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, orchestrated by Steve Shectman and shown below:
Quantitative comparison, however, to the simulation reveals the following points:
We being with the "missing" baryon problem.
The density of baryons can be estimated from models which correctly reproduce the observed abundance of light elements under and assumed range for the entropy of the Universe (photons per baryon) Calculations indicate the following: (h = H/100)
For reasonable ranges of h (0.5 -- 1.0) this means that most of the baryonic material is also "dark". Thus we also have a missing baryon problem!
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 which you have been asked to derive as part of the second homework assignment (we will derive it, from first principles, in class later).
where zh is the vertical disk scale height and Rh 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. In the case of a self-gravitating disk
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 zh and the vertical velocity dispersion SIGMAz for some well defined sample of stars. This transformation from equation 9 to observables makes use of a variant of the collisionless Boltzmann equations. Since stars are not escaping from this system, the collisionless Boltzmann equation can be combined with the equation for continuity of mass to yield (see Binney and Tremaine 1987 for details):
Measurements of the density distribution of stars in the z direction combined with the vertical velocity dispersion then constrains rho.
This is known as the Oort limit problem and its resolution is the subject of the second question in the second homework assignment.