Is GRB 110715A the Progenitor of FRB 171209?

The physical origin of fast radio bursts (FRBs) is unknown. Young magnetars born from gamma-ray bursts (GRBs) have been suggested as a possible central engine of FRBs. We test such a hypothesis by systematically searching for GRB–FRB spatial associations from 110 FRBs and 1440 GRBs. We find that one FRB event discovered by the Parkes telescope, FRB 171209, is spatially coincident with a historical long-duration GRB 110715A at z = 0.82. The afterglow of GRB 110715A is consistent with being powered by a millisecond magnetar. The extragalactic dispersion measure of FRB 171209 is in excess of that contributed by the intergalactic medium, which can be interpreted as being contributed by a young supernova remnant associated with the GRB. Overall, the significance of the association is (2.28–2.55)σ. If the association is indeed physical, our result suggests that the magnetars associated with long GRBs can be the progenitors of at least some FRBs.


INTRODUCTION
Fast radio bursts (FRBs) are mysterious radio transients with millisecond durations, extremely high brightness temperatures and large dispersion measures (DMs) (e.g., Lorimer et al. 2007; Thornton et al. 2013;Petroff et al. 2019;Cordes & Chatterjee 2019). Their DMs are in excess of the Galactic contribution and the precise localizations of the host galaxies of a few FRBs suggest that they are extragalactic (e.g., Lorimer et al. 2007; Thornton et al. 2013;Chatterjee et al. 2017;Bannister et al. 2019;Prochaska et al. 2019;Ravi et al. 2019;Marcote et al. 2020). A persistent radio emission with luminosity of L ∼ 10 39 erg s −1 at a few GHz was discovered to be spatially coincident with FRB 121102, which showed a non-thermal spectrum that deviates from a single power-law spectrum from 1 GHz to 26 GHz (Chatterjee et al. 2017). One possibility is that such a persistent radio emission source originates from a shocked nebula associated with a young magnetar born in a supernova (SN) or a gamma-ray burst (GRB) Metzger et al. 2017). On the other hand, there is no confirmed multi-wavelength transient being associated with any FRB (e.g., Petroff et al. 2015;Callister et al. 2016;Zhang & Zhang 2017;MAGIC Collaboration et al. 2018;Tingay & Yang 2019). There might be three main reasons: 1. The fluxes of the multi-wavelength counterparts of FRBs are low, e.g., for typical parameters, the FRB afterglows are very faint (Yi et al. 2014); 2. The duration of the multi-wavelength transient may be shorter than the time resolution of a detector, e.g., the prompt high-energy emission associated with the FRB itself (Yang et al. 2019b); 3. the time delay between the multi-wavelength transient and the FRB is longer than the observation time, e.g., FRBs emitted from a young magnetar born from a catastrophic event (such as a GRB or a SN) may have a long delay with respect to the event itself Metzger et al. 2017).
Lacking confirmed multi-wavelength transients associated with FRBs, the physical origin of FRBs is still unknown. The current FRB models can be divided into two categories 1 : catastrophic models (e.g., Kashiyama et al. 2013;Totani 2013;Falcke & Rezzolla 2014;Zhang 2014Zhang , 2016Wang et al. 2016) and non-catastrophic models (e.g., Murase et al. 2016;Metzger et al. 2017;Zhang 2017Zhang , 2020Dai et al. 2016;Margalit & Metzger 2018;Wang et al. 2020;Ioka & Zhang 2020). The former suggested that an FRB is directly associated with a catastrophic event, and the time delay between the FRB and the catastrophic event is short. The latter usually involved a compact star, e.g., a neutron star or a black hole, that was born in a catastrophic event as the progenitor of an FRB. Since the compact star can exist for a much longer time, the time delay between the FRB and the catastrophic event is allowed to be relatively long.
