# Are binary black hole mergers and long $\gamma$-ray bursts drawn from the
same black hole population?

Benjamin Arcier  IRAP, Université de Toulouse, CNES, CNRS, UPS, Toulouse,
France  Jean-Luc Atteia  IRAP, Université de Toulouse, CNES, CNRS, UPS,
Toulouse, France

(Received November, 2021; Revised April, 2022; Accepted April, 2022)

###### Abstract

This paper compares the population of BBH mergers detected by LIGO/Virgo with
selected long GRB world models convolved with a delay function (LGRBs are used
as a tracer of stellar mass BH formation). The comparison involves the
redshift distribution and the fraction of LGRBs required to produce the local
rate of BBH mergers.

We find that BBH mergers and LGRBs cannot have the same formation history,
unless BBHs mergers have a long coalescence time of several Gyr. This would
imply that BHs born during the peak of long GRB formation at redshift
$z\approx 2-3$ merge within the horizon of current GW interferometers. We also
show that LGRBs are more numerous than BBH mergers, so that most of them do
not end their lives in BBH mergers.

We interpret these results as an indication that BBH mergers and LGRBs
constitute two distinct populations of stellar mass BHs, with LGRBs being more
frequent than BBH mergers. We speculate that the descendants of LGRBs may
resemble galactic High-Mass X-Ray Binaries more than BBH mergers. We finally
discuss the possible existence of a sub-population of fast-spinning LGRBs
descendants among BBH mergers, showing that this population, if it exists, is
expected to become dominant beyond redshift $z\approx 1$, leading to a change
in the observed properties of BBH mergers.

Astrophysical Black Holes (98) — Gamma-Ray Burst (629) — Gravitational wave
sources (677)

††software: NumPy (van der Walt et al., 2011), SciPy (Virtanen et al., 2020),
Matplotlib (Hunter, 2007), Astropy (Astropy Collaboration et al., 2013),
LALSuite (LIGO Scientific Collaboration, 2018)

##  1 Introduction

Black hole (BH) astrophysics is a rapidly evolving field, especially thanks to
the discovery of dozens of binary black hole mergers by the LIGO Scientific
Collaboration and Virgo Collaboration (LVC). Despite the apparent simplicity
of black holes as astrophysical objects, a global vision of their origin and
evolution is still missing as well as a full understanding of their role in
the evolution of the universe and in the formation of its structures. In such
a situation, the comparison of the properties of various BH populations may
shed light on their common or diverse origins. We focus here on two
populations of stellar mass black holes, which are observed in different
contexts: long gamma-ray bursts (long GRBs or LGRBs) and the mergers of
stellar mass black holes (BBH mergers).

Long GRBs are the brightest explosions detectable in the electromagnetic
domain (e.g. Vedrenne & Atteia, 2009; Atteia et al., 2017). They occur when
the core of a massive star collapses into a black hole or an hypermassive
magnetar, subsequently ejecting a transient relativistic jet in our direction.
The association of LGRBs with stellar collapse is confirmed by the detection
of broad-line core-collapse supernovae of type Ibc following several nearby
LGRBs (z $\leq 0.3$) after a few days (Galama et al., 1998; Tagliaferri et
al., 2006). However, only a small fraction of core collapse supernovae produce
LGRBs.

This fraction has been evaluated by Soderberg et al. (2006b), based on
volumetric rates of about $9000\ \mathrm{Gpc^{-3}\,yr^{-1}}$ for supernovae
Ibc and $f_{\mathrm{b}}\times 1\ \mathrm{Gpc^{-3}yr^{-1}}$ for classical LGRBs
(e.g. Wanderman & Piran, 2010; Palmerio & Daigne, 2021), where
$f_{\mathrm{b}}$ is the GRB beaming factor. It is difficult to evaluate the
true rate of GRBs, due to large uncertainties on the measurements of the
beaming angle of GRB jets (thus on $f_{\mathrm{b}}$). With typical values of
$f_{\mathrm{b}}$ in the range $75-250$ (corresponding to jet opening angle in
the range $5$°$-9$°), the classical LGRBs represent from 1 to 3% of SNIbc.
This is compatible with another study of Soderberg et al. (2006a), which shows
that less than 10% of SNIbc are associated with a successful GRB, based on the
radio followup of 68 local supernovae of Type Ibc. We note that this fraction
concerns only classical LGRBs, excluding low-luminosity GRBs, which have also
been shown to be associated with SNIbc, or choked GRBs, whose jet does not
pierce the star’s envelope. The rate of low-luminosity GRBs is estimated to be
$\sim 100-1000\ \mathrm{Gpc^{-3}~{}yr^{-1}}$ (Soderberg et al., 2006b; Liang
et al., 2007), comparable or a few times higher than the rate of classical
LGRBs. The rate of choked GRBs has not been measured.

The conditions for GRB production by dying massive stars is far from being
fully elucidated. MacFadyen & Woosley (1999) emphasized the key role of the
angular momentum of the stellar core, which has to be sufficiently large to
permit the survival of a massive accretion disk when the BH forms. Before the
discovery of BBH mergers with low effective spins by the LVC, it was commonly
assumed that GRB progenitors could be single stars with low metallicity, whose
weak stellar winds carry away a small fraction of the star angular momentum or
stars in binary systems, which are sufficiently close to be tidally locked and
keep a high angular momentum (e.g. Woosley & Bloom, 2006; Levan et al., 2016;
Chrimes et al., 2020). Except for a few events with an associated supernova,
we have only indirect information about the progenitors of LGRBs, because the
information is mediated by the relativistic jet, whose properties are not
directly connected with the nature of the progenitor. The population of long
GRB progenitors can nevertheless be constrained via statistical studies
relying on redshift measurements and the nature of their host galaxies. This
has led to the construction of long GRB world models, which describe the
history of long GRB formation and are compatible with their observed redshift
distribution. (Salvaterra et al., 2012; Lien et al., 2014; Palmerio & Daigne,
2021)

Contrary to LGRBs, BBH mergers are mostly silent in the electromagnetic domain
(see however Farris et al. 2010; Bartos et al. 2017; McKernan et al. 2020 for
discussions about the possible emission of electromagnetic transients by BBHs
lying in a dense gaseous environment). These cataclysmic events, which have
been discovered by the LVC in 2015 (Abbott et al., 2016), release most of
their energy in gravitational waves (GW). The origin of these BBH mergers is
highly debated, and the variety of systems discovered by the LVC probably
calls for multiple populations (Zevin et al., 2021; Wang et al., 2021),
however see Bavera et al. 2020. There is nevertheless a broad consensus that
most of the mergers detected by the LVC are of astrophysical origin, even if a
marginal contribution from primordial black holes remains possible (De Luca et
al., 2021). BBH mergers of astrophysical origin result from the evolution of
massive stars, which can be in isolated binary systems or in a dense
environment prone to dynamical interactions.

Considering the different biases affecting the detection of BBH mergers and
LGRBs, the comparison of these two populations is likely to shed new light on
the origin and evolution of stellar mass black holes of astrophysical origin.

In this article, we compare basic statistics concerning these two populations.
First a $\mathrm{V/V_{\mathrm{max}}}$ test is used to compare the redshift
distribution of BBH mergers with various delayed long GRB models, where long
GRB models are used as proxies for the formation history of stellar mass black
holes. This comparison, which takes into account the shape of the redshift
evolution independent of the normalization of the two populations, leads us to
identify several models compatible which can reproduce the redshift
distribution of BBH mergers detected by LVC. In a second step, we compare the
rate of BBH mergers measured by LVC with the rate of long GRB. This comparison
makes no assumption of a physical connection between the two populations, it
only uses the rate of long GRBs as a useful reference for the comparison of
populations with different redshift evolution. We find that the progenitors of
BBH mergers are rare compared with GRBs. For these two studies the BBH mergers
are considered as a single population.
These findings are then briefly discussed in the context of models developed
to explain the BBH mergers and the GRBs. Emphasizing the potential role of
tidal spin-up for both types of sources, we show that present data allow the
existence of a minority of fast-spinning BBH mergers that quickly follow the
production of a LGRBs.

This study has been made possible by the publication of the GWTC-2 and GWTC-3
catalogs obtained with the O3 observing run (Abbott et al., 2021; The LIGO
Scientific Collaboration et al., 2021b). These catalogs contain 79 BBH mergers
with redshifts ranging from z = 0.05 to z = 0.82 with a median of z = 0.3),
enabling the comparison of the redshift distributions of BBH mergers with GRBs
over the last 7 Gyrs. It follows a first work from Atteia et al. (2018), based
on the GWTC-1 catalog (Abbott et al., 2019) and thus limited to the rates
comparison of BBHs and LGRBs.

This paper is organized as follows: in the next two sections we compare the
redshift distribution (Sect. 2) and the volumetric rate (Sect. 3) of BBH
mergers and LGRBs. Then, in Sect. 4, we briefly discuss the implication of
these results in the context of stellar evolution models developed to explain
LGRBs and BBH mergers.

In all this paper, we use a flat $\Lambda\mathrm{CDM}$ cosmological model with
the parameters measured by the Planck Collaboration:
$H_{\mathrm{0}}=67.4~{}\mathrm{km~{}s^{-1}}~{}\mathrm{Mpc}^{-1}$ and
$\Omega_{\mathrm{m}}=0.315$ (Planck Collaboration et al., 2021).

