A&A 436, 805-815 (2005)
DOI: 10.1051/0004-6361:20035593
F. Munyaneza^{} - P. L. Biermann
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany
Received 29 October 2003 / Accepted 24 February 2005
Abstract
We report on a calculation of the growth of the mass of
supermassive black holes at galactic centers from dark matter and
Eddington - limited baryonic accretion.
Assuming that
dark matter halos are made of fermions and harbor
compact degenerate Fermi balls
of masses from
to
,
we find that dark matter accretion can boost the mass of
seed black holes from about
to
black holes, which then
grow by Eddington-limited baryonic accretion to
supermassive black holes of
.
We then show that the formation of the recently
detected supermassive black hole
of
at a
redshift of z = 6.41 in the quasar SDSS J114816.64+525150.3
could be understood if the black hole completely consumes the
degenerate Fermi ball
and then grows by Eddington-limited baryonic accretion.
In the context of this model we constrain the dark matter particle masses
to be within the
range from 12
to about 450
.
Finally we investigate the black hole growth dependence
on the formation time
and on the mass of the seed black hole. We find that in order to
fit the observed data point of
and
,
dark matter accretion cannot start later
than about
years and the seed BH cannot be
greater than about
.
Our results are in full agreement with the
WMAP observations that indicate that the first onset of
star formation might have occurred at a
redshift of
.
For other models of dark matter particle masses,
corresponding constraints may be derived from the growth of black holes in the
center of galaxies.
Key words: black holes physics - galaxies: nuclei - cosmology: dark matter - galaxies: quasars: general
It has been established that the mass of the central BH is tightly correlated with the velocity dispersion of its host bulge, where it is found that (Faber et al. 1997; Magorrian et al. 1998; Ferrarese & Merritt 2000; Gebhardt et al. 2000; Ferrarese 2002; Haering & Rix 2004). This tight relation between the masses of the BHs and the gravitational potential well that hosts them suggests that the formation and evolution of supermassive BHs and the bulge of the parent galaxy may be closely related, e.g. Wang et al. (2000). In addition, the recent discovery of high redshift quasars with z > 6 (Fan et al. 2001) implies that the formation of supermassive BHs took place over fewer than 10^{9} years. In spite of the vast and tantalizing work undertaken on BHs, their genesis and evolution are not well understood (see Rees 1984, for a review). Since the discovery of quasars in the early 1960s, it has been suggested that these objects are powered by accretion of gas onto the supermassive BHs of masses (Lynden-Bell 1969). Two scenarios have been discussed in modeling the growth of BHs. One is that BHs grow out of a low mass "seed'' BH through accretion (Rees 1984), and another one is that BHs grow by merging (Barkana et al. 2001; Wang et al. 2000; Gopal-Krishna et al. 2003,2004). In a recent paper, Duschl & Strittmatter (2004) have investigated a model for the formation of supermassive black holes using a combination of merging and accretion mechanisms.
The purpose of this paper is to study the growth of BHs from dark matter and Eddington-limited baryonic accretion. Also, we would like to obtain some constraints on the DM particle masses. In the past, self gravitating neutrino matter has been suggested as a model for quasars, with neutrino masses in the range (Markov 1964). Later, neutrino matter was suggested to describe DM in clusters of galaxies and galactic halos with masses in the range of (Cowsick & McClelland 1973; Ruffini 1980). More recently, fermion balls (FBs) made of degenerate fermionic matter of were suggested as an alternative to supermassive BHs in galaxies (Viollier 1994; Bilic et al. 1999; Tsiklauri & Viollier 1998; Munyaneza & Viollier 2002). It has been also suggested that if the Galaxy harbors a supermassive BH, then there should be a density spike in which dark matter (DM) falling towards the center could annihilate and the detection of these annihilation signals could be used as a probe for the nature of DM (Bertone et al. 2002; Gondolo & Silk 1999; and Merritt et al. 2002). The current belief is that DM particles are bosonic and very massive, i.e. . However, the absence of experimental constraints on the weakly interacting massive particles (WIMPs) that probably constitute DM leaves the door open for further investigation of the hidden mass of the Universe.
