String Theory and M-Theory

536 Flux compactifications
but not necessary. This can be seen by rewriting the condition for inflation
Eq. (10.276) as
ä
a = ˙H + H 2 > 0, (10.291)
where a > 0 needs to be taken into account. This is obviously satisfied
for ˙ H > 0. From the Friedman and acceleration equations this requires in
pφ < ρφ, which is not satisfied for the scalar field described by Eqs (10.280),
(10.281). If ˙ H < 0, then the following inequality has to be satisfied
− ˙ H
< 1. (10.292)
H2 This can be rewritten in terms of ε using the slow-roll approximation
− ˙H
H2 ≈ M 2
P V ′
2 V
2
= ε. (10.293)
By the slow-roll approximation, ε ≪ 1, we observe that this condition leads
to ä > 0 and inflation. The second restriction η ≪ 1 guarantees the friction
term dominates in Eq. (10.283) so that inflation lasts long enough. The
above conditions provide a straightforward method to check if a particular
potential is inflationary. For the simple example of V (φ) = m2φ2 /2, the
slow-roll approximation holds for φ2 > 2M 2 P , and inflation ends once the
scalar field gets so close to the minimum that the slow-roll conditions break
down.
Exit from inflation
From the previous discussion, one concludes that the slow-roll conditions
provide a way to characterize the exit from inflation. The inflationary process
comes to an end when the approximations break down, which happens
for a value of φ for which ε(φ) = 1. A simple calculation shows that, for
power-law inflation, the slow-roll parameters are given by constants
ε = η/2 = 1/p, (10.294)
so that inflation never ends. In principle, this is a problem. One way of
solving it could be provided by embedding this model into string theory,
where additional dynamics might provide an end to the inflationary era.
Hybrid inflation
An inflationary model that has played a role in recent string-cosmology developments,
called hybrid inflation, was constructed in the early 1990s. This
model is based on two scalar fields: the inflaton ψ, whose potential is flat and