Fast Radio Bursts: Electromagnetic Counterparts to Extreme Mass Ratio Inspirals

Fast Radio Bursts (FRBs) are bright pulses with large dispersion measures (DMs) at radio frequency, with millisecond-duration randomly occurring at cosmological distances (Petroff et al., 2019; Xiao et al., 2021; Zhang, 2022). FRB was first discovered in 2007 (Lorimer et al., 2007), and some FRBs were observed to be active repeatedly (Spitler et al., 2016). Whether there are two intrinsically different categories, repeating one and non-repeating one, has not been confirmed yet. Up to now, over 600 FRBs have been reported (CHIME/FRB Collaboration et al., 2020). However, their physical origins are still poorly known. It has been speculated that the central engines of FRBs are magnetars (Popov & Postnov, 2013; Kulkarni et al., 2014; Murase et al., 2016; Katz, 2016; Metzger et al., 2017; Wang & Yu, 2017; Beloborodov, 2017; Lu & Kumar, 2018; Yang & Zhang, 2018; Wadiasingh & Timokhin, 2019; Wang et al., 2020). The discovery of FRB 20200428 produced by Galactic magnetar SGR J1935+2154 confirmed that at least some FRBs are from magnetars (Bochenek et al., 2020; CHIME/FRB Collaboration et al., 2020).

The variations of the RM and DM can provide us with important information about the surrounding environment of FRBs. Long-term monitoring of repeating FRBs revealed that the majority of them show large RM variations, suggesting them in dynamical magneto-ionic environments (Mckinven et al., 2023). FRB 20121102 has a large RM ($\sim 10^{5}~{}\rm{rad}~{}m^{-2}$) (Michilli et al., 2018), which decreased by 30% in about two years (Hilmarsson et al., 2021) and 70% till to 2023 (Feng et al., 2023). It may be caused by the wind nebula of a young magnetar (Margalit & Metzger, 2018; Zhao & Wang, 2021), the ejecta from the progenitors (massive stars or compact binary mergers Piro & Gaensler 2018; Zhao et al. 2021), or the outflow from an SMBH (Zhang, 2018; Katz, 2020; Yang et al., 2023). FRB 20201124A showed a short-time variation of RM (Xu et al., 2022) and the first local RM reversal (Wang et al., 2022). A magnetar/Be star binary is promising to explain the unusual features of FRB 20201124A (Wang et al., 2022; Lu et al., 2023), including the RM variation and reversal, depolarization and Faraday conversion. FRB 20190520B was found to reside in a dwarf galaxy at redshift $z=0.241$. It has a lager host DM $\sim 900~{}\rm{pc}~{}\rm{cm}^{-3}$ (Niu et al., 2022), which may be related to a young supernova remnant (Zhao & Wang, 2021; Katz, 2022). FRB 20190520B showed a large value of RM and the RM reversal (Anna-Thomas et al., 2023), which can be explained in massive binary models (Wang et al., 2022; Zhao et al., 2023; Anna-Thomas et al., 2023). FRB 20180301A also showed a varying RM with reversal, which also indicates a binary origin (Kumar et al., 2023). However, a binary origin is not necessary to interpret the RM fluctuations. As shown by Beniamini et al. (2022), if the FRB passes through a region with turbulent and even weakly magnetized plasma can result in stochastic RM fluctuations.

Extremely large RMs also have been observed in the vicinity of massive black holes. For example, PSR J1745-2900, which resides 0.12 pc from Sgr ${A^{\ast}}$ was estimated to have an RM = $-6.69\times 10^{4}~{}\rm{rad}~{}m^{-2}$ (Eatough et al., 2013). The RM variation is about 3500 $\rm{rad}~{}m^{-2}$ (Desvignes et al., 2018). PSR J1746-2850, which is close to Sgr A^∗, is reported to have RM $=-12234~{}\rm{rad}~{}m^{-2}$ with variation of $300-400$ $\rm{rad}~{}m^{-2}$ (Abbate et al., 2023). So it is possible that some FRB sources may be near SMBHs. Combining with the evidence that one magnetar can result in producing of FRB-like bursts, we consider an FRB-generating magnetar orbiting an SMBH, called EMRI.

