constant takes into account both modes of polarization [81]. In the rest of
this subsection, we obtain the PMF power spectrum for the slow-roll inflation.
Appendix D.5 contains the results for power-law and de Sitter inflation.

To obtain the solution to Eq. (5.12) in slow-roll inflation, we introduce a
new set of variables [218, 176]:

$x=\ln{\frac{aH}{kc_{A}}},~{}~{}\mathcal{A}_{k}=e^{-\frac{x}{2}}(1-\epsilon_{1})^{-\frac{1}{2}}u_{k}\,,$
(5.17)

where $\epsilon_{1}$ is the first-order slow-roll parameter defined as:

$\epsilon_{1}=-\frac{\dot{H}}{H^{2}}=1-\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}$
(5.18)

Note that the above variables are well-defined up to the exit of inflation
$(\epsilon_{1}=1)$. To solve the differential equation (5.12), we need to
evaluate $Z$ in terms of the slow-roll parameters. Rewriting Eq. (5.18), we
have [219],

$\eta=-\frac{1}{(1-\epsilon_{1})\mathcal{H}}-\int\frac{2\epsilon_{1}\,(\epsilon_{1}-\epsilon_{2})}{(1-\epsilon_{1})^{3}}\,d\left(\frac{1}{\mathcal{H}}\right)\,,$
(5.19)

where $\epsilon_{2}$ is the second slow-roll parameter (D.74). Note that the
second term in the above expression can be ignored when $\mathcal{H}$ is
approximately constant and/or $\epsilon_{1}\simeq\epsilon_{2}$. Under this
condition, we have:

$\displaystyle
aH\simeq-\frac{1}{\eta(1-\epsilon_{1})}~{}~{}\Longrightarrow~{}~{}x\simeq\ln{\frac{1}{c_{A}k\eta(\epsilon_{1}-1)}}\,.$
(5.20)

We would like to note the following points: We do not assume
$\epsilon_{1}<<1$. $\eta$ is negative during inflation, hence $x$ is a well-
defined. Using Eqs. (5.17) and (5.20), we have:

$\displaystyle\mathcal{H}=e^{x}kc_{A}\,,$
$\displaystyle\mathcal{H}^{\prime}=e^{2x}k^{2}(1-\epsilon_{1})c_{A}^{2}\,,$
(5.21)
$\displaystyle\mathcal{H}^{\prime\prime}=2e^{3x}k^{3}(1-\epsilon_{1})^{2}c_{A}^{3}\,,$
$\displaystyle\mathcal{H}^{\prime\prime\prime}=6e^{4x}k^{4}(1-\epsilon_{1})^{3}c_{A}^{4}.$
(5.22)

In the new variables, the adiabatic sound speed ($c_{A}$) and
$Z^{\prime\prime}/Z$ become:

$\displaystyle c_{A}^{2}$ $\displaystyle\simeq$ $\displaystyle
1+(d_{1}-s_{1})\frac{H}{\Lambda}+\left[s_{1}^{2}-s_{2}-s_{1}d_{1}+d_{2}+(e_{1}-b_{1})(1-\epsilon_{1})\right]\left(\frac{H}{\Lambda}\right)^{2}$
(5.23) $\displaystyle\frac{Z^{\prime\prime}}{Z}$ $\displaystyle\simeq$
$\displaystyle\frac{e^{2x}k^{2}}{4}\left\\{2s_{1}\epsilon_{1}(\epsilon-1)\frac{H}{\Lambda}\right.$
(5.24) $\displaystyle+$
$\displaystyle\left.\left[a_{1}^{2}(2+2\epsilon_{1}-3\epsilon_{1}^{2})-a_{2}(4-7\epsilon_{1}+\epsilon^{2})-b_{1}\epsilon_{1}(1-4\epsilon_{1}+3\epsilon_{1}^{2})\right]\left(\frac{H}{\Lambda}\right)^{2}\right\\}\,.$

where we have truncated the series up to $(H/\Lambda)^{2}$ and we have not
imposed any slow-roll approximation. Substituting the above expressions (5.17,
5.23, 5.24) in Eq. (5.12) leads to:

$\frac{d^{2}u_{k}}{dx^{2}}+\left[q_{1}^{2}e^{-2x}-q_{2}^{2}\right]u_{k}=0\,.$
(5.25)

where,

$\displaystyle q_{1}$ $\displaystyle=$ $\displaystyle(1-\epsilon_{1})^{-1}$
(5.26) $\displaystyle q_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}+\left\\{\frac{s_{1}\epsilon_{1}(1-2\epsilon_{1})}{2}\frac{H}{\Lambda}-\left[\frac{s_{1}^{2}}{4}(2+2\epsilon_{1}-3\epsilon_{1}^{2})-s_{2}(4-7\epsilon_{1}+\epsilon_{1}^{2})-b_{1}\epsilon_{1}(1-4\epsilon_{1}-3\epsilon_{1}^{2})\right]\left(\frac{H}{\Lambda}\right)^{2}\right\\}$
(5.27)
$\displaystyle\times(1-\epsilon_{1})^{-2}-\frac{s_{1}^{2}\epsilon_{1}^{2}}{8}\frac{(1-2\epsilon_{1})^{2}}{(1-\epsilon_{1})^{4}}\left(\frac{H}{\Lambda}\right)^{2}$

We want to note that we have not assumed $\epsilon_{1}<<1$. Assuming
$\epsilon_{1}$ is approximately constant, the solution to the above
differential equation (5.25) is Hankel functions [abramowitz+stegun]

$u_{k}(x)=\alpha\,H_{q_{2}}^{(1)}(e^{-x}q_{1})+\beta\,H_{q_{2}}^{(2)}(e^{-x}q_{1})\,.$
(5.28)

Using the relation (5.17), the mode functions of the gauge field
$\mathcal{A}_{k}$ are:

$\mathcal{A}_{k}=(-c_{A}k\eta)^{\frac{1}{2}}\left\\{\alpha
H_{q_{2}}^{(1)}(e^{-x}q_{1})+\beta H_{q_{2}}^{(2)}(e^{-x}q_{1})\right\\}.$
(5.29)

In the sub-horizon limit ($c_{A}k|\eta|\gg 1$), we assume that the modes
satisfy the Bunch-Davies vacuum, i. e., the mode approaches the Minkowski
space behavior in the asymptotic past:

$\displaystyle\lim_{k\eta\to-\infty}\mathcal{A}_{k}(\eta)=\frac{1}{\sqrt{2c_{A}k}}e^{-ic_{A}k\eta}\,.$
(5.30)

This leads to:

$\alpha=0;~{}~{}~{}~{}\beta=\sqrt{\frac{\pi}{4c_{A}k}}\,.$ (5.31)

Substituting the mode function (5.29), with the Bunch-Davies vacuum initial
condition, in Eq. (5.16), the power spectrum for the super-horizon modes
($c_{A}k|\eta|<<1$) is:

$\mathcal{P}_{B}=\frac{k^{5}}{a^{4}}\frac{c_{A}^{2}}{f_{2}}\frac{\eta}{8\pi^{3}}(\Gamma(q_{2}))^{2}\left(\frac{-c_{A}k\eta}{2}\right)^{-2q_{2}}$
(5.32)

Substituting Eq. (5.20), we have:

$\mathcal{P}_{B}=\frac{H^{4}(1-\epsilon_{1})^{4}}{8\pi^{3}}\frac{\Gamma^{2}(q_{2})}{c_{A}^{3}f_{2}}(c_{A}k\eta)^{5-2q_{2}}\,.$
(5.33)

By expanding $c_{A}$ and $f_{2}$ up to the second-order in
$(\mathcal{H}/\Lambda)$, we have:

$\mathcal{P}_{B}=H^{4}(1-\epsilon_{1})^{4}\Gamma^{2}(q_{2})(c_{A}k\eta)^{5-2q_{2}}\times\\\
\left\\{1+\frac{3s_{1}-5d_{1}}{2}\frac{H}{\Lambda}+\frac{1}{2}[3s_{2}-6s_{1}d_{1}-5d_{2}+8d_{1}^{2}+(3b_{1}-5e_{1})(1-\epsilon)]\left(\frac{H}{\Lambda}\right)^{2}\right\\}\,.$
(5.34)

This is the third key result of this chapter, regarding which we want to
discuss the following points: First, the analysis is valid for all values of
$\epsilon_{1}$, assuming that the contribution of the second term in Eq.
(5.19) can be ignored. While this is true during most of the inflation, the
above expression may not be valid at the end of inflation. Second, in Appendix
D.5, we have derived the power spectrum for de Sitter and Power-law inflation.
While the power spectrum for the de Sitter inflation is exact, we have used
the WKB approximation for the power-law case. Setting the $\epsilon_{1}=0$ in
the above power spectrum (5.33) matches with the magnetic power spectrum
(D.63) during de Sitter. Third, from Eq. (5.34), we see that the amplification
in the magnetic power spectrum — evaluated at the horizon crossing $(-k\eta
c_{A}=1$) — is possible whenever $3s_{1}-5d_{1}>0$. Specifically, we see for
sub-luminal ($d_{1}<s_{1}$) or super-luminal ($s_{1}<0$) modes, the leading
order correction term in the power-spectrum ${3s_{1}-5d_{1}}>0$. In other
words, we can have large amplifications even for super-luminal fluctuations.
To avoid EFTs with superluminal fluctuations [207], we need another physical
condition — the modes should be sub-luminal. Thus, our analysis clearly shows
that magnetogenesis models must satisfy two necessary conditions — conformal
symmetry breaking and causal propagation. In the next section, we consider a
specific model and show that this is the case.

Lastly, the above expression provides the following condition under which the
magnetic field power spectrum (5.33) will be scale-invariant:

$5-2q_{2}=0\,.$ (5.35)

From Eq. (5.27) we see that parameters $s_{1}$ and $s_{2}$ are associated with
the $(H/\Lambda)$ expansion terms, while the parameter $b_{1}$ is associated
with $\left(\mathcal{H}^{\prime}/a\Lambda\right)^{2}$. While the terms
containing the parameters $s_{1}$ and $s_{2}$ do not have time-dependent
factors, the term containing $b_{1}$ has time-dependent factors. This can be
seen by rewriting $\left(\mathcal{H}^{\prime}/a\Lambda\right)^{2}$ as
$(1-\epsilon_{1})\left(H/\Lambda\right)^{2}$. If we demand that the prefactors
in the expansion are $\mathcal{O}(1)$, then this implies that $s_{1},s_{2}$
and $((1-\epsilon_{1})b_{1})$ are all order unity during the entire
inflationary epoch. In order for this to be satisfied, in general, $b_{1}$ can
have a large value even with $\left(H/\Lambda\right)^{2}$ suppression. Hence,
in order to get a scale-invariant power spectrum, the above condition reduces
to:

$4-\frac{p\epsilon_{1}(1-4\epsilon_{1}-3\epsilon_{1}^{2})}{(1-\epsilon_{1})^{3}}\left(\frac{H}{\Lambda}\right)^{2}=0\,\quad{\rm
where}\quad p=b_{1}(1-\epsilon_{1})\,.$ (5.36)

Note that the above redefinition ensures that $p$ is $\mathcal{O}(1)$ close to
the end of inflation. 5.1 contains the allowed values of $\epsilon_{1}$ for
different values of $p$ for a fixed $({H}/{\Lambda})^{2}$. From the plot we
see that for the scale invariant power spectrum in slow-roll scenario requires
$b_{1}<0$.

