# The effect of stellar rotation on black hole mass and spin

Sohan Ghodla1, J. J. Eldridge1
1Department of Physics, University of Auckland, Private Bag 92019, Auckland,
New Zealand<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The gravitational wave signature of a binary black hole (BBH) merger is
dependent on its component mass and spin. If such black holes originate from
rapidly rotating progenitors, the large angular momentum reserve in the star
could drive a collapsar-like supernova explosion, hence substantially
impacting these characteristics of the black holes in the binary. To examine
the effect of stellar rotation on the resulting black hole mass and spin, we
conduct a 1D general relativistic study of the end phase of the collapse. We
find that the resulting black hole mass at times differ significantly from the
previously assumed values. We quantify the dependence of the black hole spin
magnitude on the hydrodynamics of the accretion flow, providing analytical
relations for calculating the mass and spin based on the progenitor’s pre-
collapse properties. Depending on the nature of the accretion flow, our
findings have implications for the black hole upper mass gap resulting from
pair-instability supernovae, the maximum mass of a maximally rotating stellar
black hole ($M_{\rm BH,max}\approx 35M_{\odot}$), and the maximum effective
spin of a BBH formed in tidally locked helium star - black hole binary
($\chi_{\rm eff,max}\approx 0.325$).

###### keywords:

accretion disks – gravitational waves – black hole physics - gamma-ray burst:
general

††pubyear: 2023††pagerange: The effect of stellar rotation on black hole mass
and spin–B

##  1 Introduction

The black hole masses resulting from rapidly rotating pre-collapse stars are
not well described by the pre-existing prescriptions that estimate the remnant
mass produced at core-collapse (e.g. Fryer & Kalogera, 2001; Eldridge & Tout,
2004; Fryer et al., 2012) . The most widely used remnant mass functions
(resulting from the supernova explosion) are based on the work of Fryer et al.
(2012). They offer two variations of supernova models to calculate the
resulting compact remnant mass, namely the _Delayed_ and _Rapid_ explosion
mechanisms. Both models rely on convection-enhanced neutrino heating (Herant
et al. 1994; see Fryer et al. 2021 for a recent review) to drive a successful
explosion. The primary difference between them lies in the instability that
could drive such an explosion. The Delayed mechanism is driven by a standing
accretion shock instability (Blondin et al., 2003) and can produce an
explosion with a delay of up to 1s after the bounce, while the Rapid mechanism
utilizes the Rayleigh-Taylor instability (e.g., Sharp 1984), and the explosion
occurs within 0.25s. Under a failed explosion, the Rapid mechanism leads to a
direct collapse, while in the Delayed mechanism, some form of explosion may
still be possible.

On the other hand, for progenitor stars with a substantial reservoir of
angular momentum at core collapse, the supernova explosion might not be
determined by convection and therefore may lead to remnant masses different
from those predicted by the convection paradigm. This is because the angular
momentum reservoir acts as an additional means of centrifugal support and the
subsequent viscous dynamics offer an efficient mechanism for energy
dissipation during the episode of the explosion.

Fryer et al. (2012) assumed that under the scenario of substantial rotation,
the occurrence of relativistic jets during core collapse (e.g., Woosley 1993;
MacFadyen & Woosley 1999) would likely disrupt most of the star and only
$\approx$ 5$M_{\odot}$ black hole would be produced. However, a number of
subsequent studies, although at times working in the paradigm of rapidly
rotating progenitor stars, have either continued to use the Fryer et al.
(2012) remnant mass function (e.g., Bavera et al. 2020; Riley et al. 2021) or
assumed a more or less direct collapse - i.e., a major mass fraction of the
pre-core collapse star ending up in the black hole, with some (typically the
envelope) mass lost during the explosion (e.g., Mandel & de Mink 2016;
Marchant et al. 2016; Qin et al. 2018; du Buisson et al. 2020). This is
primarily due to the lack of a mass function that captures the effect of
stellar rotation on the resulting black hole masses.

Another property of astrophysical black holes is their spin angular momentum,
which, together with their mass, fully determines their property (Israel,
1967; Carter, 1971; Bekenstein, 1972). Although 1D studies exist that take
into account the spin evolution of the black hole during collapse of the star
(e.g., Batta & Ramirez-Ruiz 2019; Bavera et al. 2020; Fuller & Lu 2022), they
either do not account for the effect of wind outflows (e.g., Bavera et al.
2020, a strong magnetic field and/or relativistic dynamics on the nature of
the accretion flow (Batta & Ramirez-Ruiz, 2019; Bavera et al., 2020; Fuller &
Lu, 2022). In this paper, we aim to address these shortcomings.

The collapse of a sufficiently rapidly rotating star should happen via the
formation of an equatorial accretion disk. Such a supernova explosion
paradigm, consisting of a newly formed spinning black hole penetrated by a
strong magnetic field of the surrounding accretion disk, is also known as a
collapsar (Woosley, 1993; MacFadyen & Woosley, 1999) and is believed to be
primarily responsible for the observed long-duration $\gamma$-ray bursts
(GRBs). The dynamics of the accretion flow under a collapsar scenario has been
extensively studied in the past (e.g., Gammie et al. 2004; Lee & Ramirez-Ruiz
2006; Kumar et al. 2008; Tchekhovskoy et al. 2010, 2011; Sądowski et al. 2014;
Sądowski & Narayan 2016; Fujibayashi et al. 2020, 2023; Murguia-Berthier et
al. 2020; Gottlieb et al. 2023; Jacquemin-Ide et al. 2023). The disk evolves
viscously via magnetohydrodynamics, resulting in mass outflow and a transient
energetic electromagnetic outburst. The resulting dynamics could be highly
non-linear, and a detailed study would require a full 3-dimensions magneto-
hydrodynamic simulation (e.g., Sądowski & Narayan 2016). However, conducting
such simulations is computationally expensive and time-consuming.

For the current work, we use a simpler approach based on the earlier work of
Kumar et al. (2008) \- and later numerically implemented in Fuller & Lu
(2022), which is a 1D non-relativistic, height-integrated, time-averaged
formalism, but takes into account the effect of vertical wind outflows. We
extend their work to a general relativistic framework, hence allowing us to
more accurately consider the influence on the accretion flow in the inner
region of the accretion disk around the proto-Kerr black hole. Since our
calculations are time-averaged, most of the non-linearity would be averaged
out. We adopt two variable approaches for treating the accretion flow in the
near-horizon region, namely the Novikov & Thorne (1973) accretion disk
formalism and the magnetically arrested accretion disk (Tchekhovskoy et al.,
2011) formalism. We use this framework in conjunction with the Mesa generated
stellar models to calculate the resulting black hole mass and spin relations
that would be suitable for black holes born from rotating stars. We then
compare our results with the earlier work of Fryer et al. (2012), showcasing
the crucial difference between the two. We also discuss some implications of
our work for the expected birth properties of stellar black holes.

The remainder of this paper is organized as follows. In Section 2, we discuss
details of the Mesa models used in our analysis. Here, we also develop the
formalism used for treating the collapsar dynamics. In Section 3, we present
the mass and spin functions for black holes born from rotating stars and
discuss their domain of validity. Then, in Section 4, we present some
implications of our results, followed by a brief conclusion in Section 5.

##  2 Method

###  2.1 Models for rotating stars

We use Mesa (Paxton et al., 2011; Paxton et al., 2013, 2015, 2018, 2019) to
evolve models of variably rotating single stars in the mass range 5-95
$M_{\odot}$ from the onset of helium burning to the core carbon
depletion111The inlists for the Mesa model, and the Python script consisting
of the numerical implementation of the formalism discussed in Section 2.2 can
be found in the data availability statement.. No substantial change in the
relevant structure (or the angular momentum reservoir) of the star is expected
past core carbon depletion, as is evident from the output of such evolved Mesa
models (e.g., see Fuller & Lu 2022).

Our stellar models exhibit a uniform initial rotation profile at various
angular velocities. In Mesa, rotation is treated in a shellular approximation
(Zahn, 1992; Meynet & Maeder, 1997), implying that isobaric shells have
uniform angular velocity. As the models evolve, they lose mass and hence
angular momentum. In addition, the angular momentum is redistributed
internally, causing the rotation profile to deviate from that of the initial
solid body. In Mesa, the transfer of angular momentum is modeled in a
diffusive approximation. This is accomplished by contribution from ordinary
diffusion, meridional circulations (e.g., Eddington 1925; Sweet 1950), and the
magnetic field instability-induced Spruit-Tayler dynamo (Spruit, 2002).
Although the mapping between the initial and final angular velocities is
nonlinear, we choose a sufficient number of models (900+) such that it results
in stars with different levels of final angular momentum reserves.

We use the Dutch wind mass loss scheme with the Dutch_scaling_factor = 0.25
and consider only a single metallicity with $Z=10^{-4}$. Since the results in
Section 3 are calculated so that they remain only a function of the final mass
and the angular momentum of the collapsing star, the metallicity of the star
is not the variable of interest. However, metallicity would play a direct role
in determining the amount of mass and angular momentum that is retained by the
star. For certain exceptional cases (e.g., those where angular momentum is
sourced from the tidal influence of a binary companion), the angular momentum
content can remain large even at larger metallicity; therefore, our approach
remains valid for such cases as well.

###  2.2 Formalism for collapsar dynamics

Our goal is to determine the final mass and spin of the black hole resulting
from the collapse of a rotating star of a given mass and angular momentum
distribution. For such a purpose, the detailed nature of the collapse and the
dynamics of the evolution of the accretion disk might not be necessary. As
such, we focus on the effective (i.e. time-averaged) dynamics of the accretion
flow, as detailed below. Unless explicitly stated, we use geometric units with
$G,c=1$ throughout this section.

