MODEL-DEPENDENT ESTIMATE ON THE CONNECTION BETWEEN FAST RADIO BURSTS AND ULTRA HIGH ENERGY COSMIC RAYS

In the blitzar model, the FRBs were triggered by the collapse of the SMNSs to black holes. As a result of the collapse of an SMNS, the magnetic-field lines will snap violently. Accelerated electrons from the traveling magnetic shock cause a significant fraction of the magnetosphere to dissipate and produce a massive radio burst that is observable out to cosmological distances (Falcke & Rezzolla 2014). The difference between an SMNS and a normal NS is that the former has to rotate very quickly in order to prevent itself from collapsing since the gravitational mass of an SMNS is larger than that allowed for a nonrotating NS (Friedman et al. 1986). The rapid uniform rotation can enhance the maximum gravitational mass by a factor of ~0.05 (P₀/1 ms)⁻² and can thus help make the SMNS stable (Friedman et al. 1986). In addition to the core-collapse supernovae (ccSNe) proposed in Falcke & Rezzolla (2014), the merger of some binary NSs can also produce an SMNS or even a normal NS if the equation of state of the NS material is stiff enough (e.g., Davis et al. 1994; Dai & Lu 1998a; Dai et al. 2006; Shibata et al. 2000; Baumgarte et al. 2000; Gao & Fan 2006; Metzger et al. 2008; Zhang 2013). Usually the nascent NSs formed in both ccSNe and the merger of double NSs are differentially rotating and the differential rotation is suggested to be more efficient at keeping the supernova (SN) stable. However, the differential rotation is expected to be terminated very quickly by the magnetorotational instability as well as magnetic braking (Cook et al. 1992, 1994; Hotokezaka et al. 2013). That is why we only consider the effect of uniform rotation in stabilizing the SMNS. Here we concentrate on the ccSN-formed SMNS and will discuss the merger-formed SMNS in some detail in Section 2.2.

