stricter definition of the Pop III PISN mass gap. This feature is also seen in
other BPS studies (e.g., Tanikawa et al., 2021b, 2022b). It is the mass gap
that results in the three peaks in the chirp mass distribution of NSC mergers
(dotted contour in Fig. 8): (1) The low-mass peak around $m_{\rm c}\sim 30\
\rm M_{\odot}$ is made of two BHs below the gap. (2) The intermediate-mass
peak at $m_{\rm c}\sim 100\ \rm M_{\odot}$ involves one BH below the gap and
another above the gap. (3) The massive peak at $m_{\rm c}\sim 300\ \rm
M_{\odot}$ is produced by BHs above the gap. Here the (2) mergers between one
BH below and another above the mass gap produce a peak at $q\sim 0.15$ in the
mass ratio distribution. The mass and mass ratio distributions of BHs in NSC
mergers are similar to those for all initial BBHs. The difference is that for
NSC mergers, the mass distribution is tilted to the massive end, and the peak
at $q\sim 0.15$ in the mass ratio distribution is more obvious. The reason is
that more massive BBHs are more likely to fall into NSCs by DF (see Eqs. 3 and
4). Such distinct features of the NSC-DH channel with respect to IBSE, i.e.,
large fractions of mergers with low mass ratios and massive BHs (inside/above
the mass gap), are also seen in the in-situ dynamical channel for BBH mergers
in massive Pop III clusters (Wang et al., 2022b; Liu et al., 2023; Mestichelli
et al., 2024).

Table 4: Summary statistics of BH masses in Pop III BBH mergers for the 18 models listed in Table 2. The second, third and last columns show the average chirp mass $m_{\rm c}$, primary mass $m_{1}$ and mass ratio $q$ for BBH mergers in galaxy fields/NSCs.  Model | $\bar{m}_{\rm c}^{\rm field/NSC}$ | $\bar{m}_{\rm 1}^{\rm field/NSC}$ | $\bar{q}^{\rm field/NSC}$
---|---|---|---
| $\rm[M_{\odot}]$ | $\rm[M_{\odot}]$ |
LOG1_obs | 23.4/97.2 | 31.3/160 | 0.75/0.64
TOP1_obs | 25.3/169 | 33.8/282 | 0.76/0.58
KRO1_obs | 20.4/38.4 | 28.1/56.7 | 0.71/0.71
LOG5_obs | 25.8/131 | 37/242 | 0.69/0.55
TOP5_obs | 27.9/185 | 41.7/345 | 0.68/0.48
KRO5_obs | 24.4/58.3 | 34.6/94.3 | 0.7/0.68
LOG1_full | 23.4/94.2 | 31.3/154 | 0.75/0.64
TOP1_full | 25.2/165 | 33.5/276 | 0.77/0.58
KRO1_full | 20.4/37.5 | 28/55.4 | 0.71/0.7
LOG5_full | 25.6/129 | 36.5/238 | 0.69/0.55
TOP5_full | 26.2/182 | 38.1/341 | 0.68/0.48
KRO5_full | 24.3/57.3 | 34.4/92.2 | 0.7/0.69
LOG1_low | 23.4/107 | 31.3/174 | 0.75/0.64
TOP1_low | 25.2/180 | 33.6/295 | 0.76/0.58
KRO1_low | 20.4/40.5 | 28.1/59.9 | 0.71/0.71
LOG5_low | 25.6/143 | 36.6/261 | 0.69/0.54
TOP5_low | 26.3/193 | 38.3/357 | 0.68/0.48
KRO5_low | 24.2/63.4 | 34.4/104 | 0.7/0.68

####  5.1.4 Progenitor and host system properties

Figure 11: Initial pericenter separation distribution for the progenitors of
Pop III BBH mergers in the fiducial model LOG1_obs. Shaded region: (ZAMS)
stellar progenitors of field mergers. Dash-dotted contour: stellar progenitors
of NSC mergers. Solid contour: initial BBHs that become field mergers. Dotted
contour: initial BBHs that become NSC mergers. Dashed contour: all initial
BBHs sampled in a-sloth. Hatched shaded region: initial BBHs that once fall
into NSCs but are eventually ejected and merge in galaxy fields, which make up
a tiny fraction ($\sim 0.04\%$) of field mergers.

The difference between the IBSE and NSC-DH channels in the mass distribution
of BHs can be explained by the facts that (i) only close BBHs can merge in
isolation within a Hubble time while the NSC-DH channel can also drive
initially wide BBHs to merge, and (ii) the masses and initial separations of
BHs are correlated as a result of binary stellar evolution. To illustrate the
first point (i), we plot the initial pericenter separation $r_{\rm p}$
distributions of the progenitors of Pop III BBH mergers in Fig. 11, for both
initial stellar binaries and BBHs. It can be seen that field mergers mostly
originate from BBHs with initial pericenter separations $r_{\rm p}\equiv
a(1-e)\lesssim 0.16$ AU, which are mainly produced by close stellar binaries
with $r_{\rm p}\lesssim 10$ AU initially. However, NSC mergers mostly come
from initial BBHs with $r_{\rm p}\sim 0.1-10^{3}$ AU, and the shape of the
$r_{\rm p}$ distribution for such NSC merger progenitors is similar to that of
all BBHs for $r_{\rm p}\gtrsim 0.1$ AU. The corresponding progenitor stellar
binaries also have a broad $r_{\rm p}$ distribution extending to $r_{\rm
p}\sim 10^{3}$ AU. Interestingly, there is a tiny fraction (0.04%) of field
mergers from initially wide BBHs. These binaries have once fallen into NSCs
and been processed by DH, but are ejected later and eventually merge in galaxy
fields. In general, the progenitor BBHs of field mergers and NSC mergers are
approximately separated by a critical initial pericenter separation $r_{\rm
p,crit}\sim 0.1$ AU, which is a common feature in all cases considered in this
work (Table 2). Below this critical separation, Pop III BBHs typically merge
within $~{}\rm 1$ Gyr, so there is usually not enough time for them to sink
into NSCs by DF. Above the critical separation, due to the strong dependence
of merger timescale on initial separation for isolated evolution (Eq. 8), most
BBHs cannot merge within a Hubble time in isolation.

Figure 12: Distribution of Pop III BBHs at formation in the $\log m_{1}-\log
a$ space for the LOG1 simulation by sevn from Costa et al. (2023) with the
close IBS model. To estimate which BBHs can merge by GW emission in galaxy
fields, we plot the primary mass as a function of initial separation given the
merger timescales $t_{\rm GW}=14000$ (dotted), 100 (long-dashed), and 1 Myr
(dash-dot-dotted) for equal-mass BBHs on initially circular orbits. Similarly,
for the NSC-DH channel, we derive the minimum primary mass required for a BH
binary to be hard in NSCs with $\sigma_{\star}=30$ (solid), 100 (dashed), and
$300\ \rm km\ s^{-1}$ (dash-dotted) given the criterion $a<a_{\rm
HDB}=Gm_{1}^{2}/\left[m_{\star,\rm SC}\sigma_{\star}^{2}\right]$.

