Signature of the Quantum Gravity on the CMB
31
V (ϕ) = m
4
pl
¯
m
2
ϕ
2
+ c n ϕ
n
,
(31)
where ¯
m = m/m pl and ϕ = φ/m pl are, respectively, normalized mass and the normalized inflation field with the reduced Planck mass. The first term in (31) is the
potential for the quadratic chaotic inflation in the absence of the HDO.
In principle, it is possible to expand the HDO, but for the present work, we
consider only the dominating term as c 6 , and others are set to zero because their
effect is subdominant. As noted in [13], the HDO should be a correction to the
leading term on ¯
m
2 . Therefore, c 6 = α m ¯
m
2 and it implies that the potential (31) can
be recasted as
V (ϕ) = m
4
pl ¯
m
2
ϕ
2
1 + α m ϕ
4
,
(32)
with the following condition:
| α m | ϕ
4
< 1,
(33)
the potential (32) can be expanded.
The first slow-roll parameter corresponding to the quadratic chaotic inflation
potential with HOD effect can be computed using (5) and (32), and therefore by
neglecting higher order terms, we get
V =
2m
2
pl
ϕ 2
1 + 4α m ϕ
4
.
(34)
Similarly the second slow-roll parameter of the potential is obtained using (6) and
(32), and then by neglecting higher order terms, we get
η V =
2m
2
pl
ϕ 2
1 + 14α m ϕ
4
.
(35)
Therefore, tensor-to-scalar ratio of the quadratic chaotic inflation potential with
HDO effect is
r =
32m
2
pl
ϕ 2
1 + 4α m ϕ
4
.
(36)
Inflation terminates when the first slow-roll parameter becomes unity, and therefore using (8) the inflation field can be written in terms of the e-folding number as
ϕ
2
N
N
2π
1 +
N
2
α m
6π 2
.
(37)
31
V (ϕ) = m
4
pl
¯
m
2
ϕ
2
+ c n ϕ
n
,
(31)
where ¯
m = m/m pl and ϕ = φ/m pl are, respectively, normalized mass and the normalized inflation field with the reduced Planck mass. The first term in (31) is the
potential for the quadratic chaotic inflation in the absence of the HDO.
In principle, it is possible to expand the HDO, but for the present work, we
consider only the dominating term as c 6 , and others are set to zero because their
effect is subdominant. As noted in [13], the HDO should be a correction to the
leading term on ¯
m
2 . Therefore, c 6 = α m ¯
m
2 and it implies that the potential (31) can
be recasted as
V (ϕ) = m
4
pl ¯
m
2
ϕ
2
1 + α m ϕ
4
,
(32)
with the following condition:
| α m | ϕ
4
< 1,
(33)
the potential (32) can be expanded.
The first slow-roll parameter corresponding to the quadratic chaotic inflation
potential with HOD effect can be computed using (5) and (32), and therefore by
neglecting higher order terms, we get
V =
2m
2
pl
ϕ 2
1 + 4α m ϕ
4
.
(34)
Similarly the second slow-roll parameter of the potential is obtained using (6) and
(32), and then by neglecting higher order terms, we get
η V =
2m
2
pl
ϕ 2
1 + 14α m ϕ
4
.
(35)
Therefore, tensor-to-scalar ratio of the quadratic chaotic inflation potential with
HDO effect is
r =
32m
2
pl
ϕ 2
1 + 4α m ϕ
4
.
(36)
Inflation terminates when the first slow-roll parameter becomes unity, and therefore using (8) the inflation field can be written in terms of the e-folding number as
ϕ
2
N
N
2π
1 +
N
2
α m
6π 2
.
(37)