Before discussing the high-energy cosmic ray acceleration, we examine whether or not the blitzar model can account for the observed dispersion measures. The SN outflow likely has a total mass of M_sh  ~  a fewM_☉, and the observed dispersion measures DM  ~  10²¹ cm⁻² (Thornton et al. 2013; Lorimer et al. 2007) require that the SN outflow should reach a radius of R_sh > (M_sh/4πm_p(1 + z)DM)^1/2  ~  10¹⁸ cm (M_sh/10 M_☉)^1/2 (1 + z)^−1/2, where the term (1  +  z) is introduced to address the effects of both the cosmological time dilation and the frequency shift on the measured DM. Clearly, at such a huge radius, the SN shell is also transparent for the 1.3 GHz FRB. The velocity of the SN shell can be estimated as V_sh  ~  (E_inj/M_sh)^1/2  ~  10⁹ cm s⁻¹ (M_sh/10 M_☉)^−1/2(E_inj/10⁵² erg)^1/2. The life of the SMNS should satisfy t_life > R_sh/V_sh  ~  10⁹s(1 + z)^−1/2(M_sh/10 M_☉)(E_inj/10⁵²erg)^−1/2, suggesting a dipole magnetic field strength of B_⊥ < 1.3 × 10¹²G(1 + z)^1/4(I/1.5 × 10⁴⁵ g cm²)^3/4(R_s/10⁶ cm)⁻³(P₀/1 ms)^1/2(M_sh/10 M_☉)^−1/2, where B_⊥ = B_ssin α, B_s is the surface magnetic field strength at the pole, R_s is the radius of the NS, and α is the angle between the rotational and dipole axes. Hence the SMNS should not be significantly magnetized, otherwise the FRB cannot be accounted for (see also Falcke & Rezzolla 2014). We denote such a request as Request-I. Another highly related request is that the gravitational wave radiation should be weak enough that it does not dominate over the dipole radiation of the pulsar (i.e., Request-II). In terms of the ellipticity () of the pulsar, we need < 5 × 10⁻⁶(I/1.5 × 10⁴⁵ g cm²)^−1/2 (P₀/1 ms)² (t_life/10⁹s)^1/2 (Shapiro & Teukolsky 1983). Alternatively, for a rapidly rotating pulsar, some instabilities, for example the Chandrasekhar–Friedman–Schutz instability (Chandrasekhar 1970; Friedman & Schutz 1978), may occur when the ratio ${\cal R}={\cal T}/|W|$ of the rotational kinetic energy ${\cal T}$ to the gravitational binding energy |W| is sufficiently large. In the Newtonian limit, the l = m = 2 f −mode, which has the shortest growth time of all polar fluid modes (i.e., $\tau _{\rm GW}\,{\sim}\,5\times 10^{-6} {\rm s}\, (R_{\rm s}/10^{6} \,{\rm cm})^{4}\,({\cal R}-{\cal R}_{\rm sec})^{-5}$, see Lai & Sharpiro 1995), becomes unstable when ${\cal R}\gtrsim {\cal R}_{\rm sec}\approx 0.135$. Hence ${\cal R}-{\cal R}_{\rm sec}\lesssim 10^{-3}$ is needed to satisfy τ_GW > t_life  ~  10⁹s, otherwise the SMNSs collapsed too early to get a sufficiently small DM. Finally, the contribution to the DM by the circum-burst medium is less clear. In the estimate of the wind bubble structure at the end of the Wolf–Rayet stage of the massive star, Chevalier et al. (2004) assumed that the surrounding medium has a density typical of the hot, low-density phase of a starburst galaxy (i.e., n  ~  0.2 cm⁻³). If this is the case, the contribution to the DM by the surrounding medium can be ignored. However, the ccSN may be born in a molecular cloud with a typical number density n_cloud  ~  10² cm⁻³ and a size of R_cloud  ~  (3M_cloud/4πn_cloudm_pc²)^1/3  ~  10 pc (M_cloud/10⁴ M_☉)^1/3(n_cloud/10² cm⁻³)^−1/3, which can give rise to an observed DM ≈ 10³ (1 + z)⁻¹(M_cloud/10⁴ M_☉)^1/3 (n_cloud/10² cm⁻³)^2/3 pccm⁻³. Therefore, the surrounding molecular cloud should not be very massive/dense (i.e., M_cloud < 10⁴ M_☉ for n_cloud  ~  10² cm⁻³) otherwise the central ccSNe could not be viable progenitors of the observed FRBs. Such a request is denoted as Request-III. Each of these three requests imposes some challenges on the ccSN scenario in the blitzar model. Nevertheless, the blitzar model can explain some aspects of FRBs (e.g., Falcke & Rezzolla 2014) and is still widely adopted in the literature. Below we discuss the possible high-energy cosmic-ray acceleration in such a scenario.

In view of the sensitive dependence of stabilization on P₀, the SMNS is unlikely to exist for P₀ > 2 m s unless there is some fine tuning by which the mass of the SMNS is just a little bit above that allowed by the nonrotating NS. The rotational kinetic energy of the SMNS is quite large, i.e., E_{SN, r} ≈ 3 × 10⁵² erg (I/1.5 × 10⁴⁵ g cm²)(P₀/1 ms)⁻², where I is the moment of inertia. Before collapsing into a black hole, the SMNS should have lost its rotational energy mainly via magnetic dipole radiation and possibly also gravitational wave radiation (Usov 1992; Duncan & Thompson 1992; Fan et al. 2013b). The amount of energy injected into the surrounding medium is E_inj  ~  E_{SN, r}/2  ~  1.5 × 10⁵² erg (I/1.5 × 10⁴⁵ g cm²)(P₀/1 ms)⁻², if the gravitational wave radiation is not dominant. The frequency of the electromagnetic wave is ~10³ Hz, which is much lower than the surrounding plasma's frequency $\omega _{\rm p}\,{\sim}\,5.6\times 10^{4} \,{\rm Hz} \,n_{\rm e}^{1/2}$, where n_e is the number density of the free electrons in the plasma (i.e., the SN outflow) and can be estimated as ${\sim} 10^{4} \,{\rm cm^{-3}} (M_{\rm sh}/10 \,M_\odot)({\cal R}_{\rm sh}/0.1)^{-1}(R_{\rm sh}/10^{18} \,{\rm cm})^{-3}$, where the width of the outflowing material shell is taken to be a fraction ${\cal R}_{\rm sh}\,{\sim}\,0.1$ of the R_sh. Therefore, the electromagnetic wave will be "trapped" by the plasma. The pressure of the electromagnetic wave outflow is so high that it can work on the surrounding plasma, and then the magnetic wind energy will be transferred into the kinetic energy of the outflow, as indicated by the shallow decay of some GRB afterglows (Dai & Lu 1998a, 1998b; Zhang et al. 2006) and the light curves of some superluminous SNe (Kasen & Bildsten 2010; Woosley 2010; Inserra et al. 2013; Nicholl et al. 2013). Energetic forward shocks will be driven and then accelerate protons and other charged particles to ultra high energies.⁴ Hence, in the blitzar model, FRBs may be promising sources of EeV cosmic ray protons.