In this paper, we will assume DM to be of fermionic matter and described by a Fermi-Dirac distribution with an energy cutoff in phase space (King 1966). We will then explore the limits for the DM particle masses in order to reproduce the mass distribution in the Galaxy (Wilkinson & Evans 1999) and then study the growth of a seed BH immersed at the center of the DM distribution in galaxies. We use degenerate FBs at the center of DM halos not as replacements for the BHs but as necessary ingredients to grow the BHs in galactic centers.
The resulting distribution of stars around a massive BH was studied in detail in the 1970s and early 1980s in the context of globular clusters (Hills 1975; Frank & Rees 1976; Bahcall & Wolf 1976; Duncan & Shapiro 1982; Shapiro 1985). Peebles (1972) studied the adiabatic growth of a BH in an isothermal sphere and showed that the BH would alter the matter density to an adiabatic cusp with . A few years later, Young (1980) constructed numerical models that confirmed Peebles' results and showed that the BH induces a tangential anisotropy in the velocity dispersion. In this paper, we use the results of previous calculations on the growth of BHs that are accreting stars (Lightman & Shapiro 1978) to investigate the growth of a BH that accretes DM. Here we assume that the physics driving the formation of the power law cusp in the star - star case is the same as in the case of DM particle orbits being perturbed by molecular clouds. Julian (1967) investigated a similar scenario in which the stellar orbits in our Galaxy were perturbed by molecular clouds to explain the stellar velocity dispersion dependence on the star's age. Moreover, Duncan & Wheeler (1980) investigated the anisotropy of the velocity dispersions of the stars around the BH in M87.
Given the density distribution in DM halos, we are interested in establishing how a seed BH would grow by accreting DM. Moreover, the comparison of the growth of the BH from accretion of DM and Eddington-limited baryonic matter would give us another piece of information in the debate surrounding the nature of DM. We therefore investigate how a BH seed of could grow to a BH as recently detected in quasar SDSS J1148+5251 at z=6.41 (Willot et al. 2003). Such a seed BH of a typical mass between 5 and 9 could have evolved in BH binaries (Podsiadlowski et al. 2003). Here, we note that the seed BH could be an intermediate mass black hole (IMBH) of that might have formed from collisions in dense star-forming regions (Portegies Zwart & McMillan 2002; Coleman Miller 2003; and van der Marel 2003, for a review). In fact, Wang & Biermann (1998) have established that a BH would grow exponentially with time accreting baryonic matter as long as the supply lasts. In addition, they were able to reproduce the observed correlation using standard disk galaxy parameters. Assuming a cusp - like distribution of self-interacting DM (Spergel & Steinhardt 2000), Ostriker (2000) has estimated that BHs could grow to from DM accretion. For completeness, we note that accretion of DM particles by BHs has recently been studied in Zhao et al. (2002) and Read & Gilmore (2003).
Cosmological parameters of , and are assumed throughout this paper. In Sect. 2, we establish the main equations to describe DM in galaxies. We then discuss the growth of seed BHs from DM and Eddington-limited baryonic matter accretion in Sect. 3 and conclude with a discussion in Sect. 4.
We characterize DM by its mass density
and
its velocity dispersion .
DM is assumed to be collisionless and of fermionic matter and the mass of DM particles is denoted by .
We will look for DM distributions with degenerate
cores at the center of the DM halos instead of a steep power law
with
(e.g. Navarro et al. 1997) as we are assuming here that a central
cusp is a result of BH growth.
In order to obtain bound astrophysical solutions at
finite temperature, we follow King (1966) and
introduce an energy cutoff in phase space using the Fermi-Dirac distribution function.
We therefore adopt the following prescription for the distribution function
(3) |
(5) |
(6) |
(9) |
(13) |
Figure 1: The mass density and the mass enclosed within a radius r are plotted in the upper and lower panels, respectively. Near the center of the DM halo, the fermions are completely degenerate and condensed in a FB of mass . In the outer edge of the DM halo, the density scales as 1/r^{2} (van Albada et al. 1985). The mass of fermions used in this plot is and the size of the FB is about . The data points for the mass within 50 kpc and 200 kpc are taken from Wilkinson & Evans (1999). | |
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In Fig. 1, we plot the mass density and the mass enclosed
for solutions
with a
mass and
a radius of 200 kpc, which corresponds to our Galaxy.
