------------------------------------------------------------------------
From: Max Richter  max.c.richter@gmail.com 
To: gcnews@aoc.nrao.edu
Subject: submit jcapholes3.tex JCAP, 2006, in press

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\markright{A symbiotic scenario for the rapid formation of supermassive black 
holes}


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%\runninghead{\bf A Symbiotic Scenario for Rapid Formation}

\begin{document}

\vspace*{2.75cm}

\footnotesize

%\twocolumn[
\begin{flushleft}
{
%              \\ \hline
%              \\[0.5cm]
\title{A symbiotic scenario for the rapid formation
  of supermassive black holes}

\authors{M.C. Richter, G.B. Tupper and R.D. 
Viollier}

\inst{Centre for Theoretical Physics 
and Astrophysics,
Department of Physics,\\ University of Cape Town, Private Bag, Rondebosch 
7701, South Africa\\
  }
\end{flushleft} %         \\ \hline
\vspace{0.5cm}

\begin{abstract}
  The most massive 
black holes, lurking at the centers of large galaxies, must have formed less 
than a billion years after the big bang, as they are visible today in the form 
of bright quasars at  redshift larger than six. Their early appearance is 
mysterious, because the radiation pressure, generated by infalling ionized 
matter, inhibits the rapid growth of these black holes from stellar-mass black 
holes. It is shown that the supermassive black holes may form timeously 
through the accretion of predominantly degenerate sterile neutrino dark matter 
onto stellar-mass black holes. Our symbiotic scenario relies on the formation 
of, first, supermassive degenerate sterile neutrino balls through 
gravitational cooling and, then, stellar-mass black holes through supernova 
explosions of massive stars at the center of the neutrino balls. The observed 
lower and upper limits of the supermassive black holes are explained by the 
corresponding mass limits of the preformed neutrino balls. }\\
\end{abstract}

\mbox{} \hspace{2.15cm} {\small \bf Keywords:} cosmology, black holes, sterile 
neutrinos, dark matter\\[0.5cm]
\mbox{} \hspace{2.15cm} {\small \bf ArXiv ePrint:} ???\\
\newpage

