The Dark Matter Universe
The above image is a computer simulation of the distribution of mass in a Dark Matter dominated Universe. The simulation produces much filamentary structure and is filled with voids. Structure (\eg density enhancements) clearly forms at the intersection of these filaments.
How does this compare with observations?
Qualitatively the agreement is good as can be seen by comparing against the results of the Las Campanas Redshift Survey:
Quantitative comparison, however, to the simulation reveals the following points:
Theory: if the Universe is critical then all the luminous matter seen in galaxies contributes only 0.5% of the mass required to eventually halt the expansion (note that the recent WMAP results have substantially lowered the amount of dark matter required to reach a flat/critical Universe).
How do we know its there as a function of scale size?
The general idea is to infer the existence of gravitating matter from perturbations in the motions of objects. In general, this requires application of the Virial Theorem.
where z_{h} is the vertical disk scale height and R_{h} is the radial disk scale length. This exponential form can be derived by assuming an infinitely thin disk (which is justified by the observations) together with an isothermal velocity distribution.
The self gravity in this case is provided by the sum of the stellar distribution and the dark matter distribution. The Solar Neighborhood is a region of of radius roughly 300 light years that contains a few thousand stars. This region contains thin disk, thick disk and halo stars and their normalization is important to the determination of the mass density within this region.
In a highly flattened rotating stellar system, the density distribution in the vertical (z) direction, D(z) is a measure of the surface mass density. This situation arises as Poisson's equation for a flattened system assumes the form
As the density increases, then the z-coordinate sees a larger derivative in the potential which means it experiences a larger gravitational restoring force in that direction. In practice, this gravitational restoring force can be estimated by measuring z_{h} and the vertical velocity dispersion SIGMA_{z} for some well defined sample of stars.
Measurements of the density distribution of stars in the z direction combined with the vertical velocity dispersion then constrains rho.
Results:
While a complete discussion of the Thick Disk is outside the scope of this class, the main point is that it contributes to the gravititional restoring force in a way that is sufficient to explain the observations without need to appeal to local dark matter in the thing disk.
If light traces mass in a galaxy then we expect a kelperian fall off in rotational velocity as a function of radial distance. This is never observed in any galaxy:
Some sample rotation curves:
In most all observed cases, the rotation curve is either flat or rising within the optically observed part of the galaxy. Note that, in general, spirals have an exponential light distribution:
The optical portion of a galaxy usually encompasses about 3 scale lengths (a) of light. If the mass distribution is also exponential and remembering that the circular velocity goes as (M/R)^{(1/2)} , then one can show numerically that for an exponential mass distrubtion the quantity M/R reaches a maximum at 2.15 scale lengths which is about the optical extent of a galaxy. Thus, optical rotation curves could be somewhat flat, they are not unambiguous probes of the Halo.
However, there do exist some galaxies which have large gaseous extents relative to their optical extent. In some cases, this gas has a radial extent of 10 scale lengths and hence the circular velocity is dominated by an extended halo. If not, the circular velocity would fall off rapidly once the optical edge of the galaxy is reached. An excellent example of this is shown below for NGC 3109.
In the above example, the data is fitted against mass contributions from the gas, stars and dark matter in the halo. For this case its obvious that the halo dominates but is this Universal?
Caveats: Perhaps only galaxies with extended dark matter halos are capable of retaining large gaseous extents?
In any event, the rotation curve data, in general, suggests that 90% of the mass of a galaxy is distributed in some spherical halo which contains very little light.
Historically, this was the first set of observations that hinted at dark matter. However, substructure can fool you.
The case of the Cancer cluster:
Cancer is a spiral rich cluster of about 100 bright galaxies. Below is the velocity distribution of galaxies in the Cancer cluster:
IF you treat the above distribution as belonging to one system and just apply the virial theorem you determine that the Cancer Cluster has M/L = 1000 (!). If this value is representative of clusters of galaxies then
where v is the velocity dispersion. For dispersions greater than 300 km/s, the emission is in X-rays. Most clusters of galaxies have a characteristic Temperature of 1--5 keV. But, for Cancer, there was no substantial X-ray emission observed.
Plotting contours of Galaxy density in Cancer suggested that the distribution of galaxies was not relaxed; that is secondary density maxima appeared. These are labelled A thru E below:
This brought up the issue if redshift and position were correlated in this cluster. That is, do galaxies in Group A have different velcoities than galaxies in Group B or C? After a thorough analysis it was shown that these two were highly correlated and that the Cancer cluster could in fact be decomposed into individual groups. In redshift space these groups are shown here:
There is a clear separation of these components in mean velocity and each component has an internal velocity dispersion of about 300 km/s. The M/L of these individual components is 200--300. Furthermore, the components themselves are not gravitationally bound. That is, the Cancer cluster is really an unbound collection of groups:
While Cancer is an extreme case, most all clusters do show some evidence of substructure. Hence, the measured velocity dispersion in these cases does not apply to one dynamical system but to, perhaps, several. Failure to adequately account for substructure in clusters of glaxies leads to systematic overestimates of M/L.
The observation that most clusters have significant substructure implies that cluster formation is still, in essence, occurring as small groups are assimilated into the cluster core. This has implications regarding structure formation scenarios.
Observations of galaxy rotation curves and cluster velocity dispersions suggest that M/L associated with these structures is in the range 10--500; where 500 is fairly extreme. At most, these structures then contribute 20% of the closure mass density. Hence we are driven to the following inescapable conclusion: