24 Kinematics and dynamics of an expanding universe
ε(t).The solutions, and hence the future of the universe, depend not only on the
geometry but also on the equation of state.
Problem 1.13From (1.65) and (1.67), derive the following useful relation:
H ̇ =− 4 πG(ε+p)+ k
a^2. (1.68)
Problem 1.14Show that, forp>−ε/3, a closed universe recollapses after reach-
ing a maximal radius while flat and open universes continue to expand forever.
Verify that the spatial curvature term in (1.67),k/a^2 , can be neglected asa→ 0
and give a physical interpretation of this result. Analyze the behavior of the scale
factor for the case−ε/ 3 ≥p≥−ε.
To conclude this section, let us reiterate the most important distinctions be-
tween the Newtonian and relativistic treatments of a homogeneous, isotropic uni-
verse. First, the Newtonian approach is incomplete: it is only valid (with justi-
fication from General Relativity) for nearly pressureless matter on small scales,
where the relative velocities due to expansion are small compared to the speed
of light. In Newtonian cosmology, the spatial geometry is always flat and, conse-
quently, the scale factor has no geometrical interpretation. General Relativity, by
contrast, provides a complete, self-consistent theory which allows us to describe
relativistic matter with any equation of state. This theory is applicable on arbi-
trarily large scales. The matter content determines the geometry of the universe
and, ifk=±1, the scale factor has a geometrical interpretation as the radius of
curvature.
1.3.4 Conformal time and relativistic solutions
To find particular solutions of the Friedmann equations it is often convenient to
replace the physical timetwith the conformal timeη, defined as
η≡∫
dt
a(t), (1.69)
so thatdt=a(η)dη.Equation (1.67) can then be rewritten as
a′^2 +ka^2 =8 πG
3
εa^4 , (1.70)where prime denotes the derivative with respect toη.Differentiating with respect
toηand using (1.64), we obtain
a′′+ka=4 πG
3(ε− 3 p)a^3. (1.71)