Chapter 3: THE FRIEDMANN MODELS

(3.4)
&
&&& 4πG R
RR R
3
=− ρ
2
3 R
R&
2 8πG = R
3 1
ρ + 2ξ
3 R
R&
2 8πG = ρR
2
+ 2 ξ
3
Here, ξ is a constant of integration that, in this Newtonian analysis, corresponds to the
binding energy of the system: If ξ > 0, then we have solutions with
&R 2 → 2ξ as R→ ∞. R(τ) is proportional to τ and the Universe is unbound and
expands forever. If ξ < 0, we get R & = 0 at finite R so the Universe is bound and at
some point the expansion is halted and the Universe recollapses. If ξ = 0, then we
have the asymptotic limit R & → 0 as R→ ∞
Note that this has exactly the same form as the Relativistic Friedmann equation (3.1)
with Λ= 0 if we associate the ξ binding energy with the curvature term.
Let us now return to the equation of state and consider possibilities other than the
familiar one considered above, where we assumed that the density is dominated by
dust with ρ ∝ R -3 .
An obvious possibility is that the density of the Universe is dominated by
electromagnetic radiation, e.g. the cosmic background radiation field. As the
Universe expands, the density of this radiation field will drop as R -4 . We get a factor
of R -3 from the decreasing number density of photons, but an additional factor of R -1
from the fact that all the photons lose energy due to the redshift.
There are other more exotic possibilities. As we’ll see below, we could imagine a
scalar density field that does not change its density as the Universe expands, i.e. ρ =
constant (more than that – in fact we have evidence that this is the case!). We will
refer to this as a false vacuum density.
Other possibilities exist. A few years ago there was some interest in hypothetical
Universes that were dominated by the mass from “cosmic strings” – strange entities
that have a constant density per unit length regardless of how much they are stretched.
These give ρ ∝ R -2 . It is easy to see that putting these other ρ(R) dependencies into
our Newtonian equation (3.3) will not yield the right form of (3.4) upon integration
unless we introduce a small fudge. To see how to proceed, it is useful to consider the
ρ(R) dependence in thermodynamic terms.
We can assume that the expansion is adiabatic (i.e. that there is no inflow or outflow
of heat). The 1st Law of Thermodynamics gives us that the change in the internal
energy (ρV) equals the pdV work done by the system in expanding:
2
(3.5) c d( ρV)
=−pdV
⇒
dρ
=−
( ρ + pc)
2
3
dR
R
If the pressure can be neglected, then