The Polony; problem
241
where, from (S.93),
-
1/2
IJ. '" m 3/2
(S.I04)
in the present units and f3 is given by (S.S9).
For t~ < t < iD, where the Polonyi field starts to oscillate at t = t~ when
the temperature is T~, and its vacuum energy density decays at iD,
p~(t) _ p~(t~) T~
P~(tf) T~
(S.105)
Prad(t) - Prad(t~) T = Prad(t~) T
is the ratio of the Polonyi field vacuum energy density to the radiation energy
density Prad. Since
Prad (t -) '" T~
(S.]06)
t;
t;
with the approximation (S.I 0 I), we have
p~(t)
m!~2
(S.]07)
Prad(t) '" 2T~T·
tP
Further progress requires a calculation of T~.
Between I = ID, the time at which the inflaton vacuum energy density
decays, when the temperature is TD, and t = I~. the universe is radiation
dominated. Thus, in reduced Planck-scale units,
( T t)2
=
(R)2 R = 3
I
= 3
] Prad
(
Prad
T )4 (S.lOS)
(T)
(TD) TD
where we have used the fact that RT is constant whenever the number of particle
species is constant, for conservation of entropy. This equation has the solution
t = Prad(TD)
(S.]09)
6T4 T2 .
D
At t = t~, Prad(TD) is one or two orders of magnitude larger than T~, using
(2.22) with NB + t N F = ~ for the supersymroetric standard model or ~ for
the standard model respectively. Also, since t~ '" m ~I. we have
1/2
T~ '" m~ •
(S.] 10)
Returning to (S.107), for t~ < t < iD,
p~(t)
m l
_ _ "- t;
J2;p
(8.] 11)
Prad(t)
-T~