For the second point (ii), Fig. 12 shows the distribution of all Pop III BBHs
at formation in the $\log m_{1}-\log a$ space from the LOG1 sevn catalogue
used in our fiducial model, where we also demonstrate the criteria for equal-
mass BBHs on initially circular orbits to (a) merge in galaxy fields by GW
emission for a given merger timescale $t_{\rm GW}$, and (b) survive binary-
single encounters and merge by DH in NSCs with a given velocity dispersion
$\sigma_{\star}$. On the left side of the dotted line for $t_{\rm GW}=14$ Gyr,
we have $m_{1}\lesssim 50\ \rm M_{\odot}$ for most BBHs that can merge within
a Hubble time in isolation. Such BBHs originate from close ($r_{\rm p}\lesssim
10$ AU) binaries of stars with ZAMS masses below $242\ \rm M_{\odot}$, which
lose their envelopes through mass transfer and/or CE phases, so the resulting
naked cores can only produce BHs below $\sim 50\ \rm M_{\odot}$ (Iorio et al.,
2023). Close binaries of more massive stars tend to merge during unstable mass
transfer when the donor leaves the MS as a red super-giant while the accretor
still on the MS also fills the Roche lobe due to significant expansion323232If
massive Pop III stars remain compact during MS (by, e.g., suppressed
convective overshooting or chemically homogeneous evolution), some of these
close binaries can undergo stable mass transfer without stellar mergers and
produce massive BBHs (with BHs above $130\ \rm M_{\odot}$) merging within a
Hubble time (Tanikawa et al., 2021a, b, 2022b; Hijikawa et al., 2021;
Santoliquido et al., 2023; Mestichelli et al., 2024). This indicates that the
fates of massive binary stars and the efficiency of forming massive BBH
mergers via IBSE are sensitive to the modelling of single star evolution.
(Costa et al., 2023). Initially wider ($r_{\rm p}\gtrsim 10$ AU) binaries
undergo less and even negligible mass loss during binary stellar evolution and
produce more massive BHs up to $86\ (550)\ \rm M_{\odot}$ from stars with ZAMS
masses below (above) $242\rm M_{\odot}$.

However, the resulting BBHs are mostly too wide ($a\gtrsim 0.3$ AU) to merge
within a Hubble time in isolation. Nevertheless, the majority of such massive
BBHs with wide orbits are still hard binaries in typical host NSCs with
$\sigma_{\star}=30\ \rm km\ s^{-1}$ (solid line), so a significant fraction
($\sim 45-64$%) of them can merge within a Hubble time by DH. In particular,
BBHs with at least one BH inside or above the PISN mass gap $\sim 86-242\ \rm
M_{\odot}$ originate from massive stellar binaries with $m_{\star,1}\gtrsim
242\ \rm M_{\odot}$ and form two groups in the $\log m_{1}-\log a$ space. The
first group with $m_{1}\sim 130-320\ \rm M_{\odot}$ (including all BHs inside
the PISN mass gap) and $a\sim 0.1-30$ AU is produced by stellar binaries with
initial pericenter separations of $r_{\rm p}\sim 5-30$ AU, in which the
primary star loses the H envelope by CE evolution and stable mass transfer
during the red super-giant phase before collapsing into a BH directly. The
second group with $m_{1}\sim 242-550\ \rm M_{\odot}$ (above the PISN mass gap)
and $a\gtrsim 30$ AU comes from initially wider ($r_{\rm p}\gtrsim 30$ AU)
stellar binaries in which mass transfer/loss via Roche lobe overflow never
happens, so the primary star collapses entirely into a BH (as mass loss by
stellar winds is negligible in the Pop III tracks adopted by sevn). Almost all
BBHs with BHs inside or above the PISN mass gap are massive enough to merge by
DH even in massive NSCs with $\sigma_{\star}=300\ \rm km\ s^{-1}$ (dash-dotted
line).

Beyond distinct progenitors, the two evolution channels also produce Pop III
BBH mergers in very different environments. For conciseness, here we briefly
summarize the host halo/galaxy/NSC properties of Pop III BBH mergers in our
fiducial model and comment on how they vary with model parameters. Most
($\gtrsim 90\%$) field mergers occur in small haloes with $M_{\rm h}\lesssim
5\times 10^{9}\ \rm M_{\odot}$ at $z\gtrsim 0.8$, which host low-mass, metal-
poor galaxies with $M_{\star}\lesssim 1.3\times 10^{6}\ \rm M_{\odot}$ and
$Z\lesssim 0.22\ \rm Z_{\odot}$. The typical (median) host system has $M_{\rm
h}\sim 9.4\times 10^{7}\ \rm M_{\odot}$, $M_{\star}\sim 3.5\times 10^{3}\ \rm
M_{\odot}$, $Z\sim 0.005\ \rm Z_{\odot}$, and $z\sim 6.8$. In fact, $\sim 10$%
of Pop III BBHs merge before the formation of the second generation of stars
(i.e, $M_{\star}=0$). Therefore, direct observations of the host galaxies of
Pop III BBH mergers from IBSE are very challenging. On the contrary, the
majority ($\sim 80\%$) of NSC mergers happen in more massive structures with
$M_{\rm h}\sim 6.7\times 10^{9}-1.1\times 10^{11}\ \rm M_{\odot}$,
$M_{\star}\sim 1.3\times 10^{6}-8\times 10^{8}\ \rm M_{\odot}$, $Z\sim
0.017-0.48\ \rm Z_{\odot}$, and $M_{\rm SC}\sim 9\times 10^{4}-3.4\times
10^{9}\ \rm M_{\odot}$ at $z\sim 0.1-3$. The typical host system has $M_{\rm
h}\sim 2\times 10^{10}\ \rm M_{\odot}$, $M_{\star}\sim 1.5\times 10^{7}\ \rm
M_{\odot}$, $Z\sim 0.075\ \rm Z_{\odot}$, $M_{\rm SC}\sim 4.5\times 10^{5}\
\rm M_{\odot}$, and $z\sim 0.7$. Nevertheless, the host galaxies of our Pop
III BBH mergers in NSCs are generally less massive than the galaxies hosting
NSCs in local observations (Neumayer et al., 2020, see their fig. 12),
especially for Late-Type NSC-host galaxies that mainly have $M_{\star}\gtrsim
10^{8}\ \rm M_{\odot}$. This indicates that the NSC-DH channel favours low-
mass, metal-poor, compact (Early-Type) galaxies in small haloes where the
inspiral of Pop III BBHs into NSCs by DF is efficient. We also find that among
the host NSCs of Pop III BBHs, each NSC typically only swallows a few Pop III
BBHs throughout its lifetime, so the total mass (as well as binding energy)
carried by the Pop III BBHs is minor (up to a few percent) compared with that
of the NSC itself. This justifies our ignorance of the heating effect by DH of
Pop III BBHs on NSC evolution.

In general, the host system properties of field mergers are insensitive to
model parameters, although more massive systems at lower redshifts are
slightly favoured by initially wider binaries with more bottom-heavy IMFs.
However, increasing the abundances of Pop III BBHs and NSCs will shift NSC
mergers to less massive systems with lower metallicities at higher redshifts.
For instance, in the model LOG5_full with the highest abundances of Pop III
BBHs and NSCs, the typical host system has $M_{\rm h}\sim 10^{10}\ \rm
M_{\odot}$, $M_{\star}\sim 3.6\times 10^{6}\ \rm M_{\odot}$, $Z\sim 0.05\ \rm
Z_{\odot}$, $M_{\rm SC}\sim 2.8\times 10^{5}\ \rm M_{\odot}$, and $z\sim 1.1$.
In the opposite case of KRO1_low, we have $M_{\rm h}\sim 3.7\times 10^{10}\
\rm M_{\odot}$, $M_{\star}\sim 4.9\times 10^{7}\ \rm M_{\odot}$, $Z\sim 0.17\
\rm Z_{\odot}$, $M_{\rm SC}\sim 8\times 10^{5}\ \rm M_{\odot}$, and $z\sim
0.3$. These trends can be explained by the distinct merger histories in
different models, which are discussed below.

