#### Warm inflation with an oscillatory inflaton in the non-minimal kinetic coupling model

Eur. Phys. J. C
Warm inflation with an oscillatory inflaton in the non-minimal kinetic coupling model
Parviz Goodarzi 1
H. Mohseni Sadjadi 0
0 Department of Physics, University of Tehran , Tehran , Iran
1 Department of Science, University of Ayatollah Ozma Borujerdi , Boroujerd , Iran
In the cold inflation scenario, the slow roll inflation and reheating via coherent rapid oscillation, are usually considered as two distinct eras. When the slow roll ends, a rapid oscillation phase begins and the inflaton decays to relativistic particles reheating the Universe. In another model dubbed warm inflation, the rapid oscillation phase is suppressed, and we are left with only a slow roll period during which the reheating occurs. Instead, in this paper, we propose a new picture for inflation in which the slow roll era is suppressed and only the rapid oscillation phase exists. Radiation generation during this era is taken into account, so we have warm inflation with an oscillatory inflaton. To provide enough e-folds, we employ the non-minimal derivative coupling model. We study the cosmological perturbations and compute the temperature at the end of warm oscillatory inflation.
1 Introduction
In the standard inflation model, the accelerated expansion and
the reheating epochs are two distinct eras [
1–3
]. But in the
warm inflation, relativistic particles are produced during the
slow roll. Therefore, the warm inflation explains the slow roll
and onset of the radiation dominated era in a unique
framework [
4–7
]. Warm inflation is a good model for large scale
structure formation, in which the density fluctuations arise
from thermal fluctuation [
8,9
]. Various models have been
proposed for warm inflation, e.g. tachyon warm inflation,
warm inflation in loop quantum cosmology, etc. [
10–12
].
Oscillating inflation was first introduced in [
13
], where it
was proposed that the inflation may continue, after the slow
roll, during rapid coherent oscillation in the reheating era.
An expression for the corresponding number of e-folds was
obtained in [
3
].
Scalar field oscillation in inflationary model was also
pointed out briefly in [
14
], where the decay of scalar fields
during their oscillations to inflaton particles was proposed.
A brief investigation of the adiabatic perturbation in the
oscillatory inflation can be found in [
15
]. The formalism
used in [
13
] was extended in [
16
], by considering a
coupling between inflaton and the Ricci scalar curvature. The
shape of the potential, required to end the oscillatory
inflation, was investigated in [
17
]. The rapid oscillatory phase
provides a few e-folds so we cannot ignore the slow roll era
in this formalism. Due to small few number of e-folds, a
detailed study of the evolution of quantum fluctuations has
not been performed. To cure this problem, one can consider
a non-minimal derivative coupling model. The
cosmological aspects of this model have been widely studied in the
literature [
18–39
].
The oscillatory inflation in the presence of a non-minimal
kinetic coupling was studied in [
40
] and it was shown that in
the high-friction regime, the non-minimal coupling increases
the number of e-folds and so can remedy the problem of the
smallness of the number of e-folds arising in [
13
]. Scalar
and tensor perturbations and power spectrum and spectral
index for scalar and tensor modes in oscillatory inflation were
derived in [
40
], in agreement with Planck 2013 data.
However, it is not clear from this scenario how reheating occurs
or the Universe becomes radiation dominated after the end
of inflation. For a non-minimal derivative coupling model,
the reheating process after the slow roll and warm slow roll
inflation are studied in [
41–48
], respectively.
In the present work, inspired by the models mentioned
above, we will consider oscillatory inflation in non-minimal
derivative coupling model.
We will assume that the inflaton decays to the radiation
during the oscillation, providing a new scenario: warm
oscillatory inflation. Equivalently, this can be viewed as an
oscillatory reheating phase which is not preceded by the slow
roll.
In Sect. 2, we examine conditions for warm oscillatory
inflation and study the evolution of energy density of the
scalar field and radiation. In Sect. 3, the thermal
fluctuation is considered and spectral index and power spectrum
are computed. We will consider observational constraints on
oscillatory warm inflation parameters by using Planck 2015
data [
49–53
]. In Sect. 4, the temperature at the end of warm
inflation is calculated. We will compute tensor perturbation
in Sect. 5 and in Sect. 6, we conclude our results.
