Notes on Relativity and Cosmology - Physics Department, UCSB

10.3. OUR UNIVERSE: PAST, PRESENT, AND FUTURE 295
the universe in terms of a density ρ and a pressure P. Let’s first think about
the pressure. Most matter today is clumped into galaxies, and the galaxies are
quite well separated from each other. How much pressure does one galaxy apply
to another? Essentially none. So, we can model the normal matter by setting
P = 0.
When the pressure vanishes, one can use the Einstein equations to show that the
quantity: E = 8πGρa 3 /3 is independent of time. Roughly speaking, this is just
conservation of energy (since ρ is the density of energy and a 3 is proportional
to the volume). As a result, assuming that Λ = 0 the Einstein equations can be
written:
( ) 2
1 da
c 2 − E
dt c 2 + k = 0. (10.5)
a
Recall that k is a constant that depends on the overall shape of space: k = +1 for
the spherical space, k = 0 for the flat space, and k = −1 for the Lobachevskian
space.
In the above form, this equation can be readily solved to determine the behavior
of the universe for the three cases k = −1, 0, +1. We don’t need to go into the
details here, but let me draw a graph that gives the idea of how a changes with
t in each case:
k = -1 a t
k = 0
a(t)
a
t
2/3
k = +1
proper time
Note that for k = +1 the universe expands and then recontracts, whereas for
k = 0, −1 it expands forever. In the case k = 0 the Hubble constant goes to
zero at very late times, but for k = −1 the Hubble constant asymptotes to a
constant positive value at late times.
Note that at early times the three curves all look much the same. Roughly
speaking, our universe is just now at the stage where the three curves are beginning
to separate. This means that, the past history of the universe is more
or less independent of the value of k.