Constraint on the relativistic motion of fast radio bursts based on the maximal electric field

Fast radio bursts (FRBs) were first discovered by Lorimer in 2007 as millisecond radio transient sources (Lorimer et al., 2007). They are radio signals that originate from cosmological distances. The luminosity of individual FRB event ranges from $10^{38}$ erg $\mathrm{s^{-1}}$(Bochenek et al., 2020) to $10^{46}$ erg $\mathrm{s^{-1}}$(Ravi et al., 2019). The origin of FRBs is still mysterious. For non-repeating events, some catastrophic events have been proposed, such as mergers of binary white dwarfs (Kashiyama et al., 2013). Observations by the Canadian Hydrogen Intensity Mapping Experiment (CHIME) collaboration suggest that at least a portion of FRBs is associated with magnetars (Bochenek et al., 2020). FRBs could be generated by magnetar’s charge starvation region and current sheet region (Zhang, 2022b). More recent review on the physics of FRBs can be seen in Zhang (2022a).

The brightness temperature of FRB is exceptionally high (Zhang, 2022a): $T_{{\rm{B}}}\simeq 10^{36}\ \rm{K}$. This indicates that the FRBs must be generated by a coherent process, while the exact radiation mechanism and origin are still unknown. Several models have been proposed to address this issue, as discussed in the review by Zhang (2020). Some models propose a gamma-ray burst-like mechanism (GRB-like), that is, FRBs are produced by relativistic shocks, as suggested by Lyubarsky (2014). Some models utilize a pulsar-like model, where coherent bunches of radio signals are emitted as FRBs, as described by Melikidze et al. (2000). One difference between the GRB-like and pulsar-like models is that the GRB-like models require a relativistic jet radiation region, while this is not necessary for the pulsar-like models. It would be meaningful to investigate whether the source of FRBs is relativistic or static.

Studying extreme events characterized by high luminosity and short duration can provide insights into the source’s nature. One common estimate of the size of an astrophysical source is $R\sim ct$, with $t$ being the time scale and $c$ being the speed of light. Extreme events have high luminosity and a small source size $R$, resulting in a high energy density. However, there are limits to the energy density. One of them is the strength of the corresponding electric field of the electromagnetic wave $E<E_{{\rm{s}}}\equiv m_{{\rm{e}}}^{2}c^{3}/\hbar e=1.32\times 10^{16}$ V $\mathrm{cm^{-1}}$, where $m_{{\rm{e}}}$ is the mass of the electron and $e$ is the charge of the electron and $\hbar$ is the Planck constant. Once if $E$ reaches this limit, the energy of electromagnetic field converts into kinetic and rest-mass energy of electron-position pairs (Schwinger, 1951). Zhang & Wu (2022) found a stronger constraint on $E$ based on their simulations, which is $E<3\times 10^{12}\ {\rm V\,cm^{-1}}$. With this limit they also placed constraints on the size of the radiation site. With the known duration of the FRBs, we may also constrain the Lorentz factor of the emitting region if the source is relativistic.

This paper is organized as follows. Section 2 introduces the quantum cascade process of the interaction between FRBs and high-energy photons, demonstrating the upper limit of $E$. Section 3 highlights the distinctions between a relativistic source and a non-relativistic source. In section 4, we present the selected data to examine the validity of the relativistic source hypothesis. Finally, in section 5 we draw our conclusions based on the findings presented in this paper.

From the observations, we can estimate the electric field strength of the FRBs to be greater than $10^{9}\ \rm{V\,cm^{-1}}$ (Zhang & Wu, 2022). In such a strong field, the interaction between high-energy photons and the field has been extensively studied (Kirk et al., 2009), including processes such as the Breit-Wheeler process and nonlinear inverse Compton scattering, which are quantum-electrodynamics (QED) processes. The Breit-Wheeler process involves the interaction of two high-energy photons, resulting in the creation of an electron-positron pair. The FRB GHz photons can be upscattered by high-energy electrons through inverse Compton scattering. If this scattering process generates a significant number of high-energy photons, providing sufficient energy for subsequent pair creation, the cascade starts. The presence of high-energy photons acts as a trigger, initiating the reaction. Then the Breit-Wheeler process occurs and creates positron-electron pairs. The electrons and positrons accelerate in electromagnetic field and radiate high energy photons. These high-energy photons continue to create pairs. The reaction rate is (in weak field) (Landau & Lifshitz, 1982):

with

where $\omega$ is the angular frequency of electromagnetic field, and the $\gamma$ is the Lorentz factor of electrons. There is a critical electric field $E_{s}$, which is equal to $1.32\times 10^{16}$ V $\mathrm{cm^{-1}}$. When the field strength $E$ of an electromagnetic wave reaches this critical value, the cascade process is expected to occur rapidly. This leads to a quick decrease in the energy of the electromagnetic field. Lu & Kumar (2019) investigated that this upper limit of $E$ is a constraint on the maximum luminosity of FRB. Typically, when $\eta\sim 5\times 10^{-2}$, the quantum cascade takes place. From Eq. (2), it can be inferred that higher field intensity $E$ leads to higher reaction rate, resulting in a rapid pair cascade and the formation of a dense plasma.

