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supermassive black holes ( bhs ) are now considered to be a common feature of galaxies which have a bulge . furthermore , a number of observational properties of the host galaxy correlate with the bh mass . among the strongest of these correlations is the relationship between the bh mass and velocity dispersion within the galactic bulge ( ferrarese & merritt 2000 ; gebhardt et al . 2000 ) : @xmath1 , where @xmath2 ( tremaine et al . 2002 ) . since the velocity dispersion is measured well outside of the bh `` influence , '' this correlation indicates that an intimate relationship exists between the bh and the dynamical structure of the host galaxy . we assume that a galaxy forms by the extended collapse of a `` halo '' composed of collisionless matter and that simultaneously the central black hole is growing proportionally to the halo as matter continues to fall in . this is equivalent to the assumption of multidimensional self - similarity ( carter & henriksen 1991 ; henriksen 1997 ) , and macmillan & henriksen ( 2002 ) show that , in this case , the mass @xmath3 inside any surface in the halo is related to the velocity of the particles by @xmath4 where @xmath5 is the averaged velocity . since the system is self - similar , this relation will apply at both the `` bulge '' mass scale as well as the bh mass scale . the quantities @xmath6 and @xmath7 are scales in space and time , respectively , and are related to the power - law index of the initial density perturbation @xmath8 in spherical infall models of halo growth ( henriksen & widrow 1999 ) : @xmath9    this power - law index @xmath8 depends on the spectral index @xmath10 of the primordial power spectrum , @xmath11 , by @xmath12 . there is therefore a direct link between the initial power spectrum and the predicted relationship between the bh mass and the velocity dispersion ; for agreement with observations , we must have @xmath13 , so that @xmath14 and @xmath15 . we determine below that the addition of sufficient angular momentum breaks the self - similarity in the central regions . this creates a `` mass reservoir '' around the bh that does grow in proportion to the galaxy mass , and from which the bh grows more slowly by collisional interactions between clumps of matter . provided most of this mass may be absorbed by the bh on a cosmic timescale , the proposed relation should still hold . the initial conditions for the simulations follow those of henriksen & widrow ( 1999 ) : @xmath16,\ ] ] where @xmath17 & r < r_c \\ a(1 - b)(r / r_c)^{-\epsilon }   & r \ge r_c . \end{array } \right.\ ] ] this initial density profile is that of a power - law function of radius , modified at small radii to account for suppression of small scale fluctuations . the constant @xmath18 and we choose @xmath19 . the particles are started in the unperturbed hubble flow . our initial numerical work began with a simple shell code , which is used to follow spherically symmetric shells of matter in an einstein - de sitter universe ; each shell evolves according to @xmath20 where @xmath21 is the total mass enclosed by the shell at radius @xmath22 . the angular momentum @xmath23 is given by @xmath24 , evaluated at the initial shell positions . thus the particles are allowed to have , in general , elliptical orbits , and the shells follow the evolution of their radial coordinates .    to extend the spherically symmetric numerical results , we consider black hole growth in an n - body system of @xmath25 particles . we use a modified version of piet hut s code ( http://www.ids.ias.edu/  piet / act / comp / algorithms / starter/ ) this code was modified to include a softening length to handle close encounters between particles .    in both simulations , a central bh has been included that can `` eat '' particles which fall below the schwarzschild radius , @xmath26 , and grow in mass accordingly . also , we choose a power - law index for the initial perturbation of @xmath14 , so that @xmath27 and @xmath28 , as observed . the shell code simulations are run , with an initial bh mass of @xmath29 , until @xmath30 , at which time all matter is essentially `` eaten '' by the bh in the @xmath31 case . other values of the specific angular momentum are also explored . the bh mass , as a function of time , is shown in figure  [ growth]a for j = 0 , 0.01 , 0.1 , 0.2 , and 0.5 . macmillan & henriksen ( 2002 ) predict that the bh mass should grow as @xmath32 . this is indeed the asymptotic behavior for the case where @xmath31 , as shown in the figure . this logarithmic slope , however , decreases as the specific angular momentum increases ; for @xmath33 , the mass is increasing as @xmath34 . for larger values of @xmath35 , the growth essentially `` shuts off '' at some time , indicating that shells with high initial angular momentum are never reaching the centre of the system . three different cases are explored with the n - body simulations . the first is an initial spherically symmetric system . in the second and third cases , spherical symmetry is removed by giving each of the particles a random displacement , with amplitude @xmath36 . systems are considered in which the amplitudes are about 0.8% and 1.6% of the initial extent of the system . results are shown in figure  [ growth]b for the black hole mass as a function of time . for the case with no initial random displacements ( @xmath37 ) , the growth is approximately that expected , although at later times the black hole growth slows down due to a lack of infalling particles . for the more general cases where @xmath38 , black hole growth remains self - similar , but with a much lower logarithmic slope : @xmath39 or a little greater . in these systems , the particles have enough angular momentum that they do nt fall straight into the central black hole , and rather form a `` core '' at the centre which slowly feeds the black hole growth .    