###  5.2 Exploration of parameter space

The general features of Pop III BBH mergers from the IBSE and NSC-DH channels
described above hold in all the 18 models considered in this paper. However,
the detailed properties of BBH mergers and the relative importance of the two
channels do vary greatly with the underlying assumptions on (1) Pop III IMF,
(2) IBS, and (3) occupation fraction of high-$z$ NSCs. In this subsection, we
first discuss the general trends in the BBH formation and merger efficiencies
(Sec. 5.2.1). Then we illustrate the effects of each aspect on the merger
history, SGWB, and mass distribution (Sec. 5.2.2-5.2.4).

####  5.2.1 BBH formation and merger efficiencies

The BBH formation and merger efficiencies of the 18 simulations are summarized
in Table 2 and visualized in Fig. 13. Under the wide IBS model in which close
binary interactions are rare, $\epsilon_{\rm BBH}$ is not very sensitive to
the IMF, and the fiducial log-flat IMF shows the highest $\epsilon_{\rm BBH}$.
This is produced by two competing effects: When individual stars become more
massive (with increasing $\alpha$), the number of stars/binaries is simply
smaller for a fixed total stellar mass, but meanwhile, the fraction of stars
massive enough to form BHs is larger. However, under the close IBS model where
binary interactions are important, $\epsilon_{\rm BBH}$ becomes higher when
the IMF is more bottom-heavy (with higher $\alpha$). The reason is that more
massive stars are more likely to merge during close binary interactions due to
their larger radii, so a larger fraction of stellar binaries that would form
BBHs without interactions will be lost in mergers for a larger $\alpha$.
Similarly, when there are more wide binaries in the IBS model, mergers of
massive stars are suppressed, resulting in higher $\epsilon_{\rm BBH}$ and
more massive BBHs, as shown in Fig. 14.

Figure 13: Pop III BBH formation and merger efficiencies from the 18 models
listed in Table 2. The BBH formation efficiency (crosses) is independent of
the NSC parameters, so we only show 6 data points for the 6 input sevn
catalogues. The efficiency of field mergers (diamonds) is insensitive to the
NSC parameters, and we only show the results for the fiducial NSC model (obs).
For NSC mergers, we show the results for the obs, full, and low NSC models
(Sec. 3.1) with triangles, circles, and squares, respectively. The left
(right) section of the plot shows the results for the close (wide) IBS model
(Sec. 2.1). Within each section, the IMF becomes more bottom-heavy (with
higher $\alpha$) from left to right. We have $\epsilon_{\rm GW}^{\rm
field}\simeq\epsilon_{\rm GW}^{\rm NSC}$ in the model TOP1_full, causing the
two data points to overlap.  Figure 14: Same as Fig. 12 but for the LOG5
simulation by sevn from Costa et al. (2023) with the wide IBS model.

For Pop III BBH mergers in galaxy fields, the merger efficiency $\epsilon_{\rm
GW}^{\rm field}$ is sensitive to the NSC parameters since only a small ($\sim
3-12\%$) fraction of Pop III BBHs fall into NSCs. Therefore, we focus on the
results of $\epsilon_{\rm GW}^{\rm field}$ for the fiducial NSC model (obs).
We find that $\epsilon_{\rm GW}^{\rm field}$ always increases with $\alpha$,
and the trend is stronger with the close IBS model. The evolution of
$\epsilon_{\rm GW}^{\rm field}$ with $\alpha$ is also stronger compared with
the case of $\epsilon_{\rm BBH}$, as the fraction $f_{\rm GW}^{\rm field}$ of
BBHs in galaxy fields that merge at $z>0$ increases with $\alpha$. These
trends can be explained by the fact that field mergers mostly originate from
interacting close ($r_{\rm p}\lesssim 10$ AU) binaries of relatively small
stars ($\lesssim 240\ \rm M_{\odot}$) while close binaries of massive stars
will merge, as discussed in Sec. 5.1.4 (see Fig. 12). For the same reason,
with a fixed IMF, $\epsilon_{\rm GW}^{\rm field}$ is significantly reduced (by
a factor of $\sim 6-14$) when we switch from the close IBS model to that
dominated by wide binaries. The reduction is stronger when the IMF is more
bottom-heavy because more massive stars are more likely to interact given a
fixed separation.

For NSC mergers, the variation of merger efficiency $\epsilon_{\rm GW}^{\rm
NSC}$ with IMF and IBSE is very similar to that of $\epsilon_{\rm BBH}$.
Indeed, if the probability for a BBH to merge in a NSC at $z>0$ is constant,
$\epsilon_{\rm GW}^{\rm NSC}$ should be proportional to $\epsilon_{\rm BBH}$.
In fact, the probability is slightly higher for more massive BBHs that sink
into NSCs more efficiently by DF and require less hardening to merge rapidly
via GW emission. As a result, on top of the trend driven by $\epsilon_{\rm
BBH}$, $\epsilon_{\rm GW}^{\rm NSC}$ is further enhanced by a more top-heavy
IMF or IBS with more wide binaries, which tend to produce more massive BBHs.
This point is reflected in the weak evolution of the fraction $f_{\rm GW}^{\rm
NSC}$ of BBHs in NSCs that merge at $z>0$ and the fraction $f_{\rm infall}$ of
BBHs that fall into NSCs (see the last two columns of Table 2) with IMF and
IBS. As expected, $f_{\rm infall}$ and thus $\epsilon_{\rm GW}^{\rm NSC}$
increase when the occupation fraction of NSCs is higher. Combing these
distinct features of field and NSC mergers, we find that the fraction $f_{\rm
NSC}$ of mergers in NSCs across cosmic history is higher when the IMF is more
top-heavy, the initial condition includes more wide stellar binaries, and
high-$z$ NSCs are more abundant. The most important factor here is IBS: For
the close IBS model, field mergers are more abundant than NSC mergers in most
cases with $f_{\rm NSC}\sim 8-51$% except for the TOP1_full model in which the
two populations of mergers have almost identical numbers. However, for the
wide IBS model, NSC mergers always dominate with $f_{\rm NSC}\sim 65-95$%.
Besides, the BBH merger efficiency of the NSC-DH channel is less sensitive to
the IMF and IBS compared with that of the IBSE channel.

####  5.2.2 Effects of initial binary statistics

Figure 15: Same as Fig. 4 but comparing the MRDs for two IBS models dominated
by close (blue) and wide (orange) binaries assuming the log-flat IMF and
fiducial NSC occupation fraction.

To illustrate the effects of IBS, as an example, we compare the results from
LOG1_obs (i.e., the fiducial model) and LOG5_obs with the close and wide IBS
models (see Sec. 2.1), respectively, where the IMF and NSC models are fixed to
the fiducial choices. Consistent with the trends seen in merger efficiencies
$\epsilon_{\rm GW}$ (Fig. 13), the MRDs of field and NSC mergers are reduced
and increased when there are more wide binaries in the initial condition
(LOG5_obs), respectively, as shown in Fig. 15. As a result, NSC mergers
dominate the MRD at $z\lesssim 6$ with the wide IBS model. Similarly, the
reduction (enhancement) of the MRD of field (NSC) mergers is weaker (stronger)
when the IMF is more bottom-heavy. The MRD redshift evolution for NSC mergers
is not significantly affected by IBS, which holds for other choices of IMFs
and NSC models. This indicates that the NSC-DH channel is mainly sensitive to
the abundances of Pop III BBHs rather than their initial properties, which is
also reflected in the correlation between $\epsilon_{\rm GW}^{\rm NSC}$ and
$\epsilon_{\rm BBH}$. For field mergers, the MRD evolution is insensitive to
IBS at $z\gtrsim 5$, which always shows a peak at $z\sim 15$ closely following
the peak of the Pop III SFRD (Fig. 3), since this regime is dominated by
mergers with short delay times. However, at lower redshifts when Pop III star
formation has terminated, and the MRD is dominated by mergers of long delay
times, the decrease of MRD with decreasing redshift is slower for the wide IBS
model compared with the case of the close IBS model. The reason is that the
wide IBS model produces a larger fraction of BBHs with long merger timescales
($t_{\rm GW}\gtrsim 300$ Myr) (see fig. 11 in Costa et al., 2023). This effect
is stronger when the IMF is more top-heavy. In fact, the close and wide IBS
models prefer different binary evolution pathways to BBH mergers in isolation,
which produce different delay time distributions (see fig. 12 and 17 in Costa
et al., 2023). The former favours stable mass transfer during early
evolutionary stages, while the latter is dominated by CE evolution of red
giants with large radii ($\sim 1-10$ AU). The reader is referred to Costa et
al. (2023) and Santoliquido et al. (2023) for in-depth discussions on the
roles played by different binary evolution pathways in shaping the properties
of BBH mergers via IBSE.