####  2.2.1 Background geometry

As an initial condition, we assume that a natal black hole of some mass
promptly forms during the initial collapse of the stellar core. The spacetime
of this black hole (i.e., ignoring the remaining matter) can be modeled using
the Kerr metric ($g_{ab}$) whose line element in the Boyer-Lindquist
coordinates $(x^{a}:=~{}t,r,\theta,\phi)$ reads

$\displaystyle ds^{2}$
$\displaystyle=g_{ab}dx^{a}dx^{b}=-\left(1-\frac{2Mr}{\rho^{2}}\right)dt^{2}-\frac{4Mar\sin^{2}{\theta}}{\rho^{2}}d\phi
dt$ (1)
$\displaystyle+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}+\left(r^{2}+a^{2}+\frac{2Mra^{2}\sin^{2}\theta}{\rho^{2}}\right)\sin^{2}\theta
d\phi^{2}\,,$

where $a=J/M;\;\Delta=r^{2}-2Mr+a^{2}$ and
$\rho^{2}=r^{2}+a^{2}\cos^{2}\theta$. Here, $M,J$ are the black hole’s ADM
mass and angular momentum, respectively. As the remainder of the stellar
matter falls inwards, the metric evolves due to a change in the black hole
mass and spin.

For simplicity, we assume that at any given time there is a negligible amount
of matter in the equatorial plane such that the effect of the disk mass on the
form of Eq. 1 can be ignored. Additionally, we assume that the outer shells
have a nearly spherical configuration such that the spacetime inside the
innermost infalling shell can be assumed to be flat (except for the effect due
to the black hole). There are two Killing vector fields $(\xi,\eta)$
associated with this stationary, axisymmetric spacetime namely
$\xi=(1,0,0,0)\partial_{t}$ and $\eta=(0,0,0,1)\partial_{\phi}$. They
correspond to the conservation of energy and the axial component of the
angular momentum, respectively.

####  2.2.2 The rotating fluid

The final states of a massive star evolved through Mesa constitute the initial
state of the following analysis. Prior to collapse, we assume (in accordance
with the Mesa model) that the stellar fluid comprises of matter, with only the
axial component of angular momentum being nonzero. Let $u^{a}$ be the
4-velocity of a fluid element of this star with components
($u^{t},u^{r},u^{\theta},u^{\phi}$) in Boyer-Lindquist coordinates222Latin
indices take the values 0,1,2,3 and Greek indices take the values 1,2,3.. When
the core experiences a collapse, the outer shells will initially be in a free
fall. During this phase, we model stellar matter as a perfect fluid with the
stress-energy tensor.

$T^{ab}=(\rho+\epsilon+P)u^{a}u^{b}+Pg^{ab}\,,$ (2)

where $g^{ab}$ is the “inverse” metric, $P$ is the pressure and
$\rho,\epsilon$ are the rest mass and internal energy density. We further
assume that $\rho\gg(\epsilon+P)$. This reduces Eq. 2 to a dust equation of
state. Since the pressure approaches the energy density only in the inner
region of the disk, our approximation remains valid while the matter freely
falls to form the disk.

####  2.2.3 Initial formation of the disk

Just prior to infall, the shell is located at a large $r$ value with
$u^{a}=(\gamma,0,0,\gamma\Omega)^{a}\partial_{a}$ such that $u^{t}\gg u^{\mu}$
and therefore the shell falls from a near-rest configuration. For simplicity
and compatibility with the Mesa models, we assume that the stellar matter
collapses in a shellular fashion with angular velocity $\Omega=d\phi/dt$
(measured at $r\rightarrow\infty$). The mass contained within such a shell at
time $t$ can be written as

$dM=\int_{0,0}^{2\pi,\pi}T^{ab}u_{a}n_{b}dV\,,$ (3)

where $dV=\sqrt{g_{\Sigma_{t}}}drd\theta d\phi$ and the integral is performed
over $\theta,\phi$ only. Here $g_{\Sigma_{t}}={\rm det}(g_{\mu\nu})$,
$g_{\mu\nu}$ being the induced metric on the spacelike hypersurface
$\Sigma_{t}$ at some time $t$ calculated by setting $dt=0$ in the full Kerr
metric $g_{ab}$. In addition, $T^{ab}u_{a}\,(\approx-\rho u^{t})$ represents
the energy-momentum density current and
$n_{b}=-dt/\sqrt{-g^{-1}(dt,dt)}=-dt/\sqrt{-g^{tt}}$ is the unit 1-form normal
to $\Sigma_{t}$. Hence Eq. 4 yields

$\displaystyle dM$
$\displaystyle=\int_{0,0}^{2\pi,\pi}\rho\frac{\left(a^{2}\cos^{2}\theta+r^{2}\right)^{\frac{3}{2}}\sin\theta}{\sqrt{a^{2}\cos^{2}\theta-2mr+r^{2}}}drd\theta
d\phi$ (4) $\displaystyle\approx\int_{0,0}^{2\pi,\pi}\gamma\rho
r^{2}\sin\theta\,drd\theta d\phi\,,$

where during the last approximation, we have dropped terms $\mathcal{O}(a/r)$
which is allowed if $a\ll 1$ or $r\gg M$. We are interested in finding the
stellar radius $r_{0}$ at which the infalling shell has sufficient angular
momentum to first form an accretion disk. This occurs when the (averaged)
specific angular momentum $\ell$ of the collapsed shell in the equatorial
plane equals the specific angular momentum of the innermost stable circular
orbit (ISCO) $\ell_{\rm isco}$ of the Kerr metric. At this point, the matter
first circularises at the ISCO before plunging into the black hole. The
angular momentum contained in the shell with mass $dM$ can we written as

$\displaystyle dJ$ $\displaystyle=\int_{0,0}^{2\pi,\pi}T^{ab}\eta_{a}n_{b}dV$
(5) $\displaystyle\approx\int_{0,0}^{2\pi,\pi}\gamma^{2}\rho\Omega
r^{4}\sin^{3}{\theta}\,drd\theta d\phi\,,$

where the approximation is again under the same condition as before. This
implies that

$\ell(r)=dJ/dM\,.$ (6)

Additionally, the specific angular momentum at radius $r$ of a circular orbit
in the equatorial plane of the Kerr metric assuming prograde orbits is
(Bardeen et al., 1972)

$u_{\phi}(r)=\frac{M^{1/2}\left(r^{2}-2aM^{1/2}r^{1/2}+a^{2}\right)}{r^{3/4}\left(r^{3/2}-3Mr^{1/2}+2aM^{1/2}\right)^{1/2}}\,.$
(7)

Since the above is defined per unit rest mass, we set $\gamma=1$ in Eq.4 and 5
to bring Eq. 6 to a similar form. Choosing $r=r_{\rm isco}$ in Eq. 7, the
condition

$\ell(r_{0})=u_{\phi}(r_{\rm isco})$ (8)

gives us the stellar radius $r_{0}$ at which the infalling shell has
sufficient angular momentum to first form an accretion disk.

The coordinate time interval $\Delta t~{}=t(r_{0})-t(r_{\rm isco})$ then gives
us the disk formation time post core-collapse (within an error margin of a
sound crossing time as the information of the collapse would take this much
time to reach $r_{0}$). As we discuss in Appendix A, $\Delta t$ can be well
approximated by the corresponding Newtonian free fall time. Hence, to simplify
our calculations, here we use the latter approach for calculating $\Delta t$.

####  2.2.4 Evolution of the disk

Once Eq. 8 is satisfied, the subsequent infalling matter would first be
assimilated into the disk before being accreted by the black hole or blown
away in the wind. For the accretion phase, we restrict ourselves to the
equatorial plane of the black hole, where the accretion disk will reside. The
line element in the equatorial plane $(\theta=\pi/2)$ reads

$ds^{2}=-\left(1-\frac{2M}{r}\right)dt^{2}-\frac{4aM}{r}dtd\phi+\frac{r^{2}}{\Delta}dr^{2}+\frac{A}{r^{2}}d\phi^{2}\,,$
(9)

where $A\equiv r^{4}+r^{2}a^{2}+2Mra^{2}$. At a given time $t$, let the disk
have rest mass $M_{d}$ and angular momentum $J_{d}$. The effective disk radius
$r_{d}$ can be calculated by equating

$u_{\phi}(r_{d})=J_{d}/M_{d}\,,$ (10)

where the term on the LHS is given in Eq. 7. The mass and angular momentum
evolution of the disk can be written as

$\dot{M}_{d}=\dot{M}_{\rm fb}-\dot{M}_{\rm lost};\quad\dot{J}_{d}=\dot{J}_{\rm
fb}-\dot{J}_{\rm lost}\,,$ (11)

where the dot represents derivative w.r.t. $t$. Additionally, the mass and
angular momentum assimilation rate into the disk ($\dot{M}_{\rm
fb},\dot{J}_{\rm fb}$) due to fallback can be calculated using Eq. 4 and 5,
while the mass and angular momentum dissipation rate from the disk
($\dot{M}_{\rm lost},\dot{J}_{\rm lost}$) will be calculated below.