%\baselineskip 2\baselineskip
\subsection*{1. Introduction}
Supermassive black holes of $\sim 3 \times 10^{9}M_{\odot}$ \cite{sch1}
%,sal2,zel3,lyn4,kor5,kor6,ho7,wil8,wal9} are already present at redshift
$z = 6.42$ or 850 Myr after the big bang, as is evident from the recently 
discovered \cite{wil2} quasar
%ween 10$^{6}M_{\odot}$ and $3\!\times\!10^{9}M_{\odot}^{[1-7]}$.
%%\cite{lyn1}. %Black holes at the top of this mass range are already present 
at redshift %^$z = (\lambda' - \lambda)/\lambda = 6.42$ %or 850 Myr after the 
big bang, as is evident from the recent discovery of quasar SDSS 
J114816.64+525150.3. %that is harbouring a black hole of 
$3\!\times\!10^{9}M_{\odot}^{[8,9]}$. %\cite{wil8,wal9}. Unlike the galaxies, 
which are thought to assemble through hierarchical mergers of smaller galaxies 
\cite{ell3}, the supermassive black holes form anti-hierarchically, i.e. the 
larger black holes prior to the smaller ones. %%%
This is supported by the observation \cite{com4} that the number of brighter 
quasars peaks at $z \sim 2$, while
that of lower-luminosity active galactic nuclei peaks at $z \sim 0.6$.
There is only a narrow window of about 485 Myr between the appearance of the 
first stars at $z \sim 11$, or 365 Myr after the big bang when reionization of 
the universe begins \cite{spe5}, and the appearance of the first quasars 
\cite{wil2} at $z = 6.42$ when reionization ends, during which at least some 
of the most massive black holes must have formed.\\[.2cm]
It has been known for more than three decades \cite{hil6} that the growth of a 
black hole from the $10M_{\odot}$ to the $3\!\times\!10^{8}M_{\odot}$ scale, 
through the accretion of baryonic matter radiating at the Eddington limit, 
with an efficiency between $\varepsilon \sim 0.1$ and $\varepsilon \sim 0.2$, 
would take at least 1.6 Gyr. This seems to preclude the existence of a ten 
times larger black hole at %$3\!\times\!10^{9}M_{\odot}$ black hole at $z = 
6.42$ that must have been produced in only $\sim$ 0.5 Gyr. Such a short 
accretion time can,
in the absence of frequent black hole mergers, only be achieved by drastically 
increasing either the ambient baryonic matter density, or the mass of the seed 
black hole, or both \cite{com4}. The baryonic %\newpage
%\vspace*{-4cm}
matter density would have to be much larger than the average matter density 
within 1 pc of the
%$10^{7}M_{\odot}/{\rm pc}^{3}$, which seems already uncomfortably high, as it 
is of the order of the %average matter density within 1 pc of the Galactic 
center of $\sim 10^{7} M_{\odot}/{\rm pc}^{3}$,
which is already uncomfortably large.
Alternatively, increasing the seed black hole mass by several orders of 
magnitude
is not attractive either, because, in spite of the ongoing intensive search, 
the evidence for intermediate mass black holes, with masses between 20 
$M_{\odot}$ and $10^{6}M_{\odot}$, is rather weak and controversial 
\cite{mil7}.
% \cite{mil15}. Regardless of whether the supermassive black holes were 
produced through the accretion of baryonic matter onto stellar-mass black 
holes, or through binary mergers starting from stellar-mass black holes, these 
would inevitably
have left a trail of observable intermediate mass black holes, which ran out 
of baryonic matter
or black hole supplies sometime and somewhere in this universe.\\[.2cm] Thus 
apart from the easily detectable upper limit of
$M_{\rm max} \sim 3 \times 10^{9}M_{\odot}$,
there also seems to be a lower mass limit of
%This is reflected by the observation th
%there seems to be a lower mass limit for the supermassive black holes of 
$M_{\rm min} \sim 10^{6}M_{\odot}$ of
the supermassive black holes. In this context, it is important to note that 
the
mass of the black hole at the center of M87 \cite{mac8}
is equal to that of the earliest quasar \cite{wil2} observed at $z$ = 6.42. As 
these are both archetypical examples of the most massive black holes at vastly 
different epochs of the universe, one may infer that $M_{\rm max}$ has not 
increased significantly from $z$ = 6.42 to $z$ = 0, or from 0.85 Gyr to 13.7 
Gyr after the big bang. Indeed, although Eddington-limited baryonic matter 
accretion is essential for seeing the quasars, this is a transient phenomenon 
with an estimated total lifetime of a few tens of Myr \cite{com4},
that presumably contributes very little to the final mass of the most massive 
black holes today.\\[.2cm]
