Formation of mass gap compact object and black hole binary from Population III stars

We performed population synthesis simulations of Population III binary stars with Maxwellian kick velocity distribution when MGCOs (Mass Gap Compact Objects with mass 2--5$\,M_{\odot}$) are formed. We found that for seven kick velocity dispersion models of $\sigma_{\rm k}=100$--$500$ km/s, the mean mass of black hole (BH)-MGCO binary is $\sim (30 \,M_\odot,\,2.6 \,M_\odot)$. In numerical data of our simulations, we found the existence of BH-MGCO binary with mass $(22.9 \,M_\odot,\,2.5 \,M_\odot)$ which looks like GW190814.

Virgo collaboration during the first half of the third observation period, called O3a. This is a compact binary coalescence and the parameter estimation suggests that the binary consists of a black hole (BH) with mass of 22.2-24.3 M and a compact object with mass of 2.50-2.67 M yielding a chirp mass of M chirp ∼ 6.03-6.15 M 1 . The mass of the smaller (secondary) object lies in 2-5 M , that is, between known neutron stars (NSs) and BHs. Therefore, the secondary compact object will be a NS with the maximum observed mass or a BH with the minimum observed mass, so that we define a mass gap compact object (MGCO) 2 as a compact object having mass 2-5 M . The merger rate density of this type of binaries is estimated as 1-23 Gpc −3 yr −1 [1]. So far, many proposals and discussions on the BH-MGCO binary with MGCO mass of ∼ 2.6 M were appeared after the announcement of GW190814 (see, e.g., Refs. ).
We discussed in our previous study [25] in 2016, the detection rate and the chirp mass distribution of NS-BH binaries with mass of NS below 3 M by the population synthesis simulations of Population III (Pop III) stars. We found that the merger rate density of Pop 1 M chirp = (m 1 m 2 ) 3/5 /(m 1 + m 2 ) 1/5 where m 1 and m 2 denote the mass of the primary and the secondary objects, respectively. 2 MGCO is different from "mass gap" (2.5-5 M ) used in Ref. [1] and "MassGap" (3-5 M ) defined in Ref.
c The Author(s) 2015. Published by Oxford University Press on behalf of the Physical Society of Japan. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
III NS-BH binaries is ∼ 1 Gpc −3 yr −1 although it depends on the natal kick velocity of NSs. We also found that the chirp mass distribution of Pop III NS-BH binaries that merge within the Hubble time has a peak around 6 M .
The results presented in Ref. [25] seems to be consistent with the estimation of Ref. [1] since in Ref. [25] we defined NS if its mass is below 3 M while the mass of secondary of GW190814 is 2.50-2.67 M and the chirp mass of GW190814 is evaluated as 6.03-6.15 M . Thus, it is important to revisit our previous paper using the definition of MGCO and try to answer the question raised as follows. What are "the processes by which the lightest BHs or the most massive NSs form" in Ref. [1] ?
In this letter, we summarize our previous study with additional analyses, and present a scenario to explain the BH-MGCO binary of GW190814.
2. Analysis In Ref. [25], we have calculated NS-BH formations and estimated the number of NS-BH binaries merging within the Hubble time by using the population synthesis simulations of Pop III stars [26]. The key ingredient is to introduce the NS kick velocity of 200-500 km/s that is evaluated from the observation of the proper motion of the pulsar. The effect of kick velocity is important to decrease the merging time of NS-BH binaries. For comparison, we have calculated not only the Pop III NS-BH binaries, but also Pop I and II NS-BH binaries.
To do the above analysis in our population synthesis Monte Carlo simulations, we have considered six metallicity cases from Z = 0 (Pop III) to Z = Z (Pop I) where Z is the solar metallicity, the initial mass, mass ratio, separation and eccentricity distribution functions as binary initial conditions, the Roche lobe overflow, the common envelope phase, the tidal effect, the supernova (SN) effect, and the gravitational radiation as binary interactions, and two kick velocity models with σ k = 265 km/s and σ k = 500 km/s where σ k is the dispersion of a Maxwellian distribution for kick velocity.
Next, we briefly summarize the results. The chirp mass of Pop III NS-BH binaries merging within the Hubble time is heavier than those of Pop I and II ones. The peak values of chirp mass distributions is ∼ 6 M in the case of Pop III while it is ∼ 2 M in the cases of Pop I and II. These peak values almost do not depend on the kick velocity values σ k .
The NS-BH merger rates at the present day depend on the progenitors and kick velocities (see Table 3 in Ref. [25] for the details). The sum of the merger rates of Pop I and II becomes 19.7 Gpc −3 yr −1 and 6.38 Gpc −3 yr −1 for σ k = 265 km/s and 500 km/s, respectively. For the Pop III case, we have found the NS-BH merger rates at the present day as 1.25 Gpc −3 yr −1 and 0.956 Gpc −3 yr −1 for σ k = 265 km/s and 500 km/s, respectively. In the previous study, we have assumed that the mass range of neutron stars are 1.44-3 M . Thus, GW190814 belongs to the previous Pop III NS-BH result.
