All Figures

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{0593f1.eps} \end{figure}$ 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 $\sim$ $3 \times 10^{6}~M_{\odot}$ . In the outer edge of the DM halo, the density $\rho$ scales as 1/r² (van Albada et al. 1985). The mass of fermions used in this plot is $m_{\rm f}c^{2} \approx 12 ~{\rm keV}$ and the size of the FB is about $10^{-2}~{\rm pc}$ . The data points for the mass within 50 kpc and 200 kpc are taken from Wilkinson & Evans (1999).
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In the text

$\begin{figure} \par\includegraphics[width=8.2cm,clip]{0593f2.eps} \end{figure}$ Figure 2: The mass-radius relation for degenerate FBs. The total mass scales as $M_{\rm F} \sim R_{\rm F}^{-3} m_{\rm f}^{-8}$ . 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 $M_{\rm OV}$ (see Eq. (24)) and the size of such a FB at the Oppenheimer Volkoff limit is only $4.45 R_{\rm OV}^{\rm s}$ where $R_{\rm OV}^{\rm s}$ is the Schwarzchild radius of the mass $M_{\rm OV}$ . The line of BHs and a FB at the OV limit are nearly indistinguishable in this plot.
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{0593f3.eps} \end{figure}$ 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 $12~{\rm keV} \protect \la m_{\rm f}c^{2}\protect \la 450~ {\rm keV}$ . In this plot we have used a fermion mass of $m_{\rm f} \sim 305 ~{\rm keV}/c^{2}$ 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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In the text

$\begin{figure} \par\includegraphics[width=6cm,clip]{0593f4.eps} \end{figure}$ 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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In the text

$\begin{figure} \par\includegraphics[width=7.9cm,clip]{0593f5.eps} \end{figure}$ 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 $3 \times 10^{6}~M_{\odot}$ and the corresponding fermion mass is $12~{\rm keV/}c^{2}$ . 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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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{0593f6.eps} \par\end{figure}$ 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 $3 \times 10^{6}~M_{\odot}$ . 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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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{0593f7.eps} \end{figure}$ 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 $10^{3-4}~M_{\odot}$ in about 10^7-8 years if the fermion mass is in the range between about 1 keV and 1 MeV. For fermion masses $m_{\rm f} \protect \la 1 ~{\rm keV/}c^{2}$ , the formation of a BH of $10^{3}~M_{\odot}$ would require a timescale much greater than than a Hubble time. On the other hand, for fermion masses $m_{\rm f} \protect \ga 1 ~{\rm MeV/}c^{2}$ , a $10^{3}~M_{\odot}$ 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 $t=8.47\times 10^{8}$ years is the recent detection of a BH mass of $3\times 10^{9}~M_{\odot}$ BH in the quasar SDSS SDSS J114816.64+525150.3.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{0593f8.eps} \end{figure}$ Figure 8: The mass $M_{\rm BH}$ 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 $3\times 10^{3}~M_{\odot}$ . We find that fermion masses of $m_{\rm f} \protect \ga 4~{\rm keV/}c^{2}$ suffice for stellar seed BHs to grow to about $10^{3}~M_{\odot}$ which then grow by Eddington-limited baryonic matter accretion to about $3\times 10^{9}~M_{\odot}$ . 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 $\sim$ $3\times 10^{9}~M_{\odot}$ in the quasar SDSS J114816.64+525150.3.
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{0593f9.eps} \end{figure}$ Figure 9: The same as in the last figure but for a degenerate FB mass of $3 \times 10^{6}~M_{\odot}$ .
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{0593f10.eps} \end{figure}$ Figure 10: The BH growth dependence on the starting time of accretion. We find that starting DM accretion at a time of $2\times 10^{8}~ {\rm years}$ would fit the recent discovery of a $3\times 10^{9}M_{\odot }$ 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 $m_{\rm f} = 10 ~{\rm keV}/c^{2}$ . However, any higher fermion mass would more sharply and rapidly raise the mass of a seed BH from 5 $M_{\odot }$ to $\sim$ $10^{3}~M_{\odot}$ .
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In the text

$\begin{figure} \par\includegraphics[width=7.9cm,clip]{0593f11.eps} \end{figure}$ Figure 11: We investigate how the BH growth depends on the mass of the seed BH. To fit the data point of mass $M \sim 3 \times 10^{9}M_{\odot }$ at $z \approx 6.41$ , the seed BH cannot be greater than $10^{4}~M_{\odot}$ for a FB of $10^{4}~M_{\odot}$ and the accretion starting time should be about $z \approx 17$ . In this plot, the fermion mass is fixed to $m_{\rm f} \approx 10~ {\rm keV/}c^{2}$ . Thus, any other mechanism that could create massive seed BHs would generate the same growth.
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In the text

$\begin{figure} \par\includegraphics[width=6cm,clip]{0593f4.eps} \end{figure}$	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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$\begin{figure} \par\includegraphics[width=7.8cm,clip]{0593f9.eps} \end{figure}$	Figure 9: The same as in the last figure but for a degenerate FB mass of $3 \times 10^{6}~M_{\odot}$ .
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