12 Kinematics and dynamics of an expanding universe
Fig. 1.3.special case of =1, orE=0, corresponds to parabolic expansion and flat spatial
geometry (flat universe). For both flat and open cases, the universe expands forever
at an ever-decreasing rate (Figure 1.3). In all three cases, extrapolating back to a
“beginning,” we face an “initial singularity,” where the scale factor approaches zero
and the expansion rate and energy density diverge.
The reader should be aware that the connection between
0 and the future
evolution of the universe discussed above is not universal, but depends on the
matter content of the universe. We will see later that it is possible to have a closed
universe that never recollapses.
Problem 1.3Show thata ̇→∞,H→∞andε→∞whena→ 0.
Problem 1.4Show that, for the expanding sphere of dust, (t) is equal to the
absolute value of the ratio of the gravitational potential energy to the kinetic en-
ergy. Since dust is gravitationally self-attractive, it decelerates the expansion rate.
Therefore, in the past, the kinetic energy was much larger than at present. To satisfy
the energy conservation law, the increase in kinetic energy should be accompa-
nied by an increase in the magnitude of the negative potential energy. Show that,
irrespective of its current value, (t)−→1asa−→0.
Problem 1.5Another convenient dimensionless parameter that characterizes the
expansion is the “deceleration parameter”:
q=−a ̈
aH^2