Introduction to Cosmology

A Cyclic Universe 171

Early attempts to build models with cyclically reoccurring expansion and contrac-
tion were plagued by the problem, that the entropy density would rise from cycle to
cycle. The length of cycles must then increase steadily. But, in retrospect, there must
then have been a first cycle a finite time ago, thus a beginning of time: precisely what
the cyclic model was conceived to avoid.
A cyclic model which solves the entropy problem and which appears as successful
as the ‘consensus’ inflationary model has been proposed by Steinhardt and Turok [6].
The model is described qualitatively in Figure 7.4, which depicts a potential푉(휙),
function of a scalar field휑. Unlike the inflaton field,휙does not cause an inflationary
expansion of space-time. Following the arguments of Steinhardt and Turok the uni-
verse is assumed to be a five-dimensional bulk, where our physical world with all its
observable particles is located on a four-dimensionalbraneseparated from another
brane by a microscopic gap (or rather “Planckoscopic”). Matter on the other brane can
only interact with our world gravitationally, but not through strong or electromagnetic
interactions. Such matter is therefore dark, we cannot detect it in our laboratories.
The potential푉(휙)describes a force between the two branes, and휙is a moduli field
that describes the interbrane separation. When the two branes are far apart this force
is very weak, the branes are stretched to the point where they are flat and parallel, and
we experience this as the present slow cosmic acceleration (phase 1 in Figure 7.4), the
situation where we are now. During this phase the potential energy of the scalar field
dominates over the kinetic energy of the scalar field so that dark energy drives the
expansion.
While휙decreases toward phase 2, the potential slow-rolls down the weakly sloping
positive potential and the branes draw together. At point 3 in Figure 7.4 the potential

V(φ)

Big^62

5

4

Vcrunch

3

1

V 0

Bang φ

Figure 7.4Schematic view of the potential푉(휙)as a function of the field휙for cyclic models
in a five-dimensional universe. The numbered sequence of phases is described in the text.