##  2 Redshift distribution

The fast growing number of BBH mergers detected by the LVC provides some
indications on their redshift distribution, as discussed in Abbott et al.
(2021) and The LIGO Scientific Collaboration et al. (2021a). Using a
generalization of the $\mathrm{V/V_{\mathrm{max}}}$ test developed by Schmidt
(1968), we compare here the redshift distribution of 79 BBH mergers detected
with high confidence by the LVC with three recent models of classical LGRBs
and one star formation history (SFH) model.

###  2.1 Methodology

We use a test called $\mathrm{N/N_{\mathrm{max}}}$, which is a variant of the
$\mathrm{V/V_{\mathrm{max}}}$ test developed by Schmidt (1968). These tests
are used to assess the compatibility of the volumetric distribution of
observed sources with a model. In the $\mathrm{V/V_{\mathrm{max}}}$ test, the
model is used to compute, for each source, the ratio of the volume enclosed by
the source to the maximum volume of detection for this source. If the model is
correct, the source is randomly chosen among all visible sources and the ratio
$\mathrm{V/V_{\mathrm{max}}}$ follows a uniform distribution
$\mathcal{U}(0,1)$. In its simplest form, the $\mathrm{V/V_{\mathrm{max}}}$
test considers sources without cosmological evolution. For sources with
cosmological evolution, the test can be adapted by considering the ratio of
the predicted number of sources within the volume enclosed by a source to the
number of sources within the maximum volume of detection for this source. When
the model is completely defined and the volume of detection is correctly
calculated, the $\mathrm{V/V_{\mathrm{max}}}$ and
$\mathrm{N/N_{\mathrm{max}}}$ tests are hardly affected by selection effects,
because each source is placed within its own volume of detection.

The volume of detection of each BBH merger is computed via the determination
of its horizon $z_{\mathrm{h,i}}$, which is the maximum redshift at which this
merger could be detected (see below). Knowing the merger event redshift
$z_{\mathrm{i}}$ and its horizon redshift $z_{\mathrm{h,i}}$, we can compute
$\mathrm{N_{i}}$, the number of sources closer than $z_{\mathrm{i}}$, and
$\mathrm{N_{\mathrm{max,i}}}$, the number of sources closer than the horizon
$z_{\mathrm{h,i}}$, for various source models. A model is acceptable if the
ratio $\mathrm{N_{i}/N_{\mathrm{max,i}}}$ follows a uniform distribution
$\mathcal{U}(0,1)$. The agreement of the distribution of
$\mathrm{N_{i}/N_{\mathrm{max,i}}}$ with $\mathcal{U}(0,1)$ is evaluated with
a Kolmogorov-Smirnov test (hereafter, KS test). This comparison does not take
into account the normalization of the GRB and BBH merger rates, which is
discussed in Sect. 3.

The method used to determine the horizon redshift is based on the horizon
calculation method proposed in Chen et al. (2021) and the open-source code
available at https://github.com/hsinyuc/distancetool. The equation to compute
the SNR evolution with the redshift is taken from Chen et al. (2021):

$\mathrm{SNR_{\mathrm{Ch21}}}=\sqrt{4\int_{f_{\mathrm{min}}}^{f_{\mathrm{max}}}\frac{|h^{+}(f)|^{2}}{S_{\mathrm{h}}(f)}\mathrm{d}f}$
(1)

where $h^{+}(f)$ is the plus polarization of the gravitational-wave merger and
$S_{\mathrm{h}}(f)$ is the power spectral density of the interferometer. The
calculation of $h^{+}(f)$ depends on the binary properties and its distance
relatively to the observer, in addition to the masses $m_{1}$ and $m_{2}$ of
the BBH merger components. This plus polarization is obtained using the
lalsimulation Python package with the waveform IMRPhenomD (as for the online
calculator provided by Chen et al. 2021, since $m_{1}$ or $m_{2}\geq 3\
\mathrm{M_{\odot}}$). For the term $S_{\mathrm{h}}(f)$, we took the same
strain noise curve for LIGO Handford and LIGO Livingston, for a given
observing run, already included in the Github repository from Chen et al.
(2021).

Contrary to Chen et al. (2021), who define the horizon as the highest redshift
at which a BBH merger could be detected with the best antenna pattern
(considering that the BBH system is optimally oriented relative to the GW
detectors and located in a portion of the sky where the interferometers
performances are maximum), we define the horizon as the maximum redshift at
which a BBH merger would be detected, under the conditions of its detection.
Since the antenna pattern does not depend upon distance and cosmology, it is
possible to normalize the SNR dependence on redshift with the actual signal-
to-noise ratio $\mathrm{SNR_{0}}$ obtained for a given merger detected at
redshift $z_{0}$. We obtain the SNR dependence on redshift corrected by the
antenna pattern, allowing to compute the horizon redshift for each detected
merger.

$\mathrm{SNR}(z,m_{1},m_{2})=\frac{\mathrm{SNR}_{\mathrm{Ch21}}(z,m_{1},m_{2})}{\mathrm{SNR}_{\mathrm{Ch21}}(z_{0},m_{1},m_{2})}\times\mathrm{SNR_{0}}$
(2)

Taking a threshold $\mathrm{SNR_{lim}}$, the horizon redshift is the variable
$z_{\mathrm{h,i}}$ solving the equation:

$\mathrm{SNR}(z_{\mathrm{h,i}},m_{1},m_{2})=\mathrm{SNR_{lim}}$ (3)

Figure 1: SNR dependence on redshift for the BBH mergers observed during O1,
O2, O3a and O3b. The color coding represents the total mass of the binary in
the observer frame, while the marker style represents the catalog in where the
events are listed. The dashed line for GW 170608 illustrates a SNR dependency
inversely proportional to the luminosity distance.

The SNR, mass and detected redshift are taken from the Gravitational Wave Open
Science Center (GWOSC) event list111https://www.gw-
openscience.org/eventapi/html/GWTC/, created from the GWTC catalogs (Abbott et
al., 2019, 2021; The LIGO Scientific Collaboration et al., 2021c, b). In the
following, we consider $\mathrm{SNR_{lim}}=8$ in agreement with the smallest
SNR observed for BBHs mergers (Abbott et al., 2019, 2021), and with the
theoretical sensitivities of LIGO/Virgo (Abbott et al., 2018). Recent merger
observations from the deep extended catalog from O3a (The LIGO Scientific
Collaboration et al., 2021c) and from GWTC-3 (The LIGO Scientific
Collaboration et al., 2021b) have however detected events with a network
matched filter SNR below this limit. These events are not considered in this
study. Events with mass $m_{1}$ and/or $m_{2}$ below $2.4\ \mathrm{M_{\odot}}$
have also been removed, to prevent the contamination from NS-NS or NS-BH
merger events.

Table LABEL:tab:z_tab gives the horizons of the 79 events used in this
article, computed with this method. Figure 1 illustrates the redshift
dependence and the impact of the shape of the aLIGO power-spectral density
(which is not exactly flat), which causes the departure from the inverse
luminosity distance law (shown with a faint dashed line for GW 170608).
Mergers in GWTC-1, GWTC-2 and GWTC-3 have a broad range of horizons, from z =
0.12 to z = 1.10 with a median z = 0.50. This is the reflection of their mass
range, which spans over 1 decade. This broad range of horizons implies very
different detection volumes for massive ($\sim 160~{}M_{\sun}$) and less
massive BBH mergers ($\leq 20~{}M_{\sun}$), and a mass distribution of
detected mergers which is strongly biased in favor of massive BBHs. By
construction the $\mathrm{N/N_{\mathrm{max}}}$ test takes into account this
bias by considering the individual contribution of each BBH merger within its
own volume of detection.

The $\mathrm{N/N_{\mathrm{max}}}$ test requires computing
$\mathrm{N}_{\mathrm{i}}$, the number of mergers expected up to redshift
$z_{\mathrm{i}}$ and $\mathrm{N}_{\mathrm{max,i}}$, the number of mergers
expected up to redshift $z_{\mathrm{h,i}}$. This is based on a world model of
source population that specifies $\rho(z)$, the evolution of the source
density rate with redshift. $\mathrm{N}_{\mathrm{i}}$ is computed as follows:

$\mathrm{N_{i}}=\int_{0}^{z_{\mathrm{i}}}\rho(z)\
\frac{\mathrm{d}V(z)}{\mathrm{d}z}\ \frac{1}{1+z}\ \mathrm{d}z$ (4)

Where $\rho(z)$ is the BBHs merger density rate (in $\mathrm{Gpc^{-3}\
yr^{-1}}$), $\mathrm{d}V(z)$ is the differential co-moving volume, and the
term $1/(1+z)$ accounts for the impact of time dilation when we measure a rate
of events (Mao & Paczynski, 1992).

We compare here the observed distribution of BBH mergers with several density
rate models: a constant rate model, a model of SFH and three GRB world models
which all reproduce the observed properties of GRBs detected by the Swift and
Fermi missions. These three models and the SFH are illustrated in Fig. 2a.