Thus far, no reliable γ-ray/X-ray/optical counterparts of FRBs have been identified and the nature of the central engine cannot be pinned down. A possible test of the blitzar model is to observe the very early radio emission of GRBs since, at least for some GRBs, the central engine may be SMNSs (see Zhang 2014 and references therein). Intriguingly, a tentative association between two single dispersed millisecond radio pulses and two GRBs has been reported, and these two single dispersed millisecond radio pulses were detected in the few minutes following two GRBs, and the arrival times of both pulses were found to coincide with breaks in the GRB X-ray light curves, which likely label the phase transition of the central engine, i.e., the collapse of the SMNS to the stellar black hole (Bannister et al. 2012). Such a correlation between FRBs and GRBs, if confirmed in the future, will be in support of the blitzar model.⁵

Now we discuss the cosmic-ray particle generation by the SMNS wind-accelerated outflow of the ccSNe. The progenitor star of a ccSN would experience a significant mass-loss stage and then the surrounding medium would usually not be a simple free-wind structure or a constant density structure. For simplicity, here we adopt the structure shown in Figure 1 of Chevalier et al. (2004) to estimate the maximum energy of the protons accelerated at the shock front of the SMNS wind-driven outflow. In such a scenario, significant particle acceleration takes place at R  ~  5 × 10¹⁸ cm, where the supergiant shell with a number density n  ~  10²–10³ cm⁻³ is located. Following Bell & Lucek (2001), the maximum energy of the protons accelerated by the forward shock can be estimated as

$$\begin{equation} \varepsilon _{\rm p,M}^{\rm ccSN}\,{\sim}\,10^{18} \,{\rm eV} \left({V\over 10^{9} \,{\rm cm \,s^{-1}}}\right)^{2}\left({n\over 10^{2} \,{\rm cm^{-3}}}\right)^{1/2}\left({\epsilon _{\rm B}\over 10^{-2}}\right)^{1/2}, \end{equation} \tag{ 1 }$$

where _B is the fraction of shock energy given to the magnetic field and the velocity of the SN shell has been estimated to be V_sh  ~  (E_inj/M_sh)^1/2  ~  10⁹ cm s⁻¹ (M_sh/10 M_☉)^−1/2(E_inj/10⁵²erg)^1/2. The charged particles reach energies larger by a factor of Z, the charge number. The magnetic field generated by the shock is B  ~  10⁻³ gauss (_B/10⁻²)^1/2 (n/10² cm⁻³)^1/2(V_sh/10⁹ cm s⁻¹), which is too low to effectively cool the accelerating EeV cosmic rays.

The rate of FRBs is ${\cal R}_{_{\rm FRB}} \,{\sim}\,10^{-3} \,{\rm yr^{-1}}$ per galaxy (Thornton et al. 2013), which is about one order of magnitude lower than the ccSN rate ${\cal R}_{\rm ccSN}\,{\sim}\,10^{-2} \,{\rm yr^{-1}}$ per galaxy. If FRBs are indeed the cry of the dying SMNSs, the energy released into the surrounding material of each SMNS is E_inj  ~  10⁵² erg, implying that the total energy input by FRB sources is comparable to the input by all other ccSNs, and thus FRBs should be one of the main sources of cosmic rays. In particular, as found in Equation (1), the most energetic cosmic-ray protons can reach an energy of 10¹⁸ eV, and might be the dominant components at such energies. The injected cosmic-ray density

