256
Superstring cosmology
'Green-Schwarz terms' have been omitted. The Kiihler potential for the dilaton
and moduli fields is
K = In(S + S) - 31n(T + T).
(9.37)
Then. the effective potential V derived from (2.132) is given by
-
-3
(S
V
2
+ S)(T + T)
_awnpl2
=
1
Wop - (S + S)as- - 31Wopl
- aw
I
T)-;;:r 1
2
+ Wnp -
np
3
(T +
(9.38)
The non-perturbative superpotential should be chosen such that
Re S '" 2
(9.39)
at the global minimum of the potential. because of the connection between Re S
and the value gSlring of the gauge coupling constant at the string scale
-2
Re S = 2gSlring'
(9.40)
In particular. if the ath factor in the hidden-sector gauge group is SU(Na) and
there are hidden-sector matter fields in Ma copies of N a + N a fundamental
representations. then
ba = -Na + }Ma
(9.41)
and
d a = (~Ma _ Na) (321r2e)3(M..-N,,)/(3N,,-M,,) (~a ) M,,/(3N,,-M,,) (9.42)
It is not possible to satisfy (9.39) with a single condensate but with two
condensates minimization of the effective potential gives
Re S ~ 0.17
N2Ml - NIM2
(9.43)
3N2 - Ml - 3Nl + Ml
which allows (9.39) to be satisfied for many choices of the integer parameters.
together with yielding a realistic value of the gravitino mass m3/2 given by the
value of e G / 2 at the minimum. as in section 9.5. There is also scope to tune the
parameters to obtain V = 0 at this minimum and so zero cosmological constant.
The dilaton stabilization problem [4] is a result of the peculiar shape of the
potential V illustrated schematically in figure 9.1. If Re S starts larger than 2.
there is only a very small region of Re S which allows it to roll to the desired
minimum at Re S = 2. If Re S starts smaller than 2. the very steep potential
causes it to roll over the very low barrier. failing to be trapped at Re S = 2
unless it starts very close to Re S = 2. Thus. trapping the dilaton in the desired
minimum requires fine tuning of the initial conditions. (Strictly. we should correct