The first GRB model is proposed by Salvaterra et al. (2012):

$\mathcal{R}_{\mathrm{GRB}}^{\mathrm{S}}(z)=\mathcal{R}_{\mathrm{0}}^{\mathrm{S}}\
(1+z)^{\mathrm{\delta_{n}}}\ \mathrm{\Sigma}_{\mathrm{SFH}}^{\mathrm{Li}}(z)$
(5)

based on the cosmic SFH rate from Li (2008):

$\mathrm{\Sigma}_{\mathrm{SFH}}^{\mathrm{Li}}(z)=\frac{a+bz}{1+(z/c)^{d}}$ (6)

with $\mathcal{R}_{\mathrm{0}}^{\mathrm{S}}=0.24\ \mathrm{Gpc}^{-3}\
\mathrm{yr}^{-1}$, $\delta_{n}=1.7$, $a=0.0158$, $b=0.118$, $c=3.23$ and
$d=4.66$.

The second is proposed by Lien et al. (2014):

$\mathcal{R}_{\mathrm{GRB}}^{\mathrm{L}}(z)=\mathcal{R}_{\mathrm{0}}^{\mathrm{S}}\
\left\\{\begin{array}[]{ll}(1+z)^{\mathrm{n_{1}}}&\mbox{if
}z<\mathrm{z_{1}}\\\
(1+\mathrm{z_{1}})^{\mathrm{(n_{1}-n_{2})}}(1+z)^{\mathrm{n_{2}}}&\mbox{if
}z\geq\mathrm{z_{1}}\end{array}\right.$ (7)

with $\mathcal{R}_{\mathrm{0}}^{\mathrm{L}}=0.42\ \mathrm{Gpc}^{-3}\
\mathrm{yr}^{-1}$, $n_{1}=2.07$, $n_{2}=-0.7$ and $\mathrm{z_{1}}=3.6$.

The third is proposed by Palmerio & Daigne (2021):

$\mathcal{R}_{\mathrm{GRB}}^{\mathrm{P}}(z)=\mathcal{R}_{\mathrm{0}}^{\mathrm{P}}\
\left\\{\begin{array}[]{ll}e^{az}&\mbox{if }z<\mathrm{z_{m}}\\\
e^{(a-b)\mathrm{z_{m}}}e^{bz}&\mbox{if }z\geq\mathrm{z_{m}}\end{array}\right.$
(8)

with $\mathcal{R}_{\mathrm{0}}^{\mathrm{P}}=1.0\ \mathrm{Gpc}^{-3}\
\mathrm{yr}^{-1}$, $a=1.1$, $b=-0.57$ and $\mathrm{z_{m}}=1.9$.

Figure 2:  GRB density rate models and SFH evolution with redshift, including
several time delay functions.

Since the population of BBHs mergers is probably not directly tracing the GRB
population, we also consider delayed models with a delay function between the
GRB emission (when the BH is created) and the transient GW emission (when the
BH merges with its companion). This approach is similar to the studies that
compare the rates of binary neutron star mergers and short GRBs (Nakar, 2007;
Wanderman & Piran, 2015). Considering a time delay probability density
function $f(\tau)$ and integrating over all possible time delays, the
calculation of the BBH merger rate at redshift $z_{0}$ becomes:

$\rho(z_{0})\propto\int_{z_{0}}^{\infty}\mathcal{R}_{\mathrm{GRB}}(z)\
f(T_{c}(z)-T_{c}(z_{0}))\ \frac{\mathrm{d}T_{c}}{\mathrm{d}z}\ \mathrm{d}z$
(9)

where $T_{c}(z)$ is the lookback time at redshift $z$, and
$\tau=T_{c}(z)-T_{c}(z_{0})$ is the time delay in Gyr between a GRB at
redshift $z$ and a merger produced at redshift $z_{0}<z$. In practice, a
limiting redshift $z_{\mathrm{lim}}=20$ has been chosen for the calculation of
the integral, and we have verified that changing this value from $20$ to $100$
has no significant impact on the results. The delay function $f(\tau)$ can
have several shapes, and in the following we consider three of them: a power-
law, a lognormal distribution, and a constant delay.

The power-law has two parameters: a negative slope $\alpha$ (for example
$\alpha=-1$ in Belczynski et al. 2016) and a minimum merging time
$dT_{\mathrm{min}}$. It is described by:

$f(\tau)=\left\\{\begin{array}[]{ll}0&\tau\leq dT_{\mathrm{min}}\\\
\tau^{\alpha}&\tau>dT_{\mathrm{min}}\end{array}\right.$ (10)

The lognormal distribution of width $\sigma_{\mathrm{t}}$ centered around a
time delay $t_{\mathrm{d}}$ is described by:

$f(\tau)=\frac{1}{\tau\sigma_{\mathrm{t}}\sqrt{2\pi}}\exp\left(-\frac{\ln(\tau/t_{\mathrm{d}})^{2}}{2\sigma_{\mathrm{t}}^{2}}\right)$
(11)

The constant delay is modeled by setting a small dispersion of the lognormal
distribution (typically $\sigma_{\mathrm{t}}=0.01$). This is how the constant
delay GRB density rate presented on Fig. 2b has been created. From the GWTC-1,
GWTC-2 and GWTC-3 catalogs, the maximum redshift observed for a BBHs merger is
$z=0.71$, which represents a lookback time of $\sim 6.5$ Gyr, with the
cosmology used in this paper. For this reason, we have not considered constant
delay models with a delay greater than $7$ Gyr, which would lead to produce
the most distant BBH mergers before the Big Bang. The differential of the
lookback time at redshift $z$ is calculated using the definition of the
lookback time in Condon & Matthews 2018:

$\frac{\mathrm{d}T_{\mathrm{c}}}{\mathrm{d}z}(z)=\frac{1}{H(z)(1+z)}$ (12)

Fig. 2b illustrates the impact of these different delay functions on the GRB
model of Palmerio & Daigne 2021.

###  2.2 Results

Using the $\mathrm{N/N_{\mathrm{max}}}$ test defined in the previous section,
we now compare the redshift distribution of BBH mergers in the GWTC catalogs
with various delayed GRB models. A model is considered acceptable if the KS
test gives a p-value larger than 1% and favored if the p-value exceeds $10$%.
This is illustrated in Fig. 3, which compares the cumulative
$\mathrm{N/N_{\mathrm{max}}}$ distributions of various models with the data.
The light grey zone indicates the 99% confidence region, while the dark grey
zone the 90% confidence interval: a model is accepted at a given confidence
level if its $\mathrm{N/N_{\mathrm{max}}}$ cumulative distribution function
lies entirely inside the grey zone.

Table 1: Results of the Kolmogorov-Smirnov test for various population models
(see text). The p-values of favored models are indicated in boldface.