$$\begin{equation} \dot{\epsilon }_{\rm CR}\gtrsim 10^{47}\eta \omega \left({E_{\rm inj}\over 10^{52} \,{\rm erg}}\right)\left({{\cal R}_{\rm FRB}\over 10^{4} \,{\rm yr^{-1} \,Gpc^{-3}}}\right) \,{\rm erg} \,{\rm yr^{-1}} \,{\rm Mpc^{-3}} \end{equation} \tag{ 2 }$$

per energy decade with η as the cosmic-ray acceleration efficiency, and ω = 1/ln (ε_max/ε_min) 0.1 as the fraction of the total cosmic-ray energy at each energy decade. Then the corresponding observed cosmic-ray flux at ~10¹⁸ eV is

$$\begin{eqnarray} F_{\rm EeV\hbox{--}CR} &\sim& 10^{-28} \eta \left({\omega \over 0.1}\right)\left({E_{\rm inj}\over 10^{52} \, {\rm erg}}\right)\nonumber\\ &&\times\left({R_{\rm FRB}\over 10^{4} \,{\rm yr^{-1} \,Gpc^{-3}}}\right) \,{\rm m^{-2}\, s^{-1} \,sr^{-1}\, eV^{-1}}. \end{eqnarray} \tag{ 3 }$$

Roughly, a rate of ${\cal R}_{\rm FRB}\,{\sim}\,10^{4} \,{\rm yr^{-1} \,Gpc^{-3}}$ would result in the observed flux of cosmic rays as 10⁻²⁸η m⁻² s⁻¹ sr⁻¹ eV⁻¹, which can meet the observed cosmic ray flux F_obs(ε) = C (ε/6.3 × 10¹⁸ eV)^{−3.2 ± 0.05} with C = (9.23 ± 0.065) × 10⁻³³ m⁻² s⁻¹ sr⁻¹ eV⁻¹ (Nagano & Watson 2000), for the EeV cosmic-ray acceleration efficiency η  ~  0.03.

EeV cosmic rays, in principle, can produce PeV neutrinos via interacting with the interstellar medium, but we do not expect significant PeV neutrino emission since the >10¹⁷ eV protons cannot be effectively confined and the energy loss via pion production is ignorable unless the FRBs were born in the starburst galaxies and, in particular, the so-called ultra-luminous infrared galaxies (He et al. 2013).

In the model of merging double NSs for FRBs, the radiation mechanism may be coherent radio emission, like radio pulsars, by magnetic braking when magnetic fields of NSs are synchronized to binary rotation at the time of coalescence (Totani et al. 2013). In addition to FRBs, the mergers of binary NSs may give rise to short GRBs or other kinds of violent explosions with possible central engines of magnetized millisecond NSs (e.g., Duncan & Thompson 1992; Davis et al. 1994; Dai & Lu 1998a; Baumgarte et al. 2000; Shibata et al. 2000; Duez et al. 2006; Price & Rosswog 2006; Rosswog 2007; Giacomazzo & Perna 2013). The latest numerical simulations suggest that the SMNS can be formed in the merger of an NS binary with M_tot  ~  2.6 M_☉ (note that among the ten NS binaries identified thus far, five systems have such a total gravitational mass Lattimer 2012) for reasonably stiff equations of state that are favored by current rest-mass measurements of pulsars (see Hotokezaka et al. 2013 and Fan et al. 2013a and the references therein). There are growing, though inconclusive, observational evidences for forming an SMNS or even a stable NS in NS–NS mergers. The one most widely discussed in the literature is the X-ray plateau followed by an abrupt cease in the afterglow light-curve of many short GRBs (Gao & Fan 2006; Metzger et al. 2008; Rowlinson et al. 2010, 2013). In addition, the observations on short bursts such as GRB 051221A and GRB 130603B indicate an energy injection from a highly magnetized SMNS and the injected energy is as high as 10⁵¹–10⁵²erg (Fan & Xu 2006; Rowlinson et al. 2010; Dall'Osso et al. 2011; Fan et al. 2013b). Moreover, as pointed out first by Fan & Xu (2006), the material ejected during the binary NS merger is usually not expected to be more than ~0.01 M_☉ and could be accelerated to a mildly relativistic velocity by the wind of SMNS and then produce X-ray/optical/radio afterglow emission (see Gao et al. (2013a) for a detailed numerical calculation of the lightcurves), which can account for the cosmological relativistic fading source PTF11agg, a remarkable event not associated with a high-energy counterpart (Wang & Dai 2013; Wu et al. 2014). It thus seems reasonable to assume that SMNSs, which likely collapsed at t_c  ~  10²–10⁴ s (Rowlinson et al. 2013), were formed in a good fraction of NS–NS mergers.

