A worked example
215
To avoid de Sitter fluctuations driving tP across the flat region faster than
it would roll semi-ciassically. we need a width AtP for the flat region with the
property (7.91). Combining this with (7.122) leads to
fJ >
8
N 1 / 2 ).,3I2mp.
(7.126)
'" 9./i7r3/ 2 e
The criterion for sufficient inflation (7.57) yields
fJ> 128 (32 )1/2 'A3/2mp
(7.127)
'"
97r
where IV" (tPi) I has been evaluated at tPi ...., Ho instead of tPi ...., 0 to allow for
quantum fluctuations and Ne has been taken to be about 64. The condition (7.126)
is clearly satisfied when (7.127) is.
For density fluctuations of the order required by COBE, (7.112) and (7.113)
imply that
-H2(t*)
.
'" 10-'.
(7.128)
7r 3 / 2 t/Jo(t*)
With the aid of (7.34), this leads to
10' H3(te)
(7.129)
fJ"'" 7r3/ 2tP2(t e )
if t* ~ te. the time at which slow roll ends. With tPe given by (7.122) and
2
87r -2
He ~ 3 mp Vo
(7.130)
using (7.41), and assuming that fJ/'Amp ;S I (which will turn out to be the case),
we then find that
fJ '" 8.5 x loJA 3 / 2 mp.
(7.131)
In practice, this is not a particularly good approximation because ~ is not constant
and ~(t·) < ~(te). A more careful calculation [5] gives a value of fJ several
orders of magnitude larger. Note that even for the smaller value of fJ given by
(7.131), the constraint for sufficient inflation (7.127) is satisfied with two orders
of magnitude in hand and the de Sitter fluctuation constraint (7.126) is satisfied
with four orders of magnitude in hand. Combining (7.131) with the condition
(7.125) fora to be less than mp, we find that
A < 2.5 x 10- 8 •
(7.132)
Then (7. 131) implies that
~ -I < 13
(7.133)
Amp '" .
so that fJ / Am p ;S I. as assumed earlier.