Thespec~jndex
221
Thus, using (7.58),
81TVo
Ne ~ 2 2
(7.]79)
mpm
if Vo is the dominant tenn. Ne can be adjusted to be greater than 64. For example,
for Vo ~ (10 16 GeV)4, we have Ne ~ 64 for
m :s 1013 GeV.
(7.180)
The parameter,p~ = m~/A2 gives enough freedom to obtain the observed value
for !JP/ p. (In the previous example,pc ~ 3 x 10 17 GeV.) We refer the reader to
the paper by Linde [20) and the book of Liddle and Lyth in the general references
for this chapter, for more detail.
7.12 The spectral index
The dependence of the density perturbation !Jp/ p on the scale k will eventually
allow different inflationary models to be distinguished by observations. Let us
define
P(k) == (!J P )2 = 1T-3 (~2(t*(k»)2
(7.18] )
p
t/Jo(t*(k»
where we have used (7.112). (P(k) is proportional to the 'power spectrum'.) The
spectral index n(k) is defined by
n(k) _ 1 == din P(k)
(7.] 82)
If n(k) is a constant, this reduces to
P(k) ex k n - I
(7.183)
so that n = I corresponds to a scale-independent spectrum.
The spectral index may be evaluated using the slow-roll conditions. Slowroll parameters ~(,p) and 1I(,p) may be defined by
f(,p) == ~M2 (VI(,p»)2
(7.184)
2 p V(,p)
and
1I(,p) == M2 V"(,p)
p
- ­
(7.185)
V(,p)
where
m 2
1
2 _....1..= __ .
(7.186)
Mp = 81T
81TGN