The infalling matter in the disk is accreted on a viscous timescale. To allow
for the effect of wind outflow, we model the local accretion rate as a power
law (Blandford & Begelman, 1999)

$\frac{dM_{\rm acc}(r)}{d\tau}=\frac{M_{d}}{\tau_{\rm
vis}}\left(\frac{r}{r_{d}}\right)^{s};\quad(r_{t}<r<r_{d})\,,$ (12)

where $s\in(0.3,0.8)$ is a free parameter (Yuan & Narayan, 2014) and we assume
$s=0.5$. We interpret Eq. 12 to model the accretion rate as measured from a
locally nonrotating frame of reference ($r,\theta=$ const.). Such observers
would have zero angular momentum (apart from the effect of frame dragging) and
are also known as zero angular momentum observers (ZAMO). Here, $\tau_{\rm
vis}$ is the local viscous timescale (as measured by ZAMO) and is modeled
using the Shakura & Sunyaev (1973) $\alpha$ viscosity prescription as

$\tau_{\rm vis}\approx{2}/{(\alpha\Omega)};\quad{\Omega}=\Omega_{K}-\omega\,,$
(13a)
$\Omega_{K}=\frac{u^{\phi}}{u^{t}}=\frac{M^{1/2}}{r_{d}^{3/2}+aM^{1/2}};\quad\omega=-\frac{g_{t\phi}}{g_{\phi\phi}}\,.$
(13b)

Here $\Omega_{K},\omega$ is the angular velocity of the fluid element (in a
circular orbit) and the angular velocity of the frame dragging, respectively,
as measured by an observer at radial infinity. In addition, ${\Omega}$ is the
angular velocity of the fluid as measured by a ZAMO (Bardeen et al., 1972). It
is expected that $\alpha\in[0.01,0.1]$ with theory supporting the lower value
and observation the larger one (King et al., 2007). Here, we take
$\alpha=0.01$. A variation in the value of $\alpha$ would influence the
maximal value of $a^{*}:=a/M$ (Sądowski et al., 2011) and therefore may affect
the evolution of the black hole (see Section 2.3.1). To transform Eq. 12 into
the Boyer-Lindquist frame, we convert the ZAMO proper time to the coordinate
time as

$d\tau=\frac{dt}{\sqrt{-g^{tt}}};\quad
1/\sqrt{-g^{tt}}=\sqrt{\Delta}\left(r^{2}+a^{2}+\frac{2a^{2}Mr}{\rho^{2}}\right)^{-1/2}\,.$
(14)

In Eq. 12, for $r\leq r_{t}$, where $r_{t}$ is some transitional radius, the
disk undergoes efficient neutrino cooling, resulting in no wind mass loss and
hence efficient accretion. We calculate $r_{t}$ by setting the RHS of Eq. 12
to ${10^{-2.5}r_{t}}/{(2GM)}\,M_{\odot}$/year (e.g., Kohri et al. 2005; Chen &
Beloborodov 2007; Kumar et al. 2008). When $r>r_{t}$, the disk enters the
advection-dominated accretion flow regime (ADAF, Narayan & Yi 1994; Narayan &
McClintock 2008), where most of the disk mass is blown away in the winds. One
can check that the net mass loss rate from the disk - owing to outflows and
black hole accretion - yields

$\dot{M}_{\rm lost}=\frac{M_{d}}{\tau_{\rm vis}\sqrt{-g^{tt}}}\,.$ (15)

Additionally, the net angular momentum loss rate yields

$\dot{J}_{\rm lost}=\dot{J}_{\rm
BH}+\underbrace{\int_{r_{t}}^{r_{d}}u_{\phi}(r)d\dot{M}_{\rm
acc}(r)}_{\dot{J}_{\rm wind}}\,,$ (16)

where the former term is defined in Eq. 20 and the latter term, $\dot{J}_{\rm
wind}$, represents the angular momentum lost in the outflow, assuming that the
wind removes the corresponding specific angular momentum from its location of
origin. The integral can be evaluated as

$\displaystyle\dot{J}_{\rm wind}$ $\displaystyle=\frac{M_{d}}{\tau_{\rm
vis}}\frac{s}{r_{d}^{s}}\int_{r_{t}}^{r_{d}}\frac{u_{\phi}(r)}{\sqrt{-g^{tt}}}r^{s-1}dr$
(17) $\displaystyle\approx\frac{M_{d}}{\tau_{\rm
vis}}\frac{s}{r_{d}^{s}}\left[\frac{2amr^{s-1}}{1-s}+\frac{2\sqrt{m}r^{s+\frac{1}{2}}}{2s+1}\right]_{r_{t}}^{r_{d}}$

To calculate the last term, we dropped terms $\mathcal{O}(a/r^{2})$ within the
integral. This is more accurate than assuming a Schwarzschild geometry with
the dropped terms becoming increasingly less unimportant as $a$ decreases or
$r$ increases. For the case where $a=0$, Eq. 17 reduces to the one given in
Kumar et al. (2008). Finally, to incorporate the influence of the anti-
parallel relativistic jets on the accretion flow, we assume that post disk
formation, the polar region making an angle of $\theta=30^{\circ}$ with the
rotational axis is promptly removed from the star.

###  2.3 Evolution of black hole with a Novikov-Thorne disk

In the early phase, while the disk is yet to form, we assume that the mass
freely falls into the black hole without losing energy or angular momentum.
After an initial disk formation, the subsequent disk dynamics is modeled as
discussed in Section 2.2.4. This is analogous to modeling the accretion flow
as consisting of a relativistic, radiatively efficient, geometrically thin
accretion disk that lacks the presence of large-scale magnetic field
interactions with the black hole, also known as Novikov & Thorne (1973) disk
(NTD).

For such an accretion flow, once the mass reaches $r_{\rm isco}$, it is
assumed to be directly accreted by the black hole, adding mass at a rate of

$\dot{M}_{\rm BH}=u_{t}(r_{\rm isco})\dot{M}_{\rm acc}(r_{\rm isco})\,,$ (18)

where

$u_{t}(r)=\frac{r^{3/2}-2Mr^{1/2}+aM^{1/2}}{r^{3/4}\left(r^{3/2}-3Mr^{1/2}+2aM^{1/2}\right)^{1/2}}\,,$
(19)

is the specific energy of a circular orbit at the radius $r=r_{\rm isco}$
(Bardeen et al., 1972). This also adds angular momentum to the black hole at a
rate333Note that we have added the subscript “BH” to the variables to
explicitly associate them to the black hole.

$\dot{J}_{\rm BH}=u_{\phi}(r_{\rm isco})\dot{M}_{\rm acc}(r_{\rm isco})\,,$
(20)

where, following Bardeen et al. (1972), $r_{\rm isco}$ is

$\displaystyle
r_{\rm{isco}}=M\left[3+z_{2}-\sqrt{((3-z_{1})(3+z_{1}+2z_{2})}\right]\,,$ (21)
$\displaystyle z_{1}=1+(1-a^{2}/M^{2})^{1/3}((1+a/M)^{1/3}+(1-a/M)^{1/3})\,,$
$\displaystyle z_{2}=\sqrt{3a^{2}/M^{2}+z_{1}^{2}}\,.$

####  2.3.1 Equilibrium spin of the black hole

In principle, as the black hole accretes mass (and hence angular momentum),
its spin parameter will gradually approach the limit $a=M$. However, on
considering the decelerating impact of disk-emitted photons on the NTD, Thorne
(1974) found that the dimensionless spin parameter of the black hole
$a^{*}:=a/M$ should converge to a value of $a^{*}=0.9978$. Since both Eq. 6
and 19 are functions of $a$, hence, the equilibrium value of $a^{*}$ becomes
important444E.g., for $a^{*}=0.9978,0.9994$, $u_{t}(r_{\rm isco})\approx
0.68,0.58$ respectively.. More recently, Sądowski et al. (2011) recalculated
this limit for the case of (advection dominated, optically and geometrical
thick) slim accretion disks and found $a^{*}=0.9994$ for $\alpha=0.01$ and a
$10\,\times$ super-Eddington mass accretion rate. Since, these conditions also
manifest in collapsar physics, therefore, in this work, we use the latter
equilibrium value for $a^{*}$ when modeling the accretion flow using an NTD.

###  2.4 Evolution of black hole with a magnetically arrested disk

Magnetic fields are considered to be crucial in extracting the rotational
energy of the Kerr black hole to power the anti-parallel relativistic jets. Of
particular interest are the predictions of general relativistic
magnetohydrodynamic (GRMHD) simulations of a magnetically arrested disk (MAD)
state (e.g., Tchekhovskoy et al. 2011). Here, the accretion disk is threaded
by a strong vertical magnetic field that is dragged inward by the accretion
flow resulting in an accumulation of magnetic flux near the black hole
horizon. These magnetic fields can play a central role in extracting the
rotational energy of the black hole (Penrose & Floyd, 1971; Blandford &
Znajek, 1977).

To calculate the effect of MAD on the black hole, in conjunction with Section
2.2.4, here we follow the approach of Moderski & Sikora (1996), which accounts
for the energy and angular momentum loss of the black hole due to the
Blandford-Znajek mechanism as

$\dot{J}_{\rm BH}=\ell_{\rm in}\dot{M}_{\rm
acc,in}-\frac{P_{\mathrm{EM}}}{k\Omega_{\rm H}}\,,$ (22a) $\dot{M}_{\rm
BH}=u_{t,\mathrm{in}}\dot{M}_{\rm acc,in}-\frac{P_{\mathrm{EM}}}{c^{2}}\,,$
(22b)

where $\Omega_{\rm H}=r_{\rm H}/(M_{\rm BH}a)$ is the angular frequency of the
black hole’s event horizon (measured at $r\rightarrow\infty$) and $r_{\rm
H}=M+\sqrt{M^{2}-a^{2}}$. In the MAD state, the other variables above need to
be determined from GRMHD simulations, for which we use the recent analytical
fits provided in 3D GRMHD study of Lowell et al. (2023) as

$P_{\rm EM}=\eta_{\rm EM}\dot{M}_{\rm acc,in}c^{2};\quad\eta_{\rm
EM}=1.063{a^{*}}^{4}+0.395{a^{*}}^{2}\,,$ (23a) $\ell_{\rm
in}=0.86;\quad\quad\quad u_{t,\rm in}=0.97\,,$ (23b) $k=\Omega_{\rm
F}/\Omega_{\rm H}=\rm{min}(0.1+0.5a^{*},0.35)\,.$ (23c)

Here, $P_{\rm EM}$ is the electromagnetic flux leaving the black hole, which
is primarily contained in the jets, Also, $\Omega_{\rm F}$ is the angular
frequency of the magnetic field lines at the horizon. The subscript “in”
represents the inner end of the accretion disk in the simulations of Lowell et
al. (2023), which is not necessarily at the ISCO.