In this letter, we further focus on compact object binaries which consist of a BH and a MGCO with mass of 2-5 M . We calculate 10 6 Pop III binary evolutions and the merger rates of BH-MGCO binaries, using the same setup of the previous Pop III NS-BH case [25]. For simplicity, we use the simple SN remnant model given in Refs. [26,27].  Here, we assume that the SN remnants which have a mass less than 5M experience SN kick, and we use the Maxwellian distribution for the kick velocity distribution. We calculate not only kick velocity dispersion models of the previous paper of σ k = 265 km/s and 500 km/s, but also σ k = 100, 150, 200, 300, and 400 km/s. Results are shown by Figs. 1 and 2. Figure 1 shows the chirp mass distribution of BH-MGCO binaries which merge within the Hubble time for each model, where N total = 10 6 is the total number of binaries which we have simulated. The chirp mass distribution does not strongly depend on the kick velocity and has a peak at M chirp ∼ 6M . Notice here that the chirp mass of GW190814 is ∼ 6M . Figure 2 shows the merger rate densities for each model. The peak of merger rate density depends on the kick velocity. Almost all BH-MGCO binaries cannot merge within the Hubble time without the natal kick. The natal kick changes the binary orbit and makes the binary be able to merge within the Hubble time. If the typical kick velocity is small, the binary orbits do not change so much that the typical merger time tends to be long. Thus, the smaller kick velocity the model has, the lower redshift the peak of merger rate is.
Since the typical chirp mass of BH-MGCO binary of ∼ 6M is almost the same as that of GW190814, we searched if similar BH-MGCO binary to GW190814 exists in our simulation data. As a result, we found one with mass (22.9M , 2.5M ). Figure 3 shows the evolutionary path of this binary. The binary is born as the zero age main sequence binary consisting of 54.7 M and 14.3 M stars. The primary evolves to a giant and starts a mass transfer. The merger rate densities for 7 kick models with σ k = 100, 150, 200, 265, 300, 400, and 500 km/s. We note that the smaller kick velocity model, the lower redshift the peak of merger rate is.
Next, the primary becomes so large that the secondary plunges into the primary envelope so that the binary enters common envelope phase. In the common envelope phase, the primary envelope is evaporated and the separation shrinks. Without natal kick, the mass ejection at the SN of secondary makes the binary too wide to merge within the Hubble time. However, if the direction of the natal kick is inverse to the orbital velocity, the binary is able to merge within the Hubble time. Figure. 3 shows that the final destiny of this binary is 22.9M BH and 2.5M compact object which is either the lightest BH or the most massive NS. This example might be a possible answer to the question of "What are the processes by which the lightest BHs or the most massive NSs form in Ref. [1] ?" in Introduction.
To estimate the event rate of Pop III BH-MGCO binaries, we use an inspiral-mergerringdown waveform presented in Ref. [28] which is based on Refs. [29,30], The signal-to-noise ratio (SNR) of GW events is calculated in 4 strain-noise fitted curves (see Fig. 4) which are prepared by using Ref. [31] for LIGO O3a-Livingston (O3a-L) (green) and LIGO O5 (magenta), Ref. [32] (see also Ref. [33]) for Einstein Telescope (ET-B) (red), and Ref. [34] for Cosmic Explorer (CE2) (purple). Then, for example, the maximum observable redshift z max by setting the averaged SNR = 8 for the GW190814 binary with m 1 = 23.2 M and m 2 = 2.59 M becomes z max = 0.0814 for LIGO O3a-Livingston, 0.211 for LIGO O5, 5.81 for ET-B, and 49.4 for CE2. Table 1 shows the averaged masses of BH-MGCO binaries in the solar mass M for each kick model, and the event rates in [yr −1 ] based on the maximum observable redshifts z max (shown as values in parenthesis for each detector) of this typical binaries for 4 GW detector configurations. For the O3a-L detector, the maximum event rate of GWs is 0.268 yr −1 for the kick model of σ k = 150 km/s. For the O5, ET-B and CE2 detectors, the maximum event 4/8 Fig. 3 Evolutionary path of a Pop III BH-MGCO binary similar to GW190814. MS, CHeB, HeSB, and nHe mean the main sequence, Core He burning, He shell burning, and naked He stars, respectively. The naked He star is the remnant after the common envelope phase. V orb , and V kick are the orbital velocity just before the SN, and the natal kick velocity, respectively. a (in the solar radius R ) and e denote the orbital separation and eccentricity, respectively.
rate of GWs are found as 3.62 yr −1 for the σ k = 100 km/s, 2070 yr −1 for the σ k = 265 km/s and 3830 yr −1 for the σ k = 300 km/s, respectively.
3. Discussion In a binary system, a heavier BH is formed first. After that, a lighter NS is formed. At this time, a part of the blown outer layer with a mass of ∼ 1 M falls 5/8 back, and the NS becomes a BH with a mass of ∼ 2.6 M . Normally, this binary BH takes much longer time to coalesce than the Hubble time, but considering the kick velocity of the MGCO, the binary coalescence will occur with in the Hubble time. Figure 5 shows mass distributions of MGCOs which merge within the Hubble time for each kick velocity model. As a future prediction, the lighter BHs can have various masses, and some may be NS. What seemed strange in the LIGO-Virgo paper [1] is commonplace in this scenario. The mass distribution of MGCOs (Fig. 5) does not depend on the kick velocity models, but depends on the SN remnant model. Future detections of MGCOs may give a strong constraint on the SN remnant model. As a summary, the lighter compact objects will always become BH or NS which is close to the maximum mass of NS in the scenario of this letter. Table 1 Averaged masses ( m 1 , m 2 ) of BH-MGCO binaries in the solar mass M and event rates in [yr −1 ] for 4 GW detector configurations: LIGO O3a-Livingston (O3a-L), LIGO O5 (O5), Einstein Telescope (ET-B) and Cosmic Explore (CE2) in 7 kick models with σ k = 100, 150, 200, 265, 300, 400, and 500 km/s. Given the masses of binaries, we estimate the maximum observable redshift z max (values in parenhtesis for each detector), and then the event rates for each detector are derived by using the merger rate density calculated by the population synthesis simulations of Pop III stars. Although we can observe z ∼ 40 by using the CE detector, the merger rate density is approximately 0 for z 32.75 in the cases of σ k = 100 and 150 km/s, and for z 33.30 in the cases of σ k = 200, 265, 300, 400 and 500 km/s.