1. Introduction
In the usual (cold) inflationary models, dissipative effects with decay of inflaton energy into radiation energy are neglected. However, during the evolution of warm inflation dissipative effects are important, and inflaton field energy is transformed to radiation energy. This produces heat and viscosity, which make the inflationary phase last longer. Warm inflation models were introduced and developed by Berera and coworkers [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14]. However, even earlier inflation models with dissipation of inflaton energy to radiation and particles had been considered [
15,
16,
17,
18,
19,
20,
21,
22]. Introductions to warm inflation models and references to works prior to 2009 on warm inflation are found in [
8] and [
23]. For later works, see [
9] and [
24] and references in these articles. Further developments are found in the articles [
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43].
In this scenario, there is no need for a reheating at the end of the inflationary era. The universe heats up and becomes radiation dominated during the inflationary era, so there is a smooth transition to a radiation dominated phase (
Figure 1).
In the present work, I will review the foundations of warm inflation and some of the most recent phenomenological models of this type, focusing in particular on the comparison with the experimental measurements of the scalar spectral index and the tensor to scalar ratio r by the Planck observatory.
The article is organized as follows. In
Section 2, the definition and current measurements of these quantities are given. Then, the optical parameters in the warm inflation scenario are considered. We go on and study some phenomenological models in the subsequent sections: monomial-, natural- and viscous inflation. The models are compared in
Section 7, and the results are summarized in the final section.
2. Definition and Measured Values of the Optical Parameters
We shall here briefly review a few of the mathematical quantities that are used to describe the temperature fluctuations in the CMB. The power spectra of scalar and tensor fluctuations are represented by [
44]
Here, k is the wave number of the perturbation which is a measure of the average spatial extension for a perturbation with a given power, and is the value of k at a reference scale usually chosen as the scale at horizon crossing, called the pivot scale. One often writes , where a is the scale factor representing the ratio of the physical distance between reference particles in the universe relative to their present distance. The quantities and are amplitudes at the pivot scale of the scalar- and tensor fluctuations, and and are the spectral indices of the corresponding fluctuations. We shall represent the scalar spectral index by the quantity . The quantities and are called the tilt of the power spectrum of curvature perturbations and tensor modes, respectively, because they represent the deviation of the values that represent a scale invariant spectrum.
The quantities
and
are factors representing the
k-dependence of the spectral indices. They are called the
running of the spectral indices and are defined by
They will, however, not be further considered in this article.
As mentioned above, if
the spectrum of the scalar fluctuations is said to be
scale invariant. An invariant mass-density power spectrum is called a
Harrison-Zel’dovich spectrum. One of the predictions of the inflationary universe models is that the cosmic mass distribution has a spectrum that is
nearly scale invariant, but not exactly. The observations and analysis of the Planck team [
45] have given the result
. Hence, we shall use
as the preferred value of
Different inflationary models will be evaluated against the Planck 2015 value of the tilt of the scalar curvature fluctuations,
The tensor-to-scalar ratio
r is defined by
As noted by [
46], the tensor-to-scalar ratio is a measure of the energy scale of inflation,
. From Equations (2.1) and (2.3), we have
The Planck observational data have given .
3. Optical Parameters in Warm Inflation
During the warm inflation era, both the inflaton field energy with density
and the electromagnetic radiation with energy density
are important for the evolution of the universe. The first Friedmann equation takes the form
We shall here use units so that
where
is the reduced Planck mass. In these models, the continuity equations for the inflaton field and the radiation take the form
respectively, where the dot denotes differentiation with respect to cosmic time, and
is a dissipation coefficient of a process which transforms inflaton energy into radiation. In general,
is temperature dependent. The density and pressure of the inflaton field are given in terms of the kinetic and potential energy of the inflaton field as
During warm inflation, the dark energy predominates over radiation, i.e.,
, and
and
are slowly varying so that the production of radiation is quasi-static,
and
. Note that in the slow roll era the kinetic energy of the inflaton field energy can be neglected compared to its potential energy. Then, the inflaton field obeys the equation of state
. Also, in this era, the second of Equation (3.2) gives
in the case of vanishing dissipation,
, i.e., in the warm inflation model all of the radiation is produced by dissipation of the inflaton energy. Then, the first Friedmann equation and the equation for the evolution of the inflaton field take the form
respectively. Here, a prime denotes differentiation with respect to the inflaton field
.
Defining the so-called dissipative ratio by
the last of Equation (3.4) may be written as
The quantity Q represents the effectiveness at which inflaton energy is transformed to radiation energy. If one says that there is a strong, dissipative regime, and if there is a weak dissipative regime.
