d^{4}x\,\,\sqrt{-g}\,\,\left[\,(\Box\pi)^{2}\,\nabla_{\alpha}\pi\,\nabla^{\alpha}\pi+\Box\pi\,\nabla_{\nu}\pi\,\nabla^{\nu}\nabla^{\alpha}\pi\,\nabla_{\alpha}\pi\right.$
$\displaystyle~{}~{}-\left.\,\nabla_{\mu}\nabla_{\nu}\pi\,\nabla^{\mu}\nabla^{\nu}\pi\,\nabla_{\alpha}\pi\,\nabla^{\alpha}\pi-\nabla^{\mu}\nabla^{\nu}\pi\,\nabla_{\nu}\nabla_{\alpha}\pi\,\nabla_{\mu}\pi\,\nabla^{\alpha}\pi\,\right]$
(2.18a) $\displaystyle\left.{S}_{4}^{nm}\right|_{\pi=\pi^{*}}$
$\displaystyle=-\frac{\omega}{4\,\lambda_{\pi}^{6}}\int
d^{4}x\,\,\sqrt{-g}\,\,\nabla_{\alpha}\pi\nabla^{\alpha}\pi\nabla_{\mu}\pi\nabla_{\nu}\pi
g^{\mu\nu}R$ $\displaystyle{}-\frac{\omega}{2\lambda_{\pi}^{6}}\int
d^{4}x\,\,\sqrt{-g}\,\,\nabla_{\alpha}\pi\nabla^{\alpha}\pi\nabla_{\mu}\pi\nabla_{\nu}\pi\,\,\left(R^{\mu\nu}-\frac{1}{4}g^{\mu\nu}R\right)$
(2.18b)

Although the above action looks different compared to that of Deffayet et al
[150], it is possible to show that the two actions are related by a boundary
term. In Appendix (A.5), we have explicitly shown that the above action for
real scalar field is identical to the action used in Ref. [150]. Hence, the
equations of motion from the above action matches with the equations of motion
derived in Ref. [150].

####  2.2.2 Coupling to the electromagnetic field

Like in the flat space-time, the action (2.17) is invariant under the global
transformation, i.e. $\pi\to\pi e^{-i\,e\,\theta}$, where $\theta$ is a
constant parameter and $e$ is the electric charge. In order for the action to
be invariant under the local gauge transformation, we can replace
$\nabla_{\mu}\to\mathcal{D}_{\mu}\equiv\nabla_{\mu}+ieA_{\mu}$ in action
(2.17). Beside this, the electromagnetic field will have additional Galileon
terms that vanish in the flat space-time [102]. The complete Galileon scalar
electrodynamics action in curved space time is given by

$\displaystyle{S}^{\rm{G}}_{Curved}$ $\displaystyle=\frac{1}{2}\int
d^{4}x\sqrt{-g}\mathcal{D}_{\mu}\pi\mathcal{D}^{\mu}\pi^{*}+\frac{\omega}{2\lambda_{\pi}^{6}}\int
d^{4}x\sqrt{-g}[\,\,2\mathcal{D}_{\mu}\mathcal{D}^{\mu}\pi^{*}\mathcal{D}_{\nu}\mathcal{D}^{\nu}\pi\mathcal{D}_{\alpha}\pi\mathcal{D}^{\alpha}\pi^{*}$
$\displaystyle{}-2\mathcal{D}_{\mu}\mathcal{D}_{\nu}\pi^{*}\mathcal{D}^{\mu}\mathcal{D}^{\nu}\pi\mathcal{D}_{\alpha}\pi\mathcal{D}^{\alpha}\pi^{*}+(\,\mathcal{D}_{\mu}\mathcal{D}^{\mu}\pi\mathcal{D}_{\nu}\pi^{*}\mathcal{D}^{\nu}\mathcal{D}^{\alpha}\pi\mathcal{D}_{\alpha}\pi^{*}$
$\displaystyle{}-\mathcal{D}^{\mu}\mathcal{D}^{\nu}\pi\mathcal{D}_{\nu}\mathcal{D}_{\alpha}\pi\mathcal{D}_{\mu}\pi^{*}\mathcal{D}^{\alpha}\pi^{*}+\rm{c.c.}\,\,)\,]$
$\displaystyle{}-\frac{\omega}{4\lambda_{\pi}^{6}}\int
d^{4}x\,\,\sqrt{-g}\,\,\left[\mathcal{D}_{\alpha}\pi\mathcal{D}^{\alpha}\pi\mathcal{D}_{\mu}\pi^{*}\mathcal{D}_{\nu}\pi^{*}\left(R^{\mu\nu}-\frac{1}{4}g^{\mu\nu}R\right)+\rm{c.c.}\right]$
(2.19) $\displaystyle{}-\frac{\omega}{4\lambda_{\pi}^{6}}\int
d^{4}x\,\sqrt{-g}\,\mathcal{D}_{\alpha}\pi\mathcal{D}^{\alpha}\pi^{*}\mathcal{D}_{\mu}\pi\mathcal{D}_{\nu}\pi^{*}g^{\mu\nu}\,R-\frac{1}{4}\int
d^{4}x\sqrt{-g}F_{\mu\nu}F^{\mu\nu}+S_{\rm{VEG}}$

where the last term $S_{\rm{VEG}}$ is the vector Galileon action obtained in
Ref. [102]. We have listed the terms in Appendix (A.6). We will use this
action to study the effects of the Galileon term in the early Universe.

####  2.2.3 Fixing the value of $\omega$

$\lambda_{\pi}$ is the new coupling constant of the model and can only be
fixed with observations. As mentioned in Sec. (2.1.2) $\omega$ can take either
$+1$ or $-1$. In this subsection, we fix the value of $\omega$ by evaluating
the energy density corresponding to the minimal and non-minimal terms in
(2.2.2) in the Coulomb gauge ($A^{0}=0,\partial_{i}A^{i}=0$).

To make the calculations transparent, we evaluate the energy density in the
FRW background which includes arbitrary Lapse function $N(t)$:

$ds^{2}=N^{2}(t)\,dt^{2}-a^{2}(t)(dx^{2}+dy^{2}+dz^{2})$ (2.20)

where $a(t)$ is the scale factor. To satisfy the homogeneity and isotropy of
the FRW background, the Galileon scalar electrodynamics must satisfy the
condition $A^{i}=0$. The equation of motion of $\pi$ corresponding to
$S_{4}^{\min}+S_{4}^{nm}$ is given by (using Eq.(A.4)):

$\displaystyle
2\dot{\pi}^{*}\,\ddot{\pi}^{*}\dot{\pi}\left(\frac{\dot{a}}{a}\right)^{2}-2(\dot{\pi}^{*})^{2}\,\dot{\pi}\left(\frac{\dot{a}}{a}\right)^{2}\frac{\dot{N}}{N}+33(\dot{\pi}^{*})^{2}\dot{\pi}\left(\frac{\dot{a}}{a}\right)^{3}+(\dot{\pi}^{*})^{2}\ddot{\pi}\left(\frac{\dot{a}}{a}\right)^{2}$
$\displaystyle-(\dot{\pi}^{*})^{2}\dot{\pi}\left(\frac{\dot{a}}{a}\right)^{2}\frac{\dot{N}}{N}+8(\dot{\pi}^{*})^{2}\dot{\pi}\left(\frac{\ddot{a}}{a}\right)\left(\frac{\dot{a}}{a}\right)-8(\dot{\pi}^{*})^{2}\dot{\pi}\left(\frac{\dot{a}}{a}\right)^{2}\left(\frac{\dot{N}}{N}\right)=0$
(2.21)

where $H(t)=\dot{a}(t)/a(t)$. Varying the action
${S}_{4}^{\rm{min}}+{S}^{\rm{nm}}_{4}$ with respect to the $g^{00}=N^{-2}(t)$
leads to:

$\displaystyle\delta{S}_{4}^{\rm{min}}+\delta{S}^{\rm{nm}}_{4}$
$\displaystyle=-\frac{3\omega}{2\lambda_{\pi}^{6}}\int
d^{4}x\frac{a^{3}}{N^{2}}\left[79\,({\dot{\pi}}^{*})^{2}\,\dot{\pi}\ddot{\pi}\,H(t)+79\,({\dot{\pi}})^{2}\,\dot{\pi}^{*}\ddot{\pi}^{*}\,H(t)\right.$
$\displaystyle{}\hskip
42.67912pt\left.+20\,({\dot{\pi}}^{*})^{2}\,(\dot{\pi})^{2}H^{2}(t)+23\,({\dot{\pi}}^{*})^{2}\,(\dot{\pi})^{2}\left(\frac{\ddot{a}}{a}\right)\right]\,\,\delta
N$ (2.22)

Using the definition,

$T_{00}=\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta
g^{00}}=-\frac{N^{2}}{a^{3}}\frac{\delta S}{\delta N}\,,$ (2.23)

we get,

$\displaystyle\rho=\,\,T^{0}_{0}=\rho=\frac{3\omega}{2\,\lambda_{\pi}^{6}}\,\,\left[\frac{79}{2}\,\partial_{0}N_{1}\,H+20\,N_{1}\,H^{2}+23\,\frac{\ddot{a}}{a}\,N_{1}\,\,\right]$
(2.24)

where $N_{1}=|\dot{\pi}|^{4}$. In order to get the value of $\omega$ we
simplify the analysis further by setting $N(t)=1$ and taking the limit,
$\pi=\pi^{*}$. Thus, the energy density and equation of motion reduce to

$\displaystyle\rho=\frac{3\omega}{2\lambda_{\pi}^{6}}\left[2\times
79\,({\dot{\pi}})^{3}\,\ddot{\pi}\,H+20\,(\dot{\pi})^{4}\,H^{2}+23\,(\dot{\pi})^{4}\,(H^{2}+\dot{H})\right]$
(2.25)
$\displaystyle\ddot{\pi}+\frac{41}{3}\,\dot{\pi}H+\frac{8}{3}\,\dot{\pi}\left(\frac{\dot{H}}{H}\right)=0$
(2.26)

Substituting Eq. (2.26) in Eq. (2.25), we get

$\displaystyle\rho=-\frac{\omega\dot{\pi}^{4}\,H^{2}}{2\lambda_{\pi}^{6}}\left[6349+1195\,\frac{\dot{H}}{H^{2}}\right]$
(2.27)

Note that $\dot{H}/H^{2}=-\epsilon$ which is a slow-roll parameter. During
inflation, when the non-linear terms contribution can not be ignored,
$\epsilon<1$, hence the quantity in the square bracket, is positive. The
condition that the energy density is positive implies that $\omega=-1$.

###  2.3 Applications to Early-Universe

To extract interesting features of the Galileon model, in this section, we
show that the action (2.2.2) leads to accelerated expansion in the early
Universe. We obtained the energy density of the Galileon field in (2.25).
Using the same procedure, the pressure of the Galileon field is given by:

$\displaystyle p$
$\displaystyle=\frac{\omega}{2\,\lambda_{\pi}^{6}}\,\,[\,\,17\dot{N_{1}}\,H-72\,N_{1}\,H^{2}+8\,\frac{\ddot{a}}{a}\,N_{1}+3\,\ddot{N_{1}}\,\,]$
(2.28)

Taking the case that the phase variation of the complex field is almost
constant, the Friedmann equation becomes:

$\displaystyle
1=\frac{\dot{\pi}^{4}}{6\,M_{P}^{2}\,\lambda_{\pi}^{6}}\left[6349-1195\,\epsilon\right]$
(2.29)

Inverting the above equation, we get

$\displaystyle\epsilon=\frac{6349}{1195}\left[1-\Gamma\right]\quad\mbox{where}\quad\Gamma=\frac{6\,M_{P}^{2}\,\lambda_{\pi}^{6}}{6349\,\dot{\pi}^{4}}\,$
(2.30)

In order for $\epsilon$ to be less than unity, $\Gamma\simeq 1$. To identify
the parameter space of $(\dot{\pi},\lambda_{\pi})$ that can lead to
$\Gamma\simeq 1$, we define
$\dot{\pi}\equiv\Delta_{P}\,M_{P}^{2},\lambda_{\pi}\equiv\delta_{P}\,M_{P}$
where $\Delta_{P}$ and $\delta_{P}$ are dimensionless quantities. (B.1)
contains the plot of $\log\Gamma$ as a function of $\Delta_{P}$ for different
values of $\delta_{P}$. We notice the following: First, the value of $\Gamma$
is weakly dependent on $\delta_{P}$. In other words, for different values of
$\delta_{P}$, the value of $\Gamma$ is almost the same. Second, for a large
values of $\Delta_{P}$, $\Gamma$ is almost close to unity and hence, the slow-
roll parameter is very small. More specifically, for $\Delta_{P}$ in the range
$10^{-5}$ to $0.3$, the slow-roll parameter is of the order of $0.1$. Thus,
for large parameter range, the model can lead to extended period of inflation.

Figure 2.1: Plot of $\log\Gamma(\Delta_{P})$ versus $\Delta_{P}$ for different
values of $\delta_{P}$. Note that for a range of values of the two parameters,
$\Gamma$ is also zero.

###  2.4 Conclusions and Discussions

In this chapter, we constructed Galileon scalar electrodynamics action, which
preserves Galilean symmetry and local gauge invariance. Due to the complex
scalar field, the number of Galilean symmetry invariant terms are reduced. For
an earlier result, in a different context, see Ref. [151]. In the flat space-
time, we have explicitly shown that the equations of motion are second order.
In curved space-time, due to the non-commutative nature of the covariant
derivatives, the minimal coupling of the matter and gravity term leads to
higher-derivatives in the equation of motion. We introduced non-minimal
coupling terms to the Galileon field that makes the equations of motion
second-order in an arbitrary curved space-time. The non-minimal coupling term
is different compared that the one used in the literature for the real scalar
field [150]. However, in the real scalar field limit, the complex scalar
Galileon action is identical to the real scalar Galileon action obtained by
Deffayet et al [150].

As an application of the model, we considered the case when the Galileon
scalar electrodynamics dominated the early Universe. We have shown that for a
range of parameters, the model leads to inflation. One possible application of
the model is to study the perturbations generated during the inflation
dominated by the model. Since the model breaks conformal invariance, this can
generate primordial magnetic fields [102].

The model has two free parameters – $\lambda_{\pi}$ and $\lambda_{VG}$. In the
flat FRW background, the electromagnetic field vanishes and hence the scale of
inflation fixes $\lambda_{\pi}$. The other parameter $\lambda_{VG}$ will play
a critical role in the first order perturbations. In principle, the two
parameters can be different.

The model presented here can be part of the scalar-vector-galileon (SVG)
theories. However, as mentioned earlier, not all SVG gravity theories are
gauge-invariant, while our model is gauge-invariant [147]. The
complexification of a gauge-invariant subset of the SVG theories should be
identical to our model with an appropriate choice of coefficients
$A_{i},B_{i},C_{i},D_{i}$ (for $i=1,2$) and $D_{3}$.

## Chapter 3 Helical magnetic field from Riemann coupling

Effective field theory (EFT) now forms a standard tool in early-Universe
cosmology [152, 153, 154, 155]. Effective field theories rely on the
separation of the energy scales of interest for observations and the
underlying physics of the early Universe near the singularity [156, 157, 158].
In an effective field theory approach, the usual requirement of
renormalizability is too strong. Instead, it demands that a finite number of
parameters describe the physics up to effects suppressed by $(E/M)^{n}$ where
$E$ is the energy of the particles corresponding to the quantum fields, and
$M$ is the energy scale below which the effective field theory description is
valid. As the value of $n$ increases, more parameters are required to describe
the Physics using EFT.

Effective field theory description for gravity has shown that gravity and
quantum mechanics can be compatible with the energies that have been
experimentally probed [159]. EFT has been successfully applied to inflationary
cosmology, especially for the single scalar field inflation model were the
constants in the higher derivative terms of the effective Lagrangian take
values that are powers of $M$, with coefficients roughly of order unity [154,
155]. However, these EFTs aim to obtain a generic prediction for density
perturbations in a single-scalar field inflationary models [155, 160]. Hence,
the analyses can not be extended to other fields (like electromagnetic fields)
during inflation.

Recently, the general effective field theory of gravity coupled to the
Standard Model of particle physics was constructed [161]. The authors
systematically showed that the first gravity operators appear at mass
dimension $6$ in the series expansion, and these operators only couple to the
standard model Bosons. They also showed that (i) no new gravity operators
appear at mass dimension $7$, (ii) in mass dimension $8$ the standard model
Fermions appear, and (iii) coupling between the scalar (Higgs) field and the
standard model gauge Bosons appear _only_ at mass dimension $8$. (Note that
these corrections do not include higher derivative (Galileon) terms in the
electromagnetic action [102, 103].)

In chapter (5), we will discuss a generic framework— _effective field theory_
(EFT) of magnetogenesis to generate the primordial non-helical magnetic
fields. We will show that our EFT construction can reproduce a large class of
models proposed in the literature. In this chapter, we limit to mass dimension
$6$ operators coupling to the gauge field, specifically, to the
electromagnetic field. We concentrate on the coupling of the electromagnetic
field with the dual Riemann tensor
$\tilde{R}^{\mu\nu\rho\sigma}(\equiv\epsilon^{\mu\nu\alpha\beta}R_{\alpha\beta}^{~{}~{}~{}\rho\sigma}/2$).
We show that such a term leads to the generation of a helical magnetic field
during inflation [104].

As mentioned in Chapter 1, magnetic fields have been observed at all scales in
the Universe; however, there is no compelling model of the origin of large
scale magnetic fields. Observations from Faraday rotation and synchrotron
radiation show the presence of micro-Gauss strength magnetic fields in the
galaxies and the clusters of galaxies [60, 61, 61, 62, 80, 81].

While the magnetic field measurements from Faraday rotation and synchrotron
radiation provide upper bounds of the magnetic fields, the FERMI measurement
of gamma-rays emitted by blazars provides a lower bound of the order of
$10^{-15}$ G in intergalactic voids [65]. However, inflation can not generate
these large scale magnetic as the standard 4-D electromagnetic action is
conformally invariant. To amplify the quantum fluctuations of the
electromagnetic fields in the early Universe, one needs to break the conformal
invariance of the action [97, 62, 80, 81, 162].