Figure 5.1: The allowed curves for two different values of $(H/\Lambda)^{2}$,
for which we can get a scale invariant power spectrum for the magnetic field.
The parameter $p$ is $\mathcal{O}(1)$ and this curve is only valid at the end
of inflation as we have considered the value of the first slow roll parameter
$\epsilon_{1}\rightarrow 1$. In the left panel we have set
$(H/\Lambda)^{2}=10^{-2}$ and in the right panel we have set
$(H/\Lambda)^{2}=10^{-4}$.

###  5.3 Example for EFT Magnetogenesis: Vector Galileon model

In the earlier section, using EFT, we showed that conformal invariance
breaking is not a sufficient condition for generating the primordial magnetic
field during inflation. In this section, we take a specific model and show
that the generation of primordial magnetic fields require two necessary
conditions — conformal invariance breaking and causal propogation.

In Ref. [102], two of the current authors have constructed a consistent vector
Galileon model by demanding the following conditions in action: theory be
described by vector potential $A_{\mu}$ and its derivatives, $U(1)$ gauge
invariance should be preserved, and the equations of motion must be second
order. The exciting feature of the model is that it only contains the
derivative coupling — higher derivative terms. Hence, it explicitly breaks the
conformal invariance of the action, and therefore it may lead to the
generation of magnetic fields in the early Universe. Hence, this model has a
clear advantage over other models, particularly the scalar field coupled
models [98, 200]. Moreover, due to the absence of coupling between scalar
field and electromagnetic field, the strong coupling problem is avoided
naturally; hence, the model does not require fine-tuning.

In this section, we consider the total action, i.e., standard electromagnetic
action with modification due to vector Galileon, and show that the conformal
invariance breaking is the necessary condition to amplify the magnetic field
generated during inflation, but it is not sufficient. More specifically, we
show that the amplification of order ($\sim 10^{7}$) comes with a price that
speed of sound $c_{s}>1$, without leading to the back-reaction problem. For
$c_{s}<1$, the model generates tiny magnetic fields on large scales. Thus,
despite having Lorentz-invariant Lagrangians, such theories admit super-
luminal fluctuations can not be low-energy effective field theories of
magnetogenesis.

####  5.3.1 Generation of primordial magnetic fields

This subsection will discuss the phenomenological consequences of the vector
Galileon model in the early universe. The vector Galileon action is:

$\mathcal{S}_{\rm{tot}}=S_{\rm EM}+\mathcal{S}_{\rm{VEC}}$ (5.37)

where $S_{\rm EM}$ is the standard electromagnetic action given in Eq. (5.1)
and $\mathcal{S}_{\rm{VEC}}$ is the vector Galileon part. In the flat FRW
background (D.1), with $N(\eta)=a(\eta)$, we have444 Note that in obtaining
Eq.(5.38) and Eq.(5.39), we have set $N(\eta)=a(\eta)$ in Eq. (D.32) and Eq.
(D.33), respectively :

$\displaystyle S_{EM}=\frac{1}{2}\int
d^{4}x\,\left[{A_{i}^{\prime}}^{2}-(\partial_{i}A_{j})^{2}\right]\,,$ (5.38)

The vector Galileon action in FRW background is [102]:

$\displaystyle\mathcal{S}_{VEC}=2D\,\int\,d^{4}x\Big{[}-\,\frac{a^{\prime}{}^{2}}{a^{4}}\,A_{i}^{\prime}{}^{2}+\frac{a^{\prime\prime}}{a^{3}}\,\left(\partial_{i}A_{j}\right)^{2}-\frac{{a^{\prime}}^{2}}{a^{4}}\,\left(\partial_{i}A_{j}\right)^{2}\Big{]}\,.$
(5.39)

where $A_{\mu}$ satisfies Coulomb gauge condition (D.8). $D$ is the coupling
constant with the dimensions of the square of the inverse of energy, i. e.
$D\equiv 1/\Lambda^{2}$. We define the following dimensionless parameter $J$:

$\displaystyle J=4DH^{2}=4\left(\frac{H}{\Lambda}\right)^{2}\,.$ (5.40)

Like the EFT action (5.5), the above action (5.39) is not in canonical form;
hence, we need to rewrite the action in canonical form. To do that, we define
the canonical vector field $\mathbb{A}_{i}$ as:

$\displaystyle\mathbb{A}_{i}=\left(1-J\right)^{1/2}\,A_{i}\qquad\implies\qquad\left(1-J\right)^{1/2}\,A_{i}^{\prime}=\mathbb{A}_{i}^{\prime}+\frac{J({\mathcal{H}}^{\prime}-\mathcal{H}^{2})}{\mathcal{H}(1-J)}\mathbb{A}_{i}\,.$
(5.41)

Substituting Eqs. (5.41, 5.40) in Eq. (5.37) and setting $N(\eta)=a(\eta)$
leads to:

$\displaystyle\mathcal{S}_{\rm{tot}}$
$\displaystyle=\frac{1}{2}\,\int\,d^{4}x\left[\,\,\mathbb{A}_{i}^{\prime}{}^{2}-\left[\left(\frac{\mathcal{H}^{\prime}-\mathcal{H}^{2}}{\mathcal{H}}\right)^{2}\frac{J}{(1-J)^{2}}+\frac{J\left(\mathcal{H}^{\prime\prime}-3\mathcal{H}^{\prime}\mathcal{H}+\mathcal{H}^{3}\right)}{(1-J)\,\mathcal{H}}\right]\,\mathbb{A}_{i}^{2}\right.$
$\displaystyle{}{}\qquad\left.-\left(1-\frac{J(\mathcal{H}^{\prime}-\mathcal{H}^{2})}{(1-J)\mathcal{H}^{2}}\right)\,\left(\partial_{i}\mathbb{A}_{j}\right)^{2}\right]\,,$
(5.42)

which is in the canonical form. The equation of motion for the canonical
vector field can be obtained by varying the action (5.3.1) with respect to
$\mathcal{A}_{i}$:

$\displaystyle\mathbb{A}_{i}^{\prime\prime}-c_{s}^{2}\,\nabla^{2}\mathbb{A}_{i}+\left[\left(\frac{\mathcal{H}^{\prime}-\mathcal{H}^{2}}{\mathcal{H}}\right)^{2}\frac{J}{(1-J)^{2}}+\frac{J\left(\mathcal{H}^{\prime\prime}-3\mathcal{H}^{\prime}\mathcal{H}+\mathcal{H}^{3}\right)}{(1-J)\,\mathcal{H}}\right]\,\mathbb{A}_{i}=0\,$
(5.43)

where $c_{s}$ is the propagation speed of electromagnetic fluctuations and is
given by:

$\displaystyle
c_{s}=\sqrt{1-\frac{J(\mathcal{H}^{\prime}-\mathcal{H}^{2})}{(1-J)\mathcal{H}^{2}}}=\sqrt{\frac{1-J(1-\epsilon_{1})}{1-J}}\,.$
(5.44)

In obtaining the last expression, we have used the definition (5.18).
Substituting the Fourier decomposing of the canonical vector field
$\mathbb{A}_{i}$ (5.13) in Eq. (5.43), we get:

$\displaystyle\mathbb{A}_{k}^{\prime\prime}+\left[c_{s}^{2}\,k^{2}+\left(\,\,\left(\frac{\mathcal{H}^{\prime}-\mathcal{H}^{2}}{\mathcal{H}}\right)^{2}\frac{J}{(1-J)^{2}}+\frac{J\left(\mathcal{H}^{\prime\prime}-3\mathcal{H}^{\prime}\mathcal{H}+\mathcal{H}^{3}\right)}{(1-J)\,\mathcal{H}}\right)\,\,\right]\mathbb{A}_{k}=0\,.$
(5.45)

In terms of the slow-roll parameters, the above expression reduces to:

$\displaystyle\mathbb{A}_{k}^{\prime\prime}+\left[c_{s}^{2}\,k^{2}-\frac{\mathcal{H}^{2}J\epsilon_{1}}{(1-J)^{2}}\,\left([1-J](1-3\epsilon_{1}+\epsilon_{2})-J\epsilon_{1}\right)\right]\,\mathbb{A}_{k}=0\,,$
(5.46)

where $\epsilon_{1}$ and $\epsilon_{2}$ are the slow-roll parameters defined
in Eq.(D.74). (See Appendix D.6.) This is a key expression regarding which we
would like to discuss the following: First, the above expression is exact and
valid for any cosmological scenario. Although we have expressed the equation
in terms of slow-roll parameters, we have not set $\epsilon_{1}<<1$. Second,
positive permittivity provides a condition on the value of $J$ to be $J<1$.
Hence, the transformation (5.41) is well-defined for all values of $J<1$.
Third, in the limit of $J\to 0$, $c_{s}=1$ and matches with standard
electrodynamics. However, for any non-zero value of $J$, the sound speed is
not unity; depending on the value of $J$, it can be greater or less than $1$.
In the case of inflation, $\epsilon_{1}$ is positive, and hence, for the
positive values of $J$, $c_{s}>1$. However, in the case of super-inflation
($\dot{H}>0$) [220, 199], $\epsilon_{1}<0$ leading to the speed of sound less
than 1 [199]. In the case of negative values of $J$, the sound speed is always
less than unity. 5.2 contains the plot of $c_{s}$ as a function of $J$. It
shows that $c_{s}>1$ for positive $J$ during inflation.

Figure 5.2: Plot showing the behaviour of $c_{s}$ with respect to parameter
$J$.

Our next step is to solve Eq. (5.46). It is not possible to solve the above
equation exactly for arbitrary $a(\eta)$. Hence, we consider two — sub-horizon
and super-horizon — limits of the above differential equation and match the
mode functions at the sound horizon.

In the sub-horizon limit ($c_{s}k|\eta|\gg 1$), we assume that the modes
satisfy the Bunch-Davies vacuum, i. e., the mode approaches the Minkowski
space behavior in the asymptotic past:

$\displaystyle\lim_{k\eta\to-\infty}\mathbb{A}_{k}(\eta)=\frac{1}{\sqrt{2c_{s}k}}e^{-ic_{s}k\eta}\,\,.$
(5.47)

For the physical gauge field ($A_{k}(\eta)$), using the relation (5.41), the
above condition translates to:

$\displaystyle\lim_{k\eta\to-\infty}A_{k}(\eta)=\frac{1}{\sqrt{2kc_{s}\,(1-J)}}e^{-ikc_{s}\eta}$
(5.48)

As we will see below, the above relation would be convenient to match the
solution at the sound-horizon crossing to fix the arbitrary coefficients. In
the super-horizon limit ($c_{s}k|\eta|<<1$), we have

$\displaystyle A_{k}(\eta)\simeq
C_{1}+C_{2}\,\,\frac{(1-J)^{1+\frac{2J\epsilon_{1}}{1-J}}}{1-J+2J\epsilon_{1}}\left(-k\eta\right)^{1+\frac{2J\epsilon_{1}}{1-J}}\,\,,$
(5.49)

where $C_{1}$ and $C_{2}$ are arbitrary coefficients. Since
${2J\epsilon_{1}}/(1-J)$ is positive, and hence at the end of inflation, i.e.,
$\eta\rightarrow 0$, the second term in the above expression is negligible. In
other words, the vector field freezes at the end of inflation and is
approximately constant $C_{1}$.

