DilaLon and moduli cosmology
251
This means that we need an estimate of F. at the absolute minimum of the
effective potential to estimate these masses.
First, because the dilaton and moduli fields have only gravitational strength
interactions, for any superpositions of the fields G~ ...... I in reduced Planckscale units (Mp = t). Because V = 0 at the absolute minimum (for zero
cosmological constant), the square of the F -field responsible for supersymmetry
breaking, assumed to be a superposition of the dilaton and moduli F -fields, is of
order e G . Moreover, the gravitino mass is given by
2 _
Go
m3/2 ­ e
(9.5)
where Go is the value of G at the absolute minimum. Thus,
2
2
Fj -m3/2
(9.6)
for the supersymmetry-breaking F -field superposition. If there is more than one
superposition of the dilaton and moduli F -fields with a vacuum expectation value
(VEV), we shall assume that (9.6) holds for each superposition or some are
negligible. (With the usual definition of the scale of supersymmetry breaking
msusy, (9.6) is the statement
2
4
m3/2""" mSUSY
(9.7)
in reduced Planck-scale units.)
Returning to the dilaton and moduli masses,
a2v )
m~ (
(9.8)
= at/Jat/J*
provided that the kinetic terms are minimal so that the t/J-field does not need
rescaling. With V given by (9.4) and F. of the order given by (9.6), m~ is a
sum of terms of order eGo and terms of order F. F. both of which are of order
m~/2' Thus. we might expect that m. -m3/2
(9.9)
where t/J denotes a dilaton or modulus field. Similar but somewhat more
complicated arguments can be made for the dilatino and modulinos. Detailed
calculations confirm these expectations [I J.
The cosmology of dilatinos and modulinos. which are light fermions with
masses of order m3/2 with only gravitational strength interactions. resembles that
of gravitinos. The dilatinos S will have a decay rate
r
3M-2
S -m s p
(9.10)
where m S is the dilatino mass and a decay time
t- - r- 1
(9. t t)
s s '