1. Collapse of overdensities

Remember our discussion about the
turnaround and collapse of initial overdensities in matter-only
universes. If you have a little overdense bubble of the universe
embedded in a flat, no-lambda universe, show that the density of the
overdensity at maximum size is given by

where R refers to the scale factor of the universe, R_{max} refers to the maximum size of the overdense bubble, and the "prime" mark means the value of Omega_{0} in the overdense
bubble.

where R refers to the scale factor of the universe, R

2. Baryon budgeting

First, use the literature to get estimates of
the following for the Coma cluster of galaxies:

(As usual, estimates must come from refereed journal articles or research-grade books, and you must cite your sources.)

- Total mass
- Mass of X-ray emitting gas
- Mass of stars

(As usual, estimates must come from refereed journal articles or research-grade books, and you must cite your sources.)

3. Mass Functions

We've seen that the mass function for halos can be described as a power law with a high mass exponential cutoff:

N_halos ~ C1 * (M/Mstar)**gamma * exp(-M/Mstar)

The luminosity function of galaxies has a very similar form:

N_gals ~ C2 * (L/Lstar)**alpha * exp(-L/Lstar)

This makes it awfully tempting to hope that we can have a simple model for total mass to light that would connect the two.

The halo mass function might be characterized by the terms gamma = -2 and Mstar = 10^{13} Msun, while the galaxy luminosity function would be more like alpha = -1 and Lstar=2x10^{10} Lsun (for B-band luminosity). The C's would be normalization terms to get the functions into absolute terms (ie number per cubic Mpc), but let's not worry about that part.

Find values for the total mass (including dark matter) and total B luminosity (in solar luminosities) for the Milky Way, and calculate the Milky Way's total mass-to-light ratio (including dark matter). Cite your sources.

Assume that this total mass-to-light ratio applies for all galaxies, and turn the luminosity function into a mass function: N_halos_from_gals.

Now overplot the two mass functions on the same plot, for the mass range 10^{9} to 10^{14} Msun. Make it a log-log plot, and make the y-axis a useful range to compare the two functions (that is, you don't need to plot the exponential cutoff all the way down to -infinity....). Shift your plots so that they match at Milky Way masses.

Looking at your two plots, how well do the two plots match up across the range of masses? If you wanted to make the curves match up perfectly, what would you have to do to the mass-to-light ratios for high mass halos to make them fit? What about for low mass halos? Describe physical process that might make the mass-to-light ratio for halos behave like this.

N_halos ~ C1 * (M/Mstar)**gamma * exp(-M/Mstar)

The luminosity function of galaxies has a very similar form:

N_gals ~ C2 * (L/Lstar)**alpha * exp(-L/Lstar)

This makes it awfully tempting to hope that we can have a simple model for total mass to light that would connect the two.

The halo mass function might be characterized by the terms gamma = -2 and Mstar = 10

Find values for the total mass (including dark matter) and total B luminosity (in solar luminosities) for the Milky Way, and calculate the Milky Way's total mass-to-light ratio (including dark matter). Cite your sources.

Assume that this total mass-to-light ratio applies for all galaxies, and turn the luminosity function into a mass function: N_halos_from_gals.

Now overplot the two mass functions on the same plot, for the mass range 10

Looking at your two plots, how well do the two plots match up across the range of masses? If you wanted to make the curves match up perfectly, what would you have to do to the mass-to-light ratios for high mass halos to make them fit? What about for low mass halos? Describe physical process that might make the mass-to-light ratio for halos behave like this.

4. Luminosity Functions

Take the luminosity function for galaxies given in the previous problem an plot it (again, on a single log-log plot) for value of alpha of -0.5, -1.0, -1.5, and -2.0. For each of these values of alpha calculate how many LMC-like galaxies there are for every Milky Way. Also calculate how many dwarf spheroidal galaxies like Leo I there are for each Milky Way.

Now show that for values of alpha <= -2, the total luminosity summed over all galaxies is infinite (and therefore not a very realistic model).

Now show that for values of alpha <= -2, the total luminosity summed over all galaxies is infinite (and therefore not a very realistic model).