Chapter 3: THE FRIEDMANN MODELS

k
( RA)
2
( Ω −1)
=
2
⎛ c ⎞
⎜ ⎟
⎝ H ⎠
with
Ω= 8 πG
ρ
3H
2
Here ρ is the density, be it that of normal matter density, radiation density or false
vacuum energy density, or the sum of all three if appropriate. Indeed, in the general
case we have:
k
( RA)
2
( Ωtot
=
⎛ c
⎜
⎝ H
−1)
2
⎞
⎟
⎠
with
Ω
tot
8πG
=
2
3H
∑
i
ρ
i
This inter-relationship, which can be written in several different ways, including that
of the original Friedmann equation (3.1), is the heart of the General Relativistic
Friedmann-type models.
Having been brought up since the cradle on Newtonian ideas, we tend to think of
densities and velocities as the key quantities to focus on, with gravity acting on the
density to decellerate the Universe and so on. We think intuitively of Newtonian
concepts such as escape speeds, binding energies, etc. For most of us, the curvature
is then added on as an uncomfortable afterthought. This Newtonian approach is a
useful and pragmatic way to approach questions such as the appearance of distant
objects and the physical evolution of the contents of the Universe. Indeed, as we saw,
we can derive perfectly correct expressions describing the expansion of R(τ).
However, in a fundamental way it is really the other way around. The one quantity
that can never change during the expansion of a homogeneous Universe is the
comoving curvature k/A 2 . It is the curvature that describes the Universe. Once the
curvature is defined, the density at any epoch follows from the expansion rate and
vice versa from (3.17). As the Universe expands, the equation of state tells us how the
density will change (e.g. as R -3 , R -4 and R 0 in the three cases above) and, as if by
magic (but not really of course), the expansion rate is decelerated or accelerated in the
ways that we calculated above to compensate.
In the evolution of the Universe, the effective equation of state (i.e. that of the
dominant density component) has changed and may change again. Each of these
changes brings a different form of R(τ) and of ρ(R) with smooth transitions in each so
as to preserve the correct relationship between them.
However, the topology of the Universe, represented in the homogeneous case by the
comoving curvature scalar k/A 2 , remains constant.