As mentioned earlier, we demand that the sub-horizon modes (5.48) and super-
horizon modes (5.49) are continuous and differentiable at the sound-horizon.
From Eq.(5.46), we obtain the horizon crossing time $\eta_{*}$ as

$\displaystyle\eta_{*}=-\frac{1}{k}\left[\,\frac{J_{*}\epsilon_{1}^{*}(1-J_{*}-\epsilon_{1}^{*}(3-2J_{*})+\epsilon_{2}^{*}(1-J_{*}))}{(1-J_{*})(1-J_{*}+J_{*}\epsilon_{1}^{*})}\,\right]^{1/2}\,,$
(5.50)

where $"*"$ refers to the quantities evaluated at the horizon crossing time
$\eta_{*}$. Matching the mode functions at $\eta_{*}$ leads to:

$\displaystyle C_{1}$
$\displaystyle=\frac{(1-J_{*}+2J_{*}\epsilon_{1}^{*})-i\sqrt{J_{*}\epsilon_{1}^{*}\left(1-J_{*}-\epsilon_{1}^{*}(3-2J_{*})+\epsilon_{2}^{*}(1-J_{*})\right)}}{\sqrt{2c_{s}^{*}k(1-J_{*})}\,(1-J_{*}+2\epsilon_{1}^{*}J_{*})}e^{i\sqrt{\frac{J_{*}\epsilon_{1}^{*}\left(1-J_{*}-\epsilon_{1}^{*}(3-2J_{*})+\epsilon_{2}^{*}(1-J_{*})\right)}{(1-J_{*})^{2}}}}$
(5.51) $\displaystyle C_{2}$
$\displaystyle=\frac{i\,{c_{s}^{*}}^{\frac{1}{2}+\frac{2J_{*}\epsilon_{1}^{*}}{1-J_{*}}}}{\sqrt{2k(1-J_{*})}}\left(J_{*}\epsilon_{1}^{*}\left(1-J_{*}-\epsilon_{1}^{*}(3-2J_{*})+\epsilon_{2}^{*}(1-J_{*})\right)\right)^{\frac{2J_{*}\epsilon_{1}^{*}}{J_{*}-1}}e^{i\sqrt{\frac{J_{*}\epsilon_{1}^{*}\left(1-J_{*}-\epsilon_{1}^{*}(3-2J_{*})+\epsilon_{2}^{*}(1-J_{*})\right)}{(1-J_{*})^{2}}}}$
(5.52)

####  5.3.2 Power spectrum and estimation of magnetic field strength

To study the observable effects, we evaluate the energy density of the
electromagnetic fields [80]. The $0-0$th component of the energy-momentum
tensor $T_{\mu\nu}$ in the FRW background (D.1) is

$\displaystyle T_{00}=-\frac{N^{2}}{a^{3}}\frac{\delta\mathcal{L}}{\delta
N}\,.$ (5.53)

Substituting the action (5.37) in the above expression and setting
$N(\eta)=a(\eta)$, we get:

$\displaystyle\rho=\frac{1}{2a^{4}}(1-3J)\,\delta^{ij}A_{i}^{\prime}A_{j}^{\prime}+\frac{1}{2a^{4}}(1+2J)\,\delta^{ik}\delta^{jl}\,\partial_{i}A_{j}\,\partial_{k}A_{l}+\frac{J}{H\,a^{5}}\,\delta^{ij}\,A_{i}{}^{\prime}\,\nabla^{2}A_{j}\,.$
(5.54)

The first term is the energy density of the Electric field $(\rho_{E})$. The
second and the third terms are the energy densities of the magnetic field
$(\rho_{B})$ and $(\rho_{B.B^{\prime}})$, respectively. Note that the first
two terms decay as $a^{-4}$ while the last term decays as $a^{-5}$. Hence, one
can ignore the last term in evaluating the energy density of electromagnetic
fluctuations. As we will see below, imposing the condition that the energy
density is always positive provides a condition on $J$.

Using the decomposition (5.13), the electric and magnetic part of the
perturbation spectrum per logarithmic interval can be written as:

$\displaystyle\mathcal{P}_{B}(k)\equiv\frac{\mbox{d}}{\mbox{dln}k}\langle
0|\hat{\rho}_{B^{2}}|0\rangle=\frac{(1+2J)}{2\pi^{2}}\frac{k^{5}}{a^{4}}\left|A_{k}\right|^{2}$
(5.55)
$\displaystyle\mathcal{P}_{E}(k)\equiv\frac{\mbox{d}}{\mbox{dln}k}\langle
0|\hat{\rho}_{E^{2}}|0\rangle=\frac{(1-3J)}{2\pi^{2}}\frac{k^{3}}{a^{4}}\left|A_{k}^{\prime}\right|^{2}$
(5.56)
$\displaystyle\mathcal{P}_{{}_{B.B^{\prime}}}(k)\equiv\frac{\mbox{d}}{\mbox{dln}k}\langle
0|\hat{\rho}_{{}_{B.B^{\prime}}}|0\rangle=-\frac{J}{4\pi^{2}}\frac{k^{5}}{a^{4}\mathcal{H}}(A_{k}^{\prime}A_{k}^{*}+{A_{k}^{\prime}}^{*}A_{k})~{}~{}~{}~{}.$
(5.57)

During most of the Universe’s history, the electrical conductivity of the
Universe is high. Hence, the electric fields decay and do not contribute to
the energy density. Hence, like in Sec. (5.2), we will concentrate on the
magnetic field component555In some models, the electric power spectrum
dominates over the magnetic part [170]. In our case, the electric power
spectrum does not contribute.. Imposing the condition that energy density is
non-negative implies that $J\geq-1/2$. Combining the earlier constraint, this
implies $J$ in the range $[-0.5,1)$ is well-defined.

Substituting Eq. (5.49) in Eq. (5.55) leads to:

$\displaystyle\mathcal{P}_{B}(k)=\frac{H^{4}(-k\,\eta_{f})^{4}}{4\pi^{2}}\frac{(1+2J_{f})(1-J_{*}(2-5\epsilon_{1}^{*})+J^{2}_{*}(1-5\epsilon_{1}^{*}))}{c_{A}^{*}(1-J_{*})(1-J_{*}+2J_{*}\epsilon_{1}^{*})^{2}}$
(5.58)

where $\eta_{f}$ refers to the end of inflation and $J_{f}=4DH_{f}^{2}$.

We now calculate the energy density of the generated electromagnetic fields by
integrating the power spectrum over the Fourier modes $k_{i}<k<k_{f}$ where
$k_{i}$ and $k_{f}$ refer to the initial and final modes leaving the horizon
during inflation, i. e.:

$\displaystyle\rho_{B}=\int_{k_{i}}^{k_{f}}\mathcal{P}_{B}(k)\,\mbox{dln}k=\frac{H_{f}^{4}(-k_{f}\,\eta_{f})^{4}}{16\pi^{2}}\frac{(1+2J_{f})(1-J_{*}(2-5\epsilon_{1}^{*})+J^{2}_{*}(1-5\epsilon_{1}^{*}))}{c_{A}^{*}(1-J_{*})(1-J_{*}+2J_{*}\epsilon_{1}^{*})^{2}}\,.$
(5.59)

We want to stress that the above expression is for a generic inflation model
and depends on the value of $J_{*}$. In the rest of the section, we determine
the range of $J_{*}$ for which the generated magnetic fields are sufficient
for the observable large-scale magnetic fields such that the generated
magnetic fields do not affect the background FRW metric. To avoid the back-
reaction, the energy density of the generated fields must be less than the
total background energy density during inflation. Assuming slow-roll inflation
$\epsilon_{1}<<1$, the back-reaction parameter ($\xi$) for the Galileon model
is [170, 81, 178]:

$\displaystyle\xi\equiv\frac{\rho_{\rm{B}}}{\rho_{\rm{Inf}}}\approx\frac{H_{f}^{2}}{48\pi^{2}\,M_{\rm{pl}}^{2}}\,\frac{1}{(1-\,J_{*})}<1$
(5.60)

where $\rho_{\rm{Inf}}=3H_{f}^{2}M_{\rm{pl}}^{2}$ is the background energy
density during inflation and $\rho_{B}$ is given by Eq.(5.59). As mentioned
above, we assumed $\epsilon_{1}<<1$ and set $-k_{f}\eta_{f}\sim 1$. From the
above condition, we get the following constraint:

$\displaystyle\frac{1}{1-J_{*}}\,<\,2.25\times 10^{13}$ (5.61)

Combined with the earlier constraint of $J$, we see that $J_{*}$ close to $1$
can lead to large amplification. Let us now consider two cases:

  1. ${\bf J_{*}>0}$:

When $J_{*}$ is very close to 1, we can have large amplification in the
magnetic field strength. Let us now estimate the magnetic field strength
corresponding to $(1-J_{*})^{-1}\approx 2.25\times 10^{13}$ at the comoving
wavenumber $k=1~{}{\rm Mpc}^{-1}$. The magnetic field for a mode with wave
vector $k$ at the end of inflation is given by:

$\displaystyle
B_{f}\approx\sqrt{\frac{H_{f}^{4}k^{4}\eta_{f}^{4}}{4\pi^{2}c_{s}^{*}(1-J_{*})}}\approx
1.58\times 10^{13}\,\rm{G}\,,$ (5.62)

where we have set $|\eta_{f}|=10^{-20}\rm{Mpc}$ [221] and since $J_{*}\to 1$
we have approximated $c_{s}^{*}\simeq 1$.

In the radiation-dominated (or matter-dominated) epoch, the magnetic field
decays adiabatically as $B\propto a^{-2}$. We then have the following relation
between the magnetic field strength at present and the end of inflation:

$\displaystyle B_{0}=B_{f}\left(\frac{a_{f}}{a_{0}}\right)^{2}.$ (5.63)

Assuming the instantaneous reheating and the entropy conservation, i.e.,
$gT^{3}a^{3}=\rm{constant}$ during its evolution, where $T$ is the temperature
of the relativistic fluid, and $g$ is the number of effective relativistic
degrees of freedom [81] gives

$\displaystyle\frac{a_{0}}{a_{f}}=\left(\frac{90}{8\pi^{3}}\right)^{1/4}\frac{g_{f}^{1/12}}{g_{0}^{1/3}}\frac{\sqrt{H_{f}\,M_{\rm{pl}}}}{T_{0}}\approx
0.9\times 10^{29}\,\left(\frac{H_{f}}{10^{-5}\,M_{\rm{pl}}}\right)^{1/2}$
(5.64)

where we have taken $g_{f}\approx 100$ and $g_{0}=2.64$. Substituting Eq.
(5.64) in Eq. (5.63),the present day magnetic field strength at
$\rm{Mpc}^{-1}$ scale is:

$\displaystyle B_{0}=1.5\times 10^{-45}\,\rm{G}$ (5.65)

  2. ${\bf J_{*}<0}$:

In this case, $1/(1-J_{*})\approx 1$. The magnetic field for a mode with wave
vector $k$ at the end of inflation is given by:

$\displaystyle
B_{f}\approx\sqrt{\frac{H_{f}^{4}k^{4}\eta_{f}^{4}}{4\pi^{2}c_{s}^{*}}}\approx
1.58\times 10^{7}\,\rm{G}\,,$ (5.66)

where, here again, we have set $|\eta_{f}|=10^{-20}\rm{Mpc}$ [221] and
$c_{s}^{*}\simeq 1$. Following the same analysis, the present day magnetic
field strength at $\rm{Mpc}^{-1}$ scale is:

$\displaystyle B_{0}=1.5\times 10^{-51}\,\rm{G}$ (5.67)

In estimating the magnetic field strength, we have used the standard values of
inflation, however, if inflation occurs over the energy scales such that
$10^{-10}\lesssim H_{f}/M_{\rm{pl}}\lesssim 10^{-5}$ [202], then the adiabatic
expansion rate will be much less, i. e.,

$\displaystyle 10^{26}\lesssim\frac{a_{0}}{a_{f}}\lesssim 10^{29}~{}~{}.$
(5.68)

In that case, we gain six orders of magnitude in the magnetic field strength.

