# On the growth of supermassive black holes formed from the gravitational
collapse of fermionic dark matter cores

C. R. Argüelles,1,2,3 K. Boshkayev,4,5 A. Krut,2 G. Nurbakhyt,4 J. A.
Rueda,2,3,6,7,8 R. Ruffini,2,3,9 J. D. Uribe-Suárez,2,10 and R. Yunis2
1Instituto de Astrofísica de La Plata, UNLP-CONICET, Paseo del Bosque s/n
B1900FWA La Plata, Argentina
2ICRANet, Piazza della Repubblica 10, I-65122 Pescara, Italy
3ICRA, Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro
5, I-00185 Roma, Italy
4NNLOT, Department of Theoretical and Nuclear Physics, Al-Farabi Kazakh
National University, Almaty 050040, Kazakhstan
5International University of Information Technology, Manas St. 34/1, 050040
Almaty, Kazakhstan.
6ICRANet-Ferrara, Dipartimento di Fisica e Scienze della Terra, Università
degli Studi di Ferrara, Via Saragat 1, I-44122 Ferrara, Italy
7Dipartimento di Fisica e Scienze della Terra, Università degli Studi di
Ferrara, Via Saragat 1, I-44122 Ferrara, Italy
8INAF, Istituto di Astrofisica e Planetologia Spaziali, Via Fosso del
Cavaliere 100, I-00133 Rome, Italy
9INAF, Viale del Parco Mellini 84, I-00136 Rome, Italy
10Facultad de Ciencias Básicas, Universidad Santiago de Cali, Campus
Pampalinda, Calle 5 No. 6200, 760035 Santiago de Cali, Colombia
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Observations support the idea that supermassive black holes (SMBHs) power the
emission at the center of active galaxies. However, contrary to stellar-mass
BHs, there is a poor understanding of their origin and physical formation
channel. In this article, we propose a new process of SMBH formation in the
early Universe that is not associated with baryonic matter (massive stars) or
primordial cosmology. In this novel approach, SMBH seeds originate from the
gravitational collapse of fermionic dense dark matter (DM) cores that arise at
the center of DM halos as they form. We show that such a DM formation channel
can occur before star formation, leading to heavier BH seeds than standard
baryonic channels. The SMBH seeds subsequently grow by accretion. We compute
the evolution of the mass and angular momentum of the BH using a geodesic
general relativistic disk accretion model. We show that these SMBH seeds grow
to $\sim 10^{9}$–$10^{10}M_{\odot}$ in the first Gyr of the lifetime of the
Universe without invoking unrealistic (or fine-tuned) accretion rates.

###### keywords:

galaxies: nuclei — quasars: supermassive black holes — galaxies: formation —
galaxies: structure — galaxies: high-redshift — dark matter

††pagerange: On the growth of supermassive black holes formed from the
gravitational collapse of fermionic dark matter cores–LABEL:lastpage††pubyear:
2023

##  1 Introduction

The formation, growth, and nature of the supermassive BHs (SMBHs) residing at
the galaxy centers are outstanding problems in astrophysics and cosmology.
Important unresolved questions include: how can they grow so large and so fast
to be present in the farthest distant quasars Volonteri (2012); Woods et al.
(2019); what is the nature and mass of BH seeds that grow to form the SMBHs of
$\sim 10^{8}$–$10^{9}M_{\odot}$ in the high-$z$ Universe Zhu et al. (2022);
and what is the nature of the connection between the total mass of a host
galaxy and the mass of its central SMBH Volonteri et al. (2021).

Here we propose a new paradigm for the nature and formation of SMBH seeds,
which arise from the gravitational collapse of high-density regions of dark
matter (DM) at the galaxy centers Argüelles & et al. (2021). We present
calculations on the subsequent growth of such BH seeds from an accretion disk
around a Kerr BH in a fully general relativistic framework. In this way, we
aim here to provide answers to the above three main questions.

Among the various scenarios analyzed in the literature to explain the origin
of SMBHs (see Inayoshi et al. (2020); Volonteri et al. (2021); Mirabel &
Rodríguez (2022) for recent reviews), we can divide them into two main
categories according to their formation channel: (I) channels associated with
baryonic matter, i.e., gas and stars, and (II) channels associated with early
universe cosmology. In this work, we propose a novel, third possible scenario:
(III) channels associated with DM. Before motivating this new framework, we
highlight the pros and cons of the most studied formation channels. In the
case of the baryonic channels (I), we can distinguish among (a) Population III
stars, and (b) direct collapse to a BH (DCBH). Pop. III stars are physically
motivated (though yet hypothetical) stars with an average mass of $\sim
10^{2}M_{\odot}$ Madau & Rees (2001); Hosokawa et al. (2016) originated in
metal-free clouds hosted in small halos of $\sim 10^{6}M_{\odot}$ at high $z$.
Thus, the stellar collapse of such massive stars would lead to a BH seed of
$\sim 10^{2}M_{\odot}$ which, under idealistic accretion conditions, would
reach a $10^{9}M_{\odot}$ SMBH in the first billion years. However, recent
state-of-the-art simulations show that BH seeds of $<10^{3}M_{\odot}$ fail to
grow until $10^{8}M_{\odot}$ at $z\sim 6$ because of strong radiative feedback
Zhu et al. (2022). In the DCBH scenario (b), dense gas clumps at the center of
massive halos of $\sim 10^{8}M_{\odot}$ become globally unstable and collapse
first to a supermassive star of $\sim 10^{4}$–$10^{5}M_{\odot}$, which then
undergoes core-collapse to a central BH. The newborn BH then grows fast by
accreting surrounding material ending in a larger BH seed with masses up to
$\sim$ few $10^{5}M_{\odot}$ Begelman et al. (2006, 2008); Woods et al.
(2017). Former simulations suggest that the conditions for the occurrence of
this scenario, e.g., to reach metal-free gas able to form atomic (instead of
molecular) gas, may be rare Habouzit et al. (2016). However, recent
hydrodynamic N-body simulations show that DCBH scenarios are among the most
preferred mechanisms to explain the origin of the SMBHs Zhu et al. (2022);
Latif et al. (2022), albeit numerical resolution issues and the use of
phenomenological recipes limit the generality of the results Zhu et al.
(2022).