# | Population model | Delay parameters | p-value | $\left<\mathrm{N/N_{max}}\right>$ | Fig.
---|---|---|---|---|---
$\mathcal{R}_{\mathrm{1}}$ | Constant | … | 0.035 | 0.452 | 3
$\mathcal{R}_{\mathrm{2}}$ | SFH: Li (2008) | … | 0.001 | 0.389 |  LABEL:fig:Rgrb, 3
$\mathcal{R}_{\mathrm{3}}$ | GRB: Lien et al. (2014) | … | 0.005 | 0.406 | LABEL:fig:Rgrb
$\mathcal{R}_{\mathrm{4}}$ | GRB: Salvaterra et al. (2012) | … | 0.000 | 0.356 | LABEL:fig:Rgrb
$\mathcal{R}_{\mathrm{5}}$ | GRB: Palmerio & Daigne (2021) | … | 0.014 | 0.417 |  LABEL:fig:Rgrb, LABEL:fig:Rgrb_d
$\mathcal{R}_{\mathrm{6}}$ | SFH: Li (2008) | PL, $dT_{\mathrm{min}}=0.01$ Gyr, $\alpha=-1.0$ | 0.006 | 0.412 | …
$\mathcal{R}_{\mathrm{7}}$ | GRB: Lien et al. (2014) | PL, $dT_{\mathrm{min}}=0.01$ Gyr, $\alpha=-1.0$ | 0.012 | 0.419 | 3
$\mathcal{R}_{\mathrm{8}}$ | GRB: Salvaterra et al. (2012) | PL, $dT_{\mathrm{min}}=0.01$ Gyr, $\alpha=-1.0$ | 0.003 | 0.401 | …
$\mathcal{R}_{\mathrm{9}}$ | GRB: Palmerio & Daigne (2021) | PL, $dT_{\mathrm{min}}=0.01$ Gyr, $\alpha=-1.0$ | 0.017 | 0.425 | LABEL:fig:Rgrb_d
$\mathcal{R}_{\mathrm{10}}$ | SFH: Li (2008) | PL, $dT_{\mathrm{min}}=0.01$ Gyr, $\alpha=-2.0$ | 0.001 | 0.390 | …
$\mathcal{R}_{\mathrm{11}}$ | GRB: Lien et al. (2014) | PL, $dT_{\mathrm{min}}=0.01$ Gyr, $\alpha=-2.0$ | 0.005 | 0.406 | …
$\mathcal{R}_{\mathrm{12}}$ | GRB: Salvaterra et al. (2012) | PL, $dT_{\mathrm{min}}=0.01$ Gyr, $\alpha=-2.0$ | 0.000 | 0.357 | …
$\mathcal{R}_{\mathrm{13}}$ | GRB: Palmerio & Daigne (2021) | PL, $dT_{\mathrm{min}}=0.01$ Gyr, $\alpha=-2.0$ | 0.014 | 0.417 | …
$\mathcal{R}_{\mathrm{14}}$ | SFH: Li (2008) | PL, $dT_{\mathrm{min}}=5.0$ Gyr, $\alpha=-1.0$ | 0.799 | 0.517 | …
$\mathcal{R}_{\mathrm{15}}$ | GRB: Lien et al. (2014) | PL, $dT_{\mathrm{min}}=5.0$ Gyr, $\alpha=-1.0$ | 0.144 | 0.463 | LABEL:fig:Rgrb_best
$\mathcal{R}_{\mathrm{16}}$ | GRB: Salvaterra et al. (2012) | PL, $dT_{\mathrm{min}}=5.0$ Gyr, $\alpha=-1.0$ | 0.325 | 0.476 | …
$\mathcal{R}_{\mathrm{17}}$ | GRB: Palmerio & Daigne (2021) | PL, $dT_{\mathrm{min}}=5.0$ Gyr, $\alpha=-1.0$ | 0.676 | 0.524 | …
$\mathcal{R}_{\mathrm{18}}$ | SFH: Li (2008) | Cst Delay, $\tau=6.5$ Gyr | 0.533 | 0.494 | 3
$\mathcal{R}_{\mathrm{19}}$ | GRB: Lien et al. (2014) | Cst Delay, $\tau=7.0$ Gyr | 0.014 | 0.422 | …
$\mathcal{R}_{\mathrm{20}}$ | GRB: Salvaterra et al. (2012) | Cst Delay, $\tau=7.0$ Gyr | 0.035 | 0.458 | LABEL:fig:Rgrb_best
$\mathcal{R}_{\mathrm{21}}$ | GRB: Palmerio & Daigne (2021) | Cst Delay, $\tau=6.5$ Gyr | 0.255 | 0.502 |  LABEL:fig:Rgrb_d, LABEL:fig:Rgrb_best
$\mathcal{R}_{\mathrm{22}}$ | SFH: Li (2008) | logNorm, $\mathrm{t_{d}}=7.3$ Gyr, $\sigma_{t}=0.3$ | 0.966 | 0.499 | …
$\mathcal{R}_{\mathrm{23}}$ | GRB: Lien et al. (2014) | logNorm, $\mathrm{t_{d}}=10.0$ Gyr, $\sigma_{t}=0.3$ | 0.892 | 0.519 | LABEL:fig:Rgrb_best
$\mathcal{R}_{\mathrm{24}}$ | GRB: Salvaterra et al. (2012) | logNorm, $\mathrm{t_{d}}=8.6$ Gyr, $\sigma_{t}=0.3$ | 0.990 | 0.498 | …
$\mathcal{R}_{\mathrm{25}}$ | GRB: Palmerio & Daigne (2021) | logNorm, $\mathrm{t_{d}}=7.3$ Gyr, $\sigma_{t}=0.3$ | 0.986 | 0.504 |  LABEL:fig:Rgrb_d, LABEL:fig:Rgrb_best, 3
$\mathcal{R}_{\mathrm{26}}$ | SFH: Li (2008) | logNorm, $\mathrm{t_{d}}=5.0$ Gyr, $\sigma_{t}=1.0$ | 0.023 | 0.437 | …
$\mathcal{R}_{\mathrm{27}}$ | GRB: Lien et al. (2014) | logNorm, $\mathrm{t_{d}}=5.0$ Gyr, $\sigma_{t}=1.0$ | 0.012 | 0.424 | …
$\mathcal{R}_{\mathrm{28}}$ | GRB: Salvaterra et al. (2012) | logNorm, $\mathrm{t_{d}}=5.0$ Gyr, $\sigma_{t}=1.0$ | 0.007 | 0.416 | …
$\mathcal{R}_{\mathrm{29}}$ | GRB: Palmerio & Daigne (2021) | logNorm, $\mathrm{t_{d}}=5.0$ Gyr, $\sigma_{t}=1.0$ | 0.017 | 0.436 | …

The values of $\langle\mathrm{N/N_{max}}\rangle$, the mean of the
$\mathrm{N/N_{\mathrm{max}}}$ distribution, and the p-values of the KS test
are given in Table 1 for selected models. The first 5 models are a constant
density rate evolution in addition to the SFH and the three GRB models
presented in the methodology section, without any delay. They are followed by
12 models with a power-law delay function, with two values of the power-law
index, $\alpha=-2$, and $\alpha=-1$, and two values of $dT_{\mathrm{min}}$,
$0.01$ and $5$ Gyr. The first value of $dT_{\mathrm{min}}$ is standard for
such models, while larger values of this parameter have been used to explore
its impact on the rate density distributions. Finally, we also consider 12
models with a lognormal delay function, with three values of the dispersion:
$\sigma_{\mathrm{t}}=0.01$, $0.3$ and $1.0$, and for each of them we indicate
the delay $t_{\mathrm{d}}$ that gives the largest p-value. As explained
before, the model with $\sigma_{\mathrm{t}}=0.01$ is equivalent to a constant
delay model.

According to Table 1, ten models have p-values larger than 10% (favored
models), while 11 models have a p-value between 1 and 10%. All favored models
are delayed models, with a significant delay between the GRB and the merger
event:

  * •

The SFH model and the GRB models of Salvaterra et al., Lien et al. and
Palmerio & Daigne convolved with a power-law delay function with $\alpha=-1$
and $dT_{\mathrm{min}}=5$ Gyr ($\mathcal{R}_{14}$, $\mathcal{R}_{15}$,
$\mathcal{R}_{16}$, $\mathcal{R}_{17}$).

  * •

The SFH model and the GRB model of Palmerio & Daigne convolved with a constant
delay function of 7 Gyr ($\mathcal{R}_{18}$, $\mathcal{R}_{21}$).

  * •

The SFH model and the GRB models of Lien et al., Salvaterra et al. and
Palmerio & Daigne convolved with a lognormal delay function with
$\sigma_{\mathrm{t}}=0.1$ or $0.3$ and optimal $t_{\mathrm{d}}$ values
($\mathcal{R}_{22}$, $\mathcal{R}_{23}$, $\mathcal{R}_{24}$,
$\mathcal{R}_{25}$).

Some favored models are shown in Fig. 2c, for instance the model of Lien et
al. with a delay $t_{\mathrm{d}}=10$ Gyr and a dispersion
$\sigma_{\mathrm{t}}=0.3$ ($\mathcal{R}_{23}$). A common feature of these
models is the decreasing density of mergers beyond z $\approx 0.8$. This is
due to the fact that the temporal delay between the GRB and the merger
transforms the peak of GRB production (at z $\approx 2-3$) into a peak of
merger production at z $\approx 0.5$.

Regarding models which are not-favored (rejected or simply acceptable), they
all have values of $<\mathrm{N/N_{max}}>$ below $0.5$, indicating that they
predict too many distant BBH mergers, compared with GW observations. This is
well illustrated by the cumulative distribution function of
$\mathrm{N/N_{\mathrm{max}}}$ for models $\mathcal{R}_{4}$ or
$\mathcal{R}_{11}$ in Fig. 3.

Figure 3: Comparison of $\mathrm{N/N_{\mathrm{max}}}$ cumulative distributions
with the expected $\mathcal{U}(0,1)$ distribution. The shaded areas represent
the 1% and 10% acceptance regions for the KS test. $\mathcal{R}_{1}$
represents the constant model, $\mathcal{R}_{4}$ the GRB model from Salvaterra
et al. (2012), $\mathcal{R}_{11}$ the GRB model from Lien et al. (2014)
convolved with a PL delay function (with $\alpha=-2$ and
$dT_{\mathrm{min}}=0.01\ \mathrm{Gyr}$), $\mathcal{R}_{18}$ the SFH from Li
(2008) with a constant delay of $6.5\ \mathrm{Gyr}$ and $\mathcal{R}_{25}$ the
GRB model from Palmerio & Daigne (2021) convolved with a lognormal delay
function (with $\mathrm{t_{d}}=7.3\ \mathrm{Gyr}$ and
$\sigma_{\mathrm{t}}=0.3$).  Figure 4: Evolution of the p-value with the
delay, for various delay functions applied to the three GRB world models
considered in the paper (blue, red and green for respectively Lien et al.
2014, Salvaterra et al. 2012 and Palmerio & Daigne 2021) and the SFH from Li
(2008) in orange. The shaded light-grey and dark-grey areas represent the
thresholds p-values for the accepted and favored models, respectively set to
1% and 10%.

Figure 4 provides an insight into the impact of the delay for the three delay
functions studied here, it emphasizes some trends: (i) SFH or GRB populations
with no or short delay are not favored; (ii) When a delay is included, the
three delay functions can reproduce the observed distribution if the delay is
sufficiently large: larger than $\sim 6$ Gyr for the constant and lognormal
delay functions, and $dT_{\mathrm{min}}$ larger than $\sim 4$ Gyr for the
power-law delay function; (iii) Delays with a broad distribution (e.g. a
lognormal distribution with $\sigma_{\mathrm{t}}=1$ or a power-law
distribution with $dT_{\mathrm{min}}$ smaller than 4 Gyr) are not favored, and
finally, (iv) the crucial parameter for the power-law distribution is
$dT_{\mathrm{min}}$, the slope playing a marginal role.