In summary, it seems that a consistent theory of the formation of the 
supermassive black holes should be able to explain both observational facts, 
the nearly time-independent upper and lower mass limits of the supermassive 
black holes, as well as the early formation of the most massive black holes 
around $z$ = 6.42. As both the baryonic matter accretion, as well as the black 
hole merger scenarios, do not seem to provide us with a satisfactory 
description of these observational facts, we are led to ask the pertinent 
question, whether an alternative scenario, based on the accretion of mainly 
dark matter, may do better. Galaxies are, indeed, dominated by dark matter, 
and as part of this dark matter is concentrated in the galactic centers, it 
may very well have contributed to the formation of the supermassive black 
holes. Of course, in order to make definite predictions, one needs to focus on 
a well-defined and consistent dark matter candidate. Thus, in section 2, we 
discuss the physical and cosmological properties of our sterile neutrino dark 
matter candidate, while in section 3 we explore an astrophysical consequence 
of this dark matter scenario, namely the formation of self-gravitating 
supermassive degenerate sterile neutrino balls, with masses between 
$10^{6}M_{\odot}$ and
$3 \times 10^{9}M_{\odot}$. We then discuss how these preformed neutrino balls 
convert efficiently and anti-hierarchically into supermassive black holes, 
using stellar-mass seed black holes as catalysts.
In section 4, we describe the dynamics of the accretion of degenerate sterile 
neutrinos onto a black hole in a simple nonrelativistic Thomas-Fermi field 
theory, based on the Lane-Emden equation, while our conclusions are presented 
in section 5.
%%%SUBSECT 2
\subsection*{2. Physics and cosmology of sterile neutrinos}
Recently, a set of three right-handed sterile neutrinos has been consistently 
embedded
in a renormalizable extension of the minimal standard model of particle 
physics, dubbed the $\nu$MSM \cite{asa9}.
While two of these sterile neutrinos are unstable, having masses in the 1 
GeV/$c^{2}$ to 20 GeV/$c^{2}$ range, the third one, with mass around
10 keV/$c^{2}$, is a promising quasi-stable dark matter candidate. This light 
sterile neutrino interacts with the standard model particles only through its 
tiny mixing with the active neutrinos. The bulk part of the light sterile 
neutrinos is produced $\sim 2.3\; \mu$s after the big bang, at temperatures
\begin{equation}
  T \sim 328\left(\frac{mc^{2}}{15\; {\rm keV}}\right)^{1/3}\; {\rm MeV}/k \; 
\;,
\end{equation} well ahead of the quark-gluon and chiral restoration phase 
transitions, through incoherent resonant \cite{shi10} and non-resonant 
\cite{dod11} scattering of the active neutrinos.
%in the early universe, at temperatures %$T \sim$ 328 ($mc^{2}$/15 
keV)$^{1/3}$ MeV/$k$, %well ahead of the quark-gluon and chiral restoration 
phase transitions, through
%incoherent resonant$^{[22-25]}$ and non-resonant$^{[26-28]}$ scattering of 
the active neutrinos. For a wide range of the parameters of the $\nu$MSM, the 
sterile neutrinos are generated out of thermal equilibrium, yielding a sterile 
neutrino mass fraction of the total mass-energy of this universe which is 
consistent with that of nonbaryonic dark matter. For instance, with an initial 
lepton asymmetry %$\Omega_{M} \sim$ 0.24, as derived from the WMAP 
data$^{[12]}$.
%\cite{spe12}. \begin{equation}
L_{\nu_{e}} = \frac{ n_{\nu_{e}} - n_{\bar{\nu}_{e}} }{n_{\gamma}} = 10^{-2} 
\; \; , \end{equation}
a mixing angle of the
sterile neutrino $\nu_{s}$ with the active neutrino $\nu_{e}$ given by 
$\sin^{2} \vartheta = 10^{-13}$, and a sterile neutrino mass $m$ = 15 
keV/$c^{2}$, the mass fraction of the sterile neutrinos produced in the early 
universe is indeed $\Omega_{\nu_{s}} \sim 0.24$ \cite{shi10}, i.e. equal to 
that of the nonbaryonic dark matter, derived from the WMAP data 
\cite{spe5}.\\[.2cm]
The same tiny mixing angle, which prevents the thermal equilibration and thus 
the overproduction of the sterile neutrino in the early universe, also renders 
the sterile neutrino quasi-stable
\cite{pal12}, %%%%%%%%??????
%\cite{pal29}, %%%%%%%%?????? CHECK!
with a lifetime much larger than the age of the universe. In fact, with our 
choice of the parameters, $\sin^{2} \vartheta = 10^{-13}$ and $m$ = 15 
keV/$c^{2}$, the sterile neutrino decays with a lifetime  of \begin{equation}
