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Inflationary cosmology
7.8 A worked example
The discussion of previous sections may be illustrated by the following example
of a suitable effective potential for inflation due to Steinhardt and Turner [5].
Consider the potential
v = Vo - fJ,;3 + }.,;4
(7.118)
where Vo, ,; and }. are constants with fJ, }. > O. It is assumed that the inflaton field
,; starts rolling from,; = 0, possibly because ,; was zero in a high-temperature
phase where some symmetry was restored. In that case, slow roll began at
some lower temperature after the high-temperature phase ceased to be the stable
vacuum. It is also assumed that V is zero at the absolute minimum to which ,;
rolls, corresponding to zero cosmological constant. At the absolute minimum,
3fJ _
,; = 4}" = a
(7.119)
and V(a) = 0 requires that
27fJ4
(7.120)
Vo = 256}.,3
When,; ~ 4Jo = 0, (7.41) then implies that
2
2
97rfJ4_ 2
H = Ho = 32}.3mp •
(7.121)
The slow-roll condition (7.45) implies that slow roll occurs for,; in the range
o ~ ,; < 277rfJ4 -2
(7.122)
'" 64}.,3 m p ==';e
where it has been assumed that,; « a in the slow-roll region. The second slowroll condition (7.46) is automatically satisfied whenever
V"(,;) ...... V'(,;)
(7.123)
,;
and I'; I is at least one order of magnitude less than m p. This is true here because
V"(,;) ~ -2V'(,;)
(7.124)
,;
when'; « a ~ mp. It is being assumed that a is less than mp so that we do
not have to consider the effects of quantum gravity. Following (7.119), this is
obviously arranged for
fJ $ ~}.,mp.
(7.125)