While mechanisms to generate non-helical fields were proposed four decades ago
[97, 98, 100], the generation of helical fields is recent [104, 163, 164]. One
of the interests in primordial magnetic helicity is that it can be a direct
indication of parity violation in the early Universe and may be related to the
matter-antimatter asymmetry in the early Universe [113]. Besides, the
conservation of helicity leads to an inverse cascade in the turbulent plasma
era, that can move power from small to large scales [163]. Hence, the decay
rate of energy density and coherence length is slower than the non-helical
fields during these epochs. It has been shown that helical magnetic fields
will leave distinct signatures in CMB, such as TE- and EB-cross-correlations
[165, 166].

As mentioned above, there has been a lot of interest in generating a
primordial helical field in the early Universe [104, 163, 164, 105, 167, 168,
169]. However, most models introduce non-minimal coupling of the
electromagnetic fields with a (pseudo-)scalar. While this leads to the
breaking of conformal invariance of the electromagnetic field, due to non-
minimal coupling, extra degrees of freedom are present at all energies. More
importantly, these extra degrees of freedom propagate even at the low-energy
and can potentially lead to the strong coupling problem [170]. It is possible
to overcome the strong-coupling problem for a narrow range of coupling
functions [105].

In this chapter, to avoid the strong-coupling problem and restricted coupling
functions, we propose a model that couples the electromagnetic fields with the
Riemann tensor. To our knowledge, Riemann tensor coupling has not be discussed
in the literature to generate helical fields. The model has three key
features: First, it does not require the coupling of the electromagnetic field
with the scalar field. Hence, there are no extra degrees of freedom and will
not lead to a strong-coupling problem. Second, the conformal invariance is
broken due to the coupling to the Riemann tensor. Since the curvature is large
in the early Universe, the coupling term will introduce non-trivial
corrections to the electromagnetic action. However, at late-times, the new
term will not contribute, and the theory is identical to standard
electrodynamics. The power-spectrum of the fields has slight red-tilt for
slow-roll inflation. Third, as we show explicitly, our model is free from
backreaction for a range of scale-factor during inflation. This is different
from other models where a specific form of coupling function is chosen to
avoid any back-reaction [105].

In Sec. (4.2), we introduce the model and discuss its properties. We discuss
the classical properties and define the relevant quantities. We also briefly
discuss the procedure to quantize these fields in the FRW background. In Sec.
(3.2), we explicitly evaluate the helical magnetic field generation in our
model and show that the power-spectrum is red-tilted. This is different
compared to the other models in the literature. We also show that the model
does not have a backreaction problem for a range of scale-factor. We discuss
the implications of the results in Sec. (4.4). Appendix B contains the details
of the calculations in this chapter. The results reported in this chapter are
based on Ref. [120]

###  3.1 The model

We consider the following action:

$\displaystyle S=S_{\rm{Grav}}+S_{\phi}+S_{\rm{EM}}+S_{\rm CB}$ (3.1)

where $S_{\rm{Grav}}$ is the Einstein-Hilbert action

$\displaystyle S_{\rm Grav}=-\frac{M_{\rm P}^{2}}{2}\int
d^{4}x\sqrt{-g}\,R\,,$ (3.2)

and $S_{\phi}$ is the action for the minimally coupled, self-interacting
canonically scalar field:

$\displaystyle S_{\phi}=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right].$
(3.3)

We assume that the scalar field ($\phi$) dominates the energy density in the
early Universe (during inflation) and leads to $60\,-\,70$ e-foldings of
inflation with $H\simeq 10^{14}{\rm GeV}$. $S_{\rm{EM}},S_{\rm CB}$ refer to
the standard electromagnetic (EM) and conformal breaking part of the
electromagnetic terms, respectively, and given by:

$\displaystyle S_{\rm{EM}}$ $\displaystyle=-\frac{1}{4}\int
d^{4}x\,\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu},\hskip 14.22636pt$ (3.4)
$\displaystyle S_{\rm{CB}}$ $\displaystyle=-\frac{\sigma}{M^{2}}\,\int
d^{4}x\,\sqrt{-g}\,R_{\rho\sigma}\,^{\alpha\beta}F_{\alpha\beta}\,\tilde{F}^{\rho\sigma}=-\frac{\sigma}{M^{2}}\,\int
d^{4}x\,\sqrt{-g}\,\tilde{R}^{\mu\nu\alpha\beta}F_{\alpha\beta}\,F_{\mu\nu}\,,$
(3.5)

where $R_{\rho\sigma}\,^{\alpha\beta}$ is the Riemann tensor and its dual is
$\tilde{R}^{\mu\nu\alpha\beta}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}R_{\rho\sigma}\,^{\alpha\beta}$,
$A_{\mu}$ is the four-vector potential of the electromagnetic field,
$F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$ and
$\tilde{F}^{\rho\sigma}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}$ is
the dual of $F_{\mu\nu}$.
$\epsilon^{\mu\nu\rho\sigma}=\frac{1}{\sqrt{-g}}\,\eta^{\mu\nu\rho\sigma}$ is
fully antisymmetric tensor, $\eta^{\mu\nu\rho\sigma}$ is Levi-Civita symbol
whose values are $\pm 1$ and we set $\eta^{0123}=1=-\eta_{0123}$.

The standard electromagnetic action $S_{\rm{EM}}$ is conformally invariant;
however, the presence of Riemann curvature in $S_{\rm CB}$ breaks the
conformal invariance. $M$ is the energy scale, which sets the scale for the
breaking of conformal invariance. For our discussion below, we assume that
$10^{-3}\leq(H/M)\leq 1$ [171, 172, 173, 174]. Due to the Riemann tensor, in
FRW background, $M$ appears as a time-dependent coupling [175]. To see this,
let us evaluate Riemann tensor for the flat FRW background (4.3):

$R_{\mu\nu}\,^{\sigma\gamma}\sim\frac{{a^{\prime}}^{2}}{a^{4}}~{}\rm{or}~{}\frac{a^{\prime\prime}}{a^{3}}$

Thus, the coupling function is time-dependent, i. e.,

$\displaystyle\frac{1}{M_{\rm eff}}\sim\frac{1}{M}\frac{a^{\prime}}{a^{2}}\,.$
(3.6)

Before proceeding with the analysis, we want to highlight the following
salient features of this model compared to the earlier models that introduce
non-minimal scalar field coupling in action: First, our model does not require
the coupling of the electromagnetic field with the scalar field. Hence, there
are no extra degrees of freedom and will not lead to a strong-coupling
problem. Second, the conformal invariance is broken due to the coupling to the
Riemann tensor. Since the curvature is significant in the early Universe, the
coupling term will introduce non-trivial corrections to the electromagnetic
action. However, at late-times, $S_{\rm CB}$ will not contribute, and the
model is identical to standard electrodynamics. Third, as we show explicitly,
our model is free from backreaction for a range of scale-factor during
inflation. This is different from other models where a specific form of
coupling function is chosen to avoid any back-reaction [105].

As mentioned earlier, we aim to generate the helical magnetic field during
inflation. Hence, the scalar field’s energy density dominates over the
standard electromagnetic and conformal breaking term in action (4.11). Since
the single-scalar field inflation can not generate vector perturbations, the
magnetic field generated will be due to the conformal breaking term $S_{\rm
CB}$ in action.

The variation of the action (4.11) with respect to gauge field $A_{\mu}$ leads
to the following equation:

$\displaystyle\partial_{\mu}\left(\sqrt{-g}\,F^{\mu\nu}+\frac{1}{M^{2}}\sqrt{-g}\,\epsilon^{\alpha\beta\rho\sigma}R_{\rho\sigma}\,^{\mu\nu}\,F_{\alpha\beta}+\frac{1}{M^{2}}\sqrt{-g}\,\epsilon^{\mu\nu\rho\sigma}R_{\rho\sigma}\,^{\alpha\beta}\,F_{\alpha\beta}\,\right)=0$
(3.7)

As mentioned above, we will consider a flat Friedman universe described by the
line-element:

$\displaystyle ds^{2}=a^{2}(\eta)\,(d\eta^{2}-\delta_{ij}dx^{i}dx^{j})$ (3.8)

where $\eta$ is the conformal time. For $\nu=i$, Eq. (3.7) reduces to:

$\displaystyle\partial_{0}\left(F_{0i}+\frac{2}{M^{2}}\,\eta^{0ijk}\frac{a^{\prime\prime}}{a^{3}}\,F_{jk}\,\right)-\partial_{l}\left(\,F_{li}-\frac{4}{M^{2}}\,\eta^{0ilj}\frac{a^{\prime\prime}}{a^{3}}F_{j0}\,\right)=0$
(3.9)

where we have substituted the following components of Riemann tensor:

$R_{ij}\,^{kl}=\frac{{a^{\prime}}^{2}}{a^{4}}\,\left(\delta^{l}_{i}\,\delta^{k}_{j}-\delta^{k}_{i}\,\delta^{l}_{j}\right),R_{0i}\,^{0j}=\left(\frac{{a^{\prime}}^{2}}{a^{4}}-\frac{a^{\prime\prime}}{a^{3}}\right)\,\delta^{j}_{i}\,.$

In the Coulomb gauge ($A^{0}=0,\partial_{i}A^{i}=0$), and using
$\eta^{0ijl}=\epsilon_{ijl}$ (where $\epsilon_{ijl}$ is the Levi-Civita symbol
in 3D Euclidean space), the above equation motion (of $A^{i}$) leads to the
following evolution equation:

$\displaystyle
A_{i}^{\prime\prime}+\frac{4\,\epsilon_{ijl}}{M^{2}}\,\left(\frac{a^{\prime\prime\prime}}{a^{3}}-3\frac{a^{\prime\prime}a^{\prime}}{a^{4}}\right)\partial_{j}A_{l}-\partial_{j}\partial_{j}A_{i}=0$
(3.10)

Note that the above equation differs from other models in the literature by an
overall factor in the term with $\epsilon_{ijl}\partial_{j}A_{l}$, especially,
the third derivative of the scale factor. Thus, the model can lead to
different evolution of the fluctuations in comparison to non-minimally coupled
scalar field models.

In this chapter, we consider two inflationary scenarios — power-law inflation
and the slow-roll inflation — to evaluate the power-spectrum of the
electromagnetic fluctuations. For power law inflation the scale factor (in
cosmic time) is given by $a(t)=a_{0}t^{p}$ where $p>1$ and $a_{0}$ is
arbitrary constant. In the conformal time, the scale factor is [176]:

$\displaystyle a(\eta)=\left(-\frac{\eta}{\eta_{0}}\right)^{(\beta+1)}$ (3.11)

where $\eta_{0}$ is an arbitrary constant and denotes the scale of inflation.
During inflation, $\eta\in(-\infty,0)$. $\beta$ and $\eta_{0}$ are given by:

$\displaystyle\beta=-\left(\frac{2p-1}{p-1}\right)\hskip
14.22636pt\text{and}\hskip
14.22636pt\eta_{0}=\left[(p-1)a_{0}^{1/p}\right]^{-1}.$ (3.12)

Note that $\beta\leq-2$ and $\beta=-2$ corresponds to the de Sitter. The
Hubble parameter ($\mathcal{H}\equiv{a^{\prime}(\eta)}/{a(\eta)}$) is given
by:

$\displaystyle\mathcal{H}=\frac{a^{\prime}}{a}=\frac{\beta+1}{\eta}\implies\eta=\frac{\beta+1}{\mathcal{H}}$
(3.13)

Slow-roll inflation is a generic inflationary paradigm that leads to an
accelerated expansion independent of a particular model (or potential). In
this case, we have

$\beta\approx-2-\epsilon,\quad\mathcal{H}\approx-\frac{1+\epsilon}{\eta}$
(3.14)

where $\epsilon$ is the slow roll parameter.

####  3.1.1 Physical quantities of interest

Although, the four-vector potential $A^{\mu}$ provides the covariant
description of the electromagnetic processes, to compare with observations, we
need to decompose the physical quantities in terms of the electric and
magnetic fields, that are intrinsically frame-dependent. Hence, it is always
useful to define a comoving observer with velocity
$u^{\mu}=\left(1/a(\eta),0,0,0\right)$ satisfying $u_{\mu}u^{\mu}=1$. The
electric and magnetic field four-vector for this observer is given by
projecting the EM field tensor with $u^{\mu}$ as

$\displaystyle E_{\mu}=u^{\alpha}F_{\alpha\mu},\hskip
14.22636ptB_{\mu}=\frac{1}{2}\,u^{\gamma}\,F^{\alpha\beta}\epsilon_{\gamma\alpha\beta\mu}=u^{\alpha}\tilde{F}_{\alpha\mu}.$
(3.15)

Note that the electric and magnetic field four-vectors are both three-vector
fields in a sense that they are orthogonal to the comoving observer, i.e.,
$E_{\mu}u^{\mu}=0=B_{\mu}u^{\mu}$, and we have:

$\displaystyle E_{\mu}=a(\eta)\left(0,\textbf{E}\right),\quad
B_{\mu}=a(\eta)\left(0,\textbf{B}\right)$ (3.16)

where

$\textbf{E}=\frac{A_{i}^{\prime}}{a^{2}(\eta)},\quad\textbf{B}=-\frac{1}{a^{2}(\eta)}\,\epsilon_{ijk}\,\partial_{i}A_{j}\,.$

Electromagnetic energy densities are defined as

$\displaystyle\rho_{B}\equiv-\frac{1}{2}B_{\mu}B^{\mu}=\frac{1}{2}\textbf{B}\cdot\textbf{B},\qquad\rho_{E}\equiv-\frac{1}{2}E_{\mu}E^{\mu}=\frac{1}{2}\textbf{E}\cdot\textbf{E}.$
(3.17)

and the magnetic helicity density is

$\displaystyle\rho_{h}\equiv-A_{\mu}B^{\mu}.$ (3.18)

We will evaluate these quantities for the quantum fluctuations generated
during inflation.

####  3.1.2 Quantization in the Helicity basis

In this section, we briefly discuss the evolution of the quantum fluctuations
of the electromagnetic field in the helicity basis [105]. Decomposition of the
vector potential in Fourier space, we have:

$\displaystyle
A^{i}(\vec{x},\eta)=\int\frac{d^{3}k}{(2\pi)^{3}}\sum_{\lambda=1,2}\varepsilon^{i}_{\lambda}\left[A_{\lambda}(k,\eta)b_{\lambda}(\vec{k})e^{ik\cdot
x}+A^{*}_{\lambda}(k,\eta)b^{\dagger}_{\lambda}(\vec{k})e^{-ik\cdot x}\right]$
(3.19)

where $b(\textbf{k})$ and $b^{\dagger}(\textbf{k})$ are the annihilation and
creation operators respectively for a given comoving mode k, and
$\varepsilon_{\lambda}^{i}$ is the orthogonal basis vector which in right-
handed coordinate system [105] is given by

$\displaystyle\varepsilon^{\mu}=\left(\frac{1}{a},\textbf{0}\right),\,\,\,\,\varepsilon^{\mu}=\left(0,\frac{\hat{\varepsilon}^{i}_{\lambda}}{a}\right),\,\,\,\,\varepsilon^{\mu}_{3}=\left(0,\frac{\hat{\textbf{k}}}{a}\right)\quad\text{for}\quad\lambda=1,2\,,$
(3.20)

3-vectors $\hat{\varepsilon}^{i}_{\lambda}$ are unit vectors orthogonal to
$\hat{\textbf{k}}$ and to each other. Substituting Eq. (C.2) in Eq. (C.1 ) and
defining the new variable $\bar{A}_{\lambda}=a(\eta)\,A_{\lambda}(k,\eta)$, we
have:

$\displaystyle
A_{i}(\textbf{x},\eta)=\int\frac{d^{3}k}{(2\pi)^{3}}\sum_{\lambda=1,2}\,\hat{\varepsilon}_{i\lambda}\left[\bar{A}_{\lambda}b_{\lambda}(\textbf{k})e^{i\textbf{k}\cdot\textbf{x}}+\bar{A}^{*}_{\lambda}b^{\dagger}_{\lambda}(\vec{k})e^{-i\textbf{k}\cdot\textbf{x}}\right]\,.$
(3.21)

Substituting Eq. (C.3) in Eq. (4.17), we get:

$\displaystyle\sum_{\lambda=1,2}b_{\lambda}\left[\hat{\varepsilon}_{i\lambda}\bar{A}_{\lambda}^{\prime\prime}+\frac{4i}{M^{2}}\epsilon_{ijl}k_{j}\hat{\varepsilon}_{l\,\lambda}\bar{A}_{\lambda}\,\left(\frac{a^{\prime\prime\prime}}{a^{3}}-3\frac{a^{\prime\prime}a^{\prime}}{a^{4}}\right)+k^{2}\hat{\varepsilon}_{i\lambda}\bar{A}_{\lambda}\right]=0$
(3.22)

where we have used $\partial_{j}\partial_{j}=-k^{2}$.

