**3. Observational Results**

Searching SGRB precursors has been performed in various space observatories, such as INTEGRAL [78], *Swift* [40,79], and *Fermi*/GBM [69,79]. Detecting weak signals before SGRBs will be subject to instrumental biases (the energy range and sensitivity). For example, although *Fermi*/GBM covers a broad energy range (∼8 keV–40 MeV), *Swift*/BAT has a higher sensitive in the 15–150 keV band. Thus *Swift* will be stronger to detect soft weak precursors, consistent with observations.

In the pioneer research, precursors are searched by visual inspection in binned light curves with a certain fixed bin width (e.g., [33]). Later, wavelet analysis is introduced to analyze such binned light curves [34,40]. Recently, the Bayesian block algorithm [80] has been widely applied in both the binned light curve and the time-tagged event data [69,79,81]. Yet in the Bayesian block algorithm, the false alarm probability is adopted; thus, additional analysis is required to obtain the significance of the precursor [69,79]. By applying these methods, precursors are found in both long GRBs and SGRBs, and the detection rate of precursor is higher in long GRBs (e.g., [35,81]).

For SGRBs, the fraction of precursor events is less than 0.4% for INTEGRAL [78], and is ∼8–10% for *Swift*/BAT [40]. For the combined *Swift* and *Fermi*/GBM sample, the fraction is found to be 2.7% [79]. Applying the Bayesian block algorithm in the *Fermi*/GBM sample alone, we found a fraction of 3.0% SGRBs are associated with precursor activities [69], while only a fraction of 1.2% is found in Reference [81]. The major difference in the detection fraction may arise from the selection criteria for the precursors. The precursor events provided in Reference [69] are found of the signal-to-noise ratio (SNR) & 4.5*σ*, where the SNR is obtained in the optimized energy range for the precursor. Thus, we adopt this precursor sample for *Fermi*/GBM in the following analysis. In Table 1, we list the SGRBs with precursors detected by *Fermi*/GBM [69] and/or *Swift*/BAT [40,79], where we show the duration of the precursor (*τ*pre), waiting time (*τ*wt), and the main SGRB (*τ*GRB). For the *Fermi*/GBM sample [69], the duration is provided following the common definition of *T*90, during which 90% of the total counts have been detected. However, for the *Swift*/BAT sample, the duration is directly provided by the wavelet analysis or Bayesian block analysis following References [40,79].

Precursors of SGRBs are usually too faint to perform spectral analysis. Therefore, the hardness ratio is often used to indicate the spectral properties [35,40,69,79]. Previous research found that there is no significant spectral difference between the precursors and the main GRBs for *Swift*/BAT events [35,40], while for *Fermi*/GBM events, a slight difference is found [79]. There could be two possibilities for precursor events having similar spectra to the main GRBs. On the observational side, this might be caused by the narrow bandpass of *Swift*/BAT and the lack of photon counts [40]. On the theoretical side, there is a possibility that the precursor and the main GRB are mimicked by two episodes of activities produced by collapsars with only the "tip-of-the-iceberg" of the light curve being observed, which makes them have similar spectral shapes [36,82–85]. However, the latter possibility is disfavored by the *f*-factor analysis [85] for most of the events in Table 1; thus, we focus here on the scenario that precursors have different origins from the main GRBs.

For the *Fermi*/GBM events, there are several events that have enough photon counts to do spectral analyses as shown in Table 1 of Reference [69]. The precursors of GRB111117A and GRB160804180 are found to be in favor of non-thermal spectra and can be well explained by the magnetospheric interaction model; the precursors of GRB081216 and GRB141102A favor thermal spectra and can be explained by the SBO model [69]. For the magnetospheric interaction model, the precursor duration relates to the chirp signal time of GW radiation. For the SBO model, this relates to the radius and Lorentz factor of the shock. Note that for GRB090510, there are two precursors, and the second one may be described by the thermal SBO model [69], while the first one could then originate from magnetospheric interactions.

The GRB duration is usually described as *τ*GRB ≈ *R*GRB(1 + *z*)/(2Γ 2 *c*), where *R*GRB is the jet dissipation radius, Γ is the bulk Lorentz factor at *R*GRB, and *z* is the redshift of

the source. For the magnetospheric interaction model, the waiting time consists of the jet launch time (∆*t*jet) and the jet propagation time (∆*t*GRB ∼ *τ*GRB). For the SBO model, the waiting time relates to the jet propagation from the SBO radius to the jet dissipation radius, i.e., *τ*wt = (*R*GRB − *R*SBO)(1 + *z*)/(2Γ 2 *c*). For the cases with ∆*t*jet *τ*GRB and *R*SBO *R*GRB, we would expect *τ*wt ∼ *τ*GRB. Yet there is an exception case for magnetospheric interaction model, in which the NS merger remnant is a stable NS (SNS) formed after the spin-down of the initially-formed uniform-rotation-supported supramassive NS with ∆*t*jet > *τ*GRB [19,86]. Note that for the SBO model, one can constrain the ratio of the radii *R*SBO/*R*GRB ≈ 1 − *τ*wt/*τ*GRB from observations.