###  2.3 Impact of systematics

These conclusions rely on the accuracy of the redshift measurement by LVC and
on the calculation of the merger horizon.

Figure 5: Effect of the redshift uncertainties on the p-values obtained for
the selected models. The range displayed is the 90% confidence interval on the
p-value. The shaded grey area represents the threshold below which models are
rejected, set to 1%

The effect of redshift uncertainties on the p-values has been investigated
using the redshift limits given in GWTC-1, GWTC-2 and GWTC-3. The redshifts of
BBH mergers are poorly constrained in these catalogs, with a relative median
redshift uncertainty of 39%. We have randomized the redshift values given in
the catalog (assuming that the incertitude on the redshift could be modeled by
two half-Gaussian distributions whose standard deviations are equal to the 90%
confidence interval divided by 1.65) and tested $10,000$ times the p-value
obtained for each model. Figure 5 shows the 90% confidence interval obtained
for the p-values of a given model. When the redshift uncertainty is taken into
account, the accepted models $\mathcal{R}_{5}$ $\mathcal{R}_{7}$,
$\mathcal{R}_{9}$, $\mathcal{R}_{13}$, $\mathcal{R}_{19}$, $\mathcal{R}_{20}$,
$\mathcal{R}_{27}$ and $\mathcal{R}_{29}$ can be rejected for some cases of
redshift, while models $\mathcal{R}_{6}$ and $\mathcal{R}_{28}$ become
acceptable. These models are referred as marginally acceptable in the
remaining of the paper. The list of favored models does not change, except for
model $\mathcal{R}_{15}$, $\mathcal{R}_{18}$ and $\mathcal{R}_{21}$ that
becomes only acceptable for some cases. These models are referred as
marginally favored in the remaining of the paper.

We conclude that our study is marginally sensitive to statistical
uncertainties on the measured merger redshifts. The findings of this article,
discussed in the next section, are mainly limited by the number of BBH mergers
detected until now.

##  3 The rate of BBHs

In this section, we evaluate the rate density of BBH merger progenitors
necessary to produce the BBH mergers density rate observed by LVC, considering
favored and marginally favored models identified in Sec. 2 (with a p-value
larger than 10%). This density rate will be expressed in terms of the LGRBs
density rate, the models using the SFH density rate evolution are therefore
not considered here. Using the LGRB density rate as a reference does not imply
a connection between LGRBs and BBH mergers, this is just a convenient way of
specifying the density rate. We defer to section 4 the discussion of the
possible connections between the two populations.

###  3.1 Methodology

Following the approach of Equ. 9, we write the expected rate of BBHs mergers
$\rho(z_{0})$ as a function of their birth rate
$\mathcal{R}_{\mathrm{BBHs}}(z)$ and the normalized delay distribution
$f(z,z_{0})$:

$\rho(z_{0})=\int_{\mathrm{z_{0}}}^{\infty}\mathcal{R}_{\mathrm{BBHs}}(z)\
f(z,z_{0})\frac{\mathrm{d}T_{c}}{\mathrm{d}z}\mathrm{d}z$ (13)

The best measurement of the BBH merger rate density in GWTC-3 (The LIGO
Scientific Collaboration et al., 2021a) occurs at $z=0.2$. For this reason,
the equation above will be computed at $z_{0}=0.2$ to get an estimation of the
fraction as precise as possible. Using the LGRB rate as a reference, the BBH
birth rate $\mathcal{R}_{\mathrm{BBHs}}(z)$ can be linked to the observed
LGRBs rate using two factors, $f_{\mathrm{b}}$ and $\eta_{0}$:

$\mathcal{R}_{\mathrm{BBHs}}(z)=\eta_{0}\times
f_{\mathrm{b}}\times\mathcal{R}_{\mathrm{GRB}}(z)$ (14)

The beaming factor $f_{\mathrm{b}}$ represents the ratio of total number of
LGRBs to the number of detected LGRBs pointing towards us. It is sometimes
expressed relatively to the jet opening angle $\theta$, as
$f_{\mathrm{b}}=(1-\cos\theta)^{-1}$. It is assumed to be independent of
redshift.

The parameter $\eta_{0}$ is a normalization factor to be applied to LGRBs
density rate to reproduce the observed rate of BBH mergers. $\eta_{0}$ is the
number that we aim to constrain in this section, as it permits to compare the
relative rates of BBH mergers and LGRBs. $\eta_{0}$ has been assumed constant
and independent of redshift in this analysis. However, it is also possible to
assume a difference in evolution between massive BBHs and GRBs, favoring or
disfavoring the production of BBH systems relatively to the LGRBs. This
evolution can follow a simple parametrization of the density population (e.g.
$\eta(z)=\eta_{0}(1+z)^{0.6}$, similarly as in Salvaterra et al. 2012). It can
also be assumed that the BBHs production is favored at low metallicity,
meaning that the evolution of $\eta(z)$ is inversely proportional to the
metallicity evolution ($\eta(z)=\eta_{0}10^{0.15z}$, using the metallicity
evolution from Li 2008). In both models, this factor boosts the BBHs density
evolution by a factor $\times 2$ at $z=2$ compared to LGRBs.

Given the two equations above, it is possible to compute the fraction
$\eta_{0}$ as:

$\eta_{0}=\frac{\rho(z_{0})}{f_{\mathrm{b}}}\times\left(\int_{z_{0}}^{\infty}\mathcal{R}_{\mathrm{GRB}}(z)\
f(z,z_{0})\frac{\mathrm{d}T_{c}}{\mathrm{d}z}\mathrm{d}z\right)^{-1}$ (15)

###  3.2 Results

Figure 6: The rate of BBH mergers in units of the LGRB rate, for the 7 favored
GRB models. The error reflects the incertitude on the value of
$\rho(z_{0}=0.2)$. The marker shapes represent combinations of beaming factors
($f_{\mathrm{b}}=70,250$).

Figure 6 shows the values of $\eta_{0}$ calculated for the 7 favored models in
the previous section, for two values of the GRB beaming factor,
$f_{\mathrm{b}}=250$ and $f_{\mathrm{b}}=70$, respectively corresponding to
jet opening angles of about $5$ and $9$ degrees. In the rest of the
discussion, we consider $f_{\mathrm{b}}=250$ as a typical value (Goldstein et
al., 2016; Tsvetkova et al., 2017; Lamb et al., 2021), while
$f_{\mathrm{b}}=70$ is to be considered as a lower bound to the beaming
factor. For the BBH mergers rate $\rho(z_{0})$, we adopt the value given in
the GWTC-3 catalog (The LIGO Scientific Collaboration et al., 2021a),
$\rho(z_{0}=0.2)=19-41~{}\mathrm{Gpc}^{-3}~{}\mathrm{yr}^{-1}$. Taking into
account the uncertainties on LVC BBH mergers local density rate, equation 15
gives a large range of predictions for $\eta_{0}$, ranging from $\sim 0.01$ to
$\sim 1$ (see Fig. 6). Considering the beaming factor $f_{\mathrm{b}}=250$,
the value of $\eta_{0}$ is typically of few percents for models
$\mathcal{R}_{16}$, $\mathcal{R}_{21}$, $\mathcal{R}_{23}$ and
$\mathcal{R}_{25}$, going up to ten percents for models $\mathcal{R}_{15}$ and
$\mathcal{R}_{17}$ and as low as few tenths of a percent for model
$\mathcal{R}_{24}$. The values for a beaming factor $f_{\mathrm{b}}=70$ are
about three times larger. We note that for models $\mathcal{R}_{15}$,
$\mathcal{R}_{16}$ and $\mathcal{R}_{17}$, the delay between the GRB and the
merger follows a power-law with DTmin = 5 Gyr, meaning that there is no merger
during 5 Gyr after a LGRB. These models, while acceptable from a statistical
point of view, seem little realistic from an astrophysical point of view.
Figure 6 also shows that this ratio is below $1$ for all acceptable models and
$f_{\mathrm{b}}$ larger than 70. If BBH mergers and LGRBs come from the same
BH population, BBH mergers must be rarer than LGRBs. Considering
$f_{\mathrm{b}}=250$ as previously justified, BHs in BBH mergers may represent
at most a few percent of those produced in LGRBs. The vast majority of BHs
produced in LGRBs do not end their lives in BBH mergers.