EMRI is one of the main sources expected for the future space-borne GW detector LISA to detect (Amaro-Seoane et al., 2017, 2023). When a neutron star (NS) inspirals into an SMBH, the system emits significant amounts of continuous gravitational waves (GWs) along each orbit in the frequency domain to which LISA will be most sensitive. Such GWs can be regarded as ‘standard siren’ if the information about redshift is available. In this case, it is natural to seek the EM counterpart of GW carrying the redshift information. Although it is currently believed that there are no such EM counterparts of EMRI signals in most cases, some models have proposed that EMRI can be associated with EM signals. For example, a new formation channel of EMRIs with tidal disruption flares as EM counterparts was proposed by Wang et al. (2019). Quasi-periodic eruptions generated from the Roche lobe overflows of an evolved star of white dwarf orbiting an SMBH can be detected as EMRI sources with EM counterparts (King, 2020; Zhao et al., 2022). Recently, some indications for the EM counterpart of GWs have been found. For instance, GW 190425, a binary NS merger event, was reported to be associated with a one-off FRB 20190425A with the chance probability $2.8~{}\sigma$ (Moroianu et al., 2023). However, this association was questioned by Bhardwaj et al. (2023). Sridhar et al. (2021) and Katz (2023) proposed that periodic FRBs can be produced from the extreme mass ratio binaries that contain ordinary stars and intermediate mass black holes. In this letter, we propose that FRBs may be the EM counterparts of EMRIs.

This letter is organized as follows. The relative FRB progenitor models are discussed in Section 2. The EMRI signals produced in the NS-SMBH system are given in Section 3. The event rate is estimated in Section 4. Discussions and conclusions are given in Section 5

Since the discovery of FRB 20200428, progenitors related to magnetars have received increasing attention. Several magnetar progenitor models were proposed, for more details see reviews by Xiao et al. (2021) and Zhang (2020). Here we focus only on the single magnetar model proposed by Kumar & Bošnjak (2020) and the cosmic comb model proposed by Zhang (2017).

We consider a system in which an NS moves along a highly eccentric orbit around the central SMBH, and the orbital pericenter decays owing to the GW radiation, producing the EMRI signals that may be detected by LISA and Tianqin. In such a system, except for the EMRI signals, there are two different ways to produce an FRB. Firstly, the outflow from the SMBH may comb the magnetosphere of NS, producing a bright FRB according to the cosmic comb model. Secondly, an FRB can be produced in the magnetosphere of NS according to the model of Kumar & Bošnjak (2020).

In this work, we focus on the binaries composed of an NS and an SMBH. In the single magnetar model, once an FRB-producing magnetar was captured by the SMBH, this magnetar started to enter its inspiral phase. Then it can be counted as an event. In the cosmic comb model, when an NS is captured, it is an event. So we can derive the event rate as follows. The first step is estimating the space density of SMBH in a certain mass range. The second step is estimating the rate at which each SMBH captures NS. The third step is estimating the fraction of magnetars that can produce FRBs among the NS populations. The fourth step is adding together the event rates of the two models to get the total event rate.

The $M-\sigma$ relation is (Merritt & Ferrarese, 2001)

where $\lambda=4.72,~{}M_{\rm{BH},\ast}=3\times 10^{6}M_{\odot},~{}\sigma_{\ast}=90~{}\rm{km~{}s}^{-1}$, and $\sigma$ represents the velocity dispersion. In addition, $\lambda=4.02,M_{\rm{BH},\ast}=5\times 10^{6}M_{\odot}$ was proposed by Tremaine et al. (2002). Galaxy luminosity functions can be used to constrain the galaxy velocity dispersion function. Conjoining with the $M_{\rm{BH}}-\sigma$ relation, the black hole mass function is

where $\epsilon=3.08/\lambda$, $\varphi_{\ast}$ is invariable and represents the overall number density of galaxies and $\Gamma(z)$ is the gamma function. The $\varphi_{\ast},M_{\rm{BH},\ast}$and $\gamma$ can be constrained for different types of galaxies (Aller & Richstone, 2002). For a typical SMBH mass range, $10^{5}~{}M_{\odot}\leq M_{\rm{BH}}\leq 10^{7}~{}M_{\odot}$, the total spatial density of black hole is approximately as

where $h_{70}\equiv H_{0}/70~{}\rm{km~{}s}^{-1}\rm{Mpc}^{-1}$ is the dimensionless Hubble constant. The simulation of the Milky Way result gave a prediction that the capture rate of NS is $10^{-6}~{}\rm{yr}^{-1}$ (Freitag, 2001), and the extreme mass ratio capture rate in a galaxy is (Gair et al., 2004)

where $F$ is the fraction of the total stellar mass in NS, and $t\sim 10^{10}~{}\rm{yr}$. Fixing $m=1.4~{}M_{\odot}$ and $F=0.01$, we can derive the event rate $R$, which is shown in Table 3. The typical value is about 1 $\rm{Gpc^{-3}~{}yr^{-1}}$. This is the event rate of cosmic comb model.

