242
Inflation in supergravity
If the Polonyi field were not to decay, then at T = 2.7 K :::: 1O- 3I Mp, we
would have
p~(t) _ loJlmy2~2.
(S.112)
Prad(t)
;
If we want to avoid p~(t)/Prad(t) > I, so that the (p energy density does not
dominate the energy density of the universe and produce too large an expansion
rate, we need
(p .$ (10- 15 _ 1O-16)m:1/4.
(Sol 13)
With m~ given by (S.103) and (S.93) and for
' "
m3/2 = 10- 15 _10- 16
(S.1l4)
in reduced Planck-scale units, we then require
(p .$ 10- 13 _10- 14 .
(S.lI5)
Comparing this with (8.100), this is an unnaturally small value by many orders
of magnitude. Making (8.106) more precise only increases this by an order of
magnitude.
In practice, the Polonyi vacuum energy decays but the previous discussion
suggests that there may be a problem with entropy production when it does decay.
The decay occurs at temperature iD with
t ...... iD - r:-I
(8.116)
' "
with
r:-I - m~
(8.117)
and then, from (8.111),
' "
' "
-
1/2-2
P-(tD)
m- ~
(8.11S)
Prad(iD)
iD'
We now require an estimate of the Polonyi field decay temperature iD.
Remembering that, between t = t~ and t = iD, the vacuum energy density
for (p behaves like a gas of free non-relativistic particles behaving as in (7.64),
p~(t) is growing relative to Prad(t). Consequently, we may expect p~ to dominate
the energy density of the universe by t = iD. We therefore approximate the
time dependence of the temperature by a universe dominated by the Polonyi field
energy density p~. (Recall that the inflaton vacuum energy density has already
decayed and been converted to radiation.) Then we have to solve
( t)2 (R)2 1 ( T)3
T = R = 3P~(t~) T~
(8.119)