Since the action (4.11) contains parity breaking term (helicity term), it is
always useful to work in the helicity basis. The helicity basis vectors
$\varepsilon_{+}$ and $\varepsilon_{-}$ corresponding to $h=+1$ and $h=-1$ are
defined as

$\displaystyle\varepsilon_{\pm}=\frac{1}{\sqrt{2}}\left(\hat{\varepsilon}_{1}\pm
i\hat{\varepsilon}_{2}\right).$ (3.23)

Assuming that the wave propagates in the $z-$direction, the vector potential
in the helicity basis is given by:

$\displaystyle\bar{\textbf{A}}=\bar{A}_{1}\hat{\varepsilon}_{1}+\bar{A}_{2}\hat{\varepsilon}_{2}=A_{+}\varepsilon_{+}+A_{-}\varepsilon_{-}$
(3.24)

where $A_{+}$($A_{-}$) refer to the vector potential with positive (negative)
helicity. The ground state in the helicity basis is defined as

$\displaystyle b_{h}(\textbf{k})|0\rangle=0$ (3.25)

and commutation relation are:

$\displaystyle\left[b_{h}(\textbf{k}),b^{\dagger}_{h^{\prime}}(\textbf{q})\right]$
$\displaystyle=\left(2\pi\right)^{3}\,\delta^{3}(\textbf{k}-\textbf{q})\,\delta_{hh^{\prime}}$
(3.26)
$\displaystyle\left[b_{h}(\textbf{k}),b_{h^{\prime}}(\textbf{q})\right]$
$\displaystyle=0=\left[b^{\dagger}_{h}(\textbf{k}),b^{\dagger}_{h^{\prime}}(\textbf{q})\right]\,.$
(3.27)

Rewriting (C.4) in the Helicity basis and replacing
$\epsilon_{ijl}\partial_{j}A_{l}\longrightarrow-k\sum_{h=\pm
1}hA_{h}\varepsilon_{h}$, we have:

$\displaystyle
A_{h}^{\prime\prime}+\left[k^{2}-\frac{4kh}{M^{2}}\,\Gamma(\eta)\right]A_{h}=0\,,$
(3.28)

where,

$\Gamma(\eta)=\frac{a^{\prime\prime\prime}}{a^{3}}-3\frac{a^{\prime\prime}a^{\prime}}{a^{4}}=\frac{1}{a^{2}}\left(\mathcal{H}^{\prime\prime}-2\mathcal{H}^{3}\right)\,.$
(3.29)

We would like to stress the following points regarding the above expression:
First, unlike the scalar or tensor perturbations, the mode functions contain
third-order derivatives of the scale factor. This implies that the spectrum of
perturbations may be different in our model. Second, since the perturbations
equations contain second-order derivatives of $\mathcal{H}$, the helicity
modes will be different for inflation and bounce models [177]. Third, since
$h$ takes two values, $A_{h}$ evolves differently for the two modes leading to
non-zero helicity.

The EM energy densities of the ground state with respect to the comoving
observer are:

$\displaystyle\rho_{B}\left(\eta,k\right)\equiv-\frac{1}{2}\langle
0|B_{i}B^{i}|0\rangle=\int\frac{dk}{k}\,\frac{d\rho_{B}}{d\rm{ln}k}=\int\frac{dk}{k}\frac{1}{\left(2\pi\right)^{2}}\frac{k^{5}}{a^{4}}\left(\,\,\left|A_{+}\left(\eta,k\right)\right|^{2}+\left|A_{-}\left(\eta,k\right)\,\,\right|^{2}\right)$
(3.30) $\displaystyle\rho_{E}\left(\eta,k\right)\equiv-\frac{1}{2}\langle
0|E_{i}E^{i}|0\rangle=\int\frac{dk}{k}\,\frac{d\rho_{E}}{d\rm{ln}k}=\int\frac{dk}{k}\frac{1}{\left(2\pi\right)^{2}}\frac{k^{3}}{a^{4}}\left(\,\,\left|A^{\prime}_{+}\left(\eta,k\right)\right|^{2}+\left|A^{\prime}_{-}\left(\eta,k\right)\right|^{2}\,\,\right)$
(3.31)

and the ground state helicity density as

$\displaystyle\rho_{h}\left(\eta,k\right)\equiv-\langle
0|A_{i}B^{i}|0\rangle=\int\frac{dk}{k}\,\frac{d\rho_{h}}{d\rm{ln}k}=\int\frac{dk}{k}\frac{1}{2\pi^{2}}\frac{k^{4}}{a^{3}}\left(\,\,\left|A_{+}\left(\eta,k\right)\right|^{2}-\left|A_{-}\left(\eta,k\right)\right|^{2}\,\,\right).$
(3.32)

where $d\rho_{\Upsilon}/d(\ln k)$ for $\Upsilon\in\\{E,B,h\\}$ is the spectral
energy contained in logarithmic interval in $k-$space. Note that the helicity
density is the difference between the two helicity spectrum. Hence, it is
possible to maximize the magnetic helicity density, if one helicity is
enhanced and the other helicity is suppressed [80]. For most of the
calculation, we will keep both the terms and evaluate the energy density for
both helicity modes.

###  3.2 Helical magnetic field generation

In this section, we explicitly calculate the power-spectrum and energy
densities for our model in power-law inflation. We obtain the power-spectrum
in the slow-roll limit.

Substituting the power-law scale factor (4.16) in Eq.(4.18) leads to:

$\displaystyle{A_{h}^{\prime\prime}+\left[k^{2}-\frac{8kh}{M^{2}}\frac{\beta(\beta+1)(\beta+2)}{\eta_{0}^{3}}\left(\frac{-\eta_{0}}{\eta}\right)^{(2\beta+5)}\right]\,A_{h}=0}$
(3.33)

As expected, for de-sitter case ($\beta=-2$), the helicity term
$(\Gamma(\eta)$) vanishes. This is consistent with the fact that the de Sitter
symmetry will not be preserved in the presence of helicity terms. However, it
will be non-zero for the approximately de-sitter universe i.e.,
$\beta=-2-\epsilon$.

For the power-law inflation model, the scalar and tensor perturbations can be
evaluated exactly. However, as can be seen, it is not possible to obtain an
exact expression. To obtain the solution, we consider two regions. In Region I
(sub-horizon limit), the wavelength of the mode is smaller than the Hubble
radius, i. e., $H<<k$. In this region, we can neglect $\Gamma(\eta)$ in Eq.
(C.12). In Region II (super-Horizon scales), the mode is outside the Hubble
radius i. e., $k<<H$. In this region, we can neglect $k^{2}$ in Eq. (C.12).
The constants are fixed by matching $A_{h}$ and $A_{h}^{\prime}$ at the
transition time between regions I and II at $\eta_{*}$. While evaluating the
mode-functions is trivial in Region I, it is highly non-trivial in Region II.
In the rest of this section, we obtain the mode functions and calculate the
power-spectrum.

In Region I ($\left|-k\eta\right|\gg 1$), Eq. (C.12) simplifies to:

$\displaystyle A_{h}^{\prime\prime}+k^{2}A_{h}\approx 0$ (3.34)

and assuming that the quantum field is in the vacuum state at asymptotic past
(Bunch-Davies vacuum state), we have:

$\displaystyle A_{h}=\frac{1}{\sqrt{k}}e^{-ik\eta}$ (3.35)

In Region II ($\left|-k\eta\right|<<1$), Eq. (C.12) becomes:

$\displaystyle{A_{h}^{\prime\prime}+hk\frac{\varsigma^{2}}{\eta^{2}}\left(\frac{-\eta_{0}}{\eta}\right)^{2\alpha}\,A_{h}=0}$
(3.36)

where

${\varsigma^{2}\equiv-\frac{1}{M^{2}\,\eta_{0}}(2\alpha-3)(2\alpha-1)(2\alpha+1)\,,\quad\alpha=\beta+\frac{3}{2}}$
(3.37)

$\alpha$ makes the expressions look tidier! Note that $\alpha=-\frac{1}{2}$
corresponds to de-sitter and $\alpha\leq-\frac{1}{2}$.

As mentioned above, unlike in the scalar and tensor perturbations during
inflation, the above equation is not exactly solvable. To do this, we
introduce a new dimensionless variable $\tau$, and is defined as:

$\displaystyle\tau=\left(-\frac{\eta_{0}}{\eta}\right)^{\alpha}\quad\implies\quad\eta=-\frac{\eta_{0}}{\tau^{\frac{1}{\alpha}}}\,.$
(3.38)

Note that $\tau$ is directly proportional to $\eta$ and is a positive definite
quantity ($0<\tau<\infty$). [At the start of inflation, $\tau$ is large and
vanishes at the end of inflation.] In terms of $\tau$, the scale factor for
the power-law inflation (4.16) is

$a(\tau)=\left(\frac{1}{\tau}\right)^{1-\frac{1}{2\alpha}}\,.$

Rewriting Eq. (3.36) in terms of $\tau$ (3.38), we have:

$\displaystyle{\alpha^{2}\frac{d^{2}A_{h}}{d\tau^{2}}+\frac{\alpha(\alpha+1)}{\tau}\frac{dA_{h}}{d\tau}+h\,k\,\varsigma^{2}A_{h}=0}$
(3.39)

The above equation is a Bessel differential equation, and it has a complete
solution as

$\displaystyle A_{+}(\tau,k)$
$\displaystyle=\tau^{-\frac{1}{2\alpha}}\,J_{\frac{1}{2\alpha}}\left(\frac{\varsigma\sqrt{k}}{\alpha}\tau\,\right)C_{1}+\tau^{-\frac{1}{2\alpha}}\,Y_{\frac{1}{2\alpha}}\left(\frac{\varsigma\sqrt{k}}{\alpha}\tau\right)C_{2}$
(3.40a) $\displaystyle A_{-}(\tau,k)$
$\displaystyle=\tau^{-\frac{1}{2\alpha}}\,J_{\frac{1}{2\alpha}}\left(-i\frac{\varsigma\,\sqrt{k}}{\alpha}\tau\right)C_{3}+\tau^{-\frac{1}{2\alpha}}\,Y_{\frac{1}{2\alpha}}\left(-i\,\frac{\varsigma\,\sqrt{k}}{\alpha}\tau\right)C_{4}$
(3.40b)

where $C_{1},C_{2},C_{3},C_{4}$ are arbitrary constants of dimension
$L^{1/2}$. As mentioned above, for the two helicity modes, we fix the
constants $C_{1},C_{2}$ ($C_{3},C_{4})$ by matching $A_{h}$ and
$A_{h}^{\prime}$ at the transition time between between regions I and II at
$k_{*}\sim\eta_{*}^{-1}$ where $*$ refers to the quantities evaluated at the
horizon-exit.

Although the analysis can be done for any general value of $\alpha$, to keep
the calculations tractable, we obtain the constants for $\alpha=-1$. There are
two reasons for this choice: First, in this special case, $\tau\propto\eta$
and the super-horizon modes can be written in terms of $\eta$ using the linear
relation. Second, the constants $C_{1},C_{2},C_{3},C_{4}$ have a weak
dependence of $\alpha$ and, hence, finding the value for a given value of
$\alpha$ will be accurate within a order. Thus, matching he solutions and the
derivatives at the horizon-exit, we get:

$\displaystyle C_{1}$
$\displaystyle=-e^{i}\,\sqrt{\frac{\pi\eta_{0}}{2}}\left(\frac{1}{\sqrt{\Theta}}\rm{sin}\Theta+i\sqrt{\Theta}\,\rm{cos}\Theta\right),\,\,\,\,C_{2}=-i\,e^{i}\,\sqrt{\frac{\pi\eta_{0}}{2}}\left(\frac{1}{\sqrt{\Theta}}\rm{cos}\Theta-i\sqrt{\Theta}\,\rm{sin}\Theta\right)$
(3.41) $\displaystyle C_{3}$
$\displaystyle=e^{i}\,\sqrt{\frac{\pi\eta_{0}}{2}}\left(\frac{1}{\sqrt{i\Theta}}\rm{sinh}\Theta+\sqrt{i\Theta}\,\rm{cosh}\Theta\right),\,\,\,\,C_{4}=-i\,e^{i}\,\sqrt{\frac{\pi\eta_{0}}{2}}\left(\frac{1}{\sqrt{i\Theta}}\rm{cosh}\Theta+\sqrt{i\Theta}\,\rm{sinh}\Theta\right).$

where $\Theta=\sqrt{\frac{15\eta_{*}}{M^{2}\eta_{0}^{3}}}$ is the
dimensionless constant.

To obtain the dominating helicity mode during inflation, we need to obtain the
values of the coefficients $C_{1},C_{2},C_{3},C_{4}$. To obtain these values,
we take: $\mathcal{H}\sim{\eta_{0}}^{-1}\sim
10^{14}\rm{GeV}\sim{10^{52}}\rm{Mpc}^{-1}$, and $M\sim 10^{17}\rm{GeV}$ [176].
This gives $\Theta\sim 10^{-3}$ which is small value. Approximating
trigonometric functions in Eq. (C.18), we obtain

$\displaystyle\left|C_{1}\right|\approx\left|C_{3}\right|\approx
10^{-17/2}\rm{GeV}^{-\frac{1}{2}},\hskip
14.22636pt\text{and}\,\,\,\,\,\left|C_{2}\right|\approx\left|C_{4}\right|\approx
10^{-11/2}\rm{GeV}^{-\frac{1}{2}}.$ (3.42)

Using these values in Eqs. (C.17a, C.17b), we have plotted the two modes for
$\alpha=-0.53$ and $\alpha=-1$ in 3.1. The plots show that the positive
helicity modes are growing compared to the negative helicity modes.
Specifically, from Fig.3.1 (right plot), we see that negative helicity mode is
decaying. (For $\alpha=-1$ we have $\tau\propto-\eta$ which means negative
mode is decaying from $-\infty$ to zero in conformal time). Hence, we can set
$\left|A_{-}(\tau,k)\right|=0$. The helicity density (3.32) is the difference
between the two helicity spectrum, and maximum helicity is achieved if one
helicity is enhanced compared to other. In our case, the negative helicity
mode is negligible and, has been set to zero.

Figure 3.1: Figure showing the behaviour of positive and negative helicity
mode $\alpha=-0.53$ (left plot) and $\alpha=-1$ (right plot).
$\tilde{\tau}=10^{-\frac{63}{2}}\,\tau$ and the vertical axis is in
$\rm{GeV}^{-1/2}$.

Using the series expansion of the Bessel functions (see appendix B.1 for
details), in the leading order, positive helicity mode (C.17a), takes the
following form:

$\displaystyle A_{+}(\tau,k)$
$\displaystyle=C\,k^{\frac{1}{4\alpha}}-C_{2}\frac{\mathcal{F}^{-1}}{\pi}\Gamma\left(\frac{1}{2\alpha}\right)\,k^{-\frac{1}{4\alpha}}\tau^{-\frac{1}{\alpha}}$
(3.43)

where $C,\mathcal{F}$ are constants defined in Appendix (B.1). The value of
these constants are evaluated using (3.42). Note that $\varsigma\approx
10^{-10}\rm{GeV}^{-1/2}$ the floor function $F(\tau)=1$. Fig. (B.1) contains
the plot of Floor function for the range of parameters used in evaluated
(3.42). Also, $\left|\mathcal{F}\right|\sim
10^{-\frac{5}{\alpha}}\,\,\rm{GeV}^{-1/4\alpha},\text{and}\,\left|C\right|\sim
10^{-\frac{5}{\alpha}-\frac{11}{2}}$.

####  3.2.1 Energy densities, power spectrum, and backreaction

Substituting Eq. (4.19) in the relation (3.30), the spectral magnetic energy
density is given by:

$\displaystyle\frac{d\rho_{B}}{d\rm{ln}k}=\frac{1}{\left(2\pi\right)^{2}}\frac{k^{5}}{a^{4}(\eta(\tau))}\left(\,\,\,\,\left|C\right|^{2}\,k^{\frac{1}{2\alpha}}+\left|C_{2}\frac{\mathcal{F}^{-1}}{\pi}\Gamma\left(\frac{1}{2\alpha}\right)\right|^{2}\,\left(k\,\tau^{4}\right)^{-\frac{1}{2\alpha}}\right).$
(3.44)

Using the expression:
$\tau_{*}=\left(-\frac{2\,\eta_{0}k_{*}}{2\alpha-1}\right)^{\alpha}$ at the
exit of inflation, spectral energy density at horizon exit is given by:

$\displaystyle\left.\frac{d\rho_{B}}{d\rm{ln}k}\right|_{k_{*}\sim\mathcal{H}}=\frac{(-\eta_{0})^{4\alpha-2}}{\left(2\pi\right)^{2}}\left[\,\left|C\right|^{2}\,k_{*}^{3+4\alpha+\frac{1}{2\alpha}}+\left|C_{2}\frac{\mathcal{F}^{-1}}{\pi}\Gamma\left(\frac{1}{2\alpha}\right)\right|^{2}\,\frac{(2\alpha-1)^{2}}{4\eta_{0}^{2}}\,k_{*}^{1+4\alpha-\frac{1}{2\alpha}}\right]$
(3.45)

Let us understand the properties of the spectral energy density obtained
above: First, it has two branches. The first branch (setting $C_{2}=0$) has
scale-invariant spectrum for $\alpha=-\frac{1}{2},-\frac{1}{4}$. Similarly,
the second branch (setting $C=0$) has scale invariant spectrum for
$\alpha=-\frac{1}{2},\frac{1}{4}$. Note that the physically allowed values of
$\alpha\leq-1/2$. Hence, $\alpha=\pm 1/4$ is ruled out. Thus, the two branches
has scale-invariant power-specrum for exact de Sitter ($\alpha=-\frac{1}{2}$).
Second, for slow-roll type of inflation $\alpha=-\frac{1}{2}-\epsilon$, the
two branches scale differently — $k_{*}^{-2\epsilon}$ (first branch) and
$k_{*}^{-6\epsilon}$ (second branch). Since $\epsilon$ is positive, this
implies that our model produces more power on the large scales. Thus, our
model predicts a red spectrum for the helical modes for slow roll inflation.
In the next subsection, we compare the results of our model with other models.