Throughout our paper, we will study solutions of type 3 which have
degenerate FBs in the center.
For the case of our galaxy,
in order to have a degenerate FB
of
at the center and fit the rotation curve of the Galaxy, i.e.
reproduce the 1/r^{2} law,
the mass of fermions should be about
.
For a FB of
,
one would need
a fermion mass of
.
Thus under the assumption of a degenerate FB of
to
at the center of a DM halo of
with a density scaling as 1/r^{2} in the outer edge of the halo, the fermion mass
is constrained in the range
The solutions shown in Fig. 1 have been obtained using the fact that at large distances the Fermi gas is non degenerate. The degree of non-degeneracy is described by Eq. (7) which contains the chemical potential . Thus, the use of the chemical potential in the Fermi-Dirac ditribution allows to constrain the DM particles in the range from 12 to 450 keV, much below the Lee-Weinberg lower limit of about 2 GeV (Lee & Weinberg 1977) which was established under the assumptions that DM particles were in thermodynamic equilibrium at the freeze - out temperature. We also note that a suitable candidate for the DM particle as defined by the range given in Eq. (14) could be either the gravitino, postulated in supergravity theories with a mass in the to range (Lyth 2000), or the axino, with a mass in the range between 10 and , as predicted by supersymmetric extensions of the Peccei-Quinn solution to the strong CP problem (Goto & Yamaguchi 1992). For a recent review on DM particle candidates, the reader is referred to Bertone et al. (2005) and Baltz (2004).
We define the local dynamical (orbital) time scale
and the mass accretion flow as
(15) |
(17) |
Near the center of the DM halo, the core becomes
completely degenerate and the Pauli condition can be written as
(19) |
(20) |
Here we note that the mass
of nonrelativistic
degenerate FBs scales with radius as
(22) |
In Fig. 2 we plot the mass-radius relation for FBs for
different values of the fermion mass. In the same plot we have
drawn the BH line and two horizontal lines that
correspond to a mass of
and
.
From this plot, we find an upper limit for the fermion mass of
Figure 2: The mass-radius relation for degenerate FBs. The total mass scales as . In the same plot, we have shown the BH line and two horizontal lines for the lower and upper limits for the total mass of the degenerate Fermi balls. Relativistic effects make the mass-radius relation curve have a maximum mass (see Eq. (24)) and the size of such a FB at the Oppenheimer Volkoff limit is only where is the Schwarzchild radius of the mass . The line of BHs and a FB at the OV limit are nearly indistinguishable in this plot. | |
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The choice of the lowest value for the FB mass is motivated by the fact that once the BH has consumed a FB of , it would grow by Eddington-limited baryonic matter accretion to as observed in quasars at redshifts . The choice of the second value of , is due to increasing evidence for the existence of a BH of mass at the center of our galaxy. Thus we will assume throughout this paper that galaxies which have BHs had degenerate FBs of masses ranging from to . The fermion mass range given by Eq. (14) is of course well below the upper limit for the fermion mass from Eq. (23).
For completeness we note here that stable degenerate FBs exist up
to a maximum mass also called the Oppenheimer Volkoff limit
From Eq. (24), the fermion mass needed to form a degenerate FB of
at the OV limit is about
.
This value is in agreement with the lower limit on the mass of fermions obtained from fitting the
Galactic DM halo with a FB of
at the center and 1/r^{2} density law from about 3 kpc.
The most compact supermassive object of
has been
identified as a BH at the center of M87 (Macchetto et al. 1997).
Figure 3: Fit of the rotation curve of dwarf galaxies DD0154. The data points can be fitted by a non-degenerate Fermi-Dirac distribution function for DM particles in the mass range discussed in this paper, i.e. for . In this plot we have used a fermion mass of to fit the rotation curve of the dwarf galaxy DD0154. Here we also note that some small addition of baryonic matter would probably allow a closer fit to the data. | |
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Apart from fermions as candidates of DM, bosons have also been considered
as an alternative to the DM constituents in galaxies. Self-gravitating bosons can form boson stars and they are prevented from complete gravitational collapse by the Heisenberg
uncertainty principle
(Schunck & Liddle 1977).