The cascade process causes the transfer of energy from the electromagnetic field to the plasma, resulting in an increased background plasma density and a decrease in the electromagnetic field strength. To ensure the propagation of electromagnetic waves in the medium, the condition $\omega>\omega_{\rm{p}}$ must be satisfied for electromagnetic waves to propagate, where the $\omega_{{\rm{p}}}$ is the plasma frequency. We have:

where $n$ represents the plasma density, which must be sufficiently low for electromagnetic waves to propagate. However, the cascade process increases plasma density $n$. Therefore, it imposes a constraint on the electric field $E$ of the electromagnetic wave.

Zhang & Wu (2022) used simulations to determine that the electric field $E$ of an FRB must be smaller than $3\times 10^{12}$ V $\mathrm{cm^{-1}}$ (in the laboratory reference frame), ensuring that the FRBs can propagate in the medium. This new constraint is much smaller than the $E_{s}$. There should be a new constraint on the physical properties, such as the radiation radius, the Lorentz factor of the emitting region, and the maximum luminosity. Zhang & Wu (2022) has already placed a lower limit of the radiation radius. In this work, we focus on the Lorentz factor.

In some models, it is predicted that FRB sources exhibit GRB-like behavior and must be in relativistic motion, with a Lorentz factor $\Gamma\sim 10^{2}$ (Lyubarsky, 2014). These are the so-called GRB-like models, which radiate FRB from relativistic jet. In this section, we aim to explore the disparities between static and relativistic sources with a focus on their geometries. Notably, highly relativistic motion leads to the beaming effect. Considering FRBs as plane waves, we denote the energy flux density as: $S_{0}=E^{2}c/4\pi$ (in cgs units). Therefore, for an isotropic static source, the FRB luminosity is:

For the static and relativistic sources, the Eq. (4) has different forms (see the discussion below).

Assuming the source is isotropic and the radiation is simultaneous, the duration of FRB $t$ is the result of the time delay during propagation. The scale of the source is:

Due to the differences in geometry, the scale of the source in a static configuration differs from that of a relativistic source.

In the case of relativistic sources, the radiation is concentrated within a cone of angle $1/\Gamma$, because of the relativistic beaming effect. Therefore, in areas where the angle between the line of sight and the line connecting the observer’s position and the source center is larger than $1/\Gamma$, the radiation will not propagate to observer. Consequently, not all parts of the source are observable. This geometry differs from that of static sources. In the observer’s line of sight, the source appears as a cone. In this scenario, the radius and time $t$ obeys:

The difference between relativistic sources and non-relativistic sources lies in its geometry, as shown in Eq. (5) and Eq. (6).

In Section 3, it is described that by imposing constraints on $\Gamma$, it is possible to determine whether $\Gamma$ of the GRB-like model can be confirmed. Applying Eq. (5) to Eq. (4), we have

Eq. (7) shows the radiation luminosity of a static source and can be plotted on a log-log scale. If the static assumption is not valid (breakthrough the $E_{\max}\equiv 3\times 10^{12}\rm{V\;cm^{-1}}$), the Eq. (7) should be swapped to ( isotropic equivalent luminosity) ¹¹1 Here we assume the relativistic jet opening angle is greater than $1/\Gamma$. Because of the beaming effect, the main emission is from the region inside $1/\Gamma$. Therefore, we do not need a jet angle correction for the observed the luminosity.:

with the same maximum $E$, this relativistic source could supply higher luminous events. The difference can be shown in a diagram $L-t$, which will be shown later.

In this work, we examined whether some bright, short duration FRBs could originate from relativistic sources, based on a quantum cascade limit on the electric field strength. The luminosity-duration distribution for 828 FRB events were analyzed. Most FRBs fell below the limit expected for a static, non-relativistic source. However, the two bursts had a higher luminosity and a shorter duration, suggesting they may originate from relativistic sources. This could support GRB-like radiation models for part of the FRBs. However, more extreme FRBs are needed to strengthen this conclusion.

If confirmed, the Lorentz factor distribution could distinguish GRB-like and pulsar-like models. The progenitor environments and emission physics may also differ between relativistic and non-relativistic FRB sources. For the GRB-like model, it predicts multiwavelength afterglow. Further multi-wavelength follow-up will shed light on the origin. The radio spectra and polarization properties could also diagnose relativistic boosting effects. Ultimately, identifying more high luminosity, short duration FRBs and modeling their radiation mechanisms will help unravel the origin of these enigmatic signals.

We thank the helpful discussions with Enping Zhou, Hao Wang, Shiyan Tian, Lin Zhou, Yuanyuan Zuo and Ruihang Dong. The English is polished with ChatGPT. This work is supported by the National Natural Science Foundation of China (Grant No. 12041306).

The data used are publicly available in the FRBSTATS catalog (https://www.herta-experiment.org/frbstats/catalogue).