We use units h¯ = c = 1 throughout this paper.
2 Oscillatory warm inflation
In this section, based on our previous work [
40–42
], we will
introduce the rapid oscillatory inflaton decaying to radiation
in a non-minimal kinetic coupling model. We start with the
action [
54
]
S =
M2
2P R − 21 μν ∂μϕ∂ν ϕ − V (ϕ) √−gd4x + Sint + Sr, (1)
where μν = gμν + M12 Gμν , Gμν = Rμν − 21 Rgμν is the
Einstein tensor, M is a coupling constant with mass
dimension, MP = 2.4 × 1018 GeV is the reduced Planck mass,
Sr is the radiation action and Sint describes the interaction
of the scalar field with radiation. There are no terms
containing more than two times the derivative, so we have no
additional degrees of freedom in this theory. We can
calculate the energy-momentum tensor by variation of action with
respect to the metric,
1
Tμν = Tμ(ϕν) + M 2 μν + Tμ(rν).
The energy-momentum tensor for radiation is
Tμ(rν) = (ρr + Pr)uμuν + Pr gμν ,
where uμ is the four-velocity of the radiation and Tμ(ϕν) is
the minimal coupling counterpart of the energy-momentum
tensor,
1
Tμ(ϕν) = ∇μϕ∇ν ϕ − 2 gμν (∇ϕ)2 − gμν V (ϕ).
The energy-momentum tensor corresponding to the
nonminimal coupling term is
1 1
μν = − 2 Gμν (∇ϕ)2 − 2 R∇μϕ∇ν ϕ + Rμα∇αϕ∇ν ϕ
+Rνα∇αϕ∇μϕ + Rμανβ ∇αϕ∇β ϕ + ∇μ∇αϕ∇ν ∇αϕ
−∇μ∇ν ϕ ϕ − 21 gμν ∇α∇β ϕ∇α∇β ϕ + 21 gμν ( ϕ)2
−gμν ∇αϕ∇β ϕ Rαβ .
Energy transfer between the scalar field and radiation is
assumed to be
Qμ = − uν ∂μϕ∂ν ϕ,
where
∇μTμ(rν) = Qν and
∇
μ
(2)
(3)
(4)
(5)
(6)
(8)
1
Tμ(ϕν) + M 2 μν
= −Qν . (7)
The scalar field equation of motion, in the Friedmann–
Lemaître–Robertson–Walker (FLRW) metric, is
1 +
3H 2
where H = aa˙ is the Hubble parameter, a dot is
differentiation with respect to cosmic time t , and a prime denotes
differentiation with respect to the scalar field ϕ.
is a positive constant, first introduced in [
55
] as a
phenomenological term which describes the decay of ϕ to the
radiation during reheating era. This term was vastly used in
the subsequent literature studying the inflaton decay in the
reheating era (see [
56,57
] and the references therein), where
like our model the inflaton experiences a rapid oscillation
phase. In [
58,59
], it is shown that the production of
particles during high-frequency regime in reheating era can be
expressed by adding a polarization term to the inflaton mass.
To do so, a Lagrangian comprising the inflaton field and its
interactions with bosonic and fermionic fields was employed.
It was shown that the phenomenological term proposed in
[
56
] can be derived in this context. The precise form of the
dissipative term depends on the coupling between the inflaton
and the relativistic particles it decays to, and also on the
interactions of relativistic particles. As the nature of the inflaton
and these relativistic particles are not yet completely known,
the precise form of is not clear.
However, one may employ a phenomenological effective
field theory, or also thermal field theory [
60
], to study the
effective dependency of on temperature and dynamical
fields. Thermal effect can also be inserted by including the
thermal correction in the equations of motion [
60
]. In the
period where the inflaton is dominant over relativistic thermal
particles, it is safe to approximately take as = T =0
(like [
55
]), as explained in [
61
].