Thus, the magnetic energy density can be obtained as:

$\displaystyle\rho_{B}=\int_{\frac{\mathcal{H}}{100}}^{\mathcal{H}}\frac{dk}{k}\frac{d\rho_{B}}{d\rm{ln}k}=\frac{1}{\left(2\pi\right)^{2}a^{4}(\eta(\tau))}\left[\left|C\right|^{2}\,\frac{\mathcal{H}^{5+\frac{1}{2\alpha}}}{5+\frac{1}{2\alpha}}+\left|C_{2}\frac{\mathcal{F}^{-1}}{\pi}\Gamma\left(\frac{1}{2\alpha}\right)\right|^{2}\,\tau^{-\frac{2}{\alpha}}\frac{\mathcal{H}^{5-\frac{1}{2\alpha}}}{5-\frac{1}{2\alpha}}\right]$
(3.46)

where we have used
$\mathcal{H}^{5+\frac{1}{2\alpha}}-\left(\frac{\mathcal{H}}{100}\right)^{5+\frac{1}{2\alpha}}\approx\mathcal{H}^{5+\frac{1}{2\alpha}}$.
Total magnetic energy density at the exit of inflation can be obtained as

$\displaystyle\left.\rho_{B}\right|_{k_{*}\sim\mathcal{H}}=\frac{(-\eta_{0})^{4\alpha-2}}{\left(2\pi\right)^{2}}\left[\,\left|C\right|^{2}\,\frac{k_{*}^{3+4\alpha+\frac{1}{2\alpha}}}{5+\frac{1}{2\alpha}}+\left|C_{2}\frac{\mathcal{F}^{-1}}{\pi}\Gamma\left(\frac{1}{2\alpha}\right)\right|^{2}\,\frac{(2\alpha-1)^{2}}{4\eta_{0}^{2}}\,\frac{k_{*}^{1+4\alpha-\frac{1}{2\alpha}}}{5-\frac{1}{2\alpha}}\right]$
(3.47)

Using the fact that at super-horizon scales, we can approximate
$\partial_{\eta}\sim\mathcal{H}$ (see, for instance, Ref. [106]), the total
electric energy density at horizon exit is given by:

$\displaystyle\left.\rho_{E}\right|_{k_{*}\sim\mathcal{H}}$
$\displaystyle=\frac{(-\eta_{0})^{4\alpha-2}}{\left(2\pi\right)^{2}}\left[\,\left|C\right|^{2}\,\frac{k_{*}^{3+4\alpha+\frac{1}{2\alpha}}}{3+\frac{1}{2\alpha}}+\left|C_{2}\frac{\mathcal{F}^{-1}}{\pi}\Gamma\left(\frac{1}{2\alpha}\right)\right|^{2}\,\frac{(2\alpha-1)^{2}}{4\eta_{0}^{2}}\,\frac{k_{*}^{1+4\alpha-\frac{1}{2\alpha}}}{3-\frac{1}{2\alpha}}\right].$
(3.48)

Thus, using the above two expressions, the total energy density at horizon
exit is given by

$\displaystyle\left.(\rho_{B}+\rho_{E})\right|_{k_{*}\sim\mathcal{H}}$
$\displaystyle=\rho_{T}^{(1)}+\rho_{T}^{(2)}$ (3.49)

where

$\displaystyle\rho_{T}^{(1)}$
$\displaystyle=\frac{(-\eta_{0})^{4\alpha-2}}{\left(2\pi\right)^{2}}\,\left|C\right|^{2}\frac{4\alpha(8\alpha+1)}{(10\alpha+1)(6\alpha+1)}k_{*}^{3+\alpha+\frac{1}{2\alpha}}$
(3.50) $\displaystyle\rho_{T}^{(2)}$
$\displaystyle=\frac{(-\eta_{0})^{4\alpha-4}}{\left(2\pi\right)^{2}}\left|C_{2}\right|^{2}\,\,\left|\frac{\mathcal{F}^{-1}}{\pi}\Gamma\left(\frac{1}{2\alpha}\right)\right|^{2}\frac{4\alpha(8\alpha-1)(2\alpha-1)^{2}}{(10\alpha-1)(6\alpha-1)}k_{*}^{1+4\alpha-\frac{1}{2\alpha}}$
(3.51)

Since the power-spectrum is red-titled, the power in the long wavelengths is
more than in the short wavelengths. Thus, there is a possibility that these
helical modes can backreact on the metric. Since the effect is cumulative, all
fluctuation modes contribute to the change in the background geometry.
Consequently, the backreaction effect can be large, even if the amplitude of
the fluctuation spectrum is small. To identify whether these modes lead to
backreaction on the metric, we define $R$, which is the ratio of the total
energy density of the fluctuations and background energy density during
inflation [178]:

$\displaystyle
R=\frac{\left.(\rho_{B}+\rho_{E})\right|_{k_{*}\sim\mathcal{H}}}{6M_{P}^{2}H^{2}}\,.$
(3.52)

Using $M_{P}=10^{19}\rm{GeV}$ and $H=10^{15}~{}\rm{GeV}$, the background
energy density during inflation ($M_{P}^{2}H^{2}$) is
$10^{68}~{}\rm{GeV}^{4}$. The table below contains estimates of the total
energy density at the horizon exit for different values of $\alpha$.

$\alpha$ |  $\rho^{(1)}_{T}$ (in $\rm{GeV}^{4}$) |  $\rho^{(2)}_{T}$ (in $\rm{GeV}^{4}$) | Total (in $\rm{GeV}^{4}$) |  $R$
---|---|---|---|---
$-\frac{1}{2}-\epsilon$ | $\sim 10^{64}$ | $\sim 10^{52}$ | $\sim 10^{64}$ | $\sim 10^{-4}$
$-\frac{3}{4}$ | $\sim 10^{62}$ | $\sim 10^{54}$ | $\sim 10^{62}$ | $\sim 10^{-6}$
$-1$ | $\sim 10^{61}$ | $\sim 10^{55}$ | $\sim 10^{61}$ | $\sim 10^{-7}$
$-3$ | $\sim 10^{59}$ | $\sim 10^{57}$ | $\sim 10^{59}$ | $\sim 10^{-9}$
Table 3.1: The total energy density at the exit of inflation for different
values of $\alpha$. To estimate $\rho^{(1)}_{T}$ and $\rho^{(2)}_{T}$, we
take: $\mathcal{H}\sim{\eta_{0}}^{-1}\sim
10^{14}~{}\rm{GeV}\sim{10^{52}}~{}\rm{Mpc}^{-1}$, and $M\sim
10^{17}~{}\rm{GeV}$.

We see from the above table that for varied values of $\alpha$, $R<<1$,
implying that the backreaction of the helical modes on the background metric
during inflation is negligible. The ratio $R$ is maximum ($10^{-4}$) for slow-
roll inflation. Thus, while our model produces helical magnetic fields with
more power at large length-scales, the backreaction of these on the metric is
negligible, and these modes do not stop inflation.

Although spectral helicity density can not be directly measured, for
completeness, we give the expression for the spectral helicity density:

$\displaystyle\frac{d\rho_{h}}{d\rm{ln}k}$
$\displaystyle=\frac{1}{2\pi^{2}}\frac{k^{4}}{a^{3}(\eta(\tau))}\left(\,\,\,\,\left|C(\tau)\right|^{2}\,k^{\frac{1}{2\alpha}}+\left|C_{2}\frac{\mathcal{F}(\tau)^{-1}}{\pi}\Gamma\left(\frac{1}{2\alpha}\right)\right|^{2}\,\left(k\,\tau^{4}\right)^{-\frac{1}{2\alpha}}\right).$
(3.53)

####  3.2.2 Comparison of our model with scalar-field
($f^{2}(\phi)\,F\tilde{F}$) coupling models

Often in the literature, the breaking of conformal invariance of the
electromagnetic action is through the non-minimal coupling of the
electromagnetic field ($f^{2}(\phi)\,F\tilde{F}$) with scalar field (possibly
inflaton). For a suitable choice of the coupling parameters, it has been shown
that a sufficient amount of large-scale magnetic fields can be generated [104,
163, 164, 105, 167, 168, 169].

In contrast, our model does not rely on the fine-tuning of the extra coupling
parameter to the electromagnetic field and depends on the background
quantities through the Riemann tensor. Due to this, the mode functions (4.18)
contain higher derivatives of ${\cal H}$ compared to scalar field coupled
models. In Appendix B.2, we use a naive approximation of our model for power-
law and slow-roll inflation, which effectively ignores higher-derivatives of
${\cal H}$. Under this approximation, we show that the model leads to a blue-
tilt spectrum. Thus, the presence of a higher-derivative of ${\cal H}$ leads
to the red-tilt. This is an important difference between our model compared to
scalar-field coupled models.

To further understand this, we define the overall coupling function in our
model by a dimensionless coupling function:

$\displaystyle I=\frac{R_{\mu\nu}\,^{\sigma\gamma}}{M^{2}}\qquad$ (3.54)

In the flat FRW line-element, Riemann tensor
$R_{\mu\nu}\,^{\sigma\gamma}\sim\frac{{a^{\prime}}^{2}}{a^{4}}$ and
$\sim\frac{a^{\prime\prime}}{a^{3}}$. Let us now compare our model with the
two specific forms of scalar-field coupled models [106, 105].

In our model, for power-law inflation, the coupling function $I$ is

$\displaystyle
I\sim\frac{{a^{\prime}}^{2}}{M^{2}a^{4}}~{}~{}\left(\mbox{or}~{}\frac{{a^{\prime\prime}}}{M^{2}a^{3}}\right)\propto\frac{1}{\eta^{2\beta+4}}\propto\eta^{\delta}\qquad(\delta>0)$
(3.55)

As mentioned above, for the scalar field coupled models, many authors have
used different forms of $f^{2}(\phi)$:

$f_{1}(\phi)\propto e^{\frac{\phi}{m_{P}}}~{};~{}f_{2}(\phi)\propto a^{2}$
(3.56)

Note that $f_{1}$ was used in Ref. [106] while $f_{2}$ was used in Ref. [105]
. In both the cases, the coupling function is of the form:

$f(\phi(\eta))\propto\frac{1}{\eta^{\sigma}}\qquad(\sigma>0)$ (3.57)

We want to make the following remarks regarding the two coupling forms (3.55)
and (3.57): First, the functional form of the coupling function in our case is
different compared to the scalar field coupling models. Since $\eta$ is large
at early times, the Riemann coupling term contributes significantly at early
times, and hence, the modes that leave the horizon at early times will have
large helicity. In the scalar field coupled models, the modes generated at
early times will not have significant helicity modes. In contrast, the modes
generated close to the end of inflation will have significant helicity. This
provides a qualitative understanding of the red-tilt power-spectrum in our
case. Second, since most of the helical modes are generated at early times,
unlike in Ref. [105], the generated helical fields are not sensitive to the
reheating dynamics. Thus, the helical modes generated evolve similar to the
scalar and tensor perturbations generated during inflation. Today’s observable
scales (in the CMB and LSS) span roughly three orders in the comoving
wavenumber $k$. The largest observable wavelength $\lambda_{\rm max}$,
associated with the wavenumber $k_{\rm max}$, corresponds to the horizon
radius. For a model with 60 e-foldings of inflation, the observable
cosmological wavelengths exit the Hubble radius around 30 e-foldings before
the end of inflation. This will also apply to the helical modes generated in
our model. Third, the total energy density of the helical fields in our model
is larger compared to the scalar-field coupled models. Specifically, the
energy density of the helical fields in our model is at least of an order of
magnitude larger than for the coupling function $f_{2}(\phi)$ in Ref. [105].
However, as shown in the previous subsection, our model is free from the
backreaction problem for a range of scale-factor during inflation.

####  3.2.3 Estimating the strength of the helical magnetic fields

As mentioned above, the model generates helical fields around $30$ e-folding
before the end of inflation. To estimate the current value of the helical
fields, we assume instantaneous reheating, and the Universe becomes radiation
dominated after inflation. Due to flux conservation, the magnetic energy
density will decay as ${1}/{a^{4}}$, i. e.:

$\rho_{B}(0)=\rho_{B}^{(f)}\left(\frac{a_{f}}{a_{0}}\right)^{4}$

where $a_{0}$ is the present day scale-factor, $\rho_{B}^{(f)}$ and $a_{f}$
refer to the magnetic energy density and the scale-factor at the end of
inflation, respectively. Using the entropy conservation i.e.,
$g\,T^{3}\,a^{3}=constant$ where $g$ refers to the effective relativistic
degrees of freedom and $T$ is the temperature of the relativistic fluid, we
get ${a_{0}}/{a_{f}}\approx
10^{30}\left({H_{f}}/{10^{-5}M_{\rm{Pl}}}\right)^{1/2}$ [81].

Using the fact that the relevant modes exited Hubble radius around 30
e-foldings of inflation, with energy density $\rho_{B}\approx
10^{64}\rm{GeV}^{4}$, the primordial helical fields at ${\rm GPc}$ scales is:

$\displaystyle B_{0}\approx 10^{-20}\rm{G}$ (3.58)

where we have used $1G=1.95\times 10^{-20}\rm{GeV}^{2}$ and
$H_{f}=10^{-5}M_{\rm{Pl}}$ is the Hubble parameter during inflation. Our model
predicts the following primordial helical fields that re-entered the horizon
at two different epochs:

$\left.B\right|_{50~{}{\rm MPc}}\sim 10^{-18}~{}G~{}(z\sim
20)~{};~{}\left.B\right|_{1~{}{\rm MPc}}\sim 10^{-14}~{}G~{}(z\sim 1000)\,.$

Thus, the model generates sufficient primordial helical magnetic fields at all
observable scales.

###  3.3 Conclusions and Discussions

We have proposed a viable scenario for the generation of helical magnetic
fields during inflation, which does not require coupling to the scalar field.
The generation of the helical fields is due to the coupling of the
electromagnetic fields with the Riemann tensor. To our knowledge, Riemann
tensor coupling has not been discussed in the literature to generate helical
fields.

The model has many key features: First, it does not require the coupling of
the electromagnetic field with the scalar field. Hence, there are no extra
degrees of freedom and will not lead to a strong-coupling problem. Second, the
conformal invariance is broken due to the coupling to the Riemann tensor.
Since the curvature is large in the early Universe, the coupling term will
introduce non-trivial corrections to the electromagnetic action. However, at
late-times, the new term will not contribute, and the theory is identical to
standard electrodynamics. Third, the power spectrum of the helical fields
generated has a slight red-tilt for slow-roll inflation. This is different
compared to the scalar field coupled models where the power-spectrum has a
blue-tilt. We have also identified the reason for this difference. Fourth, our
model is free from backreaction for a range of scale-factor during inflation.
This is different from other models where a specific form of coupling function
is chosen to avoid any back-reaction [105].

In this chapter, we did not discuss the generation of non-helical magnetic
fields. The generation of the non-helical magnetic field with Riemann coupling
has been discussed in the seminal paper by Turner and Widrow [97]. An analysis
including parity preserving term can be done straightforwardly, and we can
obtain total energy density of the non-helical ($\rho_{B}$) and helical energy
density ($\rho_{H}$). As shown recently, the two energy densities must satisfy
the realizability condition [179], i. e., $\rho_{H}\leq 2\,\xi_{M}\rho_{B}$,
where $\xi_{M}$ is the magnetic correlation length. Assuming that the non-
helical and helical power spectra are a power-law:

$P_{B}=A_{B}k^{n_{B}},\quad P_{H}=A_{H}k^{n_{H}}\,,$

for maximal helicity, it was shown that the helical magnetic fields must have
red-tilt. More specifically, for the WMAP nine-year data, using the cross
power-spectrum between the temperature and B-mode polarization they set $95\%$
confidence level upper limit on the helicity amplitude to be
$10\rm{nG}^{2}\,\rm{Gpc}$ for the helical spectral index $n_{H}=-1.9$ and for
a cosmological magnetic field with effective field strength of $3~{}\rm{nG}$
and $n_{B}=-2.9$. PLANCK 2015 data placed constraints on the strength for
causally generated magnetic fields with spectral index $n_{B}=2$ and fields
with almost scale-invariant spectrum with $n_{B}=-2.9$ are
$B_{1\rm{Mpc}}<.011~{}\rm{nG}$ and $B_{1\rm{Mpc}}<0.9~{}\rm{nG}$ at $95\%$
confidence level [76]. Thus, the PLANCK 2015 data also prefers $n_{H}$ to be
negative. With improved B-mode polarization measurements, helicity modes can
be better constrained and put our model’s prediction to test with the CMB
data. We hope to address this soon.

The perturbations equations (4.17) contain second-order derivatives of
$\mathcal{H}$. Since, ${\cal H}$, and ${\cal H}^{\prime\prime}$ are different
for inflation and bounce models [177], the helicity modes may provide
signatures to distinguish the two paradigms.

In this chapter, we have focussed on the modes which exit the horizon around
40-50 e-foldings and shown that they can potentially explain the large scale
magnetic fields observed in galaxies. In the next chapter 4, we will discuss
how the same mechanism can help us to understand the baryogenesis problem. We
will show that focusing on the modes which leave the horizon at around 5-10
e-foldings during inflation can lead to the creation of matter-antimatter
asymmetry at the beginning of radiation dominated era.

## Chapter 4 Helical magnetic field from Riemann coupling lead to baryogenesis

Understanding the physical processes in the very early Universe is a crucial
ingredient for deciphering the physics at energies that we cannot currently
probe in terrestrial experiments. While most observables have been washed away
by the thermal bath of the pre-recombination era and do not have observational
consequences, _three observables_ provide crucial information of the physics
at high-energies. These are the spectrum of energy density fluctuations [22,
23, 24, 21], excess of baryons over antibaryons (baryon asymmetry) [107, 108,
109, 110, 180, 181, 182], and coherent large-scale magnetic fields [61, 62,
80, 81, 78, 87].

As mentioned in Chapter 1, the inflationary paradigm provides an attractive
mechanism to generate the primordial density perturbations that lead to
anisotropies in the cosmic microwave background (CMB) and the formation of
large-scale structures [22, 23, 24, 21]. During inflation, the early Universe
underwent an accelerated expansion, stretching quantum fluctuations to super-
horizon scale density perturbations. Besides providing a causal mechanism to
density perturbations, inflation also solves the standard cosmological model’s
long-standing puzzles, such as the horizon, flatness, and monopole problems.

As mentioned in Chapter 1, the predictions of inflation are in good agreement
with the present-day observations of CMB anisotropies and polarization [183].
However, within the standard electrodynamics, inflation cannot provide a
mechanism to generate large-scale B fields. This is because in 4-dimensions
electromagnetic field is conformally invariant. Since FRW models are
conformally flat, the electromagnetic field vacuum in FRW is the same as the
Minkowski space-time. Hence, the standard electromagnetic fields generate
negligible magnetic fields. More importantly, even if the baryon asymmetry or
cosmological magnetic fields existed before the epoch of inflation, these
would have been diluted by a factor of $e^{-3N}$, where $N$ is the number of
e-foldings of inflation [114, 184, 119].