During warm inflation, the second of the Equation (3.2) reduces to
In the warm inflation scenario, a thermalized radiation component is present with
, where both
T and
H are expressed in units of energy. Then, the tensor-to-scalar ratio defined in Equation (2.3), is modified with respect to standard cold inflation, so that [
12]
Hence, the tensor-to-scalar ratio is suppressed by the factor compared with the standard cold inflation.
Hall, Moss and Berera [
9] have calculated the spectral index in warm inflation for the strong dissipative regime with
or
. We shall here follow Visinelli [
47] and permit arbitrary values of
. Differentiating the first of the Equation (3.4) and using Equation (3.6) gives
Hence .
We define the potential slow roll parameters
and
by
These expressions are to be evaluated at the beginning of the slow roll era. Using Equations (3.4), (3.6) and (3.9) and the first of Equation (3.10) we get
Differentiation of Equation (3.6) and using that
gives
Dividing by
and using the first of Equation (3.4) in the two last terms leads to
Defining
and using Equation (3.12) we get
in agreement with Equation (3.14) of Visinelli [
47] .
It follows from Equation (3.6) that
From Equation (3.5) and the first of Equation (3.4) we have
Using Equations (3.14), (3.16) and (3.17) can be written as
During slow roll the second of the Equation (3.2) reduces to
Inserting Equations (3.11), (3.15) and (3.18) into Equation (3.20) gives
We now define
, where
is the scalar spectral index. Visinelli [
48] has deduced
where
Since
we have that
Differentiating this we get
Differentiating Equation (3.5) gives
Using Equations (3.11) and (3.18) then leads to
Inserting Equations (3.11), (3.21) and (3.27) into Equation (3.25) gives
Visinelli has rewritten this as follows
Inserting the expressions (3.11), (3.15) and (3.29) into Equation (3.22) gives
The usual cold inflation is found in the limit
and
, i.e.,
. Then,
In the strong regime of warm inflation,
,
we get
In the weak regime,
, Equation (3.16) leads to
It may be noted that in warm inflation the condition for slow roll is that the absolute values of and are much smaller than .
Visinelli has found that the tensor-to-scalar ratio in warm inflation is
In the cold inflation limit, this reduces to
In the strong dissipation regime warm inflation gives in general
Hence, all the warm inflation models predict an extremely small tensor-to-scalar-ratio in the strong dissipation regime with and .
4. Warm Monomial Inflation
Visinelli [
48] has investigated warm inflation with a polynomial potential which we write in the form
since the potential and the inflaton field have dimensions equal to the fourth and first power of energy, respectively. Here,
M represents the energy scale of the potential when the inflaton field has Planck mass. Furthermore he assumes that the dissipative term is also monomial
He considered models with
and
. However, in the present article, we shall also consider polynomial models with
. From Equations (3.3) and (3.4) we have
The constant
represents the strength of the dissipation. For
the dissipative ratio is constant,
. We shall here consider the strong dissipative regime where
. Then, the second of Equation (3.3) reduces to
Inserting Equations (4.1) and (4.2) gives
Integration leads to
where
K is a constant of integration. The initial condition
gives
.
The special cases (i)
, i.e.,
and (ii)
, i.e.,
, both with the initial condition
, i.e.,
, have been considered by Sharif and Saleem (2015). For these cases, the condition
requires
. In the first case, Equation (3.6) reduces to
Note that the time has dimension inverse mass with the present units, so that is dimensionless.
Visinelli, however, has considered polynomial models with
. Then, we have to change the initial condition. The corresponding solution of Equation (4.5) with
and the inflaton field equal to the Planck mass at the Planck time gives
It may be noted that
gives a different time evolution of the inflaton field. Then, Equation (3.5) with the boundary condition
has the solution
In this case, the inflaton field decreases or increases exponentially, depending upon the sign of .
Inserting Equations (4.1) and (4.2) into Equations (3.9) and (3.13), the slow-roll parameters are
With these expressions Equation (3.32) valid in the regime of strong dissipation,
, gives
The slow-roll regime ends when at least one of the parameters (4.10) is not much smaller than
. In the strong dissipative regime
and
. Using Equations (4.3) and (4.10) we then get
The number of e-folds,
N, in the slow roll era for this model has been calculated by Visinelli [
48] . It is defined by
Using Equations (3.3) and (3.5) we get
Inserting the potential (4.1), performing the integration and considering the strong dissipative regime gives
The time dependence of the inflaton field is given by Equation (4.6) when
showing that
in this case, and by Equation (4.8) when
implying
in that case, showing that
in both cases (not dot here)
Inserting this into the first of Equations (4.10) and (4.3) gives
Inserting these expressions into Equation (4.11) gives
Note that with
, i.e., a constant value of the dissipation parameter
, Equation (4.18) reduces to
for all values of
p. Then
gives
which is smaller than the preferred value from the Planck data,
. Inserting
in Equation (4.18) and solving the equation with respect to
p gives,
The Planck values give and .
