Reading: Chapters 17 and 29 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. [8 pts] Using SuperMongo, plot the parametric solutions to the Newtonian open and closed models of the universe. Set the scale so that R = 1 at the present time.
2. [6 pts] A galaxy is observed with a redshift of 0.69. How long did light take to travel from the galaxy to us if we assume that we live in an Einstein-de Sitter universe with Ho = 75 km/s Mpc-1? Relative to the present size, what was the size of the universe when the light was emitted?
3. [10 pts] Starting from first principles, calculate the age of a "dust-filled" universe with qo = 2. Hint: start from (dR/dt)2 + 2 R (d2R/dt2) = - k c2 and derive the t(z) relation given in class, but using q instead of omega.
4.
[6 pts]
In the Newtonian framework applicable to our local neighborhood, the isotropic
Hubble law may be expressed as the velocity distance relation:
V(r) = Ho r,
where r is the position vector of a galaxy relative to the origin. If
the observer at the origin has peculiar velocity w, he or she observes
an anisotropic velocity distance relation given by:
V'(r) = V(r) - w = Ho r - w.
Show that the effective Hubble constant H(theta) in a direction making an angle
theta with the direction of the observer's peculiar velocity is given by
H(theta) = Ho - w cos (theta)/ r.
Thus, H(theta) is maximum at antapex (theta = pi) and minimum at apex (theta = 0).
5. [20 pts] Continue the "shape of the universe" project.