As mentioned in Sections (1.4,1.8), the present Universe is observed to
contain essentially only matter and no antimatter, except for the rare
antiparticles produced by cosmic rays. The asymmetry between baryons and
antibaryons, referred to as Baryon Asymmetry of the Universe (BAU), can be
expressed as [185, 183]

$\eta_{B}=\frac{n_{b}-n_{\bar{b}}}{n_{\gamma}}=\left\\{\begin{array}[]{r}{[5.8-6.6]\times
10^{-10}}~{}~{}\text{(from BBN)}\\\ (6.09\pm 0.06)\times
10^{-10}~{}~{}\text{(from CMB)}\end{array}\right.$ (4.1)

where $n_{b},n_{\bar{b}},n_{\gamma}$ refer to the density of baryons,
antibaryons and photons, respectively. Magnetic fields permeate the Universe.
Coherent magnetic fields in spiral galaxies and clusters of galaxies have a
magnitude of the order of $\mu$Gauss [61, 80, 81, 62]. There is also indirect
evidence of a lower limit of order $10^{-16}~{}$G for the magnetic field
contained in the voids between galaxies and clusters of galaxies [65].

As mentioned in Sections (1.4,1.8), the origin of primordial magnetic fields
and baryon asymmetry of the Universe are still unresolved issues and require
physics beyond the standard models of cosmology and particle physics. This
leads to the following questions: As the Universe cooled, from the early
Universe to today, what were the processes responsible for generating baryon
asymmetry and large-scale magnetic fields? Are these processes cosmological or
particle physics or both? Since both require physics beyond the standard
model, there is a _tantalizing possibility_ that the same new physics can
solve both. In this chapter, we consider such a possibility and show that the
mechanism that leads to primordial helical magnetic fields also leads to
baryogenesis at the beginning of the radiation-dominated epoch. Interestingly,
our mechanism _also requires_ stretching of the primordial helical magnetic
fields to super-horizon scales during inflation — the same mechanism that
leads to primordial density perturbations.

Before we discuss the model itself, it is necessary to understand the key
ingredients to generate baryon-asymmetry and magnetic fields and why the same
new physics can potentially solve both these problems [111, 113]. In 1967,
Sakharov listed three necessary conditions for creating the BAU [111, 107]:
(1) baryon number violation, (2) charge ($C$) and charge parity ($CP$)
violation, and (3) departure from thermal equilibrium. All three of the
Sakharov conditions are satisfied in the Standard Model; however, the
electroweak phase transition is not sufficiently strong in the first order
[107, 108, 109, 110]. The CP-violating effects are not sufficiently pronounced
to account for as large a BAU as we observe. As a result, there must have been
additional physics beyond the standard model to produce it. This physics could
have been operating anywhere between the weak scale and the GUT scale.
Corresponding to out-of-equilibrium conditions, the baryogenesis scenarios are
divided into two categories: (a) by the universe expansion itself or (b) by
fast phase transition and bubble nucleation. In particular, the latter
concerns the electroweak baryogenesis schemes, while the former is typical for
a GUT type baryogenesis or leptogenesis [107, 108, 109, 110].

More than two decades ago, Davidson pointed out an interesting relation
between the primordial magnetic field and Sakharov’s conditions [113]. She
argued that the presence of background magnetic fields in the early Universe
could lead to the breaking of $C,CP,SO(3)$ symmetries and thermal equilibrium.
Specifically, she argued that the presence of the magnetic fields leads to the
following three conditions: (1) There should be some moderately out-of-
thermal-equilibrium dynamics because in equilibrium, the photon distribution
is thermal, and there are no particle currents to sustain a "long-range"
field, (2) Since B is odd under $C$ and $CP$, the presence of magnetic field
will lead to $CP$ violation, (3) Since the magnetic field is a vector
quantity, it chooses a particular direction hence breaks the isotropy
(rotational invariance). Thus, Davidson provided a possible link between the
presence of magnetic fields to the conditions required for baryogenesis [113].

Davidson’s conditions are necessary _but not_ sufficient. One key missing
ingredient, as shown in this chapter, is the requirement of _primordial
helical magnetic fields_ (details in Sec. 4.1). Primordial helical magnetic
fields are generated by the terms that break conformal invariance and parity
symmetry [104, 163, 164, 105, 167, 168, 169, 120]. If we could measure them,
primordial helical magnetic fields provide evidence of CP violation in the
early Universe. Interestingly, the presence of primordial helical fields leads
to non-zero Chern-Simons number [114, 116, 117] and, eventually, the change in
the Fermion number.

In chapter 3, we discussed a simple model of inflationary magnetogenesis that
couples the electromagnetic fields with the Riemann tensor [120]. We have seen
that the model leads to a primordial helical magnetic field where one helical
mode is enhanced while the other mode is suppressed. The model has two key
advantages over other models [164, 105, 167, 168, 169]: First, it does not
require the coupling of the electromagnetic field with any scalar field.
Hence, unlike Ratra model [98, 186, 170], there is no strong-coupling problem
caused by the extra degrees of freedom. Second, the model is free from
backreaction for generic slow-roll inflation models [120]. In Ref. [105],
authors have shown the strong-coupling problem in Ratra model [98] can be
avoided by choosing a particular coupling function. In chapter 3, we used the
general effective field theory of gravity coupled to the Standard Model of
particle physics framework to obtain leading order gravity terms that couple
to the standard model Bosons [161]. As we have done in the previous chapter,
we limit to mass dimension 6-operators coupling to the gauge field Lagrangian,
specifically, to the electromagnetic field.

Like in Chapter 3, in this chapter, we limit to mass dimension 6-operators
coupling to the gauge field, specifically, to the electromagnetic field. We
show that the generation of primordial helical magnetic fields in the previous
chapter 3 leads to baryogenesis. Since the model produces helical fields over
large length scales, we show that the Chern-Simons (CS) number density is non-
zero (details in Sec. 4.1). Considering that the model generates primordial
helical modes at all length scales, we focus on the last ten e-foldings of
inflation. As we have discussed in chapter 1, the modes that leave the Hubble
radius during the last 10 e-foldings of inflation will reenter the Universe
just after the reheating (see Fig. 4.1); these primordial helical modes will
lead to baryogenesis just at the beginning of the radiation-dominated epoch.
Furthermore, we show that the BAU is independent of inflation models and
depends _only on_ the energy scale at the exit of inflation and reheating
temperature.

Figure 4.1: Modes exiting the horizon around last 10 e-foldings (gray shaded
circle) will re-enter the Universe early and lead to baryogenesis.

In Sec. (4.1), we discuss the relation between primordial helical magnetic
fields and baryogenesis, in particular, the chiral anomaly in the presence of
the magnetic field, and obtain the expression for Chern-Simon number density.
In Sec. (4.2), we discuss the generation of primordial helical modes and show
that primordial helical modes lead to a non-zero CS number density. Then we
evaluate the baryon asymmetry parameter in Sec.(4.3). Sec. (4.4) contains the
implications of the results. Appendix C contains the details of the
calculations in this chapter. The results reported in this chapter are based
on Ref. [121].

###  4.1 Conditions on baryogenesis in the presence of primordial magnetic
field

As we mentioned in the introduction, Davidson’s conditions are necessary but
not sufficient. One key missing ingredient is the requirement of _primordial
helical magnetic fields_. In this section, we briefly discuss this.

In the very early Universe, just after the exit of inflation, the energy scale
of the Universe was close to $10^{14}~{}{\rm GeV}$. All particles, including
Fermions, are highly relativistic and can be treated as massless. Although the
massless Dirac equation is invariant under chiral transformations in the
classical theory, the chiral symmetry is broken due to quantum mechanical
effects in the presence of the external electromagnetic fields. This
phenomenon, known as the quantum axial anomaly, affects the transport
properties of the chiral medium, leading to experimentally accessible
signatures such as the chiral magnetic effect [187] and the chiral separation
effect [188].

In the early Universe, the generation of the non-zero primordial helical
magnetic fields leads to a chiral anomaly resulting from the imbalance between
left and right-handed fermions. In the presence of an electromagnetic field in
curved space-time, the chiral anomaly is given by the following equation [189,
190]:

$\displaystyle\nabla_{\mu}J_{A}^{\mu}=-\frac{1}{384\pi^{2}}\epsilon^{\mu\nu\rho\sigma}R_{\mu\nu\alpha\beta}R^{\alpha\beta}\,_{\rho\sigma}+\frac{e^{2}}{16\pi^{2}}\epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}$
(4.2)

where $J^{\mu}_{A}$ is the chiral current, $R_{\rho\sigma}\,^{\alpha\beta}$ is
the Riemann tensor and $A_{\mu}$ is the four-vector potential of the
electromagnetic field, $F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$.
$\epsilon^{\mu\nu\rho\sigma}=\frac{1}{\sqrt{-g}}\,\eta^{\mu\nu\rho\sigma}$ is
a fully antisymmetric tensor, $\eta^{\mu\nu\rho\sigma}$ is Levi-Civita symbol
whose values are $\pm 1$ and we set $\eta^{0123}=1=-\eta_{0123}$. It is easy
to see from the above equation that the anomaly contribution from the
electromagnetic field and the gravity act independently and, for most parts,
can be treated independently.

In the case of flat FRW background in conformal time ($\eta$):

$\displaystyle ds^{2}=a^{2}(\eta)\,(d\eta^{2}-\delta_{ij}dx^{i}dx^{j})$ (4.3)

the contribution of the first term in the RHS of Eq. (4.2) vanishes, i. e.,

$\displaystyle\epsilon^{\mu\nu\rho\sigma}R_{\mu\nu\alpha\beta}R^{\alpha\beta}\,_{\rho\sigma}=0\,.$
(4.4)

It can be shown that even at the first-order, the gravitational contribution
vanishes, and the non-zero contribution arises only at second order [191]. Due
to the presence of the antisymmetric tensor, the gravitational fluctuations
lead to gravitational birefringence and can lead to net chiral current.

In the flat FRW background, the second term in the RHS of Eq.(4.2) is given
by:

$\displaystyle\frac{e^{2}}{16\pi^{2}}\epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}=\frac{e^{2}}{4a^{4}}\epsilon_{ijk}\partial_{j}A_{k}\,\partial_{0}A_{i}\,.$
(4.5)

In the presence of the magnetic field, this term is non-zero and hence leads
to a net chiral current. Thus, if we consider only up to the first-order in
perturbations, _only_ the second term in the RHS of Eq. (4.2) contributes and
the chiral anomaly equation reduces to:

$\displaystyle\partial_{\mu}\left(\sqrt{-g}J_{A}^{\mu}\right)=\frac{e^{2}}{16\pi^{2}}\eta^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}\,,$
(4.6)

where we have used

$\nabla_{\mu}J^{\mu}_{A}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}J^{\mu}_{A}\right),\quad\epsilon^{\mu\nu\alpha\beta}=\frac{1}{\sqrt{-g}}\eta^{\mu\nu\alpha\beta}\,.$

Note that during inflation, LHS in Eq. (4.6) is zero, and due to the
exponential expansion, standard model particles are diluted. However, if we
can generate non-zero primordial helical fields during inflation, then these
non-zero primordial helical fields can lead to chiral current at the
radiation-dominated epoch (or during reheating when the standard model
particles are created). To see this, we rewrite Eq. (4.6) using
$\eta^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}=4\partial_{\mu}\left(\eta^{\mu\nu\alpha\beta}A_{\nu}\partial_{\alpha}A_{\beta}\right)$,
i.e.,

$\displaystyle\partial_{\mu}\left(\sqrt{-g}J^{\mu}_{A}\right)=\frac{e^{2}}{4\pi^{2}}\partial_{\mu}\left(\eta^{\mu\nu\alpha\beta}A_{\nu}\partial_{\alpha}A_{\beta}\right)=\frac{e^{2}}{4\pi^{2}}\partial_{\mu}\left(\sqrt{-g}K^{\mu}\right)$
(4.7)

where

$K^{\mu}=\frac{\eta^{\mu\nu\alpha\beta}}{\sqrt{-g}}A_{\nu}\partial_{\alpha}A_{\beta}$

is the topological current. For FRW background, the components are given by

$\displaystyle
K^{0}=a^{-4}(\eta)\,\epsilon_{ijk}A_{i}\partial_{j}A_{k}\qquad\text{and}\qquad
K^{i}=a^{-4}(\eta)\,\epsilon_{ijk}A_{j}\partial_{0}A_{k}.$ (4.8)

Solving Eq. (4.7), we get,

$J^{\mu}_{A}=\frac{e^{2}}{4\pi^{2}}K^{\mu}\,.$

Thus, the net baryon number density, $n_{B}=n_{b}-n_{\bar{b}}=a(\eta)\langle
0|J^{0}_{A}|0\rangle$ is related to Chern-Simon number density $n_{CS}=\langle
0|K^{0}|0\rangle$ as [190],

$\displaystyle n_{B}\equiv\frac{e^{2}}{4\pi^{2}}a(\eta)n_{CS}.$ (4.9)

Note that $n_{CS}=0$ at the start of inflation, and due to the absence of
standard model particles $n_{B}=0$ during inflation. Using the expression for
$K^{0}$, we can write the Chern-Simon number density as

$\displaystyle n_{CS}=\frac{1}{a^{4}}\epsilon_{ijk}\langle
0|A_{i}\,\partial_{j}A_{k}|0\rangle=\frac{1}{a^{4}}\int_{\mu}^{\Lambda}\frac{dk}{k}\frac{k^{4}}{2\pi^{2}}\left(|A_{+}|^{2}-|A_{-}|^{2}\right)\,,$
(4.10)

where $\Lambda$, and $\mu$ set the possible energy range (or epoch) during
which baryon asymmetry is generated after inflation, and $A_{\pm}$ refer to
the positive and negative helicity modes of the electromagnetic field. The
above expression is key in illuminating a useful relation between primordial
helical magnetic fields generated during inflation and baryogenesis: First, we
see that the contribution to $n_{CS}$ is from all the modes that reenter the
horizon at the beginning of the radiation-dominated epoch. Thus, the value of
$n_{CS}$ depends on the upper cut-off $\Lambda$. Second, the expression
corresponds to the total Chern-Simons number density generated from the modes
in the energy range $[\mu,\Lambda]$ — when these helical modes re-enter during
the radiation-dominated epoch. The helicity modes $A_{+}$ and $A_{-}$ are
generated during inflation, and ${a^{-4}(\eta)}$ is the dilution due to the
expansion of the Universe during this epoch. Finally, $n_{CS}$ vanishes if the
primordial magnetic fields are non-helical, i. e. $|A_{+}|=|A_{-}|$. Hence, as
mentioned at the beginning of this section, the generation of non-helical
magnetic fields will not lead to baryogenesis. Thus, the key missing
ingredient of Davidson’s argument is the requirement of primordial helical
magnetic fields.

In the following two sections, we explicitly evaluate the Chern-Simons number
for our model and show that it is not sensitive to inflationary and reheating
dynamics.

###  4.2 The model and the primordial helical fields

Like in chapter 3, we consider the following action [120] :

$\displaystyle S=S_{\rm{Grav}}+S_{\phi}+S_{\rm{EM}}+S_{\rm CB}$ (4.11)

where $S_{\rm{Grav}}$ is the Einstein-Hilbert action

$\displaystyle S_{\rm Grav}=-\frac{M_{\rm P}^{2}}{2}\int
d^{4}x\sqrt{-g}\,R\,,$ (4.12)

and $S_{\phi}$ is the action for the minimally coupled, self-interacting
canonical scalar field:

$\displaystyle S_{\phi}=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right].$
(4.13)

$S_{\rm{EM}},S_{\rm CB}$ refer to the standard electromagnetic (EM) and
conformal breaking part of the electromagnetic terms, respectively, which are
given by:

$\displaystyle S_{\rm{EM}}$ $\displaystyle=-\frac{1}{4}\int
d^{4}x\,\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu},\hskip 14.22636pt$ (4.14)
$\displaystyle S_{\rm{CB}}$ $\displaystyle=-\frac{1}{M^{2}}\,\int
d^{4}x\,\sqrt{-g}\,R_{\rho\sigma}\,^{\alpha\beta}F_{\alpha\beta}\,\tilde{F}^{\rho\sigma}=-\frac{1}{M^{2}}\,\int
d^{4}x\,\sqrt{-g}\,\tilde{R}^{\mu\nu\alpha\beta}F_{\alpha\beta}\,F_{\mu\nu}\,,$
(4.15)

where
$\tilde{R}^{\mu\nu\alpha\beta}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}R_{\rho\sigma}\,^{\alpha\beta}$
is the dual of Riemann tensor and
$\tilde{F}^{\rho\sigma}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}$ is
the dual of $F_{\mu\nu}$. The standard electromagnetic action $S_{\rm{EM}}$ is
conformally invariant; however, the presence of Riemann curvature in $S_{\rm
CB}$ breaks the conformal invariance. $M$ is the energy scale, which sets the
scale for the breaking of conformal invariance. Note that the signs of
$S_{\rm{EM}}$ and $S_{\rm{CB}}$ are chosen with respect to the positive
electromagnetic energy density.

In Ref. [161], the authors systematically showed that the first gravity
operators appear at mass dimension 6 in the series expansion of the coupling
between gravity and the standard model of particle physics. These operators
only couple to the standard model Bosons. They also showed that (i) no new
gravity operators appear at mass dimension 7, (ii) in mass dimension 8, the
standard model Fermions appear, and (iii) coupling between the scalar (Higgs)
field and the standard model gauge Bosons appear only at mass dimension 8.
Since mass dimension 8 operators are highly suppressed, like in Ref. [120], we
limit ourselves to mass dimension 6 operators. Due to Riemann coupling, $M$
appears as a time-dependent coupling in the FRW background i.e., $1/M_{\rm
eff}\sim H/M$. At the current epoch where $H_{0}\approx 10^{-42}\rm{GeV}$ and
assuming the parameter $M\approx 10^{17}\rm{GeV}$, we obtain ${H_{0}}/{M}\sim
10^{-59}$. Therefore, the coupling (Riemann tensor) is tiny and the non-
minimal coupling term in the electromagnetic action will have significant
contribution only in the early universe. We also would like to point that the
coupling term ($S_{\rm CB}$) is tiny near the Schwarzschild radius of a solar
mass black-hole (for details, see appendix C.4).

