A Channel to Form Fast-spinning Black Hole--Neutron Star Binary Mergers as Multi-messenger Sources

After the successful detection of a gravitational-wave (GW) signal and its associated electromagnetic (EM) counterparts from GW170817, neutron star--black hole (NSBH) mergers have been highly expected to be the next type of multi-messenger source. However, despite the detection of several of NSBH merger candidates during the GW third observation run, no confirmed EM counterparts from these sources have been identified. The most plausible explanation is that these NSBH merger candidates were plunging events mainly because the primary BHs had near-zero projected aligned-spins based on GW observations. In view that NSs can be easily tidally disrupted by BHs with high projected aligned-spins, we study an evolution channel to form NSBH binaries with fast-spinning BHs, the properties of BH mass and spin, and their associated tidal disruption probability. We find that if the NSs are born firstly, the companion helium stars would be tidally spun up efficiently, and would thus finally form fast-spinning BHs. If BHs do not receive significant natal kicks at birth, these NSBH binaries that can merge within the Hubble time would have BHs with the projected aligned-spins $\chi_{z}\gtrsim0.8$ and, hence, can certainly allow tidal disruption to happen. Even if significant BH kicks are considered for a small fraction of NSBH binaries, the projected aligned-spins of BHs are $\chi_z\gtrsim0.2$. These systems can still be disrupted events unless the NSs are very massive. Thus, NS-first-born NSBH mergers would be promising multi-messenger sources. We discuss various potential EM counterparts associated with these systems and their detectability in the upcoming fourth observation run.


INTRODUCTION
Neutron star-black hole (NSBH) mergers are prime search targets for the ground-based gravitational wave (GW) detectors, e.g., LIGO (LIGO Scientific Collaboration et al. 2015), Virgo (Acernese et al. 2015) and KAGRA (Aso et al. 2013). Recently, two highconfidence GW events (GW200105 and GW200115) were announced by the LIGO-Virgo-KAGRA (LVK) Collaboration, which was, for the first time, identified to come from mergers of NSBH binaries (Abbott et al. 2021a;Nitz et al. 2021). Furthermore, two lower mass-gap sources (GW190814 and GW200210 092254) that could either be from a NSBH or a binary BH, and several marginal NSBH candidates were discovered in GW during the third observation run (O3) of LVK (Abbott et al. 2020b(Abbott et al. , 2021bThe LIGO Scientific Collaboration et al. 2021a,b). The analysis of these NSBH candidates displayed that their effective inspiral spins could be near-zero or very low, suggesting that their primary BHs would plausibly have negligible projected spins aligned to the direction of the orbital angular momentum (abbreviated as projected aligned-spins, hereafter). Among these candidates, only the BH component of GW200115 showed a non-zero BH spin with an apparent BH spin-orbit misalignment angle (Abbott et al. 2021a), which may require a strong natal kick for the BH or the NS Gompertz et al. 2021;Zhu 2021). Conversely, Mandel & Smith (2021) argued that GW200115 could potentially be a merger between a non-spinning BH and a typical mass of NS by reanalysing its GW signal with the consideration of astrophysically motivated priors.
NSBH mergers have long been proposed to be progenitors of some fast-evolving electromagnetic (EM) counterparts, including short-duration gamma-ray bursts (sGRBs; Paczynski 1991;Narayan et al. 1992;Zhang 2018) and kilonovae (Li & Paczyński 1998;Metzger et al. 2010). As main GW sources of ground-based GW detectors, it was thus expected that the follow-up searches after GW triggers could help find out the associated EM signals of NSBH mergers. However, despite many efforts toward the follow-up observations, no confirmed EM counterpart candidate was identified (e.g., Anand et al. 2021;Gompertz et al. 2020b;Kasliwal et al. 2020;Coughlin et al. 2020a,b;Page et al. 2020;Becerra et al. 2021). The tidal disruption probability of NSBH mergers and the brightness of EM signals depend on NS mass, NS equation of state (EoS), BH mass, and especially BH projected aligned-spin (e.g., Kyutoku et al. 2015;Kawaguchi et al. 2016;Foucart et al. 2018;Barbieri et al. 2019;Raaijmakers et al. 2021;Zhu et al. 2020Zhu et al. , 2021a. Disrupted events associated with brighter EM signals tend to occur only if a NSBH binary has a low-mass NS component with a stiff EoS and a low-mass BH component with a high projected aligned-spin. The most promising explanation for the lack of EM counterparts is that these NSBH candidates were plunging events without forming any bright EM signals, mainly due to their near-zero BH projected aligned-spins (Zhu et al. 2021b;Fragione 2021;D'Orazio et al. 2021).
The most widely accepted formation channel for the majority of NSBH binaries in the universe is the classical Common Envelope (CE) scenario (e.g., Giacobbo & Mapelli 2018;Belczynski et al. 2020;Drozda et al. 2020;Shao & Li 2021;. In this scenario, the immediate progenitor of a NSBH binary just after the CE phase is a close binary system, which consists of a compact object (NS or BH) and a helium star. On the one hand, if the first-born compact object is a BH, its spin has been found to be negligible (Qin et al. 2018;Belczynski et al. 2020). This was supported by the finding of the near-zero distribution of BH spin in the O3 GW NSBH candidates (Zhu et al. 2021a), which implies that the NSs in these NSBH candidate systems may be directly plunged into the BHs. On the other hand, for a fraction of NSBH progenitor systems in which the NS was first born, the companion helium star can be spun up by the NS and finally form a BH with a high projected aligned-spin. Since a fast-spinning BH can easily tidally disrupt the NS and produce bright EM signals (i.e., sGRBs and kilonovae), NSBH mergers formed via this formation channel can be potential multi-messenger sources that allow us to discover their associated bright EM counterparts with a high probability after GW triggers (Zhu et al. 2021c). Furthermore, as studied recently by Román-Garza et al. (2021), corecollapse physics plays a critical role in the observability of the EM signals produced by NSBH mergers. In viewing of the lack of relevant researches on the BH spin properties of such NS-first-born formation channel for NSBH mergers, we investigate this formation channel of NSBHs with fast-spinning BHs in detail and study their corresponding tidal disruption probability.
In this work, we investigate the detailed binary evolution process of forming fast-spinning BHs in NSBH binaries by taking into account the accretion feedback of the core-collapse processes, supernova (SN) kicks of newlyformed BHs, and different NS EoSs. The main methods adopted in the stellar and binary evolution models are shown in Section 2. We then present in Section 3 our findings of NSBH mergers without and with natal kicks, respectively, along with their associated probability for tidal disruption. The multi-messenger observational signatures of this channel are discussed in Section 4. Finally, main conclusions are summarized in Section 5 with some discussion.

