Problem set due in class, Apr 30. Show your work, neatly.

1) For a Milne Universe, what is our current age?

2) Consider a `circle' universe, i.e. where space is reduced to a single dimension along the circumference of a circle. The universe has a radius, R, and each point in the universe is given by its co-moving coordinate, theta. Notice that as the universe expands (R increases), the distance between points, s, increases but the co-moving coordinate remains the same. Show that if the universe expands with rate dR/dt, that

ds/dt = (1/R)(dR/dt)s

Why is this Hubble's law?

3) If the speed of light is 1 in this universe, show that for uniform cosmic expansion (dR/dt = constant), that there is a distance s where light cannot reach an observer (this is called a horizon).

4) Consider a universe where you live on the surface of a sphere of radius R. For an object of size x (x << 2*pi*R), what is the angular size of the object as a function of s, the distance from you on the surface of the sphere. What is its behavior as s goes to zero and pi*R? Where is it a minimum? (Hint: put yourself at the north pole and let x lie along a line of latitude similar to the side of a spherical triangle)