Pre-big-bang cosmology
269
appears in the spectrum. Then, inflation ends in a version of hybrid inflation. In
some orbifold models a brane-antibrane pair then annihilates to produce a single
brane with one dimension more than the original brane, wrapped around this extra
dimension, which is stable at a smaller radius. The reheating is then controlled by
the difference in tension between the brane-antibrane pair and the single brane.
As to the number of e-folds of inflation, it does not appear possible [13] in the
models described earlier to obtain a sufficient number of e-folds, except for large
values of the Ns;. In that case, the approximation of retaining only the tension
term in the potential breaks down and even including exchange of single bulk
states (e.g. the graviton) in the brane-antibrane interaction may not be sufficient.
However, this problem may be overcome [13] when the compactified dimensions
form an orientifold rather than an orbifold. Then, there still remains the difficulty
that we have had to freeze all but one of the T; moduli or dilaton fields arbitrarily.
An alternative type of brane model (see, for example, [141 and references
therein) is based on intersecting D-branes with chiral matter living on (some of)
the intersections, in the sense that it is associated with open strings that begin and
end at a particular intersection of two D-branes. This type of model also provides
a satisfactory model of in flation [15] up to a point but with the same difficulty of
having to freeze all but one of the moduli discussed earlier.
9.7 Pre-big-bang cosmology
The presence of the dilaton in the heterotic string theory action allows for a
possible alternative origin for inflation in a period of evolution of the universe
before the big bang [16]. The basic idea is that it may be possible to join together
two solutions of the cosmological field equations, one for t < 0 and one for
t > 0, with the following properties. The t < 0 solution is chosen to have the
dilaton tP growing to produce a growing Hubble parameter H, so that the universe
expands rapidly (dilaton-driven inflation). This is an even more rapid expansion
than the more familiar inflation driven by an approximately constant H. The
I > 0 solution, conversely, has 14>1 decreasing and 4> rolling into a minimum of its
potential. Thereafter, there is FRW cosmology (with a constant dilaton), possibly
higher-dimensional, with compactification to three spatial dimensions to follow.
As t = 0 is approached from t < 0, we shall see that it is possible for the
value of 4> to be positive and diverge logarithmically, which pushes the universe
into a strongly-coupled regime because e U controls the strength of the gauge
and gravitational interactions in heterotic string theory. At some point, weaklycoupled string theory breaks down and non-perturbative effects may allow the
transition between the t < 0 and t > 0 weakly-coupled solutions. We shall now
fill in a little of the detail of this idea.
The effective action for the gravitational field and dilaton is as in (9.63).
We shall leave the number of spatial dimensions N arbitrary, allowing for the
possibility that the cosmology starts higher-dimensional with compactification to
269
appears in the spectrum. Then, inflation ends in a version of hybrid inflation. In
some orbifold models a brane-antibrane pair then annihilates to produce a single
brane with one dimension more than the original brane, wrapped around this extra
dimension, which is stable at a smaller radius. The reheating is then controlled by
the difference in tension between the brane-antibrane pair and the single brane.
As to the number of e-folds of inflation, it does not appear possible [13] in the
models described earlier to obtain a sufficient number of e-folds, except for large
values of the Ns;. In that case, the approximation of retaining only the tension
term in the potential breaks down and even including exchange of single bulk
states (e.g. the graviton) in the brane-antibrane interaction may not be sufficient.
However, this problem may be overcome [13] when the compactified dimensions
form an orientifold rather than an orbifold. Then, there still remains the difficulty
that we have had to freeze all but one of the T; moduli or dilaton fields arbitrarily.
An alternative type of brane model (see, for example, [141 and references
therein) is based on intersecting D-branes with chiral matter living on (some of)
the intersections, in the sense that it is associated with open strings that begin and
end at a particular intersection of two D-branes. This type of model also provides
a satisfactory model of in flation [15] up to a point but with the same difficulty of
having to freeze all but one of the moduli discussed earlier.
9.7 Pre-big-bang cosmology
The presence of the dilaton in the heterotic string theory action allows for a
possible alternative origin for inflation in a period of evolution of the universe
before the big bang [16]. The basic idea is that it may be possible to join together
two solutions of the cosmological field equations, one for t < 0 and one for
t > 0, with the following properties. The t < 0 solution is chosen to have the
dilaton tP growing to produce a growing Hubble parameter H, so that the universe
expands rapidly (dilaton-driven inflation). This is an even more rapid expansion
than the more familiar inflation driven by an approximately constant H. The
I > 0 solution, conversely, has 14>1 decreasing and 4> rolling into a minimum of its
potential. Thereafter, there is FRW cosmology (with a constant dilaton), possibly
higher-dimensional, with compactification to three spatial dimensions to follow.
As t = 0 is approached from t < 0, we shall see that it is possible for the
value of 4> to be positive and diverge logarithmically, which pushes the universe
into a strongly-coupled regime because e U controls the strength of the gauge
and gravitational interactions in heterotic string theory. At some point, weaklycoupled string theory breaks down and non-perturbative effects may allow the
transition between the t < 0 and t > 0 weakly-coupled solutions. We shall now
fill in a little of the detail of this idea.
The effective action for the gravitational field and dilaton is as in (9.63).
We shall leave the number of spatial dimensions N arbitrary, allowing for the
possibility that the cosmology starts higher-dimensional with compactification to