1. Introduction
Magnetic fields are observed over a wide range of scales from within galaxy clusters to intergalactic voids [
1,
2,
3]. From a theoretical perspective, there are two approaches to understanding the origin of such magnetic fields: (1) the astrophysical origin of the fields which are amplified by some dynamo mechanism [
4,
5,
6] and (2) the primordial origin of the magnetic fields from the inflationary scenario [
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36] or from the alternative bouncing scenario [
37,
38,
39].
Among all the proposals discussed so far, inflationary magnetogenesis has particularly earned significant attention due to its simplicity and elegance. Inflation is one of the cosmological scenarios that successfully describes the early stage of the universe; in particular, it resolves the flatness and horizon problems, and more importantly, inflation can predict an almost scale invariant curvature power spectrum to be well consistent with the recent Planck data [
40,
41,
42,
43,
44]. Thus, it would be nice if the same inflationary paradigm could also describe the origin of the observed magnetic fields, which is the essence of inflationary magnetogenesis. However, in the standard Maxwell’s theory, the electromagnetic (EM) field does not fluctuate over the vacuum state due to the conformal invariance of the EM action, and thus a sufficient amount of magnetic field cannot be generated at the present epoch of the universe. The way to boost the magnetic energy from the vacuum state is to break the conformal invariance of the EM action, and this can suitably be done by introducing a non-minimal coupling of the EM field with the background inflaton field or with the background spacetime curvature [
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36]. Moreover, depending on the nature of the electromagnetic coupling function, the parity symmetry of the EM field may or may not be violated, and thus the EM field can either be helical or non-helical, respectively. However, this simple way of inflationary magnetogenesis may be riddled with some problems, such as the backreaction issue and the strong coupling problem. The backreaction issue arises when the EM field energy density dominates (or becomes comparable) over the background energy density, which in turn spoils the background inflationary expansion of the universe. On other hand, the strong coupling problem is related when the effective electric charge becomes strong during inflation. Therefore, the backreaction and the strong coupling problems need to be resolved in a successful inflationary magnetogenesis scenario (see [
12,
24,
36,
45]). Additionally, during the inflation, the occurrence of a prolonged reheating phase after the inflation has been proven to play a significant role in the magnetic field’s power spectrum (for studies of various reheating mechanisms, see [
46,
47,
48,
49,
50,
51,
52,
53,
54,
55]). Such effects of the reheating phase having a non-zero e-fold number in the realm of inflationary magnetogenesis have been addressed in the context of curvature coupling as well as the scalar coupling magnetogenesis scenario [
7,
15,
16,
17,
18]. In fact, the existence of a strong electric field at the end of inflation induces the magnetic field during the reheating phase from Faraday’s law of induction, which in turn enhances the magnetic strength at the current epoch.
Recently, we proposed a curvature coupling helical magnetogenesis model where the conformal and parity symmetries of the electromagnetic field are broken through its non-minimal coupling to the background
gravity via the dual field tensor, so that the generated magnetic field is helical in nature [
7]. This is well motivated from the rich cosmological consequences of
gravity (see [
56,
57,
58,
59,
60,
61,
62,
63,
64] for various perspectives of
cosmology). After the end of inflation, the universe enters a reheating phase, and depending on the reheating mechanism, we have considered two different reheating scenarios in [
7], namely (a) instantaneous reheating where the universe instantaneously converts to the radiation era immediately after the inflation and (b) the Kamionkowski reheating scenario characterized by a non-zero reheating e-fold number and a constant equation of state parameter. The proposed magnetogenesis scenario shows the following features: (1) for both the reheating cases, the model predicts sufficient magnetic strength over the large scale modes at the present universe for a suitable range of the model parameter; (2) the model is free from the backreaction and the strong coupling problems; and (3) due to the helical nature, the magnetic field of strength
over the galactic scales predicts the correct baryon asymmetry of the universe that is consistent with the observation. However in the realm of inflationary magnetogenesis, these requirements are not enough to argue for the viability of a magnetogenesis model; in particular, one needs to examine some more important requirements in order to argue for the viability of the model. In this regard, one may recall that the calculations that we use to determine the magnetic field’s evolution and its power spectrum are based on the perturbative quantum field theory: therefore, it is important to examine whether the predictions of such perturbative QFT are consistent with the observational bounds of the model parameter. Such a perturbative requirement in the context of the axion magnetogenesis scenario was studied earlier in [
9,
65]. On other hand, the generated EM field may source the curvature perturbation during inflation at super-Hubble scales. Therefore, by considering that the curvature perturbation observed through the Planck data is mainly contributed from the slow-roll inflaton field, we need to investigate whether the curvature perturbation induced by the EM field does not exceed that induced by the background inflaton field in order to be consistent with the recent Planck observation. The authors of [
66,
67,
68,
69] addressed the induced curvature perturbation from the EM field and determined the necessary constraints in the scalar coupling inflationary magnetogenesis scenario. However, in the context of the curvature coupling magnetogenesis scenario, the investigation of such a perturbative requirement and the induced curvature perturbation from the EM field have not yet been given proper attention.
