5/14f_ξ,-3^-26/105ν_ GHz^43/210ℰ_ r,40^11/42 t_ ms^3/35
  
  which scales weakly with FRB properties and model parameters.
  
  Figure <ref> shows an example calculation of the characteristic energies versus time (top panel) and the build-up of the neutrino spectrum with time as the shock propagates outwards (bottom panel).  Figure <ref> 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_ FRB, ℰ_ radio, ν_ obs) and known distances.  Approximate IceCube sensitivity curves[These are oobtained by scaling the max-burst sensitivity of  for a power-law spectrum E^-2 with the detector effective area to northern-sky events.]  are shown for comparison. 
  
  
  
  
  
  
  
  § DETECTION PROSPECTS
  
  
  The upshot of our work is the prediction of an efficiency of up to η_ν∼ 10^-8-10^-5 (eq. <ref>; Fig. <ref>) 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  from the Galactic magnetar SGR 1935+2154.  Taking ℰ_ flare∼ 2E_ X/γ∼ 10^40 erg, we predict a neutrino burst of energy ℰ_ν≲ 10^32 erg, corresponding to a fluence (given an assumed source distance of 10 kpc) of F_ν∼ 10^-12 GeV cm^-2 (Fig. <ref>).  This is ∼ 11 orders of magnitude smaller than the IceCube upper limit of 5× 10^-2 GeV cm^-2 <cit.>.  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 <cit.>, 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 fortuitious geometry of a future flare and scaling ℰ_ flare up by a factor of ∼ 10^6 to the ∼ 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 ≈ 2.7 kpc; ) could reach ∼ 10^-2 GeV cm^-2.  In this case the detection of a neutrino from a giant flare becomes potentially feasible (see also ), particularly given proposed future upgrades to neutrino detectors (e.g. ).  Note that an FRB energy ≳ 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 Fig. <ref>).  However, thanks to new surveys like ASKAP () and CHIME/FRB (), the rate of FRB discoveries is rapidly growing and may reach thousands or more over the coming years.  
  
  The distribution of FRB energies dN/dℰ_ radio∝ℰ_ radio^-γ is roughly flat or increasing with energy (γ≲ 2; ; though see e.g.  ), i.e. the total radiated energy is dominated by the rarest, most energetic bursts.  Given the neutrino radiative efficiency of the shocks η_ν∝ℰ_ radio^19/42 (eq. <ref>), the total radiated neutrino energy is also dominated by the high-ℰ_ν bursts.
  
  A key question then becomes the upper cut-off energy to the FRB energy distribution, ℰ_ r,max.  Modeling the ASKAP FRB sample, <cit.> find a range of allowed values ℰ_ r,max≈ 3× 10^42-2× 10^44 erg for a burst frequency width ν∼ 1.4 GHz.  In what follows, we optimistically assume[A burst of energy ℰ_ r,max∼ 10^44 erg corresponds to a flare of energy ℰ_ flare∼ 3× 10^48 erg (eq. <ref>), approaching the entire magnetic energy budget of a magnetar.] ℰ_ r,max≈ 10^44 erg, bursts for which the local volumetric rate is estimated as ℛ(z = 0) ∼ 6 Gpc^-3 yr^-1 (adopting a γ = 1.6 power-law luminosity function; ).  At this rate, the closest FRB of this energy per year occurs at a distance D_ min∼ 0.34 Gpc.  Given its predicted neutrino energy ℰ_ν∼ 5× 10^43 erg (eq. <ref>), its neutrino flux would be F_ν≈ℰ_ν/(4π D_ min^2) ∼ 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. ).  Unfortunately, given the relatively long duration t_ max∼ 3000 s for the most energetic ℰ_ radio≈ 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∝ℰ_ radio^4/7, their sharply lower radiated energies ℰ_ν∝ℰ_ radio^61/42 largely cancel out these benefits.    
  
  In addition to a burst of neutrinos (from π^± decay), we predict a burst of ∼ TeV-PeV gamma-rays from neutral pion decay.  As shown in Figure <ref>, the TeV thermal neutrinos fall in the sensitivity range of ground-based water Cerenkov detectors, including  HAWC <cit.>, and upcoming and future detectors LHAASO <cit.> and SWGO <cit.>.  However, again for fiducial parameters we require a nearby source ≲ 0.1 kpc for a detection.  Furthermore, the pion decay signal could be overwhelmed by non-thermal leptonic emission, depending on the electron acceleration efficiency of the shocks.  
  
  
  
  
  
  § CONCLUSIONS
  
  
  With the discovery of , 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 <cit.> are neutrinos generated by ions heated or accelerated at the shock interact with thermal synchrotron shock photons via the photo-hadronic process.  
  
  We predict a burst of ∼ TeV-PeV neutrinos of total energy ℰ_ν≈ 10^35-10^44 erg (depending most sensitively on the FRB isotropic energy; eq. <ref>) which lasts for a timescale t_ max∼ 0.1-1000 s (eq. <ref>) 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) non-thermal 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≡ r[t_ max] ∼ (3-30)r_ FRB beyond the radius r_ FRB probed by the FRB emission itself.[We have additionally assumed, for simplicity, a constant density profile for this upstream medium, however our results are not particularly sensitive to this choice. E.g. for fiducial FRB parameters and k=-2 instead of k=0 (where n_ ext∝ r^-k) we find that the peak neutrino efficiency η_ν [t_ max] is only increased by a modest factor of ∼ 5. Similarly, a positive k=2 would decrease the efficiency by a factor of ∼ 4.]
  
  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.  BDM acknowledges support from the Simons Foundation (grant 606260). Support for KF 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. BM 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.
  
  
  natexlab#1#1
  
  
  
  
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