We assume that the scalar field ($\phi$) dominates the energy density in the
during inflation and leads to $60\,-\,70$ e-foldings of inflation with $H_{\rm
Inf}\sim 10^{14}{\rm GeV}$. Specifically, we consider power-law inflation in
which the scale factor (in conformal time) is [176]:

$\displaystyle a(\eta)=\left(-\frac{\eta}{\eta_{0}}\right)^{(\beta+1)}$ (4.16)

where, the constant $\eta_{0}$ denotes the scale of inflation and
$\beta\leq-2$. $\beta=-2$ corresponds to exact de Sitter. During inflation,
$\eta\in(-\infty,0)$. For slow-roll inflation $\beta\approx-2-\epsilon$ and
$\mathcal{H}\equiv a^{\prime}/{a}\approx-(1+\epsilon)/{\eta}$, where
$\mathcal{H}$ is the Hubble parameter in conformal time and $\epsilon$ is the
slow roll parameter. For our discussion below, we also assume that
$10^{-3}\leq(H_{\rm Inf}/M)\leq 1$ [171, 172, 173, 174].

Equation of motion of the gauge field can be obtained by varying the action
(4.11) with respect to $A^{\mu}$. In the Coulomb gauge
($A^{0}=0,\partial_{i}A^{i}=0$), we have:

$\displaystyle
A_{i}^{\prime\prime}+\frac{4\,\epsilon_{ijl}}{M^{2}}\,\left(\frac{a^{\prime\prime\prime}}{a^{3}}-3\frac{a^{\prime\prime}a^{\prime}}{a^{4}}\right)\partial_{j}A_{l}-\partial_{j}\partial_{j}A_{i}=0$
(4.17)

where $\epsilon_{ijl}$ is the Levi-Civita symbol in the 3-D Euclidean space.
The above equation is different from other models in the literature and leads
to distinct evolution of the magnetic field fluctuations in comparison to non-
minimally coupled scalar field models [120]. In the helicity basis, the above
equation reduces to (see appendix C.1):

$\displaystyle
A_{h}^{\prime\prime}+\left[k^{2}-\frac{4kh}{M^{2}}\,\left(\frac{a^{\prime\prime\prime}}{a^{3}}-3\frac{a^{\prime\prime}a^{\prime}}{a^{4}}\right)\right]A_{h}=0\,.$
(4.18)

For the two helicity states ($h=\pm$), the above expression leads to two
different evolution equations [cf. Eqs. (C.17a, C.17b)]. From Eq. (4.10) we
see that to obtain appreciable value of Chern-Simons number ($n_{CS}$), the
difference between the two helicity states should be non-zero, and it is
maximum if one helicity mode is enhanced compared to other.

In the previous chapter 3, we showed that for a range of parameters of
interest, negative helicity mode decays while the positive helicity mode is
enhanced [120]. Hence, negative helicity mode ($A_{-}$) will have negligible
contribution and can be set to zero, i. e., $|A_{-}|=0$. Using the series
expansion of the Bessel functions, in the leading order, the positive helicity
mode takes the following form (C.17a):

$\displaystyle A_{+}(\tau,k)$
$\displaystyle=C\,k^{\frac{1}{4\alpha}}-C_{2}\frac{\mathcal{F}^{-1}}{\pi}\Gamma\left(\frac{1}{2\alpha}\right)\,k^{-\frac{1}{4\alpha}}\tau^{-\frac{1}{\alpha}}$
(4.19)

where,

$\displaystyle|C|\approx\varsigma^{-1}|C_{2}|\approx\frac{M^{3/2}\eta_{0}}{\sqrt[4]{\eta_{end}10^{45}GeV^{3}}},~{}~{}\mathcal{F}\approx|\varsigma|^{-1}\approx\sqrt{M^{2}\eta_{0}},~{}~{}\alpha=-\frac{1}{2}-\epsilon$
(4.20)

For details, see Appendix (C.2).

Our model generates primordial magnetic fields through the non-minimal
coupling of the electromagnetic field. The model requires inflation. Inflation
generates density perturbations at all scales and provides a causal mechanism
to generate the structure formation. Similarly, our model generates magnetic
fields at all length scales, including the current Horizon radius [61, 80, 81,
62, 78]. This has to be contrasted from the models where the magnetic field is
generated during recombination. In these models, the coherence scale of the
generated fields cannot exceed the size of the horizon radius at that time.

In Appendix C.2, we have plotted the power spectrum of the present-day helical
magnetic field ($B_{0}$) as a function of $k$. Assuming $M=10^{17}GeV$, our
model predicts the primordial helical magnetic fields of strength
$10^{-20}\rm{G}$ on Gpc scales at the current epoch. From C.1 we can see that
our model predicts the present-day helical magnetic field of strength
$10^{-15}G$ on Mpc scales. The primordial fields generated from our model are
within the upper bounds on the strength of the seed magnetic fields needed to
explain the current galactic magnetic fields [192]. These primordial fields
are amplified by the dynamo mechanism and can lead to the observed magnetic
fields; hence our model requires the dynamo mechanism.

###  4.3 Baryon Asymmetry of the Universe

In this section, we compute the baryon asymmetry parameter due to the
primordial helical magnetic fields. Specifically, we compute it for the
maximum helicity modes — one mode is enhanced compared to the other.
Substituting Eq (4.19) in Eq. (4.10), we obtain

$\displaystyle
n_{CS}=\frac{1}{2\pi^{2}\,a^{4}(\eta)}\int^{\Lambda}_{\mu}dk\left(\left|C\right|^{2}\,k^{3+\frac{1}{2\alpha}}+\left|C_{2}\frac{\mathcal{F}^{-1}}{\pi}\Gamma\left(\frac{1}{2\alpha}\right)\right|^{2}\,k^{3-\frac{1}{2\alpha}}\tau^{-\frac{2}{\alpha}}\right).$
(4.21)

Integrating the above expression, we get

$\displaystyle
n_{CS}=\frac{1}{2\pi^{2}\,a^{4}(\eta)}\left[\left.\left|C\right|^{2}\,\frac{k^{4+\frac{1}{2\alpha}}}{4+\frac{1}{2\alpha}}\right|^{\Lambda}_{\mu}+\left.\left|C_{2}\frac{\mathcal{F}^{-1}}{\pi}\Gamma\left(\frac{1}{2\alpha}\right)\right|^{2}\,\frac{k^{4-\frac{1}{2\alpha}}}{4-\frac{1}{2\alpha}}\tau^{-\frac{2}{\alpha}}\right|^{\Lambda}_{\mu}\,\,\right].$
(4.22)

We want to make the following remarks regarding the above expression: First,
the BAU is generated similarly to the inflationary mechanism of the generation
of density perturbation. During inflation, the primordial helical magnetic
field fluctuations are stretched exponentially and exit the horizon. The modes
that reenter during the radiation-dominated epoch are responsible for the
generation of baryon asymmetry. Second, the generation of baryon asymmetry
does not strongly depend on the reheating dynamics since only the modes that
reenter the Hubble radius during the radiation-dominated epoch are relevant.

Assuming a de-Sitter (or approximately de-Sitter) Universe, from Eq. (4.20),
we have $\tau^{-\frac{2}{\alpha}}=a^{-2}(\eta)$. Substituting this in the Eq.
(4.22), we see that the the second term in the RHS decays faster compared to
the first term by $a^{-2}(\eta)$. Hence, we can neglect the second term.
Substituting the resulting form of $n_{CS}$ in Eq. (4.9) leads to:

$\displaystyle
n_{B}=\frac{e^{2}}{4\pi^{2}}\frac{1}{2\pi^{2}\,a^{3}(\eta)}\left.\left|C\right|^{2}\,\frac{k^{4+\frac{1}{2\alpha}}}{4+\frac{1}{2\alpha}}\right|^{\Lambda}_{\mu}.$
(4.23)

To obtain the ranges of $\Lambda$ and $\mu$, we need to know the modes exited
during inflation. For the density perturbations, the largest scales observed
in the CMB are produced around 40 - 60 e-foldings before the end of inflation
[193]. This is because the adiabatic quantum fluctuations responsible for the
density perturbations reenter the Hubble radius around $z\sim 1500$. Hence, in
Ref. [120], the current authors only looked at primordial helical fields
generated around 40 - 60 e-foldings before the end of inflation. However, in
this case, we will concentrate on the primordial helical fields that renter
the horizon very early (at the beginning of the radiation-dominated epoch) to
generate the required BAU. This means that the modes that left the horizon
around the last 5 to 10 e-foldings of inflation are only relevant. Since these
modes have already left the Hubble radius during inflation, the reheating
dynamics do not alter these primordial helical modes. Hence, the model is
insensitive to the reheating dynamics.

Our focus now shifts to explicitly evaluating BAU for our model. First step is
to evaluate the dilution factor $a^{-3}$ in Eq. (4.23). To do this, we define
$a_{\Lambda}$ (and $a_{\mu}$) as the scale factor at the time when the maximal
helicity mode with energy $\Lambda$ (and $\mu$) left the Hubble radius during
inflation. Assuming an instant reheating, and following the calculations given
in Appendix (C.3), we have $a_{\mu}=10^{6}a_{\Lambda}$. Taking into account
that these modes exited the Hubble radius during inflation in the last 5
e-foldings, the the dilution factor [prefactor in Eq. (4.23)] becomes
$a^{-3}\sim 10^{-24}$.

The second step is to obtain the constant $C$. As discussed in previous
section, for slow-roll inflation, $|C|$ is given by Eq. (4.20). Thus, Eq.
(4.23) reduces to:

$\displaystyle n_{B}\approx\quad\frac{10^{-24}\cdot|C|^{2}\cdot
e^{2}}{24\pi^{4}}\left(\Lambda^{3}-\mu^{3}\right)\,.$ (4.24)

Third step is to compare the theoretically derived quantity ($n_{B}$) with
observations Eq. (4.1). However, $n_{\gamma}$ is not constant in the early
Universe (since the photon chemical potential is zero) and is approximately
constant only after the last scattering surface. Since entropy density per
comoving volume is conserved, the quantity $n_{B}/s$ is better suited for
theoretical calculations [22]. Assuming that there was no significant entropy
production after reheating phase, entropy density in the radiation-dominated
epoch is:

$s\simeq\frac{2\pi^{2}}{45}g\,T^{3}_{\rm{RH}}\,,$ (4.25)

where $T_{\rm{RH}}$ is the reheating temperature and the effective
relativistic degrees of freedom $g\sim 100$ at reheating. From Eqs. (4.24,
4.25), we can define the following dimensionless BAU parameter:

$\displaystyle\eta_{B}=\frac{n_{B}}{s}\approx 10^{-24}\frac{|C|^{2}\cdot
e^{2}}{24\pi^{4}}\left(\Lambda^{3}-\mu^{3}\right)\frac{45}{2\pi^{2}gT_{RH}^{3}}\approx
10^{-29}|C|^{2}\frac{\Lambda^{3}}{T^{3}_{\rm{RH}}}$ (4.26)

where in the last expression we have neglected $\mu^{3}$ i.e.,
$\Lambda^{3}-\mu^{3}\approx\Lambda^{3}$. Appendix (C.3) contains plots for
different values of $\Lambda$ and $\mu$. From these plots, we infer that the
results do not strongly depend on the exact value of $\mu$.

Finally, substituting the value of $|C|^{2}$ (from Eq. (4.20) and using the
values in Appendix C.2) in Eq. (4.26), we obtain:

$\displaystyle\eta_{B}\approx\frac{10^{-29}\cdot\eta_{0}^{2}}{\sqrt{\eta_{end}\cdot
10^{45}GeV^{3}}}\frac{M^{3}\Lambda^{3}}{T^{3}_{\rm{RH}}}\approx
10^{-2}\left(\frac{M}{M_{P}}\right)^{3}\left(\frac{\Lambda}{T_{\rm{RH}}}\right)^{3}$
(4.27)

This is one of the crucial expressions in this chapter regarding which we
would like to stress the following: First, the BAU parameter depends on three
quantities — $M$ (the conformal invariance breaking scale), $T_{\rm{RH}}$
(reheating temperature scale) and $\Lambda$ (the largest helical mode that
catalyses baryogenesis). Second, the BAU parameter is inversely proportional
to the reheating temperature. This behavior is different from the results of
Ref. [191, 190, 119, 114]. In some of these models, BAU is linearly dependent
on the reheating temperature. The difference in the relationship is because
the detailed reheating dynamics is not required, only the information about
the entropy production is required in our model. In other models, the exact
detailed reheating dynamics is required, which is avoided in our approach.
Third, the BAU parameter is linearly proportional to $M$ and $\Lambda$. For
smaller $M$, the contribution of the conformal breaking term (4.15) will be
much larger, and hence, more primordial helical fields are produced during
inflation. However, for the same reheating temperature, $\Lambda$ has to be
larger to produce the same amount of BAU. Fourth, to get a better
understanding of the dependence of BAU on various parameters, we use the
following parametrization:

$\displaystyle\eta_{B}=n\times 10^{-10},\quad M=m\times
10^{14}GeV,\quad\Lambda=\delta\times 10^{12}GeV,\quad T_{RH}=\gamma\times
10^{12}GeV$ (4.28)

where $n,m,\delta,\gamma$ are dimensionless parameters. The maximum reheating
corresponds to the inflation scale [193]. With supersymmetry, the requirement
that not too many gravitinos are produced after inflation provides a stringent
constraint on the reheating temperature, $T_{\rm RH}\sim
10^{10}-10^{11}~{}$GeV [194, 195]. Hence, we consider the range of $\gamma$ to
be $\\{10^{-2},1000\\}$. Since the value of $M$ should be between the GUT and
Planck scale, we consider the range of $m$ to be $\\{1,1000\\}$. We assume
that the modes that reenter during radiation epoch is around $10^{12}~{}$GeV.
Hence, we consider the range of $\delta$ to be $\\{1,100\\}$. Using the above
parametrization in Eq. (4.27), we get:

$\displaystyle\frac{m^{3}\times\delta^{3}}{\gamma^{3}}\approx n\,10^{7}.$
(4.29)

Figure 4.2: Plot of the rescaled reheating temperature $T_{RH}$ with the
rescaled conformal symmetry breaking parameter $M$, for different values of
$n$. Here, we have set $\Lambda=10^{14}GeV,\mu=10^{10}GeV$.

B.1 and 4.3 contain the plots of $\gamma$ versus $m$ for different values of
$n$ and fixed $\delta$. In Appendix (C.3) we have plotted the same for other
values of $\delta$. From these plots, we deduce the following: First, for a
range of values of $\gamma,$ $\delta$, and $m$, BAU can have values between
$10^{-10}$ to $10^{-9}$. Thus, the model can lead to the observed amount of
baryon asymmetry of the Universe consistent with the Planck data [183].
Second, the model does not depend on the nature of the reheating dynamics. As
can be seen from the plots, for a range of values of $m,\delta$, the model can
lead to BAU for a range of reheating temperatures. This has to be contrasted
with other models in the literature [119, 114] which requires detailed
knowledge of the reheating phase of the Universe. Third, the unknown parameter
in the model is $M$. In Ref. [120], we showed that for the model to be
consistent with the lower limit of $10^{-16}$ Gauss magnetic fields in the
voids [65], then $M\sim 10^{17}GeV$. The current analysis shows that $M\sim
10^{17}GeV$ is consistent with baryogenesis. Thus, the model is _tantalizingly
close_ to solving baryogenesis and magnetogenesis using the same causal
mechanism that solves the origin of density perturbations.

Figure 4.3: Plots showing the behaviour of reheating temperature $T_{RH}$
(vertical axis) with parameter $M$ (horizontal axis), for lower energy scales
of $\Lambda$ and $\mu$.

###  4.4 Conclusions and Discussions

In this chapter, we have proposed a viable baryogenesis scenario in the early
Universe that does not require any extension to the Standard Model of particle
physics. The crucial ingredient is the generation of primordial helical
magnetic fields due to Riemann coupling. The advantage of the primordial
helical fields is that the non-zero helicity suggests a non-zero contribution
in the CP violation term. An interesting feature of our model is the
stretching of the primordial helical magnetic fields to super-horizon scales
during inflation — the same mechanism that leads to primordial density
perturbations. While the helical modes generated around 40 - 60 e-foldings
before the end of inflation lead to the observed large-scale magnetic fields,
the helical modes that renter the horizon very early (at the beginning of the
radiation-dominated epoch) lead to the baryon asymmetry. Thus, our mechanism
provides _possible testable evidence_ for the entire inflationary epoch.

More than two decades ago, Davidson pointed out an interesting relation
between the primordial magnetic field and Sakharov’s conditions [113]. In this
chapter, we have explicitly shown that Davidson’s conditions are necessary
_but not_ sufficient. The key missing ingredient is the requirement of
_primordial helical magnetic fields_. While the helical and non-helical fields
break the isotropy and lead to CP violation, only the modes with maximal
helicity contribute significantly to the Chern-Simon number density. We have
shown that the BAU parameter predicted by our model is independent of any
specific inflation model and reheating dynamics; however, it depends on the
scale at which inflation ends and reheating temperature.

The BAU parameter (4.27) obtained in our model is inversely proportional to
reheating temperature. Assuming the exit of inflation at $10^{14}$ GeV, for
the observed amount of baryon asymmetry $\eta_{B}\sim 10^{-10}$, we obtained
that the reheating temperature should be in the range $10^{12}-10^{14}$ GeV,
which is consistent with the constraints on the reheating temperature [193,
194, 195]. This means that our model _does not_ prefer a very low-energy
reheating temperature [195].

