AST204 February 6, 2008
LECTURE II: THE REDSHIFT AND THE WHISPERS OF CREATION:
THE COSMIC MICROWAVE BACKGROUND
I. The Past and Future of the Universe
Last time we derived the Friedman equation, which describes the expansion of the universe:
˙
R
2

8
πG
¯
ρR
2
/
3 = 2
ε
(2)
and is exact for the whole universe, because it is the same for all shells and is exact for
small, nearby shells. This result is consistent with (and indeed demands) the cosmological
principle. Shells of radius smaller than some radius of interest expand more slowly, and
shells larger expand more rapidly, so shells never cross. We may take as origin anywhere
in the universe as we have seen, and the expansion is spherically symmetric about that
place, so the density is just inversely proportional to the volume of a sphere centered on
the (arbitrary) origin:
m
= 4
πr
3
¯
ρ/
3 = 4
πR
3
u
3
ρ/
3 =
const.,
so, since u is constant (and, of course, is 4
π/
3) we can write
R
3
¯
ρ
=
const,
and the density is proportional to 1
/R
3
, a function of time alone, and is spatially constant
for all time if it is at any time. Thus the density, Hubble constant, and anything else one
can measure about the expansion are homogeneous for all time.
II. Looking Back: How old is the universe?
If we look at the Friedman equation, it is clear that the second term is proportional to
1
/R
, since we just showed that the density is proportional to 1
/R
3
. So as we look back in
time to times when R was smaller, the gravitional term becomes larger and larger. Since
the
difference
between the gravitational and kinetic terms is constant, the kinetic term
also gets larger and larger, and so the time derivative
˙
R
gets larger and larger. Thus the
recession velocity of any particle, which is just proportional to
˙
R
, decreases with time,
which is exactly what one expects with gravitation; the universe is, in this simple model,
decelerating
. The other side of that coin is that the velocities get higher and higher as one
goes into the past, and so the scale factor
MUST
go to zero at some finite time in the past,
and indeed at a time
less long ago
than 1
/H
, since the motion is accelerated. Thus the
conclusion is inescapable that there was indeed a Big Bang, a time when the density was
infinite and the scale factor zero.
It is also clear that since the kinetic and potential terms both become very large in the
distant past, their difference, the total energy, becomes negligible in comparison to either,
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