Our analysis shows that appreciable magnetic field strength can be achieved
when $J_{*}\to 1$. However, as can be see in (5.2), $J_{*}>0$ leads to super-
luminal electromagnetic perturbations. Let us now compare the above result
with the EFT analysis. Following the discussion in Appendix (D.2.4), we find
that $s_{2}$ and $e_{1}$ are the non-zero parameters that describe vector
Galileon model. Substituting these in Eq. (5.23), we have:

$c_{A}=1-\frac{1}{2}\left[s_{2}-e_{1}(1-\epsilon_{1})\right]\left(\frac{H}{\Lambda}\right)^{2}.$
(5.69)

Using the relation (D.22), we have:

$c_{A}=1+\frac{J\epsilon_{1}}{8}\,.$ (5.70)

The EFT analysis shows that $J<0$ avoids super-luminal propagation. Hence, we
need another physical condition — the modes should be sub-luminal. This
conclusion is identical to the one we obtained from the EFT analysis. Thus,
our analysis shows that conformal invariance breaking is necessary but not
sufficient condition to generate sufficient magnetic fields in the early
Universe.

Another interesting thing to note is that by setting the above parameters in
the EFT power spectrum (5.33), we can see that the vector Galilean model will
never lead to a scale-invariant power spectrum and is consistent with the
analysis in this section.

###  5.4 Conclusions and Discussions

The origin of primordial magnetic fields is still unresolved and requires
physics beyond the standard models of cosmology and particle physics. Although
inflation provides a causal mechanism for the generation of primordial density
perturbations, it can not generate the appreciable primordial magnetic field
in the early Universe. It has been argued that conformal invariance breaking
is a sufficient condition to generate primordial magnetic fields during
inflation. In this chapter, using EFT based on expansion about the Hubble
parameter $(H)$ and its derivatives, we show that the generation of primordial
magnetic fields requires two necessary conditions — conformal invariance
breaking and causal propagation. We have also shown that 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,
$HH^{\prime}/\Lambda^{3}$, $HH^{\prime\prime}/\Lambda^{4}$, $\cdots$.

Like the EFT of inflation, the EFT of magnetogenesis requires the inclusion of
fluctuations in the matter and metric degrees of freedom. As shown in Appendix
A, the gauge field is the relevant gauge-invariant variable for the EFT.
Demanding that the EFT of magnetogenesis breaks conformal invariance, however,
satisfies local Lorenz invariance and gauge invariance, we obtained the
general second-order EFT action. This action depends on two expansion scalar
functions — $f_{1}(H,a,\Lambda)$ and $f_{2}(H,a,\Lambda)$. We then showed that
the EFT action (5.5) could reproduce all the known 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).

By truncating the expansion scalar functions to $\Lambda^{-2}$, we derived the
EOM of the gauge field and obtained the power spectrum in the slow-roll
inflation scenario. From Eq. (5.34), we see that the amplification in the
magnetic power spectrum — evaluated at the horizon crossing $(-k\eta c_{A}=1$)
— is possible whenever $3s_{1}-5d_{1}>0$. Specifically, we see for sub-luminal
($d_{1}<s_{1}$) or super-luminal ($s_{1}<0$) modes, the leading order
correction term in the power-spectrum ${3s_{1}-5d_{1}}>0$. In other words, we
can have large amplifications even for super-luminal fluctuations. To avoid
EFTs with superluminal fluctuations [207], we need another physical condition
— the modes should be sub-luminal.

We then considered a specific model of inflationary magnetogensis where the
vector Galileon breaks the conformal invariance. We extensively studied the
magnetic field generation during inflation, considering the total action. Due
to the absence of coupling between scalar field and electromagnetic field, the
model does not lead to a strong coupling problem [102]; hence, the model does
not require fine-tuning, which is an interesting feature of our model.
Furthermore, the evolution of vector modes is frozen on the super-horizon
scale. We showed that the model predicts the present-day magnetic field of
strength $B_{0}\sim 10^{-45}{\rm G}$ on the cosmological (Mpc) scale which is
several orders higher compared to the standard electromagnetic action, and
there is no back-reaction problem. For low-scale inflationary models i.e.,
$10^{3}-10^{4}\,{\rm GeV}$, the model generates $B_{0}\sim 10^{-33}\rm{G}$.
However, this comes with the price — super-luminal electromagnetic
perturbations. In other words, the model generates large magnetic fields and
does not violate any other conditions; it leads to super-luminal
perturbations.

Modifications of general relativity provide an alternative explanation for
cosmological inflation [222]. Modified gravity theories have extra degrees of
freedom that might have interesting physical consequences in the early
Universe. For instance, Stelle gravity [223] contains massive tensor modes,
and these modes carry more energy than the scalar modes in $f(R)$ gravity
models [224]. While the analysis presented in this chapter can be extended to
$f(R)$ gravity models, it is not straightforward to extend to Stelle gravity.
Investigating the EFT of inflation and magnetogenesis in these modified
gravity models is interesting. Also, it will be interesting to extend the EFT
of magnetogenesis to bouncing models. The analysis presented in this chapter
can be extended to helical fields.

## Chapter 6 Gertsenshtein-Zel′dovich effect: A plausible explanation for fast
radio bursts?

Technological advancement has fuelled research in high-energy astrophysical
phenomena at larger redshift ranges, and we are in a position to address some
unresolved questions starting from pulsar emission mechanism [225] to short
bursts such as Gamma-ray bursts (GRBs) [226], Fast radio bursts (FRBs) [227,
228, 229]. To date, more than 600 FRBs have been reported in various
catalogues [230, 229, 231, 232]. $99\%$ of these FRBs have the following three
characteristic features: observed peak flux ($S_{\nu}$) varies in the range
$0.1~{}{\rm Jy}<S_{\nu}<700~{}{\rm Jy}$, coherent radiation and the pulse
width is less than a second [232, 230]. These observations have posed the
following questions: What causes these extreme high-energy transient radio-
bursts from distant galaxies, lasting only a few milliseconds each [227, 228,
229]? Why do some FRBs repeat at unpredictable intervals, but most do not
[229]? Does strong gravity provide an active role?

In the previous chapters, we have looked at the generation and effects of
large scale magnetic fields. In this chapter, we will discuss the interesting
consequences of small-scale strong magnetic fields, for example, magnetar (see
Fig. 1.4 in chapter 1). As we have briefly discussed in chapter 1, the origin
of FRBs is one of the open problems in modern cosmology and astrophysics. In
this chapter, we will attempt to address the origin of the FRBs and their
coherent nature of radiation. More specifically, what is the physical
mechanism that results in large amount of coherent radiation in a short time
[227, 228, 230, 229, 231, 232].

Naturally, many models have been proposed to explain the origin of FRBs. All
these models try to provide a physical mechanism that results in large amount
of coherent radiation in a short time [227, 228, 230, 229, 231, 232]. Since
the time scale of these events is less than a second, and the emission is
coherent, the astrophysical processes that explain these events _cannot_ be
thermal [227].

Broadly, these models can be classified into two categories [233]: FRBs
created by interaction of an object with a pulsar/magnetar and FRBs created
from the magnetar/pulsar itself [227, 228, 230, 229, 231, 232]. The first
category can further be classified into two broad classes. In the first class,
the energy powering FRBs comes from the neutron star magnetosphere/wind
themselves, and an orbiting object converts this energy into radiation. In the
second class, the object falls onto the neutron star, and its gravitational
energy partly gets converted to FRBs. In the second class, many models
involving non-thermal processes such as Synchrotron radiation [234], black
hole super-radiance [235], evaporating primordial black hole [236, 237], spark
from cosmic strings [238], Quark Novae [239], synchrotron maser shock model
[240], radiation from reconnecting current sheets in the far magnetosphere
[241], curvature emission from charge bunches [242] have been proposed.
Several classes of FRB models predict prompt multiwavelength counterparts and
specify the ratio between the energy emitted by the counterpart and by the FRB
[243, 244].

However, despite the use of exotic new physics, no single model has provided a
universal explanation for the enormous energy released in these events. It is
important to note that all these mechanisms require electromagnetic
interaction to generate FRBs. Due to the nature of electromagnetic
interaction, small-scale emission mechanisms usually predominate over large-
scale coherent electromagnetic processes (like astrophysical masers and pulsar
radio emission). In this chapter, we provide an alternative framework that
overcomes this and can explain the observed coherence in FRBs.

As shown below, one key missing ingredient is the _dynamics of strong-
gravity_. The Spatio-temporal changes in the strong-gravity regime —
oscillons, phase transitions, plasma instability, primordial black holes,
reheating — generate gravitational waves (GWs) in a broad range of frequencies
($10^{-15}-10^{15}$ Hz) [245, 246, 247, 248, 249, 250, 251]. Like EM waves,
GWs are generated by the time-varying quadrupole moment [252, 253]. Since all
masses have the same gravitational sign and tend to clump together, they
produce large coherent bulk motions that generate _energetic, coherent GWs_
[254]. Thus, if a mechanism that converts incoming coherent GWs to EM waves
exists, we can explain the extremely energetic, coherent nature of FRBs [255,
256]. In this chapter, we construct a model that uses this feature.

Since FRBs are highly energetic, an attentive reader might wonder do incoming
GWs carry such large energies. GWs carry an enormous amount of energy. For
example, typical GWs from a compact binary collapse with amplitude $h\sim
10^{-22}$ carry the energy of the order of $10^{20}~{}\rm{Jy}$ [252, 253]. If
the GWs indeed carry a lot of energy, can this energy transform into other
observable forms of energy?

Currently, there is evidence of GWs in the frequency range
$10^{-9}-10^{4}~{}{\rm Hz}$ from LIGO-VIRGO-KAGRA and PTA observations [257,
258]. While most of the current effort has focused on these frequency ranges,
there is a surge in activity for the possibility of detecting GWs in the MHz-
GHz frequency range [249]. New physics beyond the standard model of particle
physics, like an exotic compact object, can produce observable GW signals in
this frequency ranges [249, 250, 251].

Figure 6.1: Schematic depiction of GZ effect. The externally generated GWs at
distance $D$ from the NS is converted to EM Waves in the magnetosphere.The
blue region corresponds to NS with radius $r_{*}$ and the yellow region around
NS corresponds to magnetosphere of radius $R_{\rm LC}$. The black curves
correspond to the magnetic field lines.

A physics maxim is that energy can be transformed between different forms.
Although the total energy is conserved, the efficiency of the transformation
depends on the energy scale, background dynamics, and external conditions
(parameters). Energy transformation is one way to probe strong gravity regions
like the early universe, black-holes, and NS. In this chapter, we propose a
novel approach that uses the energy conversion from incoming, coherent GWs to
electromagnetic (EM) waves that can _explain milli-second bursts like FRBs_.