Figure 16: Same as Fig. 7 but comparing the energy spectra of SGWB for the two
IBS models dominated by close (blue) and wide (orange) binaries assuming the
log-flat IMF and fiducial NSC occupation fraction.

Fig. 16 shows the contributions of Pop III BBH mergers to the SGWB for the two
IBS models. Compared with LOG1_obs with the close IBS model, the contribution
from field (NSC) mergers is lower (higher) for LOG5_obs with the wide IBS
model, where NSC mergers completely dominate the Pop III SGWB at $\nu\lesssim
200$ Hz. This dominance of NSC mergers is a common feature of the wide IBS
model regardless of the chosen IMF and NSC parameters. This can be understood
by the fact that the SGWB (observed at $z=0$) captures the summation of the GW
energies from all mergers across cosmic history and more massive mergers at
lower $z$ contribute higher energies to the SGWB (Sec. 5.1.2). As mentioned in
the last subsection and shown in Fig. 15, the merger efficiency and local MRD
of NSC mergers are always larger than those of field mergers under the wide
IBS model. Moreover, NSC mergers are more massive than field mergers, and thus
release more energies in GWs (Sec. 5.1.3). Therefore, in LOG5_obs, the
contribution from NSC mergers is higher than that from field mergers by about
100 times in the Pop III SGWB, which is as strong as the SGWB from PBH mergers
predicted by Bavera et al. (2022) at $\nu\lesssim 10$ Hz.

Figure 17: Same as Fig. 8, but comparing the chirp mass distributions of Pop
III BBH mergers in galaxy fields (middle) and NSCs (bottom) for the two IBS
models dominated by close (blue shaded region) and wide (orange contour)
binaries assuming the log-flat IMF and fiducial NSC occupation fraction. Here,
we only show the results for Pop I/II BBH mergers in GCs (dashed curve) and
PBH mergers (dash-dot-dotted curve) from the best-fit model in Franciolini et
al. (2022b) adopted by Bavera et al. (2022, see their fig. 1), and the
intrinsic chirp mass distribution (thick solid curve) inferred from GWTC-3
events from (Abbott et al., 2023a, see their fig. 2).

Finally, we compare the chirp mass distributions between the two models in
Fig. 17 (see also Table 4). We find that both field and NSC mergers become
more massive with the wide IBS model, and the effect is stronger for NSC
mergers. The average chirp mass increases by 10% and 35% for field and NSC
mergers given the wide IBS model compared with the case of the close IBS
model. On the one hand, mass loss and stellar mergers during close binary
interactions are suppressed in the wide IBS model, which enhances the
formation of massive BBHs (see Fig. 12 and 14) and thus greatly increases the
masses of NSC mergers (whose mass distribution closely correlates with the
mass distribution of all BBHs at formation, see Fig. 9). Besides, the fraction
of NSC mergers with small mass ratios $q\lesssim 0.4$ is higher in the wide
IBS model. On the other hand, since close binary interactions are required to
produce BBHs that can merge within a Hubble time in isolation, field mergers
mostly involve low-mass BHs ($\lesssim 50\ \rm M_{\odot}$) regardless of the
IBS model. Therefore, changing from the close IBS model to the wide IBS model
only shifts the primary mass distribution of field mergers to the massive end
by a few solar masses (but still limited by $m_{1}\lesssim 50\ \rm
M_{\odot}$), leading to slightly larger chirp masses.

####  5.2.3 Effects of IMF

Figure 18: Same as Fig. 4, but comparing the MRDs for the three IMFs with
power-law slopes of $\alpha=1$ (blue) $0.17$ (orange) and 2.3 (green),
assuming the close IBS and fiducial NSC models. Figure 19: Same as Fig. 3 but
comparing the SFRDs of Pop III (dashed) and Pop I/II (solid) stars from
LOG1_obs (blue), TOP_obs (orange), and KRO1_obs (green) with Pop III IMF
slopes of $\alpha=1$, $0.17$, and 2.3 under the close IBS model and the
fiducial NSC model.

Next, we look into the effects of IMF variations using the results from
LOG1_obs, TOP_obs, and KRO1_obs with IMF slopes of $\alpha=1$, $0.17$, and 2.3
(see Sec. 2.1), respectively, where we adopt the close IBS model and fiducial
NSC occupation fraction. As shown in Fig. 18, when the IMF is more bottom-
heavy (with a larger $\alpha$), the MRD of field (NSC) mergers is
significantly (mildly) increased. For both field and NSC mergers, the MRD
redshift evolution is insensitive to the IMF, while the normalization of the
MRD curve is approximately proportional to the merger efficiency (Fig. 13),
whose dependence on the IMF is explained in Sec. 5.2.1. These trends reflect
that the IMF has minor effects on the delay time distributions for both field
and NSC mergers, and that the Pop III formation history does not vary greatly
with $\alpha$, as shown in Fig. 19. In fact, the total stellar mass density of
Pop III stars ever formed in our simulation box is $4.6$, $5.6$ and $5.7\times
10^{4}\ \rm M_{\odot}\ Mpc^{-3}$ for $\alpha=1$, 0.17, and 2.3, respectively.
This small variation of the Pop III star formation history with $\alpha$
justifies our simple modelling of Pop III stellar feedback (Sec. 2.1).

The SGWB contributed by Pop III BBH mergers in galaxy fields also increases
with $\alpha$, as shown in Fig. 20. The shape of the energy spectrum at
$\nu\sim 15-200$ Hz varies slightly with the IMF. Compared with the flat
spectrum for $\alpha=1$, $\Omega_{\rm GW}$ slowly increases (decreases) with
$\nu$ for $\alpha=2.3$ (0.17). The reason is that BBH mergers become slightly
more massive when the IMF is more top-heavy (see Fig. 21 and Table 4), which
enhances GW emission at lower frequencies. However, for NSC mergers, the SGWB
increases with $\alpha$ at $\nu\gtrsim 30$ Hz, but decreases with $\alpha$ at
$\nu\lesssim 30$ Hz. Here, the high-frequency regime is dominated by low-mass
mergers with $m_{\rm c}\lesssim 60\ \rm M_{\odot}$ whose fraction increases
with $\alpha$ as shown in Fig. 21. Since the merger efficiency and local MRD
of NSC mergers also increase with $\alpha$ (Fig. 13 and 18), it is reasonable
that the SGWB from such low-mass mergers is stronger. On the other hand, the
low-frequency part of the SGWB is produced by massive mergers with $m_{\rm
c}\gtrsim 60\ \rm M_{\odot}$ whose fraction is reduced when the IMF is more
bottom-heavy (see Fig. 21 and Table 4). This reduction overcomes the increase
of merger number with $\alpha$ and drives the decrease of SGWB with $\alpha$.