Figure 1: The masses of the black hole as a function of the final mass of
their progenitor star (latter evolved till core carbon depletion). Figure (a)
is for an NTD state, while Figure (b) is for the MAD state. Figure 2: Same as
Fig. 1 but now the y-axis shows the ratio of the black hole’s mass to the pre-
explosion mass of its progenitor star and the legend shows the isotropic-
equivalent energy $E_{\rm iso}$ of the GRB.

###  2.5 Electromagnetic energy of explosion

The energy radiated during a GRB may arise either due to the viscous dynamics
of the accretion flow e.g., via the Blandford & Payne (1982) mechanism (as is
the case for an NTD) or a combination of the former plus the Blandford-Znajek
mechanism, the latter robbing the black hole of its energy and angular
momentum (e.g., for the case of a MAD). This energy would have to climb the
gravitational potential well of the black hole in order to be observed by a
distant observer.

####  2.5.1 Gravitational redshift of the emitted energy

We account for the energy lost due to the gravitational redshift of the
radiation when measured by an observer at radial infinity as follows. For a
stationary observer at radius $r$, its 4-velocity is
$u^{a}=\sqrt{-1/g_{tt}}\cdot\xi^{a}$. The energy of a radially moving photon
as measured by this observer at $r$ is

$E(r)=-g_{ab}p^{a}u^{b}=-\sqrt{-1/g_{tt}}g_{ab}p^{a}\xi^{b}\,,$ (24)

where $p^{a}$ is the 4-momentum of the photon. In addition,
$g_{ab}p^{a}\xi^{b}$ should remain conserved along the photon’s geodesic,
given $\xi$ is Killing. As such, the observed energy at $r\rightarrow\infty$
is given by $E_{\infty}=-g_{ab}p^{a}\xi^{b}$. Hence, the energy of radiation
at infinity is related to its value at some radius $r$ as

$E_{\infty}=E(r)\sqrt{-g_{tt}}\,,$ (25)

where $g_{tt}$ is calculated at the radius $r$ and $E(r)=1-u_{t}(r)$ is the
total specific energy emitted by the fluid element during its decent to radius
$r$ from infinity. For computational ease, we evaluate the $g_{tt}$ term in
Eq. 25 assuming $r=2r_{\rm acc}$, where $r_{\rm acc}$ is the radius at which
the element is accreted (e.g., $r_{\rm acc}=r_{\rm isco}$ for an NTD). This is
a fair approximation since roughly half of the energy would be emitted on each
side of this value. On the other hand, if the energy is sourced from the spin
of the black hole (e.g., in the MAD state), assuming that the photons move
radially outward from the ergosphere (as we did in Eq. 25), the latter becomes
a surface of infinite redshift. However, actual photons would contain angular
momentum and hence be able to escape the ergosphere to infinity. For our
current purpose we somewhat ad hocly set $r=1.01\times r_{\rm ergo}$ if the
energy is sourced from the spin of the black hole, where $r_{\rm
ergo}=M+\sqrt{M^{2}-a^{2}\cos^{2}{\theta}}$ is the outer radius of the
ergosphere of the black hole, and we assume $\theta=\pi/2$.

####  2.5.2 Isotropic equivalent energy of explosion

We classify the collapsar explosion as a potential source of GRB if the
isotropic equivalent energy $E_{\rm iso}$ that is received at the radial
infinity is larger than $10^{51}$ erg (Perley et al., 2016), i.e.

$E_{\rm iso}=\frac{\beta}{f}\int_{t_{\rm
exp}}E_{\infty}dt>10^{51}\,\rm{erg}\,.$ (26)

Above $\beta=0.01$ is the fraction of energy generated that is released in the
form of radiation (with the rest lost in neutrinos). In addition, we set
$\beta=1$ if the radiation energy is sourced from the spin of the black hole.
The term $f:=d\Omega/4\pi$ represents the fractional probability that the jet
aligns with the observer’s line of sight, where $d\Omega$ is the total solid
angle spanned by the jet cones. Following the calibration of Bavera et al.
(2022) we take $f=0.05$.

##  3 Result

By numerically implementing the formalism discussed in Section 2, here, we
assess the impact of the progenitor star’s mass and angular momentum reservoir
on the properties of the subsequently produced black hole, such as its mass
and spin. For the latter, we provide fitting functions that can be readily
used by future studies (in particular population synthesis) where the pre-
collapse nature of the progenitor star is known.

###  3.1 Masses of black holes born from rotating progenitors

Fig. 1 and 2 show the masses of black holes born from the Mesa evolved stellar
models. For comparison, we use the Fryer Delayed remnant mass function as a
source of reference. We find that there is a threshold explosion energy
$E_{\rm iso}$ beyond which there is a sudden change in the masses of black
holes as a function of the pre-explosion progenitor’s mass $M_{\rm exp}$. The
magnitude of such an $E_{\rm iso}$ and the amount of depreciation in the
resulting black hole mass is a function of the hydrodynamical nature of the
accretion flow. In the case of the NT accretion disk, the resulting drop in
the mass of black holes can be as large as $50\%$ \- Fig. 2(a). The MA
accretion flow in Fig. 2(b), tends to produce black holes with relatively
larger masses as compared to the NT flow. However, these are still lower than
those resulting from the convection-enhanced neutrino heating explosion, as
showcased by the Fryer delayed mechanism.

We note that, unlike the case of NTD, for the MAD state, the threshold $E_{\rm
iso}$ does not guarantee a downward shift in the black hole masses. For
example, in Fig. 2(b), we find very luminous GRBs with $E_{\rm iso}\gtrsim
3\times 10^{53}$ erg, that also produce relatively massive black holes. Such
explosions result from progenitor stars that rotate sufficiently rapidly to
experience a strong GRB, but not enough to experience a complete spin-down. To
achieve the latter, a near maximally rotating black hole needs to accrete
$\approx 20\%$ of its initial mass (e.g., Jacquemin-Ide et al. 2023). However,
the aforementioned luminous GRBs with larger black hole masses only develope
an accretion disk during the later stage of their evolution, hence minimizing
their chances of a complete spin-down.

For stars with relatively smaller reservoirs of angular momentum, the collapse
is semi-direct, without much dependence on the nature of the accretion flow.
For such stars, we find the masses of the black holes to be typically larger
than those predicted by the Fryer delayed mechanism. Although for such slow
rotators, one might expect convection-enhanced neutrino heating to
additionally contribute towards explosion energetics. However, the scarcity of
observations of supernova explosions of stars with mass $\gtrsim
18\,M_{\odot}$ suggests that direct collapse might be possible (Smartt, 2015).
Numerical studies (e.g., Murguia-Berthier et al. 2020) also align with the
observations, suggesting that massive stars might experience a direct collapse
without an ensuing electromagnetic counterpart.

To better quantify the dependence of the black hole properties on the
progenitor’s structure, we now consider the full parameter space constituting
the mass of the star prior to explosion $M_{\rm exp}$ and its (averaged
specific) angular momentum reservoir $\ell_{\rm exp}$. As shown in Fig. 3, in
such a case, there appears to be a definite relation between $M_{\rm exp}$ and
$\ell_{\rm exp}$ (i.e., the black dashed line):

$\bar{\ell}_{\rm exp}>(4\times 10^{3}M_{\rm exp})^{1/2};\quad\text{ for
\,\,}M_{\rm exp}\gtrsim 5M_{\odot}\,,$ (27)

where we first expect a GRB to take place and the black dotted line:

$\bar{\ell}_{\rm exp}>(7.5\times 10^{4}M_{\rm exp})^{1/2};\quad\text{ for
\,\,}M_{\rm exp}\gtrsim 15M_{\odot}\,,$ (28)

where we expect to see a strong deviation in the black hole masses $M_{\rm
BH}$. Here, for readability, we define ${\ell}_{\rm exp}:=\bar{\ell}_{\rm
exp}\times 10^{14}\text{ [gcm}^{2}{\rm s}^{-1}]$. The corresponding fit
function for the masses of the black holes can be approximated within $3\%$
error (with $90\%$ credibility; see Fig. 7) as

