216
Inflationary cosmology
7.9 Complex inflaton field
In previous sections. we have been assuming that the inflaton field f/J is a real
scalar field. A simple extension is to take r/J to be a complex scalar field [18] with
the Lagrangian density
c = aJLf/JaJLf/J* - V (f/J. f/J*).
(7.134)
For a homogeneous field. the Euler-Lagrange equations are then
..
. av
f/J+3Hf/J+-=O
(7.135)
af/J*
with the Hubble constant given by
H2 = 81r m;2 V (f/J, f/J*).
(7.136)
3
When the energy is dominated by the potential energy,
.
I av
f/JI = - - -
(7.137)
6H af/JI
.
I av
f/J2 = - - -
(7.138)
6H af/J2
where we have separated f/J into its real and imaginary parts
f/J = ~I + if/J2.
(7.139)
For the slow-roll approximation to be valid,
~I «I
(7.140)
I 3H~1
(7.141)
1 -£1 «l.
3Hf/J2
These may be cast as the sufficient conditions (exercise 4)
21 VII VI + V2 VI21 48
(7.142)
mp
VVI
«1r
21 V22 V2 + VI Vl21 48
(7.143)
mp
VV2
«1r
V2 + \1,21
m 2 1
2 « 961r
(7.144)
p 1 V2
where Va == av /a~a(a = 1,2) etc. The last condition ensures that the kinetic
tenn may be neglected compared with the potential tenn in the vacuum energy
density.