COSMOLOGY
Problem Set 2
April 7, 2010
Problem 1
: Collapse of a Uniform Sphere
This is really a problem in
Newtonian
mechanics.
Consider a uniform (homoge
neous) sphere of mass
M
whose initial radius (at
t
= 0) is
R
0
. The sphere is released
from rest at
t
= 0, collapsing under its own gravity.
a) In terms of
M
and
R
0
, find the time,
t
coll
, for the sphere to collapse to
R
= 0.
[10]
b) Suppose the
same
uniform sphere of mass
M
were released from
R
′
0
=
αR
0
. Find
the ratio of the new collapse time
t
′
coll
to
t
coll
(from part a).
[5]
c) In the limit
R
≪
R
0
, where
R
=
R
(
t
) is the radius of the collapsing sphere (
t > t
0
),
find how rapidly the collapse is accelerating; i.e., find the
de
celeration parameter
q
.
(Recall that
q
≡
−
1
H
2
[
1
R
(
d
2
R
dt
2
)].)
[5]
Problem 2
: PowerLaw Expanding Universe
There is an interesting class of cosmological models for which the 3space curvature
may be neglected (
k
= 0) and the timeevolution of the scale factor is a power law
(
a
∝
t
α
, where 0
< α <
1). For this class of models find (in terms of
α
):
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 Spring '10
 STEIGMAN
 Mass, General Relativity, Big Bang, Redshift, Physical cosmology, deceleration parameter, uniform sphere

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