to test the prediction that @xmath28 , the velocities of some fraction of the particles are averaged ( equating @xmath40 ) . the predictions of macmillan & henriksen ( 2002 ) apply to a fixed scaled position vector @xmath41 , implying that we should average all particles within some comoving radius . results are shown in figure  [ msigma]a for a comoving radius of @xmath42 . although the case where @xmath37 may have a regime where it follows the predicted @xmath43 relation , the cases with @xmath38 clearly follow no relation at all ; their average velocity is about @xmath44 for most black hole masses .    of interest , however , are the results if we take a fixed radius , rather than a comoving one , and average the velocities of the particles within that radius . the plot of @xmath45 versus @xmath5 is shown in figure  [ msigma]b . all three cases clearly show a trend that goes like @xmath28 .        for systems in which there is no angular momentum or deviation from spherical symmetry initially , the numerical work , with both the spherical shell code and the more general n - body code , confirms the predicted black hole growth : @xmath46 . however , for the simulations in which there was some deviation from a simple spherical infall of matter , results were contrary to what was expected . in particular , the n - body results show a black hole growth approximately proportional to the time .    however , it is apparent from the simulations that the black hole growth in these cases is not dominated by the infalling matter , as it is in the initially spherical case . rather , the particles form a `` core '' about the centre , and so the growth is fed in a different manner . dimensional arguments suggest that the schwarzschild radius should scale with time as @xmath47 , where @xmath48 is the speed of light , a constant of the system . if this is the case , then we get that the black hole mass will grow as @xmath49 , which is approximately what is observed . note , however , that this breaks the symmetry of the system , since the mass on a larger scale is predicted to grow as @xmath50 . the numerically derived @xmath0 relationship , which here compared the black hole growth with the average velocity of particles within some radius @xmath51 , only gave the expected relation , @xmath28 , when a fixed , rather than comoving radius was considered . furthermore , the cases for which the growth rate was much shallower than expected still showed the same trend as the other case ; that is , @xmath28 regardless of how the black hole grew with time . as stated above , however , this result is for a fixed radius . from the definition of @xmath41 and @xmath52 in macmillan & henriksen ( 2002 ) , we see that @xmath53 . fixing @xmath51 requires @xmath41 to then depend on time . recalling that we re considering a density profile that goes as @xmath54 , and @xmath8 is related to the quantity @xmath55 by equation ( [ eq : relate ] ) , we can write the mass as a function of @xmath56 : @xmath57 thus , as a function of the ( fixed ) radius , we have @xmath58 . similarly , we can write the self - similar velocity , assuming it is in its virialized state , as a function of @xmath56 : @xmath59 so that the radial dependence of the velocity takes the form @xmath60 . combining these two equations for mass and velocity and eliminating @xmath51 gives us the familiar relation given by equation ( [ eq : predict ] ) . thus , taking the mass and velocity at a fixed radius @xmath51 gives the same relation regardless of the time dependence of either . we consider black hole growth in a system which forms as the black hole is growing . if this is the case , then the system has multidimensional self - similarity , and the black hole should grow as a power - law function of time , @xmath61 . furthermore , the velocities of the particles of the system will also evolve in a similar manner , and we predict that @xmath62 . numerically , we confirm that the black hole does indeed grow as a power - law function of time ; however , it only follows the predicted relationship if the initial system has little or no angular momentum . for systems with non - spherical initial conditions , the black hole grows as @xmath39 or a little greater . furthermore , we test the@xmath0 relation with n - body simulations . although results do not show a strong relation if we take an averaged velocity within a comoving radius , the predicted relation , that @xmath28 , is clearly present if we take a fixed radius and average the particles within that . this can be explained by considering the self - similar variable @xmath41 to be a function of time such that the time dependence of the radial coordinate is fixed .
this poster discusses a possible explanation for the relationship between the mass of the central supermassive black hole and the velocity dispersion in the bulge of the host galaxy . 
 we suppose that the black hole and the dark matter halo are forming simultaneously as matter falls in , and a self - similar system then exists in which the mass and the velocities of the system evolve as power - law functions of time . 
 this leads naturally to a relationship between the black hole mass and the velocities in the halo which , with a reasonable choice of cosmological parameters , is in good agreement with the observed relationship . 
 we also confirm this relationship with more robust numerical results . 
 [ 1996/06/01 ]    a&a