Similarly, in the framework of slow roll warm inflation,
the possibility that is a function of ϕ and temperature was
discussed in the literature [
7,62
].
The Friedmann equations are given by
H 2 = 3 M1 2 ρϕ + ρr
P
1
H˙ = − 2M 2 ρϕ + ρr + Pϕ + Pr .
P
The energy density and the pressure of the inflaton can be
expressed as
9H 2
1 +
and
Pϕ =
1 −
3H 2 2H˙
M 2 − M 2
ϕ˙22 − V (ϕ) − 2HMϕ2˙ϕ¨ ,
respectively. Energy density of radiation is ρr = 43 T S [
7
]. S is
the entropy density and T is the temperature. The equation of
state parameter for radiation is 13 , hence the rate of radiation
production is given by
ρ˙r + 4Hρr =
ϕ˙2.
We assume that the potential is even, V (−ϕ) = V (ϕ), and
consider a rapid oscillating solution (around ϕ = 0) to (8),
which in the high-friction regime, H 2 M 2, reduces to
ϕ¨ + 3H
2 H˙
1 + 3 H 2
ϕ˙ +
M 2V,ϕ M 2
3H 2 + 3H 2 = 0.
In our formalism the inflation has a quasi-periodic evolution,
ϕ(t ) = φ (t ) cos
A(t )dt ,
with time dependent amplitude φ (t ). The rapid oscillation
(or high-frequency oscillation) is characterized by
H
˙
H
A,
φ
˙
φ
ρ˙ϕ
ρϕ
A.
The existence of such a solution is verified in [
41
]. It is worth
to note that for a power-law potential V (ϕ) = λϕq , (15) holds
provided that
A,
,
1
q2 MP4 M 2 q+2
λ
1), the adiabatic index
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
which is opposite to the slow roll condition ϕq+2
MP4λM2 [
41
].
The period of oscillation is
τ (t ) = 2
φ dϕ(t )
−φ ϕ˙(t )
,
Therefore the Hubble parameter in the inflaton dominated
era can be estimated as H ≈ 3γ2 t . In the rapid oscillation
and the rapid oscillation occurs for H τ1 and HH˙ τ1 . The
inflaton energy density may estimated as ρϕ = V (φ (t )). In
this epoch ρϕ and H change insignificantly during a period
of oscillation in the sense indicated in (15).
In the rapid oscillatory phase, the time average of the
adiabatic index, defined by γ = ρϕρ+ϕPϕ , is given by γ = ρϕρ+ϕPϕ ,
where the bracket denotes time averaging over one
oscillation,
O(t ) =
tt+τ O(t )dt
τ
2
∝ t 3γ .
By this assumption the radiation may be still in equilibrium,
and besides we can neglect the third term in (21). But as H
decreases, and the radiation production term becomes more
relevant, this approximation fails and the third terms in (21)
get the same order of magnitude as the second term at a time
trh. Note that at trh the radiation and inflaton densities have the
same order of magnitude, ρϕ (trh) ∼ ρr(trh) [
41,57
]. When
t < trh, the average of the energy density of the scalar field
can be approximated as
By using Eq. (23) and the Friedmann equation (H 2 ≈
3 M1P2 ρϕ ), in the ϕ dominated era, we can easily obtain
phase and with the power-law potential (19) we can write the
amplitude of the oscillation as
We can calculate the temperature at the end of warm inflation
by [
4–6
]
(25)
π 2
ρr(tRD) = gRD 30 TR4D,
Our formalism is similar to methods used in the papers
studying the reheating era after inflation in the minimal case
[
57
]. But in the minimal case, for << 3H until ∼ H ,
where the Universe is dominated by the oscillating inflaton,
2
instead of (24), we have a(t ) ∝ t 3 .
In the high-friction limit, time averaging over one
oscillation gives
2M 2
ϕ˙2(t ) = 9H 2(t ) ρϕ (t ) − V (ϕ(t )) ,
where we have used the fact that the Hubble parameter
changes insignificantly during one period of oscillation. But
ρϕ (t ) − V (ϕ(t )) =
therefore
ϕ˙2 ≈ γ MP2 M 2.