GWs get converted to EM waves in the presence of strong transverse magnetic
fields — Gertsenshtein-Zel’dovich (GZ) effect [259, 260, 261, 262]. To
understand the GZ effect, consider coherent GWs with a frequency $\omega_{g}$
passing through a region with a high transverse static magnetic field ${\bf
B}$. The propagation of GWs leads to compression and stretching of the
magnetic field proportional to $h{\bf B}$ ($h$ is the amplitude of GWs), which
acts as a source leading to the generation of EM waves [259, 260]. The induced
(resultant) EM waves generated will have maximum amplitude at resonance, i.e.,
the frequency of EM waves is identical to $\omega_{g}$. (For details, see
appendix E.1)

In quantum mechanical language, the GZ effect is analogous to the mixing of
neutrino flavors — the external field _catalyzes_ a resonant mixture of photon
and graviton states [263]. The external magnetic field provides the extra
angular momentum necessary for the spin-1 (photon) field to mix with the
spin-2 (graviton) field. Thus, the GZ mechanism involves the transfer of
energy from the incoming, coherent GWs to emitted, coherent EM radiation in
the presence of the background magnetic field. The maximum efficiency of this
conversion can be achieved if the background field is strong at the resonance
frequency (when the frequency of the emitted EM radiation is the same as the
incoming GWs). Hence, the background magnetic field acts as a catalyst in this
mechanism.

We show that the _GZ effect_ may provide an interpretation for a class of non-
repeating FRBs. To explain the energy bursts in FRBs, we propose a model with
the following two realistic assumptions: (i) the astrophysical object is
compact and has a strong gravity environment, and (ii) the object possesses a
small time-dependent magnetic field on top of the large, effective static,
transverse magnetic field. These two assumptions principally lead us to
stellar remnants, such as NS and magnetars with magnetic field strength
ranging from $10^{8}-10^{15}~{}\rm{G}$ [264, 265, 228, 227]. The small time-
dependent magnetic fields arise due to the rotation of the NS about its axis
with frequency $\omega_{B}$ [265, 266, 229, 267, 268]. We consider
$\omega_{B}$ in the range $[1,10^{3}]~{}{\rm Hz}$ [234]. As a result, the
effective magnetic field at a given point in the NS magnetosphere is ${\bf
B}(t)={\bf B}^{(0)}+\delta{\bf B}\sin(\omega_{B}t)$. It has been noted that
$|\delta{\bf B}/{\bf B}^{(0)}|$ can be as large as $0.1$ [268]. Here, we take
$|\delta{\bf B}/{\bf B}^{(0)}|\sim 10^{-2}$.

###  6.1 Model

6.1 gives the schematic depiction of the physical model to explain the energy
burst in FRBs. Consider GWs generated due to exotic compact objects (such as
Boson stars, Oscillons, gravastars) [249, 250, 251] passing through the
magnetosphere of NS at a distance $D$. In the figure, the magnetosphere is
depicted as a cylinder. GZ effect converts GWs to EM waves as they pass
through the magnetosphere [259, 260]. This conversion occurs at all points in
the magnetosphere. Therefore, a faraway observer will see the integrated
effect happening in the entire magnetosphere in this short duration. For
example, the light cylinder radius ($R_{\rm LC}$) for a typical NS is $\sim
10^{7}-10^{9}~{}{\rm cm}$, implying that the GWs take less than one second to
cover the entire magnetosphere. This is one of the primary ingredients
supporting our analysis for the FRB observations.

To compute the GZ effect at a point in the magnetosphere, we consider source-
free Maxwell’s equations on the background space-time with GW fluctuations
(see details in Appendix (E.1)). The linearized Einstein’s equations, up to
first order in the space-time perturbations are highly accurate. In this
approximation, the effects of GWs on the stress-tensor (Riemann tensor) are
negligible. Hence, we consider background space-time to be Minkowski (in
cartesian coordinates) [96]). The two polarizations of GW (with frequency
$\omega_{g}$ and wave-vector $k_{g}$) propagating along the z-direction are:

$\displaystyle
h_{+}=A_{+}\,e^{i\left(k_{g}z-\omega_{g}t\right)},h_{\times}=iA_{\times}\,e^{i\left(k_{g}z-\omega_{g}t\right)},$
(6.1)

where $A_{+}$ and $A_{\times}$ are the constant amplitudes of the GWs. We
assume that both the modes of GWs are generated with an equal amount of
energy, i.e., $|A_{+}|=|A_{\times}|$ — the isospectrality condition in general
relativity [269]. Taking the distance between GW source (possibly an exotic
compact object) and NS to be $D=1\,\rm{kpc}$, gives $h\simeq A_{+}=10^{-20}$
at $1\,\rm{GHz}$ (see Appendix E.5). In this chapter, we have assumed $h$ to
be three orders smaller ($\sim 10^{-23}$) near the NS.

As mentioned above, the key requirement of the GZ-effect is the presence of
the transverse magnetic field to the direction of propagation of coherent GWs.
Besides, the time taken by the GWs to pass through the entire magnetosphere is
much smaller than the rotation period of the millisecond pulsar/magnetar
($\omega_{B}^{-1}$). Given the direction of propagation of GWs along the
z-axis, the effective time-dependent transverse magnetic field is taken to be
$\textbf{B}(t)=\left(0,B^{(0)}_{y}+\delta B_{y}\sin(\omega_{B}t),0\right)$
[267, 268].

Although the effective magnetic field depends on the distance from the surface
of the object [227, 265, 228, 264], we assume that ${\bf B}(t)$ is independent
of the distance from the surface up to $R_{\rm LC}$ [234]. In appendix (E.2),
we explicitly show that the above assumption that the background magnetic
field can be treated as a constant in the entire magnetosphere gives
_identical results_ to that of the background field decreasing radially, i.
e.,

$\displaystyle\left(B_{r},B_{\theta},B_{\phi}\right)=B_{*}\,\left(\frac{r_{*}}{r}\right)^{3}\,\left(2\cos\theta,\sin\theta,0\right)$
(6.2)

where $B_{*}$ is the magnetic field on the NS surface. More specifically,
assuming that the NS magnetic field is dipolar, we show in appendix (E.2) that
we can approximate the average magnetic field at any point in the
magnetosphere to be constant. In other words, the total conversion factor we
obtain using the above assumption mimics the realistic NS regions.

Given the above setup, we now evaluate the GZ-effect in the magnetosphere of
the NS and compare it with observational quantities in two steps:

  1. 1.

First step involves evaluating the conversion from coherent GWs to EM waves at
a typical point inside the magnetosphere. This is referred to as _conversion
factor_ ($\alpha$). We then obtain the total conversion factor ($\alpha_{\rm
tot}$) inside the entire magnetosphere _at resonance_ (the frequency of EM
waves is identical to $\omega_{g}$). This conversion factor includes only
those contributions that are along the direction of line-of-sight and
coinciding with the incoming GWs. Also, only the transverse magnetic field
component to the direction of propagation at each point of the magnetosphere
will contribute to the emitted EM waves. This can potentially explain the
coherent nature of FRBs [270, 271].

  2. 2.

For a given conversion factor, we obtain the Poynting vector of the resultant
EM waves along the direction of propagation ($S_{z}$). Then, we compare the
theoretically derived Poynting vector with the observation of peak flux with
the reported FRBs.

The Poynting vector is a well-defined quantity for photons that travel from
the source to the observer without any hindrance [272, 273, 274]. More
specifically, assuming there is no absorption/emission of the photons during
the entire journey, the Poynting vector (of the EM waves) $S_{z}$ is conserved
(independent of the distance between the magnetosphere and observer) and is
valid for emissions from compact sources. Hence, the Poynting vector estimated
at a small angle (along the direction of the incoming gravitational waves)
remains the same at the source and the detector. Furthermore, as shown in 6.1,
the incoming, coherent GW is along the $z-$axis in the entire magnetosphere.
Therefore, the cumulative effect of the emitted EM waves is in the same
direction.

To evaluate the conversion factor, solving the linearized covariant Maxwell’s
equations leads to the following electric and magnetic fields induced due to
GWs, i.e., $\tilde{E}_{x}$ and $\tilde{B}_{y}$ as111We thank Archana Pai for
pointing out a mistake in $\tilde{E}_{x}$ where $1/2$ factor was missing in
the first term.:

$\displaystyle\tilde{E}_{x}$
$\displaystyle\simeq-\frac{A_{+}}{4}\,B^{(0)}_{y}\left(1-\xi\,\omega_{B}t\,\,\right)\,\,e^{i\left(k_{g}z-\omega_{g}t\right)}$
(6.3) $\displaystyle\tilde{B}_{y}$
$\displaystyle\simeq-\frac{A_{+}}{4}\,B^{(0)}_{y}\left(1+2\xi\,\omega_{g}t\,\,\right)e^{i\left(k_{g}z-\omega_{g}t\right)}\,,$
(6.4)

where $\xi\equiv\delta B_{y}/B_{y}^{(0)}$. Note that the amplitude of
$\tilde{B}_{y}$ has a dependence on $\omega_{g}$, while the amplitude of
$\tilde{E}_{x}$ has $\omega_{B}$ dependence on $\omega_{g}$. This is because
the induced electric field arises due to the time-varying magnetic field (see
appendix E.1).

The conversion factor ($\alpha$) — ratio of the energy density of EM wave and
GWs — gives the efficiency of the process at resonance. $\alpha$ for this
process is

$\displaystyle\\!\\!\\!\alpha\equiv\frac{\rho_{\rm EM}}{\rho_{\rm
GW}}\simeq\frac{G|B^{(0)}_{y}|^{2}}{4c^{2}}\left[2\left(\frac{\xi
z}{c}\right)^{2}+2\frac{\xi}{\omega_{g}}\frac{z}{c}+\frac{1}{\omega_{g}^{2}}\right]$
(6.5)

where $z$ refers to the radial distance in the magnetosphere. For details, see
appendix E.3.

The above expression is the conversion factor at a single point on the
magnetosphere. Assuming that there is no cross-correlation of GZ-effect at two
distinct points in the magnetosphere, we obtain the total conversion by
integrating over the entire magnetosphere (from the surface of the compact
object to the light cylinder $R_{\rm LC}$). (The cross-correlation corresponds
to the induced EM waves at two distinct points affecting each other. This
physically corresponds to higher-order effects
($A_{+}\partial_{z}\tilde{B}_{y}$) which are neglected.). Thus, the total
conversion factor is

$\displaystyle\alpha_{\rm{tot}}\simeq\frac{\pi
G|B^{(0)}_{y}|^{2}}{c^{2}}\left[\frac{2}{3}\left[\frac{\xi R_{\rm
LC}}{c}\right]^{2}\\!\\!+\frac{\xi R_{\rm
LC}}{\omega_{g}c}+\frac{1}{\omega_{g}^{2}}\right].$ (6.6)

See appendix E.3 for details. It is important to note that $\alpha_{\rm tot}$
is _independent of the amplitude_ of GWs. To understand the variation of
$\alpha_{\rm tot}$ with $\omega_{g}$, in 6.2, we have plotted the conversion
factor for magnetar and NS/milli-second pulsar.

Figure 6.2: Log-Log plot of $\alpha_{\rm tot}$ versus $\omega_{g}$ for typical
magnetar (left plot) and typical NS (right plot). For magnetar, we set
$B^{(0)}_{y}=10^{15}\,{\rm G},R_{\rm LC}=10^{9}~{}\,{\rm
cm},\omega_{B}=1\,{\rm Hz}$. For NS, we set $B^{(0)}_{y}=10^{10}{\rm G},R_{\rm
LC}=10^{7}{\rm cm},\omega_{B}=1{\rm kHz}$.