The prospect of forming SMNS in the mergers can also be roughly estimated as the following. On the one hand, the gravitational mass (M) of the isolated NS is related to the baryonic mass (M_b) as M_b ≈ M + αM², where $\alpha \approx 0.08 \,M_\odot ^{-1}$ (Lattimer & Yahil 1989; Timmes et al. 1996). On the other hand, in the numerical simulations of mergers of binary NSs performed in full general relativity incorporating the finite-temperature effect and neutrino cooling, Sekiguchi et al. (2011) found that the effect of the thermal energy is significant and can increase the maximal gravitational mass M_max by a factor of 20%–30% for a high-temperature state with T ≥ 20 MeV. Since they are not supported by differential rotation, the supermassive remnants that were predicted to be stable until neutrino cooling, with a luminosity of 3 × 10⁵³ erg s⁻¹, removed the pressure support in a few seconds (Sekiguchi et al. 2011). After neutrino cooling, the supermassive remnant is still stable if

$$\begin{equation*} \alpha M_{\rm r,max}^{2}+M_{\rm r,max}-(M_{1}+M_{2})-\alpha \left(M_{1}^{2}+M_{2}^{2}\right)+m_{\rm loss}>0, \end{equation*}$$

where m_loss is the baryonic mass loss of the system during the merger, M_{r, max} ≈ [1 + 0.05(P₀/1 ms)⁻²]M_max, and M₁ and M₂ are the gravitational masses of the binary NSs, respectively. m_loss  ~  10⁻³ to 10⁻² M_☉ has been inferred in the numerical simulation (Rosswog et al. 1999). In the modeling of the infrared bump of short GRB 130603B, m_loss  ~  a few × 10⁻² M_☉ is needed (Tanvir et al. 2013; Berger et al. 2013; Fan et al. 2013b). As a conservative estimate, we take m_loss = 0.01 M_☉. In order to estimate the mass distribution of the NSs in the NS–NS binary systems, Özel et al. (2012) divided the sample into one of pulsars and one of the companions (For the double pulsar system J0737-3039A, they assigned the faster pulsar to the "pulsar" and the slower to the "companion" categories). Repeating the above inference for these two subgroups individually, Özel et al. (2012) found that the most likely parameters of the mass distribution for the pulsars are M₀ = 1.35 M_☉ and σ = 0.05 M_☉, whereas for the companions they are M₀ = 1.32 M_☉ and σ = 0.05 M_☉. Hence in our simulation, the distributions of gravitational masses of NSs as dN_NS/dM∝exp [ − (M − M₀)²/2σ²] with these parameters are adopted. The possibility distribution of the gravitational masses of supermassive remnants (i.e., ${\cal M}\approx {(-1+ \sqrt{{{1+4\alpha [M_1+M_2+\alpha (M_1^{2}+M_2^{2})-m_{\rm loss}}}} ]/2\alpha }$) formed in the simulated double NS mergers is presented in Figure 1. We find for M_max ≥ 2.36 M_☉, about half of the mergers will produce SMNSs with P₀ = 1 ms. Observationally, the pulsar PSR J0348+0432 has an accurately measured gravitational mass of 2.01 ± 0.04 M_☉ (Antoniadis et al. 2013) and J1748-2021B has a gravitational mass of ≈2.74 ± 0.21 M_☉ (Lattimer 2012). Hence M_max  ~  2.36 M_☉ is still possible, with which a sizeable fraction of NS–NS mergers may produce SMNSs.