−φφ(t()t) ρϕ − V (ϕ)dϕ
φ(t) dϕ
−φ(t) √ρϕ−V (ϕ)
1 √1 − xq dx
= λφq (t ) 0 01 √1d−xxq
= λφq (t )
q
q + 2
,
This relation shows that, for non-minimal derivative coupling
model and in the rapid oscillation phase, when the Universe
is ϕ dominated, ϕ˙2 is approximately a constant. By inserting
(28) into Eq. (12) we obtain
ρr =
(8 + 3γ )
3 γ 2 M 2 M 2 ⎡
P t ⎣ 1 −
t0
t
where t0 is the time at which ρr = 0. The number of e-folds
from a specific time t∗ ∈ (t0, tRD) in inflation until radiation
dominated epoch is given by
tRD
t∗
NI =
H dt ≈
t∗
tRD 2
2
3γ t dt ≈ 3γ ln
tRD
t∗
,
(30)
where tRD is the time at which the Universe becomes
radiation dominated and inflation ceases. At this time
ρr(TRD) ≈ ρϕ (tRD).
(32)
(33)
(34)
(35)
(36)
where gRD is number of degree of freedom of relativistic
particles and TRD is the temperature of radiation at the beginning
of radiation dominated era.
3 Cosmological perturbations
In this section, we study the evolution of thermal
fluctuation during oscillatory warm inflation. We use the
framework used in [
8
] and ignore the possible viscosity terms and
shear viscous stress [
65
]. To investigate cosmological
perturbations, we split the metric into two components: the
background and the perturbations. The background is described
by homogeneous and isotropic FLRW metric with oscillatory
scalar field and the perturbed sector of the metric determines
anisotropy. We assume that the radiation is in thermal
equilibrium during warm inflation. The thermal fluctuations arising
in warm inflation evolve gradually via cosmological
perturbations equations. Until the freeze out time, the thermal noise
has not a significant effect on perturbations development [
8
].
We consider the evolution equation of the first order
cosmological perturbations for a system containing inflaton and
radiation. In the longitudinal gauge the metric can be written
as [
63
].
ds2 = −(1 + 2 )dt 2 + a2(1 − 2 )δi j dxi dx j .
As mentioned before, the energy-momentum tensor splits
into radiation part Trμν and inflaton part Tϕμν as
T μν = Trμν + Tϕμν .
The unperturbed parts of four-velocity components of the
radiation fluid satisfy uri = 0 and ur0 = −1. By using
normalization condition gμν uμuν = −1, the perturbed part
of the time component of the four-velocity becomes
δu0 = δu0 =
The space components, δui , are independent dynamical
variables and δui = ∂i δu [
63
]. Energy transfer is described by
[
64
]
(26)
(27)
(28)
(29)
(31)
Qμ = − uν ∂μϕ∂ν ϕ.
(37)
(38)
(39)
(40)
(41)
ϕ˙δ˙ϕ −
+ V (´ϕ)δϕ
2H ϕ˙ ∇2(δϕ)
M 2 a2
18H 2
ϕ2
˙
+ δρr ,
(44)
Similarly, for the i th component we derive
4ρrδu˙i +4ρ˙rδui +20Hρrδui =−[3 ϕ˙∂i δϕ+∂i δρr+4ρr∂i ].
(42)
The equation of motion for δϕ, computed by variation of
(38), is δ(∇μTϕμν ) = −δ Qν . The zero component of this
equation is
−
+
3H 2
1 + M2
3H 2
δ¨ϕ + (1 + M2 + 2MH˙2 )3H +
3H 2 2H˙
+ϕ˙δ − 1 + M2 + M2
∇2δϕ
a2
= − 2V (ϕ) + 3 ϕ˙ − 6MH2ϕ˙ (3H 2 + 2H˙ )
6H 2ϕ¨
M2
9H 2
+ 1 + M2
ϕ˙ ˙
2H ϕ˙ ∇2 9H 2 2H˙
+ M2 a2 + 3 1 + M2 + M2
4
δρ˙r+4H δρr+ 3 ρr∇2δu−4 ˙ ρr = −
ϕ˙2+δ ϕ˙2+2 δ˙ϕϕ˙.
