It is twenty-five years since the discovery that galaxies are surrounded by extended massive halos of dark matter. A variety of observational probes -- disk rotation curves, stellar kinematics, gas rings, motions of globular clusters, planetary nebulae and satellite galaxies, hot gaseous atmospheres, gravitational lensing effects -- are now making it possible to map halo mass distributions in some detail. These distributions are intimately linked to the nature of the dark matter, to the way halos formed, and to the cosmological context of halo formation. Indeed, early analytic and numerical work suggested a clear dependence between halo structure and cosmogony. Unfortunately, the numerical limitations of these early simulations precluded probing the central structure of halos, precisely the region where theoretical predictions can actually be compared with observations.
In a series of recent papers we have examined in detail the structure of dark matter halos formed in different cosmogonies using high-resolution N--body simulations[16,24]. These studies have revealed a remarkable similarity in the structure of halos formed in hierarchically clustering scenarios[28]. Independent of halo mass, the spectrum of initial density fluctuations, or the value of the cosmological parameters, the density profiles of virialized systems can be well fit by scaling a simple ``universal'' profile (Figure 5a) of the for
rho(r)/rho_crit= delta_c/(r/r_s)(1+r/r_s)^2},
where r_s is a scale radius, \delta_c is a characteristic (dimensionless)
density, and \rho_{crit}=3H^2/8 \pi G is the critical density for closure.
The radial scale determines the mass of the system, so that at fixed mass
this profile has a single free parameter, the characteristic density (\delta_c)
of a halo. (The characteristic density of a halo can, equivalently, be
expressed as a ``concentration'' by the ratio between virial and scale
radii, c=r_{200}/r_s). Our simulations reveal that halo mass and \delta_c
are strongly correlated, a relation that can be reproduced by assuming
simply that the characteristic density is just proportional to the mean
density of the universe at the time of halo collapse, ie. \delta_c(M) \propto
\Omega_0 (1+z_{coll}(M))^3 (Figure 5b). Constraints on the shape of the
mass profile of dark halos therefore translate into constraints on their
collapse redshifts, which in turn depend on the cosmological parameters.
(Figure 5
a)-left: The typical density profiles of CDM
halos. The leftmost (rightmost) system has a mass M_{200}=3\times 10^{11}
M_{\odot} (M_{200}=3 \times 10^{15} M_{\odot}). Arrows indicate the
value of the gravitational softening in each simulation. (Figure
5b--right) The characteristic density as a
function of collapse redshift, defined as the time when half of the final
mass is in collapsed progenitors more massive than 10 \% of the final mass.
Solid circles correspond to runs with \Omega_0=1 and different power spectra,
open circles to runs with \Omega_0=0.1 and \Lambda=0, and starred symbols
to runs including a cosmological constant (\Omega_0=0.25, \Lambda=0.75).
The solid line shows the ``natural'' scaling, \delta_c \propto \Omega_0
(1+z)^3, expected if the characteristic density of a halo is directly proportional
to the mean matter density of the universe at the time of collapse.
The characteristic density, or concentration, of dark halos can be constrained
by detailed disk-halo decomposition of the rotation curves of disk galaxies
and by gravitational lensing analysis of the mass profiles of galaxy cluster
halos[26,27]. The mass profile of
eq.~1 is consistent with measured rotation curves, as shown in Figure 6a
for NGC3198, a spiral galaxy with high-quality HI velocity data. Halo parameters
derived from such fits for sixteen galaxies are shown in Figure 6b, where
we show, as a function of circular velocity, the range of allowed halo
concentrations (solid circles). The open circles in Figure 6b correspond
to fits to the projected mass distribution of galaxy clusters determined
by weak gravitational lensing techniques. For comparison, we show the predictions
of our numerical models for two popular cosmological scenarios, the standard
biased CDM model (SCDM: \Omega_0=1, h=0.5, \sigma_8=0.6) and a flat, low
density CDM model normalized to match the COBE measurements of fluctuations
in the CMB (\LambdaCDM: \Omega_0=0.3, \Lambda=0.7, h=0.5). Clearly, the
observations rule out halos as concentrated as expected in the SCDM model,
and favour the low density \LambdaCDM model. This illustrates nicely how
the similarity in structure of dark matter halos unveiled by our numerical
experiments can be combined with observations to derive strong constraints
on the cosmological parameters.
(Figure 6a--left) Disk-halo fit to the rotation curve of NGC3198, using the halo density profile in eq.~1. The contribution of the halo is shown by the dashed line, that of the disk with the dotted line. The contribution from neutral hydrogen is given by the dot-dashed line. (Figure 6b--right) Halo mass versus concentration, as derived from disk-halo fits to the rotation curves of late-type disk galaxies (filled circles) or from weak gravitational lensing analyses of galaxy clusters (open circles).
Other testable predictions from this modeling concern systematic
changes in the {\it shape} of the rotation curves as a function of
galaxy surface brightness[25,26].
Since near the center the halo density scales like r^{-1}, its contribution
to the circular velocity increases outwards, V_c(r)=(GM/r)^{1/2}
\propto r^{1/2}. Flat rotation curves are, therefore, a direct result
of the contribution of the galaxy disk to the circular velocity.
This implies that disks of similar mass but different sizes, e.g.
those of high and low-surface brightness galaxies of the same luminosity,
should have rotation curves whose shapes differ systematically; normal,
high-surface brightness galaxies should have fast-rising rotation
curves while those of low-surface brightness galaxies should rise more
slowly, usually out to the outermost measured point. Observations of
the shape of the rotation curves of low surface brightness galaxies
can then provide important clues regarding the effects of the assembly
of the disk on dark matter halos. I am currently involved in a vigorous
observational program that aims to measure optical rotation curves of about
50 low-surface brightness galaxies to test this
prediction in detail.
We also plan to extend our N--body study in order to address issues
such as the magnitude and source of scatter in the characteristic
density of halos of a given mass, as well as the degree of substructure
in galaxy clusters and the timescale on which it is erased. The latter
is especially important for gravitational lensing studies, since
these only probe the projected mass density profiles and are therefore
extremely sensitive to substructure and deviations from spherical
symmetry. Both scatter and substructure are expected to depend markedly
on the cosmological parameters and, therefore, a systematic study
of this kind is likely to yield observational predictions that can
be used directly to constrain cosmological models further.