##  4 Discussion

###  4.1 Constraints on the BBH merger population

Section 2 shows that BBH mergers and LGRBs cannot have the same formation
history, unless BBH mergers have a long coalescence time of several Gyr. We
briefly discuss this assumption here. Since the LGRB formation rate peaks
about 2-3 Gyr after the big bang, models that consider a delay of
$t_{\mathrm{d}}$ Gyr between the GRB and the merger predict a peak of mergers
at 3+$t_{\mathrm{d}}$ Gyr after the big bang. For values of $t_{\mathrm{d}}$
smaller than 5 Gyr, this peak occurs beyond the horizon of GW interferometers,
leading to a rate of BBHs that steadily increases with the redshift in the
range of current GW interferometers ; such models are not compatible with the
data. On the other hand, models with a longer delay ($t_{\mathrm{d}}\approx
5-6$ Gyr), which lead to a peak of BBH mergers formation at redshift $z\approx
0.5-0.6$, appear favored by the data. These models imply a decrease of the BBH
mergers beyond the peak (Fig. 2c), providing a direct mean to test the
assumption that BBH mergers and LGRBs have progenitors with the same formation
history.
Most evolutionary models of BBH mergers, however, predict a delay function
following a power-law with steep index $\alpha$ and low $dT_{\mathrm{min}}$,
with a mean delay time close to $\sim 1\ \mathrm{Gyr}$ (e.g. Dominik et al.,
2012), even if other studies suggest that a delay time of several Gyr might
also be possible for BH-BH systems (Peters, 1964; Abbott et al., 2016;
Broekgaarden et al., 2021). Since coalescence times of several Gyr are hardly
credible, we conclude that BBH mergers and LGRBs do not have the same
formation history. This conclusion needs, of course, to be confirmed with
larger BBH merger samples, which will become available in a near future, with
runs O4 and O5.

In Sec. 3, we compare the rate of BBH mergers measured by the LVC
collaboration with the rate of LGRBs. Assuming again that BBH mergers and
LGRBs have the same formation history, we show that BBH mergers are rarer than
LGRBs: taking for example the model $\mathcal{R}_{25}$ and a standard value
for the beaming factor $f_{\mathrm{b}}=250$, the normalization factor
$\eta_{0}$ is between $2$ and $4$%. If the two populations share the same
formation history, several percent of the LGRBs suffice to explain the
measured rate of BBH mergers, and if not, this is an indication of a lack of
connection between LGRBs and BBH mergers. In both cases, the large majority of
the BHs born in LGRBs do not end their lives in BBH mergers.
Taken together these two studies suggest that BBH mergers and LGRBs have
distinct progenitors, and that LGRBs are more frequent than BBH mergers.

###  4.2 The nature of LGRB progenitors

One of the surprises of LVC detections was the dominance of BHs with a low
spin. This tells us that the cores of massive stars have lost most of their
angular momentum when they collapse into a BH. The core angular momentum has
been transferred to the stellar envelope which has been ejected. This is the
indication of an efficient transport of angular momentum in the progenitors of
BBH mergers and possibly in most massive stars (Olejak & Belczynski, 2021;
Bavera et al., 2022; Fuller & Lu, 2022). If this scenario is correct, LGRBs,
which require stellar cores with a large angular momentum at BH creation,
cannot be emitted by single massive stars, nor by massive stars in wide binary
systems. According to Kushnir et al. (2017); Piran & Piran (2020); Bavera et
al. (2020); Belczynski et al. (2020); Marchant et al. (2021) the best way for
a massive stellar core to keep a large angular momentum until it becomes a BH
is through tidal spin-up in a compact system, which is bound to merge after
some time. In this model, the population of LGRBs and BBHs (or at least a sub-
sample of the latter) would therefore share common progenitors.

Bavera et al. (2022) have recently published a study using binary stellar
evolution and population synthesis calculations to explain the sub-population
of spinning merging BBHs and the emission of LGRBs at the BH formation. They
reach the conclusion that the progenitors of fast-spinning BBH mergers, formed
via isolated binary evolution, are likely a major contribution to the observed
rate of luminous LGRBs.

This conclusion differs from the results obtained in this paper, as we propose
different origins for BBH mergers and LGRBs. This tension can be explained as
follow:

  * •

The population of BH progenitors generated by Bavera et al. (2022) for the
production of LGRBs is based on a value close to the upper limit of the local
rate measured by LVC,
$\rho(z_{0}=0)=38.3~{}\mathrm{Gpc}^{-3}~{}\mathrm{yr}^{-1}$. We have taken the
logarithmic mean of the upper- and lower- limits
($\rho(z_{0}=0.2)=28~{}\mathrm{Gpc}^{-3}~{}\mathrm{yr}^{-1}$), which is
smaller.

  * •

The SHOALS survey used as a reference for the LGRBs distribution in Bavera et
al. (2022) gives a local rate of observed GRBs $\sim
0.2~{}\mathrm{Gpc^{-3}~{}yr^{-1}}$, while the ones used in this papers are
larger, and closer to $1~{}\mathrm{Gpc^{-3}~{}yr^{-1}}$.

  * •

Bavera et al. (2022) use a beaming fraction of $0.05$ (corresponding to a
beaming factor of $f_{\mathrm{b}}=20$ or a jet opening angle of
$\theta_{\mathrm{j}}\sim 18$°), while we consider a larger beaming factor, by
a factor $\sim 3-12$. Goldstein et al. (2016) show that the distribution of
jet opening angles derived from the prompt emission of 638 GRBs observed by
the Fermi Gamma-ray Burst Monitor (GBM) peaks at $\sim 6$° or
$f_{\mathrm{b}}=180$, and Tsvetkova et al. (2017) find a median opening angle
of $\theta_{\mathrm{j}}\sim 4$° ($f_{\mathrm{b}}=400$), based on breaks in the
afterglows of 43 bright GRBs detected by KONUS. Independent measurements
obtained by Lamb et al. (2021) on long GRB afterglows also seem to point
towards a jet opening angle around $\sim 5$° ($f_{\mathrm{b}}=260$). As the
beaming factor is directly connected to the true GRB rate, the different
values taken by Bavera et al. (2022) and us explain our different results.

These three factors, which all go in the same direction, lead to a ratio of
$\sim 100$ between the long GRB rate used by Bavera et al. (2022) and the one
we use. This explains why we find that at most a small fraction of LGRBs could
be associated to BBH mergers, while Bavera et al. (2022) propose to associate
all long GRBs with the sub-population of BBH mergers with non-zero spin. In
the rest of this discussion we consider that our LGRB models (including the
beaming factor) are realistic, and we discuss our findings in this context.

If the majority of LGRBs do not end their lives in BBH mergers, they cannot be
produced in compact binary systems and must be emitted by single stars (e.g.
Aguilera-Dena et al., 2020) or in binary systems that do not merge. It may be
natural to associate them with systems like the X-ray Binaries (LMXBs, HMXBs)
found in the galaxy, such as Cygnus X-1 (Krawczynski & Beheshtipour, 2022;
Zhao et al., 2021), GRS 1915+105 (Shreeram & Ingram, 2020) or EXO 1846-031
(Draghis et al., 2020) which are rapidly spinning and could be the descendants
of LGRBs (although Belczynski et al. (2021) show that BH spins determination
can be subject to systematic effects). The detection of such systems,
especially HMXBs for which accretion spin-up is limited by the lifetime of the
secondary star (e.g. Wong et al., 2012), requires the birth of highly spinning
BHs (Qin et al., 2019; Miller-Jones et al., 2021), an expected characteristics
of LGRB descendants. Nonetheless, it is also possible that LGRB descendants
are partitioned in several populations, X-ray binaries being only a subset of
them.

###  4.3 Short-lived LGRB descendants in the BBH merger population ?

While our analysis suggests that the majority of BBH mergers have no direct
connection with LGRBs, it does not preclude the existence of a small fraction
of GRB descendants, with non-zero spin and short merging times, among BBH
mergers. Such systems could be produced during a phase of tidal spin-up in a
compact system of two massive stars (or a BH and a massive star) as proposed
by Bavera et al. (2022). We show here that the addition of a minority fraction
of short-lived GRB descendants to a favored model does not change its
acceptability.
For illustration we have constructed a mixed model, made of 85% of sources
following model $\mathcal{R}_{25}$ and 15% of sources following the GRB
distribution of Palmerio & Daigne (2021) (model $\mathcal{R}_{5}$). The
fraction used roughly corresponds to the fraction of BBH mergers in our sample
with a $\chi_{\mathrm{eff}}$ consistent with being positive. This mixed model,
shown in Figure 7, is acceptable for the $\mathrm{N/N_{\mathrm{max}}}$ test (p
= 0.90). It requires the following fractions of the GRB population to explain
the two components of the model: 3% for sources following the
$\mathcal{R}_{25}$ distribution and 2% for sources following the Palmerio &
Daigne (2021) distribution ($\mathcal{R}_{5}$).

Figure 7: Illustration of the potential two-population: the rate density
presented on top is composed at $85$% by a high-delay (low-spin) population
and $15$% by a low-delay (high-spin) population, the sum of the two
illustrated by the dashed grey line. The bottom panel illustrates the
evolution of the high-spin fraction of BBHs mergers relatively to redshift.

The addition of a sub-population of GRB descendants with short merging times
has an interesting consequence: this population becomes dominant beyond a
certain redshift ($z\approx 0.7$ in Fig. 7). This is the consequence of the
weak evolution of BBH mergers (cf. Sec. 2) compared with the strong evolution
of GRBs. This provides a test of the existence of a fraction of direct GRB
descendants with short merging times among BBH mergers. If this population
exists, it may become dominant at intermediate redshifts and start to be
detectable in runs O4 and O5.

Another prediction of having GRBs exploding in compact binary systems is the
possibility of precessing black holes and precessing GRB jets, that may leave
an imprint on the prompt emission or the afterglow of some LGRBs. This
possibility has been discussed by various authors, in different contexts
(Blackman et al., 1996; Fargion & Grossi, 2006; Huang & Liu, 2021).