Motivated by the above arguments, in the present work, we will study the following points in the curvature coupling helical magnetogenesis model proposed in [
7]:
Is the model consistent with the perturbative requirement?
What about the power spectrum for the curvature perturbation sourced by the EM field during inflation? Is it compatible with the Planck observation?
For the perturbative requirement, we will examine whether the condition is satisfied, where and are the canonical and the conformal breaking action of the EM field, respectively. This condition indicates that the loop contribution in the EM two-point correlator is less than the tree propagator of the EM field, as the loop contribution in the EM propagator arises due to the presence of the action . In regard to the second requirement, we will calculate the power spectrum of the curvature perturbation induced by the EM field during inflation and will determine the necessary constraints in order to have a consistent model with the Planck data. The model parameter(s) will be critically scanned so that both the above requirements, along with the large scale observations of the magnetic field, are concomitantly satisfied.
The paper is organized as follows: in
Section 2, we will briefly describe the essential features of the magnetogenesis model that we will use in the present work. In
Section 3,
Section 4 and
Section 5, we will determine the cut-off scale, the perturbative requirement, and the induced curvature perturbation of the model, respectively, and will reveal the necessary constraints. The paper ends with some conclusions. Finally, we would like to clarify the notations and conventions that we will use in the subsequent calculations. We will work with an isotropic and homogeneous Freidmann Robertson Walker (FRW) spacetime, where the metric is
with
being the scale factor of the universe, and
t being the cosmic time. The conformal time and the e-folding number will be denoted by
and
N, respectively. An overdot and an overprime will indicate
and
, respectively. A quantity with a suffix ’f’ will represent the quantity at the end of inflation: for example,
is the total inflationary e-folding number,
represents the mode that crosses the Hubble horizon at the end of inflation, etc. Moreover the cosmic Hubble parameter will be represented by
, and the conformal Hubble parameter will be
.
2. Essential Features of the Magnetogenesis Model
Here, we consider the higher curvature helical magnetogenesis scenario that we proposed in [
7], where the electromagnetic dual field tensor couples with the background Ricci scalar as well as with the Gauss–Bonnet scalar. The action is given by
where
is the gravitational action that serves the inflationary agent during the early universe and is given by
Here,
is a scalar field under consideration, and
R and
are the background Ricci scalar and the background Gauss–Bonnet terms, respectively. At this stage, we do not propose any particular form of
for the background gravitational action. In fact, we will give some suitable forms of
which lead to successful inflation, and thus, any of such forms of
are allowed in the context of the magnetogenesis scenario. In this work, we consider the power law inflationary scenario to evaluate the power spectrum of the electromagnetic fluctuations. For power law inflation, the scale factor is given by
with
. In the conformal time (represented by
), the scale factor reads as [
70]
and
is a constant having mass dimension
, and
denotes the scale of inflation. Moreover, an overprime denotes
, and
is the conformal Hubble parameter defined by
. Using the above expression of
, we obtain
In the subsequent calculations, the e-folding number will be represented by
N, and
indicates the beginning of inflation, i.e., the e-folding number increases as the inflation goes on. For the above scale factor, the cosmic Hubble parameter (defined by
with an overdot representing the derivative with respect to cosmic time
t) is given by
in terms of the e-folding number, where
is a constant that represents the Hubble parameter at the beginning of inflation. Here, we would like to mention that for the scale factor of Equation (
3), the slow roll parameter comes as
and thus
. However, due to
, the slow roll parameter is slightly different than
: for example,
leads to
and
.
