258
Superstring cosmology
Then
H2 = i(V + !2 + p£)
(9.47)
is the generalization of (7.40) and (7.41). in reduced Planck-scale units. Using
(9.47) and (9.44),
.
1 '2
I
H=--tP +-Pt!
(9.48)
2
6H'
Also, from the energy-momentum conservation equation (1.45),
P£ = -3H(pt! + p£)
(9.49)
so that
H
•
1'2
)
= -'1(tP + Pt! + Pf •
(9.50)
This ignores any interaction other than gravitational of tP with matter fields. These
equations can be solved analytically for the asymptotic form of tP in the case of
very steep potentials
).2> 3E.
(9.51 )
The field l/J increases at first but the 'friction' corresponding to the Hubble
constant then freezes l/J at a near-constant value i >o for a time, where
-
../6 In (I+XO)
=
- -
(9.52)
t/Io t/Io + 3(2 - E)
I - Xo
and
o
(9.53)
Xo == ../6Ho
with t/Io, o and Ho denoting initial values. Finally, l/J approaches the asymptotic
form
I (
2).2Vo
)
+
3E
l/J(t) = - In
2
- In R(t)
(9.54)
).
9H o (2 - f)f
).
where R(t) is the scale factor ('radius') of the universe. The approach to
the minimum is then slow and, after oscillations about the minimum with an
exponentially damped amplitude, l/J settles to its minimum.
Let us now apply these considerations to Re S, taken for the moment to have
minimal kinetic terms. In the region Re S < 2 but not too close to Re S = 2
where the minimum has been produced by the balancing of two terms, the nonperturbative superpotential may be approximated by a single term W~ with Sdependence:
W: p ,...., e-A..S
(9.55)
where
24:n- 2
(9.56)
Il. a = Na -}Ma