GRBs are the most luminous catastrophic events, and are produced by core collapse of massive stars or binary compact star mergers (Mészáros 2006;Zhang 2018b). Although GRBs are much rarer than FRBs, the following reasons have been raised to suggest that a fraction of FRBs could be associated with GRBs: 1. an FRB might occur when a supermassive magnetar born in a GRB collapses to a black hole, so called "blitzar" scenario (Falcke & Rezzolla 2014;Zhang 2014); 2. an FRB might be related to the merger of a binary neutron star which produces a short GRB (Totani 2013;Wang et al. 2016); 3. a GRB as the source of astrophysical stream could interact with the magnetosphere of a neutron star to produce an FRB (Zhang 2017); 4. a GRB could produce a young magnetar that emits FRBs at a much later epoch (e.g., Metzger et al. 2017;Margalit et al. 2019;Wang et al. 2020). In the first three scenarios, an FRB could occur from milliseconds before to a few thousand seconds after the GRB. The fourth scenario allows a much longer delay of FRBs with respect to the GRB. Related to this, recently Eftekhari et al. (2019) discovered a radio source coincident with the SLSN PTF10hgi, similar to the persistent radio emission of FRB 121102 (Chatterjee et al. 2017), about 7.5 years post-explosion, which might be emitted by the magnetar born in the SLSN. However, no FRB was detected from the source yet.
In particular, Men et al. (2019) recently performed dedicated observations of the remnants of six GRBs with Platts et al. (2019). evidence of having a magnetar central engine, but these observations did not lead to detection of any FRB from these remnants during a total of ∼ 20h of observations.
In this work, we adopt a different approach to test the hypothesis that GRBs can be the progenitor of FRBs. We systematically search for GRB-FRB association events based on the precise localization of GRB afterglows, allowing a few years of time delay between a GRB and an FRB. Observationally, GRBs are typically localized by the Neil Gehrels Swift Observtory, i.e. the burst is detected by Swift /BAT and quickly localized by Swift /XRT with a several-arcsecond error bar, and later further localized by Swift /UVOT or groundbased telescopes to sub-arcsecond precision. Based on the archival Swift /XRT and optical observational data, we search for possible GRB-FRB spatial association candidates. We detect one possible association between FRB 171209 (Os lowski et al. 2019) and GRB 110715A at z = 0.82 (Sánchez-Ramírez et al. 2017). This paper is organized as follows. In Section 2, we present the search method and result. The GRB 110715A -FRB 171209 association is discussed in Section 3 in detail. The results are summarized with discussion in Section 4. Throughout the paper, we adopted a concordance cosmology with parameters H 0 = 71 kms −1 Mpc −1 , Ω M = 0.30, Ω Λ = 0.70, and temporal and spectral slopes of GRB afterglow emission are defined as F ∝ t −α ν −β . Moreover the convention Q = 10 n Q n is adopted in cgs units.  Figure 1 shows the sky distribution of our samples (110 FRBs and 1440 GRBs) in celestial coordinates. The GRBs in our sample show a large-scale isotropic distribution, which is well known from the BATSE observations (Briggs et al. 1996). Although the sample size of FRBs is smaller than that of GRBs, the FRBs in our sample also show an isotropic distribution, consistent with their cosmological origin. As shown  in Figure 2, the distributions of the FRB fluence (log N − log S) 4 show a tendency with N (> S) ∝ S −3/2 at high S values. The deviation from the 3/2 power law is evident at low S values, which may be related to the spatial inhomogeneity effect and likely also observational biases and instrumental effects. We perform a systematic search for GRBs that satisfy the following three criteria: (1) the GRB position is consistent with that of an FRB; (2) the GRB occurred earlier than the FRB if a position coincidence is discovered; (3) the redshift of the GRB is lower than the maximum FRB redshift derived from its DM.
The redshift of GRB 110715A was measured to be z = 0.82 (Piranomonte et al. 2011).