Now, we will propose some suitable forms of , which indeed leads to power law inflation:
The action with a non-minimally coupled scalar field, where the
is given by [
71],
results in a viable power law inflation described by
with
. Here,
G is the Newton’s constant,
is the non-minimal coupling of the scalar field, and
is the scalar field potential which has the following form:
where
,
A, and
B are constants, and
n is related to the exponent of the scale factor (
p) as
. The authors of [
71] showed that the inflationary quantities lie within the observational constraints for
.
The
f(
R) model given by [
72],
allows a power law inflationary solution
(with
) when
p and
are related by
. It has been shown in [
72] that the inflationary quantities in the context of such power law inflation satisfy the recent Planck constraints for
.
In the context of k-Gauss–Bonnet inflation, the Gauss–Bonnet term is coupled with the kinetic term of a scalar field under consideration. In particular, the
is given by [
73]:
where
X is the kinetic term of the scalar field. A stable power law inflationary solution of the form
(with
) can be obtained from the above model for
, where
n and
p are related by a suitable fashion given in [
73]. Here, it deserves mentioning that in absence of scalar field potential, the power law inflation in the k-Gauss–Bonnet model leads to the stability of the primordial tensor perturbation [
73].
Based on the above arguments, if we consider the the background action of Equation (
6), then the exponent of the power law inflationary scale factor should lie within
, or if we consider the gravitational action of Equation (
7), then we need to choose
in order to obtain a viable power law inflation. Keeping this in mind, we consider
in the present context, for which one obtains
,
, or
(see Equations (
3) and (
5) for the expressions of
and
, respectively). We will demonstrate that with this value of
, the current magnetogenesis scenario predicts a sufficient magnetic strength for suitable values of other model parameters.
The
and
in Equation (
1) are the canonical kinetic term and the non-minimal coupling of the EM field, respectively. In particular,
and
respectively. Here,
represents the EM field tensor, and
is the corresponding EM field. Moreover,
where
is the four dimensional Levi–Civita tensor defined by
, and the
symbolizes the completely antisymmetric permutation with
. Equation (
10) reveals that the EM field couples with the background Ricci scalar as well as with the Gauss–Bonnet scalar through the non-minimal coupling function
. The form of
is considered to be a power law of
R and
, particularly
with
q being a parameter of the model, and
, where
G is the Newton’s constant. The parameter
q plays an important role with regard to the estimation of the magnetic field at the current universe. The presence of
spoils the conformal invariance, yet preserves the U(1) symmetry, of the EM action. Furthermore, Equation (
10) depicts that the EM field couples with the background spacetime curvature via its dual tensor (
), which further breaks the parity symmetry of the EM field, and consequently, the generated EM field turns out to be helical in nature. With Equations (
3) and (
4), the explicit form of
from Equation (
11) becomes
Varying the action Equation (
1) with respect to
, we obtain
We will work with the Coulomb gauge, i.e.,
and
, due to which the temporal component of Equation (
13) becomes trivial, while the spatial component of the same becomes
where
, and
. It is evident that the presence of the
modifies the EM field equation in comparison to the standard Maxwell’s equation. At this stage, we quantize the EM field, so that one does not need an initial seed magnetic field at the classical level, and we may argue that the EM field generates from the quantum vacuum state. For this purpose, we use
where
is the EM wave vector,
runs along the polarization index with
and
being two polarization vectors, and
is the
k-th mode function for the EM field. In the present context, since the magnetic field is helical in nature, we work with the helicity basis set where the polarization vectors are given by
and
, respectively. Consequently,
follows:
where
and
have the following forms:
Therefore, the photon dispersion relation in the present context is given by
which, due to the presence of the factor ’
’, is different than the axion magnetogenesis-like model where a (pseudo) scalar field is coupled linearly with the Chern–Simons term [
74,
75,
76]. We will show below that the presence of
is crucial, due to which the present curvature-coupled magnetogenesis scenario predicts a sufficient magnetic strength at the current universe.