##  5 Conclusion and perspectives

With the increasing number of stellar mass BH detections, population studies
become effective for the comparison of their various sub-classes. Section 2
shows the comparison between the BBH merger population and a BH population
traced by LGRBs, showing that their redshift distributions are compatible only
if we consider a delay of several Gyr between the occurrence of the LGRB and
the BBH merger. Section 3 shows that for most of the compatible models, the
BBH merger progenitors rate density represents less than $10$% of the LGRBs
rate density. These two results are interpreted in Sec. 4.1 as an indication
that BBH mergers and LGRBs have distinct progenitors, with the proposal that
the descendants of LGRBs could resemble more BHs found in galactic X-ray
binaries than those of BBH mergers (Sec. 4.2).

After the publication of GWTC-2, various authors have underlined the fact that
BBH mergers may not be a single population (e.g. Bouffanais et al., 2021;
Hütsi et al., 2021; Zevin et al., 2021), but a mix of BH populations
originating from isolated stellar binaries, dynamic mergers, and possibly
primordial BHs. This claim, however, has been challenged by other authors
(e.g. Belczynski et al., 2022; Broekgaarden et al., 2021). The same is true
for LGRBs, which can have different “engines”, a black hole or a fast rotating
magnetar (e.g. Bernardini, 2015), and which encompass various sub-classes like
X-ray Flashes, low-luminosity GRBs, choked GRBs, ultra-LGRBs or LGRBs with a
plateau in their afterglow. Future samples of BBH mergers with better
statistics will permit to sort out these sub-populations, which are expected
to have different mass, spin and redshift distributions. It will then be
possible to correlate GRBs with theoretically or empirically defined BBH
merger sub-classes (e.g. BBH mergers with non-zero effective spin, within a
given mass range), getting more insight into the physical connections between
these phenomena.

In this context, larger samples of BBH mergers and GRBs with a redshift
originating from various sub-classes will permit population studies of these
sub-classes, providing insight into the zoo of astrophysical BHs. Section 4.3
has shown for example a test for the existence of a sub-population of BBH
mergers that would be descendants from LGRBs. Larger samples of BBH mergers
are expected with the increasing sensitivity of GW interferometers (Abbott et
al., 2018), and larger samples of GRBs with a redshift will be provided by
current and future GRB missions doing fast localizations like Swift (Gehrels
et al., 2004; Barthelmy et al., 2005), Fermi (Meegan et al., 2009), INTEGRAL
(Winkler et al., 2003; Lebrun et al., 2003; Mereghetti et al., 2003), GECAM
(Lv et al., 2018) or the coming SVOM (Wei et al., 2016; Godet et al., 2014;
Arcier et al., 2020).

In parallel with these studies, it is essential to develop models of stellar
evolution designed to follow the evolution of massive stars up to the
production of BHs and their manifestations like GRBs, and BBH mergers (see for
instance Bavera et al., 2022). Only with such models guiding us, we will be
able to get all the knowledge that can be extracted from the diversity of BH
manifestations that are being revealed by present day instruments.

The authors thank the referee for helpful comments on the evolution of binary
systems of massive stars that contributed to improve this paper. The authors
are grateful to N. Leroy for helpful discussions regarding the operations of
the LIGO and Virgo GW detectors. This work is supported by CNES and the Paul
Sabatier University.

##  Appendix A Horizon redshift calculation

Table LABEL:tab:z_tab contains the BBH mergers properties used to obtain the
results presented in this paper.