On different physical grounds, channels of BH formation and growth associated
with the early Universe (II) would take place before galaxy formation and
include primordial BHs Carr & Kühnel (2020), or even more exotic candidates
such as topological defects in forms of cosmic string-loops Bramberger et al.
(2015). However, there is no observational evidence (nor direct or indirect)
of such processes since they are associated with very early cosmological
epochs poorly constrained by observations.

This work proposes an SMBH formation channel in the high $z$ Universe
conceptually different from (I) and (II) discussed above. A crucial
qualitative difference with the channels mentioned above is that it does not
rely either on specific pristine gas assemblies or on the very early epoch of
the Universe. Instead, it depends on the gravitational collapse and subsequent
growth of dense fermionic DM cores that originate at the center of the halos
as they form. Such novel dense core-diluted halo DM density distributions
(profiles) are natural consequences of maximum entropy production principle
(MEPP) scenarios of halo formation, in which the fermionic (quantum) nature of
the particles is duly accounted for Argüelles & et al. (2021); Argüelles et
al. (2022a).

This paper is organized as follows. In section 2, we discuss the underlying
physics behind our new SMBH seed formation scenario and give specific examples
of how such seeds may arise in the high $z$ Universe. In section 3, we present
relativistic calculations of the time evolution of the mass and spin of the
recently born BH seed with a surrounding accretion disk. We summarize and
conclude in section 4.

Figure 1: Caloric curves for the critical solutions of three examples with a
darkino mass of $mc^{2}=100$ keV: (1) $M_{\rm vir}=5\times 10^{10}M_{\odot}$
and $r_{s}=30$ kpc, (2) $M_{\rm vir}=5\times 10^{11}M_{\odot}$ and $r_{s}=80$
kpc, (c) $M_{\rm vir}=5\times 10^{12}M_{\odot}$ and $r_{s}=250$ kpc. The
necessary constraints ($\mu={\rm e}^{W_{0}-\theta_{0}}$ and $N$) to compute
the caloric curves are extracted from the corresponding critical solutions.
For the x-axis, $1-M/(mN)$ is (minus) the binding energy normalized with $mN$.
For the y-axis, $T$ is the normalized temperature of the system as measured by
an observer placed at infinity. Thin lines are unstable solutions, while thick
lines are meta-stable. A black dot marks the change in stability. For example
(3), there are no stable core-halo solutions. An equivalent solution with the
same core mass as for the other examples ($C_{1}$ and $C_{2}$) is marked by
the black diamond. Figure 2: Density profiles of the solutions $C_{1}$,
$C_{2}$ and $C_{3}$ from fig. 1. Thin lines are unstable solutions, while
thick lines are meta-stable. The dotted line shows the corresponding fully
degenerate core solution, providing the core radius $r_{c}$ where the density
falls to zero (e.g., surface). The black point is labeled with the value of
core mass $M_{\rm crit}=M(r_{c})$. Figure 3: Surface density curves for three
examples as described fig. 1. Thin lines are unstable solutions, while thick
lines are meta-stable. The shaded regions mark the maximal extent of
1-$\sigma$ and 3-$\sigma$ errors as obtained in Donato et al. (2009)).

##  2 The dark matter channel of SMBH formation

The DM channel proposed for forming early BH seeds takes place within the more
general and broad theory of DM halo formation as developed in Argüelles & et
al. (2021) within WDM cosmologies. Unlike the (zoom-in) hydrodynamical
simulation approaches applied in baryonic BH-seed formation channels Zhu et
al. (2022) (such as (Ia) and (Ib) introduced above), the problem of DM halo
formation is here assessed using a thermodynamical approach for systems of
self-gravitating fermions which maximize its coarse-grained entropy at the end
of relaxation Argüelles & et al. (2021). This mechanism’s most general density
profile develops a dense and compact DM core (supported against gravity by
Pauli degeneracy pressure) surrounded by a dilute halo. The first early
studies of this kind of fermionic core-halo solutions dates back to the
${}^{\prime}80$s (Chau et al., 1984; Ingrosso & Ruffini, 1988), followed by a
series of further and recent developments Gao et al. (1990); Chavanis &
Sommeria (1998); Bilic et al. (2002); Chavanis (2006); Destri et al. (2013);
Argüelles & Ruffini (2014); Ruffini et al. (2015); Chavanis et al. (2015);
Argüelles & et al. (2018, 2019); Becerra-Vergara et al. (2020); Becerra-
Vergara & et al. (2021); Chavanis (2022); Argüelles et al. (2022b); Krut et
al. (2023). The more realistic version of this model, which includes particle
evaporation and central (fermion) degeneracy, was developed in General
Relativity in Argüelles & et al. (2018) and is referred to as the (extended)
Ruffini-Argüelles-Rueda (RAR) model111It also called in the literature as the
general relativistic fermionic King model (Chavanis, 2022).. The model’s
fermionic halo explains the galaxy rotation curves, while the degenerate
fermion core has key implications for galactic centers: it can mimic a central
BH or eventually collapse into one (Argüelles & et al., 2018, 2019; Becerra-
Vergara et al., 2020; Argüelles & et al., 2021; Becerra-Vergara & et al.,
2021; Argüelles et al., 2022b, a). One key advantage of this semi-analytical
approach is that it allows for a detailed description of the relaxed halos
from the very center to the periphery, not possible in N-body simulations (due
to limitations in resolution at inner-halo scales). Moreover, the
thermodynamic approach here applied includes richer physical ingredients than
those involved in traditional simulations: (i) general relativity (GR) —
necessary for a proper gravitational DM core-collapse towards an SMBH seed;
(ii) the quantum nature of the particles — allowing for an explicit fermion
mass dependence in the profiles; (iii) the Pauli principle self-consistently
included in the phase-space DF at relaxation — giving place to novel core-halo
profiles. Interestingly, this theoretical framework allows linking the
behavior and evolution of the dark matter particles from the early Universe to
the late stages of non-linear structure formation at virialization. That is,
the DM halo profiles are obtained by first calculating the linear matter power
spectrum for $\mathcal{O}(\text{keV})$ DM fermions, to then use the
corresponding extended Press-Schechter formalism to obtain the virial halo
mass, $M_{\rm vir}$, with associated redshift $z_{\rm vir}$ (see Appendix, and
Argüelles & et al. (2021) for a previous result). Finally, we obtain the
fermionic halos by assuming that a MEPP takes place at the end of relaxation
and agrees with the virial mass constraints. Such a MEPP, originally
introduced in Chavanis (1998) generalizing Lynden-Bell results, allows
obtaining a most likely coarse-grained DF of Fermi-Dirac type222Approaches of
this kind based on statistical mechanics may present potential difficulties:
the relatively short time-scale involved during violent relaxation may not be
enough for the system to explore the full phase-space to reach a most likely
final state (see Chavanis, 2022). [see Eq. (1) in Argüelles & et al. (2021)]
that depends on four free parameters, $m$ the particle mass, $\beta$ the
(dimensionless) temperature, $\theta$ the degeneracy parameter, and $W$ the
cut-off particle energy. All parameters are set at the center of the
configuration (denoted with the subscript $0$) to fully solve the system of
equilibrium differential equations of the RAR model [see Eqs. (8)–(12) in
Argüelles & et al. (2021)]. The RAR model can be applied to form a DM halo and
their central SMBH seeds.