In the literature, various mechanisms have been discussed to solve the BAU
problem using the primordial helical magnetic fields [190, 119, 114, 116,
184]. In Ref. [114], the authors obtained the required BAU by assuming the
presence of helical magnetic fields of present-day strength
$10^{-14}G<B_{0}<10^{-12}G$ and coherence length $1\rm{pc}<\lambda<1\rm{Mpc}$,
and taking into account of the MHD effects. In Ref. [119], authors studied the
generation of a primordial magnetic field in conjunction with the BAU
generation through leptogenesis; however, the predicted value of the present-
day coherence length of such magnetic fields is very small$\sim 10$ pc.

In Refs. [116, 184], the authors consider pseudoscalar inflation (axion
inflation) model with a dimension five couplings. In these models, the authors
assumed the scale of the baryogenesis to be electroweak scale, and they
obtained the required BAU assuming the scale of inflation to be
$10^{10}\rm{GeV}$ — $10^{12}\rm{GeV}$ [184, 116]. In Ref. [190], the authors
considered the extension of the Standard Model with anomalous gauge symmetry.
They obtained the required BAU for $H_{Inf}\sim 10^{14}\rm{GeV}$ and reheating
temperate at $10^{16}\rm{GeV}$. In Ref. [196], the authors argued that to
generate the observed baryon asymmetry, some asymmetry in the initial
conditions of either $\bf{B}$ or scalar field $\phi$ is required, which can be
induced from temperature-dependent potential or asymmetry in quantum
fluctuations. Our model is robust to inflationary/reheating dynamics and uses
the same success of inflationary perturbations to generate BAU. Thus, our
model is _tantalizingly close_ to solving baryogenesis and magnetogenesis
using the same causal mechanism that solves the origin of density
perturbations.

In this chapter, we did not consider the gravity contribution to the chiral
anomaly equation. In Ref. [191], the authors considered the phenomenon of
gravitational birefringence to show that the gravitational fluctuations
generated during inflation can give the Universe’s observed amount of baryon
asymmetry. However, as we showed in. Sec. (4.1), $R\tilde{R}$ contributes only
in the second-order, and hence we have ignored it in this analysis. It may be
interesting to look at the second-order corrections and analyze the parameter
constraints.

In this chapter, we have used the general effective field theory of gravity
coupled to the Standard Model of particle physics framework to obtain leading
order gravity terms that couple to the standard model Bosons [161]. We have
considered only the mass dimension 6-operators coupling to the gauge field
Lagrangian, specifically, to the electromagnetic field. The coupling to the
Fermions arises at the mass dimension 8. Thus, coupling of Fermion-anti-
Fermion with $U(1)$ field will play a role only at this order. While these are
expected to be suppressed compared to mass-dimension 6 operators, they are
relevant at Planck scale. We plan to look at the effects of mass dimension 8
operators on the baryogenesis.

In this chapter, we focused on the electromagnetic fields and the effects of
the helical fields on baryogenesis. It will be interesting to extend the
analysis to Gluons and study the effects on the asymmetry generated in quarks
and the Baryons. It is particularly important, and we will discuss this in
detail in chapter 7.

## Chapter 5 Effective field theory of magnetogenesis

As discussed in Section 1.5, various observations have confirmed the existence
of magnetic fields in the Universe [60, 61, 65]. In galaxies and galaxy
clusters, the typical magnetic field strength is found to be on the order of
micro-Gauss with a coherence length of kpc to Mpc [60, 61, 62, 78, 80, 81].
Existing data on the magnetic fields in these regions cannot directly
constrain the properties and origin of cosmic-scale magnetic fields.
Therefore, it is unknown whether their origin is astrophysical or primordial.
However, magnetic field measurements from Faraday rotation and Synchrotron
radiation provide an upper bound for magnetic fields. In contrast, FERMI
measurements of gamma-rays emitted by Blazars provide a lower bound of the
order of $10^{-15}~{}{\rm G}$ in the intergalactic voids [65].

As mentioned in Chapter 1, according to the widely accepted paradigm, magnetic
fields in these regions are produced by the dynamo amplification of the weak
primordial magnetic field [81]. There are various mechanisms for producing the
primordial magnetic field, but in most of them, either the produced magnetic
field is too weak to be amplified via dynamo or the coherence length is too
short to be sustained due to Universe expansion [90]. Inflation provides a
causal mechanism to generate the magnetic field over a large scale [97].
However, one of the pre-requisites for generating the primordial magnetic
field during inflation is breaking the conformal invariance of the 4-D
electromagnetic action.

As discussed in previous chapters (2,3,4), Several models have been proposed
to break the conformal invariance of the action without breaking the gauge
invariance. Broadly they can be classified into two categories — coupling
electromagnetic fields with other matter (scalar) fields and higher-derivative
terms in the electromagnetic action leading to the non-minimal coupling of the
electromagnetic field with curvature [97, 98, 100, 178, 197, 198, 199, 102,
103, 120]. Due to simplicity, the first class of models, especially, scalar
field coupled models are extensively studied [98, 178]. However, these models
suffer from strong coupling and back-reaction problems that necessitate
parameter tweaking. For the possible resolution of these issues, see Refs.
[170, 200, 201, 202].

The second class of models is more natural as higher derivative terms are
expected to arise when quantum gravitational effects are taken into account
[203, 204]. Demanding that the theory be Lorentz invariant in flat space-time,
the field action (in Fourier space) can only be a function of $k^{2}(\equiv
k_{\mu}k^{\mu})$ [130, 128, 129]. Besides, the divergence structure of quantum
field theory is expected to vastly improve when the quantum gravitational
effects are taken into account [130]. For instance, higher-derivative
electromagnetic theory by Podolsky-Schwed [124] removes the divergence of the
Coloumb potential. One problem with higher derivative theories is the
appearance of negative energy states. Although they can be traded by negative
norm states (or ghosts), they normally lead to non-unitary theories. However,
vector Galileons do not have ghosts [102, 103].

Ideally, we require a fundamental theory of quantum gravity to obtain a
generic magnetic field power spectrum generated in the early Universe.
However, since we do not have such a consistent model of quantum gravity yet,
we aim to obtain an _effective field theory_ (EFT) description of primordial
magnetogenesis during inflation (based on expansion about the Hubble parameter
$(H)$ and its derivatives). In this chapter, we obtain a generic magnetic
field power spectrum from a low-energy effective field theory of
magnetogenesis. While broken conformal invariance is a common requirement for
primordial magnetogenesis, for the first time, we show that causal propagation
is also a necessary condition.

While EFT of inflation has been systematically analyzed following Ref. [155],
there is no such systematic analysis for magnetogenesis. See, for instance,
Refs. [205, 206]. However, two key differences exist between the EFT of
magnetogenesis and inflation. First, the EFT of inflation is a model-
independent framework for studying scalar (and tensor) perturbations. In this
setup, time-translation invariance needs to be broken as inflation ends at a
finite time. In the case of EFT of magnetogenesis, the conformal invariance of
the gauge fields also needs to be broken in the cosmological background
besides breaking time diffeomorphism. Second, in EFT of inflation, one makes a
specific gauge choice (unitary gauge) where the scalar-field (inflaton)
fluctuations are zero, add gravitational operators to the Lagrangian that
preserves spatial-diffeomorphism and breaks the time diffeomorphism. As the
system breaks time diffeomorphism, one can write down a Lagrangian for the
Goldstone Boson associated with the broken symmetry with the Stuckleberg
trick. Now, this Goldstone Boson $\pi$ is related to the gauge-invariant
quantity the curvature perturbation $\zeta$ as $\zeta=-H\pi$ with $H$ being
the Hubble parameter. So by analyzing the dynamics of $\pi$, one can analyze
the scalar mode of perturbation through the gauge-invariant quantity $\zeta$
produced during inflation. In the case of magnetogenesis, we do not have to
make any specific gauge choice where the perturbed gauge fields vanish. This
is because, unlike the scalar perturbations, the vector modes of the
gravitational operators do not dynamically affect the gauge field. Hence, we
do not need to construct any gauge-invariant variable out of the gauge field
and the vector modes of perturbation, and for the EFT expansion, we need only
to consider the gauge field $A_{\mu}$. In Appendix (D.1), we show this
explicitly for general relativity.

In this chapter, we systematically write the EFT of magnetogenesis in the
early Universe in terms of the Hubble parameter $H(t)$ and its derivatives.
More specifically, we expand the Lagrangian in the powers of the cut-off scale
$\Lambda$ and consistently analyze the conditions for generating a primordial
magnetic field without any other assumptions. These terms containing $\Lambda$
break the conformal invariance of the gauge field and satisfy one of the key
criteria. Our approach is different from the approaches in Ref. [206, 205]. In
Ref. [205] the effective action terms only consist of four derivatives
associated with functions of background inflaton. This expansion leads to
different susceptibilities for electric and magnetic fields, but the results
are produced with a particular parametrization of these susceptibilities. In
Ref. [206], the EFT Lagrangian is written in second-order with all possible
contractions of electromagnetic tensor $F_{\mu\nu}$ with itself associated
with time-dependent analytic functions $f_{i}(\eta)$. The functions
$f_{i}(\eta)$ is chosen to be proportional to either
$\left({a(\eta)}/{a_{f}(\eta_{f})}\right)^{2}$ or higher powers where
$\eta_{i}$ and $\eta_{f}$ are (conformal) time at beginning and end of
inflation. More specifically, in both these cases, the authors did not include
terms that can be proportional to $({H}/{\Lambda})$ where in our expansion
scheme, it naturally arises.

The EFT method we use to study the physics of magnetogenesis in model-
insensitive. We explicitly show that the conformal invariance breaking is
_only_ a necessary condition, not a sufficient condition. We show that we can
have large amplifications even for super-luminal fluctuations. To avoid EFTs
with superluminal fluctuations, we need another physical condition — the modes
should be sub-luminal. As explained in Ref. [207], local quantum field
theories contain a Lorentz-invariant concept of causality and satisfy the
typical S-matrix axioms. This should also be satisfied by the expansion scalar
functions (cf. Eq. (5.5)) describing effective field theory of magnetogenesis.
By construction, these functions are arbitrary, one need to impose additional
conditions for a well-defined relativistic field theory. For a recent
discussion in the context of black holes, see Ref. [208]. Thus, there are two
necessary conditions for the generation of primordial magnetic fields —
conformal invariance breaking and causal propagation. We confirm this by
considering a specific model of primordial magnetogenesis. In this chapter, we
will only look at the generation of non helical magnetic field, the
modification terms for the helical magnetic fields, for example as discussed
in chapters (3,4) can be obtained in straightforward manner.

The rest of the chapter is organized as follows: In section 5.1 we write down
the EFT action of magnetogenesis and compare it with models proposed in the
literature. Section 5.2 obtains the magnetic power spectrum for the EFT in
slow-roll inflation and identifies the loophole in the magnetogenesis model
building. In section 5.3, we take a specific example and show that magnetic
field amplification is possible with the cost of super-luminal propagation.
Appendix D contains the details of the calculations in this chapter. The
results reported in this chapter are based on Ref. [122].

###  5.1 EFT action of magnetogenesis

Like any effective field theory, the EFT of magnetogenesis includes two
components: Symmetries and degrees of freedom [209, 210]. The vector
perturbations do not influence the dynamical evolution of the gauge field. See
Appendix D.1 for details. Hence, the vector perturbations and gauge fields are
decoupled, and the only relevant degree of freedom is the gauge field. Having
identified the relevant degrees of freedom, our next step is to write down the
action.

In principle, effective action can have infinite terms. Therefore, we need to
identify (broken) symmetries that describe the physics to expand the action
about the expansion parameter. Because of the conformal invariance of the
standard electromagnetic action

$\displaystyle S_{\rm EM}=-\frac{1}{4}\int
d^{4}x\sqrt{-g}F_{\mu\nu}F^{\mu\nu}$ (5.1)

it is impossible to produce a detectable amount of magnetic field from this
setup111Note that $F_{\mu\nu}\tilde{F}^{\mu\nu}$ is a total derivative term in
FRW background and does not contribute to the dynamics.. Splitting the gauge-
field in the background and perturbations (D.2), we have:

$A_{\mu}=\bar{A}_{\mu}+\delta A_{\mu}=\delta A_{\mu}.$ (5.2)

Due to the background symmetry, we have considered the background field
$\bar{A}_{\mu}=0$ and if we use the well-known choice of Coulomb gauge (see
Appendix D.1 for details),

$\delta A_{0}=0,\partial_{i}\delta A^{i}=0\,,$ (5.3)

the Lagrangian for the fluctuation field $\delta A_{\mu}$ from (5.1) can be
written as,

$\displaystyle S_{\rm EM}=\int d^{4}x\left[(\delta
A_{i}^{\prime})^{2}-(\partial_{i}\delta A_{j})^{2}\right]$ (5.4)

This Lagrangian for the fluctuation is also conformally invariant and thus,
the generation of primordial magnetic fields in the early Universe requires
conformal invariance breaking of the electromagnetic action [97, 98, 100]. In
other words, the terms in the effective action must break the conformal
invariance. Here, we demand that the effective action satisfies the following
symmetries:

  1. 1.

Local Lorentz invariance: In the Minkowski limit, we demand that the effective
action is Lorentz invariant. In other words, in the limit, $a(\eta)\to{\rm
constant}(H(t)\to 0)$, the EFT action of magnetogenesis reduces to the
standard electromagnetic action.

  2. 2.

Gauge invariance: As mentioned in detail in Ref. [2001-Jackson.Okun-RMP], an
implicit assumption behind the formalism of the gauge invariance is that the
field equations must have unique solutions. We impose the validity of this
condition in the early Universe. Thus, the EFT action _do not_ contain terms
proportional to $A_{\mu}A^{\mu}$. Note that from now on, we will use $A_{i}$
instead of $\delta A_{i}$ to denote the gauge field fluctuation.

While the above symmetry requirements will restrict the form of the EFT action
of magnetogenesis, we need to identify the expansion parameters of the action.
Although the two expansion parameters are well-defined in the time-independent
system, it is not straightforward for time-dependent systems like in the early
Universe [152]. In general, it is not possible to construct an EFT without
energy conservation, as EFTs divide states based on energy. However, if the
time-evolution of the degrees of freedom $(A_{i})$ is adiabatic — they vary
sufficiently slowly compared to the UV scales of interest $\Lambda$ — we can
treat the Hamiltonian obtained from the EFT action to be approximately
conserved Hamiltonian with an approximate time-dependent low/high energy split
[211, 152].

In the case of slow-roll inflation, the energy scale during inflation $H$ is
approximately constant. Let us define energy scale $\Lambda$ that represents a
cutoff scale for which the effects of high-scale physics are described by non-
renormalizable operators, which can be thought of as originating from
integrating out all particles with mass $m>\Lambda$ [158, 210]. Thus, at these
energies, the scale dependence between the characteristic energy scale $H$ and
the cutoff scale $\Lambda$ is given by the expansion parameter $H/\Lambda$.
Like in general EFT, we fix the level of precision and sort by $H/\Lambda$ at
a given order all terms that contribute to the action. Note that we cannot use
standard perturbation theory to quantize, and the problem of non-
renormalizability becomes an actual problem at energies close to $\Lambda$. We
assume $\Lambda$ to be at least _one order_ higher than $H$ and less than
$M_{\rm Pl}$.

Thus, the second-order EFT action of magnetogenesis, based on expansion about
the Hubble parameter $(H)$ and its derivatives, is

$\mathcal{S}_{\rm EFT}=\int
d^{4}x\left[f_{1}(H,a,\Lambda)({A}_{i}^{\prime})^{2}-f_{2}(H,a,\Lambda)(\partial^{j}A_{i}\,\partial_{j}A_{i})\right]$
(5.5)

where, the _expansion scalar functions_ — $f_{1}(H,a,\Lambda)$ and
$f_{2}(H,a,\Lambda)$ — depend of Hubble parameter $H$, scale factor $a(\eta)$,
and cutoff scale $(\Lambda)$. This is a crucial expression regarding which we
want to discuss the following: First, as explained in Ref. [207], local
quantum field theories contain a Lorentz-invariant concept of causality and
satisfy the typical S-matrix axioms. This should also be satisfied by the
functions $f_{1}$ and $f_{2}$ describing effective field theory of
magnetogenesis. By construction, these functions are arbitrary, one need to
impose additional conditions for a well-defined relativistic field theory. For
a recent discussion in the context of black holes, see Ref. [208]. Second, we
have chosen the Coulomb gauge condition for the EM fields and hence resulting
in only two physical degrees of freedom222The Coulomb gauge allows us to
evaluate the observables like magnetic and electric power-spectrum
efficiently.. In principle, we can have separate expansion scalar functions in
front of $A_{1}^{\prime 2},A_{2}^{\prime 2},A_{3}^{\prime 2}$ but this will
lead to different propagation speeds for the three components. (See Appendix
D.3 for details). In this chapter, we focus on action (5.5).

Also, note that one can add parity breaking term
$\epsilon^{ijk}A_{i}^{\prime}\partial_{j}A_{k}$ in the EFT action (5.5) with
additional scalar expansion function $f_{3}$. In the limit of
$\Lambda\to\infty$ and $a(\eta)\to{\rm constant}$ (or $H(t)\to 0$),
$f_{3}\to~{}\rm{constant}$. Since the parity symmetry only determines the
nature of primordial magnetic field, we will not consider the parity breaking
term in the action. Third, in the limit of $\Lambda\to\infty$,
$f_{1}=f_{2}\simeq{\rm constant}$. In other words, the above expression will
reduce to Eq. (5.1). Also, in the limit of $a(\eta)\to{\rm constant}$ or
$H(t)\to 0$), $f_{1}=f_{2}\simeq{\rm constant}$ and reduce to Eq. (5.1).
Fourth, $f_{1}(H,a,\Lambda)$ and $f_{2}(H,a,\Lambda)$ capture all possible
interactions of electromagnetic field that leads to breaking of conformal
invariance.