By using −H ∂i − ∂i ˙ = 4π G(ρ + P)∂i δu, we can
obtain (from the 0i component of the field equation)
2H ϕ¨
M2
˙ +
6H ϕ˙
M2 ¨ −
2(ϕ¨ + H ϕ˙) ∇2
M2 a2 .
(43)
δu = − ak veikx .
and its i i component is
(3H 2 + 2H˙ )
= 4π G
+ H (3 ˙ + ˙ ) + ∇2(
+ ¨
(3H 2 + 2H˙ ) 2Mϕ˙22 − ϕ˙2 +
8H ϕ˙ϕ¨
M2
+ 3HMϕ2˙2 ˙
Using (41–46) we can calculate the perturbation parameters.
Depending on the physical process, e.g. thermal noise,
expansion, curvature fluctuations, three separate regimes for
the evolution of the scalar field fluctuations may be
considered [
8
]. But one can generalize this approach, by adding
stochastic noise source and viscous terms to cosmological
perturbations equations [
65
].
During inflation the background has two components,
oscillatory scalar field and radiation. The energy density
of the scalar field decreases due to expansion and radiation
generation. Quantities related to the scalar field in the
background have oscillatory behaviors. So we replace the
background quantities with their average values over oscillation.
Also, we consider non-minimal derivative coupling at the
high-friction limit.
By going to the Fourier space, the spatial parts of
perturbational quantities get eikx where k is the wave number. So
∂ j → i k j and ∇2 → −k2. Also we define
We have also
∇μTrμν = Qν ,
and
∇μTϕμν = −Qν .
Equation (36) gives Q0 = ϕ˙2 and the unperturbed Eq. (37)
becomes Q0 = ρ˙r + 3H (ρr + Pr), which is the continuity
equation for the radiation field. In the same way Eq. (38)
becomes −Q0 = ρ˙ϕ + 3H (ρϕ + Pϕ ). Perturbations to the
energy momentum transfer are described by (there is no
perturbation for the dissipation factor , which we have assumed
to be a constant)
δ Q0 = −δ ϕ˙2 +
ϕ˙2 − 2 ϕ˙δ˙ϕ
and
δ Qi = − ϕ˙∂i δϕ.
The variation of Eq. (37) is δ(∇μTrμν ) = δ Qν , so its (0-0)
component is
(45)
ϕ˙δϕ
(46)
(47)
So (41) becomes 4
δρ˙r + 4H δρr + 3 kaρrv − 4ρr ˙ = −
M 2 MP2 ,
and (42) becomes
a
4 ((ρ˙rv) + 4H (ρrv)) = −δρr − 4ρr .
k
(43) reduces to
3H 2
δ¨ϕ +
3H 2 2H˙
M 2 + M 2
= −2V (ϕ)
+ 3
9H 2 2H˙
M 2 + M 2
˙ .
From (44) we have
−3H ˙
3γ
1 − 2
1
= 2M 2 (V (ϕ)δϕ + δρr),
P
and we rewrite (45) as
− 3H 2 (1 − 3γ )
(3H 2 + 2H˙ ) (1 − γ ) + H ˙ (4 − 3γ ) + ¨
1
= 2M 2 (−V (ϕ)δϕ + δ Pr).