\tau (\nu_{s} \rightarrow \nu_{e} \nu \bar{\nu}) = \frac{192 
\pi^{3}}{G_{F}^{2} m^{5} \sin^{2} \vartheta} = 1.21 \times 10^{19}\; {\rm 
yr}\; \;,
\end{equation}
predominantly into a $\nu_{e}$ and a neutrino-antineutrino pair, $\nu_{i}$ and 
$\bar{\nu_{i}}$, carrying the flavours $i = e, \mu$ or $\tau$ \cite{pal12}. 
These standard neutrinos are all virtually unobservable because their energy 
is too small. But there is also a subdominant radiative decay mode, with a 
branching ratio %\vspace{-1.25cm}
\begin{equation}
%\begin{array}{lcl}
\frac{\tau \left( \nu_{s} \rightarrow \nu_{e} \nu \bar{\nu} \right)}{\tau 
\left( \nu_{s} \rightarrow \nu_{e} \gamma \right)} = \frac{2 \alpha}{8 \pi} = 
0.784 \times 10^{-2}
\end{equation}
into a
%\end{array}
%\end{eqnarray}
%\vspace{-0.5cm}
potentially observable photon and a $\nu_{e}$ \cite{pal12}.
%\cite{pal29}. However, due to the smallness of its partial decay width of 
\begin{equation}
\left[\tau (\nu_{s} \rightarrow \nu_{e} \gamma)\right]^{-1} = 0.649 \times 
10^{-21} {\rm yr}^{-1}\; \; ,
\end{equation}
these photons of energy $mc^{2}$/2, which may be the ``smoking gun'' of the 
sterile neutrino, are  difficult to observe as well. In fact, for the chosen 
model parameters, a sterile neutrino dark matter concentration of mass $M$ has 
a luminosity of merely \begin{equation}
L_{X} = \frac{Mc^{2}}{2 \tau (\nu_{s} \rightarrow \nu_{e} \gamma)} = 1.84 
\times 10^{25} (M/M_{\odot})\;\; {\rm erg/s} \; \; ,
\end{equation}
in photons of $mc^{2}$/2 = 7.5 keV energy. Thus the best places to look for 
these photons are the diffuse extragalactic X-ray background, as well as the 
X-rays emitted by large galaxy clusters, low-surface-brightness and dwarf 
galaxies that are dominated by nonbaryonic dark matter
\cite{vie13}.
%It is interesting to note that sterile neutrinos may also be responsible for 
the pulsar kicks in %excess of 1000 km/s which magnetars acquire in supernova 
explosions \cite{kus13}. In summary, %a sterile neutrino, having the chosen 
model parameters, meets all the constraints which %an acceptable dark matter 
particle must fulfil \cite{wil2,vie12}, but it has several remarkable 
%additional properties that make it a rather unique candidate for dark matter, 
as we will see further %below.\\[.2cm]
For an initial lepton asymmetry $L_{\nu_{e}} \sim 10^{-10}$, which is of the 
order of the baryon asymmetry \cite{spe5} %\cite{spe12} \begin{equation}
B = \frac{n_{b}}{n_{\gamma}} \sim 6\!\times\!10^{-10} \; \;,
\end{equation}
mainly non-resonant neutrino scattering contributes to the production of 
sterile neutrino dark matter. These sterile neutrinos inherit a nearly thermal 
energy spectrum from the active neutrinos
\cite{dod11}, which allows them to play the role of warm dark matter in the 
large-scale structure of the universe, the clusters of galaxies and the 
galactic halos.
They may also erase the undesirable excessive substructure on the galactic 
scales. \\[.2cm]
%%%
The initial lepton asymmetry does not need to be of the same order of 
magnitude as the baryon asymmetry. However, for a larger initial lepton 
asymmetry, like $L_{\nu_{e}} \sim 10^{-2}$, there is, in addition to 
non-resonant neutrino scattering, also resonant or matter-enhanced neutrino 
scattering contributing to the production of dark matter \cite{shi10}. The 
latter yields cool sterile neutrinos that have a distorted quasi-degenerate 
spectrum, with an average energy of about two-thirds of that of the warm 
sterile neutrinos, due to the resonant Mikheyev-Smirnov-Wolfenstein (MSW) 
oscillations
\cite{mik14}.\\[.2cm] %In fact, for our model parameters
%$m$ = 15 keV/$c^{2}$, $\sin^{2} \vartheta = 10^{-13}$ and $L_{\nu_{e}} = 
10^{-2}$, cool (or resonant) %dark matter is about three times more abundant 
than warm (or non-resonant) dark matter \cite{shi9}.
A relatively large initial lepton asymmetry of
$L_{\nu_{e}} \sim 10^{-1}$ to 10$^{-2}$ is also what seems to be required, to 
bring the observed light element abundances in line with the number of three 
active neutrinos at nucleosynthesis \cite{ste15}. Sterile neutrinos may as 
well be responsible for the pulsar kicks of up to $\sim$ 1600 km/s, which 
magnetars acquire in supernova explosions \cite{kus16}. Thus our
sterile neutrino meets all the constraints which an acceptable dark matter 
particle must fulfil \cite{shi10},\cite{vie13}, but it has several remarkable 
additional properties that make it a rather unique candidate for dark matter.