Once having such a DF at the end of the relaxation, we calculate the full
family of fermionic density profiles at equilibrium in GR, all with given
total particle number N and thus with the same total (Newtonian) halo mass
($M_{\rm tot}\equiv M_{\rm vir}=mN$). For such an endeavor, we follow the
thermodynamic approach applied in Argüelles & et al. (2021) within the
microcanonical ensemble to further calculate the important problem of
(thermodynamical and dynamical) stability of such a family of equilibrium
solutions. That is, not all the equilibrium solutions of self-gravitating
fermions will be thermodynamically stable333While the problem of equilibrium
involves the extremization of entropy $\delta S=0$ (at fixed energy and N),
the problem of stability has to do with second-order variations of entropy
$\delta^{2}S=0$ Chavanis (2020).. This is done following the Katz criterium
(see Appendix A in Argüelles & et al. (2021) and Appendix C in Alberti &
Chavanis (2020) for updated summaries, and Katz (1978, 1979) for the original
works) for which is necessary to calculate the caloric curves in GR, given as
the inverse temperature of the system $1/\hat{T}\equiv\beta^{-1}$ Vs. (minus)
the binding energy $-E_{b}=-(M-mN)c^{2}$ (with $\hat{T}$ the normalized
temperature such that $\beta=kT/(mc^{2})$ and $k$ the Boltzmann constant). A
distinctive characteristic of such general relativistic caloric curves for
fermions (at difference with the Newtonian case), is the existence of a last
stable configuration located at the turning point (see points $C_{i}$ in fig.
1), followed by a second spiral feature of relativistic origin. Based on the
Katz criterium, this important result was first shown in Chavanis & Alberti
(2020); Chavanis (2020); Alberti & Chavanis (2020); Chavanis (2022) for a
self-gravitating system of fermions bounded in a box, together with a detailed
characterization of the caloric curves and the gravitational phase transitions
occurring to the Fermi gas. Remarkably, as shown as well in Argüelles & et al.
(2021) for the more realistic RAR (or relativistic fermionic King) model, the
existence of such a last stable configuration located at point $C_{i}$,
implies the onset of a thermodynamical instability of the core-halo solutions,
where the fermion-core collapses towards an SMBH.

Based on the above (relativistic) thermodynamical analysis, we demonstrate,
for the first time, the existence of a critical fermion-core (located at
$C_{i}$ in the caloric curve) which is surrounded by a DM halo of realistic
astrophysical application (see the DM density profiles in fig. 2, concerning
fig. 1 and the fulfillment with observations in fig. 3). We do it for a
typical particle mass in the range $50$–$345$ keV, i.e., $m=100$ keV, to then
explore (see section 3.2) other particle masses in that range. The relevance
of such a narrow window of particle masses is taken from Argüelles & et al.
(2018, 2019), where it was shown it is possible to find core-halo RAR
solutions where the outer halo agrees with the galaxy rotation curves while
the DM core (not necessarily critical) can mimic the central BH (see also
Becerra-Vergara et al. (2020); Becerra-Vergara & et al. (2021); Argüelles et
al. (2022b) for a tailored analysis about the Milky Way and Sgr A*).

In fig. 1, we give three specific examples of caloric curves for different
(Newtonian) halo masses $M_{\rm vir}$ covering the relevant range between
$5\times 10^{10}$–$5\times 10^{12}M_{\odot}$. Among all the equilibrium core-
halo solutions along each caloric curve, only the ones placed within the
branches $B_{i}-C_{i}$ ($i=1,2,3$) are thermodynamically and dynamically
stable within cosmological time-scales, as clearly explained in Argüelles & et
al. (2021) (see also (Chavanis & Alberti, 2020; Chavanis, 2020; Alberti &
Chavanis, 2020; Chavanis, 2022) for analogous results obtained for fermionic
systems bounded in a box). We recall that the core-halo solutions located at
$C_{i}$ (see fig. 2) correspond to the last stable configuration where the DM-
core achieves the onset of gravitational collapse (of relativistic origin)
towards a BH. Interestingly, at the fixed mass of $m=100$ keV, the theory
predicts a threshold total halo mass $M_{\rm vir}\sim 5\times
10^{12}M_{\odot}$ above which the stable branch $B_{3}-C_{3}$ disappeared (see
right panels of fig. 1). The shrinking of the metastable branch ($B-C$) as the
total mass $M_{\rm tot}$ (or $N$) increases; together with the existence of a
threshold particle number $N^{*}$ above which the meta-stable branch
disappear, was first shown in Chavanis & Alberti (2020); Alberti & Chavanis
(2020) for box confined systems. This remarkable result, when applied to
realistic halos as in this work, may explain why we do not observe single
virialized galaxies above such an order of magnitude (i.e., above
$\mathcal{O}(10^{12})M_{\odot}$), indicating how powerful the thermodynamics
of self-gravitating systems can be.

Moreover, in such a halo-mass window, we further show in fig. 3 that the halo
regime of the RAR solutions has the required morphology in the sense of being
able to fulfill the DM surface density relation. We also show in fig. 2 the
density profiles of such core-halo astrophysical solutions at the onset of DM
core-instability, all having a typical SMBH seed of $M_{\rm crit}\approx
6.3\times 10^{7}M_{\odot}$. We have defined the SMBH seed mass at the core
radius $r_{c}$ of the core-halo solution, i.e., $M_{\rm crit}=M(r_{c})$, with
$r_{c}$ coinciding with the surface radius of the corresponding fully
degenerate solution (e.g., where the density falls to zero, see dotted line in
fig. 2). Such a numerical value of the critical mass can be well approximated
with the semi-analytic eq. 1 (only valid within the fully degenerate regime),
which is no other than the Oppenheimer-Volkoff (OV) mass limit Oppenheimer &
Volkoff (1939)

$M_{\rm crit}\approx 0.384\frac{m_{\rm Pl}^{3}}{m^{2}}\approx 6.274\times
10^{9}\left(\frac{\text{10 keV}}{mc^{2}}\right)^{2}M_{\odot},$ (1)

where $m_{\rm Pl}=\sqrt{\hbar c/G}\approx 2.176\times 10^{-5}$ g is the
Planck’s mass and $m$ is the darkino mass.