Fifth, since $A_{\mu}$ has mass dimension $1$, both $f_{1}(H,a,\Lambda)$ and
$f_{2}(H,a,\Lambda)$ have mass dimension zero [212]. Formally, the effective
action can be written as [212]:

$S\sim\int
d^{4}x~{}\mathcal{O}_{p,q}\sim\left(\frac{E}{\Lambda}\right)^{p+q-4},$

where, the operator $\mathcal{O}$ is made up of $p$ fields and $q$
derivatives. Since, all the observable quantities in cosmology are related to
the Hubble parameter $H(t)$ and it determines the energy scale of the epoch.
To compare the inflationary scale with EFT scale, a broad class of
magnetogenesis models can be reproduced from the EFT that is a sum of series
in $H/\Lambda$, time-derivatives of $H$ — ($H^{\prime}/\Lambda^{2}$),
($H^{\prime\prime}/\Lambda^{3}$), $\cdots$ — and their products, for instance,
$HH^{\prime}/\Lambda^{3}$, $HH^{\prime\prime}/\Lambda^{4}$, $\cdots$. Since,
$p$ and $q$ are integers, the effective field theory can be expanded only as a
series in terms of the Hubble parameter ($\mathcal{H}$) in conformal
time333Higher derivatives of $\mathcal{H}$ are also present, however, not
shown in the expansion.:

$\displaystyle\begin{split}\\!\\!\\!\\!\\!\\!\\!f_{1}(H,a,\Lambda)&=\sum_{n=0}^{\infty}s_{n}\frac{1}{a^{n}}\left(\frac{\mathcal{H}}{\Lambda}\right)^{n}+\sum_{m=1}^{\infty}b_{m}\frac{1}{a^{2m}}\left(\frac{\mathcal{H}^{\prime}}{\Lambda^{2}}\right)^{m}+\sum_{m,n=1}^{\infty}v_{n,m}\frac{1}{a^{m+2n}}\left(\frac{\mathcal{H}}{\Lambda}\right)^{m}\left(\frac{\mathcal{H}^{\prime}}{\Lambda^{2}}\right)^{n}\\\
\\!\\!\\!\\!\\!\\!\\!f_{2}(H,a,\Lambda)&=\sum_{n=0}^{\infty}d_{n}\frac{1}{a^{n}}\left(\frac{\mathcal{H}}{\Lambda}\right)^{n}+\sum_{m=1}^{\infty}e_{m}\frac{1}{a^{2m}}\left(\frac{\mathcal{H}^{\prime}}{\Lambda^{2}}\right)^{m}+\sum_{m,n=1}^{\infty}w_{n,m}\frac{1}{a^{m+2n}}\left(\frac{\mathcal{H}}{\Lambda}\right)^{m}\left(\frac{\mathcal{H}^{\prime}}{\Lambda^{2}}\right)^{n}\end{split}$
(5.6)

where $s_{n},b_{m},d_{n},e_{n},v_{n,m},w_{n,m}$, are the unknown real (postive
or negative) parameters and can be fixed for a particular magnetogenesis
model. $s_{0}$ and $d_{0}$ correspond to the values in standard
electrodynamics satisfying the local Lorenz invariance. Note that
$H^{\prime}=(\mathcal{H^{\prime}}-\mathcal{H}^{2})/a$, hence the expansion in
either of the two variables are equivalent. In the above expression, we have
not included series in higher derivatives of $\mathcal{H}$ like
$(\mathcal{H}^{\prime\prime}/\Lambda^{3})$. In principle, these terms should
also be included in the EFT and appear in higher-order gravitational coupling.
As mentioned above, since the coefficients
$s_{n},b_{m},d_{n},e_{n},v_{n,m},w_{n,m}$ are unknown, one need to impose
additional conditions, like causality, for a well-defined relativistic field
theory [207, 208].

Sixth, in the literature, the odd powers of ${\mathcal{H}}/{\Lambda}$ are not
included and the first-order correction is taken to be
${\mathcal{H}^{2}}/{\Lambda^{2}}$ or ${\mathcal{H}^{\prime}}/{\Lambda^{2}}$.
However, we have included odd powers in the expansion parameter to keep the
analysis general. As mentioned above, the above expansion is valid only for
$H<\Lambda$.

Lastly, Table (I) identifies the early Universe magnetogenesis models and the
corresponding EFT parameters. The list is not exhaustive but gives a good
representation of the various magnetogenesis models discussed in the context
of inflation. Thus, we see that the EFT action (5.5) can reproduce most of the
known magnetogenesis models. For most models, it is sufficient to consider up
to the second order in the expansion parameter. Appendix D.2 contains detailed
calculations that provide a one-to-one mapping between the magnetogenesis
model and EFT parameters.

Magnetogenesis models | Non-zero EFT parameters
---|---
Class of Ratra Model: $f(\phi)F_{\mu\nu}F^{\mu\nu}$ [98] |  $s_{n},d_{n}$ (depending on $f(\phi)$)
Higgs Starobinsky Inflation [213] | $s_{n},d_{n}$
Vector Galileon Model [102] | $s_{2},e_{1}$
Gravitational Coupling: $RF_{\mu\nu}F^{\mu\nu}$ [97] | $s_{2},b_{1},d_{2},e_{1}$
Gravitational Coupling: $R_{\mu\nu}F^{\mu\alpha}F^{\nu}_{\alpha}$ [97] | $s_{2},b_{1},d_{2},e_{1}$
Gravitational Coupling: $R_{\mu\nu\alpha\beta}F^{\mu\nu}F^{\alpha\beta}$ [97] | $b_{1},d_{1}$
Higher order Gravitational Coupling: | $s_{6},b_{3},v_{4,1},v_{2,2}$
$R^{3}F_{\mu\nu}F^{\mu\nu}$ [214] | $d_{6},e_{3},w_{4,1},w_{2,2}$
Table 5.1: One-to-one mapping between the magnetogenesis model and the EFT
parameters. See Appendix D.2 for details.

Before we proceed with the rest of the analysis, we compare the above EFT
action (5.5) with the ones recently proposed in the literature [205, 206]. In
Ref. [205] the effective action terms only consist of four derivatives
associated with functions of background inflaton. This expansion leads to
different susceptibilities for electric and magnetic fields, but the results
are produced with a particular parametrization of these susceptibilities. In
Ref. [206], the EFT Lagrangian is written in second-order with all possible
contractions of electromagnetic tensor $F_{\mu\nu}$ with itself associated
with time-dependent analytic functions $f_{i}(\eta)$. The functions
$f_{i}(\eta)$ are chosen to be proportional to either
$\left({a(\eta)}/{a_{f}(\eta_{f})}\right)^{2}$ or higher-order where
$\eta_{i}$ and $\eta_{f}$ are time at beginning and end of inflation. More
specifically, in both these cases, the authors did not include terms that can
be proportional to $({H}/{\Lambda})$ where in our expansion scheme, it
naturally arises.

###  5.2 Generic magnetic field power-spectrum from EFT action

In the previous section, we constructed EFT action (5.5) of magnetogenesis
based on symmetries and degrees of freedom. We also constructed a form of the
expansion scalar functions ($f_{1}(H,a,\Lambda)$ and $f_{2}(H,a,\Lambda)$) and
showed that the EFT parameters in this generic form indeed correspond to the
various magnetogenesis models. To make the computation of the power-spectrum
tractable and to highlight the importance of speed of perturbations, we
truncate the series (5.6) up to second order. However, the truncation of the
series to compute the power spectrum has no bearing on the EFT expansion
(5.6).

####  5.2.1 Equation of motion from EFT action

Truncating the expansion scalar functions ($f_{1}(H,a,\Lambda)$ and
$f_{2}(H,a,\Lambda)$) in Eq. (5.6) to $\Lambda^{-2}$ order, we have,

$\begin{split}&f_{1}(H,a,\Lambda)\simeq
s_{0}+\frac{s_{1}}{a(\eta)}\left(\frac{\mathcal{H}}{\Lambda}\right)+\frac{s_{2}}{a^{2}(\eta)}\left(\frac{\mathcal{H}}{\Lambda}\right)^{2}+\frac{b_{1}}{a^{2}(\eta)}\left(\frac{\mathcal{H}^{\prime}}{\Lambda^{2}}\right),\\\
&f_{2}(H,a,\Lambda)\simeq
d_{0}+\frac{d_{1}}{a(\eta)}\left(\frac{\mathcal{H}}{\Lambda}\right)+\frac{d_{2}}{a^{2}(\eta)}\left(\frac{\mathcal{H}}{\Lambda}\right)^{2}+\frac{e_{1}}{a^{2}(\eta)}\left(\frac{\mathcal{H}^{\prime}}{\Lambda^{2}}\right).\end{split}$
(5.7)

As mentioned earlier, in the limit of $\Lambda\to\infty$, the expansion should
reduce to Eq. (5.1). Hence, we have $s_{0}=d_{0}=1/2$. (See Appendix (D.4) for
more details.)

To obtain the equation of motion corresponding to the action (5.5), we first
need to rewrite the action in canonical form. To do that, we define
$A_{i}=\mathcal{A}_{i}/Z$ in the effective action (5.5) and we have:

$\mathcal{S}_{\rm EFT}=\int
d^{4}x\left[(\mathcal{A}_{i}^{\prime})^{2}+\frac{Z^{\prime\prime}}{Z}\mathcal{A}_{i}^{2}-\frac{f_{2}}{f_{1}}(\partial_{j}\mathcal{A}_{i})^{2}\right].$
(5.8)

where $Z=f_{1}^{1/2}$. Using Eq. (5.7), we have:

$\frac{Z^{\prime\prime}}{Z}=\frac{s_{1}}{a(\eta)}\left(\frac{\mathcal{H}}{\Lambda}\right)\left[\frac{\mathcal{H}^{2}}{2}-3\mathcal{H}^{\prime}+\frac{1}{2}\frac{\mathcal{H}^{\prime\prime}}{\mathcal{H}}\right]+\frac{1}{a^{2}(\eta)}\left(\frac{\mathcal{H}}{\Lambda}\right)^{2}\left[\left(\frac{s_{1}^{2}}{4}+2s_{2}\right)\mathcal{H}^{2}\right.\\\
\left.+\left(s_{1}^{2}-5s_{2}+2b_{1}\right)\mathcal{H}^{\prime}+\left(\frac{s_{1}^{2}}{4}+s_{2}-b_{1}\right)\left(\frac{\mathcal{H}^{\prime}}{\mathcal{H}}\right)^{2}-\left(\frac{s_{1}^{2}}{2}+s_{2}+2b_{1}\right)\frac{\mathcal{H}^{\prime\prime}}{\mathcal{H}}+b_{1}\frac{\mathcal{H}^{\prime\prime\prime}}{\mathcal{H}^{2}}\right]$
(5.9)

Note that we have only kept terms up to $1/\Lambda^{2}$ and ignored higher-
order $\Lambda$ contributions. Like in non-canonical scalar fields,
$f_{2}/f_{1}$ in the effective action (5.8) can be identified as the adiabatic
sound speed:

$\displaystyle c_{A}^{2}$ $\displaystyle=$
$\displaystyle\frac{1+\frac{d_{1}}{a(\eta)}\left(\frac{\mathcal{H}}{\Lambda}\right)+\frac{d_{2}}{a^{2}(\eta)}\left(\frac{\mathcal{H}}{\Lambda}\right)^{2}+\frac{e_{1}}{a^{2}(\eta)}\left(\frac{\mathcal{H}^{\prime}}{\Lambda^{2}}\right)}{1+\frac{s_{1}}{a(\eta)}\left(\frac{\mathcal{H}}{\Lambda}\right)+\frac{s_{2}}{a^{2}(\eta)}\left(\frac{\mathcal{H}}{\Lambda}\right)^{2}+\frac{b_{1}}{a^{2}(\eta)}\left(\frac{\mathcal{H}^{\prime}}{\Lambda^{2}}\right)}$
(5.10) $\displaystyle\simeq$ $\displaystyle
1+\frac{d_{1}-s_{1}}{a(\eta)}\left(\frac{\mathcal{H}}{\Lambda}\right)+\frac{1}{a^{2}}\left(\frac{\mathcal{H}}{\Lambda}\right)^{2}\left[s_{1}^{2}-s_{2}-s_{1}d_{1}+d_{2}+(e_{1}-b_{1})(1-\epsilon_{1})\right]$
(5.11)

where in arriving at the above expression, we have assumed that $H/\Lambda$ is
small and higher-order terms are negligible. This is the second key result of
this chapter regarding which we want to stress the following points: First, as
mentioned above, since the coefficients $s_{1},s_{2},d_{1},d_{2},b_{1},e_{1}$
are unknown, one need to impose additional conditions, like causality, for a
well-defined relativistic field theory [207, 208]. More specifically,
considering both $d_{1}$ and $s_{1}$ to be positive, the above expression
implies that $c_{A}^{2}>1$ if $d_{1}>s_{1}$, irrespective of the value of
$H/\Lambda$. Thus, such models violate standard causality condition [215].
Earlier, effective field theories have been rejected based on super-luminal
fluctuations as such propagation generally leads to a global breakdown of
causality [207]. Second, models with $s_{1}<0$ and $d_{1}>0$, will always lead
to superluminal modes. Third, when $d_{1}=s_{1}$ and assuming all EFT
parameters are positive, the causality condition implies
$s_{2}>d_{2}+(e_{1}-b_{1})(1-\epsilon_{1})$ during the entire inflationary
epoch [$\epsilon_{1}$ is the first-order slow-roll parameter defined in Eq.
(5.18)]. Lastly, in general, $c_{A}$ is a function of time. Since $H/\Lambda$
is small, one can assume that $c_{A}$ has a weak time dependence. Note that,
by construction, the EFT action (5.5) is locally Lorenz invariant, and the
causality condition imposes restrictions on the EFT parameters.

Fourth, in Appendix (D.4) we have obtained the energy density (D.45)
corresponding to the EFT action (5.5). We infer the following from the energy-
density (D.45): $\rho_{\rm mixing}$ decays faster than $\rho_{\rm E}$ and
$\rho_{\rm B}$. Hence, we can ignore $\rho_{\rm mixing}$ contribution in
evaluating the energy density of the EFT. Take the extreme scenario where
$\rho_{\rm E}$ and $\rho_{\rm B}$ contribute equally, imposing the condition
that the energy density is always positive provides a condition that
$s_{1}>d_{1}$ (assuming $s_{1},d_{1}$ are positive). This is consistent with
the causality condition we obtained earlier. We show that all these features
are satisfied for the specific Galileon vector model in Sec. (5.3).

Lastly, the equation of motion corresponding to the action (5.8), in the
Fourier domain $(k)$, is

$\mathcal{A}_{k}^{{}^{\prime\prime}}+\left[c_{A}^{2}k^{2}-\frac{Z^{{}^{\prime\prime}}}{Z}\right]\mathcal{A}_{k}=0\,,$
(5.12)

where, $k=|{\bf k}|$ and ${\bf k}$ is the comoving wave vector. For brevity,
we have defined $\mathcal{A}_{k}=\mathcal{A}_{i}^{(k)}$. In the rest of this
section, we now compute the magnetic field power spectrum during inflation.

####  5.2.2 Generic magnetic power spectrum during inflation

In this section, we quantize the effective gauge field given by the action
(5.8), and obtain the general expression for the primordial magnetic field
(PMF) power spectra. On quantization, the gauge field $\mathcal{A}$ can be
expressed as follows:

$\displaystyle\mathcal{\hat{A}}_{i}(\eta,{\bf x})=\int\frac{d^{3}{\bf
k}}{(2\pi)^{3/2}}\sum_{\lambda=1}^{2}\epsilon_{\lambda i}({\bf
k})\Big{[}\hat{b}_{\bf k}^{\lambda}\mathcal{A}_{\bf k}e^{i{\bf
k.x}}+\hat{b}_{\bf k}^{\lambda\dagger}\mathcal{A}_{\bf k}^{*}(\eta)e^{-i{\bf
k.x}}\Big{]}\,,$ (5.13)

where $\lambda$ corresponds to two orthonormal transverse polarizations,
$\epsilon_{\lambda i}$ are the polarization vectors and the creation
(${\hat{a}}_{\bf k}$) and the annihilation (${\hat{a}}_{\bf k}^{{\dagger}}$)
operators obey the usual commutation relations. Like scalar and tensor
perturbations [40], the power spectrum as well as the statistical properties
of the gauge field is characterized by the Wightman function of the gauge
field. The power spectrum (the two-point correlation in Fourier space) is
[80]:

$\left\langle
B_{i}^{*}(\mathbf{k})B_{j}\left(\mathbf{k}^{\prime}\right)\right\rangle=(2\pi)^{3}\delta^{3}\left(\mathbf{k}-\mathbf{k}^{\prime}\right)P_{ij}\mathcal{P}_{B}(k)\,,$
(5.14)

where $P_{ij}$ is a projector onto the transverse plane and is given by

$P_{ij}=\delta_{ij}-\frac{\mathbf{k}_{i}\mathbf{k}_{j}}{k^{2}},~{}~{}P_{ij}P_{jk}=P_{ik},~{}~{}P_{ij}\mathbf{k}^{j}=0\,,$
(5.15)

and $\mathcal{P}_{B}(|\mathbf{k}|)$ is the gauge field power spectrum. Since
$B$ is statistically homogeneous and isotropic, the correlation depends only
on the distance $|\mathbf{x}-\mathbf{y}|$. Using the decomposition (5.13), the
PMF spectrum per logarithmic interval can then be written in terms of the
modes $\mathcal{A}_{k}$ as

${\cal
P}_{B}(k)=\frac{k^{5}}{2\pi^{2}a^{4}}\left|\frac{\mathcal{A}_{k}}{Z}\right|^{2}$
(5.16)

and the expression on the right-hand side is to be evaluated when the physical
wavelength $(k/a)^{-1}$ of the mode corresponding to the comoving wavenumber
${\bf k}$ equals the _effective sound horizon_ of the electromagnetic
fluctuations $c_{A}\,H^{-1}$ [216]. The above condition translates to
$(-k\eta)c_{A}=1$ corresponding electromagnetic fluctuations exiting the sound
horizon during inflation. This sound horizon is not the same sound horizon of
the scalar perturbations during inflation [216, 217]. The normalization