On Due to the existence of large-scale structures, the uncertainty of the DM contributed by the IGM is about σ IGM ∼ 300 pc cm −3 . Thus the maximum redshift is constrained in the range of z < (1.08 − 1.78). This is larger than the redshift of GRB 110715A.
To calculate of chance possibility for the putative GRB 110715A -FRB 171209 association, we assume that the spatial distribution of GRBs is isotropic and the number of GRBs within a specific sky area and time interval satisfies the Poisson distribution. The chance probability of having at least one GRB in the error circle of one FRB can then be written as where λ = ρS is the expected number of GRBs in the FRB error region S. To estimate the p-value of the chance coincidence, we adopt two approaches. First, for a conservative estimate, we use the uncertainty of 7.5 arcmin defined by the error bar of the FRB position, i.e. δR = 0.125 • . We obtain the chance probability of having at least one (out of 1440) GRB whose distance to FRB 171209 is smaller than 0.125 • , which gives P 1 ≈ 0.0017. The chance probability of having only one such association for all 110 FRBs can be estimated as P = 1 − (1 − P 1 ) 110 ≈ 17.1%. We verify this simple estimate through Monte Carlo sim-ulations. We randomly generate 1440 GRBs and 110 FRBs in the sky. Based on 10 5 simulations, the chance probability of having a GRB/FRB pair with a separation smaller than 0.125 • is 17.4%, consistent with the analytical estimate.
One also needs to consider two other criteria for an association, i.e. the timing criterion (the GRB needs to occur before the FRB) and the redshift criterion (the maximum redshift derived from the FRB DM is larger than that of the GRB). To do this, we use the observed distributions of the detection time and redshift for both GRBs and FRBs to perform the simulations. Since most GRBs were detected earlier than FRBs (FRBs were discovered much later than GRBs), adding the timing criterion does not reduce the chance probability significantly, i.e. ∼ 14.1%. However, since the average redshift of GRBs is higher than the average maximum redshift of FRBs, adding the redshift criterion reduces the chance probability significantly to ∼ 2.3%, which corresponds to a significance of 2.28σ.
Second, since GRB 110715A is well located inside the error circle of FRB 171209,one may use the angular distance between the centers of the error boxes of the two events, 0.0836 • , as δR 5 One can obtain the chance probability of having at least one (out of 1440) GRB whose distance to FRB 171209 is smaller than 0.0836 • , i.e. P 1 ≈ 0.0007. We also randomly generate 1440 GRBs and 110 FRBs in the sky. Based on 10 5 simulations, the chance probability of having one GRB/FRB pair with a separation smaller than 0.0836 • is 7.6%. Considering the timing criterion, we obtain P = 6.3%. When the redshift criterion is also considered, the final chance probability is 1.1%, which corresponds to a 2.55σ confidence level for the GRB 110715A/FRB 171209 association.
Even though statistically one cannot establish a firm association between GRB 110715A and FRB 171209, it is nonetheless interesting to investigate whether physically such an association makes sense.

Magnetar as central engine of GRB 110715A
The Swift /BAT time-integrated spectrum of GRB 110715A can be well fitted with a Band function (Band et al. 1993), with E peak = 92.8 ± 18.1 keV, α = −1.23 ± 0.12, and β = −2.05 ± 0.19 and χ 2 =0.98 (as shown in Figure 3). We obtained the isotropic γ-ray energy E γ,iso = 1.06 ± 0.10 × 10 53 erg in the 1-10 4 keV band. The results from the time-resolved spectral analysis show the "flux-tracking" pattern for E p . To fit the GRB 110715A afterglow lightcurves, we employed a broken power-law function where F 1 is the flux normalization, α 1 and α 2 are the afterglow flux decay indices before and after the break time (t b ), respectively, and ω is a smoothness parameter which represents the sharpness of the break. Figure 4 shows the X-ray and optical light curves of GRB110715A. The X-ray light curve can be well fitted by a broken power-law function, with the best-fit power-law slope α X,1 = 0.70 +0.04 −0.05 (shallow decay) before the break (t b = T 0 + 2.0 +0.4 −0.3 ks) and α X,2 = 1.60 +0.11 −0.09 (normal decay) after the break, respectively. There is a re-brightening component appearing at ∼ T 0 +50 +0.4 −0.3 ks. For the optical light curve, there is an early steep decay phase, which may be interpreted as the reverse shock emission as the ejecta is decelerated. This is followed by a shallow decay phase (with α O,1 = 0.70 +0.13 −0.12 ) breaking at t b and further decays with α O,2 = 1.60 +0.15 −0.11 . The re-brightening component also appeared in the optical afterglow. The result of the temporal analysis suggests that the X-ray and optical afterglow show an achromatic behavior (Wang et al. 2015).
