 
Summary: Astrophysics (Physics 489) Exam 2
Please show all significant steps clearly in all problems.
1. Fun with dark energy. Since the dark energy is currently a total mystery, many theoretical
cosmologists have had fun assuming some bizarre form of matter with an equation of state which is not
exactly the same as that of vacuum energy namely,
p = w! , w = "1± #
where ! is positive but must be small because of the limits already imposed by observations (of Type Ia
supernovae, the cosmic microwave background radiation, and the large scale structure of galaxies in the
universe). Recall that p is the pressure, which is also important in Einstein gravity.
Since you worked hard in the homework to help derive the following equations, we will simply assume
them in doing this problem:
d
dt
!a3
( )= " p
d
dt
a3
( ) da / dt
a