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We have briefly mentioned in Section 2.1 that in the blitzar model a small fraction of FRBs may be relevant to the mergers of double NSs that produce SMNS remnants. In such a scenario, some FRBs are expected to be detected in the afterglow emission phase of short GRBs. In the model of merging double NSs, FRBs are generated at the time of coalescence of double NSs and are expected to precede short GRBs or other kinds of violent explosions (i.e., no FRB is expected to occur in the afterglow phase of short GRBs) and the three requests outlined in Section 2.1 do not apply. The follow-up observations of FRBs and short GRBs would be crucial to distinguish between the blitzar model and the merging NS model. The origin of merging double NSs for FRBs, if confirmed in the future, has interesting implications on the sources of EeV cosmic-ray protons. This is because there is some tentative evidence for the formation of SMNSs in a sizeable fraction of NS–NS mergers and such a kind of long-lived central engine can accelerate the materials ejected during the merger to very high velocities (Fan & Xu 2006; Gao et al. 2013a; Wu et al. 2014). The physical reason for converting the SMNS wind energy into the kinetic energy of the forward shock is the same as that in the case of ccSNe. Within a radius of R_sh ct_c  ~  3 × 10¹⁴ cm (t_c/10⁴ s), the SMNS wind is likely Poynting-flux dominated rather than electron/positron pair dominated (Vlahakis 2004). The frequency of the electromagnetic wave of the SMNS (  ~  10³ Hz) is much smaller than the surrounding plasma's frequency $\omega _{\rm p}\,{\sim}\,5.6\times 10^{4} \,{\rm Hz\,} n_{\rm e}^{1/2}$, where $n_{\rm e}\,{\sim}\,3\times 10^{11} \,{\rm cm^{-3}}\, (M_{\rm ej}/0.01 \,M_\odot)({\cal R}_{\rm sh}/0.1)^{-1}\,(R_{\rm sh}/3\times 10^{14} \,{\rm cm})^{-3}$ is the number density of the free electrons in the merger outflow with a rest mass of M_ej  ~  0.01 M_☉. As a result, the electromagnetic wave is "trapped" by the plasma. The high magnetic pressure works on and hence accelerates the surrounding plasma (see Yu et al. 2013 and the references therein). The SMNS-driven outflow, almost isotropic, will generate energetic forward shocks and then accelerate ultra high energy cosmic rays.

The maximum energy of the cosmic rays accelerated by the wide outflow driven by the SMNS can be estimated as

$$\begin{eqnarray} \varepsilon _{\rm p,M}^{\rm NS-NS} &\sim& \beta Z e B_{\rm d} R_{\rm d} \sim 1.5\times 10^{18} \,{\rm eV} Z\left({\beta \over 0.5}\right)^2\left({E_{\rm inj}\over 10^{52}\; {\rm erg}}\right)^{1\over 3}\nonumber \\ &&\times \left({n\over 10^{-2}\; {\rm cm^{-3}}}\right)^{1\over 6}\left({\epsilon _{\rm B}\over 10^{-2}}\right)^{1\over 2}{\Gamma }^{1\over 3}, \end{eqnarray} \tag{ 4 }$$

where β is the velocity of the ejecta in units of the speed of light c, e is the electron's charge, R_d  ~  1.2 × 10¹⁸ cm(E_inj/10⁵² erg)^1/3(n/10⁻² cm⁻³)^−1/3(Γ/5)^−2/3 is the deceleration radius, and B_d = 0.02 gaussβ (Γ/5) (n/0.01 cm⁻³)^1/2(_B/0.01)^1/2 is the magnetic field strength at R_d. The β has been normalized to 0.5 since, in the presence of a highly magnetized SMNS, the material ejected during the NS–NS merger is expected to be accelerated to a trans-relativistic or even mildly relativistic velocity (Fan & Xu 2006). Different from the ccSN scenario, we normalize n to the value of 10⁻² cm⁻³ to address the fact that some mergers of NSs are expected to take place in the low-density medium. Again the cooling of the accelerating EeV cosmic rays do not suffer significant energy loss via synchrotron radiation due to the low B_d.