$\begin{array}[]{l}M_{\rm BH}^{\rm
NT}=\left\\{\begin{array}[]{ll}0.38\sqrt{M_{\rm exp}\bar{\ell}_{\rm
exp}}+0.41\sqrt{M_{\rm exp}}-3.4\sqrt{\bar{\ell}_{\rm exp}}\\\ -0.72M_{\rm
exp}+0.006\bar{\ell}_{\rm exp}+81;\,\,\text{if Eq.~{}\ref{eq: threshold AM}
holds}.\\\ \\\ 0.019\sqrt{M_{\rm exp}\bar{\ell}_{\rm exp}}-0.065\sqrt{M_{\rm
exp}}-0.051\sqrt{\bar{\ell}_{\rm exp}}\\\ +0.986M_{\rm
exp}-0.0024\bar{\ell}_{\rm exp}+0.55;\,\,\text{
otherwise},\end{array}\right.\end{array}$ (29a) $\begin{array}[]{l}M_{\rm
BH}^{\rm MA}=\left\\{\begin{array}[]{ll}-0.014\sqrt{M_{\rm exp}\bar{\ell}_{\rm
exp}}-0.076\sqrt{M_{\rm exp}}-0.077\sqrt{\bar{\ell}_{\rm exp}}\\\ +0.874M_{\rm
exp}+0.0006\bar{\ell}_{\rm exp}+1.083;\,\,\text{ if Eq.~{}\ref{eq: threshold
AM} holds},\\\ \\\ 0.029\sqrt{M_{\rm exp}\bar{\ell}_{\rm
exp}}-0.424\sqrt{M_{\rm exp}}-0.055\sqrt{\bar{\ell}_{\rm exp}}\\\ +1.002M_{\rm
exp}-0.0041\bar{\ell}_{\rm exp}+1.67;\,\,\text{
otherwise},\end{array}\right.\end{array}$ (29b)

where the superscripts on $M_{\rm BH}$ $[M_{\odot}]$ represent the nature of
the accretion flow during the collapse, and the constant coefficients have
appropriate units to make the LHS and RHS dimensionally consistent. We note
that the region plotted in gray dotted lines in Fig. 3(b) corresponds to the
very luminous systems in Fig. 2(b) that have $E_{\rm iso}\gtrsim 3\times
10^{53}$ erg but do not satisfy Eq. 28 to experience a complete spin-down.

Figure 3: The masses of the black holes as a function of their progenitor
star’s mass and angular momentum reservoir. The figures are drawn from the
fits provided in Eq. 29b. The x-axis represents the mass of the star and the
y-axis is the specific angular momentum of the star at the point of core
carbon depletion. Figure (a) is for the NT accretion flow, while Figure (b) is
for the MA accretion flow. The region in gray dotted lines in Figure (b)
corresponds to the very luminous systems in Fig. 2(b) that have $E_{\rm
iso}\gtrsim 3\times 10^{53}$ erg but do not satisfy Eq. 28 to experience a
complete spin-down.

.

Figure 4: Same as Fig. 3 but now showing the dimensionless spin parameter
$a^{*}$ of black holes. Figures are drawn using the fits provided in Eq. 30b.

###  3.2 Spin of black holes born from rotating progenitors

Similarly, Fig. 4 shows the dependence of black hole spin on the progenitors
$M_{\rm exp}$ and $\ell_{\rm exp}$. In the case of an NTD - see Fig. 4(a), for
any given value of $M_{\rm exp}$ the black hole spin increases monotonically
as a function of $\ell_{\rm exp}$. Beyond a certain threshold value of
$\ell_{\rm exp}$ (dotted red line in Fig. 4, see Eq. 28), the black hole
typically acquires a spin of $a^{*}\gtrsim 0.9$. This is the same region where
the mass of the black hole also experiences a sudden downward shift in trend.
This shift can be attributed to the large reservoir of angular momentum of the
progenitor, which causes the accretion disk to form promptly. After sufficient
time, the accretion flow of such a collapse transforms into the ADAF regime
(Blandford & Begelman, 1999), resulting in a strong mass loss. Moreover, such
NTD-hosting black holes also lose a considerable amount of mass in neutrinos
from the inner region (owing to the URCA processes, Gamow & Schoenberg 1941),
hence efficiently losing mass-energy to outgoing neutrinos.

While in the case of an NTD, the GRB energetics fully result from the
accretion dynamics, for a MAD, the spin angular momentum of the black hole
plays a direct role in powering the jets. Consequently, we expect a dramatic
variation in the final spin of the black holes depending on the nature of the
accretion flow. This can be seen in Fig. 4(b), where the maximum attainable
spin value is $a^{*}\lesssim 0.65$. For a MAD, we find that the strongest GRBs
(systems lying above the dotted red line in Fig. 4(b)) rapidly spin down and
result in relatively lower mass black holes. However, as discussed previously
there is a region (plotted in dull-red lines) that results in strong GRBs but
the $a^{*}$ still remains large. The resulting spin evolution of the black
hole can be approximated within 12% error (with 90% credibility) as

$\begin{array}[]{l}a_{\rm
NT}^{*}=\left\\{\begin{array}[]{ll}-0.0021\sqrt{M_{\rm exp}\bar{\ell}_{\rm
exp}}+0.0024\sqrt{M_{\rm exp}}+0.052\sqrt{\bar{\ell}_{\rm exp}}\\\
+0.004M_{\rm exp}-0.0005\bar{\ell}_{\rm exp}+1.19\times 10^{-6}M_{\rm
exp}\bar{\ell}_{\rm exp}\\\ -0.12;\,\,\text{ if Eq.~{}\ref{eq: threshold AM}
holds},\\\ \\\ -0.0016\sqrt{M_{\rm exp}\bar{\ell}_{\rm exp}}+0.046\sqrt{M_{\rm
exp}}+0.02\sqrt{\bar{\ell}_{\rm exp}}\\\ -0.0048M_{\rm
exp}+0.0004\bar{\ell}_{\rm exp}-2.6\times 10^{-6}M_{\rm exp}\bar{\ell}_{\rm
exp}\\\ -0.125;\,\,\text{ otherwise},\end{array}\right.\end{array}$ (30a)
$\begin{array}[]{l}a_{\rm MA}^{*}=\left\\{\begin{array}[]{ll}0.01;\qquad\text{
if Eq.~{}\ref{eq: threshold AM} holds},\\\ \\\ -0.0006\sqrt{M_{\rm
exp}\bar{\ell}_{\rm exp}}+0.1162\sqrt{M_{\rm exp}}+0.0254\sqrt{\bar{\ell}_{\rm
exp}}\\\ -0.0137M_{\rm exp}-0.0002\bar{\ell}_{\rm exp}+2.43\times
10^{-6}M_{\rm exp}\bar{\ell}_{\rm exp}\\\ -0.276;\,\,\text{
otherwise}.\end{array}\right.\end{array}$ (30b)

The source of the larger error can be attributed to some data points that
deviate from the general trend.

###  3.3 Domain of validity

Although it is challenging to measure the spin of individual components in a
merging BBH, the mass-weighted effective spin parameter $\chi_{\rm eff}$ of
the binary can be relatively well estimated from the observed gravitational
wave signal. For a BBH comprising masses $M_{1}$ and $M_{2}$ with spins
$a^{*}_{1}$ and $a^{*}_{2}$ aligned with their orbital angular momentum,

$\chi_{\rm eff}=\frac{M_{1}a^{*}_{1}+M_{2}a^{*}_{2}}{M_{1}+M_{2}}\,.$ (31)

Interestingly, recent findings suggest that most stellar mass black holes
should be born with a negligible spin (Eggenberger et al., 2019; Fuller et
al., 2019) with values around $a^{*}\approx 0.01$ (Fuller & Ma, 2019).
Therefore, binaries with $\chi_{\rm eff}>0$ might have a distinct formation
history. The prevalent methods for generating spinning black holes in isolated
binaries involve either (i) tidal spin-up of a progenitor star within a tight
binary, with a fraction of the acquired angular momentum subsequently
inherited by the ensuing black hole (e.g., Mandel & de Mink 2016; Marchant et
al. 2016; Qin et al. 2018; Bavera et al. 2020), or (ii) sufficient mass (and
hence angular momentum) accretion onto the progenitor star (e.g., Cantiello et
al. 2007; Ghodla et al. 2023) or the first-born black hole (e.g., Zevin &
Bavera 2022).

Figure 5: The maximum period at which the star would acquire and retain
sufficient angular momentum to satisfy Eq. 28 by core carbon depletion. The
blue and orange curves assume tidal locking at the onset of the helium main-
sequence and core helium depletion, respectively. The x-axis shows the mass of
the star at the onset of tidal locking, and the y-axis shows the period of the
binary assuming that the other component is a black hole with mass ratio
$q=1$. The grey band of uncertainty results from the variance in the value of
$r_{g}$.

####  3.3.1 Spin-up via tidal interaction

Tidal spin-up can occur while both the stars are on their hydrogen main-
sequence due to the occurrence of chemically homogeneous evolution (CHE e.g.,
Mandel & de Mink 2016; Marchant et al. 2016) or when the primary star first
evolves into a black hole with now having a tidally locked secondary star that
is on its helium main-sequence with negligible hydrogen envelope (e.g., Qin et
al. 2018; Bavera et al. 2020; Fuller & Lu 2022). Assuming that the star’s spin
period promptly synchronizes with its orbital period $P$, the value of such a
star’s (averaged) specific angular momentum $\ell$ can be estimated as

$\ell=\frac{I\omega}{M_{\rm star}}=\frac{2\pi r_{g}^{2}R^{2}}{P}\,,$ (32)

where $I,\omega,M_{\rm star},r_{g},R$ is the moment of inertia, orbital
angular velocity, mass, radius, and radius of gyration of the star,
respectively. Using Eq. 32 one can estimate the domain of validity of the fit
functions provided in Eq. 29b and 30b. For example, the blue curve in Fig. 5
shows the threshold minimum orbital period of the binary at the onset of
helium main-sequence required to sufficiently spin-up the star - with
$Z\approx\mathcal{O}(10^{-4})$ \- for it to satisfy Eq. 28 by the time it
reaches core carbon depletion. The orange curve, on the other hand, considers
tidal synchronization till core helium depletion and therefore is applicable
to high metallicity stars. This is because, post core helium depletion, the
star with even larger metallicities will undergo relatively less mass loss and
hence spin angular momentum loss.

