The spectral index
223
and n(k) differs little from l. The deviation of n(k) from I at values of k of
interest (k- I between the present Hubble radius of 3000 Mpc and the smallest
scale for large-scale structure observations of I Mpc) is determined by the slowroll parameters e(tP) and T/(tP) for the given model. The variation of n(k) with k
is also easily calculated using (7.192) and (7.193) to be [22]
d Inn(k) = 16E7I- 24E2 _ 2e.
(7.198)
dlnk
The spectral index distinguishes models of inflation. Consider, for example,
an inflationary potential of the form
v (tP) = AtP P (p > 0).
(7.199)
(The chaotic-inflation potential (7.156) is a special case with p = 4.) Then
e(tP) = !M~p2tP-2 and 71(tP) = M~p(p - l)tP- 2
(7.200)
so that
n(k) -I = -M~p(p+2)tP-2
(7.201)
with tP to be evaluated at t = t·(k) for scales k of interest. The key thing is the
number of e-folds of inflation that occurs after cosmologically interesting scales
leave the horizon. From (7.51), what we require is
Ne(tP(t·» = _M;2 f~ V(tP) dtP
(7.202)
J~(t·) V'(tP)
where tPe corresponds to the end of slow roll. In the present model,
V(tP)
_I
(7.203)
V'(tP) = p tP
leading to
tP 2 (t·) - tP~ = 2Ne(tP(t·»pM~.
(7.204)
Slow roll ends when e(tP) ..... I and, from (7.200), we see that this happens when
tP = tPe ..... pMp.
(7.205)
Thus,
tP 2 (t·) :::: p2M~ + 2Ne(tP(t·»pM~.
(7.206)
For modest values of p and large values of Ne(tP(t·»,
tP(t·) :::: J2Ne (tPW»pMp
(7.207)
so that from (7.20 I)
2+p
n(k) - I ::::
(7.208)
2Ne(tPW»
For example [23], if Ne (tP(t·» = 50, we have
2+p
n(k) - 1 :::: - - ­
(7.209)
lOO
and n(k) at observable scales differs from 1 by a few percent.