Figure 20: Same as Fig. 7, but comparing the energy spectra of SGWB for the
three IMFs with power-law slopes of $\alpha=1$ (blue) $0.17$ (orange) and 2.3
(green), under the close IBS model and the fiducial NSC model. Figure 21: Same
as Fig. 8 and Fig. 17, but comparing the chirp mass distributions of Pop III
BBH mergers in galaxy fields (middle) and NSCs (bottom) for the three IMFs
with power-law slopes of $\alpha=1$ (blue shaded region) $0.17$ (orange
contour) and 2.3 (green contour), again under the close IBS and fiducial NSC
models.

####  5.2.4 Effects of NSC occupation fraction

Finally, we consider the effects of the NSC occupation fraction in Fig. 22-23
with the wide IBS model and log-flat IMF. We find that the properties of field
mergers are hardly affected by the NSC parameters, which is reasonable as only
a small ($\lesssim 12$%) fraction of Pop III BBHs ever fall into NSCs.
Therefore, we focus on NSC mergers in this subsection. The chirp mass
distributions of NSC mergers are very similar in the three NSC models and not
shown here for conciseness (see Table 4 for the average chirp masses). As
expected, the MRD of NSC mergers is enhanced when more galaxies host NSCs.
Unlike the fiducial NSC model where the MRD of NSC mergers flattens at
$z\lesssim 4$, the MRD of NSC mergers shows a peak at $z\sim 4$ in the
optimistic NSC model (where every galaxy with $M_{\star}>10^{6}\ \rm
M_{\odot}$ hosts a NSC). The MRD keeps increasing rapidly towards lower
redshifts in the conservative NSC model, producing a local MRD that is
slightly higher than that in the fiducial NSC model. The difference between
the fiducial and conservative NSC models in the SGWB is very small, which can
be understood with their similar merger efficiencies and local MRDs, although
the peak value of the SGWB in the conservative NSC model is slightly ($\sim
16\%$) higher than that in the fiducial NSC model due to the slightly higher
local MRD. The SGWB is stronger in the optimistic NSC model as expected from
its higher merger efficiency and local MRD. In this case (LOG5_full), the Pop
III SGWB becomes comparable ($\sim 84$%) to the SGWB from Pop I/II and PBH
mergers predicted by Bavera et al. (2021) at $\nu\lesssim 10$ Hz.

Figure 22: Same as Fig. 4 but comparing the MRDs for the fiducial (blue),
optimistic (orange), and conservative (green) models of NSC occupation
fraction under the wide IBS model and log-flat IMF. Figure 23: Same as Fig. 7
but comparing the energy spectra of SGWB for the fiducial (blue), optimistic
(orange), and conservative (green) models of NSC occupation fraction under the
wide IBS model and log-flat IMF.

###  5.3 Observational perspective

We estimate the detection rates $\dot{\mathcal{N}}_{\rm ET,SNR>8}$ of Pop III
BBH mergers by ET for all the 18 models (Table 2) given the detection
probability of each source with $\rm SNR>8$ derived from the python package
gwtoolbox (Yi et al., 2022a; Yi et al., 2022b), using the formalism of Eq. 24.
For simplicity, we do not distinguish high-$z$ and low-$z$ sources but sum up
all events detectable by ET. We also do not consider eccentricities in this
calculation as in the case of SGWB (Sec. 5.1.2). The results are summarized in
Fig. 24, where we only show the detection rates of field mergers with the
fiducial NSC model since the results for the other two NSC models are almost
identical. As discussed in Sec. 5.1.1, most ($\gtrsim 90$%) BBH mergers with
$z<20$ will be detectable by ET thanks to its high sensitivity across a broad
frequency range, so the all-sky merger rate $\dot{\mathcal{N}}(z<20)$ within
$z<20$ is a good approximation of the ET detection rate (for $\rm SNR>8$).
Therefore, we derive the fraction of $\dot{\mathcal{N}}(z<20)$ contributed by
Pop III BBHs based on the reference rates for Pop I/II and PBH mergers from
Franciolini et al. (2022b) to estimate the chance of finding Pop III BBHs by
ET. The results are summarized in Table 3.

In general, the ET detection rate of all Pop III BBH mergers is
$\dot{\mathcal{N}}_{\rm ET,SNR>8}\sim 50-1370\ \rm yr^{-1}$, and we have
$\dot{\mathcal{N}}_{\rm ET,SNR>8}\sim 30-1230\ \rm yr^{-1}$ and
$\dot{\mathcal{N}}_{\rm ET,SNR>8}\sim 6-230\ \rm yr^{-1}$ for field and NSC
mergers, respectively. The highest rate is achieved by KRO1_full, in which
case Pop III BBHs contribute $\sim 6.5$% of the all-sky merger rate within
$z<20$. The dependence of detection rates on model parameters follows similar
trends as those seen in merger efficiencies (Fig. 13), which are explained in
Sec. 5.2.1. However, here the effects of NSC parameters are stronger, and
field mergers become more important. The reason is that the detection rate is
more sensitive to the high-$z$ MRD than the merger efficiency. IBS still plays
the most important role in determining the relative importance of the NSC-DH
and IBSE channels. Given the close IBS model, field mergers always dominate
the total detection rate, as the contribution of NSC mergers remains low
($\sim 0.7-22$%). With the wide IBS model, the NSC mergers account for $\sim
13-88$% of the total detection rate. Their contributions are only higher than
those of field mergers in 5 of the 9 models with relatively high NSC
occupation fractions and large $\alpha$, although they always dominate the
total number of mergers across cosmic history with $f_{\rm NSC}\sim 65-95\%$.

Figure 24: Detection rates of Pop III BBH mergers by ET with $\rm SNR>8$ for
the 18 models listed in Table 2. The detection rates of field mergers
(diamonds) are insensitive to the NSC parameters, and we only show the results
for the fiducial NSC model (obs). For NSC mergers, we show the detection rates
for the obs, full, and low NSC models (Sec. 3.1) with triangles, circles, and
squares, respectively. The left (right) section of the plot shows the results
for the close (wide) IBS model (Sec. 2.1). Within each section, the IMF
becomes more bottom-heavy (with higher $\alpha$) from left to right.  Figure
25: Same as Fig. 24 but for the detection rates of Pop III BBH mergers
including at least one IMBH ($m_{1}>100\ \rm M_{\odot}$) in NSCs. The
detection rates of such mergers in galaxy fields remain below $0.1\ \rm
yr^{-1}$ and are therefore not shown here.

As discussed in Sec. 5.1.3, an important feature of the NSC-DH channel is the
ability to produce Pop III BBH mergers with massive ($\gtrsim 50\ \rm
M_{\odot}$) BHs, especially those above $130\ \rm M_{\odot}$. In light of
this, we further look into the ET detection rates $\dot{\mathcal{N}}_{\rm
ET,IMBH,SNR>8}$ of Pop III mergers with at least one IMBH333333Because of
PISNe, all IMBHs produced by Pop III stars in our models are above $130\ \rm
M_{\odot}$ (see Fig. 9), which is a common feature of the adopted BPS
catalogues from sevn (Costa et al., 2023), and also found in other BPS studies
(Tanikawa et al., 2021a, 2022b; Hijikawa et al., 2021). ($m_{1}>100\ \rm
M_{\odot}$). We find that such mergers are extremely rare in galaxy fields,
with ET detection rates below $0.1\ \rm yr^{-1}$ in all the 18 models
considered here. However, we have $\dot{\mathcal{N}}_{\rm ET,IMBH,SNR>8}\sim
0.5-200\ \rm yr^{-1}$ for NSC mergers as shown in Fig. 25. The completeness of
detection of such mergers by ET is also very high ($\gtrsim 95$%). Here, the
detection rate is sensitive to all model parameters on Pop III IMF, IBS, and
occupation fraction of high-$z$ NSCs, implying that BBH mergers with BHs of
$\sim 100-1000\ \rm M_{\odot}$ are valuable for constraining the properties of
Pop III binaries and high-$z$ SCs with GW observations.