P
Note that we have replaced φ˙ 2 and φ˙ by their average
values i.e. < φ˙ 2 >= γ M 2 MP2 and < φ˙ >= 0. We restrict
ourselves to the high-friction regime MH22 1 and the modes
H and the zero-shear gauge [
8
] are
=
During the rapid oscillation, the Hubble parameter is H =
3γ2 t , therefore (55) becomes
∝ t α± , therefore
where C is a numerical constant. Thus the density
perturbation, from Eq. (60), becomes [
66,67
]
α s are the roots of this quadratic equation. We denote the
positive root by α+. From Eqs. (51) and (52), we deduce
In this relation δϕ is the scalar field fluctuation during
the warm inflation, which instead of quantum fluctuation,
are generated by thermal fluctuation [
4–6,60
]. Due to the
thermal fluctuations, ϕ satisfies the Langevin equation with
a stochastic noise source, using which one finds [
43–45
]
kF =
H + 3H 2 1 +
3H 2
M 2
.
In the minimal case MH22 = 0, and (63) gives the well known
result [
60
].
δϕ2 =
√ H + 3H 2T
2π 2
,
which reduces to δϕ2 = √23πH2T [
4–6
] in weak dissipative
regime H , and to δϕ2 = √2πH2T , in the strong
dissipative regime H . For a more detailed discussion as
regards the scalar field fluctuations (64), based on quantum
field theory first principles; see [
14
]. In our case, as we are
restricted to the high-friction regime MH22 1 and also use
the approximation (22) before the radiation dominated era,
we have
δϕ2
3H 2T
= 2M π 2 .
Note that our study is restricted to the region H <
9HM22 H . By using (65), the density perturbation
16π
5MP2+α+
2
t4−2α+
V 2δϕ2
3H 2
M
T
V 2
(63)
(64)
(65)
2
(66)
We can now calculate power spectrum from relation Ps(k0) =
245 δH2(k0) [
66,67
]. k0 is a pivot scale. The spectral index for
scalar perturbation is
4.1 Oscillatory warm inflation
ns − 1 =
d ln δ2H .
d ln k
where kF is the freeze-out scale, containing also terms
corresponding to the non-minimal coupling. To compute kF, we
must determine when the damping rate of Eq. (52) becomes
less than the expansion rate H . At tF (freeze-out time [
8
]),
k
the freeze-out wave number kF = a(tF) is given by [
43–45
]
The derivative is taken at horizon crossing k ≈ a H . The
spectral index may be written as
(69)
(70)
(71)
(72)
H
H + H˙
d ln δH2
dt
.
d ln δ2H
ns − 1 = d ln (a H ) =
From H = 3γ2 t we have
ns − 1 ≈
therefore
ns − 1 ≈
t
,
In this section, by using our previous results, we intend to
calculate the temperature of warm inflation as a function of
observational parameters for the power-law potential (19)
and a constant dissipation coefficient , in the high-friction
limit. For this purpose we follow the steps introduced in [
72
],
and we divide the evolution of the Universe from t∗ (a time
at which a pivot scale exited the Hubble radius) in inflation
era until now into three parts
I from t until the end of oscillatory warm inflation,
denoted by tRD. in this period energy density of the
oscillatory scalar field is dominated.
II from tRD until recombination era, denoted by trec.
III from trec until the present time t0.
Therefore the number of e-folds from horizon crossing until
now becomes
N = ln
= ln
a0
a
a0
arec
+ ln
arec
aRD
+ ln
aRD
a
During the warm oscillatory inflation, the scalar field
oscillates and decays into the ultra-relativistic particles. In this
2
δH ≈
34γ γ2 − 5 + γ + 32γ 7 − 152γ α+ + 1 − 21 γ α+(α+ − 1)
From equations (73) and (29) we can calculate energy density
of radiation at tRD
ρr(t = tRD) ≈ MP2 12 2γ 2 M 4 31 .
(8 + 3γ )2
Note that tRD ∼ trh, where trh is defined after (22). The
temperature of the Universe at the end of oscillatory warm
inflation becomes
30M 2
TRD4 ≈ π 2gRD
P
4.2 Radiation dominated and recombination eras
At the end of the warm inflation the magnitude of
radiation energy density equals the energy density of the scalar
field. Thereafter the Universe enters a radiation dominated
era. During this period, the Universe is filled with
ultrarelativistic particles which are in thermal equilibrium. In this
epoch the Universe undergoes an adiabatic expansion where
the entropy per comoving volume is conserved: dS = 0 [
57
].