How large is E_inj? The answer is somewhat uncertain since the SMNSs that formed in double NS mergers might suffer significant energy loss in addition to the regular dipole magnetic radiation. For GRB 130603B displaying a SMNS signature, E_inj 2 × 10⁵¹ erg is needed to account for the multi-wavelength afterglow data. In a good fraction of short GRBs with distinguished X-ray afterglow plateaus, as reported in Rowlinson et al. (2013), the energy release by the central SMNS is found to be E_inj < 10⁵² erg. The inferred E_inj  ~  a few × 10⁵¹ erg is also favored by the weak radio afterglow emission of most short GRBs and has been suggested to be the signature of the significant gravitational wave radiation of the SMNS (Fan et al. 2013a, 2013b), that may be possible if the interior toroidal magnetic field is high up to ~10¹⁷ gauss which can give rise to a sizeable deformation   ~  0.01 of the magnetar or the secure instability occurring with a ${\cal R}\,{\sim}\,{\cal R}_{\rm sec}+0.03 \,{\sim}\,0.165$. While in the modeling of GRB 051221A and PTF11agg, E_inj  ~  10⁵² erg is needed, on average, E_inj is likely   ~  quiteafew × 10⁵¹ erg. With Equation (3), it is straightforward to show that η  ~  0.1(E_inj/3 × 10⁵¹ erg)⁻¹ is needed, otherwise the accelerated ultra high energy cosmic rays cannot account for a sizeable fraction of the observed EeV ones. We then suggest that in the model of merging double NSs (Totani et al. 2013), the FRBs may still have a significant connection with ultra high-energy cosmic-ray sources; though, the argument is less direct than in the blitzar model.⁶ Possible byproducts are high-energy neutrinos if there are dense/energetic seed photons. Even with very optimistic assumptions, the resulting PeV neutrinos are likely too weak to give rise to significant detections for IceCube-like detectors (see Gao et al. (2013b) for a relevant estimate).

In the model of the merger of binary degenerate white dwarfs, the FRBs were produced by coherent emission from the polar region of a rapidly rotating, magnetized massive white dwarf, which formed after the merger (Kashiyama et al. 2013). The energy budget of the nascent massive white dwarf can be estimated as $E_{\rm bug} \,{\sim}\,{GM}_{_{\rm WD}}^2/R_{_{\rm WD}} \,{\sim}\,3\times 10^{50}\; {\rm erg} (M_{_{\rm WD}}/1 \,M_\odot)^{2}(R_{_{\rm WD}}/10^{9} {\rm cm})^{-1}$. Magnetic activity of the post-merger object has been demonstrated by recent numerical simulations (Ji et al. 2013), in which the magnetic energy of the remnant at its peak is found to exceed 10⁴⁸ erg (the corresponding volume-averaged magnetic field strength is $\bar{B}\,{\sim}\,10^{11} \, {\rm gauss}$) and about $M_{\rm ej,_{\rm WD}} \,{\sim}\,10^{-3} \,M_\odot$ mass is ejected from the system over the run time of the simulations, i.e., t = 2 × 10⁴ s. With a spin-down luminosity of the magnetized massive white dwarf $L_{\rm dip,_{\rm WD}} \,{\sim}\,2\times 10^{38} (B_{\perp,_{\rm WD}}/10^{9} \,{\rm G})^{2}$, the ejected material, as well as the swept circum medium with a density of n  ~  0.01 cm⁻³, cannot be accelerated to a velocity larger than ~ 10⁹ cm s⁻¹, where we have normalize $B_{\perp,_{\rm WD}}$ to a value of 10⁹ G, the surface magnetic field strength of the highly magnetized white dwarfs observed thus far. With Equation (4), we find that the cosmic-ray protons that are more energetic than ~10¹⁶ eV cannot be accelerated, and are therefore not of interest to us. If some FRBs are associated with type Ia SNe, as suggested in Kashiyama et al. (2013), the SN outflow can also accelerate cosmic rays. However, it is widely known that the SNe Ia outflow can, at most, accelerate protons to the so-called "knee" of the cosmic-ray spectrum (i.e., ~3 PeV). Therefore, in the model of the merger of binary white dwarfs, EeV cosmic rays are not expected.