218
Inflationary cosmology
the flat region is in the vicinity of a maximum or turning point of the effective
potential. some fine tuning of the initial conditions may be required if tfJ is to start
out in the flat region.
An alternative possibility [19] is that the initial conditions are provided by a
chaotic quantum state which existed for times t ~ Ip. where
-I
tp =mp
(7.154)
is the Planck time. A simple chaotic inflation model can be developed using the
Lagrangian density for the inflaton field tfJ.
c = !a"t/>a"t/> - V(t/»
(7.155)
with
V(t/» = ~At/>4
(7.156)
4
and A « 1. so that the potential is flat. At the Planck time. the uncertainty
principle implies that V(t/» can only be measured with an accuracy ofm~. Thus.
instead of tfJ being fixed at the minimum of V (t/J) at t/J = 0 in all regions of space.
we should expect t/> to take values in the range
_~<
< mp
(7.157)
AI/4 '" t/> '" AI/4
in various regions of space (domains). These are the initial conditions for the
domains. The evolution of tfJ for t > t p will permit a classical description,
provided
V(t/» ~m~
a ""a""" < m 4
(7.158)
"."
." "" P
in all domains. Since V(tfJ) is. in general. non-zero, the various domains will
undergo varying amounts of exponential expansion (inflation).
Consider one such domain with an initial homogeneous field t/>(tp). (As
observed after (7.94), spatial dependence of t/> is in any case damped out rapidly
by the exponential growth of R(t).} For the potential (7.156), the Hubble constant
of (7.41) is
2 )1/2
H = ( 3:7rA t/J 2m p l •
(7.159)
Neglecting the ~ term in (7.33) for slow roll
,p = _ (~)1/2
(7.160)
6:7r
mpt/J
so that
~ (I,)exp [ - (6~)'" m,(1 - Ip) l (7.161)