\subsection*{3. A symbiotic black hole formation scenario}
For our model parameters $m$ = 15 keV/$c^{2}$, $\sin^{2} \vartheta = 10^{-13}$ 
and $L_{\nu_{e}} = 10^{-2}$, cool (or resonant) dark matter dominates over 
warm (or non-resonant) dark matter by a factor of about three \cite{shi10}. 
The cool sterile neutrinos become nonrelativistic $\sim\;22\;$min after the 
big bang, well after nucleosynthesis, and they begin, together with the warm 
sterile neutrinos and baryonic matter, to dominate the expansion of the 
universe $\sim$ 79 kyr after the big bang, well ahead of recombination. Thus 
the primordial density fluctuations of sterile neutrino dark matter have 
enough time to grow nonlinear and form degenerate sterile neutrino balls
\cite{mar17}, through a process called gravitational cooling \cite{bil18}, 
prior to the appearance of the first quasars, 850 Myr after the big bang. This 
collapse process may start ahead of reionization, perhaps as early as $\sim$ 
320 Myr after the big bang. Initially, the free-falling sterile neutrino dark 
matter, dominating baryonic matter by about a factor of six \cite{spe5}, drags 
the baryonic matter  along towards the center of the collapse. The baryonic 
gas will get heated, reionized and evaporated, but the free fall of the 
neutrinos will not be inhibited by the Eddington radiation limit.
Eventually, the quasi-degenerate
sterile neutrino dark matter hits, $\sim$ 640 Myr after the big bang, the 
degeneracy pressure, bouncing off a number of times,
while ejecting a fraction of the dark matter at every bounce. The neutrino 
ball finally settles in a condensate of degenerate sterile neutrino matter at 
the center of the collapsed object, as has been shown in calculations based on 
time-dependent Thomas-Fermi mean field theory \cite{bil18}.\\[.2cm] The 
smallest mass that may collapse is the mass contained within the 
free-streaming length at matter-radiation equality, $\sim$ 79 kyr after the 
big bang. For $m$ = 15 keV/$c^{2}$, this free-streaming mass is $M_{\rm warm} 
\sim 7 \times 10^{6} M_{\odot}$ in the case of warm (or non-resonant) sterile 
neutrinos, and
$M_{\rm cool} \sim 2 \times 10^{6} M_{\odot}$ in the case of the dominant cool 
(or resonant) sterile neutrinos \cite{shi10}.
As part of the neutrino dark matter is ejected during the collapse process, 
the
minimal mass of a degenerate sterile neutrino ball may be somewhat smaller 
than
$M_{\rm cool}$, perhaps
$M_{\rm min} \sim 10^{6} M_{\odot}$,
consistent with the lower mass limit of the observed supermassive black 
holes.\\[.2cm]
%%%START HERE
The maximal mass that a self-gravitating degenerate neutrino ball can support 
gravitationally, is the Oppenheimer-Volkoff (OV) limit \cite{opp19}
%%%EQUATION 2
\begin{equation}
%\begin{array}{lcl}
M_{\rm max} = 0.5430 \; M_{\rm Pl}^{3} \; m^{-2} \; g^{- 1/2}\; \;.
\end{equation}
% &=& 2.782 \times 10^{9} M_{\odot} \;
%\left( \frac{15\; {\rm keV}}{m c^{2}} \right)^{2} \; \left( \frac{2}{g} 
\right)^{1/2}  \; \; ,
%\end{array}
%\end{eqnarray}
For $m$ = 15 keV/$c^{2}$, and spin degeneracy factor $g = 2$, this mass
$M_{\rm max} = 2.789 \times 10^{9} M_{\odot}$ is consistent with that of the
most massive black holes observed in our universe \cite{sch1},\cite{wil2}.
As such a supermassive degenerate neutrino ball has
a radius of only 4.45 Schwarzschild radii \cite{opp19},
it is almost a black hole.
Thus the maximal mass scale of these objects may be linked to the existence of 
a sterile neutrino of $\sim$ 15 keV/$c^{2}$ mass, in a similar fashion as the 
maximal mass scale of the neutron stars is linked to the effective mass of the 
neutron \cite{baa20}.\\[.2cm] %%%%%%%%%%%%%
Since the gravitational potential in the interior of a neutrino ball is nearly 
harmonic,
these objects, in particular those near the upper mass limit, are ideal 
breeding
grounds for stars of mass $M \simgt 25 M_{\odot}$. As soon as such a central 
massive star is formed
from a collapsing molecular hydrogen cloud that was attracted to the neutrino 
ball, it may be kicked out through close binary encounters with intruding 
stars. However, before that happens, a star of 25 $M_{\odot}$ will evaporate 
large portions of its hydrogen and helium envelope and become a Wolf-Rayet 
star. And about 3 Myr after its formation, the star undergoes a core collapse 
supernova explosion of type Ic, leaving a black hole of 3 to 4 $M_{\odot}$ at 
the center of the neutrino ball. Some of these most massive supernova 
explosions, occurring in high-mass neutrino balls, may be observable as 
long-duration $\gamma$-ray bursts \cite{che21}. As in contrast to
pulsars, black holes do presumably not acquire ``black hole kicks'' during a 
supernova explosion, the velocity of the stellar-mass black hole will be small 
compared to the escape velocity from the center of the neutrino ball.
For a ball of 3$\times$10$^{6} M_{\odot}$ mass and 25 light-days radius 
\cite{mar17}, consisting of degenerate sterile neutrinos of 15 keV/$c^{2}$ 
mass and degeneracy factor $g = 2$, the escape velocity from the center is 
1700 km/s, while for a ball of the same sterile neutrinos at the OV-limit, 
with 2.8$\times$10$^{9}$ $M_{\odot}$ mass and 1.4 light-days radius, the 
escape velocity from the center is the velocity of light \cite{opp19}.\\[.2cm] 
%%%%%%
The supernova explosion of the massive star,
giving birth to a stellar-mass black hole at the center, sparks the rapid
growth of the black hole through nearly radiationless, and therefore, 
Eddington-unlimited accretion of mainly degenerate sterile neutrino dark 
matter from the surrounding neutrino ball, until the supplies dry up. In this 
symbiotic scenario, the anti-hierarchical formation of the bright 
quasi-stellar objects and low-luminosity active galactic nuclei may be 
explained by the fact that the escape velocity from the center of a 3 $\times 
10^{6} M_{\odot}$ neutrino ball, is 176 times smaller than that of a 3 $\times 
10^{9} M_{\odot}$ neutrino ball. The low-mass neutrino ball may, therefore, 
have difficulty capturing a molecular hydrogen cloud that is able to produce a 
massive star. In particular, a  low-mass neutrino ball may experience a large 
number of unsuccessful attempts, leading to ordinary low-mass stars, or 
neutron stars after supernova explosion, prior to delivering the expected 
stellar-mass black hole. These unwanted stellar-mass objects will eventually 
be ejected from the neutrino ball through pulsar kicks or close binary 
encounters with intruder stars from the surrounding star cluster, thus 
clearing the scene for the next attempt at forming this stellar-mass black 
hole. The randomness of this process  may very well delay the formation of a 
stellar-mass black hole at the center of a low-mass neutrino ball by several 
Gyr,
while a neutrino ball at the top of the mass scale may easily deliver the 
stellar-mass black hole on its first attempt in less than 10 Myr. Of course, 
these sketchy ideas will have to be tested in  realistic numerical 
simulations. However, if this scenario is correct, some low-mass neutrino 
balls may still be around at some galactic centers. For instance, a 10$^{6} 
M_{\odot}$ neutrino ball would reveal itself through its X-ray emission of 2 
$\times$ 10$^{31}$ erg/s at 7.5 keV, for our model parameters.

