204
Inflationary cosmology
If we assume that the energy density is dominated by the potential energy, then
2
811' -2
H = "3mp V (t/».
(7.41 )
We shall see shortly that this is a consistent approximation when the effective
potential Vet/»~ is flat enough to satisfy the conditions for slow roll. Returning to
(7.36) with if estimated from (7.41), we see that
-"'-.
:i.
=
I
81rm- 2
--V'(t/» + -_P-H-4(V'(t/>))2.
(7.42)
3Ht/>
9H2
54
To satisfy (7.35), we require that
1V'(t/» I «1
(7.43)
9H2
and
81rm 2
54 p (V'(t/>))2 H -4« I.
(7.44)
With H2 given by (7.41), these slow-roll conditions are
m~IV'(t/»1 «2411'
(7.45)
and
2 (V'(t/»)2 «48Jr.
(7.46)
mp Vet/»~
Slow roll occurs in the range of t/> for which Vet/»~ is flat enough to satisfy these
two conditions.
It can now be seen that, when the slow-roll conditions are satisfied, the
vacuum energy density is dominated by the potential energy. Using (7.34), the
kinetic tenn in (7.40) is
~,;,2 = (V:~~!)2
(7.47)
so that. using (7.41) and (7.46),
!,;,2 m~ (V'(t/»)2
Vet/»~ =
(7.48)
9611' vet/»~ « I.
If the slow roll occurs between times ti and t I, and the value of the scalar
field evolves from t/>i to t/> I during this time, then the amount of inflation.
measured as the number of e-folds, is given by
Ne == In - RI =
Ri
1' / - R dt = 1'/ H dt.
(7.49)
' 1 R
' 1