Panotopoulos and Videla [
24] have investigated the tensor-to-scalar ratio in warm in inflation for inflationary models with an inflaton field given by the potential
where
M is the energy scale of the potential when the inflaton field has Planck mass,
. Let us choose
in the monomial models above. Inserting this in Equation (3.18) gives
. With
we get
.
In this case
for cold inflation. For
this corresponds to
which is an acceptable number of e-folds. Then, the tensor-to-scalar ratio is
, which is much larger than allowed by the Planck observations [
45]. Panotopoulos and Videla found the corresponding
relation in warm inflation with
, where a is a dimensionless parameter. They considered two cases.
(A) The weak dissipative regime. In this case
and Equation (3.7) reduces to
. They then found
With the Planck values and this requires . However, they also found that in this case giving which is too small to be compatible with the standard inflationary scenario.
(B) The strong dissipative regime. Then,
and
. They then found
Then and , so this is a promising model.
5. Warm Natural Inflation
Visinelli [
47] has also investigated warm natural inflation with the potential
where
, and
M is the spontaneous symmetry breaking scale, and
in order for inflation to occur. The constant
is a characteristic energy scale for the model. The potential
has a minimum at
. Inserting the potential (5.1) into the expressions (3.9) we get
From Equation (3.3) with the potential (5.1) we have
Equations (3.4) and (5.3) then give
During the slow roll era we must have
. Using the expressions (5.2) and (5.4) we find that this corresponds to
Inserting Equations (5.2) and (5.4) into Equation (3.31) with
gives in the strong dissipative regime
We shall now express the
in terms of the number of e-folds of expansion during the slow roll era for this inflationary universe model, again following Visinelli. Assuming that the dissipation parameter
is independent of
, i.e., that
, the number of e-folds is given by
Differentiating the potential (5.1) and inserting Equation (5.3) we get
Visinelli has argued that
giving
Inserting this into Equation (5.9) gives
Applying the trigonometric identity
in the expression (5.12) and inserting the result into Equation (5.6) we finally arrive at
Here, we must have
in order to give the Planck value
for
. Hence, Equation (5.12) gives
. A good approximation for
is therefore
Inserting and gives .
Visinelli (2011) further found that the tensor-to-scalar ratio for this inflationary model is
Differentiating the expression (5.3) gives
Combining this with Equation (3.8) in the strong dissipative regime and using Equation (3.4) gives
The energy density of the radiation is
where
a = 7.5657 × 10
−16 J·m
−3·K
−4 = 4.69 × 10
−6 GeV·m
−3·K
−4 is the radiation constant. Combining with Equation (3.6) we get
Equations (5.15), (5.18) and (5.19) give
Visinelli [
47] has evaluated the constant B and concluded that for this type of inflationary universe model the expected value of
r is extremely low. If observations give a value
this model has to be abandoned. On the other hand, the predictions of this model are in accordance with the observations so far.
6. Warm Viscous Inflation
As noted by del Campo, Herrera and Pavón [
29], it has been usual, for the sake of simplicity, to study warm inflation models containing an inflaton field and radiation, only, (comma here) ignoring the existence of particles with mass that will appear due to the decay of the inflaton field. However, these particles modify the fluid pressure in two ways: (i) The relationship between pressure and energy density is no longer
as it is for radiation. A simple generalization is to use the equation of state
, where w is a constant with value
; (ii) Due to interactions between the particles and the radiation there will appear a bulk viscosity so that the effective pressure takes the form
where
is a coefficient of bulk viscosity.
We shall now consider isotropic universe models corresponding to the anisotropic models considered by Sharif and Saleem [
37]. Equation (3.8) can be written
For these models, the time dependence of the scale factor during the inflationary era may be written
where
is the value of the scale factor at
before the slow roll era has started, and
is the Hubble time of the corresponding De Sitter model having
. The Hubble parameter and its rate of change with time is
Note that
for
. Inserting the second expression into Equation (6.2) gives
Sharif and Saleem considered two cases. In the first one
. Equations (3.3) and (3.4) then gives
. Furthermore, for several reasons, they restricted their analysis to the strong dissipative regime where
. Equation (6.5) then reduces to
Integrating with the initial condition
and assuming that
we get
Hence,
is an increasing function of time. Inserting the first of the expressions (6.4) into the first of the Equation (3.3) gives
Combining this with Equation (6.7) leads to
Sharif and Saleem used the Hubble slow roll parameters,
Note that
, where
q is the deceleration parameter. In the present case and in the strong dissipative regime, we can replace
by
. Then
and
. Differentiating the expression (6.9) then gives
The slow roll era ends when the inflaton field has a value
so that
, corresponding to
, which gives
The number of e-folds is given by Equation (4.15), which in the present case takes the form
Inserting the potential (6.9) and integrating gives
Since the left hand side is positive, this requires that or . For this means that .
