Physics 310 – Cosmology
Homework Set P
1. In class we found a formula for the age of the Universe assuming there is only matter
with a density parameter  . The formula was:
t0  H
1
0
1

0
dx
 x 1 
The assumptions was that matter has a density proportional to a 3 . If the universe is
filled only with radiation, this formula must be changed. We will assume the
universe contains only radiation, so   a 4
(a) Find a formula for the current value of 83  G  and k a 2 in terms of H 0 and  .
These will be identical to the ones found in class.
(b) Find a formula for the value of 83  G  and k a 2 at arbitrary time in terms of
H 0 and  , and the relative scale factor a a0 . This will not be identical.
(c) Find an integral equation for the age of the universe in terms of H 0 and  . It will
be similar to, but not identical to before.
(d) Perform the integration in the special case   1 .
(d) For extra credit, do it for   1 and   1 as well. Sketch t0 H 0 as a function of
 in the range 0 to 2.
2. If the universe is closed, then it must be finite.
(a) Experimentally, I told you that  tot  1.0023  0.0055 . Assume that the central
value is exact, i.e., that  tot  1.0023 . Based on this, determine the current value
of k and a, the latter in terms of Gpc. Hint: use the equation
 kc 2 a 2  H 2 1   tot 
(b) Determine the distance r to the antipodal point in terms of the scale factor a. The
antipodal point is the farthest point from here. This can be determined using our
formula for the area of a sphere of radius r. The antipodal point is the point
where the area of the sphere vanishes. Also, write it in Gpc, using the value from
part (a) for a.