Reheating after inflation
207
where NB + IN F = ~ for the supersyrrunetric standard model and ~ for the
standard model. If we define a scale M by
M4 == Pv(t = ti),
(7.66)
where li is the time that slow roll begins, then we still have
Pv (I = lose) ~ M4
(7.67)
because Pv ~ V(t/» during slow roll and V(t/» does not change significantly over
the flat region in which slow roll occurs. The initial value Ho~ of the Hubble
constant when the oscillatory period starts is then given by
H 2 _ 81l' -2 M 4
o~ -
3 mp
.
(7.68)
The time lose when the oscillatory period starts is then of order
to~ '" Ho~ = VS;
{3 mpM- 2
(7.69)
(or one or two orders of magnitude greater). Then, from (7.64), we see that
PV(I = ril)
2
---~ = (r ~/ose)
(7.70)
pv(t = lose)
and, using (7.69) and (7.67), we have
3
Pv (I = r",
_I
2
) = -81l' (7.71)
Consequently, the reheating temperature T R of (7.65) is
TR _ (
45
)1/4
(7.72)
41l'3(N8 + ~NF) Note that this is not, in general, of order M.
This discussion depends upon (7.63) for the time development of the (timeaveraged) vacuum energy density. This is known to be correct for the oscillatory
period if the t/> particles decay only into fermions. However, when the t/> particles
decay into pairs of bosons t/> -+ X X, then it is possible for very rapid decay via
parametric resonance to occur [10], a process which generates very large numbers
of X particles. This is referred to as 'preheating'. However, the (radiation) energy
density in light X particles rapidly becomes small compared to the (matter) energy
density remaining in the oscillatory t/> vacuum. Thereafter, the reheating process
occurs as before and the estimate of T R is not much altered.