From the plots, we infer that the total conversion factor is almost
insensitive at higher frequencies ($>1~{}{\rm MHz}$) for both types of compact
objects. This is because the second and third terms in RHS of Eq. (6.6) are
inversely proportional to $\omega_{g}$ and, hence, the contribution to
$\alpha_{\rm tot}$ is only from the first term. To elaborate, 6.3 plots each
of these terms in Eq. (6.6) which shows a clear distinction between milli-
second Pulsar and magnetar. In both cases, the cross-over occurs below
$1~{}{\rm MHz}$. Since we are interested in radio frequency in the GHz range,
the total conversion factor (6.6) is independent of the incoming, coherent GW
frequency.

This leads to the important question: What is the efficiency of the GZ-effect
near magnetar and NS? Table (6.1) lists the total conversion factor (4th
column) for magnetar and NS for different frequencies. We want to emphasize
the following points: First, as mentioned earlier, we have assumed the
amplitude of the GWs, i.e., $A_{+}=A_{\times}=1.4\times 10^{-23}$ [247, 249].
However, as we show below, even with this conservative value, the model
explains the peak flux of FRBs. Second, $\alpha_{\rm tot}$ is very high near
the magnetar for low-frequency GWs. Specifically, the conversion is $0.06\%$
and $0.25\%$ at 1000 Hz and 500 Hz, respectively (for the magnetar
$B_{y}^{(0)}=10^{15}\,{\rm G}$). For high-frequency GWs, as we show below,
even a total conversion factor $(\alpha_{\rm tot})$ of $10^{-19}$ can lead to
appreciable energy in the radio frequency (GHz range).

Figure 6.3: Log-Log plot of the three terms in the RHS of Eq. (6.6) versus
$\omega_{g}$ for typical magnetar (top plot) and typical NS (bottom plot). For
magnetar, we have set $B^{(0)}_{y}=10^{15}{\rm G},R_{\rm LC}=10^{9}~{}{\rm
cm},\omega_{B}=1{\rm Hz}$. For NS/milli-second pulsar, we have set
$B^{(0)}_{y}=10^{10}{\rm G},R_{\rm LC}=10^{7}{\rm cm},\omega_{B}=1{\rm kHz}$.

Further, to compare the generated EM energy density in the entire
magnetosphere with the observations, we compute the flux of the induced EM
waves (6.3, 6.4) by calculating the Poynting vector [275]. Rewriting Eq. (6.6)
as a quadratic equation in $\xi R_{\rm LC}/c$ provides the functional
dependence in-terms of $\alpha_{\rm tot}$. This leads to:

$\displaystyle
S_{z}\simeq\frac{A_{+}^{2}|B_{y}^{(0)}|^{2}c}{256\pi}\left[\sqrt{\frac{24c^{2}\omega_{g}^{2}\alpha_{\rm
tot}}{\pi G|B^{(0)}_{y}|^{2}}-15}-\frac{12c^{2}\omega_{g}\omega_{B}\alpha_{\rm
tot}}{\pi G|B^{(0)}_{y}|^{2}}-1\right].$ (6.7)

Note that $\alpha_{\rm{tot}}$ in RHS of the above expression is a function of
$\omega_{g}$ and the parameters of magnetar/NS. See appendix E.4 for details.
Thus, the Poynting vector of the induced EM waves in the vicinity of
magnetar/NS can be obtained by substituting the the parameters of magnetar/NS,
$\omega_{g}$ and $A_{+}$. To compare with the peak flux of FRBs, 6.4 contains
the plot of the Poynting vector per unit frequency $(S_{z}/\omega_{g})$ as a
function of $\omega_{g}$ for magnetar and NS.

Figure 6.4:  Log-Log plot of $S_{z}/\omega_{g}$ versus $\omega_{g}$ for
typical magnetar (top plot) and typical NS (bottom plot). For magnetar, we
have set $B^{(0)}_{y}=10^{15}{\rm G},R_{\rm{LC}}=10^{9}~{}\,{\rm
cm},\omega_{B}=1\rm{Hz}$. For NS/milli-second pulsar, we have set
$B^{(0)}_{y}=10^{10}\,{\rm G},R_{\rm{LC}}=10^{7}{\rm
cm},\omega_{B}=1\,\rm{kHz}$. For both plots we have set $A_{+}=1.4\times
10^{-23}$ corresponding to a typical GW source [249].

The last column in Table (6.1) contains the spectral flux density (Poynting
vector per unit frequency) for generic parameter ranges of $R_{\rm LC}$ and
$B_{y}^{(0)}$. Given the parameters listed in the table (6.1), our model
predicts a range of spectral flux density that can be as small as $0.1~{}{\rm
Jy}$ (milli-second Pulsar) and can be as large as $10^{11}~{}{\rm Jy}$
(Magnetar). We are now in a position to compare the results of the model with
radio observations in GHz frequency.

###  6.2 A plausible explanation for FRBs

As mentioned earlier, more than 600 (non-repeating) FRBs are reported in
various catalogues [230, 229, 231, 232] $99\%$ of these FRBs were found to
have the following three characteristic features: peak flux ($S_{\nu}$) varies
in the range $0.1~{}{\rm Jy}<S_{\nu}<700~{}{\rm Jy}$, the pulse width is less
than one second and coherent radiation [232, 230]. In Ref. [276], the authors
classified the quantities associated with FRBs into two types — observed and
derived. Peak flux and width are observed quantities, while Luminosity and
Luminosity distance are derived quantities. The uncertainty in the Dispersion
Measure versus redshift relation can have a cascading effect on the derived
quantities of the FRBs, leading to uncertainty in the origin of these events
[277]. Hence, to reduce the systematic bias, we focus on observed Peak flux
and estimate the same using the Poynting vector of the emitted EM waves
($S_{z}/\omega_{g}$).

$R_{\rm LC}$ | $B_{y}^{(0)}$ | $\omega_{g}$ | $\alpha_{\rm tot}$ | $\rho_{\rm EM}\,$ | $\frac{S_{z}}{\omega_{g}}$
---|---|---|---|---|---
(cm) | (Gauss) | (MHz) |  | ($\rm{Jy\,cm^{-1}\,s\,Hz}$) | (Jy)
$10^{9}$ | $10^{15}$ | $1$ | $1.74\times 10^{-5}$ | $9.3\times 10^{10}$ | $9.95\times 10^{11}$
$10^{9}$ | $10^{12}$ | $500$ | $1.72\times 10^{-11}$ | $2.3\times 10^{10}$ | $9.94\times 10^{5}$
$10^{8}$ | $10^{11}$ | $1400$ | $1.72\times 10^{-15}$ | $1.8\times 10^{7}$ | $961.57$
$10^{7}$ | $10^{10}$ | $1400$ | $1.72\times 10^{-19}$ | $1.8\times 10^{3}$ | $0.99$
$10^{8}$ | $10^{9}$ | $1400$ | $1.72\times 10^{-19}$ | $1.8\times 10^{3}$ | $0.09$
Table 6.1: The table contains numerical values of the total conversion factor
($\alpha_{\rm tot}$), energy density of EM waves ($\rho_{\rm EM}$), and
spectral flux density (Poynting vector per unit frequency). The first two rows
are for a typical Magnetar and the last three rows are for a typical NS. We
have set $A_{+}=1.4\times 10^{-23}$ corresponding to a typical GW source
[249].

As mentioned earlier, $R_{\rm LC}$ of a typical NS is $\sim
10^{7}-10^{9}~{}\rm{cm}$ and it takes less than one second for the GWs to pass
through the entire magnetosphere. This directly implies that the induced EM
waves due to GZ-effect will appear as a burst lasting for less than one
second. Thus, our model provides a natural explanation for FRBs lasting less
than a second. Further, we see from the last column of Table (6.1) that our
model predicts the burst of EM wave with the flux $<1000~{}\rm{Jy}$. Thus, our
model naturally explains the observed peak flux and pulse width of $99\%$ of
the reported FRBs.

Having estimated the peak flux and pulse width for our model, we can now
calculate other quantities associated with FRBs. In particular, we calculate
two such quantities — Fluence (${\cal F})$ [278, 279] and Isotropic Equivalent
Luminosity (IEL) [280]. The Fluence is defined as the product of the burst
width $\Delta t=(2R_{LC})/c$ and the peak flux ($S_{z}/\omega_{g}$) [278,
279]:

${\cal F}=(S_{z}/\omega_{g})\times\Delta t.$

Using $S_{z}$ from the table 6.1, we see that our model predicts the Fluence
in the range $1<{\cal F}({\rm Jy~{}ms})<10^{7}$. We now rewrite the energy
density $(\rho_{\rm EM})$ of emitted EM waves in terms of the isotropic
equivalent luminosity ($L_{\rm IEL}$). Assuming that the progenitor is at a
distance $d$ from the observer, $L_{\rm IEL}$ is given by [280]:

$\displaystyle\rho_{\rm EM}=\frac{L_{\rm IEL}}{4\pi d^{2}c}.$ (6.8)

The above expression assumes that $d\gg$ size of the progenitor is justified
for the reported FRBs. Due to the lack of confirmed detection in other
wavelengths, the distance of the FRBs is not well constrained [233]. The
dispersion measure of the FRB gives one indirect estimation [281, 282]. It is
expected that the distance of the FRBs is in the range $[10~{}{\rm
kpc}-1~{}{\rm Gpc}]$.

From the FRB catalogs [230, 229, 231, 232], we see that FRBs have a peak flux
of around $100~{}{\rm Jy}$. For our model, this corresponds to the energy
density of the emitted EM waves to be $\sim 9.2\times 10^{5}~{}{\rm
Jy\,cm}^{-1}{\rm s\,Hz}$. Taking the above distance range ($[10~{}{\rm
kpc}-1~{}{\rm Gpc}]$), the above energy density translates to $L_{\rm IEL}$ in
the range $[3.1\times 10^{39}-10^{49}]~{}{\rm erg/s}$. Thus, our model can
explain the inferred luminosity of FRBs [283].

Given this, we can now identify the possible progenitors of FRBs. To identify
this, we consider two FRBs — FRB120127 [284] and FRB011025 [285]. These two
sources represent a typical FRB in the catalogs. The observations of these two
FRBs at $1.5~{}\rm{and}~{}1.3~{}\rm{GHz}$, show a typical peak flux of
$0.62^{+0.35}_{-0.10}~{}\rm{Jy}$ and $0.54^{+0.11}_{-0.07}~{}\rm{Jy}$,
respectively. From the last row of the Table (6.1), we see that our model
predicts the progenitor should be a millisecond pulsar with an effective
magnetic field strength of $10^{10}~{}{\rm G}$ and $R_{\rm{LC}}\sim
10^{7}~{}\rm{cm}$. Fluence for these two FRBs is around $0.66~{}\rm{Jy\,ms}$
and isotropic equivalent luminosity $(L_{\rm IEL})$ at a distance
${d=10~{}{\rm Mpc}}$ is $3\times 10^{42}\rm{erg\,s^{-1}}$.