The reason for the validity of this critical mass approximation of our DM
cores can be easily understood when realizing that the core-halo fermionic
solutions under consideration here (see fig. 2) encompass two different
regimes: a highly degenerate (quantum) regime of the fermionic-core (i.e.,
$\theta_{0}>10$) close to the fully-degenerate case, which monotonically
transitions to the classical regime at larger distances from the center,
leading to the (Boltzmannian) halo region (where $\theta(r)\ll-1$). Further
detailed explanations about the equivalence between the traditional turning-
point instability criterium of core-collapse Schiffrin & Wald (2014), to that
of the last (dynamical and thermodynamical) stable solution at point $C_{i}$
in the caloric curves are given in Section 4 of Argüelles & et al. (2021) and
references therein.

##  3 BH mass and spin evolution

We follow the treatment of a nearly geodesic thin accretion disk around a Kerr
BH in Novikov & Thorne (1973); Page & Thorne (1974); Thorne (1974). Matter and
radiation transfer energy and angular momentum to the BH during the accretion.
In particular, it is essential to account for the feedback of
radiation/photons onto the BH since they exert a counter-torque Godfrey (1970)
that avoids the BH from reaching the extreme regime $a=M$. This implies that
the accretion of massive particles and radiation does not lead the BH to
become a naked singularity Thorne (1974). We denote by $dm$ the rest-mass
accreted by the BH in a coordinate time interval $dt$, so $\dot{m}=dm/dt$ is
the rest-mass accretion rate, and $\dot{M}_{\rm rad}$ and $\dot{J}_{\rm rad}$
are the rate of energy and angular momentum transfer by radiation to the BH.
We refer the reader to the appendix B for details of the equations governing
the evolution of the mass and angular momentum of the BH. We use geometric
units $c=G=1$ unless otherwise specified.

###  3.1 Accretion rate and luminosity

We calculate the rate at which rest-mass flows inward through the local
balance between the tidal gravitational acceleration and the radiation
pressure along the $z$ coordinate. This condition is

$zR^{\tilde{z}}_{\tilde{0}\tilde{z}\tilde{0}}=\kappa F(r),$ (2)

where $\kappa=0.34$ cm2 g-1 is the Thomson electron scattering opacity,
$\bm{R}$ is the Riemann tensor. Using the change of variable $x=\sqrt{r/M}$,
Eq. (2) becomes

$\frac{3\,\kappa\,\dot{m}}{8\pi z}f(x,\alpha)=1,$ (3)

where

$f(x,\alpha)=\frac{x^{3}\,g(x,\alpha)}{x^{4}-4\alpha x+3\alpha^{2}},$ (4)

and

$g(x,\alpha)=x-x_{0}-\frac{3}{2}\alpha\ln\left(\frac{x}{x_{0}}\right)-3(A_{1}+A_{2}+A_{3}),$
(5)

with

$A_{1}=\frac{(x_{1}-\alpha)^{2}}{x_{1}(x_{1}-x_{2})(x_{1}-x_{3})}\ln\left(\frac{x-x_{1}}{x_{0}-x_{1}}\right).$
(6)

Here, $x_{1},x_{2},x_{3}$ are the roots of the polynomial $x^{3}-3x+2a$. The
terms $A_{2}$ and $A_{3}$ can be obtained from $A_{1}$ through a cyclic order
of the set $\\{1,2,3\\}$. For a given value of $\alpha$ and an appropriate
expression $z=z(r)$, the value of $\dot{m}$ such that Eq. (3) has exactly one
solution defines a critical accretion rate $\dot{m}_{\rm crit}$. Above this
rate, the disk enters the super-Eddington regime, and the thin disk
approximation breaks down. The tidal gravitational pull reaches a maximum
value at $z=H$. Additionally, the thin disk approximation is reasonably
accurate insofar as the half thickness of the disk obeys $H<r$ Chen &
Beloborodov (2007); Liu et al. (2017). Consequently, by introducing a
parameter $0<\beta\leq 1$ and setting $z=r$ (see Abolmasov & Chashkina (2015)
for details), we calculate the accretion rate by

$\dot{m}\equiv\beta\,\dot{m}_{\rm crit}=\frac{8\pi\beta M}{3\kappa\,{\rm
max}\left\\{\frac{f(x,\alpha)}{x^{2}}\right\\}},$ (7)

with ${\rm max}\left\\{\frac{f(x,\alpha)}{x^{2}}\right\\}$, the maximum value
can take the given function at each radius for given $\alpha$. To calculate
the power emitted by the system, we only consider the photons that leave the
disk and do not fall into the BH. The procedure is equivalent to calculating
the rate of energy transfer to the BH, i.e., Eq. (27), but using the factor
$1-C$ instead of $C$

$L_{\rm
source}=-\frac{2}{\pi}\int_{r_{0}}^{\infty}\\!\int_{0}^{\pi/2}\\!\\!\int_{0}^{2\pi}\\!\\!\left(1-C\right)k_{t}\\!F(r)dS.$
(8)

We do not consider the possible photon recapture by the disk.

###  3.2 Growth of Kerr BH seeds

Figure 4: BH evolution in time for different BH seeds and values of $\beta$.
Initial conditions are $\alpha_{i}=0$ and darkino masses $56$ keV, $100$ keV,
$200$ keV, and $350$ keV. The spin parameter does not depend on the BH mass.
The initial redshift is $z_{0}=5.5$ with $t_{0}=1022$ Myr for a halo mass
$M_{vir}=5\times 10^{11}M_{\odot}$.

Figure 5: Accretion rate and Luminosity at source as functions of $t$ for
different BH seeds with $\beta=1$. Initial conditions are $\alpha_{i}=0$ and
darkino masses $56$ keV, $100$ keV, $200$ keV, and $350$ keV. The initial
redshift is $z_{0}=5.5$ with $t_{0}=1022$ Myr for a halo mass $M_{vir}=5\times
10^{11}M_{\odot}$.