Boson stars are described by Einstein's field equations coupled with the non-linear Klein Gordon equation
for a complex field
with self-interaction of the form
where
is the boson mass
and
is a coupling constant.
It has been shown that boson stars have a maximum mass called the Kaup limit given by
In Fig. 3, we fit the rotation curve of dwarf galaxies using a Fermi-Dirac distribution. The rotation curve data points are well fitted in the outer region. In the inner region, i.e. around 1 kpc, baryonic matter is perhaps needed to get a better fit of the rotation curve. In order to fit the rotation curve of dwarf galaxies, a non-degenerate FD distribution is needed. The data points are taken from Carignan & Purton (1998) and the mass of the dwarf galaxy DD0154 is taken to be with a size of (Kravtsov et al. 1998). The merging of dwarf galaxies would generate another galaxy with a degenerate FB, and the BH would grow by the same mechanism considered in this paper in the merged galaxy. The Fermi-Dirac distribution studied in this paper allows two types of solutions: one which fits dwarf galaxies with a non-degenerate core and the other solution with 1/r^{2} drop off in the outer edge of the halo. Degenerate FBs are consistent with data on the rotational curves only in the second type of solution with a 1/r^{2} density profile at large distances.
From Eq. (26), we can estimate the time
needed for the BH mass to be infinite
to be
(28) |
(29) |
The mechanism of consumption of DM particles by the BH can be understood using a quantum cascade mechanism. First, low angular momentum DM particles at the inner Fermi surface in phase space will be consumed by the BH. Due to a high degeneracy pressure, high angular momentum particles will be pushed to the inner Fermi surface and taken up by the BH. Outside the FB, fermions will hit the outer Fermi surface and for the FB to be hit continuously by DM particles, we use molecular clouds as perturbers of DM orbits. Thus, we have a three body interaction between the FB, the molecular clouds and DM particles. For our mechanism of consumption of DM by the BH to work, we will assume that the dark matter particles interact with the FB through inelastic collisions. In hitting the FB, some of the DM particles will be excited to a slightly higher energy level and get stuck in high momentum segments of phase space. This process will continue until the FB has been completely consumed by the BH and thereafter baryonic matter accretion will control the BH growth.
In this paper, we assume that there is a compact degenerate FB at the center of galaxies that have BHs. The seed BH first consumes the degenerate FB and grows to a mass of about and then grows by Eddington-limited baryonic matter accretion to higher masses of .
We will be mostly concerned with the calculation of the time needed to refill the BH loss cone using molecular clouds as DM perturbers. To this effect, we denote by r the radial distance from the BH to the DM particle position, the distance from the BH to the molecular clouds, is the angle between the direction from the BH to the DM particle and its projection in the disk and is the angle between the direction from the BH to the molecular clouds and to the foot of the DM particle location in the disk. The diagram for the geometry used is shown in Fig. 4.
In order to get the expression for the
time
needed to refill the loss
cone
which is due to multiple small-angle
gravitational (Coulomb) encounters,
we evaluate the
random walk integral (Spitzer 1987)
(31) |
Figure 4: Geometry for the system BH (BH), DM (DM) particles and molecular clouds (MC). FD stands for the foot (projection) of the DM particle in the disk. The angles BH-FD-DM and MC-FD-DM are both right angles. | |
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(34) |
(35) |
The FB is consumed from inside by the BH with an accretion rate
(36) |
(37) |
(38) |
In Fig. 5, we plot the dynamical and the refilling timescales as functions of the radius. At small radii, i.e. , the time to refill the loss cone is too long for the DM to feed the BH. Near the BH, we use the dynamical timescale to feed the BH. For radii greater than the size of the FB, we will use the slowest of the two timescales to feed the FB. Thus the FB feeds from the outside DM particles via the loss cone mechanism while the BH feeds from DM inside the FB via the quantum cascade mechanism.