####  3.3.2 Spin-up via mass accretion

Stellar spin-up via mass accretion might occur while the star is on its
hydrogen main-sequence (Cantiello et al., 2007; Ghodla et al., 2023). For such
systems, we expect the stars to undergo a GRB if they satisfy Eq. 27 at core
carbon depletion. Additionally, the properties of the subsequently formed
black hole could be highly affected if such systems also satisfy Eq. 28 (see
Fig. 3 and 4). The latter is likely to be only valid in rapidly rotating low
metallicity stars, i.e., $Z\lesssim\mathcal{O}(5\times 10^{-4})$ (Ghodla et
al., 2023). However, in contrast to black holes born in tight binaries, we
find that black holes born from stars that undergo accretion-induced spin-up
do not merge efficiently unless the supernovae kick results in compact and/or
significantly eccentric orbits. This is because to efficiently spin-up a star
one likely requires the formation of an accretion disk around the mass gainer
(e.g., Ghodla et al. 2023). However, for the formation of an accretion disk
around the mass gainer, one needs to allow for a relatively larger separation
between the binary components which subsequently results in longer
gravitational wave merger time. As such, although produced in relatively large
numbers (e.g., Ghodla et al. 2023), such black holes might seldom feature in
the BBH mergers detected on Earth by the current generation detectors.

##  4 Discussion

In this work, we have adopted two types of accretion flows that could occur
near the region surrounding the black hole during collapsar dynamics. While a
MA accretion flow can be invoked to produce the anti-parallel GRB jets via the
Blandford & Znajek (1977) mechanism, jets can also be produced in
astrophysical objects lacking a black hole at the center (e.g., Metzger et al.
2011; Metzger et al. 2017; Smith 2012). Hence, the energy of the jets need not
come from the rotational energy of the black hole. For example, one can expect
the formation of anti-parallel jets purely from the energy released during the
accretion flow via the Blandford & Payne (1982) mechanism. Therefore, the
approach of employing an NT accretion flow for the collapse dynamics is also
relevant. However, a hydrodynamical flow (such as in the case of an NTD) would
still require the presence of magnetic field lines to channel the outflow
(Blandford & Payne, 1982). In the following, we discuss some implications of
the results detailed in Section 3 depending on whether an NTD or MAD operates
to produce the GRB jets followed by some caveats and sources of uncertainty in
our work.

###  4.1 Impact of stellar rotation on the lower end of PISN mass-gap

Stars with helium cores in the mass range of $\approx
60M_{\odot}-130M_{\odot}$ may develop the right condition for $e^{-}e^{+}$
pair-production induced pair-instability supernova (PISN). This results in a
(local) upper bound on the value of $M_{\rm exp}$ that could survive during
core oxygen fusion (Fowler & Hoyle, 1964; Rakavy & Shaviv, 1967) without being
completely disrupted due to the aforementioned process. Consequently, one also
expects the formation of an upper mass-gap in the black hole mass spectrum.

However, rapid rotation can shift the lower boundary of PISN to larger $M_{\rm
exp}$ values555In this section we assume the largest $M_{\rm exp}$ that
survives PISN is $65M_{\odot}$, cf. Farmer et al. (2019); Marchant & Moriya
(2020) (Glatzel et al., 1985). Marchant & Moriya (2020) found that for such
systems, the lower edge of the black hole mass-gap can move upwards by
$4\%-15\%$ depending on the efficiency of angular momentum transport within
the progenitor stars during their evolution. In contrast, the present work
suggests that if the collapse of these rapid rotators666We allow for efficient
angular momentum transport within the stellar interior aided by the Spruit
(2002) dynamo. Turning this off would preserve more angular momentum within
the star and result in favorable conditions for producing rapid rotators.
operates under the NTD formalism, such that the $a^{*}\gtrsim 0.9$, then the
maximum black hole mass should converge around $M_{\rm BH}\approx
35M_{\odot}$. The latter value is much lower than the expected lower end of
the mass-gap near $M_{\rm BH}\in~{}\approx~{}[45M_{\odot},55M_{\odot}]$, e.g.,
Farmer et al. (2019); Marchant & Moriya (2020). Consequently, rapid rotators
under an NT accretion flow seem unlikely to extend the lower end of the PISN
mass-gap.

###  4.2 Case for a massive and maximally spinning stellar black hole

We find that for MAD, the maximum black hole spin accumulates near
$a^{*}\approx 0.65$ while for an NTD it takes the value $a^{*}\approx 1$.
However, for the latter, when $a^{*}\approx 1$, the collapsar energetic places
an upper limit on the black hole mass as $M_{\rm BH}\lesssim 35M_{\odot}$.
Hence, irrespective of the nature of the accretion flow, the present work
suggests that a near maximally rotating black hole (i.e., $a^{*}\approx 1$)
with $M_{\rm BH}\gtrsim 35M_{\odot}$ might not result from stellar collapse.
Instead, such a black hole could either be a second-generation black hole
(i.e., a merger product of an earlier BBH system, e.g., Pretorius 2005;
Fishbach et al. 2017; Gerosa & Berti 2017) or a product of subsequent mass
(and angular momentum) accretion (e.g., Gammie et al. 2004; Sądowski et al.
2011). For the latter, super-Eddington accretion would be the likely scenario,
as sub-Eddington accretion would make it difficult to spin up black holes to
critical within the lifetime of a massive mass donor star.

###  4.3 A possible upper bound on $\chi_{\rm eff}$ in black hole - helium
star binaries

From Fig. 4, we see that for a given $M_{\rm exp}$, a higher value of
$\ell_{\rm exp}$ does not always lead to a larger $a^{*}$. Instead, once Eq.
28 is satisfied, the assumption of the nature of the accretion flow becomes
crucial. For the case of MAD, a rapid spin-down is observed with $a^{*}\approx
0$ (cf. Gottlieb et al. 2023; Jacquemin-Ide et al. 2023). This results in an
upper bound on the spin with $a^{*}\lesssim 0.65$. In such a case, under the
scenario that only the second-born black hole acquires spin in a tidally
locked helium star-black hole binary, an upper bound on $\chi_{\rm eff}$ can
be estimated as

$\chi_{\rm eff,max}=\frac{a^{*}_{2}}{q^{-1}_{\rm max}+1}\approx 0.325\,,$ (33)

where $q:=M_{2}/M_{1}\leq 1$ and we set $q_{\rm max}=1$.

On the other hand, the LIGO-Virgo-KAGRA collaboration (Abbott et al., 2021b,
a) has identified more than half a dozen candidate BBHs that do not respect
this equality (e.g., see Table 1). If MAD is the mechanism behind GRBs, this
hints that an alternative scenario might be behind the formation of such BBHs.
Tidal CHE can produce systems with large $\chi_{\rm eff}$ values, since, in
this case, both black holes can acquire large spin. However, the latter
requires nearly equal mass-ratio binaries (Marchant et al., 2016), which is
not the case for the black holes in Table 1. Alternatively, super-Eddington
accretion onto the first formed black hole in an ordinary binary system can
also generate a rapidly spinning (and massive) black hole. But the systems in
Table 1 likely result from massive stellar progenitors that have a lifespan of
only a few million years. This allows a tiny window for mass transfer, which
implies the requirement of a rapid mass transfer rate supported by a very
strong super-Eddington accretion. Additionally, the period of the binary
should be such that they could subsequently merge within a few billion years.
It is uncertain as to how likely these conditions could be met simultaneously.

A more plausible formation pathway for BBHs in Table 1 could be via
hierarchical merger (e.g., Fishbach et al. 2017; Gerosa & Berti 2017; Kimball
et al. 2021), e.g., in a triple system or dynamical capture in stellar
clusters or in the disk of active galactic nuclei (AGN, e.g., Gerosa &
Fishbach 2021; Arca Sedda et al. 2023 and reference therein). This is because
the merger of even non-rotating black holes can result in second-generation
black holes with $a^{*}\approx 0.7$ (e.g., Pretorius 2005). Additionally, BBHs
living in the disks of AGN also have the possibility of accreting mass and
angular momentum from the disk.

###  4.4 Comparison to the non-relativistic approach

A non-relativistic variant of the NTD formalism described in Section 2 is
presented in Kumar et al. (2008) and later numerically implemented in Fuller &
Lu (2022). We find the relativistic treatment to have a negligible effect on
the final properties of the black hole. This is primarily because, during the
initial stage of collapse when near-horizon physics is important, most of the
disk mass is efficiently accreted by the black hole due to the Urca processes
(Gamow & Schoenberg, 1941). Consequently, this hides the effect of a
relativistic treatment in our time-averaged approach. During later stages, the
disk circularizes at a relatively larger distance from the horizon, thus
diminishing the effect of the Kerr geometry. However, the work presented in
Section 2.5 is missing in the treatment of Kumar et al. (2008). Including this
reduces the magnitude of $E_{\rm iso}$, as measured by an observer at radial
infinity by $79\pm 2\%$, $85\pm 5\%$ for an NTD and MAD state, respectively.

###  4.5 Caveats and uncertainties

  * •

The final structure of the star has been deemed crucial for the following
explosion, with some massive stars producing a neutron star instead of a black
hole, leading to the so-called islands of explodebility (O’Connor & Ott, 2011;
Sukhbold et al., 2016; Sukhbold et al., 2018). Although this might be true for
stars with low reservoirs of angular momentum, it would unlikely be the case
for systems that are rapidly rotating as the dynamics of core-collapse will be
overpowered by the collapsar energetics. However, there is a possibility for a
magnetar-like explosion (Woosley, 2010; Kasen & Bildsten, 2010) where the
energetics are dictated by the magnetic fields of a proto-neutron star instead
of a black hole. We ignore such a scenario in this work. As such in our
analysis a black hole always forms during the collapse of the core of a
massive star.

