32
P. K. Suresh
The initial value of the inflaton at the beginning stage of inflation depends on α m N
2
but (33) implies that
| α m | ϕ
4
N
N
2
| α m |
4π 2
1.
(38)
It is possible to estimate an effective field theoretical bound of the HDO by considering a large e-folding number and is
| α m |
E FT
2 × 10
−2
.
(39)
The first slow-roll parameter of the quadratic inflation with HDO effect can be
obtained in terms of e-folding number by (34) and (37) as
V =
1
2N
1 +
5
6
N
2
α m
π 2
,
(40)
hence the obtained tensor-to-scalar ratio of quadratic inflation with HDO effect is
r =
8
N
1 +
5
6
N
2
α m
π 2
.
(41)
4.1.2 Quartic Chaotic Inflation
By considering the effective theory approach, the potential for the quartic chaotic
inflation can be written as [13]
V (ϕ) = m
4
pl
λϕ
4
+ c n ϕ
n
,
(42)
where ϕ = φ/m pl is the normalized inflation field for the quartic inflation with the
reduced Planck mass. The first term in (42) is the potential corresponding to the
standard quartic chaotic inflation model.
Here also we consider only the dominant term as c 6 in (42), and hence the HDO
correction is taken as the leading term on λ as c 6 = α λ λ, which implies that the
quartic chaotic inflation with HDO effect is
V (ϕ) = m
4
pl λϕ
4
1 + α λ ϕ
2
,
(43)
with the condition
| α λ | ϕ
2
< 1,
(44)
hence the quartic chaotic inflation potential (43) with HDO effect can be expanded.