FRB 200428 is confirmed to be a galactic FRB [286]. Recent observations
indicate that the magnetar SGR 1935+2154 residing in the Milky Way is
associated with the FRB 200428 with a fluence of $>1.5\times 10^{6}~{}{\rm
Jy~{}ms}$ in the $1.28-1.4$ GHz band detected by the STARE2 radio array [286,
287]. It is reported that this magnetar has a surface dipole magnetic field of
$\sim 2.2\times 10^{14}$ G, which can be deduced from the period slow-down
rate of $3.24~{}{\rm s}$. Substituting the constant magnetic field
approximation $B_{y}(0)\sim 5\times 10^{11}~{}{\rm G}$ with $R_{\rm LC}\sim
10^{9}~{}{\rm cm}$ and $A_{+}\sim 10^{-26}$ in Eq.(6.7), our model predicts
peak flux $\sim 4.8\times 10^{4}~{}{\rm Jy}$ and Fluence to be $1.65\times
10^{7}{\rm Jy~{}ms}$. Thus our model predicts isotropic equivalent luminosity
at $~{}10\ {\rm kpc}$ to be $L_{\rm IEL}\sim 1.4\times
10^{43}\rm{erg\,s^{-1}}$. Consequently, our model has the potential to explain
these observations and play a crucial role in any future FRB and progenitor
association (if any).

We can do a similar analysis for all the FRBs in the catalog with a pulse
width of less than one second [232, 230]. Our model predicts that the
progenitor can be a NS with an effective magnetic field strength in the range
$10^{9}-10^{11}~{}{\rm G}$ and rotation frequency $1<\omega_{B}<1000$. Our
model can provide an explanation for the observed peak flux for a class of
_non-repeating_ FRBs and predicts FRBs with extremely low and high peak flux.
Note that our model is not sensitive to the galactic environment. Such future
detections will further strengthen the model.

As mentioned earlier, the model uses the GZ mechanism to explain the origin of
FRBs. GZ mechanism involves the transfer of energy from the incoming GWs to
emitted EM radiation in the presence of the background magnetic field.
Although our model falls in the first category where FRBs are generated from
the magnetar/NS, electromagnetic radiation is generated when GWs pass through
the magnetosphere. In other words, the background magnetic field acts as a
catalyst in this mechanism. This can be verified by evaluating the ratio of
the energy density of the outgoing EM waves ($\rho_{\rm EM}$) with the energy
density of the background magnetic field ($\rho_{B}$). For a magnetic field
strength of $10^{10}{\rm G}$, this ratio is minuscule:

$\frac{\rho_{\rm EM}}{\rho_{B}}\sim 2.2\times 10^{-39}.$

Thus, the proposed mechanism is not expensive compared to the magnetic energy
present in the magnetosphere.

###  6.3 Discussions

Our mechanism requires that the emitted GWs pass through the NS, resulting in
non-repeating FRBs along the line of sight of the observer. In other words, we
have assumed that the EM emission due to the GZ-mechanism in the pulsar
magnetosphere is directional dependent. The intervening medium does not impact
the radiation from this mechanism. More importantly, the mechanism requires
that the emitted GWs pass through the NS, resulting in FRBs along the line of
sight of the observer. The probability of this event is a product of the
probability that the GW passes through the NS and the probability that the
emitted EM is along the line of sight of the observer. Hence, the probability
of seeing such an event is not very high. More importantly, any existing
NS/Magnetar in any galaxy can produce FRB.

Further, it is estimated that around $10^{8}-10^{9}$ NSs are present in the
Milky Way galaxy, roughly $1\%$ of the total number of stars in the galaxy
[288, 289]. Also, it is estimated that the magnetar formation rate is
approximately $1-10$ percent of all pulsars [290, 291]. One of the basic
assumption of our model is that, given GW signal in GHz frequency, all NSs can
act as source of FRBs at all times. This assumption translates to the fact
that the maximum FRB events per day can be $10^{8}-10^{9}$. However, the
observed FRB rate is $10^{3}$ for the entire sky per day. This can be
attributed to the fact that the probability of this event is a product of the
probability that the GW passes through the NS times the probability that the
emitted EM is along the line of sight of the observer. Consequently, our model
predicts the observed FRB event rate and the coherent nature of FRBs [270].
Since, GWs can be generated up to 14 GHz [292, 293], our model naturally does
not have high-energy counterpart.

As mentioned in the Introduction, the co-planar property of the EM wave
emitted due to the GZ mechanism is due to the coherent nature of the incoming
GWs [254]. Due to the co-planar property, it can maintain the flux [256].
Hence, the three key features of FRBs are naturally explained in our model.

A variety of processes generate GWs in a broad range of frequencies [245, 246,
247, 248, 249, 251]. However, it is possible to detect these waves only in a
limited range. The proposed model provides an indirect mechanism to detect GWs
at high frequencies. Interestingly, our model also provides a way to test
modified theories of gravity. In this chapter, we have focused on GWs in
general relativity. Certain modified theories like Chern-Simons gravity lead
to birefringence [191] which can explain the polarized nature of FRBs.
However, exact rotating solutions in these theories are unknown and require
sophisticated numerics. Thus, the high-frequency GWs will provide a unique
view of the Universe — it’s birth and evolution.

## Chapter 7 Conclusions and future outlook

Due to technological advancements and the availability of vast amounts of data
from various observations, we are currently in an era of precision cosmology.
Although precision cosmology enhances our understanding of the Universe, it
also challenges established theoretical ideas, giving rise to intriguing new
questions. To seek answers to these questions, new ideas are introduced, often
requiring going beyond the standard model of cosmology. These issues are
commonly referred to as _open problems_ (see sec 1.4). Finding solutions to
these problems requires either extending the standard model or considering the
fine-tuning of model parameters. In this thesis, we looked at three problems

  1. 1.

What is the origin of large-scale magnetic fields?

  2. 2.

Why there is more matter than antimatter and how to create the observed amount
of matter-antimatter asymmetry in the Universe?

  3. 3.

What is the origin of Fast radio bursts in the sky?

and proposed novel mechanisms to resolve them.

Chapter 1 discussed these open problems in detail. The common theme of this
thesis is the generation of cosmological magnetic fields and using these
fields as a tool to probe the Universe at very large (scales of galaxies) and
small scales (scales of compact objects). As discussed in chapter 1, the
magnetic fields are present at all length scales, from small-scale but very
strong magnetic fields of magnetars to large-scale weak magnetic fields in
galaxies. We also saw that understanding the origin and evolution of the
magnetic fields on different epochs provides novel probes to the Universe at
relevant scales. Unfortunately, the origin of magnetic fields on large scales
is still unresolved because it is difficult to generate sufficiently strong
primordial seed magnetic fields that are coherent over large scales. This
requires physics beyond the standard models of cosmology. Although inflation
provides a causal mechanism for generating primordial density perturbations,
it can not generate the appreciable primordial magnetic field in the early
Universe. It has been argued that conformal invariance breaking is necessary
to generate sufficiently strong primordial magnetic fields during inflation.
Various mechanisms and models have been proposed in the literature to break
the conformal invariance of the electromagnetic fields; however, they mostly
involve fine-tuning the coupling parameters. In this thesis, we have proposed
novel mechanisms to generate the primordial magnetic field during inflation
and looked at the effect on matter-antimatter asymmetry. We have also
constructed a generic framework—effective field theory of magnetogenesis which
can reproduce a large class of magnetogenesis models proposed in the
literature.

In chapter 2, we constructed Galileon scalar electrodynamics action, which
preserves Galilean symmetry in field space and local gauge invariance. Due to
the complex scalar field, the number of Galilean symmetry invariant terms is
reduced. In 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. As an
application of the model, we considered the case when the Galileon scalar
electrodynamics dominated the early Universe. We have shown that the model
leads to inflation for a range of parameters. Since the model breaks conformal
invariance, one possible application of the model is to study the
phenomenological consequences of generating primordial magnetic fields during
inflation.

In chapter 3, we have proposed a viable scenario for generating helical
magnetic fields during inflation, which does not require coupling the
electromagnetic field to the scalar field. We have shown that the model leads
to the generation of the helical fields due to the coupling of the
electromagnetic fields with the dual of Riemann tensor. To our knowledge,
Riemann tensor coupling has not been discussed in the literature to generate
primordial helical magnetic fields. We have shown the key features of the
model: First, it does not require the coupling of the electromagnetic field
with the scalar field. Hence, there are no extra degrees of freedom, and it
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, the modification term will
not contribute at late-times, and the theory is identical to standard
electrodynamics. Third, the power spectrum of the helical magnetic fields
generated has a slight red-tilt for slow-roll inflation. This differs from 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.

After discussing the generation of the helical magnetic field, in chapter 4,
we have studied 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 proposed in chapter 3. The advantage of the primordial
helical magnetic fields is that the non-zero helicity suggests a non-zero
contribution in the CP violation term. An interesting feature of the 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. Also, we have
explicitly shown that Davidson’s conditions (see Ref. [113]) are necessary
_but not_ sufficient in order to map the symmetries of the universe in the
presence of the magnetic field with Sakharov’s conditions for baryogenesis.
The key missing ingredient is the requirement of _primordial helical magnetic
fields_. 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.
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].

In chapter 5, using effective field theory, we showed that the generation of
primordial magnetic fields requires two necessary conditions — conformal
invariance breaking and causal propagation. We explicitly showed that the
metric vector perturbations do not have temporal evolution and do not affect
the dynamics of the gauge field. Hence, the gauge field is the relevant gauge-
invariant variable for the EFT. We obtained the general second-order EFT
action by demanding that the EFT of magnetogenesis breaks conformal
invariance; however, it satisfies local Lorentz invariance and gauge
invariance. This action depends on two expansion scalar functions —
$f_{1}(H,a,\Lambda)$ and $f_{2}(H,a,\Lambda)$. We then showed that the EFT
action could reproduce a large class of the known magnetogenesis models. For
most models, it is sufficient to consider up to the second order in the
expansion parameters. We also compared our approach to previous approaches and
showed the advantages of our approach. We considered a specific model of
inflationary magnetogensis where the vector Galileon breaks the conformal
invariance. We extensively studied the magnetic field generation during
inflation, considering the total action, i.e., standard electrodynamics action
with vector Galileon modification terms. Due to the absence of coupling
between the scalar and electromagnetic fields, the model does not lead to a
strong coupling problem [102]; hence, the model does not require fine-tuning,
which is an interesting feature of our model. Furthermore, the evolution of
vector modes is frozen on the super-horizon scale. We showed that the model
predicts the present-day magnetic field of strength $B_{0}\sim 10^{-45}{\rm
G}$ on the cosmological (Mpc) scale, which is several orders higher compared
to the standard electromagnetic action, and there is no back-reaction problem.
However, this comes with the price — super-luminal electromagnetic
perturbations. In other words, the model generates large magnetic fields and
does not violate any other conditions; it leads to super-luminal
perturbations. Thus, our EFT framework suggests that to construct a field
theoretical consistent model of inflationary magnetogenesis, we need two
necessary conditions— breaking of conformal invariance and causal propagation
of perturbations.