  * •

At times, the stochasticity of the core structure can influence the nature of
remnant masses. However, any sort of stochasticity would be lost in our
approach due to the averaging performed in carrying out the analysis. As
discussed in Fryer et al. (2022), the effect of stochasticity would
nevertheless average out for a large enough sample size, which is always true
in the context of population studies. This would also be the case with
observations in the coming year as the number of gravitational wave detections
grows.

  * •

In the case of non/slowly rotating stars, our formalism results in a direct
collapse. Although this might not necessarily be the case, the lack of
detections of supernova explosions from high-mass stars (Smartt, 2015) as well
as the theoretical findings that more massive stars might disappear without
leaving any observation feature (Murguia-Berthier et al., 2020), suggest that
black holes forming more massive stars might experience direct collapse.

  * •

For the case of an NTD, the accretion flow is circular and the disk truncates
at the ISCO. However, numerical studies find that at large accretion rate the
inner edge of the disk moves closer to the black hole but cannot be uniquely
defined (Abramowicz et al., 2010). The flow also deviates from a Keplerian
profile (Abramowicz et al., 2010; Sądowski et al., 2011), becoming super-
Keplerain for our choice of $\alpha=0.01$. We do not consider these aspects
here. For rapid rotators, the ISCO will approach very close to the black hole,
hence reducing our margin of error.

##  5 Conclusion

Black holes born from rapidly rotating progenitor stars can experience a very
different supernova explosion than those born from non/slowly rotating stars.
This is because a rapidly rotating star contains a large reservoir of angular
momentum that it would need to lose to collapse below a certain characteristic
radius. Here, we studied the resulting masses and spins of the black hole when
dealing with the collapse of rotating stars. We showed that for rapid
rotators, the black hole mass and spin is a function of assumption on the
accretion flow and can be significantly different from that expected from a
non/slowly rotating progenitor star (Section 3). This discrepancy could
influence the merger time, luminosity distance, and gravitational wave
properties of the BBH. We find that rapid rotators undergoing a collapsar
explosion using an NT accretion flow might not extend the lower edge of the
black hole upper mass-gap (Section 4.1) and that a maximally rotating black
hole born directly from stellar collapse might have $M_{\rm BH}\approx
35M_{\odot}$ as its maximum possible mass (Section 4.2). For the case of a MA
accretion flow, we find the maximum black hole dimensionless spin of
$a^{*}\approx 0.65$. For black holes born in helium star black hole binaries,
this puts an upper limit on the efficient spin of the BBH as $\chi_{\rm
eff}\lesssim 0.325$ (Section 4.3).

## Acknowledgements

SG is supported by the University of Auckland doctoral scholarship. JJE
acknowledges support of Marsden Fund Council managed through Royal Society Te
Apārangi. This work utilized NeSI high performance computing facilities.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author. The numerical work discussed in Section 2 can be found
at https://github.com/SohanGhodla/Collapsar-Formalism.

## References

  * Abbott et al. (2021a) Abbott The LIGO Scientific Collaboration the Virgo Collaboration the KAGRA Collaboration 2021a, arXiv e-prints, p. arXiv:2111.03606
  * Abbott et al. (2021b) Abbott The LIGO Scientific Collaboration the Virgo Collaboration the KAGRA Collaboration 2021b, Physical Review X, 11, 021053
  * Abramowicz et al. (2010) Abramowicz M. A., Jaroszyński M., Kato S., Lasota J. P., Różańska A., Sądowski A., 2010, A&A, 521, A15
  * Arca Sedda et al. (2023) Arca Sedda M., Naoz S., Kocsis B., 2023, Universe, 9, 138
  * Bardeen et al. (1972) Bardeen J. M., Press W. H., Teukolsky S. A., 1972, Astrophysical Journal, Vol. 178, pp. 347-370 (1972), 178, 347
  * Batta & Ramirez-Ruiz (2019) Batta A., Ramirez-Ruiz E., 2019, arXiv e-prints, p. arXiv:1904.04835
  * Bavera et al. (2020) Bavera S. S., et al., 2020, A&A, 635, A97
  * Bavera et al. (2022) Bavera S. S., et al., 2022, A&A, 657, L8
  * Bekenstein (1972) Bekenstein J. D., 1972, Phys. Rev. D, 5, 1239
  * Bičák & Stuchlik (1976) Bičák J., Stuchlik Z., 1976, Monthly Notices of the Royal Astronomical Society, 175, 381
  * Blandford & Begelman (1999) Blandford R. D., Begelman M. C., 1999, MNRAS, 303, L1
  * Blandford & Payne (1982) Blandford R. D., Payne D. G., 1982, MNRAS, 199, 883
  * Blandford & Znajek (1977) Blandford R. D., Znajek R. L., 1977, MNRAS, 179, 433
  * Blondin et al. (2003) Blondin J. M., Mezzacappa A., DeMarino C., 2003, The Astrophysical Journal, 584, 971
  * Cantiello et al. (2007) Cantiello M., Yoon S. C., Langer N., Livio M., 2007, A&A, 465, L29
  * Carter (1971) Carter B., 1971, Phys. Rev. Lett., 26, 331
  * Chen & Beloborodov (2007) Chen W.-X., Beloborodov A. M., 2007, ApJ, 657, 383
  * Eddington (1925) Eddington A. S., 1925, The Observatory, 48, 73
  * Eggenberger et al. (2019) Eggenberger P., den Hartogh J. W., Buldgen G., Meynet G., Salmon S. J. A. J., Deheuvels S., 2019, A&A, 631, L6
  * Eldridge & Tout (2004) Eldridge J. J., Tout C. A., 2004, MNRAS, 353, 87
  * Farmer et al. (2019) Farmer R., Renzo M., de Mink S. E., Marchant P., Justham S., 2019, ApJ, 887, 53
  * Fishbach et al. (2017) Fishbach M., Holz D. E., Farr B., 2017, ApJ, 840, L24
  * Fowler & Hoyle (1964) Fowler W. A., Hoyle F., 1964, ApJS, 9, 201
  * Fryer & Kalogera (2001) Fryer C. L., Kalogera V., 2001, ApJ, 554, 548
  * Fryer et al. (2012) Fryer C. L., Belczynski K., Wiktorowicz G., Dominik M., Kalogera V., Holz D. E., 2012, ApJ, 749, 91
  * Fryer et al. (2021) Fryer C. L., Karpov P., Livescu D., 2021, Astronomy Reports, 65, 937
  * Fryer et al. (2022) Fryer C. L., Olejak A., Belczynski K., 2022, ApJ, 931, 94
  * Fujibayashi et al. (2020) Fujibayashi S., Shibata M., Wanajo S., Kiuchi K., Kyutoku K., Sekiguchi Y., 2020, Phys. Rev. D, 102, 123014
  * Fujibayashi et al. (2023) Fujibayashi S., Sekiguchi Y., Shibata M., Wanajo S., 2023, ApJ, 956, 100
  * Fuller & Lu (2022) Fuller J., Lu W., 2022, MNRAS, 511, 3951
  * Fuller & Ma (2019) Fuller J., Ma L., 2019, ApJ, 881, L1
  * Fuller et al. (2019) Fuller J., Piro A. L., Jermyn A. S., 2019, MNRAS, 485, 3661
  * Gammie et al. (2004) Gammie C. F., Shapiro S. L., McKinney J. C., 2004, ApJ, 602, 312
  * Gamow & Schoenberg (1941) Gamow G., Schoenberg M., 1941, Physical Review, 59, 539
  * Gerosa & Berti (2017) Gerosa D., Berti E., 2017, Phys. Rev. D, 95, 124046
  * Gerosa & Fishbach (2021) Gerosa D., Fishbach M., 2021, Nature Astronomy, 5, 749
  * Ghodla et al. (2023) Ghodla S., Eldridge J. J., Stanway E. R., Stevance H. F., 2023, MNRAS, 518, 860
  * Glatzel et al. (1985) Glatzel W., Fricke K. J., El Eid M. F., 1985, A&A, 149, 413
  * Gottlieb et al. (2023) Gottlieb O., Jacquemin-Ide J., Lowell B., Tchekhovskoy A., Ramirez-Ruiz E., 2023, arXiv e-prints, p. arXiv:2302.07271
  * Herant et al. (1994) Herant M., Benz W., Hix W. R., Fryer C. L., Colgate S. A., 1994, ApJ, 435, 339
  * Israel (1967) Israel W., 1967, Phys. Rev., 164, 1776
  * Jacquemin-Ide et al. (2023) Jacquemin-Ide J., Gottlieb O., Lowell B., Tchekhovskoy A., 2023, arXiv e-prints, p. arXiv:2302.07281
  * Kasen & Bildsten (2010) Kasen D., Bildsten L., 2010, ApJ, 717, 245
  * Kimball et al. (2021) Kimball C., et al., 2021, ApJ, 915, L35
  * King et al. (2007) King A. R., Pringle J. E., Livio M., 2007, MNRAS, 376, 1740
  * Kohri et al. (2005) Kohri K., Narayan R., Piran T., 2005, ApJ, 629, 341
  * Kumar et al. (2008) Kumar P., Narayan R., Johnson J. L., 2008, MNRAS, 388, 1729
  * Lee & Ramirez-Ruiz (2006) Lee W. H., Ramirez-Ruiz E., 2006, ApJ, 641, 961
  * Lowell et al. (2023) Lowell B., Jacquemin-Ide J., Tchekhovskoy A., Duncan A., 2023, arXiv e-prints, p. arXiv:2302.01351
  * MacFadyen & Woosley (1999) MacFadyen A. I., Woosley S. E., 1999, ApJ, 524, 262
  * Mandel & de Mink (2016) Mandel I., de Mink S. E., 2016, MNRAS, 458, 2634
  * Marchant & Moriya (2020) Marchant P., Moriya T. J., 2020, A&A, 640, L18
  * Marchant et al. (2016) Marchant P., Langer N., Podsiadlowski P., Tauris T. M., Moriya T. J., 2016, A&A, 588, A50
  * Metzger et al. (2011) Metzger B. D., Giannios D., Thompson T. A., Bucciantini N., Quataert E., 2011, MNRAS, 413, 2031
  * Metzger et al. (2017) Metzger B. D., Berger E., Margalit B., 2017, ApJ, 841, 14
  * Meynet & Maeder (1997) Meynet G., Maeder A., 1997, A&A, 321, 465
  * Moderski & Sikora (1996) Moderski R., Sikora M., 1996, MNRAS, 283, 854
  * Murguia-Berthier et al. (2020) Murguia-Berthier A., Batta A., Janiuk A., Ramirez-Ruiz E., Mandel I., Noble S. C., Everson R. W., 2020, ApJ, 901, L24
  * Narayan & McClintock (2008) Narayan R., McClintock J. E., 2008, New Astron. Rev., 51, 733
  * Narayan & Yi (1994) Narayan R., Yi I., 1994, ApJ, 428, L13
  * Novikov & Thorne (1973) Novikov I. D., Thorne K. S., 1973, in Black Holes (Les Astres Occlus). pp 343–450
  * O’Connor & Ott (2011) O’Connor E., Ott C. D., 2011, The Astrophysical Journal, 730, 70
  * Paxton et al. (2011) Paxton B., Bildsten L., Dotter A., Herwig F., Lesaffre P., Timmes F., 2011, ApJS, 192, 3
  * Paxton et al. (2013) Paxton B., et al., 2013, ApJS, 208, 4
  * Paxton et al. (2015) Paxton B., et al., 2015, ApJS, 220, 15
  * Paxton et al. (2018) Paxton B., et al., 2018, ApJS, 234, 34
  * Paxton et al. (2019) Paxton B., et al., 2019, ApJS, 243, 10
  * Penrose & Floyd (1971) Penrose R., Floyd R. M., 1971, Nature Physical Science, 229, 177
  * Perley et al. (2016) Perley D. A., et al., 2016, ApJ, 817, 7
  * Pretorius (2005) Pretorius F., 2005, Phys. Rev. Lett., 95, 121101
  * Qin et al. (2018) Qin Y., Fragos T., Meynet G., Andrews J., Sørensen M., Song H. F., 2018, A&A, 616, A28
  * Rakavy & Shaviv (1967) Rakavy G., Shaviv G., 1967, ApJ, 148, 803
  * Riley et al. (2021) Riley J., Mandel I., Marchant P., Butler E., Nathaniel K., Neijssel C., Shortt S., Vigna-Gómez A., 2021, MNRAS, 505, 663
  * Sądowski & Narayan (2016) Sądowski A., Narayan R., 2016, Monthly Notices of the Royal Astronomical Society, 456, 3929
  * Shakura & Sunyaev (1973) Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337
  * Sharp (1984) Sharp D. H., 1984, Physica D: Nonlinear Phenomena, 12, 3
  * Sądowski & Narayan (2016) Sądowski A., Narayan R., 2016, MNRAS, 456, 3929
  * Sądowski et al. (2011) Sądowski A., Bursa M., Abramowicz M., Kluźniak W., Lasota J. P., Moderski R., Safarzadeh M., 2011, A&A, 532, A41
  * Sądowski et al. (2014) Sądowski A., Narayan R., McKinney J. C., Tchekhovskoy A., 2014, MNRAS, 439, 503
  * Smartt (2015) Smartt S. J., 2015, Publ. Astron. Soc. Australia, 32, e016
  * Smith (2012) Smith M. D., 2012, Astrophysical Jets and Beams
  * Spruit (2002) Spruit H. C., 2002, A&A, 381, 923
  * Sukhbold et al. (2016) Sukhbold T., Ertl T., Woosley S. E., Brown J. M., Janka H. T., 2016, ApJ, 821, 38
  * Sukhbold et al. (2018) Sukhbold T., Woosley S., Heger A., 2018, The Astrophysical Journal, 860, 93
  * Sweet (1950) Sweet P. A., 1950, MNRAS, 110, 548
  * Tchekhovskoy et al. (2010) Tchekhovskoy A., Narayan R., McKinney J. C., 2010, ApJ, 711, 50
  * Tchekhovskoy et al. (2011) Tchekhovskoy A., Narayan R., McKinney J. C., 2011, MNRAS, 418, L79
  * Thorne (1974) Thorne K. S., 1974, ApJ, 191, 507
  * Woosley (1993) Woosley S. E., 1993, ApJ, 405, 273
  * Woosley (2010) Woosley S. E., 2010, ApJ, 719, L204
  * Yuan & Narayan (2014) Yuan F., Narayan R., 2014, ARA&A, 52, 529
  * Zahn (1992) Zahn J. P., 1992, A&A, 265, 115
  * Zevin & Bavera (2022) Zevin M., Bavera S. S., 2022, ApJ, 933, 86
  * du Buisson et al. (2020) du Buisson L., et al., 2020, MNRAS, 499, 5941

