Upper Field Strength Limit of Fast Radio Bursts

Since first discovered in 2007 (Lorimer et al., 2007), fast radio bursts (FRBs) have been recognized as real astronomical events and gain much research interest (Katz, 2018; Platts et al., 2019; Zhang, 2020). Although event reports and theoretical models of FRBs have exploded over the past decade, the origin of FRBs remains unclear. Due to large dispersion measures with hundreds or even thousands of $\textrm{cm}^{-3}$pc, these radio transients have cosmological origins (Xu & Han, 2015), which has been confirmed by several events with located host galaxies (Thornton et al., 2013; Petroff et al., 2016). Therefore FRBs can be served as novel probes for interstellar and intergalactic matters (Prochaska et al., 2019; Macquart et al., 2020).

Assuming an isotropic emission, the luminosity of FRBs ranges from $10^{38}$ to $10^{45}$erg/s (Thornton et al., 2013; Zhang, 2018), many orders of magnitude more powerful than radio pulsars. Meanwhile, the ultrahigh brightness temperature $\sim 10^{35}$K (Katz, 2018; Zhang, 2020) indicates that FRB radiations must be coherent. The millisecond duration implies that the source is limited to hundreds of kilometers in size, which points to compact objects in universe, such as white dwarfs, neutron stars, or black holes. The observed FRB200428 (Bochenek et al., 2020; CHIME/FRB et al., 2020; Lin et al., 2020) in the Milky Way associated with a hard X-ray burst from magnetar SGR1935+2154, suggest magnetars can generate FRBs.

Many models of FRBs (Katz, 2018; Platts et al., 2019; Zhang, 2020) have been proposed. Possible FRB sources are located in- or out-side of the magnetospheres of neutron stars. In the magnetosphere, the radiation mechanisms include plasma maser emissions from relativistic plasmas or plasma instabilities (Lyubarsky, 2020), and curvature radiation of charged bunches (Kumar et al., 2017; Katz, 2014; Yang et al., 2020; Lu & Kumar, 2018). Outside of the magnetosphere, relativistic shocks driven by outflows from neutron stars may also induce FRBs (Lyubarsky, 2014; Waxman, 2017; Metzger et al., 2019; Beloborodov, 2020). Although the emission region of the FRBs in neutron stars is still being debated, most theoretical models show that FRBs are accompanied by energy comparable or even more powerful counterparts in x and $\gamma$ rays bands of keV-to-TeV (Chen et al., 2020). The observed x-rays from FRB200428 are four orders of magnitude more energetic than the radio emission.

FRBs have been detected in the range of 0.3-8 GHz (Zhang, 2020), with a bandwidth of hundreds of MHz limited by the detection band of radio telescopes. The coherent GHz radiation implies an emitter in the sub-meter scale. In the immediate vicinity of the emitters, FRB corresponds to an extremely strong microwave. Research on this extreme microwave is rare (Wu, 2016). In this paper we simulate the interaction between ultrastrong GHz radio waves and GeV gamma photons. It is found that at field-strength of $3\times 10^{12}$V/cm, dense pair plasmas are produced by quantum cascades that significantly dampen the radio pulses. This process should occur in the FRB emission region and constrains the radiation intensity near the emitters.

The first simulation is conducted for $E_{p}=2.68\times 10^{12}$V/cm, which corresponds to $a_{0}=eE_{p}/m_{e}c\omega_{0}\simeq 2.5\times 10^{7}$ and magnetic field of $E_{p}/c\simeq 0.89\times 10^{10}$ Gauss. The simulation box is $2\lambda$ in length with a spatio/temporal resolution of 10000 grids per wavelength/cycle. Ten 1 GeV photons simultaneously incident to the FRB pulse with $\theta=120^{\circ}$.

Figure 2 shows photon-triggered pair plasma sparks in the comoving frame of the radio pulse. Obvious QED cascades occur at $E_{y}\approx 0.7E_{p0}$, where annihilation of incident photons generates hundreds of $\chi\sim 0.1$ pair particles, which are in turn violently accelerated by the intense FRB fields to ultra-relativistic energies and emit high-energy photons. The latter are further annihilated into electron-positron pairs by the FRB fields. As shown in Figure 2, three distinct plasma clumps appear, they grow and finally merge into a plasma sheet within $\sim 1$ ns. The plasma density increases exponentially before the clump merges and saturates at $t=1.6\lambda/c$ with the peak density of $7.2\times 10^{6}n_{c}$. Here, $n_{c}\simeq 1\times 10^{13}(\mathrm{cm}/\lambda)^{2}\mathrm{cm^{-3}}=1.11\times 10^{10}\mathrm{cm^{-3}}$ is the critical density for 1 GHz waves. After $t=1.6\lambda/c$, the pair plasma sheet comoves with the field, and hence quantum cascades cease. For an ultra-relativistic field, the plasma density for screening the field is $\sim a_{0}n_{c}$. The plasma sheet has a density lower than $a_{0}n_{c}=2.5\times 10^{7}n_{c}$ and its thickness is $0.01\lambda$. Thus it causes only a small distortion on the FRB field.