We also analyze the spectral energy distributions (SEDs) of GRB 110715A afterglow, by jointly fitting the optical and XRT data with the Xspec package (Arnaud 1996) and the optical data that are corrected for Galactic extinction based on the burst direction, with A V = 0.030, A R = 0.119 and A I = 0.016 (Schlafly & Finkbeiner 2011). The extinction in the host galaxy is also taken into account assuming an extinction curve similar to that of Small Magellanic Cloud (SMC) with its Standard value of the ratio of total to se- lective extinction R v,SMC = 2.93 (Pei 1992). The equivalent hydrogen column density of our Galaxy is N H = 4.33 × 10 21 cm −2 . The equivalent hydrogen column density of the host galaxy N host H = (4.22 ± 2.95) × 10 21 cm −2 is derived from the time integrated XRT spectrum. We fix these values in our time-resolved spectral fits. We subdivided the broadband data into four temporal ranges (as marked in Figure 4). The SEDs of the joint optical and X-ray spectra can be well fitted with a single absorbed power-law function. The photon indices Γ (the spectral index β = Γ − 1) are 1.69, 1.70, 1.87 and 1.89 for the Slice 1 (T 0 + [200, 500] s), Slice 2 (T 0 + [3 × 10 3 ,8 × 10 3 ] s), Slice 3 (T 0 + [2 × 10 4 ,1 × 10 5 ] s), and Slice 4 (T 0 + [2 × 10 5 ,1 × 10 6 ] s), respectively. There is no obvious spectral evolution observed in the afterglow phase. The temporal slopes of the normal decay phase (α X,II and α O,II ) are well consistent with the closure relations (α − β) of the fireball external shock model α = 3β/2 + 0.5 = 1.54 ± 0.08, which are located in spectral regime (ν m < ν < ν c ) in the wind stellar medium (e.g. Gao et al. 2013). For the shallow decay phase closure relation α = q/2 + (2 + q)β/2 (Zhang et al. 2006), we obtained the energy injection parameter q = 0 for α X,I = 0.70 +0.04 −0.05 and α O,I = 0.70 +0.13 −0.12 , which is consistent with the energy injection from the spin-down of a millisecond magnetar (Dai & Lu 1998;Zhang & Mészáros 2001). Ç ıkıntoglu et al. (2019) also argued that the millisecond magnetar could be the central engine of GRB 110715A.