METHODS
We use the release 15140 of MESA stellar evolution code (Paxton et al. 2011(Paxton et al. , 2013(Paxton et al. , 2015(Paxton et al. , 2018(Paxton et al. , 2019 to perform all of the binary evolution calculations in this work. A metallicity of Z = Z , where the solar metallicity is Z = 0.0142 (Asplund et al. 2009), is adopted. We create single helium stars at Zero-Age Helium Main-Sequence (ZAHeMS) following the method in Qin et al. (2018) and then relax the created helium stars to reach the thermal equilibrium, where the Heburning luminosity just exceeds 99% of the total luminosity. We model convection using the mixing-length theory (Böhm-Vitense 1958) with a mixing-length parameter α = 1.93. The Ledoux criterion is used to treat the boundaries of the convective zone, while we consider the step overshooting as an extension given by α p = 0.1H p , where H p is the pressure scale height at the Ledoux boundary limit. Semiconvection (Langer et al. 1983) with an efficiency parameter α = 1.0 is also included in our model. The network of approx12.net is chosen for nucleosynthesis.
Stellar winds are modelled with the standard "Dutch" scheme, calibrated by multiplying with a scaling fac-tor of 0.667 to match the recently updated modeling of helium stars' winds (Higgins et al. 2021). We model angular momentum transport and rotational mixing diffusive processes (Heger & Langer 2000), including the effects of Eddington-Sweet circulations, the Goldreich-Schubert-Fricke instability, as well as secular and dynamical shear mixing. We adopt diffusive element mixing from these processes with an efficiency parameter of f c = 1/30 (Chaboyer & Zahn 1992;Heger & Langer 2000). Mass transfer is modeled following the Kolb scheme (Kolb & Ritter 1990) and the implicit mass transfer method (Paxton et al. 2015) is adopted.
We model helium stars until the carbon depletion in the center. The baryonic remnant mass is calculated following the "delayed" supernova prescription in Fryer et al. (2012). As shown in Batta & Ramirez-Ruiz (2019), the newly formed BH might be unable to accrete all of the available stellar material. Therefore, in order to calculate the final mass and spin of the BH from the direct collapse, we follow the framework given in Batta & Ramirez-Ruiz (2019), which has already been implemented in a recent work (Bavera et al. 2020). The neutrino loss as in Zevin et al. (2020) is taken into account. Furthermore, the maximum NS mass is assumed to be 2.5 M in this work.
BHs formed through direct collapse are considered to receive no mass loss and thus no natal kick (Belczynski et al. 2008). Recently, an estimation of the NSBH merger rate from the LIGO-Virgo O3a run showed that negligible kicks imparted on the low-mass BHs is favored (Román-Garza et al. 2021). Consequently, calculation with no natal kick included is considered as our fiducial model. Additionally, we also take into account natal kicks onto BHs formed through direct collapse. We follow the parametrised recipes of Mandel & Müller (2020) to calculate the natal kicks of BHs. The binary properties just after the natal kick is given based on the framework of Kalogera (1996); Wong et al. (2012) and a recent work by Callister et al. (2021). For binaries surviving after the kick, we estimate the merger time given from Peters (1964) through GW emission, with the updated fitting formula for eccentricity in Mandel (2021).