%%%REWRITTEN by GARY%%%
\subsection*{4. Accretion of a neutrino halo onto a black hole}
%statics of the neutrino ball}
%%%RDEV's changes to Gary's chapter 4:
A particle that is initially at rest at the surface of a $3 \times 10^{6} 
M_{\odot}$ neutrino ball reaches the center in the free-fall time $\tau_{F} 
\sim 35$ yr. This is also the time frame in which the accretion process onto 
the central black hole reaches a steady-state flow. In the steady-state 
approximation, the flow is governed by Bernoulli's eq.
\begin{equation}
\phi(r) + \frac{1}{2}(u^{2}(r) + v_{F}^{2}(r)) = \phi(r_{H}) = {\rm const} \; 
\; . \end{equation}
Here $u(r)$ is the flow velocity of the infalling degenerate sterile neutrino 
fluid,
$v_{F}(r)$ its Fermi velocity, $\phi(r)$ the gravitational potential and 
$r_{H}$ the radius of the halo. Assuming that the flow makes the gravitational 
potential $\phi$ extremal for all values of the radius $r$, with respect to 
variations that satisfy the constraint of mass conservation,
\begin{equation}
\rho u = \frac{m^{4}\;g\;v_{F}^{3}\;u}{6 \pi^{2} \hbar^{3}} = {\rm const}\; \; 
,
\end{equation}
we obtain
\begin{equation}
u^{2}(r) = \frac{1}{3}v_{F}^{2} (r) = c_{S}^{2} (r) \; \;,
\end{equation}
which means that the inflow of the sterile neutrinos is trans-sonic, i.e. it 
flows at the local velocity of sound $c_{S} (r)$. Thus Bernoulli's eq. is now
\begin{equation}
\frac{2}{3} v_{F}^{2} (r) = \phi (r_{H}) - \phi (r) = GM_{\odot} 
\frac{v(x)}{bx}\; \;,
\end{equation}
which defines the quantity $v(x)$. Using Poisson's eq., one can readily verify 
that $v(x)$ fulfils the Lane-Emden eq.
%%%EQ 1
\begin{equation}
\frac{1}{x} \frac{d^{2}v(x)}{dx^{2}} = - \left( \frac{v(x)}{x} \right)^{3/2} 
\; \; ,
\end{equation}
provided the length scale is
%%EQ2
\begin{equation}
b = \frac{4}{3} \; \left( \frac{3 \pi \hbar^{3}}{4 \sqrt{2} gm^{4} G^{3/2} 
M_{\odot}^{1/2}} \right)^{2/3} \; \; .
\end{equation}
%%%%%%%%%%%%%%%%%
Thus for $m$ = 15 keV/$c^{2}$ and $g = 2$ we have $b = 2.587$ lyr. The
total mass enclosed within a radius $r = bx$ is \cite{vio22}
\begin{equation}
M(x) = M_{\odot} \left[v (x) - x v\,' (x)\right]\; \;.
\end{equation}
Various solutions $v(x)$ of the Lane-Emden eq.(13), all having the total mass 
$M$ = 2.714 $M_{\odot}$, are shown in Fig.1. \begin{figure}[h]
%{\bf Figure 1}
\begin{center}
\mbox{} \hspace{-0.843cm}
\epsfysize 7.5cm
\epsfbox{fig1f.png}
\caption{\footnotesize Various solutions of the Lane-Emden equation, all 
having total mass $M = M_{P} + M_{H} = 2.714\;M_{\odot}$. The solid, dashed 
and dash-dotted lines represent the $E-$, $F-$ and $M-$ solutions, 
respectively.}
\end{center}
\end{figure}
There are three distinct classes of solutions.
The M-solutions exhibit a central point mass $M_{P} = M_{\odot} v$(0), 
surrounded by a self-gravitating degenerate sterile neutrino halo. The 
F-solutions describe shells of self-gravitating degenerate neutrino matter 
that are gravitationally unstable. The E-solution, with
$v (0) = 0$ and $v\,'(0) = 1$, stands for a pure neutrino ball with $M_{P} 
=M_{\odot} v$(0) = 0.
Our focus is on the M-solutions of the Lane-Emden eq.(13) with
$M_{P} =M_{\odot} v$(0) $>$ 0, because these describe, in the steady-state 
approximation, the various stages of the accretion history of a black hole 
surrounded by a degenerate sterile neutrino halo.
%%%%%%%%%%%%%%%%%%%%
%%%HERE COMES FIGURE 1
%%%%%%%%%%%%%%%%%%%%
%\vspace{-0.5cm}
%%%START HERE%%%%%
%FIG 1 COMES HERE
%\begin{center}
%\begin{figure}[h]
%%{\bf Figure 1}
%\begin{center}
%\mbox{} \hspace{-0.843cm}
%\epsfysize 7.5cm
%\epsfbox{fig1f.eps}
%\caption{\footnotesize Various solutions of the Lane-Emden equation, all 
having total mass %$M = M_{P} + M_{H} = 2.714\;M_{\odot}$. The solid, dashed 
and dash-dotted lines represent the $E-$, %$F-$ and $M-$ solutions, 
respectively.}
%\end{center}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%
%%HERE COMES FIGURE 2
%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.5cm}
%\begin{figure}[h]
%\begin{center}
%%{\bf Figure 2}
%\mbox{} \hspace{1cm}
%\epsfysize 7.5cm
%\epsfbox{fig2f.eps}
%\caption{\footnotesize Various shut-off parameters $f(\mu)$ as a function of 
the mass ratio $\mu = %M_{C}/M$. The neutrino ball masses vary from $M = 
