Review of what we established so far:

1. A geometric framework based on the Roberston-Walker Metric and General Relativity which describes the Universe via the rate of change of the scale factor (R(t)):
   • Its first derivative is the expansion rate, what we call H_o
   • Its second derivative is the deceleration rate of the Universe which depends on the total mass density, pho

2. Measurements of H_o obtained by estimating distances to galaxies (say via the Tully-Fisher relation) or generally in the range 60 - 90 km s^-1 Mpc^-1. Systematic error in this estimate could come from:
   • systematic error in the distance to the LMC
   • small differences in the ratio of dark-to-light matter in galaxies that are systematically dependent on environment or formation conditions
   • failure to adequately account of substructure in the Virgo Cluster
   • zeropoint errors in the distance to the Hyades and Pleideas star clusters

3. This range of 60 - 90 km s^-1 Mpc^-1 is marginally to strongly inconsistent with the expansion age of the Universe for OMEGA = 1. This is the first indication that LAMBDA may be non-zero.

4. Next we considered the evidence for dark matter on various scales to conclude that
   • After the thick disk mass contribution is taken into account there is very little evidence for substantial amounts of dark matter in the Solar neighborhood

   • Flat rotation curves obtained from extended gas distributions in disk galaxies, however, provides unambiguous evidence (modulo MOND) for substantial unseen mass which is distributed in a spherical halo around the galaxy.

   • X-ray halos around elliptical galaxies also suggest the hot halo gas is being found by a larger dark matter halo

   • Analysis of the dynamics of clusters of galaxies, even after substructure is taken into account, suggests that much of the cluster binding mass must be non-luminous

5. Amazingly, flat rotation curves are the natural result of a simple consideration for the structure of a potential --> that of an isothermal sphere which is in hydrostatic equilibrium and whose phase-space density distribution is Maxwellian in character.

   Now we turn to the issue of structure formation via gravitational instability and do a very bit of Physics . The frist 20 pages of Chapter 5, referenced in the previous link, will come in particularly handy.

   Important note: When we talk about galaxy formation we are talking about two main issues:

   • The actual formation of the potential (which is likely dark) --> this is what we will do in detail
   • The formation of luminous material in that potential --> this we will handwave.

   So On to the Equation of Continuity, the Euler Equation and the Poisson Equation and you can blame this all on Jeans who was quite an accomplished physicist.