We further investigate the afterglow data with the standard forward shock model with energy injection (q = 0). A Markov Chain Monte Carlo (MCMC) method is adopted to search for the best fitting pa- rameters. The results are shown in Figure 4. One can see that the model can well reproduce the data. The best fitting parameters are: the isotropic kinetic energy E K,iso = 2 × 10 53 erg, the initial Lorentz factor Γ 0 = 45, the fraction of shock energy to electrons ǫ e = 0.268, the fraction of shock energy to magnetic fields ǫ B = 1.1 × 10 −6 , wind density parameter A * = 0.25, the energy injection luminosity L 0 = 1×10 50 erg s −1 , and the duration of energy injection t b = 2000 s. The fitting parameters are consistent with the statistical properties of a large sample of GRBs (e.g., Wang et al. 2015). Since the energy injection q = 0 is well consistent with the magnetar spin-down model, we can derive the magnetar parameters of GRB 110715A based on the data. The maximum energy is the total rotational energy of a millisecond magnetar and is defined as where I is the moment of inertia, P 0 is the initial spin period, Ω 0 = 2π/P 0 is the initial angular frequency of the neutron star, M 1.4 = M/1.4M ⊙ , and R is the radius of the magnetar. The isotropic γ-ray and kinetic energies are larger than this value, suggesting that the outflow is beamed, with a beaming factor f b = 1 − cos θ j < 0.1, where θ j is the half opening angle of the jet. Based on the characteristic spin down luminosity L 0 and the spindown timescale τ of a magnetar as shown in Equation (6) and (8) in Zhang & Mészáros (2001), one can calculate the surface polar cap magnetic field strength B p and the initial spin period P 0 : Observationally, the spindown luminosity L 0 can be generally written as (Lü & Zhang 2014) where L X,iso is the X-ray luminosity due to internal dissipation of the magnetar wind, which is negligible in our case.
Since no jet break is observed in GRB 110715A, we can use the epoch of the last observational data point to set a lower limit on θ j (Wang et al. 2018b), i.e. θ j > 6.2 o . Using E K,iso = 2 × 10 53 erg, L X,iso = 3.28 × 10 47 erg s −1 , and τ = t b /(1+z) = 2000/(1+0.82) = 1099 s, we obtain P 0 < 3.59 ms and B p < 4.95 × 10 15 G, respectively. These parameters fall into the regime of typical young magnetars for GRB central engines. Such a magnetar is believed to power repeating FRBs when the environment becomes clean Metzger et al. 2017;Margalit & Metzger 2018).

Is the magnetar the progenitor of FRB 171209?
As reported by Os lowski et al. (2019), FRB 171209 has a duration of ∆t ∼ 0.138 ms and a fluence of f ν 3.7 Jy ms at ν ∼ 1 GHz. If FRB 171209 is indeed associated with GRB 110715A, according to the redshift z = 0.82 of GRB 110715A, the luminosity distance is d L ≃ 5 Gpc. The isotropic energy of FRB 171209 is about E FRB ∼ 4πd 2 L νf ν 1.1 × 10 41 erg. If this energy is provided by the magnetic energy of the underlying magnetar, one may place a most demanding constraint on the strength of the magnetic field of the underlying magnetar assuming isotropic FRB radiation. The emission radius can be approximately estimated as r e ∼ c∆t ≃ 4.1 × 10 6 cm. The magnetic field strength at r e should satisfy where B = B p (r e /R) −3 . Therefore, the observation of FRB 171209 demands that the surface polar cap magnetic field strength is which is consistent with the observation constraints derived from the afterglow emission of GRB 110715A. According to the redshift z = 0.82 of GRB 110715A, the DM contribution from the IGM is given by (Zhang 2018a), where µ m = 1.2 is the mean molecular weight for a solar composition in the SNR ejecta, and η is the ionization fraction of the medium in the SNR. We can see that for a typical SN with a few times solar masses, the corresponding DM contribution could reach the required host-galaxy DM of FRB 171209. One should check the the free-free absorption in the SN. For a young SNR, the free-free optical depth through the ejecta shell is where n e and n i are the number densities of electrons and ions, respectively, and n e = n i and Z = 1 are assumed for an ejecta with a fully ionized hydrogendominated composition, L ∼ r ∼ vt is the ejecta thickness, andḡ ff ∼ 1 is the Gaunt factor. If the SNR ejecta is transparent for FRB, i.e., τ ff 1, one gets the SNR age (e.g. Yang et al. 2019b) t 5 yr M M ⊙ 9/10 E SN 10 51 erg where ν ∼ 1 GHz and T ∼ 10 4 K are taken. This is consistent with the 6.4 yr time delay between FRB 171209 and GRB 110715A.