Then we present the estimation for the event rate of single magnetar model. We firstly adopt the result that the fraction of NSs being born as magnetars is $\sim 10\%$ (Popov et al., 2010), but Beniamini et al. (2019) also estimated the fraction to be $\sim 40\%$. Secondly, we consider the requirement that the FRB-producing magnetar should be in its active lifetime when it is captured. The active lifetime that the magnetar spends in the EMRI inspiral phase during its whole life can be roughly calculated. The age of an NS can easily reach Gyr, and the active phase only lasts for $\sim 10^{4}~{}\rm{yr}$. So the fraction of active magnetars can be calculated as $10^{4}\rm{yr}/10^{9}\rm{yr}\sim 10^{-5}$. In such a situation, the event rate is so tiny: $10^{-6}~{}\rm{Gpc^{-3}~{}yr^{-1}}$ or $4\times 10^{-6}~{}\rm{Gpc^{-3}~{}yr^{-1}}$.

We should add together the event rates corresponding to the two models to get the total event rate. Eventually, the total event rate is estimated to be $\sim 1~{}\rm{Gpc^{-3}~{}yr^{-1}}$, which is similar to the event rate of the cosmic comb model. This is because the contribution from the cases that FRBs produced in magnetosphere is so tiny. Captures of compact objects by SMBHs serve as the standard formation channel of EMRIs. Other processes including the tidal separation of compact binaries and formation or capture of NSs in accretion disks can also form EMRIs (Amaro-Seoane et al., 2007; Maggiore, 2018). Beyond that, ‘fake plunges’ (Amaro-Seoane et al., 2013) and Hills binary disruption (Sari & Fragione, 2019) also contribute to the event rate of EMRIs significantly. Therefore, the event rate is not grossly overestimated.

Based on the detection distances of the detectors shown in Table 1 and Table 2, few events fall within the detection range in a 4-yr mission time of LISA. However, there are uncertainties in both the spatial density of SMBHs and the capture rate. The SMBH space density in our range was found to be magnitude lower using SDSS data (Sheth et al., 2003). However this extrapolation should not be reliable on account of the insufficient resolution of SDSS spectra.

The host galaxies of a few FRBs were confirmed by arcsecond-scale localization by modern radio interferometers (Gordon et al., 2023). Recently, the potential host galaxies of FRB 20200223B and FRB 20190110C were proposed (Ibik et al., 2023). These observations found that most of FRBs are not located in the centers of their host galaxies. However, those FRBs with less than a few kpc offset can still be potential candidates for such systems because SMBHs may have an offset of kpc scale from the center of galaxies (Reines et al., 2020).

For scenario 1, due to the limits of the cosmic comb model, the magnetar must approach the SMBH closely (e.g. $r\sim 10^{-6}~{}\rm{pc}$). We use the upper limit of $a$ to calculate the detection range. Theoretically, we can detect the event with a smaller semi-axis $a$ as long as the lower limit of the orbit is met. However, binary systems with extremely small semi-axis are actually rare in the universe. For scenario 2, when the semi-axis $a$ is the order of $10^{-3}~{}\rm{pc}$, it seems hard to detect the EMRI signals. Unless the orbit has a large eccentricity (e.g.$~{}e>0.995$), the peak of signal amplitude can reach the most sensitive frequency band of the detectors. In general, the maximum horizon of such events is $z\sim 0.04$ in the most optimistic case. If the magnetar gets closer to the SMBH, it can be extended to a further detection distance but is not likely to be detected due to the short lifetime. Under the assumption that cosmic comb model proves to be a viable model for FRB production, the total event rate can be as high as 1 Gpc^-3 yr^-1.

Even though there are various potential candidates for FRB progenitors, the origin of FRBs still remains mysterious. In the future, the detection of EMRIs with FRBs as the EM counterpart will help us to study FRBs. These sources are also important for cosmological applications, such as measuring the Hubble constant (Abbott et al., 2017; Yu et al., 2018; Wu et al., 2022).

We thank the anonymous referee for constructive comments. We thank Yuan Feng, Chen Xian, Chen-Ran Hu, Xuan-Dong Jia, Hao-Tian Lan, Jian-Guo He and Yun-Qing Wang for their helpful discussions. This work was supported by the National Natural Science Foundation of China (grant Nos. 12273009, 12041304 and 12288102), the National SKA Program of China (grant No. 2022SKA0130100), and the China Manned Spaced Project (CMS-CSST-2021-A12).