222
Inifationary cosmology
The derivatives dE/dlnk and d'7/dlnk are required for the discussion of n(k).
From (7.34), during slow roll,
~dtP.
(7.187)
dl = - V'(tP)
Moreover. the right-hand side of (7.181) is to be evaluated, as discussed after
(7.112), when
k = R(t*(k»H(t*(k».
(7.188)
During slow-roll inflation, the rate of change of H is small compared with the rate
of change of R and so
dlnk = Hdt.
(7.189)
Combining this with (7.187) gives
d
1 ,
d
(7.190)
dlnk = -3H2 V (tP)dtP·
Then, using (7.41) for H2,
d
2 V'(tP) d
(7.191)
dlnk = -Mp V(tP) dtP·
With the aid of this, we may now evaluate the required derivatives of the slow-roll
parameters (7.184) and (7.185), with the result
-
dE
=
2
-2(E'7 - 2E )
(7.192)
dlnk
and
d'7
(7.193)
din k = 2E'7 - ~2
where
~2 == M4 V'(tP)V"(tP)
(7.194)
p
V (fP)2
Returning to (7.181) and (7.182), we may first simplify P(k) using (7.34)
and (7.41) to obtain
P(k) = _1_ V(tP).
(7.195)
611"3E M4 p
Then differentiating with the aid of (7.191) gives [21]
n(k) - 1 = -6E + 2'7.
(7.196)
Being slow-roll parameters. E(tP) and '7(tP) are small. Consequently,
In(k) - 11« 1
(7.197)