SUMMARY AND DISCUSSIONS
Lacking multi-wavelength observational data of FRBs, it is hard to constrain their physical origin. It has been suggested that at least some FRBs may be physically associated with GRBs (Zhang 2014;Metzger et al. 2017). The GRB may leave behind a long-lived magnetar, which may produce FRBs through ejecting magnetosphere upon collapse Falcke & Rezzolla (2014), or more likely, produce repeated bursts through crust cracking or magnetic reconnection (e.g. Popov & Postnov 2010;Katz 2016;Beloborodov 2017;Kumar et al. 2017;Yang & Zhang 2018;Wang et al. 2018a).
We searched for possible GRB-FRB associations based on the localization data of 110 FRBs and the precise localization data of 1440 GRB afterglows. We found that the long-duration GRB 110715A is within the error box of FRB 171209 and the redshift of the GRB 110715A is lower than the maximum redshift derived from the DM of the FRB 171209. Taking the factors of spatial location, time of occurrence, and the redshift criterion, we derive a chance probability of 2.3% to 1.1%, corresponding to a 2.28σ to 2.55σ confidence level for the association.
Even though the chance coincidence probability cannot establish a firm association between GRB 110715A and FRB 171209, we nonetheless investigated whether there exists a self-consistent physical picture to make a connection between the two. We modeled the afterglow of GRB 110715A and identified a shallow-decay signature, which is consistent with energy injection by a millisecond magnetar with P 0 < 3.59 ms and B p < 4.59 × 10 15 G. With the Milky Way and IGM contributions subtracted, the observed DM of FRB 171209 has an excess of ∼ 950 pc cm −3 , which is consistent with the DM contribution of a young (∼ 6.4 yr old) SNR associated with GRB 110715A with a few solar masses and kinetic energy E SN ∼ 10 51 erg. The requirement that the free-free optical depth τ ff 1 suggests that FRBs can be observable only a few years after the explosion, consistent with the observed 6.4 yr delay between GRB 110715A and FRB 171209. FRB 171209 so far does not show a repeating behavior. Its lightcurve shows one single pulse without a noticeable temporal structure (Os lowski et al. 2019). The intrinsic duration is sub-millisecond. In principle, the burst could be an oneoff event. If it is associated with GRB 110715A, it may be related to the collapse of the supramassive neutron star at such a late epoch (Falcke & Rezzolla 2014;Zhang 2014). However, contrived conditions are needed to allow the collapsing time to be at such a late stage after the spindown timescale. More likely, FRB 171209 may be one of many repeating bursts powered by the magnetar harbored in GRB 110715A Metzger et al. 2017). Searching for repeating bursts from FRB 171209 would be essential to test this possibility.
Observationally, no SN was reported for GRB 110715A (Sánchez-Ramírez et al. 2017). This is not surprising, since GRB 110715A is not nearby and is a highluminosity long GRB with a bright optical afterglow. The SN signature is likely outshone by the afterglow. It is well known that essentially every long GRB is accompanied by a Type Ic SN (Woosley & Bloom 2006), so that invoking a SN to account for the extra DM from FRB 171209 is justified.
XGW, DBL and EWL acknowledges support from the National Natural Science Foundation of China (grant No.11673006, U1938201, 11533003, 11773007), the Guangxi Science Foundation (grant  No. 2016GXNSFFA380006, 2017GXNSFBA198206, 2018GXNSFFA281010, 2017AD22006, 2018GXNS-FGA281007), the One-Hundred-Talents Program of Guangxi colleges, and High level innovation team and outstanding scholar program in Guangxi colleges. BZ and JWL acknowledge the UNLV Top-Tier Doctoral Graduate Research Assistantship (TTDGRA) grant for support. We also acknowledge the use of public data from the Swift data archives and the FRB catalog (http://frbcat.org).