The equations of evolution are

$\displaystyle\dot{M}$ $\displaystyle=\dot{m}\epsilon_{0}+\dot{M}_{\rm rad},$
(9) $\displaystyle\dot{\alpha}$
$\displaystyle=\frac{1}{M^{2}}\left(l_{0}\dot{m}+\dot{J}_{\rm
rad}-2M\dot{M}\alpha\right).$ (10)

It is helpful to make the $M$ dependence explicit and express the equations in
terms of the system’s parameters and dimensionless variables. To that end, let
us write

$\displaystyle M$ $\displaystyle=M_{i}\tilde{M},$ (11) $\displaystyle\dot{m}$
$\displaystyle=M_{i}\tilde{M}\beta\dot{\tilde{m}},$ (12)
$\displaystyle\dot{M}_{\rm rad}$
$\displaystyle=M_{i}\tilde{M}\beta\dot{\tilde{m}}\tilde{M}_{\rm rad},$ (13)
$\displaystyle\dot{J}_{\rm rad}$
$\displaystyle=M_{i}^{2}\tilde{M}^{2}\beta\dot{\tilde{m}}\tilde{J}_{\rm rad},$
(14) $\displaystyle l_{0}$ $\displaystyle=M_{i}\tilde{M}\tilde{l}_{0},$ (15)

where $\dot{\tilde{m}}$, $\tilde{M}_{\rm rad}$, $\tilde{J}_{\rm rad}$ and
$\tilde{l}_{0}$ are functions of $\alpha$ only. With these definitions, the
ratio between Eq. (9) and Eq. (10) leads to the solution

$\tilde{M}=\exp\left\\{\\!\int_{\alpha_{i}}^{\alpha}\\!\frac{\epsilon_{0}+\tilde{M}_{\rm
rad}}{\tilde{l}_{0}+\tilde{J}_{\rm
rad}-2\alpha_{*}\\!\left(\epsilon_{0}+\tilde{M}_{\rm
rad}\right)}d\alpha_{*}\\!\\!\right\\}\\!,$ (16)

for the BH mass as a function of the dimensionless spin parameter. From Eq.
(10) we obtain

$\Delta
t=\frac{1}{\beta}\\!\int_{\alpha_{i}}^{\alpha}\frac{1}{\dot{\tilde{m}}\left[\tilde{l}_{0}+\tilde{J}_{\rm
rad}-2\alpha_{*}\left(\epsilon_{0}+\tilde{M}_{\rm
rad}\right)\right]}d\alpha_{*}.$ (17)

From Eq. (16) and Eq. (17), we can deduce some properties of the BH’s temporal
evolution: The solution $\alpha(t)$ is independent of $M$, but it does depend
on the parameter $\beta$. Moreover, the general solution $\alpha(t,\beta)$
obeys the relation $\alpha(t,\beta)=\alpha(\beta t,1)$. The BH mass inherits
this property through Eq. (16), and we get $M(t,\beta)=M(\beta t,1)$. The same
will happen with any $\alpha$ and $M$ function. This behavior suggests that
knowing the solution for $\beta=1$ is enough to produce other solutions by
properly scaling the variable $t$ and multiplying it by an adequate power of
$\beta$. For example, the accretion rate and the power emitted obey the
relations $\dot{m}(t,\beta)=\beta\dot{m}(\beta t,1)$ and $L_{\rm
source}(t,\beta)=\beta L_{\rm source}(\beta t,1)$.

The properties of $M(t,\beta)$ and $\alpha(t,\beta)$ become evident when
inspecting the panels in fig. 4, where we show the evolution of several BH
seeds for values $\beta=1$, $\beta=0.1$, and $\beta=0.01$ as a function of
time and the cosmological redshift. The initial redshift $z_{0}$ corresponds
to the typical collapse $z^{*}$ of a gravitationally bound object with mass
$M_{\rm vir}$ in the extended Press-Schechter formalism (see appendix A). The
standard $\Lambda$CDM cosmology sets the relation between $t$ and $z$. Note
that $\alpha(t,\beta)$ does not depend on the BH mass, so the top left panel
contains only three curves, one for each value of $\beta$.

In fig. 5, we show the evolution of the accretion rate and the power emitted
by the BH. The slight reduction in the accretion rate for small values of $t$
occurs because the factor ${\rm max}\left\\{f(x,\alpha)/x^{2}\right\\}$ in Eq.
(7) increases by a factor of $\sim 5$ as $\alpha$ grows. The spin-up of a BH
from $\alpha_{i}=0$ to $\alpha=0.99775$ takes $\sim 37/\beta$ Myr. The BH mass
grows from $M_{i}$ to $\sim 3M_{i}$. Once $\alpha$ reaches the state
$\dot{\alpha}\approx 0$, the BH mass grows exponentially. By solving Eq. (9),
we find a relation that allows us to estimate the time $\Delta t$ needed for
any BH seed to grow up to a final mass $M_{f}>3M_{i}$, starting from
$\alpha_{i}=0$:

$\Delta t=6.2\times
10^{7}\ln\left\\{\frac{3M_{f}}{5M_{i}}\right\\}\beta^{-1}\;\text{yr}.$ (18)

Our result differs from Eq. (1) in Haiman & Loeb (2001), viz.,

$\Delta t_{\rm HL}=4\times
10^{8}\varepsilon\ln\left\\{\frac{M_{f}}{M_{i}}\right\\}\beta^{-1}\text{ yr},$
(19)

where $\varepsilon=L_{\rm source}/\dot{m}c^{2}$ is the radiative efficiency of
the accretion process. The differences arise for two reasons: first, Haiman &
Loeb (2001) did not consider the BH spin, while our treatment accounts for it
in a self-consistent manner: fig. 4 shows that during the transition to a
saturated spin, the BH growth is faster than exponential. Second, their
definition of the accretion rate rests upon a global balance between Newtonian
gravity and spherically symmetric radiation pressure. Consequently, the
radiative efficiency appears as a constant factor in eq. 19. Our definition
adopts a relativistic local balance between radiation pressure and vertical
gravity, which increases the accretion rate Abolmasov & Chashkina (2015) and
allows the evolution of the efficiency with the BH parameters $M$ and
$\alpha$.