In the sub-Hubble scale when the relevant modes lie within the Hubble horizon, one can neglect the term containing
in Equation (
16), and thus, both the EM mode functions remain in the Bunch–Davies vacuum state. However, in the super-Hubble scale when the modes move outside the Hubble horizon, the term containing
in Equation (
16) dominates over the
term, and thus,
has the following solution in the super-Hubble scale:
Here,
(
) are integration constants that can be determined from the Bunch–Davies initial condition; the explicit forms of
are shown in the
Appendix A. In the expressions of
and
, the arguments inside the Bessel functions are complex, as opposed to that of
and
where the Bessel functions contain real arguments. This makes
, or equivalently
, i.e., the amplitude of the positive helicity mode during inflation is much larger than that of the negative helicity mode. Consequently,
are given by
With the above expressions of
and
, the electric and magnetic power spectra during inflation are given by [
7]:
and
respectively, where we consider the contribution from the positive helicity mode only, due to
. It is evident that both the
and
tend to zero as
(i.e., near the end of inflation), which indicates that the EM field has negligible backreaction on the background spacetime (for detailed analysis of the backreaction issue in the present magnetogenesis model, see [
7]). Moreover, the helicity power spectrum during the inflation is given by
After the inflation ends, the universe enters a reheating phase, and depending on the reheating mechanisms, we consider two different reheating scenarios: (a) instantaneous reheating, in which case the universe instantaneously converts to the radiation era immediately after the inflation, and hence the e-folding number of the instantaneous reheating is zero; (b) the Kamionkowski reheating proposed in [
46], which has a non-zero e-fold number and is characterized by a reheating equation of state (EoS) parameter (
) and a reheating temperature (
). In the instantaneous reheating case, the magnetic field energy density redshifts by
from the end of inflation to the present epoch. However, in the Kamionkowski reheating case, the scenario becomes different; in particular, the magnetic energy density follows a non-trivial evolution during the reheating phase and then goes by the usual redshift
from the end of reheating to the present epoch of the universe. During the Kamionkowski reheating era, the magnetic power spectrum is controlled by the two factors:
and
, respectively (
H is the Hubble parameter during the reheating era), where the latter factor encodes the information of the prolonged reheating stage. At this stage it deserves mentioning that the effect of
depends on the hierarchy between the electric and the magnetic field at the end of inflation. In particular, if the electric field at the end of inflation becomes much stronger than that of the magnetic field (nearly
, where
is the total inflationary e-fold number), the effect of
becomes dominant over the other one, and then the reheating phase shows an important role in the magnetic field’s evolution.
In the present context of the higher curvature helical magnetogenesis scenario, we showed that (1) the EM field has negligible backreaction on the background spacetime and does not jeopardize the inflationary expansion; (2) the model is free from the strong coupling problem; (3) for both the reheating cases, the model predicts a sufficient magnetic strength at the current epoch of the universe for a suitable range of
q given by
for the instantaneous reheating scenario and
for the Kamionkowski reheating case, respectively [
7]; and (4) due to the helical nature, the magnetic field of strength
over the galactic scales predicts the correct baryon asymmetry of the universe that is consistent with the observation. Here, we would like to mention that the related results of baryogenesis can be obtained when the EM field dual tensor couples to an axion field with cosmological time dependence, which leads to tachyonic instabilities and results in a growth of the magnetic field [
77]. It is evident that the viable range of
q is almost same for both the reheating cases. This is due to the reason that the electric and the magnetic fields do not have enough hierarchy at the end of inflation, which in turn makes the instantaneous and Kamionkowski reheating scenarios almost similar in respect to the EM field’s evolution.