##  Appendix A Freefall timescale

For a shell made of perfect fluid, the collapse is such that the coordinate
$\theta$ remains unchanged. This means that the trajectory of the particles
would not cross each other during infall. Solving for the geodesic motion
(assuming an infall from rest at radial infinity), one can show that for such
a case

$\frac{dr}{dt}=-\frac{(r^{2}-2Mr+a^{2})[2Mr(r^{2}+a^{2})]^{1/2}}{(r^{2}+a^{2})^{2}-(r^{2}-2Mr+a^{2})a^{2}\sin^{2}{\theta}}\,.$
(34)

Eq. 34 depends on the coordinate $\theta$, implying that in a Kerr geometry,
the equatorial part of a spherical shell falls faster than the polar region
(e.g., Bičák & Stuchlik 1976). Integrating Eq. 34 when $a\neq 0$ is difficult,
and the resulting level of precision is not required here. For example, in
Fig. 6, we compare the infall coordinate time (resulting from integrating Eq.
34 with $a=0$) with the infall proper time, assuming that both time
coordinates are synchronized at some fixed distance $r_{0}=30M$ (which we
approximate as the minimum distance from where an infalling shell would first
circularize at ISCO in our work). We find that upon reaching ISCO, there is
$\approx 12\%$ disagreement in the two time intervals. As $r_{0}$ moves to
larger values, this disagreement becomes smaller. E.g., for $r_{0}=200M$, it
is reduced to $\approx 2\%$. Finally, when calculating the disk formation time
$\Delta t$ in Section 2.2.3, we note that the matter falls from a finite
radius $r_{0}$, from a near rest configuration, which also needs to be taken
into account.

Figure 6: The deviation in the infall coordinate time and the infall proper
time (in a Schwarzschild geometry), assuming that both time coordinates are
synchronized at some fixed distance $r_{0}=30M$, which we approximate as the
minimum distance from where an infalling shell would first circularize at ISCO
($r=6M$) in our work.

##  Appendix B Additional Material

Table 1: LVK inferred parameters for systems with $\chi_{\rm eff}\geq 0.32$ (Abbott et al., 2021b, a) Name | $M_{1}\,[M_{\odot}]$ | $M_{2}\,[M_{\odot}]$ | $\chi_{\rm eff}$
---|---|---|---
GW190403_051519 | $85.0^{+27.8}_{-33.0}$ | $20.0^{+26.3}_{-8.4}$ | $0.68^{+0.16}_{-0.43}$
GW190517_055101 | $39.2^{+13.9}_{-9.2}$ | $24.0^{+7.4}_{-7.9}$ | $0.49^{+0.21}_{-0.28}$
GW200208_222617 | $51^{+103}_{-30}$ | $12.3^{+9.2}_{-5.5}$ | $0.45^{+0.42}_{-0.46}$
GW170729 | $50.2^{+16.2}_{-10.2}$ | $34.0^{+9.1}_{-10.1}$ | $0.37^{+0.21}_{-0.25}$
GW190805_211137 | $46.2^{+15.4}_{-11.2}$ | $30.6^{+11.8}_{-11.3}$ | $0.37^{+0.29}_{-0.39}$
GW190620_030421 | $58.0^{+19.2}_{-13.3}$ | $35.0^{+13.1}_{-14.5}$ | $0.34^{+0.22}_{-0.29}$
GW190519_153544 | $65.1^{+10.8}_{-11}$ | $40.8^{+11.5}_{-12.7}$ | $0.33^{+0.20}_{-0.24}$
GW200306_093714 | $28.3^{+17.1}_{-7.7}$ | $14.8^{+6.5}_{-6.4}$ | $0.32^{+0.28}_{-0.46}$
Figure 7: Figure (a) and (b) show the fitting error in the black hole masses
given in Eq. 29b, where Figure (a) is for NTD and Figure (b) for MAD state.
Figure (c) and (d) show the fitting error in the black hole spins given in Eq.
30b, where Figure (c) is for NTD and Figure (d) for MAD state.