Previous results based on *Fermi*/GBM events have indicated that *τ*wt ∼ *τ*GRB can be generally satisfied [69]; here, in Figure 2, we show the updated *τ*wt − *τ*GRB diagram, which includes both *Fermi*/GBM and *Swift*/BAT events. The fitting of the data (red line) shows that *τ*wt ≈ 1.9*τ*GRB, largely consistent with theories and previous results [69]. However, note here that the fitting errors are not provided, as the errors of the data points are not available for the *Swift* events [40,79]. However, there are two outliers, GRB090510 (the first precursor event) and GRB191221802, with *τ*wt *τ*GRB. This might suggest that SNSs are formed in these two events.

**Table 1.** The durations of the precursor (*τ*pre), waiting time (*τ*wt), and the main SGRB (*τ*GRB) are taken from [69] for *Fermi*/GBM detected bursts, and from [40,79] for *Swift* detected bursts (marked with ' <sup>+</sup>'). *<sup>a</sup>* For the events only detected by *Fermi*/GBM, their names are provided following the Fermi GBM Burst Catalog. The redshift is 0.287 for GRB060502B, and GRB090510 for 0.903.


**Figure 2.** The waiting time and GRB duration are taken from [69] for *Fermi* detected bursts, and from [40,79] for *Swift* detected bursts. The black is the *τ*wt = *τ*GRB line, and the red line is the fitting of the data.

#### **4. Discussion and Prospects**

Precursors have been detected for a small fraction of SGRBs. Here, we briefly review the models for precursors, mainly focusing on the magnetospheric interaction model and the SBO model, while the crust crack model and the fireball photospheric radiation model are found to be less likely based on current observations [69]. We focused on the explanation of the major physical processes in these models. To directly compare with observations, we estimated the luminosity, spectrum, duration, and opening angle for these models.

For the magnetospheric interaction model, the precursor will be produced simultaneously with GWs. A cutoff-power-law spectrum is expected with a photon index ∼−2/3 and a cutoff at MeV. Moreover, fast radio bursts (FRBs) are suggested to be produced during the magnetospheric interaction (e.g., [60,63]). It should be noted that for NS–BH binaries with mass ratio <0.2, the NS would be swallowed by the BH without producing a GRB and, thus, only the precursor is available.

While for the SBO model, the precursor is produced after the merger, but before the main GRB. Although GRB 170817A was classified as an SGRB, with a duration ≈0.5 s, it was fainter than the faintest SGRB previously detected by roughly three orders of magnitude, with the isotropic equivalent energy of *<sup>E</sup>γ*,iso <sup>=</sup> <sup>3</sup> <sup>×</sup> <sup>10</sup><sup>46</sup> erg. The delay time between the GW signal and the *γ*-rays, *τ*GW,*<sup>γ</sup>* = 1.7 s. The peak energy of the observed integrated spectrum is *E<sup>p</sup>* = 185 ± 62 eV [2,3]. The breakout layer parameter that could produce the observables of GRB 170817A are *<sup>R</sup>*SBO <sup>≈</sup> <sup>6</sup> <sup>×</sup> <sup>10</sup><sup>11</sup> cm and <sup>Γ</sup>SBO <sup>≈</sup> 4. SBO emissions from the jet-cocoon system seems to provide a natural explanation for this observed event because of the low radiation efficiency and the wide emission angle. However, the event rate for cocoon-SBO-induced GRBs should be very small, considering that most GRBs are observed at cosmological distances [87].

Compared with the main GRBs, we found that precursors are usually much weaker, but with a larger opening angle. Thus, for the NS mergers that occurred within several hundred Mpc, the detection of precursors is very likely. This will greatly benefit the search for gamma-ray counterparts of GW events and FRBs, which can be well tested by the current and near-future observatories, e.g., *Fermi*/GBM and *Swift*/BAT, GECAM [68], and the space-based multi-band astronomical variable objects monitor (SVOM) [88]. Furthermore, the time delay between precursors and GRBs or GW can be used to constrain the jet launch mechanism and post-merger remnant [19,69]. For the magnetospheric interaction model, photon splitting could be important, and it might significantly change the polarization state of emitted photons [89,90]. This can also be tested by the future gamma-ray polarimeter detector POLAR-2 [91].

**Funding:** J.W. acknowledges the sponsorship of the Alexander von Humboldt Foundation.

**Data Availability Statement:** No new data were created or analyzed in this study.

**Acknowledgments:** The authors thank the reviewers for the valuable comments and suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.
