Signature of the Quantum Gravity on the CMB
25
¨
φ + 3H ˙
φ + V
= 0,
(1)
where V is the inflaton potential, prime means derivative with respect to the inflaton
field and H =
˙
a
a
(a is the scale factor of the FLRW metric) is the Hubble parameter
governed by the Friedmann equation,
H
2
=
1
3m
2
pl
1
2
˙
φ
2
+ V (φ)
,
(2)
where m pl is the reduced Planck mass and term inside the bracket is the energy
density of the inflaton.
In a wider sense, inflation means a phase of accelerated expansion of the universe.
Actually, the dynamical systems corresponding to (1) and (2) do not necessarily lead
to an accelerated expansion always; however, the inflation can take place when the
potential energy of the inflaton dominates over its kinetic energy, that is, under the
slow-roll regime. Hence, under the slow-roll condition, that is, when
˙
φ
2
2
V , (2)
takes a simple form:
H
2
V
3m
2
pl
,
(3)
and the Klein–Gordon equation (1) becomes
3H ˙
φ + V
0.
(4)
Therefore, using Eq. (4) with slow-roll condition gives the first slow-roll parameter
[18–20]
V =
m
2
pl
2
V
V
2
,
(5)
and when ¨
φ V
with (4) and (2) gives the second slow-roll parameter
η V = m
2
pl
V
V
,
(6)
and so on, with the conditions V , η V 1. The inflation terminates when (φ end ) = 1
but it should last long enough to resolve at least some of the shortcomings of the
hot Big Bang model. The duration of inflation is usually characterized by a number
known as e-folding number N , defined by
N = ln
a end
a in f
,
(7)
where a end and a in f are, respectively, the values of the scale factor at the end of the
inflation and during the inflation. According to the definition, N decreases during the
25
¨
φ + 3H ˙
φ + V
= 0,
(1)
where V is the inflaton potential, prime means derivative with respect to the inflaton
field and H =
˙
a
a
(a is the scale factor of the FLRW metric) is the Hubble parameter
governed by the Friedmann equation,
H
2
=
1
3m
2
pl
1
2
˙
φ
2
+ V (φ)
,
(2)
where m pl is the reduced Planck mass and term inside the bracket is the energy
density of the inflaton.
In a wider sense, inflation means a phase of accelerated expansion of the universe.
Actually, the dynamical systems corresponding to (1) and (2) do not necessarily lead
to an accelerated expansion always; however, the inflation can take place when the
potential energy of the inflaton dominates over its kinetic energy, that is, under the
slow-roll regime. Hence, under the slow-roll condition, that is, when
˙
φ
2
2
V , (2)
takes a simple form:
H
2
V
3m
2
pl
,
(3)
and the Klein–Gordon equation (1) becomes
3H ˙
φ + V
0.
(4)
Therefore, using Eq. (4) with slow-roll condition gives the first slow-roll parameter
[18–20]
V =
m
2
pl
2
V
V
2
,
(5)
and when ¨
φ V
with (4) and (2) gives the second slow-roll parameter
η V = m
2
pl
V
V
,
(6)
and so on, with the conditions V , η V 1. The inflation terminates when (φ end ) = 1
but it should last long enough to resolve at least some of the shortcomings of the
hot Big Bang model. The duration of inflation is usually characterized by a number
known as e-folding number N , defined by
N = ln
a end
a in f
,
(7)
where a end and a in f are, respectively, the values of the scale factor at the end of the
inflation and during the inflation. According to the definition, N decreases during the