RESULTS
Here we present a parameter space study of the initial binary properties for a close binary system composed of a helium star and a NS. The parameters include the initial helium star masses and the initial orbital periods. We cover the helium star mass range 8 − 40 M , the NS star mass range 1.2−2.5 M , as well as the orbital period from 0.2 to 2 days. We also take into account the natal kicks imparted onto BHs in the core-collapse models.
We describe in detail our findings for the NSBH formation without and with natal kicks, respectively, along with their associated tidal disruption probability.
3.1. Fast-spinning BHs originated from tidal spin-up in NSBH binaries Figure 1 presents various outcomes of the detailed binary evolution for helium stars at different initial masses orbiting around a NS of 1.4 M in close orbits. We choose 2 days as the upper limit of the orbital period, beyond which tides are not important (Qin et al. 2018). We do not take into account the parameter space where the initial orbital periods are shorter than 0.2 days, as the binary would experience either initial overflow or dynamically unstable mass transfer. NSs with other masses have similar results, which are not shown in the paper.
First, we note from the left panel of Figure 1 that helium stars with the initial mass higher than 10 M and initial orbital period larger than ∼ 0.4 days collapse to form BHs. Furthermore, helium stars in tighter orbits tend to lose more mass due to rotationally enhanced mass loss (Heger & Langer 1998;Langer 1998). This is why we can see that helium stars in short orbits end up forming low-mass BHs. For low-mass (< 10 M ) helium stars, including a small fraction with initial orbital periods less than ∼ 0.3 days, NSs are formed instead. A further investigation of binary NS systems formed in our grid is postponed in a future work.
In the right panel of Figure 1, we show the magnitudes of the BH projected aligned-spin for different initial binary properties. As shown in Qin et al. (2018), the spins of the resulting second-born BHs are exclusively dependent on the tidal interaction and stellar winds of the helium stars. For initial orbital periods considered in this gird, the BHs have the entire range of the spin from zero to value allowed by maximally spinning. For more massive helium stars, their stellar winds are stronger, which widen the binaries and thus make NSBHs undetectable for GW emission within Hubble Time. Interestingly, we note that for helium stars with different initial masses, the BH projected aligned-spins have a similar trend in decreasing magnitudes. After the formation of the second-born BH, the GW emission removes the orbital angular momentum of the NSBH, shrinks the orbit and eventually leads to merger of the two compact objects. The merger time is calculated as given in Peters (1964). It is worth noting that all NSBHs with their merger time less than the Hubble time have extremely fast spins. This finding is mainly attributed to the strong tidal force in close binaries. However, BHs formed through direct core-collapse may receive SN kicks, which can change the direction of the BH spins and, hence, their projected aligned-spins. The impact of SN kicks is discussed as follows.
3.2. Impact of SN kicks on BH projected aligned-spins and the probability of merging NSBHs The SN kicks imparted onto BHs are considered to cause a misalignment of the BH spin to the orbital angular momentum. In addition, the post-SN orbital separation and the eccentricity of the binary can also be modified. We briefly describe these calculations in Appendix and more details can be found in Kalogera (1996); Wong et al. (2012); Callister et al. (2021). Similar to Mandel & Müller (2020), kicks are drawn from a Gaussian distribution and we repeat 10 5 times to obtain the average value of the kick and also its associated angle θ. The corresponding averaged BH projected aligned-spins (χ z ) are shown in the left panel of Figure 2. We note thatχ z values are slightly less than those obtained assuming no kicks (see the right panel of Figure 1). This is because the SN kicks we apply to the BHs are typically small (see the red line for BH kicks in Figure 4 of Mandel & Smith 2021). Accordingly, we then obtain on average a small misalignment for BH spins.
Let us see how kicks change the fraction of merging NSBHs. As shown in the right panel of Figure 2, with the post-SN-explosion binary properties updated, for each initial system we can then calculate the fraction of the post-SN-explosion NSBH systems that can survive and merge within Hubble time. The BH masses of these systems against their BH projected aligned-spins are presented in Figure 3. As the kick is considered, massive BHs with M BH 7.6 M do not experience natal kicks (the last three columns from the right side) as they are heavier than the carbon-oxygen core of their progenitors at the pre-SN state. Some NSBH binaries with relatively low spins, which can not merge in a wide orbit, can instead merge within Hubble time caused by the SN kicks. However, the merger fraction deceases since the systems have lower-mass BHs with lower projected aligned-spins. More specifically, as shown in Figure 3, BHs with M BH 5 M have a wide range of the projected aligned-spin χ z from ∼ 0.2 to ∼ 1.0, while more massive BHs are found to be extremely spinning, i.e., χ z 0.8.
The first-born BHs in NSBH mergers are expected to have near-zero spin distributions (e.g., Qin et al. 2018;Belczynski et al. 2020). ; Kinugawa et al. (2022) suggested that GW200105 and GW200115 are expected to be formed via the classical BH-first-born isolated formation channel. Their posterior distributions of BH projected aligned-spin versus BH mass from Abbott et al. (2021a) are plotted in Figure 3. It is clear that the distributions of the BH projected aligned-spin for GW200105 and GW200115 are quite different from our calculated distributions for NSfirst-born NSBH mergers.

