Reading: Sections 1.4 - 1.5 and 7.2 of Sparke & Gallagher
and Chapter 17 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] Evaluate the critical density of the universe (rho_crit) at the present time and compare the baryonic density of the universe with the critical density.
2. [6 pts] One of the "self-evident" truths on which Euclid's geometry is based is the so-called parallel postulate. This states that given a straight line l and a point P not on it, only one line parallel to l can be drawn through P (that is, a line that does not meet l if both lines are extended indefinitely). What happens to this postulate in the geometry on the surface of a sphere and on a saddle-shaped surface?
3. [8 pts] It might be argued that the inverse square law for light would provide a solution to Olber's paradox. To see that this is not so, consider a uniform distribution of stars with n stars per unit volume, each of luminosity L. Imagine that two thin spherical shells of stars with radii r1 and r2 are centered on Earth; let the thickness of each shell be delta r. Show that the same energy flux reaches Earth from each shell.
4. [12 pts] Show that R is proportional to t2/3 in the early universe for both open and closed Newtonian models.
5. [20 pts] Begin the "shape of the universe" project.