10^{6} M_{\odot}$ for the box function, through $M = 10^{7}, 10^{8}$ to 
10$^{9}\;M_{\odot}$ for the curve with the largest peak value.}
%\end{center}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.5cm}
For instance, decreasing the halo radius $r_{H} = bx_{H}$, while keeping the 
total mass $M$ fixed, causes the central point mass $M_{P}$ to increase, as 
seen in Fig.1. The solutions of the Lane-Emden eq.(13) with $M_{P} = 
M_{\odot}v$(0) $>$ 0,
for arbitrary mass $\simM$ can be obtained noting that, if $v(x)$ is a 
solution, $\simv(x)=A^{3} v(Ax)$ with $A > 0$ is a solution as well.
Thus all the masses and radii scale as $\simM=A^{3} M$ and
$\simr=r/A$ \cite{vio22}. The mass accretion rate into a sphere, containing 
the mass $M_{C}$ within a radius $r_{C}$ from the center, is thus given by 
\begin{equation}
\frac{dM_{C}}{dt} = 4 \pi r_{C}^{2} \rho (r_{C}) u (r_{C})
\end{equation}
  or
%%%%%%%%%%%%%%%%%%%%%%%
%%%EQ3
\begin{equation}
\frac{d M_{C}}{dt} \; = \; \frac{ \sqrt{3} g m^{4} G^{2} M_{\odot}^{2}}{2 \pi 
\hbar^{3}} \; \left[ v(x_{C}) \right]^{2} \; = \;
\frac{f(\mu)}{\tau M_{\odot}} \; M_{C}^{2} \; \; ,
\end{equation}
where we have introduced the universal time-scale
%%%EQ4
\begin{equation}
\tau = \frac{2 \pi \hbar^{3}}{\sqrt{3} g m^{4} G^{2} M_{\odot}} =
1.488\!\times\!10^{7} \; \frac{2}{g} \; \left( \frac{15\;{\rm keV}}{mc^{2}} 
\right)^{4} \; {\rm yr} \; \; .
\end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The shut-off parameter
\begin{equation}
f(\mu) = \frac{v \left( x_{C} \right)^{2}}{ \left[ v (x_{C}) - x_{C} v\,' 
(x_{C}) \right]^{2}}
\end{equation}
is a function of the mass ratio $\mu = M_{C}/M$.
We now choose $r_{C} = bx_{C}$ to be the radius at which the escape velocity 
reaches the velocity of light, i.e. $x_{C}$ is given by
\begin{equation}
\frac{1}{x_{\odot}} = \frac{v (x_{C})}{x_{C}} - v\,'(x_{H})\; \; ,
\end{equation}
where $r_{\odot} = bx_{\odot}$ is the Schwarzschild radius of the sun with
%%%%%%%%%%%%%%%%%%%%%
%%%EQ5
\begin{equation}
x_{\odot} = \frac{2 GM_{\odot}}{bc^{2}} =
{\rm 1.207} \times 10^{-13}
\; \left( \frac{g}{2} \right)^{2/3} \;
\left( \frac{mc^{2}}{15\;{\rm keV}} \right)^{8/3} \; \; .
\end{equation}
%%%FIG2
\begin{figure}[h]
\begin{center}
%{\bf Figure 2}
\mbox{} \hspace{1cm}
\epsfysize 7.5cm
\epsfbox{fig2f.png}
\caption{\footnotesize Various shut-off parameters $f(\mu)$ as a function of 
the mass ratio $\mu = M_{C}/M$. The neutrino ball masses vary from $M = 10^{6} 
M_{\odot}$ for the box function, through $M = 10^{7}, 10^{8}$ to 
10$^{9}\;M_{\odot}$ for the curve with the largest peak value.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%
%HERE COMES FIGURE 3
%%%%%%%%%%%%%%%%%%%
%\vspace{0.5cm}
%%\begin{center}
%\begin{figure}[h]
%\begin{center}
%\mbox{} \hspace{-0.94cm}
%\epsfysize 7cm
%\epsfbox{fig3f.eps}
%\end{center}
%\caption{\scriptsize The growth of a central black hole as a function of 
time, in halos of between %$10^{6}$ and $10^{9} M_{\odot}$. For small values 
of $\mu$, the growth curves match the Bondi  growth %curve too closely to 
distinguish. For larger $\mu$, differences show up.  Larger neutrino ball 
masses %result in a faster central black hole growth.}
%\end{figure}
%\vspace{0.5cm}
As for $M \ll M_{\rm max}$ and $\mu < 1$, we may approximate $v (x_{C}) \sim 
v$(0) and
$f( \mu) \sim 1$, eq.(17) agrees well with standard Bondi accretion theory 
\cite{bon23}. Integrating eq.(17) using this approximation, the growth of the 
black hole is given by
\begin{equation}
M_{C} (t) \sim \frac{M_{C}(0)}{(1 - t/ \tau_{A})}\; \; ,
\end{equation} yielding an
accretion time-scale of
$\tau_{A} = \tau M_{\odot}/M_{C}$(0).
During the accretion process, both $M_{C}$ and $x_{C}$ grow, while $x_{H}$
shrinks as a function of time, eventually causing $x_{C}$ and $x_{H}$ to 
converge and $v(x_{C})$ to vanish.
The shut-off parameter $f(\mu)$, shown in Fig.2 as a function of the mass 
ratio $\mu = M_{C}/M$, is for a neutrino ball of mass $M = 10^{6}\;M_{\odot}$ 
a simple Heavyside function. As $M$ increases towards $M_{\rm max}$, this 
curve starts deviating from the simple box form, thus signalling
the breakdown of our nonrelativistic theory. %Since, for $M \ll M_{\rm max}$, 
\sim M_{\rm max}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%USE THIS SECTION