In this era the entropy density, s = Sa−3, is [
57
]
In the recombination era, grec corresponds to degrees of
freedom of photons, hence grec = 2. Thus
NII = ln
TRD
Trec
1
gRD 3
2
.
By the expansion of the Universe, the temperature
decreases via T (z) = T (z = 0)(1+z), where z is the redshift
parameter. Hence Trec in terms of TCMB is
gT 3.
s =
2π 2
45
So we have
arec TRD
aRD = Trec
1
gRD 3
grec
.
Trec = (1 + zrec)TCMB.
We have also
= (1 + zrec).
a0
arec
4(8 + 3γ )
tRD3 = 9 γ 4 M 2 .
period the energy density of oscillatory scalar field is
dominated and the Universe expansion is accelerated. The
beginning time of radiation dominated era is determined by the
condition ρr(tRD) ρϕ (tRD), which gives [
41,42
]
Therefore
In this relation T is the temperature of the Universe at the
horizon crossing. By Eq. (29) we can calculate temperature
at horizon crossing as a function of t∗
T∗ =
T∗ =
90 γ 2 M 2 M 2 41 t 14 .
P
(8 + 3γ )π 2g∗ ∗
1
π γ gR2D
2√10MP
1
4 3 1
TR2Dt∗4 .
We can remove
M 2 in Eq. (86) by (75)
TRD
TCMB
1
gRD 3
2
.
By using relation Ps(k0) = 245 δH2(k0) and Eq. (67), power
spectrum becomes
(73)
(74)
(75)
(76)
(77)
(78)
(79)
(80)
(81)
(82)
(83)
(84)
2
(85)
(86)
(87)
4.3 Temperature of the warm oscillatory inflation
To obtain temperature of the warm inflation we must
determine N in (72). We take a0 = 1, so the number of
efolds from the horizon crossing until the present time is
= exp(N ), where
1
= a
∗
gRD
gRD
V
= qλ −φφ ϕq−1 d˙ϕ
ϕ
φ dϕ
−φ ϕ˙
1 xq−1dx
= qλφn−1 0 √1−xq
01 √1d−xxq
= 2qλ
2+q
2q
1
q
φq−1.
V (ϕ∗) = 12λ
and H 2 ≈ 9γ42t2 , therefore
In the inflationary regime we have H 2 ≈ 3 M1P2 ρϕ ≈ 3 M1P2 λφq
We have taken gRD ∼ g∗. The time average of the potential
derivative may computed as follows:
The number of e-folds during warm oscillatory inflation
becomes
1
4(8 + 3γ ) 3
9 γ 4 M 2
× ⎝
⎛
We set gRD = 106.75, which is for the ultra-relativistic
degrees of freedom at the electroweak energy scale. Also,
from Planck 2015 data, at the pivot scale k0 = 0.002 Mpc−1
and in one sigma level, we have Ps(k0) = (2.014 ±
0.046) × 10−9 and ns = 0.9645 ± 0.0049 (68% CL, Planck
TT, TE, EE + low P) [
49–53
]. By using γ = 0.55902, M =
10−16 MP, λ = (10−8 MP)4−q , and = 10−4 MP in Eq. (93)
the temperature of the Universe at the end of warm inflation
and the number of e-folds become Tend ≈ 3.83 × 1012 GeV,
and N = 61.42 respectively.
5 Tensorial perturbation
In this section, we follow the method used in [
68
] to study
tensorial perturbation. The power spectrum for tensorial
perturbation is given by [
68
]
From Eq. (90), we derive t∗ as
1
Ps(k0)gR4D
M 25 − 34γ −2α+ λ 32γ −1 M 2 4 β
1 1
P
gRD
3γ k0
By substituting t∗ from Eq. (92) into Eq. (83), the
temperature at the end of warm oscillatory inflation or beginning of
the radiation domination is obtained:
2 31 2
,
where vk can be calculated from the Mukhanov equation
[
69–71
]
d2vk
dη2 +
2 2 1 d2z
c k − z dη2
vk = 0.