Using the limiting efficiency for a Kerr BH, $\varepsilon=0.3$, we obtain
$\Delta t<0.52\Delta t_{\rm HL}$. Thus, the growth from a possible BH seed
within our framework of $M_{i}=5\times 10^{6}M_{\odot}$ to $M_{f}=5\times
10^{9}M_{\odot}$ takes only $\Delta t\approx 0.4$ Gyr. In particular, our DM
channel for SMBH formation predicts that typical DM halos of $M_{\rm vir}\sim
10^{11}M_{\odot}$ formed at $z_{0}\sim 7.5$ (see appendix A) can harbor SMBH
seeds of $M_{i}=6.3\times 10^{7}M_{\odot}$ (i.e., for $m=100$ keV), which can
grow (within standard accretion rates) up to $M_{f}\equiv M=3\times
10^{9}M_{\odot}$ in $\Delta t\approx 0.2$ Gyr, thus in agreement with most
distant (i.e., $z\sim 6$) and most massive quasars observed (see, e.g.,
Mirabel & Rodríguez (2022)). This result provides a new channel for SMBH
formation from DM which can overcome traditional baryonic scenarios such as
Pop. III stars, whose light BH seeds of $<10^{3}M_{\odot}$ fail to grow even
to $\sim 10^{8}M_{\odot}$ by $z\sim 6$ (Zhu et al., 2022).

##  4 Conclusions

We have proposed a novel channel for SMBH formation in the high redshift
Universe, which is not associated with baryonic matter (massive stars) or
primordial cosmology. Instead, it relies on the gravitational collapse into a
BH of fermionic dense DM cores that arise at the center of DM halos as they
form and on the subsequent growth of the newborn BH by accretion. The
formation of dense core-dilute halo density distributions of DM form when
accounting for a fermionic (quantum) nature of the DM particles in the
structure of the DM halos, which is not feasible in traditional N-body
simulations Argüelles & et al. (2021); Argüelles et al. (2022a). For fermion
masses in the range of $50$–$345$ keV, this alternative non-linear structure
formation approach predicts stable DM halos that agree with observations and
harbor dense DM cores at the brink of gravitational collapse, with masses on
the range of $10^{6}$–$10^{8}M_{\odot}$ (see Section 2 for a WDM cosmology
with $m=100$ keV). Thus, it offers a whole new range of SMBH seeds that are
considerably larger than the ones predicted by baryonic formation channels,
including the DCBH scenario (see Section 1).

In this article, we assessed the mass and angular momentum evolution of such
massive BH seeds using a standard, geodesic general relativistic disk
accretion model. We self-consistently account for the feedback of
radiation/photons onto the BH (see Section 3). We have explicitly shown in
Section 4 that these SMBH can grow to masses in the range $\sim
10^{9}$–$10^{10}M_{\odot}$ within the first Gyr of the life of the Universe,
in good agreement with the farthest quasars observed, without invoking
unrealistic (or fine-tuned) accretion rates. A relevant advantage of the
present framework is that it does not require star formation within such a
short cosmological time scale as in traditional baryonic channels. At the same
time, it naturally connects the total mass of a host galaxy and the mass of
its central SMBH observed today, all in terms of DM and its cosmological
evolution.

## Acknowledgements

C.R.A. was supported by CONICET of Argentina, the ANPCyT (grant
PICT-2018-03743), and ICRANet. K.B. acknowledges partial support from the
Science Committee of the Ministry of Science and Higher Education of the
Republic of Kazakhstan (Grant No. AP19680128).

## DATA AVAILABILITY

The article’s data will be shared with the corresponding author upon
reasonable request.

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##  Appendix A DM halo formation scales within the Press-Schechter paradigm

In the context of the calculation of solutions that represent fermionic DM
halos via the MEPP, as shown in section 2 and initially developed in Argüelles
& et al. (2021), a key parameter comes from the boundary conditions for the
solutions, for these general quasi-equilibrium solutions can only describe DM
halos if their masses and radii correspond to the ones characteristic of such
structures Argüelles & et al. (2018, 2019). Indeed, the conclusions of such
analysis regarding the thermodynamic stability of this system are susceptible
to such contour conditions, with other authors reaching different conclusions
by using the same analysis when considering systems smaller than galactic
halos Chavanis et al. (2015); Alberti & Chavanis (2020). Thus, appropriately
choosing the mass and radius corresponding to a galactic halo becomes
important. While DM halo masses are usually a well-estimated observable for
most observed structures Karukes et al. (2019), defining the radius of such
structures can depend on the particular cosmological model considered (see,
e.g., Mo et al. (2010) for a thorough explanation). A common measure for such
radii is the virial radius, obtained from applying the virial theorem to the
gravitationally bound structure. However, due to differences arising from the
particular collapse models used, it is common to use instead the $r_{200}$
radius, defined as the radius where the density of the system is $200$ times
the background DM density Binney & Tremaine (2008), which is close to the
value of the virial radius $r_{\rm vir}$ for most models Bryan & Norman
(1997). Thus, the relation between mass and $r_{200}$ radius for these
structures is relatively straightforward:

$M_{\rm vir}=200\frac{4}{3}\pi
r_{200}^{3}\rho_{c}(t)=100\frac{H_{0}^{2}r_{200}^{3}[1+z(t)]^{3}}{G}\ ,$ (20)

where $M_{\rm vir}$ stands for the (virial) mass of the object. We see,
however, that this definition is now time-dependent, as the background density
of the universe evolves with time. In a cosmological context, we can also
reinterpret this time dependence as the expansion of the spatial scales
involved in the problem with time.