In Fig. 6, we plot the mass flow as a function of the radius from the center.
To this end, we fix the mass of the FB to
and
use a fermion mass of
.
Using the BH accretion rate Eq. (21), the
accretion rate becomes
for a fermion mass of
and a
BH mass of
.
For fermions of mass
and
a BH mass of
,
the accretion rate is
.
It is seen than the BH accretion rate is independent on the radius and drops
significantly when the size of the FB is reached. Thus the BH grows much faster from
the inner DM particles than the FB from the outer DM particles.
After the BH has reached a mass equal to that of the FB, i.e.
,
its growth is
controlled by Eddington baryonic matter accretion.
Figure 5: The refilling and orbital timescales plotted as a function of the radius from the center of the DM halo. In this plot, the mass of the FB is and the corresponding fermion mass is . For small r, the time to refill the loss cone is much greater than the age of the Universe. For the growth of the BH, we use the orbital time scale and the BH feeds on DM from inside the FB. For the DM inside the FB to be fully consumed by the BH, we introduce a quantum cascade effect which is a result of the existence of a high degeneracy pressure that pushes high angular momentum particles to the inner Fermi surface. The particles at the inner Fermi surface in 6D phase space will then be consumed by the BH. This process continues until the BH has consumed the entire FB. The BH then grows by Eddington-limited baryonic matter accretion. The FB is fed from DM from outer orbits but its accretion rate is much lower than that of the BH inside the FB as seen in the next figure. The vertical thick line shows the size of the FB. | |
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Figure 6: The mass accretion flow as a function of distance from the center. For a given mass of the BH, the accretion rate is independent of the radius until the Bondi radius where the BH dominates the gravitational potential of the fermions. As the BH mass approaches that of the Fermion ball, the BH accretion rate becomes constant until the radius of the FB and as the size of the FB is reached, the DM accretion rate onto the FB drops dramatically. Thus the growth rate of the FB is much lower than that of the BH which grows by feeding on DM from inside the FB. Similarly to the last plot, we have used here a FB mass of . The left side of the plot shows the BH accretion rate and the FB consumption rate is shown on the right side of the figure. | |
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Figure 7: The black hole mass growth from only DM accretion. The fermion mass is varied as shown on the plot. It can be seen that the BH mass can grow to in about 10^{7-8} years if the fermion mass is in the range between about 1 keV and 1 MeV. For fermion masses , the formation of a BH of would require a timescale much greater than than a Hubble time. On the other hand, for fermion masses , a BH would have formed in less than than one year after the beginning of the Universe. Thus the degeneracy pressure of fermions allows for the capture of a large amount of mass if the fermion mass is constrained in a range between about 1 keV and 1 MeV. The data point shown at years is the recent detection of a BH mass of BH in the quasar SDSS SDSS J114816.64+525150.3. | |
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In Fig. 7, we plot the BH mass as a function of time for different values of the fermion mass. The BH mass - time dependence is given by Eq. (26). It is shown that for fermion masses lower than about 1 keV, it would take more than 10^{10} years for the BH to grow to . However, due to the Pauli principle used in the derivation of Eq. (26), if the fermion mass is constrained in the mass range between about 1 keV and 1 MeV, a seed BH of 5 can grow to a BH in about 10^{7-8} years, which would then grow by Eddington-limited baryonic accretion to at a redshift of (see next subsection). Fermion masses greater than about 1 MeV would make the BH grow much faster and the BH could be as large as in less than about one year after the beginning of the Universe.
In the next section, we will discuss the Eddington baryonic matter accretion and discuss the plots of BH growth from both DM and Eddington baryonic matter accretion.
