COSMOLOGY
Problem Set 3
April 12, 2010
Problem 1
: Matter Dominated Evolution
Consider a “Matter Dominated” (MD:
p
= 0, Λ = 0) RobertsonWalkerFriedman
cosmological model.
a) Find the density parameter as a function of the redshift, Ω(
z
), and its present
value Ω
0
. In the limit of high redshift (e.g., Ω
0
z >
1) find Ω(
z
).
[10]
b) In terms of
H
0
and Ω
0
, find the Hubble parameter as a function of redshift,
H
(
z
).
In the limit of high redshift (e.g., Ω
0
z >
1) find
H
(
z
).
[10]
c) For Ω
0
<
1, find an analytic expression for the ratio of the present age to the
Hubble age,
H
0
t
0
, as a function of Ω
0
. How old is this Universe at present if Ω
0
= 0
.
3
(you may assume
H
0
= 70 km/s/Mpc)?
[10]
d) How old would this Universe be at present if Ω
0
= 1?
[5]
Problem 2
: MD, High Density
For a MD, RWF model with Ω
0
= 2 find:
[Note: You may assume here that
a
0
≡
c/H
0
.]
a) The present comoving radial coordinate of the horizon, Θ
H
(
t
0
). What fraction of
the entire Universe is observable (in principle) at
t
0
? [Note that this is a closed, finite
universe.]
[10]
b) The comoving radial coordinate of the horizon at the time of maximum expansion,
Θ
H
(
t
max
) (where
a
max
=
a
(
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 Spring '10
 STEIGMAN
 Dark Energy, General Relativity, Big Bang, Redshift, Physical cosmology

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