To marginalize this time dependence, a first approach can be to estimate the
most likely collapse time of a structure of size $M_{\rm vir}$ and obtain the
physical radius corresponding to such scale. While a deep study of this
collapse time would typically involve a full study on nonlinear cosmology and
structure formation (such as, e.g., Navarro et al. (1997); Macciò et al.
(2012); Fitts et al. (2019)), it is sufficient for this study to use the
Press-Schecter formalism Press & Schechter (1974), based on the results from
linear cosmology for a given model. This is a well-studied theory, and a full
description can be found, for example, in Mo et al. (2010); Binney & Tremaine
(2008) among many other works. Here, we will limit ourselves to a summary of
the assumptions and results. To study this formalism, it is necessary first to
study how the overdensities collapse in the late, matter-dominated universe.
For this, we can use one of the simpler nonlinear collapse models, the
spherical collapse (see, e.g., Mo et al. (2010)). Under the assumptions of
this model, any given overdensity collapses to form a virialized structure,
and at that point, the linear theory of perturbation evolution will predict an
overdensity of $\delta_{c}\sim 1.69$. So, according to this theory, we can
assume that given an overdensity field evolving according to linear dynamics,
a peak of density $>\delta_{c}$ would instead correspond to a collapsed halo.
The core assumption of Press-Schechter’s formalism regards how to relate this
observation to the halo mass function. The formalism proposes that the
fraction of the universe’s total mass that is in halos of masses greater than
$M$ is equal to the probability of a given overdensity is greater than the
critical $\delta_{c}$, where the overdensity field $\delta_{M}$(t) is averaged
with a filter of characteristic mass $M$. For this study, knowing the exact
mass fraction of a given mass $M$ is not important. Instead, we are interested
in the characteristic timescale at which overdensities of mass $M$ are most
likely to collapse. In this case, we can assume most overdensities will start
to collapse when the standard deviation $\sigma_{M}$ of the Gaussian field
$\delta_{M}$ (known as the mass variance) crosses the threshold of
$\delta_{c}$. Thus, for an overdensity of mass $M$, we define a characteristic
collapse time as

$\sigma_{M}(z^{*})=\delta_{c},$ (21)

where

$\sigma^{2}_{M}(t)=\frac{1}{2\pi^{2}}\int_{0}^{\infty}P(k)D^{2}(t)W^{2}(k,R)k^{2}{\rm
d}k,$ (22)

being $D(t)$ the linear growth rate of perturbations, $P(k)$ the linear matter
power spectrum, $W(k,R)$ a window function of characteristic radius $R(M)$
(taken as a top hat function here, see e.g. Mo et al. (2010)), and
$\delta_{c}\simeq 1.69$ according to spherical collapse Mo et al. (2010).

We note that to obtain this characteristic collapse mass, it is necessary to
know the linear matter power spectrum, typically obtained through a study on a
particular cosmological model. In the preceding sections, we have seen a
particle mass-dependent description of DM halo formation, where these masses
are of order $\mathcal{O}$ 100 keV. This particle mass is significantly
lighter than most CDM models suggest Kolb & Turner (1990), and instead points
to an extension of the standard $\Lambda$CDM theory known as warm dark matter
(WDM) Bode et al. (2001); Lovell et al. (2012); Boyarsky et al. (2019). These
types of extensions, typically characterized by DM particle masses in the keV
ranges, show a considerably higher initial velocity dispersion when compared
to their standard CDM counterparts, which in turn predicts a smaller number of
small-scale structures in the universe and may alleviate some existing
tensions with CDM Bullock & Boylan-Kolchin (2017). The simplest production
scenario for WDM is that these particles are created via thermal processes
very early in the universe’s history, and indeed this production scenario
requires some degrees of freedom in the initial plasma well above $10^{3}$ Mo
et al. (2010). In any case, this scenario is an interesting benchmark for
other production scenarios, as most result in a similar suppression feature in
the matter power spectrum. An interesting class of models that can realize
this WDM scenario can be found in sterile neutrino WDM, where non-equilibrium
production of these particles can account for the observed DM fraction
Boyarsky et al. (2019); Adhikari & Others (2017), and particle self-
interaction can reconcile the parameter space of these models with observation
Yunis et al. (2021).

Figure 6: Characteristic collapse redshift $z^{*}$ as a function of
substructure mass $M$. CDM models are represented in black lines, and WDM
thermally produced relic models with $mc^{2}=2.5$ keV and $mc^{2}=10.0$ keV in
red, blue, and green, respectively. The models considered here are taken at
the $3\sigma$ threshold in the sense of equation (22), as an estimation of the
earliest possible halos. Figure 7: Virial radius $r_{200}$ at virialization
time as a function of substructure mass $M$. CDM models are represented in
black lines and WDM thermally produced relic models with $mc^{2}=2.5$ keV,
$mc^{2}=5.0$ keV, and $mc^{2}=10.0$ keV in red, blue, and green, respectively.
The models considered here are taken at the $3\sigma$ threshold in the sense
of equation (22), as an estimation of the earliest possible halos.

We can see the results for the $r_{200}$ as a function of the halo mass scale
$M$ in figure fig. 7 according to the Press-Schechter formalism, calculated
for three thermally produced WDM models as well as a standard $\Lambda$CDM
model. For reference, these $2.5$ and $10$ keV thermal models have similar
suppression features as $\sim 15$ and $100$ keV non-resonantly produced
sterile neutrinos, according to the criteria of Viel et al. (2013). We also
include a line representing a CDM model, but where eq. 22 has been refactored
with $3\sigma$ instead to represent the time when earliest collapsed
structures in the formalism are formed. There are thought to be produced
roughly when rare $3\sigma$ overdensities cross the $\delta_{c}$ barrier and
enter nonlinear collapse. However, in the mass scales relevant to these
studies, the differences in $r_{200}$ between WDM and CDM models are minimal,
and these cosmologies can be used interchangeably to calculate DM halo
solutions.

##  Appendix B Geodesic disk accretion

We here present the equations of the evolution of a Kerr BH during the
accretion of matter from a geodesic thin disk. The treatment closely follows
the formulation in Novikov & Thorne (1973); Page & Thorne (1974); Thorne
(1974).

In the equatorial plane ($\theta=\pi/2$) and close above it, the Kerr
spacetime metric can be written as

$ds^{2}=-e^{2\nu}dt^{2}+e^{2\psi}(d\phi-\omega dt)^{2}+e^{2\mu}dr^{2}+dz^{2},$
(23)

where $z$ is the height above the equatorial plane, and $\nu$, $\psi$, $\mu$,
$\omega$ are functions of the radial coordinate $r$:

$e^{2\nu}=\frac{r^{2}\Delta}{A},\quad e^{2\psi}=\frac{A}{r^{2}},\quad
e^{2\mu}=\frac{r^{2}}{\Delta},\quad\omega=\frac{2Mar}{A},$ (24)

being $\Delta=r^{2}-2Mr+a^{2}$, and $A=(r^{2}+a^{2})^{2}-\Delta a^{2}$, with
$M$ and $a=J/M$, respectively, the BH mass and angular momentum per unit mass.