Figure 3 shows the electrons and photons energy spectra. During the active cascades at $t\sim 1.3\lambda/c$, the particles and photons are synergetic with each other and have almost the same maximum energy. However, due to the wave field acceleration, the charged particle energy are always more concentrated than the photon energy. After the saturation at $t\sim 1.5\lambda/c$, the charged particles are free from radiation damping, and their energy quickly overtake that of the photons. They are more monoenergetic than before, and have an average energy $\sim 0.2$ TeV at $t=1.8\lambda/c$.

We increase the peak wave field to $E_{p}=3\times 10^{12}$ V/cm with $a_{0}=2.8\times 10^{7}$. Figure 4 shows the electric field $E_{y}$ and pair plasma density $n_{e}$. Due to the generated pair plasmas, partial field screening starts at $t\sim 1\lambda/c$. Then the plasma density dramatically increases and exceeds the relativistic critical density $a_{0}n_{c}\sim 2.8\times 10^{7}n_{c}$. The field screening then becomes complete in the plasma with $n_{e}>a_{0}n_{c}$, and the pair plasma expands towards the frontside and further grows to the backside. Figure 5 shows the electron energies distribution. At the front of the plasma, electrons/positrons are further accelerated along the $x$ axis to higher energies. According to Equation (3) the QED effects becomes weak and pairs are no longer produced. On the backside, the oscillating charged particles on the vacuum-plasma boundary continues to collide with the right-going field and produce new pairs. However, Fig. 5(b) and (c) show that due to the continuous radiation damping, the energy of these particles are suppressed, as compared with that of the accelerated particles on the right front. In Figure 5(c), one can see that a plasma spark appears downstream in the second half cycle. It is triggered by the emitted photons from the QED-active backside of the pair plasma region. Such sparks could dampen the FRB field in the more extended space.

Energy spectra of electrons and photons are given in Figure 6. During the active quantum cascades at $t=1\lambda/c$, charged particles and photons have similar spectra as that in figure 3. After the complete field screening, the high-energy part of the spectra at $t=1.4\lambda/c$ and $1.8\lambda/c$ in Figure 6(a) comes from the field accelerated particles at the plasma front. Depletion of GeV photons at $1\lambda/c<t<1.4\lambda/c$ can be attributed to photon annihilation and production of pair plasmas.

Gamma photons above 1 GeV are observed to more easily cause the field depletion. We also carried out the simulations for charged particles in GeV-TeV range and found that tens GeV are required to trigger significant radiation damping. In 3D space, pair plasmas will fill the main radio-field volume, as demonstrated in multi-dimentional PIC simulation of ultraintense laser breakdown triggered by a single electron in vacuum (Nerush et al., 2011). On the other hand, here we have only tens photons interacting with the FRB. Existing models invoke photons with comparable high energy (GeV-TeV) co-existing with the FRBs. The results here indicate that such high-energy photons will fully deplete the FRB.

For FRBs propagating cross the neutron star magnetic field $\boldsymbol{B_{NS}}$, incoherent scattering by magnetospheric plasmas could be significant and gamma rays produced in this process would trigger severe cascades in the FRB(Beloborodov, 2021). When radio waves propagate along the background magnetic field $B_{NS}$, gamma-photon annihilation due to ${B_{NS}}$ is negligible due to the vanishing term $\boldsymbol{v}\times\boldsymbol{B_{NS}}=0$ in Eq. (2). Therefore, our results are more applicable for FRBs escaping from the polar regions of neutron stars.

As an extremely efficient radio burst, the plasma/beam emitter must be able to coexist with its self-field and radiation field, and will not induce severe field breakdown within both emitter and FRB bodies. According to our results, this requires that the plasma density in or nearby the emitter should be lower than the level of $a_{0}n_{c}$ for less field screening, and the plasma/beam driver could mainly propagate along the radiation direction for negligible pair cascades. The driver can produce high-energy gamma rays along its momentum direction. By Compton backscattering from background plasmas, these gamma rays could turn back to scatter the FRB field as studied here.

We have investigated radiation dampings of FRBs triggered by high-energy photons in GeV. At the field amplitude of $3.0\times 10^{12}$V/cm, dense pair plasmas are generated within sub-cycle of these radio transients. The plasma generation and radiation absorption lead to breakdown of FRBs fields. This energy depletion can critically limit the field-strength of FRBs around their emitters. Similar QED effects can also be expected during other stages of FRB propagation. Our work also implies that QED effects (Philippov et al., 2020; Katz, 2017) could be indispensable near the FRB emitter.

This work was supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No.XDA17040503). We thank W. M. Wang and M. Y. Yu for helpful discussions.