Thus, as a whole, the present magnetogenesis model with is found to be viable with regard to the CMB observations of the current magnetic field as well as free from the backreaction and the strong coupling issues. However, these requirements are not sufficient to claim that a magnetogenesis model is a viable model, particularly we need to investigate some more important requirements in this regard. Here, one needs to recall that the calculations regarding the magnetic field’s evolution and its power spectrum are based on perturbative QFT: therefore, it is important to examine whether the magnetogenesis model under consideration is consistent with the predictions of such perturbative QFT. On other hand, the generation of a primordial EM field may source the curvature perturbation in the super-Hubble scales, and thus, we need to investigate whether the curvature perturbation induced by the EM field does not exceed the curvature perturbation contributed from the background inflaton field in order to be consistent with the Planck data. Thus, in the present higher curvature helical magnetogenesis scenario, our aim is to investigate the following points: (a) whether the underlying theory of the model is consistent with perturbative QFT, and (b) whether the curvature perturbation induced by the EM field does not exceed that coming from the inflaton field. As mentioned earlier, the range leads to the correct magnetic field in the present context; thus, we will examine the above-mentioned requirements in this range of q in order to keep the generation of the EM field intact.
However, before moving to examine the perturbative validity, we first determine the cut-off scale of the present model by using the power counting analysis as demonstrated in [
78,
79,
80] and check whether the relevant energy scales lie below the cut-off scale. This in turn will provide a hint for the perturbative validity of the model.
3. The Cut-Off Scale of the Model
To estimate the cut-off scale, we expand the metric around the background FRW spacetime:
where
is the FRW metric, and
are metric perturbations with mass dimension
. Consequently, the determinant of the metric obtains the following expressions (in the leading order of
) around its background value:
The variations of the Ricci scalar and the Gauss–Bonnet scalar are given by
and
respectively. Therefore, the conformal breaking Lagrangian (see Equation (
10)) is expanded as
where the overbar with a quantity indicates the respective quantity formed by the FRW metric
. The first two terms in the above expression, i.e., ∼
, encode the backreaction of the gauge fields on the background dynamics, while the rest of the above expression forms the interaction part between
and
; in particular,
It may be observed from Equation (
28) that the interaction Lagrangian acquires dimension 5 operators (such as
) and dimension 7 operators (such as
); in particular, we individually express such dimension 5 (represented by
) and dimension 7 (
) interaction operators as follows:
and
respectively. Equations (
29) and (
30) immediately suggest that the dimension 5 and dimension 7 operators come with the following interaction coefficients:
which have mass dimension [−1] and [−3], respectively, as expected. We now estimate the cut-off of the present magnetogenesis model by power counting of the operators present in the expression of the interaction Lagrangian [
78,
79,
80]. In particular, the presence of the dimension 5 interaction operators introduces the cut-off scale (
), which can be estimated by
where we use Equation (
31), and recall,
, and
H is the Hubble parameter during inflation. Similarly, the cut-off introduced by the
is given by
Clearly,
, as
and also
are suppressed by the exponent
. Thereby, we may argue that the cut-off scale of the present model is given by
which is obtained in Equation (
33). Having obtained the cut-off scale, we now investigate whether the relevant energy scale of the proposed model lies below the cut-off. During the inflationary stage, the typical momentum of the relevant excitations is equal to the Hubble parameter. Thus, we determine the ratio
in order to examine the validity of the present theory as an effective field theory, as follows:
As we have mentioned earlier, the present magnetogenesis scenario predicts sufficient magnetic strength at the current universe when the parameter
q lies within
. With this information, we give the plots of
with respect to
q in the range
(see
Figure 1). The blue curve and yellow curve represent the respective
at the beginning of inflation (when
) and at the end of inflation (when
, with
being the inflationary e-folding number), respectively. In the
Figure 1, we take
.
Figure 1 clearly demonstrates that the ratio
during the inflation remains less than unity for the aforementioned range of
q, which also leads to the correct magnetic field over the large scale modes at the present epoch of the universe. The following points can be further argued from
Figure 1: (a)
at the end of inflation obtains a lower value compared to that at the beginning of inflation, and (b) the quantity
seems to decrease as the value of
q increases. The fact that
remains less than unity, i.e., the relevant energy scale of the present model lies well below the cut-off scale, suggests the validity of the proposed theory as an effective field theory. Therefore, the regime of the parameter
q, which makes the model viable in regard to the CMB observations of the current magnetic strength and also makes the relevant energy scale of the model below the cut-off scale, is given by
.
