The Polonyi problem
243
with the result that
t - (p-;tr
12
r-'12.
(8.120)
At I = iD, and with iD given by (8.116), we get
T -
2 -1/3
) T
D - m ~p~ (t~. ~
(8.121)
Using (8.111), and with p~(t~) = P~(lf) given by (8.101), we have
i _ 11/6;;-2/3
D
m~
Y'
•
(8.122)
Substituting this into (8.118) gives
P~(iD)
-4/3 -S/3
-.!:.-_--m- t/J •
(8.123)
Prad(tD)
~
Consequently, the (p vacuum energy density dominates the density of the universe
at the moment of decay provided
-
1/2
t/J > m~ .
(8.124)
With m~ given by (8.103), this condition is
(p> 10- 8
(8.125)
in reduced Planck-scale units. With the estimate (8.1 (0) of (p, (8.125) is satisfied
with ease.
Provided that P~ does dominate the energy density of the universe at the
moment of decay, there is a further reheating of the universe (in addition to the
reheating that occurred when the inflaton decayed) to a temperature
iR _m~/2
(8.126)
q,
where r ~ is given by (8.117). For successful nucleosynthesis, we must have
iR> I MeV = 1O- 2I Mp.
(8.127)
This requires
m~ > 10- 14
(8.128)
in reduced Planck-scale units. The value of m~ used here (- 10- 8 ) satisfies this
bound with ease. There is then an entropy increase of
6. = (~R)3 '" m -: I ~2 _ 108 ~2.
(8.129)
TD
~