Tidal disruption probability
The tidal disruption probability of NSBH mergers is determined by NS mass, NS EoS, BH mass, and BH projected aligned-spin. By considering two representative EoSs, i.e., AP4 (Akmal & Pandharipande 1997) and DD2 (Typel et al. 2010), in which AP4 is one of the most likely EoSs while DD2 is one of the stiffest EoSs constrained by GW170817 (Abbott et al. 2018(Abbott et al. , 2019, the parameter space where the NS can be tidally disrupted using the formula from Foucart et al. (2018) is shown in Figure 3. A NSBH binary system that has a lowermass NS with a stiffer EoS and a lower-mass BH with a higher projected aligned-spin can more easily make tidal disruption. The Tolman-Oppenheimer-Volkov (TOV) mass for EoSs of AP4 and DD2 is M TOV = 2.22 M and 2.42 M , respectively. On one hand, for almost all of the parameter space of the NS mass, Figure 3 reveals that NSBH mergers formed via NS-first-born formation scenario should be disrupted events if the remnant BHs are heavier than ∼ 5 M . Furthermore, for the BH mass located in the range of M BH 5 M , NSBH mergers whose BH projected aligned-spins are χ z > 0.8 are expected to make tidal disruption as shown in Figure 3. On the other hand, some mergers of NSBH binaries with relatively low BH projected aligned-spins (i.e., χ z 0.8) caused by the impact of SN kicks on low-mass BH com-ponents could still be plunging events, if the mass of the NS companions is M NS 1.5 M (M NS 1.7 M ) with the adoption of an EoS of AP4 (DD2). However, since the fraction of NSBH binaries with low BH projected aligned-spins that can merge within Hubble time after the SN kick is limited as discussed in Section 3.2, plunging events may account for only a small part of NSBH mergers formed via NS-first-born formation scenario. Therefore, regardless of the EoS we choose, it is expected that most NSs can be tidally disrupted by the BHs in NS-first-born NSBH merger systems and they would be plausible multi-messenger sources, which may be discovered in the future.

Multi-messenger signals
In Section 3.3, we showed that most of NS-first-born NSBH mergers can easily make tidal disruption. Disrupted NSBH mergers are believed to drive some bright EM counterparts. After a NSBH merger, there might be a pair of relativistic jets launched along the polar axis by the accreting remnant BH via the Blandford-Znajek effect (Blandford & Znajek 1977;Gompertz et al. 2020a), which can power a sGRB and its broad-band afterglow emission as the jet interacts with the interstellar medium. The radioactive decays of the rapid neutroncapture-formed heavy nuclei can effectively heat the ejected materials in the dynamically ejecta ejecta or the wind outflows launched from the disk around the rem- Both GW200105 and GW200115, which are believed to form via the BH-first-born formation scenario Kinugawa et al. 2022), are most likely plunging events. nant BH, which would power the fast-evolving kilonova emission (e.g., Kawaguchi et al. 2016;Barbieri et al. 2019;Zhu et al. 2020;Darbha et al. 2021). The brightness of the emission may be further enhanced by the energy injection from the central engine (e.g., Ma et al. 2018;Qi et al. 2021).
In the fourth observing run (O4), the number of detected NSBH GWs is believed to increase significantly (e.g., Abbott et al. 2020a;Zhu et al. 2021c). A higher accuracy in both localization and distance measurement by the GW observations is expected, which make the post-GW-trigger follow-up search observations more efficiently. Furthermore, during the same GW period, several more powerful telescopes, e.g., the Space Variable Objects Monitor (Wei et al. 2016) in gamma-rays, the Einstein Probe (Yuan et al. 2016) in X-rays, the Large Synoptic Survey Telescope (LSST Science Collaboration et al. 2009) and the Wide-Field Infrared Transient Explorer (Frostig et al. 2021) in optical-infrared bands, will start to operate and join the GW follow-up observational campaigns. It is possible that multi-messenger signals from NSBH mergers, especially for NS-first-born NSBH mergers, could be discovered in the forthcoming O4.