4. Constraint from Perturbative Requirement
In this section, we derive a bound on the parameter space of the conformal breaking coupling function
such that the theory can be treated perturbatively, and the perturbative QFT makes sense. If we expand the metric as
, where
is the background FRW metric and
are the metric perturbations, then the conformal breaking action
of Equation (
10) introduces non-minimal interaction terms between the graviton and photon. Such interaction Lagrangian is obtained in Equation (
28) as
where
and
are obtained in Equations (
25) and (
26), respectively. The above interaction terms contribute to the Feynman–Dyson series of the two-point correlator of the EM field, and from the perturbative requirement, we demand that the first terms in the Feynman–Dyson series be small. In particular, the constraint on the coupling function from the perturbative requirement can be derived by either of the following two conditions:
- 1.
The ratio of the actions for the conformal breaking term to the canonical electromagnetic term should be less than unity [
9], i.e.,
- 2.
The loop contribution in the EM field propagator should be less than that of the tree propagator [
65]. In particular,
where
represents the tree propagator of the EM field, and
indicates the loop correction in the EM two-point correlator.
Here we would like to mention that these two conditions are equivalent, as the loop contribution in the EM propagator arises due to the presence of the action .
To examine the first condition in the present context, we start with the following expression of the canonical EM Lagrangian:
where
and
are the electric and the magnetic energy density, respectively. Consequently, the canonical EM action takes the following form:
with
denoting the average over a spatial volume
V and being considered to be equivalent to the vacuum expectation value over the Bunch–Davies state (defined in Equation (
20) or in Equation (
21)). In particular,
For the purpose of determining the
, we express
in the language of differential forms as
Therefore, the conformal breaking action turns out to be
To arrive at the second equality of the above expression, we use the integration by parts. Considering the comoving observer (having four velocity
or
) for measuring the electric and magnetic fields, we find
(with
being the helicity density) [
9]. Accordingly, the
becomes
where
is given by
Plugging the above expressions back into the left hand side of Equation (
37), we arrive at the following equation:
Now, for the condition
to be satisfied, it is sufficient to require
Let us denote the ratio in the left hand side of Equation (
47) by
. Equation (
12) immediately leads to
as
where
is given in Equation (
17), due to which
can be equivalently expressed as
Moreover, from Equation (
20) and Equations (
21) and (
22), we have the following expressions:
respectively, where
(
) are shown in the
Appendix A. The integration limit in Equation (
50) is taken from
to
, i.e., from the mode that crosses the horizon at the beginning of inflation to the mode which crosses the horizon at the instance
. Now, we identify the beginning of inflation when the horizon is of the same size as the CMB scale mode, i.e., we may write
. Furthermore, we have
, with
being any time during the inflation, and thus
. The quantity
is the e-folding number up to
measured from the beginning of inflation, i.e.,
with
and
. Having obtained the necessary ingredients, we now examine whether the condition
is satisfied during inflation. However, due to the dependence of
(
), the integrations in Equation (
50) may not be obtained in analytic form(s), and thus, we numerically approach to integrate
,
and
(at
) present in Equation (
50). For this purpose, we consider
,
and
, respectively, and perform the numerical integrations of Equation (
50). Consequently, we depict the plot of
with respect to the parameter
q in the range
(see
Figure 2). Recall this range of
q results to the correct magnetic strength at the present epoch of the universe, and thus, we are using such range of
q to examine the perturbative condition in order to keep the generation of the EM field intact. We consider different values of
in
Figure 2; in particular, we consider
and
in the left and right plot of
Figure 2, respectively. Here, we would like to mention that the mode
crosses the horizon near the end of inflation, i.e.,
, while the mode
crosses the horizon near
, i.e., near the beginning of inflation.
Figure 2 clearly demonstrates that the perturbative condition
is satisfied for
, which also leads to the viability of the model with regard to the CMB observations of the current magnetic strength. Therefore, the predictions of perturbative QFT in the model are found to be consistent with the observational bound of the model parameter required to obtain sufficient magnetic strength at the current stage of the universe.
5. Curvature Perturbation Sourced by Electromagnetic Field during Inflation
The produced electromagnetic field during inflation may induce the curvature perturbation [
66,
67,
68,
69], and the power spectrum of the induced curvature perturbations should satisfy the recent Planck constraints. Thereby, in the present magnetogenesis scenario where the electromagnetic field couples with the background curvature terms via the dual field tensor, it is important to examine the viability of the sourced curvature perturbations in respect to the Planck constraints.
