Reading: Chapter 8 of Sparke & Gallagher; Section 26.2 of Carroll & Ostlie
The rules: Collaborative work is encouraged. This homework can be done in consultation with your fellow classmates, the AI, or the professor. However, everyone must submit their own solutions to get credit, and all help should be acknowledged (a single sentence mentioning the others in your study group is sufficient). Show your work.
1. [4 pts] Explain the significance of the Jeans mass in relation to the perturbations that can or cannot grow in the expanding universe.
2. [8 pts] Based on the motions of the Milky Way and M31, the mass of the Local Group is approximately 3 x 1012 Msun. Assuming its radius is approximately 1 Mpc, what is its average density. Show that this is only about 2 h-2 of the critical density.
3. [8 pts] The time taken for a galaxy or cluster to grow to density rho must be at least as great as the free-fall time, tff = 1/sqrt(G rho), since cosmic expansion must first be halted locally. The Milky Way has an average density that is approximately 105rhocrit, while a cluster of galaxies has an average density of approximately 100 rhocrit. How fast can the collapse occur for the Milky Way? For a cluster of galaxies?
4. [4 pts] If the Local Group's motion of 600 km/s relative to the CBR represents a typical peculiar velocity, how long would it take for a galaxy to move from the center to the edge of a void 60 Mpc across?
5. [6 pts] Problem 8.6 in Sparke & Gallagher.
6. [12 pts] Problem 26.11 in Carroll & Ostlie
7. [8 pts] Problem 26.12 in Carroll & Ostlie