Cosmology
Examples Sheet 3: Observational Cosmology and the age of the universe
1. A galaxy emits light of a particular wavelength. As the light travels, the expansion
of the Universe slows down and stops. Just after the Universe begins to
recollapse, the light is received by an observer in another galaxy. At the time the
light is received, is the galaxy moving towards or away from the observer? Does
the observer see the light redshifted or blueshifted?
2. Throughout this question, assume a matter-dominated Universe with k = 0. By
considering light emitted at time te and received at the present time t0, show that
the coordinate distance travelled by the light is given by:

1 
r0  3ct 0 1

1 z 

Derive a formula for the apparent angle subtended by an object of length l at
redshift z. Find the behaviour in the limits of small and large z, and provide a

physical explanation.
Show that the object appears the smallest if it is located at
redshift z = 5/4.
3. What is the maximum possible physical separation in a closed Universe?
4. The galaxy’s age can be estimated by radioactive decay of Uranium. Uranium is
produced as an r-process element in supernovae, and on this basis the initial
abundances of U235 and U238 are expected to be:
U 235
 1.65
U 238 initial
The decay rates are:

(U235) = 0.97  10-9 yr-1
(U238) = 0.17  10-9 yr-1
Finally, the present abundance ratio is
U 235
U 238
 0.0072
final
U 235
U 238
 0.0072
final


Please turn over
Use the decay law U(t) = U(0) exp (-t) to estimate the age of the galaxy.
Assuming the galaxy took a minimum of an additional 109 yr to form in the first
place, obtain an upper limit on the value of the Hubble parameter h assuming a flat
matter-dominated Universe (2001 exam, Q2).
matter-dominated open universe, the present age of the Universe is given
by the appalling formula:
H0t0 


1
0
1 2  0

cosh


1 0 2(1 0 ) 3 / 2
 0 
Demonstrate that in the limiting case of an empty universe 0  0 we get H0t0 = 1
and in the limiting case of a flat universe 0  1 we get H0t0 = 2/3.

[Useful formulae: cosh-1(x)  ln(2x) for large x
cosh-1[(1+x)/(1-x)]  2 x + 2x3/2/3 for small x]