The details of Eddington-limited baryonic accretion have been studied by many authors,
here we
refer the reader to Wang & Biermann (1998) and provide just the main formulae:
Figure 8: The mass of the BH is plotted as a function of time t for DM and Eddington-limited baryonic matter accretion. It is seen from this plot that DM dominates the BH growth at early times. The mass of fermions is varied as shown on the plot. The total mass of the degenerate FB has a mass of . We find that fermion masses of suffice for stellar seed BHs to grow to about which then grow by Eddington-limited baryonic matter accretion to about . The obtained constraint on the fermion mass covers the range of dark matter masses discussed earlier. The data point shown at z=6.41 corresponds to the BH mass of in the quasar SDSS J114816.64+525150.3. | |
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Figure 9: The same as in the last figure but for a degenerate FB mass of . | |
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It is seen from Figs. 8 and 9 that
only baryonic matter accretion
cannot fit this data point and the
curve for the growth of the BH due to baryonic
matter lies on the right side of the data point.
Using
Eq. (41) the total mass
of the BH from Eddington-limited baryonic accretion at
years
i.e. z=6.41 is
which is about ten times less
than the inferred BH mass of
,
clearly outside the error
range.
The growth of a seed BH from 5 to
is accomplished in two steps:
First, the BH completely consumes the degenerate Fermi core with
masses between
and
.
In the second step, the
BH feeds on
baryonic matter accretion to reach the mass
of
at a redshift of
.
Figure 10: The BH growth dependence on the starting time of accretion. We find that starting DM accretion at a time of would fit the recent discovery of a BH in the quasar SDSS J1148+5251 at a redshift of z=6.41, which is the most distant known quasar, observed only 840 million years after the beginning of the Universe (Willot et al. 2003). The fermion mass used in this plot is . However, any higher fermion mass would more sharply and rapidly raise the mass of a seed BH from 5 to . | |
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In Fig. 10, we investigate how the growth of the mass of the BH depends on the accretion starting time. To this effect, we fix the total mass of the FB to and the fermion mass to and vary the initial time when the DM accretion starts as shown on the graph. We find that the data point of at is well fitted for an accretion starting time of for any fermion greater than . Here we also note that Eddington baryonic matter accretion could also allow the seed BH to grow to a BH if the efficiency is . The accretion starting time of corresponds to a redshift of reionization of as obtained by WMAP observations (Krauss 2003).
Figure 11: We investigate how the BH growth depends on the mass of the seed BH. To fit the data point of mass at , the seed BH cannot be greater than for a FB of and the accretion starting time should be about . In this plot, the fermion mass is fixed to . Thus, any other mechanism that could create massive seed BHs would generate the same growth. | |
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Finally in Fig. 11, we investigate how the growth of the BH is affected by the mass of the seed BH at different starting times of accretion. In order to constrain the mass of the seed BH, we vary the mass of the FB from to and also the mass of the seed BH from to at initial times of and years. These values correspond to the earliest and latest plausible times for the formation of the first stellar seed BHs in order to grow into supermassive BHs at redshift . From recent WMAP observations (Spergel et al. 2003), BHs of masses might have formed at a redshift . In Fig. 11, we have plotted only the limiting curves which show that the mass of the FB should be and the mass of the seed BH cannot be greater than about . The best fit is obtained for a BH formation time a reionization time of , i.e. .
It is interesting to estimate the maximum luminosity attainable by an accreting BH of this mass. This is usually considered to be the Eddington luminosity at which the outward radiation pressure equals the inward gravitational attraction. Using Eq. (42), the luminosity of a BH is of the order of a few and the corresponding accretion rate is about . However, such high luminosity values (Fan et al. 2001) would be obtained only for a standard efficiency of which as seen from Figs. 8 and 9 does not fit the data point for only baryonic matter accretion.
A gravitational system in which molecular clouds provide the
interaction path between DM particles and feeding the BH leads to a strong
limit on the maximum mass of the BH.
For an initial distribution of DM in the halo
without a BH,
the
total energy of the system is given by
(43) |
(44) |
(45) |
(46) |
(47) |
(48) |
(49) |
If we use heavy fermions with masses of about , then according to Eq. (24) the maximum mass allowed for the degenerate Fermi core would only be of about , which is not enough to grow a stellar mass BH to in about 10^{8} years. On the other hand, very light fermions i.e. would generate very massive degenerate FBs of masses greater than and the BH would grow to in a very short time, i.e .