η is the conformal time, ct is the sound speed for the tensor
mode and k is the wave number for the mode function vk
[
69–71
] and z is given by
(91)
(95)
(96)
z = a(t )Mp
!eiλj eiλj √
2
1 − α.
(97)
The polarization tensor is normalized as eiλj eiλj = 2δλλ . For
our model, with a quasi-periodic scalar inflaton background,
ϕ˙2
we have α = 2M2 Mp2 and c is given by relation
(106)
(107)
(108)
At(q) =
ck=a H
.
c2 = 11 +− αα .
Therefore
1 d2z
z dη2 =
Well within the horizon, the modes satisfy k
can be approximated by flat waves. Therefore
a H , and they
6 Conclusion
√
2
vk (η) ≈
π ei ν+ 21 π2 (−η) 21 Hν(1)(−ckη).
On the other hand, when we want to compute power
spectrum, we need to have modes that are outside the horizon. So
by taking the limit akH → 0, we obtain the asymptotic form
of mode function as
vk (η) → ei ν+ 21 π2 2 ν− 23
(ν) 1
( 23 ) √2ck
(−ckη) −ν+ 21 .
(98)
where
(99)
(100)
(101)
(102)
(103)
(104)
(105)
At the horizon crossing csk = a H , we can write (104) as
Pt = At2(q)
H
|ck=a H ,
From Eq. (108) by using γ = 0.55902, gRD = 106.75,
M = 10−16 Mp and = 10−4 Mp, the ratio of tensor to scalar
at the pivot scale k∗ = 0.002 Mpc−1 becomes r ≈ 0.081,
which is consistent with Planck 2015 data, r0.002 < 0.10
(95% CL, Planck TT, TE, EE + lowP).
In the standard model of inflation, the inflaton begins a
coherent rapid oscillation after the slow roll. During this stage, the
inflaton decays to radiation and reheats the Universe. In this
paper, we considered a rapid oscillatory inflaton during the
inflation era. This scenario does not work in minimal
coupling model due to the fewness of e-folds during rapid
oscillation. But the non-minimal derivative coupling can remedy
this problem in the high-friction regime. Therefore we
proposed a new model in which inflation and rapid oscillation
are unified without considering the slow roll. The number of
e-folds was calculated. We investigated cosmological
perturbations and the temperature of the Universe was determined
as a function of the spectral index.
We used a phenomenological approach to describing the
interaction term between the inflaton and the radiation, but a
precise study of thermal radiation productions must be
performed based on quantum field theory principles. An attempt
in this regard may be found in [
14
].
To complete our study one may consider quantum and
thermal corrections to the parameters of the system such as
the inflaton mass and its coupling to the radiation. In the
slow roll model, the role of theses corrections on the observed
spectrum is studied in the literature [
73,74
]. Recently, in [75],
it was shown that in warm slow roll inflation, it is possible
to sustain the flatness of the potential against the thermal
By using this relation we can write the power spectrum as
k3 2(2ν−3) ⎛
Pt(k) = 2π2 β2a2 ⎝
(ν) ⎞ 2 1
3 ⎠ 2ck (−ckη)(−2ν+1) coth
2
k
2T
.
In the rapid oscillation epoch = HH˙2 = 32γ (see (24)), so
we can write the conformal time as
1 1
η = − a H 1 − .
and loop quantum corrections. In the non-minimal
derivative coupling model, by power counting analysis, and
unitar1
ity constraint which implies H , where = (H 2 MP) 3
is the cutoff of the theory, it was shown that for power-law
potentials quantum radiation corrections are subleading [
54
].
However, it may be interesting to study in detail the effect of
loop quantum corrections and the corresponding
renormalization on the behavior of our model and on its spectral index
and power spectrum.
Note that our model is an initial study of warm oscillatory
inflation. Further studies may be performed by considering
thermal correction to the effective potential [
76
], and also
by taking into account the temperature dependency of the
dissipative factor and checking all consistency conditions.
We leave these problems for future work.
Open Access This article is distributed under the terms of the Creative
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