A Possible Connection Between Long-duration and Short-duration GRBs
One interesting inference from this channel is that there could be a common progenitor system for a small fraction of long-duration GRBs (lGRBs) and sGRBs. As shown in Section 3, the resulting BHs in close binaries are found to be highly spinning due to tidally spin-up. Such a fast-spinning BH can in principle launch a relativistic jet and produce a lGRB through the Blandford-Znajek mechanism (Blandford & Znajek 1977) during the formation of the second-born BH. After the formation of the BH, the NSBH binary would merge long time later after losing orbital energy and angular momentum due to GW emission. Eventually, a sGRB may be produced at the merger.
We show for each NSBH system the merger time, namely the time delay between the lGRB and the following sGRB, on the left side of the symbol in the right panel of Figure 1. The time delay varies from a few tens to hundred billion years. When the SN kicks imparted onto the newly-formed BHs are considered, the average merger time 1 drops by roughly one to two orders of magnitude (see the right panel of Figure 2. In any case, the delay time is too long and it is impossible to directly test observationally the existence of both a lGRB and a sGRB from the same progenitor system. One possible approach to test this scenario is through host galaxy studies. It has been well established that lGRBs and sGRBs have statistically very different host properties (e.g. Fruchter et al. 2006;Fong et al. 2010;Li et al. 2016). Cross comparing the host properties with the consideration of galaxy evolution in the timescale of the delay between the SN explosion and merger may offer some clues. This is beyond the scope of this paper.

CONCLUSIONS AND DISCUSSION
In this work, we first present a detailed binary evolution modeling of the formation of NSBH systems, concentrating on a special channel in which NSs are first formed and BHs are born with high natal spins. With this formation scenario considered, we explore the tidal disruption probability of NSBH mergers, taking into account the impact of accretion feedback of direct corecollapse modeling on the mass and spin of the newlyformed BHs, SN kicks, as well as different NS EoSs. We find that these NSBH mergers produce BHs with extremely high natal spins. With no natal kicks for BHs, we note that NSBH binaries that can merge within Hubble time would have BHs with the projected alignedspins χ z 0.8, and therefore, can definitely have tidal disruption of the NSs at the merger. On the other hand, when natal kicks are taken into account, BHs with M BH 5 M and with projected aligned-spins χ z down to 0.2 (see the right panel of Figure 2) can still merge with a NS within Hubble time, but with a low probability. These systems can still be disrupted events to produce bright EM signals unless the NSs are very massive.
Very recently, Olsen et al. (2022) reported an marginal candidate, GW190920 113516, whose secondary could be a heavy NS. With an effective inspiral spin of χ eff = 0.60 +0.26 −0.07 , this event could be a potential NSBH merger with the NS first born. Román-Garza et al. (2021) claimed that the fraction of the NS-first-born systems with different SN engines is ∼ 10% (see also Kinugawa et al. 2022). The event rate and system parameter distribution for NS-first-born NSBH mergers are subject to further studies in the future. With the upgrade of GW observatories and the update of large survey telescopes, it is foreseen that one may detect high-confidence GW signals with associated EM counterparts from NSBH mergers, especially for NS-first-born NSBH mergers, in the near future GW-led multi-messenger era.

ACKNOWLEDGMENTS
YQ acknowledges the support from the Doctoral research start-up funding of Anhui Normal University and from Key Laboratory for Relativistic Astrophysics in Guangxi University. EWL is supported by the National Natural Science Foundation of China (Grant Nos. 12133003, U1731239) and the Guangxi Science Foundation (Grant No. AD17129006). This work is supported by the National Natural Science Foundation of China (Grant Nos. 12192220, 12192221) and by the Natural Science Foundation of Universities in Anhui Province (Grant No. KJ2021A0106).