Physics 410: Cosmology
Test 1
April 28, 2017

NAME _____________________________________________________________________

Do two of the three problems.

Question 1

Einstein noted that in order for a static universe filled with non-relativistic matter to exist, there must be some unknown repulsive force at play in the universe.

a. Show that Λ, the measure of the repulsive force suggested by Einstein, must have value 4πGρ in order to counteract gravity.

b. What is the curvature of a static universe with the given Λ?

c. Suppose matter is converted into photons, say in stars, will the universe expand? Will it contract? Explain your answer.

Question 2

The cosmological redshift is given by 1+z = [a(t_o)/a(t_e)]

where z is the redshift, t_o is the time of observation, and t_e is the time the galaxy emitted the light we observe at t_o.

a. For an empty universe with k = -1, find the time at which a galaxy emitted the light which led to the redshift measurement of z. Express your answer in terms of H_o and z.

b. What was the distance to the galaxy when it emitted the light?

c. Show that your expression leads to Hubble's law for small z.

Question 3

a. Show that the single-component Friedmann equation with k = 0 can be written

ಠ= {(8πG/3c²)ε_o a_o^3(1+w)} a^-(1+3w)

where a is the scale factor, G is the gravitational constant, ε_o is the energy density at the current time, c is the speed-of-light, and w is the coupling coefficient for the pressure and energy density, P = wε.

b. Solve for a. Guess a solution of the form

a = αt^β

where α and β are constants.

c. Find an expression for the age of the universe in this case. For what values of w are the ages greater than and/or less than the Hubble time, τ_H = H_o^-1?