Finally, we estimate the detection rate of Pop III BBH mergers by the LVK
network during the O4 run with $\rm SNR>8$ as $\dot{\mathcal{N}}_{\rm
LVK,SNR>8}\sim 0.9-9\ \rm yr^{-1}$. This small detection rate is consistent
with the small local MRD $\dot{n}(z=0)\sim 0.034-0.17\ \rm yr^{-1}\ Gpc^{-3}$
that is much lower than the observed local MRD $\dot{n}_{\rm
obs}(z=0)=19.3_{-9}^{+15.1}\ \rm yr^{-1}\ Gpc^{-3}$ (Abbott et al., 2021).
This indicates that Pop III mergers are overwhelmed by other populations of
BBH mergers in the low-$z$ Universe (see also Table 3). Although Pop III stars
can still be important for the most massive mergers, such events are rare in
LVK observations. For instance, the LVK O4 detection rate of Pop III mergers
with IMBHs (in NSCs) is $\dot{\mathcal{N}}_{\rm LVK,IMBH,SNR>8}\sim 0.04-1.5\
\rm yr^{-1}$, and the expected number of events detected in 18
months343434https://observing.docs.ligo.org/plan/ of observations is only
$\sim 0.07-2.2$. Considering the typical horizon redshift $z_{\rm horizon}\sim
1$ of the LVK network during O3 for BBH mergers with total masses $\sim
100-300\ \rm M_{\odot}$ (see fig. 8 of De Luca et al., 2021), we derive the
MRD of Pop III mergers with IMBHs for $z<1$ as $\dot{n}_{\rm IMBH}(z<1)\sim
0.003-0.15\ \rm yr^{-1}\ Gpc^{-3}$, which is lower than the upper limits
derived from O3 data under most conditions (Abbott et al., 2022, see their
fig. 2). For Pop III BBH mergers with both BHs in the standard PISN mass gap
$50-130\ \rm M_{\odot}$ like GW190521 (detected at
$z=0.82_{-0.34}^{+0.28}\lesssim 1$, Abbott et al., 2020b), we predict the MRD
to be $\dot{n}_{\rm gap}(z<1)\sim 0.0013-0.034\ \rm yr^{-1}\ Gpc^{-3}$ for
$z<1$, also consistent with the rate $0.08_{-0.07}^{+0.19}\ \rm yr^{-1}\
Gpc^{-3}$ estimated for GW190521-like mergers from observations (Abbott et
al., 2022). In fact, we find that GW190521 can be marginally explained by our
Pop III BBH mergers (in NSCs) assuming the log-flat or bottom-heavy IMF and
the wide IBS model. In general, the results from all the 18 models are
consistent with current observations of BBH mergers, but it will be
challenging to derive strong constrains on Pop III BBH mergers from current
and upcoming observations by the LVK network that is only sensitive to low-$z$
events among which Pop III mergers are expected to be sub-dominant (unless Pop
III star formation is much more efficient than that modelled by a-sloth in our
case). This highlights the importance of the 3rd-generation GW detectors like
ET in observing Pop III BBH mergers at high $z$.

##  6 Summary and Discussion

We study the roles played by two evolution channels in producing binary black
hole (BBH) mergers from Population III (Pop III) stars using the public semi-
analytical model a-sloth (Hartwig et al., 2022; Magg et al., 2022a) applied to
the halo merger trees produced by the cosmological simulation from Ishiyama et
al. (2016). The first channel considers isolated binary stellar evolution
(IBSE) in which interactions of close ($\lesssim 10$ AU) binary stars shrink
binary orbits to form tight ($\lesssim 0.1$ AU) BBHs that can merge within a
Hubble time by gravitational wave (GW) emission in galaxy fields (i.e.,
effectively in isolation). In the second channel, Pop III BBHs fall into
nuclear star cluster (NSCs) by dynamical friction (DF) and are subsequently
driven to merge via dynamical hardening (DH) by binary-single encounters,
which also works for initially wide (up to $10^{5}$ AU) Pop III BBHs given
their massive nature. We combine the binary population synthesis (BPS) results
of sevn (Costa et al., 2023) with customized routines for the formation,
dynamics, and internal orbital evolution of Pop III BBHs in galaxy fields and
NSCs (Fig. 1), the formation and evolution of NSCs (Fig. 2), as well as halo
growth and galaxy evolution at lower redshifts ($z\lesssim 5$, Sec. 4) in
a-sloth based on the earlier work by Liu & Bromm (2021). We explore 18 models
(Table 2) under different assumptions on Pop III initial mass function (IMF),
initial binary statistics (IBS, see Sec. 2.1), and occupation fraction of
high-$z$ NSCs (see Sec. 3.1). Our main findings are summarized as follows.

  1. 1.

Although only a small fraction ($3-12\%$) of Pop III BBHs fall into NSCs, a
significant fraction ($\sim 45-64\%$) of the BBHs in NSCs merge at $z>0$. The
NSC-DH channel can be as efficient as the classical IBSE channel for producing
Pop III BBH mergers at $z\lesssim 8$, while the IBSE channel always dominates
the merger rate density at $z\gtrsim 10$, producing a peak at $z\sim 15$ that
closely follows the peak of Pop III star formation. The NSC-DH channel
contributes $f_{\rm NSC}\sim 8-95\%$ of Pop III BBH mergers across cosmic
history, and is generally more important at lower redshifts, accounting for
$\mathcal{F}_{\rm NSC}\sim 34-98$% ($0.7-88$%) of the all-sky merger rates of
Pop III BBH mergers within $z<1$ (20). Higher contributions from the NSC-DH
channel are achieved by initially wider binary stars, more top-heavy IMFs, and
higher occupation fractions of NSCs.

  2. 2.

The most important factor that determines the relative importance of the two
channels is IBS. When the IBS is dominated by close binaries, the NSC-DH
channel produces $f_{\rm NSC}\sim 8-51$% of all Pop III mergers across cosmic
history and explains $\mathcal{F}_{\rm NSC}\sim 0.7-22$% of the all-sky merger
rate within $z<20$. However, when the IBS is dominated by wide binaries, we
have $f_{\rm NSC}\sim 65-95$% and $\mathcal{F}_{\rm NSC}\sim 12-88$% for
$z<20$. These outcomes are mainly driven by the strong dependence of the IBSE
channel on IBS. The NSC-DH channel is relatively insensitive to the initial
properties of Pop III binary stars (i.e., IMF and IBS).

  3. 3.

Pop III BBH mergers in NSCs are more massive than those from IBSE. The latter
mostly involve BHs below $\sim 50\ \rm M_{\odot}$ from the collapse of naked
cores of stellar progenitors that have relatively low initial masses
($\lesssim 240\ \rm M_{\odot}$) and lose their hydrogen envelopes during
binary interactions. As pointed out in Mestichelli et al. (2024), binaries of
more massive stars that can potentially form more massive BHs either merge
during unstable mass transfer when they are initially too close or do not
produce tight enough BBHs that can merge within a Hubble time in isolation
when they are initially too far apart. However, the NSC-DH channel can produce
BBH mergers from wide ($\gtrsim 0.1$ AU) binaries of more massive BHs (in the
mass ranges $50-86\ \rm M_{\odot}$ and $130-550\ \rm M_{\odot}$) whose stellar
progenitors start on wide ($\gtrsim 5$ AU) orbits and experience less and even
negligible mass loss during binary stellar evolution. For instance, a
significant fraction ($\sim 4-84\%$) of Pop III BBH mergers in NSCs involve at
least one intermediate mass BH (IMBH) above $100\ \rm M_{\odot}$, while such
mergers are extremely rare from IBSE.