The induced curvature perturbation (represented by
) from the electromagnetic field is expressed as [
66]
where
is the background inflaton energy density, and
denotes the EM field energy density. Here, it may be mentioned that the contribution from the electromagnetic anisotropic stress is suppressed compared to the contribution written in the r.h.s of Equation (
51) (see [
69]), and thus, the electromagnetic anisotropic stress in the curvature perturbation is not taken into account in Equation (
51). The lower limit of the integral, i.e.,
, represents the time at which the EM production effectively starts.
The EM energy density can be expressed by
, where
and
are the energy density for electric and magnetic fields, respectively. However, from Equations (
20) and (
21), the ratio of the electric to magnetic power spectrum during inflation comes as
. In particular, we give the plot of
with respect to
q in the range
in which we are interested (see
Figure 3).
The figure clearly depicts that the electric field during inflation is ∼
times stronger than that of the magnetic field strength. This in turn indicates that the main contribution of the EM energy density comes from the electric field, and thus, we may write
. Consequently, the EM field energy density in Fourier space is given by
where the electric field is defined as
with respect to the comoving observer. Thereby, Equation (
19) immediately leads to the electric field as
With the above expression of
, we evaluate the two-point correlator of
in the present context as [
66]:
where
and
have the following forms:
and
respectively. Here,
in Equation (
54) represents the mode that crosses the horizon at the end of inflation. Moreover, to derive
, we use
. Such expression of
holds true as the EM field provides a negligible backreaction on the background spacetime in the present magnetogenesis scenario. We may perform the
integral of Equation (
54) to obtain
where we use the integral
. For the momentum variable
in the above integral, the corresponding lower limit of the
integral is taken as
i.e., when the mode
crosses the horizon. This is due to the reason that the EM fluctuations of momentum
start to effectively produce from the horizon crossing of
. In particular, the energy density stored in a certain mode of the gauge field is maximal (compared to the background energy density) at the horizon crossing of the corresponding mode and then redshifts almost like radiation. Therefore, a certain EM mode is mainly produced near the horizon crossing of that mode in the present magnetogenesis scenario. The consideration of
indeed takes care of the horizon crossing region of the mode variable
. With
and
, we evaluate the
integral of Equation (
57) and obtain
We will eventually evaluate the two point correlator at the CMB scale, and thus,
. The above expression of the two-point correlator yields the power spectrum of the curvature perturbation (at
) induced by the EM field as
where the functional forms of
or
are shown in Equation (
56).
Having obtained the theoretical expression of the induced power spectrum at hand, we now confront the model with the Planck results, which put constraint on curvature perturbation as
We consider that the dominant component of the power spectrum of the curvature perturbation is generated by the background slow-roll inflaton field. As a consequence, the theoretical prediction of
does not exceed the aforementioned Planck constraint; in particular,
In order to investigate
in the present context, we need to evaluate the
integral of Equation (
60). However, due to the aforementioned form of
, this integral may not be obtained in a closed form, so we perform the integration by numerical analysis. This is depicted in
Figure 4 where we take the following set of parameters:
,
,
, and
. In particular, we plot the ratio of
with respect to the parameter
q in
Figure 4.
Figure 4 clearly demonstrates that in order to satisfy
, the parameter
q should lie within
. Moreover, we recall that the magnetogenesis model under consideration predicts the correct magnetic strength at the present universe for
. Therefore, it turns out that the whole range of
q which gives the correct magnetic strength does not obey the condition of the induced curvature perturbation, i.e.,
. In particular, the range of
q which leads to a sufficient magnetic strength at the present universe and also ensures
is given by
.
Before concluding, we would like to mention that some recent studies have argued that non-linear enhancement of the magnetic fields at the end of inflation, inverse cascade of helical photons after inflation, and/or a simultaneous coupling to the photon kinetic term
could help increase the strength of the magnetic field [
81,
82]. Such considerations in the present curvature-coupled helical magnetogenesis scenario will be examined in future work.