172 Cosmic Inflation
has decreased to the point where푉(휙)=^12 휙^2 where the field crosses zero potential,
the total energy density is zero and the Hubble expansion halts. As the branes draw
closer together, the potential energy decreases from positive to negative values and
the universe switches from expansion to a rapid contraction with equation of state
푤≫1, towards a minimum at point 4 in Figure 7.4. The spatial contraction occurs in
the extra dimension휙, rather than in our three dimensions.
As the scalar field rolls down into this minimum (phase 4), it picks up speed on
the way and causes a Hubble blueshift of the scalar field kinetic energy. With this
speed the field just continues beyond the minimum, climbs out of it on the other side
and continues towards the bounce at negative infinity in a finite time. the potential
disappearing exponentially. In the approach towards휙=−∞, the scale factor푎(푡)goes
to zero (phase 5), but the coupling of the scalar field to radiation and matter conspires
in such a way as to keep the temperature and the energy density finite. Space does not
disappear completely since this scenario takes place in a five-dimensional space-time,
and푎(푡)has to be interpreted as the effective scale factor on the brane.
Each cycle ends and begins with a crunch turning into a bang at the field value
휙=−∞. The bang is a turn-around or bounce from the pre-existing contracting phase
with a decreasing field. At the bounce, the branes collide and this is the big bang of
a new cycle. After the branes bounce apart there immediately follows an expanding
phase and an acceleration of the field towards positive values (phase 6 in Figure 7.4),
when matter and radiation are created at large but finite temperature from the kinetic
energy of the field.
This phase is quite similar to the corresponding post-inflationary epoch in the con-
ventional inflationary scenario, and therefore the predictions are the same. But, unlike
the conventional model, here a subdominant dark energy field is required. During
radiation and matter domination the scalar dark energy field is effectively frozen in
place by the Hubble redshift of its kinetic energy and the branes slow down to essen-
tially a halt. The potential energy of this field starts to dominate only when radiation
and matter energy densities have been sufficiently diluted by the expansion; then a
slow cosmic acceleration commences (phase 1).
The heat from the collision dominates the universe for a few billion years, but even-
tually, on the way from phase 6 to phase 1, it is diluted enough that the positive inter-
brane potential energy density dominates. This acts as a source of dark energy that
causes the expansion of the branes to accelerate. Although the acceleration due to
dark energy is very slow, causing the Universe to double in size every 15 billion years
or so, compared with the enormous expansion in Equation (7.45) during 10−^35 s, this
is enough to empty the Universe of its matter and radiation. The matter, radiation,
and large scale structure dilute away over the next trillion years or so, and the branes
become nearly perfect vacua at the end of each cycle (at phase 1), preparing the way
for a new cycle of identical duration.
The homogeneity and flatness of the Universe and the density perturbations
are established during long periods of ultra-slow accelerated expansion, and the
conditions are set up during the negative time prior to a bang. In contrast, standard
inflationary theories have very little time to set up large-scale conditions, only about
10 −^35 s until the inflationary fluctuations have been amplified and frozen in, at a length