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:

- The simulations have more filamentary structure in them -
though the statistical characterization of filamentary structure is
a real numerical challange. The eye-brain is extrememly good at
identifying it compared to machine algorithms.
- The structure which forms at the intersections of filaments or
sheets in the simulations is much richer than the observations show
- The distribution of void sizes in simulation is wrong - there
are too many large voids.

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!

What is Dark Matter?

- Dark matter is material that gravitates but does not emit very much light.
- More specifically, it is material which has a high ratio of mass-to-luminosity

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 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.
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 z_{h} and the vertical
velocity dispersion SIGMA_{z} 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.

Results:

- 1932 - Oort derives local mass density of 0.15 solar masses per cubic parsec
- Mid 80's Bahcall gets 0.18 -- 0.21 solar
masses per cubic parsec
- Testing for the presence of dark matter in the solar neighborhood now
becomes an accounting problem The
possible sources of this mass in the solar neighborhood are 1)
luminous stars, 2) interstellar gas, 3) stellar remnants (mostly
white dwarfs) and 4) dark matter:
- Luminous stars --> 0.044 solar masses per cubic pc
- gas --> 0.042 (not surprising its the same as stars)
- white dwarfs --> difficult to really measure 0.01 --0.03;
reasonable upper limit is 0.044 as galaxy not yet old enough
to have more mass in remnants than luminous stars
- The above implies that 0.05 solar masses per cubic pc is missing
This is known as the Oort limit problem and its resolution is the subject of the second question in the second homework assignment.