  4. 4.

We find that the Einstein Telescope (ET, Punturo et al., 2010; Maggiore et
al., 2020) is able to detect most ($\gtrsim 90$%) Pop III BBH mergers in our
models with a signal-to-noise ratio (SNR) above 8. The total detection rate is
estimated as $\dot{\mathcal{N}}_{\rm ET,SNR>8}\sim 50-1370\ \rm yr^{-1}$, and
we have $\dot{\mathcal{N}}_{\rm ET,SNR>8}\sim 30-1230\ \rm yr^{-1}$ and
$\dot{\mathcal{N}}_{\rm ET,SNR>8}\sim 6-230\ \rm yr^{-1}$ for the IBSE and
NSC-DH channels, respectively. Moreover, ET will detect $\sim 0.5-200$ Pop III
BBH mergers involving at least one IMBH ($m_{1}>100\ \rm M_{\odot}$) per year
from the NSC-DH channel (mostly at $z\lesssim 10$). The detection rate of such
mergers from IBSE is below $0.1\ \rm yr^{-1}$. In our simulations, Pop III
stars only make up a tiny fraction ($\lesssim 10^{-4}$) of the total mass
budget of stars ever formed in the universe. However, we estimate that Pop III
BBH mergers will make up a much larger fraction ($\sim 0.3-6.5$%) of the BBH
mergers that the 3rd-generation GW detectors will observe, given the
conservative predictions on the merger rates of BBHs from metal-enriched
Population I/II (Pop I/II) stars and primordial BHs (PBHs) by Franciolini et
al. (2022b) as the reference.

  5. 5.

Our results are consistent with current observations of BBH mergers by the LVK
network, especially for the merger rates of IMBHs and BHs with masses in the
standard PISN mass gap ($50-130\ \rm M_{\odot}$) like those in GW190521
(Abbott et al., 2020b, 2022). However, it will be challenging to identify and
characterize Pop III mergers with observations of the LVK network that are
mostly sensitive to the low-$z$ regime ($z\lesssim 1$) where Pop III mergers
in our models are completely sub-dominant (accounting for $\lesssim 1\%$ of
the total merger rate). Even for massive BBH mergers with IMBHs for which Pop
III stars are expected to make higher contributions, the estimated detection
rate of such mergers from Pop III remnants (mostly in NSCs) during the O4 run
of the LVK network is very low ($\dot{\mathcal{N}}_{\rm LVK,IMBH,SNR>8}\sim
0.04-1.5\ \rm yr^{-1}$). This indicates that the 3rd-generation detectors like
ET are required to efficiently constrain Pop III star formation (and high-$z$
SCs) with GW observations (Iwaya et al., 2023; Franciolini et al., 2024;
Santoliquido et al., 2024).

  6. 6.

The stochastic GW background (SGWB) produced by Pop III BBH mergers has a peak
value of $\Omega_{\rm GW}\sim 10^{-11}-8\times 10^{-11}$ around observer-frame
frequencies $\nu_{\rm peak}\sim 10-100$ Hz. Mergers in NSCs dominate the Pop
III SGWB at $\nu\lesssim 20$ Hz in most cases due to their massive
nature353535Interestingly, our predictions on the SGWB from Pop III BBH
mergers at $\nu\lesssim 20$ Hz (dominated by the NSC-DH channel in most cases)
are similar to previous results (Périgois et al., 2021; Martinovic et al.,
2022) that only consider the IBSE channel and assume much higher Pop III star
formation rates. This indicates that significant degeneracy exists in the
large parameter space of Pop III BBH mergers to explain a single observable..
The SGWB from Pop III BBH mergers is equal to $\sim 2-32\%$ of the total SGWB
of compact object mergers inferred from observations (Abbott et al., 2023a) at
$\nu\lesssim 10\ \rm Hz$. Therefore, it is very difficult to observe such a
sub-dominant signal from Pop III BBHs in the total SGWB. However, the
contribution of Pop III mergers can be larger in the residual background
observed by the 3rd-generation detectors like ET where individual detected
sources can be properly subtracted (Zhong et al., 2024), under certain
conditions for the other BBH merger populations (Périgois et al., 2021;
Martinovic et al., 2022; Kouvatsos & Sakellariadou, 2024). Moreover, if the
cosmic star formation rate density of Pop III stars is higher than that
predicted by a-sloth in our case by a factor of a few, Pop III BBHs in NSCs
can dominate the SGWB at $\nu\lesssim 10\ \rm Hz$ if the Pop III IMF is top-
heavy like the TOP models considered here (see Sec. 2.1).

In general, the IBSE and NSC-DH channels produce Pop III BBH mergers of
distinct properties and shape the Pop III contributions to the GW signals from
compact object mergers in different ways. In particular, an important feature
of the NSC-DH channel is that it can produce mergers with IMBHs ($>100\ \rm
M_{\odot}$) and BHs in the (standard) PISN mass gap ($50-130\ \rm M_{\odot}$)
efficiently. This is consistent with previous studies showing that such
massive BBH mergers form efficiently via dynamical processes in SCs (Rodriguez
et al., 2015; Rodriguez et al., 2019; Di Carlo et al., 2020; Fragione & Silk,
2020; Fragione et al., 2020; Fragione et al., 2022; Arca Sedda et al., 2020;
Arca-Sedda et al., 2021; Khan & Holley-Bockelmann, 2021; Gerosa & Fishbach,
2021; Kimball et al., 2021; Liu & Lai, 2021; Mapelli et al., 2021; Mapelli et
al., 2022; Anagnostou et al., 2022; Wang et al., 2022b; Bruel et al., 2023;
Chattopadhyay et al., 2023; Dall’Amico et al., 2023; Fragione & Rasio, 2023;
Liu et al., 2023; Mestichelli et al., 2024), which can also provides seeds for
supermassive BHs (e.g., Askar et al., 2022; Rose et al., 2022; Kritos et al.,
2022; Kritos et al., 2023; Rantala et al., 2024).

Our results show strong, complex dependence on the largely unknown properties
of Pop III binary stars and high-$z$ NSC clusters. In addition to the
uncertain aspects explored in the current paper, there are other
uncertainties/caveats in our modelling that may have significant effects on
our results. Below we discuss the ones known to date in the hope that they
provide directions for future work.

  * (1)

The properties of Pop III BBHs at birth are crucial inputs in our model, which
not only depend on initial conditions but are also highly sensitive to the
parameters/algorithms governing binary interactions and single stellar
evolution (see, e.g., Iorio et al., 2023). Our conclusions are based on the
BPS results from Costa et al. (2023) using sevn with a specific set of
parameters, and thus can be altered if other BPS results are considered. For
instance, the finding that the IBSE channel hardly produces mergers with BHs
above $100\ \rm M_{\odot}$ results from stellar mergers caused by significant
expansion of massive Pop III stars during the MS phase in the stellar
evolution tracks adopted by sevn (see Sec. 5.1.3). However, as shown in
Tanikawa et al. (2021b), if massive Pop III stars remain compact during MS
evolution, massive BBH mergers (involving BHs above $130\ \rm M_{\odot}$) can
still form via IBSE.