The growth of seed BHs from DM accretion is investigated using the quantum cascade mechanism upon which low angular momentum DM particles at the inner Fermi surface are first consumed by the BH and then due to a high degeneracy pressure, higher angular momentum particles are pushed inwards and the process continues until the entire degenerate FB is consumed by the BH. Moreover, molecular clouds have been used as perturbers of DM particle orbits outside the FB and we have shown that the BH grows faster than the FB. After the BH has consumed the entire FB, it then grows by Eddington-limited baryonic accretion to higher masses of at redshifts . We also point out that molecular clouds of mass have also been detected in the host galaxy of the same quasar at a redshift of (Walter et al. 2003).
We have also constrained the possible starting time of accretion, i.e. the time of BH seed formation. From our analysis, the mass of a BH in the quasar SDSS sJ114816.64+525150.3 at a redshift of z=6.41 can be fitted exactly if the accretion process starts at a time of about years, which corresponds to the reionization time.
The seed BH mass is found to be in the range from a few solar masses up to an upper limit of . For a seed BH mass of , Eddington baryonic matter accretion would be enough to cause the seed BH to grow into a supermassive BH of mass. The data point at a redshift of z=6.41 can be fitted by only Eddington baryonic matter accretion with an efficiency of .
Our model provides a method to find the DM particles mass. If it is found that there is a clear lower mass cut of 10^{3} to in the distribution of BH masses, then this mass can be used for the mass of the FB to find the corresponding mass of the fermions which will be in the range of 12 keV to 450 keV. If on the other hand the BH mass distribution is a continuous function, then our model of BH growth with DM will probably be ruled out.
The postulated DM particles in this paper were non-relativistic at the decoupling time and are usually called cold dark matter particles (CDM). The latter have to be neutral, stable or quasi-stable and have to weakly interact with ordinary matter. As mentioned in Sect. 2.1, these particles could be axions which have been investigated by DAMA/NAI (Bernabei et al. 2001). The heavier particles of mass above could also be of the class of DM candidates named WIMPS (Weakly Interactive Massive particles). However, in the standard model of particle physics, CDM cannot be suitable candidates for particles. Thus, a new window beyond the standard model of particle physics has to accommodate these particles for our model of BH growth to work. The DAMA/NAI experiment (see Bernabei et al. 2004, for a review) which aims at the verification of the presence of DM particles in the Galactic halo, will be able to confirm whether GeV WIMPS or axions of masses of 12 to 450 keV could exist in nature.
Observations have shown that the masses of supermassive BHs at galactic centers correlate with the masses of the host bulges, i.e. (Haering & Rix 2004). This result is obtained in our model as long as the Eddington limited accretion dominates the final growth of the BH. This happens for BH masses of more than about (Wang et al. 2000).
Mergers of dwarf galaxies as well as the spinning of BHs would play an important role in the growth of the BHs. The consideration of these two effects will be the subject of further investigations. In addition, it would be of great interest to study the growth of BHs from boson DM particles. Although it has been shown that bosons could provide a good fit to the rotation curves in dwarf galaxies, it is not yet clear whether an analogous mechanism could work in galaxies with a 1/r^{2} density fall off.
While it takes only years to grow supermassive BHs in most distant quasars, the Galactic center might have grown to its current mass of with only DM accretion in a Hubble time. In the following paper, we will address the growth of the Galactic center BH.
Acknowledgements
We thank Yiping Wang for providing us with the data file for baryonic matter growth. We are also grateful to Heino Falcke and Rainer Spurzem for useful discussions. F.M. research is supported by the Alexander von Humboldt Foundation. Work with P.L.B. has been supported through the AUGER theory and membership grant 05 CU1ERA/3 through DESY/BMF (Germany). Further support for the work with PLB has come from the DFG, DAAD, Humboldt Foundation (all Germany), grant 2000/06695-0 from FAPESP (Brasil) through G. Medina-Tanco, a grant from KOSEF (Korea) through H. Kang and D. Ryu, a grant from ARC (Australia) through R. J. Protheroe, and European INTAS/ Erasmus/ Sokrates/Phare grants with partners V. Berezinsky, L. Gergely, M. Ostrowski, K. Petrovay, A. Petrusel, M. V. Rusu and S. Vidrih.