Sharif and Saleem have calculated the scalar spectral index with the result
Using Equation (6.15) we get
This equation can be written
Inserting the Planck value and , give corresponding to . This value of is not allowed by Equation (6.15).
In the second case, Sharif and Saleem assumed that
. Equations (3.3) and (3.4) then give
. Using Equations (6.2) and (6.4) and integrating with the initial condition
, leads to
In this case
and
becomes
The final value of
is given by
The scalar spectral index is
which can be written
Inserting the Planck value and gives outside the range which requires . However, in the anisotropic case considered by Sharif and Saleem, one may obtain agreement with the Planck data for . As noted above, the tensor to scalar ratio has a very small value in these models. The time evolution of the inflaton field is given by Equation (6.7).
7. Comparison of Models
The models of Sharif and Saleem are a class of the monomial models. Comparing Equations (4.1) and (6.9) we have or . Hence, for we must have while Visinelli considered models with . Furthermore, in the first case of Sharif and Saleem with we have and in the case with we have . Also, it should be noted that Visinelly has deduced the expression for the spectral parameters from the potential slow roll parameters, while Sharif and Saleem have used the Hubble slow roll parameters, and they have got slightly different expressions.
Let us consider an isotropic monomial model with scale as given in Equation (6.3). Then, we have two formulae for the potential—Equations (4.1) and (6.9). Hence
where
is the Planck time. As mentioned above in Sharif and Saleem’s first case
. Combining this with the first Equation (3.3) we get
. Furthermore they considered the strong dissipative regime with
. Hence
. The slow roll era begins at a point of time,
, when the inflaton field is given by Equation (6.23). This leads to
The Hubble parameter is given by the first equation in (6.4) with a maximal value at the beginning of the inflationary era. Hence, the condition
requires that
Inserting the expression (7.2) for
we arrive at
Hence in this model with for example
and
the inflationary era begins much earlier than at around 30 Planck times. Inserting the inequality (7.4) into Equation (7.1) we get
Hence , so these models are large field inflation models.
V. Kamali and M. R. Setare [
49] have considered warm viscous inflation models in the context of brane cosmology using the so-called chaotic potential (3.1) with
, i.e.,
. We have considered the corresponding models in ordinary (not brane) spacetime which corresponds to taking the limit that the brane tension
in their equations. They first considered the case
, i.e.,
. Then, the time evolution of the inflaton field is given by Equation (4.9) with
. As noted above, in this case
which is smaller than the preferred value from the Planck data. It may be noted that Kamali and M. R. Setare got a different result. Letting
in their Equation (68) gives
, i.e., a scale invariant spectrum.
Next, they considered the case
. With
this corresponds to the first case considered by Sharif and Saleem [
37].
8. Conclusions
Warm inflation is a promising model of inflation, taking account of dissipative processes that are neglected in the usual, cold inflationary models. In warm inflation, radiation is produced by dissipation of the inflaton field, and reheating is not necessary. This type of inflationary model was introduced and developed initially by Berera and coworkers. Also, interactions between the inflaton field and the radiation provide a mechanism for producing viscosity.
In this article, I have given a review of some recent models with particular emphasis on their predictions of optical parameters, making it possible to evaluate the models against the observational data obtained by the Planck team. In particular, power law potential inflation, PI, and natural inflation, NI, in the warm inflation scenario have been considered.
I have emphasized that there are some interesting differences between the predictions of these models and the corresponding cold inflation models. The first thing to be noted is that the warm inflation models in general predict a vanishingly small value of the tensor-to-scalar ratio, r. I the present paper I have parametrized the scalar spectral index by . The Planck data favor the value , and a number of e-folds .
Cold PI with the potential (4.1) predicts and . Inserting and gives and . The corresponding warm PI model with constant value of the dissipation parameter predicts, according to Equation (6.24), giving . The corresponding model with predicts giving . However, according to Equation (6.15), this model is only consistent for . Hence, this model is in conflict with the Planck data.
Cold natural inflation predicts
Inserting
and
gives
or
, giving
. Since
this is large field inflation according to the standard definition of this classification (Lyth [
50], Dine and Pack [
51]). The corresponding warm natural inflation model has two parameters,
and
, contained in
in the expression for
. Hence, some assumption concerning the relationship between
and
, is needed to make a prediction of the value of
in this model.