Figure 8: Left panel: Angular velocity of test particles versus radial
distance $r$ in units of the BH mass, $M$. Right panel: Angular momentum
$L^{*}$ of test particles versus radial distance $r$ in units of the BH mass,
$M$

Figure 9: Left panel: Energy $E^{*}$ of test particles versus radial distance
$r$ in units of the BH mass, $M$. Right panel: radiative flux
$\mathcal{F}^{*}$ multiplied by $10^{5}$ of the accretion disk versus radial
distance $r$ in units of the BH mass, $M$.

Matter and radiation transfer energy and angular momentum to the BH during the
accretion. In particular, it is essential to account for the feedback of
radiation/photons onto the BH since they exert a counter-torque Godfrey (1970)
that avoids the BH from reaching the extreme regime $a=M$. This implies that
the accretion of massive particles and radiation does not lead the BH to
become a naked singularity Thorne (1974). We denote by $dm$ the rest-mass
accreted by the BH in a coordinate time interval $dt$, so $\dot{m}=dm/dt$ is
the rest-mass accretion rate, and $\dot{M}_{\rm rad}$ and $\dot{J}_{\rm rad}$
are the rate of energy and angular momentum transfer by radiation to the BH.
Therefore, we can write the BH energy and angular momentum conservation
equations as Thorne (1974)

$\displaystyle\dot{M}$ $\displaystyle=\dot{M}_{\rm matter}+\dot{M}_{\rm rad},$
(25) $\displaystyle\dot{J}$ $\displaystyle=\dot{J}_{\rm matter}+\dot{J}_{\rm
rad},$ (26)

where

$\displaystyle\dot{M}_{\rm matter}$ $\displaystyle=\epsilon_{0}\,\dot{m},$
(27) $\displaystyle\dot{J}_{\rm matter}$ $\displaystyle=l_{0}\,\dot{m},$ (28)
$\displaystyle\dot{M}_{\rm rad}$
$\displaystyle=-\frac{2}{\pi}\int_{r_{0}}^{\infty}\\!\int_{0}^{\pi/2}\\!\\!\int_{0}^{2\pi}\\!\\!Ck_{t}F(r)dS,$
(29) $\displaystyle\dot{J}_{\rm rad}$
$\displaystyle=\frac{2}{\pi}\int_{r_{0}}^{\infty}\\!\int_{0}^{\pi/2}\\!\\!\int_{0}^{2\pi}\\!\\!Ck_{\phi}F(r)dS,$
(30)

being $\epsilon_{0}$ and $l_{0}$ the specific (i.e., per unit mass) energy and
angular momentum of the matter accreted, i.e., at the radius $r_{0}$ of the
innermost stable circular orbit (ISCO). Assuming that radiation emitted from
the disk’s surface is isotropic, the normalized photon four-momentum, as
measured by a comoving observer, is
$k^{\tilde{\mu}}=p^{\tilde{\mu}}/p^{\tilde{0}}=(1,\sin\Theta\cos\Phi,\sin\Theta\sin\Phi,\cos\Theta)$
and $k^{\mu}=p^{\mu}/p^{\tilde{0}}$, where $p^{\mu}$ is the normalized photon
four-momentum in the coordinate frame. The expressions for $k^{\mu}$ can be
found in Appendix A of Thorne (1974) (see also Appendix C in Rueda et al.
(2022)). The factor $C=C(r,\Theta,\Phi)$ is the capture function that takes
the value $1$ or $0$ when photons emitted at radius a $r$ with local direction
$(\Theta,\Phi)$ are captured by the BH or escape to infinity, respectively.
The surface area element is $dS=2\pi r\sin\Theta\cos\Theta d\Phi d\Theta dr$.
The function $F(r)$ is the radiation flux emitted from the disk measured by
the comoving observer Novikov & Thorne (1973); Page & Thorne (1974)

$F(r)=-\frac{\dot{m}}{4\pi\sqrt{-g}}\frac{\Omega_{,r}}{(\epsilon-\Omega
l)^{2}}\int_{r_{0}}^{r}(\epsilon-\Omega l)l_{,\bar{r}}d\bar{r},$ (31)

where $\sqrt{-g}=e^{\nu+\psi+\mu}=r$, $\epsilon$, and $l$ is the specific
energy and angular momentum of circular geodesics of radius $r$ in Kerr-metric
(measured at infinity), and $\Omega=u^{\phi}/u^{t}$ is their angular velocity
measured in the coordinate frame, being $u^{\mu}$ the fluid four-velocity
Bardeen (1973). Using the change of variable $x=\sqrt{r/M}$ we can write

$\displaystyle\epsilon$
$\displaystyle=\frac{x^{3}-2x\pm\alpha}{x^{3/2}\sqrt{x^{3}-3x\pm 2\alpha}},$
(32) $\displaystyle l$ $\displaystyle=\pm\frac{M\left(x^{4}\mp 2\alpha
x+\alpha^{2}\right)}{x^{3/2}\sqrt{x^{3}-3x\pm 2\alpha}},$ (33)
$\displaystyle\Omega$ $\displaystyle=\frac{1}{M}\frac{1}{\alpha\pm x^{3}},$
(34)

where $\alpha=a/M$ is the dimensionless spin parameter. The upper/lower sign
corresponds to co-rotating/counter-rotating circular orbits. Clearly, we have
$\epsilon_{0}=\epsilon(r_{0})$ and $l_{0}=l(r_{0})$. The radius of the ISCO is
given by Bardeen (1973)

$\displaystyle r_{0}$
$\displaystyle=M\left[3+Z_{2}\mp\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})}\right],$ (35)
$\displaystyle Z_{1}$
$\displaystyle=1+(1-\alpha^{2})^{1/3}\left[(1+\alpha)^{1/3}+(1-\alpha)^{1/3}\right],$
(36) $\displaystyle Z_{2}$ $\displaystyle=\sqrt{3\alpha^{2}+Z_{1}},$ (37)

In fig. 8, we plot the orbital angular velocity $\Omega^{*}=M\Omega$ (left
panel) and angular momentum $L^{*}=l/M$ (right panel) of test particles as a
function of radial coordinate in the Kerr metric, for selected values of the
spin parameter. Counter-rotating test particles possess larger angular
velocity and momentum than co-rotating particles (for details see Boshkayev et
al. (2020); Boshkayev et al. (2021)).

In fig. 9, the energy per unit mass $\epsilon$ of test particles is shown as a
function of radial coordinate (left panel) and the radiative flux
$F^{*}=10^{5}M^{2}F/\dot{m}$ emitted from the accretion disk as a function of
radial coordinate (right panel) for different values of the spin parameter.