Table 2: BBHs mergers used in this paper from Abbott et al. (2019, 2021); The LIGO Scientific Collaboration et al. (2021b), with the original SNR and the calculated horizon redshift $z_{\mathrm{h}}$ thanks to Equation 3. Name | $m_{1}\left(\mathrm{M_{\odot}}\right)$ | $m_{2}\left(\mathrm{M_{\odot}}\right)$ | SNR | z | $z_{\mathrm{h}}$ | $V_{\mathrm{max}}/V$
---|---|---|---|---|---|---
GW200316_215756◆ | 13.1 | 7.8 | 10.3 | $0.22^{+0.08}_{-0.08}$ | $0.28^{+0.10}_{-0.10}$ | 2.05
GW200311_115853◆ | 34.2 | 27.7 | 17.8 | $0.23^{+0.05}_{-0.07}$ | $0.53^{+0.12}_{-0.17}$ | 9.59
GW200302_015811◆ | 37.8 | 20.0 | 10.8 | $0.28^{+0.16}_{-0.12}$ | $0.38^{+0.22}_{-0.17}$ | 2.33
GW200225_060421◆ | 19.3 | 14.0 | 12.5 | $0.22^{+0.09}_{-0.10}$ | $0.35^{+0.15}_{-0.16}$ | 3.55
GW200224_222234◆ | 40.0 | 32.5 | 20.0 | $0.32^{+0.08}_{-0.11}$ | $0.86^{+0.24}_{-0.33}$ | 12.34
GW200220_124850◆ | 38.9 | 27.9 | 8.5 | $0.66^{+0.36}_{-0.31}$ | $0.71^{+0.39}_{-0.33}$ | 1.18
GW200219_094415◆ | 37.5 | 27.9 | 10.7 | $0.57^{+0.22}_{-0.22}$ | $0.79^{+0.32}_{-0.32}$ | 2.19
GW200216_220804◆ | 51.0 | 30.0 | 8.1 | $0.63^{+0.37}_{-0.29}$ | $0.64^{+0.37}_{-0.29}$ | 1.03
GW200210_092254◆ | 24.1 | 2.8 | 8.4 | $0.19^{+0.08}_{-0.06}$ | $0.20^{+0.08}_{-0.06}$ | 1.14
GW200209_085452◆ | 35.6 | 27.1 | 9.6 | $0.57^{+0.25}_{-0.26}$ | $0.70^{+0.32}_{-0.33}$ | 1.64
GW200208_130117◆ | 37.8 | 27.4 | 10.8 | $0.40^{+0.15}_{-0.14}$ | $0.55^{+0.21}_{-0.20}$ | 2.30
GW200202_154313◆ | 10.1 | 7.3 | 10.8 | $0.09^{+0.03}_{-0.03}$ | $0.12^{+0.04}_{-0.04}$ | 2.41
GW200129_065458◆ | 34.5 | 28.9 | 26.8 | $0.18^{+0.05}_{-0.07}$ | $0.64^{+0.20}_{-0.28}$ | 30.53
GW200128_022011◆ | 42.2 | 32.6 | 10.6 | $0.56^{+0.28}_{-0.28}$ | $0.76^{+0.39}_{-0.39}$ | 2.13
GW200112_155838◆ | 35.6 | 28.3 | 19.8 | $0.24^{+0.07}_{-0.08}$ | $0.62^{+0.20}_{-0.23}$ | 12.76
GW191230_180458◆ | 49.4 | 37.0 | 10.4 | $0.69^{+0.26}_{-0.27}$ | $0.92^{+0.35}_{-0.36}$ | 1.95
GW191222_033537◆ | 45.1 | 34.7 | 12.5 | $0.51^{+0.23}_{-0.26}$ | $0.83^{+0.39}_{-0.44}$ | 3.28
GW191216_213338◆ | 12.1 | 7.7 | 18.6 | $0.07^{+0.02}_{-0.03}$ | $0.16^{+0.05}_{-0.07}$ | 11.78
GW191215_223052◆ | 24.9 | 18.1 | 11.2 | $0.35^{+0.13}_{-0.14}$ | $0.50^{+0.19}_{-0.21}$ | 2.56
GW191204_171526◆ | 11.9 | 8.2 | 17.5 | $0.13^{+0.04}_{-0.05}$ | $0.29^{+0.09}_{-0.11}$ | 9.48
GW191204_110529◆ | 27.3 | 19.3 | 8.8 | $0.34^{+0.25}_{-0.18}$ | $0.38^{+0.28}_{-0.20}$ | 1.31
GW191129_134029◆ | 10.7 | 6.7 | 13.1 | $0.16^{+0.05}_{-0.06}$ | $0.26^{+0.08}_{-0.10}$ | 4.11
GW191127_050227◆ | 53.0 | 24.0 | 9.2 | $0.57^{+0.40}_{-0.29}$ | $0.66^{+0.47}_{-0.34}$ | 1.45
GW191126_115259◆ | 12.1 | 8.3 | 8.3 | $0.30^{+0.12}_{-0.13}$ | $0.31^{+0.13}_{-0.14}$ | 1.11
GW191109_010717◆ | 65.0 | 47.0 | 17.3 | $0.25^{+0.18}_{-0.12}$ | $0.55^{+0.39}_{-0.26}$ | 8.22
GW191105_143521◆ | 10.7 | 7.7 | 9.7 | $0.23^{+0.07}_{-0.09}$ | $0.28^{+0.09}_{-0.11}$ | 1.73
GW191103_012549◆ | 11.8 | 7.9 | 8.9 | $0.20^{+0.09}_{-0.09}$ | $0.22^{+0.10}_{-0.10}$ | 1.36
GW190930_133541■ | 12.3 | 7.8 | 9.8 | $0.15^{+0.06}_{-0.06}$ | $0.18^{+0.07}_{-0.07}$ | 1.82
GW190929_012149■ | 80.8 | 24.1 | 9.9 | $0.38^{+0.49}_{-0.17}$ | $0.47^{+0.59}_{-0.21}$ | 1.74
GW190926_050336■ | 39.8 | 23.2 | 9.0 | $0.62^{+0.40}_{-0.29}$ | $0.71^{+0.47}_{-0.34}$ | 1.37
GW190925_232845■ | 21.2 | 15.6 | 9.9 | $0.19^{+0.07}_{-0.07}$ | $0.24^{+0.09}_{-0.09}$ | 1.83
GW190924_021846■ | 8.9 | 5.0 | 13.2 | $0.12^{+0.04}_{-0.04}$ | $0.20^{+0.07}_{-0.07}$ | 4.23
GW190916_200658■ | 44.3 | 23.9 | 8.2 | $0.71^{+0.46}_{-0.36}$ | $0.73^{+0.47}_{-0.37}$ | 1.07
GW190915_235702■ | 35.3 | 24.4 | 13.1 | $0.30^{+0.11}_{-0.10}$ | $0.50^{+0.19}_{-0.18}$ | 3.96
GW190910_112807■ | 43.9 | 35.6 | 13.4 | $0.28^{+0.16}_{-0.10}$ | $0.48^{+0.29}_{-0.18}$ | 4.27
GW190909_114149■ | 45.8 | 28.3 | 9.0 | $0.62^{+0.41}_{-0.33}$ | $0.71^{+0.47}_{-0.38}$ | 1.39
GW190828_065509■ | 24.1 | 10.2 | 11.1 | $0.30^{+0.10}_{-0.10}$ | $0.42^{+0.14}_{-0.14}$ | 2.50
GW190828_063405■ | 32.1 | 26.2 | 16.0 | $0.38^{+0.10}_{-0.15}$ | $0.81^{+0.24}_{-0.36}$ | 6.76
GW190814■ | 23.2 | 2.6 | 22.2 | $0.05^{+0.01}_{-0.01}$ | $0.14^{+0.02}_{-0.03}$ | 19.29
GW190805_211137■ | 48.2 | 32.0 | 8.3 | $0.82^{+0.48}_{-0.40}$ | $0.85^{+0.50}_{-0.41}$ | 1.08
GW190803_022701■ | 37.3 | 27.3 | 8.6 | $0.55^{+0.26}_{-0.24}$ | $0.60^{+0.29}_{-0.26}$ | 1.22
GW190731_140936■ | 41.5 | 28.8 | 8.5 | $0.55^{+0.31}_{-0.26}$ | $0.58^{+0.33}_{-0.28}$ | 1.16
GW190728_064510■ | 12.3 | 8.1 | 13.6 | $0.18^{+0.05}_{-0.07}$ | $0.31^{+0.09}_{-0.12}$ | 4.58
GW190727_060333■ | 38.0 | 29.4 | 12.3 | $0.55^{+0.21}_{-0.22}$ | $0.89^{+0.36}_{-0.38}$ | 3.16
GW190725_174728■ | 11.5 | 6.4 | 9.1 | $0.21^{+0.10}_{-0.09}$ | $0.24^{+0.11}_{-0.10}$ | 1.43
GW190720_000836■ | 13.4 | 7.8 | 11.7 | $0.16^{+0.12}_{-0.06}$ | $0.23^{+0.18}_{-0.09}$ | 2.97
GW190708_232457■ | 17.6 | 13.2 | 13.1 | $0.18^{+0.06}_{-0.07}$ | $0.30^{+0.10}_{-0.12}$ | 4.07
GW190707_093326■ | 11.6 | 8.4 | 13.0 | $0.16^{+0.07}_{-0.07}$ | $0.26^{+0.12}_{-0.12}$ | 4.00
GW190706_222641■ | 67.0 | 38.2 | 12.3 | $0.71^{+0.32}_{-0.27}$ | $1.10^{+0.45}_{-0.38}$ | 2.70
GW190701_203306■ | 53.9 | 40.8 | 11.6 | $0.37^{+0.11}_{-0.12}$ | $0.54^{+0.16}_{-0.18}$ | 2.75
GW190630_185205■ | 35.1 | 23.7 | 15.6 | $0.18^{+0.10}_{-0.07}$ | $0.36^{+0.21}_{-0.15}$ | 6.81
GW190620_030421■ | 57.1 | 35.5 | 10.9 | $0.49^{+0.23}_{-0.20}$ | $0.68^{+0.32}_{-0.28}$ | 2.28
GW190602_175927■ | 69.1 | 47.8 | 12.1 | $0.47^{+0.25}_{-0.17}$ | $0.71^{+0.36}_{-0.25}$ | 2.86
GW190527_092055■ | 36.5 | 22.6 | 8.9 | $0.44^{+0.34}_{-0.20}$ | $0.49^{+0.39}_{-0.23}$ | 1.33
GW190521_074359■ | 42.2 | 32.8 | 24.4 | $0.24^{+0.07}_{-0.10}$ | $0.78^{+0.25}_{-0.36}$ | 22.02
GW190521■ | 95.3 | 69.0 | 14.4 | $0.64^{+0.28}_{-0.28}$ | $1.08^{+0.40}_{-0.40}$ | 3.36
GW190519_153544■ | 66.0 | 40.5 | 12.0 | $0.44^{+0.25}_{-0.14}$ | $0.67^{+0.37}_{-0.21}$ | 2.90
GW190517_055101■ | 37.4 | 25.3 | 10.2 | $0.34^{+0.24}_{-0.14}$ | $0.44^{+0.32}_{-0.19}$ | 2.00
GW190514_065416■ | 39.0 | 28.4 | 8.3 | $0.67^{+0.33}_{-0.31}$ | $0.70^{+0.35}_{-0.33}$ | 1.10
GW190513_205428■ | 35.7 | 18.0 | 12.3 | $0.37^{+0.13}_{-0.13}$ | $0.58^{+0.22}_{-0.22}$ | 3.25
GW190512_180714■ | 23.3 | 12.6 | 12.3 | $0.27^{+0.09}_{-0.10}$ | $0.42^{+0.14}_{-0.16}$ | 3.31
GW190503_185404■ | 43.3 | 28.4 | 12.1 | $0.27^{+0.11}_{-0.11}$ | $0.41^{+0.17}_{-0.17}$ | 3.21
GW190426_190642■ | 106.9 | 76.6 | 9.6 | $0.70^{+0.41}_{-0.30}$ | $0.82^{+0.46}_{-0.34}$ | 1.45
GW190424_180648■ | 40.5 | 31.8 | 10.1 | $0.39^{+0.23}_{-0.19}$ | $0.50^{+0.30}_{-0.25}$ | 1.91
GW190421_213856■ | 41.3 | 31.9 | 10.6 | $0.49^{+0.19}_{-0.21}$ | $0.66^{+0.27}_{-0.29}$ | 2.12
GW190413_134308■ | 47.5 | 31.8 | 9.0 | $0.71^{+0.31}_{-0.30}$ | $0.81^{+0.36}_{-0.34}$ | 1.37
GW190413_052954■ | 34.7 | 23.7 | 8.6 | $0.59^{+0.29}_{-0.24}$ | $0.63^{+0.32}_{-0.26}$ | 1.20
GW190412■ | 30.1 | 8.3 | 18.9 | $0.15^{+0.03}_{-0.03}$ | $0.35^{+0.07}_{-0.07}$ | 11.12
GW190408_181802■ | 24.6 | 18.4 | 14.7 | $0.29^{+0.06}_{-0.10}$ | $0.55^{+0.12}_{-0.20}$ | 5.48
GW170823O | 39.5 | 29.0 | 11.5 | $0.35^{+0.15}_{-0.15}$ | $0.51^{+0.23}_{-0.23}$ | 2.75
GW170818O | 35.4 | 26.7 | 11.3 | $0.21^{+0.07}_{-0.07}$ | $0.30^{+0.10}_{-0.10}$ | 2.69
GW170814O | 30.6 | 25.2 | 15.9 | $0.12^{+0.03}_{-0.04}$ | $0.24^{+0.06}_{-0.08}$ | 7.36
GW170809O | 35.0 | 23.8 | 12.4 | $0.20^{+0.05}_{-0.07}$ | $0.31^{+0.08}_{-0.11}$ | 3.51
GW170729O | 50.2 | 34.0 | 10.2 | $0.49^{+0.19}_{-0.21}$ | $0.63^{+0.25}_{-0.28}$ | 1.92
GW170608O | 11.0 | 7.6 | 14.9 | $0.07^{+0.02}_{-0.02}$ | $0.13^{+0.04}_{-0.04}$ | 6.20
GW170104O | 30.8 | 20.0 | 13.0 | $0.20^{+0.08}_{-0.08}$ | $0.33^{+0.14}_{-0.14}$ | 4.01
GW151226O | 13.7 | 7.7 | 13.1 | $0.09^{+0.04}_{-0.04}$ | $0.15^{+0.07}_{-0.07}$ | 4.22
GW151012O | 23.2 | 13.6 | 10.0 | $0.21^{+0.09}_{-0.09}$ | $0.26^{+0.11}_{-0.11}$ | 1.89
GW150914O | 35.6 | 30.6 | 24.4 | $0.09^{+0.03}_{-0.03}$ | $0.28^{+0.10}_{-0.10}$ | 25.65

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