32 An introduction to the physics of cosmology
model, a particle initially at rest with respect to the origin falls towards the origin,
passes through it, and asymptotically regains its initial comoving radius on the
opposite side of the sky. This behaviour can be understood quantitatively using
only Newtonian dynamics.
Two further cases are worth considering. In an empty universe, the equation
of motion isr ̈=0, so the particle remains atr=r 0 , while the universe expands
linearly witha∝t. In this case,H= 1 /t,sothatδv=−Hr 0 , which declines
as 1/a, as required. Finally, models with vacuum energy are of more interest.
Providedv>m/2,r ̈is initially positive, and the particle does move away
from the origin. This is the criterion forq 0 <0 and an accelerating expansion. In
this case, there is a tendency for the particle to expand away from the origin, and
this is caused by the repulsive effects of vacuum energy. In the limiting case of
pure de Sitter space (m=0,v=1), the particle’s trajectory is
r=r 0 coshH 0 (t−t 0 ),which asymptotically approaches half ther=r 0 expH 0 (t−t 0 )that would have
applied if we had never perturbed the particle in the first place. In the case of
vacuum-dominated models, then, the repulsive effects of vacuum energy cause all
pairs of particles to separate at large times, whatever their initial kinematics; this
behaviour could perhaps legitimately be called ‘expanding space’. Nevertheless,
the effect stems from the clear physical cause of vacuum repulsion, and there
is no new physical influence that arises purely from the fact that the universe
expands. The earlier examples have proved that ‘expanding space’ is in general a
fundamentally flawed way of thinking about an expanding universe.
2.5 Inflationary cosmology
We now turn from classical cosmology to aspects of cosmology in which quantum
processes are important. This is necessary in order to solve the major problems
of the simple big bang:
(1) The expansion problem. Why is the universe expanding att =0? This
appears as an initial condition, but surely a mechanism is required to lauch
the expansion?
(2) The flatness problem. Furthermore, the expansion needs to be launched at
just the correct rate, so that is is very close to the critical density, and can
thus expand from perhaps near the Planck era to the present (a factor of over
1030 ).
(3) The horizon problem. Models in which the universe is radiation dominated
(witha∝t^1 /^2 at early times) have a finite horizon. There is apparently no
causal means for different parts of the universe to agree on the mean density
or rate of expansion.The list of problems with conventional cosmology provides a strong hint that
the equation of state of the universe may have been very different at very early