Since, for $M \ll M_{\rm max}$, the black hole growth is approximately given 
by the Bondi formula, $\dot{M}_{C} \propto M_{C}^{2}$, we expect the mass 
growth curves to match the Bondi solution closely,
%given by eqs.(30) and (31) closely, with the attractive feature that the 
solution of eq.(17) eventually brings the growth to a halt. %In fact, the 
larger the neutrino ball mass, the quicker this maximum is reached.
The mass ratio $\mu$ is shown in Fig.3 as a function of time in units of the 
accretion time-scale $\tau_{A}$.
% = \tau M_{\odot}/M_{C}(0)$ For low-mass central black holes, these curves 
are indistinguishable from the Bondi solution, represented by the dotted 
curve. Here we start with $\mu(0) = 0.1$ at $t = 0$ to illustrate the 
differences for halo masses between $10^{6}$ and $10^{9} M_{\odot}$.
The universal time-scale is $\tau$ = 14.88 Myr, and the accretion time-scales 
for a $3\!\times\!10^{6} M_{\odot}$
neutrino ball onto 3 and 4 $M_{\odot}$ seed black holes are, therefore, 
$\tau_{A} = 4.96$ Myr and $\tau_{A} = 3.72$ Myr, respectively.
%\vspace{0.25cm}
%\begin{center}
\begin{figure}[h]
\begin{center}
\mbox{} \hspace{-0.94cm}
\epsfysize 7cm
\epsfbox{fig3f.png}
\end{center}
\caption{\footnotesize The growth of a central black hole as a function of 
time, in units of the accretion time $\tau_{A}$, for halos of between $10^{6}$ 
and $10^{9} M_{\odot}$, as in Fig.2. For $\mu < 0.1$, the growth curves match 
the Bondi growth curve too closely to distinguish.} %Larger neutrino ball 
masses result in faster central black hole growth.}
\end{figure}
%\end{center}
%%%%%%%%%%%%%%%%%%%%%
%CHAPTER 5 COMES HERE
\subsection*{5. Conclusions}
In summary, the neutrino balls are almost entirely swallowed by the seed black 
holes in an accretion time scale $\tau_{A} <$ 5 Myr, thus converting these 
rapidly into supermassive black holes with negligible residual sterile 
neutrino halos. The most massive black holes may, therefore, form between 650 
Myr and 840 Myr after the big bang. Although the nonrelativistic Thomas-Fermi 
mean field theory breaks down for $M \sim M_{\rm max}$ and $r_{H} \sim r_{C}$, 
we expect these basic results to persist in a relativistic theory of the 
accretion process.\\[.2cm]
There are three main features which distinguish the accretion of degenerate 
sterile neutrino dark matter
from that of baryonic matter, playing a decisive role in the rapid growth of a 
stellar-mass black hole to the supermassive scale.
%%%
Firstly, in contrast to the clumping of baryonic matter, neither the neutrino 
ball formation nor the neutrino halo accretion onto a black hole is inhibited 
by the Eddington radiation limit.
%%%
Secondly, the matter densities of the degenerate sterile neutrino balls are 
much larger than those of any known form of baryonic matter having the same 
total mass, leading to much faster growth of the black holes through neutrino 
dark matter.
%%%
Thirdly, the preformed degenerate sterile neutrino balls have, for $m \sim$ 15 
keV/$c^{2}$ and $g$ = 2, masses in the same range as the observed supermassive 
black holes, which sets a natural limit to the growth of the black 
holes.\\[.2cm] %%%
We, therefore,
conclude that supermassive neutrino balls, with stellar-mass black holes at 
their center, indeed offer an intriguing symbiotic scenario, in which baryonic 
matter conspires with degenerate sterile neutrino dark matter, to form these 
galactic supermassive black holes, with masses between 10$^{6}\;M_{\odot}$ and 
$3 \times 10^{9}\;M_{\odot}$, rapidly and efficiently.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage

%\vspace{0.3cm}

%%%ACKNOWLEDGEMENTS COME HERE
\noindent
{\large \bf Acknowledgements}\\[.2cm]
\noindent
This research is supported by the Foundation for Fundamental Research and the 
National Research Foundation of South Africa. %Max Richter is supported by an 
NRF Masters bursary.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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%\vspace{2cm}

%{\bf Author Information}\\[.2cm]
%The authors declare no competing financial interests.\\[.2cm]
%Correspondence should be addressed to RDV: viollier@physci.uct.ac.za.
\end{document} %%%%%%%%%%%%%%%%%%%%%%%%%%