  * (2)

Our modelling of the dynamics of Pop III BBHs in host galaxies and during halo
mergers under spherical symmetry is highly idealized. The adopted assumptions
on the galaxy size and structure (Arca-Sedda & Capuzzo-Dolcetta, 2014) are
based on local observations and may not hold at high redshifts. Besides, we
assume that Pop III BBHs are always on their own, i.e., not embedded by larger
structures (e.g., satellite galaxies, bound clusters of Pop III stars and
remnants) that can sink into galaxy centres more efficiently by dynamical
friction, so the efficiency of the NSC-DH channel may be underestimated.
Similarly, we assume that the smaller galaxies hosting NSCs during galaxy
mergers are immediately destroyed, leaving behind naked NSCs.

  * (3)

We use a phenomenological model of NSCs with adiabatic evolution based on
empirical scaling relations from local observations (Neumayer et al., 2020).
These relations may not be valid at high redshifts and cannot capture
dynamical NSC assembly by mergers of globular clusters and young star clusters
and in-situ star formation, which may affect the evolution of BBHs in NSCs.
Moreover, we only consider dynamical hardening, softening/disruption and
ejection by binary-single encounters to follow the internal orbital evolution
of BBHs in NSCs, ignoring higher-order processes (e.g., relativistic phase
space diffusion, tides-driven eccentricity excitation, and Kozai–Lidov
mechanism) involving tidal fields, general relativity effects, and
interactions with other components (e.g., single/binary Pop I/II BHs, central
massive BHs and disks of active galactic nuclei) in NSCs (see Sec. 2.2.2).

  * (4)

The merger trees adopted in our simulations only cover the high-$z$ regime
($z\gtrsim 4.5$) where the limited simulation volume is cosmologically
representative. At lower redshifts, we use a simple galaxy evolution model to
keep track of Pop III BBHs, calibrated to reproduce the observed star
formation rate density within a factor of 2. This model only considers smooth
halo growth and star formation without taking into account halo/galaxy mergers
which are important for the infall of BBHs into NSCs. How exactly this
simplification affects our results is unclear and related to the assumptions
on the dynamics of BBHs and NSCs during galaxy mergers.

  * (5)

We use a stochastic model for the formation of Pop III BBHs that is not self-
consistently connected to the stellar feedback routine, since we do not use
the same population of binary and single stars to model stellar feedback,
which is instead derived from a separate population of single stars. Besides,
we use the fiducial star formation and stellar feedback parameters of a-sloth
(calibrated against observations) in all our simulations. These parameters may
need re-calibrations to be consistent with observations when we change the IMF
and include binary stars. In general, the cosmic star formation rate density
of Pop III stars is poorly constrained without direct observations, and
current theoretical predictions have a scatter of $\sim 1.5$ dex (see, e.g.,
Liu & Bromm, 2020b; Hartwig et al., 2022; Klessen & Glover, 2023). Considering
other Pop III star formation histories will change the merger histories of Pop
III BBHs, especially for the IBSE model (Santoliquido et al., 2023).

Despite these uncertainties, our results indicate that valuable information on
the initial properties of Pop III binary stars and the evolution
channels/environments of BBHs is encoded in the GWs from Pop III BBH mergers.
The former can be fundamentally different from the observed properties of Pop
I/II binaries in the local Universe, and capture the elusive interplay between
fragmentation, (proto)stellar feedback, competitive accretion, migration,
gravitational scatters, and mergers of protostars during Pop III star
(cluster) formation, as well as the dynamics of Pop III star clusters (Sakurai
et al., 2017; Hirano & Bromm, 2018; Chon & Hosokawa, 2019; Sugimura et al.,
2020, 2023; Liu et al., 2021a; Wang et al., 2022b; Riaz et al., 2022; Park et
al., 2023, 2024; Franchini et al., 2023; Liu et al., 2023; Mestichelli et al.,
2024). The latter are governed by key processes in galaxy evolution at Cosmic
Dawn, such as star cluster formation and galactic dynamics which have been
increasingly revealed by JWST observations (e.g., Baggen et al., 2023;
Ferreira et al., 2023; Kartaltepe et al., 2023; Langeroodi & Hjorth, 2023; Ito
et al., 2023; Ormerod et al., 2024; Ji et al., 2024a; Adamo et al., 2024).
Although distinguishing Pop III BBHs from other populations of mergers on an
individual basis is non-trivial (Franciolini et al., 2022a; Costa et al.,
2023), identifying a subgroup of BBH mergers of Pop III origins from a large
enough sample of mergers is feasible and promising given the unique features
of Pop III BBHs and the non-proportionally high contributions to GW signals by
Pop III stars (Tanikawa et al., 2022b; Iwaya et al., 2023; Franciolini et al.,
2024; Santoliquido et al., 2024). In the next decades, the 3rd-generation GW
detectors like ET and the Cosmic Explorer (Reitze et al., 2019; Evans et al.,
2023) will observe thousands of events per year reaching $z\sim 30$. The
‘gravitational-wave archaeology’ in synergy with other observations of Cosmic
Dawn will provide us new insights into the first stars, star clusters, and
galaxies in the early Universe.

## Acknowledgements

The authors thank Michela Mapelli for providing the input sevn catalogues
underlying Costa et al. (2023) and valuable discussions. We thank Giuliano
Iorio for developing and making sevn available. We thank Benedetta Mestichelli
for useful discussions on Pop III binary stellar evolution. We thank Mattis
Magg and Li-Hsin Chen for their help in code development with a-sloth. We
thank Simone S. Bavera for providing the data underlying Bavera et al. (2022).
We thank Sebastian Ljung for feedback on the BBH and NSC modules in a-sloth.
The authors acknowledge the Texas Advanced Computing Center (TACC) for
providing HPC resources under FRONTERA allocation AST22003. BL is supported by
the Royal Society University Research Fellowship. NS gratefully acknowledges
the support of the Research Foundation - Flanders (FWO Vlaanderen) grant
1290123N. GC acknowledges support from the Agence Nationale de la Recherche
grant POPSYCLE number ANR-19-CE31-0022. FS acknowledges financial support from
the AHEAD2020 project (grant agreement n. 871158). GC and FS acknowledge
financial support from the European Research Council for the ERC Consolidator
grant DEMOBLACK, under contract no. 770017. RSK acknowledges financial support
from the European Research Council via the ERC Synergy Grant ‘ECOGAL’ (project
ID 855130), from the German Excellence Strategy via the Heidelberg Cluster of
Excellence (EXC 2181 - 390900948) ‘STRUCTURES’, and from the German Ministry
for Economic Affairs and Climate Action in project ‘MAINN’ (funding ID
50OO2206). RSK thanks for computing resources provided by the Ministry of
Science, Research and the Arts (MWK) of the State of Baden-Württemberg through
bwHPC and the German Science Foundation (DFG) through grants INST 35/1134-1
FUGG and 35/1597-1 FUGG, and also for data storage at SDS@hd funded through
grants INST 35/1314-1 FUGG and INST 35/1503-1 FUGG. This work excessively used
the public packages numpy (van der Walt et al., 2011), matplotlib (Hunter,
2007), and scipy (Virtanen et al., 2020). The authors wish to express their
gratitude to the developers of these packages and to those who maintain them.

## Data availability

a-sloth is publicly available at https://gitlab.com/thartwig/asloth. The
version used in this paper (including the new modules for compact object
mergers and NSCs) is public at https://gitlab.com/Treibeis/a-sloth-cob. sevn
is publicly available at https://gitlab.com/sevncodes/sevn. The data
underlying this paper will be shared on reasonable request to the
corresponding author.

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