In chapter 6, we proposed a novel mechanism that explains the Fast radio
bursts that utilize the strong magnetic field of a compact object. The energy
conversion from GWs to electromagnetic waves occurs when GWs pass through the
magnetosphere of compact objects (neutron stars or magnetars) due to the
Gertsenshtein-Zel′dovich effect. This conversion produces bursts of
electromagnetic waves in the GHz range, leading to FRBs. The mechanism
requires that the incoming GWs pass through the neutron star with a magnetic
field transverse to the direction of propagation of GWs, resulting in FRBs
along the line of sight of the observer. The probability of this event is a
product of the probability that the GW passes through the NS and the
probability that the emitted EM is along the line of sight of the observer.
Hence, the probability of seeing such an event is not very high. We showed
that our model explains why the FRB rates are low and why certain FRBs are
random. Furthermore, our model predicts that the progenitor of these events is
NS in any galaxy. Moreover, importantly, our model predicts that FRBs can also
be extragalactic in origin. We have shown that the mechanism presented in
chapter 6 can explain three key features of FRBs—peak flux, pulse width, and
coherent nature of radiation. We have also discussed the implications of the
mechanism in providing the indirect detection of high-frequency gravitational
waves.

###  7.1 Future outlook

As discussed in chapters 3 and 4, the Riemann coupling to the electromagnetic
fields generates the primordial helical magnetic fields and has interesting
consequences for understanding the creation of baryon asymmetry in the
universe. We saw that the perturbations equation (4.17) contains second-order
derivatives of $\mathcal{H}$. Since ${\cal H}$, and ${\cal H}^{\prime\prime}$
are different for inflation and bounce models [177], the helical modes may
provide signatures to distinguish the two paradigms. Also, in chapter 4, we
have seen that small-scale modes of the helical fields which exit the horizon
near the end of inflation lead to baryogenesis. However, it will be
interesting to extend the analysis to Gluons and study the effects on the
asymmetry generated in quarks and the baryons. This could be done by
considering the coupling the similar mass dimension 6 operator (Riemann
coupling) to the non-abelian SU(2) gauge fields. Another interestingly aspect
of the small scales modes discussed in chapter 4 would be to see if these
modes have sufficiently large power, because they could also provide a
mechanism for the formation of primordial black holes in the early Universe.

Modifications of general relativity provide an alternative explanation for
cosmological inflation [222]. Modified gravity theories have extra degrees of
freedom that might have interesting physical consequences in the early
Universe. For instance, Stelle gravity [223] contains massive tensor modes,
and these modes carry more energy than the scalar modes in $f(R)$ gravity
models [224]. While the analysis in chapter 5 can be extended to $f(R)$
gravity models, it is not straightforward to extend to Stelle gravity.
Investigating the EFT of inflation and magnetogenesis in these modified
gravity models would be interesting to explore. Also, it will be interesting
to extend the EFT of magnetogenesis to bouncing models. The EFT action
constructed in chapter 5 can also be extended to include the parity-breaking
term, leading to the helical magnetic field. In chapter 5, we have seen that
causal propagation must be considered while constructing the inflationary
magnetogenesis models; exploring this issue in other magnetogenesis scenarios
would be interesting. In chapter 6, we have explained the properties of the
non-repeating FRBs and focussed on the transient incoming GWs. It would be
interesting to see if the Gertsenshtein-Zel’dovich mechanism can also explain
the repeating nature of FRBs and also if we can incorporate the incoming
continuous GWs to explain the FRBs.

##  Appendix A Galileon scalar electrodynamics: Details

In this Appendix, we provide more details on the results from chapter 2.

###  A.1 Conformal transformation in 4-D curved space-time

Conformal transformation is a local change of scale, which is given by scaling
the metric by a smooth non-vanishing function of space-time $\Omega(x^{\mu})$
[93, 94]:

$\tilde{g}_{\mu\nu}=\Omega^{2}(x)g_{\mu\nu}$ (A.1)

Let us see how this transformation affects the electromagnetic field tensor
$F^{\mu\nu}$ in arbitrary curved space-time.

$F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}=\partial_{\mu}A_{\nu}-\Gamma^{\lambda}_{\mu\nu}A_{\lambda}-\partial_{\nu}A_{\mu}+\Gamma^{\lambda}_{\nu\mu}A_{\lambda}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$
(A.2)

Therefore, $F_{\mu\nu}$ is invariant under the conformal transformation (A.1),
and is not affected by geometry of the space.
Now we will see how other geometrical quantities which are derived from the
metric, change under this transformation. The Christoffel symbol changes as

$\tilde{\Gamma}^{\lambda}_{\mu\nu}=\Gamma^{\lambda}_{\mu\nu}+C^{\lambda}_{\mu\nu}$
(A.3)

where
$C^{\lambda}_{\mu\nu}=\Omega^{-1}\left(\delta^{\lambda}_{\mu}\nabla_{\nu}\Omega+\delta^{\lambda}_{\nu}\nabla_{\mu}\Omega-
g_{\mu\nu}g^{\rho\lambda}\nabla_{\rho}\Omega\right)$
Putting this value of Christoffel connetion in the Riemann tensor in the
conformal frame (Jordan frame) we get,

$\tilde{R}^{\lambda}_{\sigma\mu\nu}=R^{\lambda}_{\sigma\mu\nu}+\nabla_{\mu}C^{\lambda}_{\nu\sigma}-\nabla_{\nu}C^{\lambda}_{\mu\sigma}+C^{\lambda}_{\mu\rho}C^{\rho}_{\nu\sigma}-C^{\lambda}_{\nu\rho}C^{\rho}_{\mu\sigma}$
(A.4)

Therefore, Riemann tensor changes under the conformal transformaton,
contracting with the metric gives the Ricci tensor

$\tilde{R}_{\mu\nu}=R_{\mu\nu}-[2\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}+g_{\mu\nu}g^{\alpha\beta}]\Omega^{-1}(\nabla_{\alpha}\nabla_{\beta}\Omega)+[4\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}-g_{\mu\nu}g^{\alpha\beta}]\Omega^{-2}(\nabla_{\alpha}\Omega)(\nabla_{\beta}\Omega)$
(A.5)

and Ricci curvature scalar as

$\tilde{R}=\Omega^{-2}R--
6g^{\alpha\beta}\Omega^{-3}(\nabla_{\alpha}\nabla_{\beta}\Omega)$ (A.6)

Hence, we can see that Christoffel connections, Riemann tensor, Ricci tensor
and Ricci scalar transform according to the relation given above. So,
introducing these quantities in the action break the conformal invariance of
the action. It is important to note that for higher derivative field theories
( e.g., Galileon fields) the conformal invariance is explicitly broken as

$\tilde{\nabla}_{\mu}\phi=\nabla_{\mu}\phi=\partial_{\mu}\phi$ (A.7)
$\displaystyle\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}\phi=\nabla_{\mu}\nabla_{\nu}\phi-(\;\delta^{\alpha}_{\mu}\;\delta^{\beta}_{\nu}\;+\;\delta^{\beta}_{\mu}\;\delta^{\alpha}_{\nu}\;)\;\Omega^{-1}\;(\nabla_{\alpha}\Omega)(\nabla_{\beta}\Omega)$
(A.8)

where $\phi$ is any real scalar field.

###  A.2 Fixing the coefficients of $\mathcal{L}_{4}$

In this Appendix, we fix all the coefficients in the Lagrangian (2.1.2).
Demanding that the Lagrangian is invariant under the Galileon transformation
(2.2) leads to the following constraint:

$\displaystyle
2A_{1}\,\partial_{\mu}\partial^{\mu}\pi^{*}\partial_{\nu}\partial^{\nu}\pi^{*}\partial_{\alpha}\pi
a^{\alpha}+A_{2}\,\partial_{\mu}\partial^{\mu}\pi^{*}\partial_{\nu}\partial^{\nu}\pi\partial_{\alpha}\pi^{*}a^{\alpha}$
$\displaystyle+B_{1}\,(\partial_{\mu}\partial^{\mu}\pi^{*}\partial^{\nu}\partial^{\alpha}\pi\partial_{\nu}\pi^{*}a_{\alpha}+\partial_{\mu}\partial^{\mu}\pi^{*}\partial^{\nu}\partial^{\alpha}\pi\partial_{\alpha}\pi{a_{\nu}}^{*})+2B_{2}\,\partial_{\mu}\partial^{\mu}\pi^{*}\partial^{\nu}\partial^{\alpha}\pi^{*}\partial_{\nu}\pi
a_{\alpha}$
$\displaystyle+2C_{1}\,\partial_{\mu}\partial_{\nu}\pi^{*}\partial^{\mu}\partial^{\nu}\pi^{*}\partial^{\alpha}\pi
a_{\alpha}+C_{2}\,\partial_{\mu}\partial_{\nu}\pi^{*}\partial^{\mu}\partial^{\nu}\pi\partial^{\alpha}\pi^{*}a_{\alpha}$
$\displaystyle+2D_{1}\,\partial^{\mu}\partial^{\nu}\pi^{*}\partial_{\nu}\partial_{\alpha}\pi^{*}\partial_{\mu}\pi
a^{\alpha}+D_{2}\,\partial^{\mu}\partial^{\nu}\pi^{*}\partial_{\nu}\partial_{\alpha}\pi\partial_{\mu}\pi^{*}a^{\alpha}+D_{3}\,\partial^{\mu}\partial^{\nu}\pi\partial_{\nu}\partial_{\alpha}\pi^{*}\partial_{\mu}\pi^{*}a^{\alpha}+c.c.=0\,.$
(A.9)

Here, c.c. denotes the complex conjugate, we have considered only first order
terms in $a^{\mu}$ and have ignored the quadratic terms of the infinitesimal
shift parameters. All the first order terms in Eq. (A.2) either do not
contribute or contribute total derivative terms. After integrating by parts
(and ignoring the total derivative terms as they do not contribute to the
action) and collecting similar terms, we get

$\displaystyle\left[(2A_{1}-B_{1})\,(\Box\pi^{*})^{2}\partial_{\alpha}\pi+(A_{2}-2B_{2}-B_{1})\,\Box\pi^{*}\Box\pi\partial_{\alpha}\pi^{*}-(B_{1}+2C_{1})\,\partial^{\nu}\Box\pi^{*}\partial_{\nu}\pi^{*}\partial_{\alpha}\pi\right.$
$\displaystyle\left.-(B_{1}+D_{3})\,\partial^{\mu}\Box\pi\partial_{\mu}\pi^{*}\partial_{\alpha}\pi^{*}+(C_{2}-2D_{1}-D_{3})\,\partial_{\mu}\partial_{\nu}\pi^{*}\partial^{\mu}\partial^{\nu}\pi\partial_{\alpha}\pi^{*}\right.$
$\displaystyle\left.-(2D_{1}+2B_{2})\,\partial^{\mu}\Box\pi^{*}\partial_{\mu}\pi\partial_{\alpha}\pi^{*}-(2C_{1}-D_{2})\,\partial_{\mu}\partial_{\nu}\pi^{*}\partial^{\nu}\partial_{\alpha}\pi\partial^{\mu}\pi^{*}\right]a^{\alpha}=0$
(A.10)

Only way to satisfy the above equation for an arbitrary $\pi$ and $\pi^{*}$ is
that all the coefficients vanish. This leads to the following relations:

$\displaystyle B_{1}$ $\displaystyle=2A_{1};~{}C_{1}=-A_{1}$ $\displaystyle
B_{2}$ $\displaystyle=-A_{1}+\frac{A_{2}}{2}=-E$ $\displaystyle C_{2}$
$\displaystyle=-A_{2};~{}D_{2}=-2A_{1}$ $\displaystyle D_{1}$
$\displaystyle=A_{1}-\frac{A_{2}}{2}=E$ $\displaystyle D_{3}$
$\displaystyle=-2A_{1}\,.$ (A.11)

It is straightforward to see from the above expressions that all the
parameters can be written in terms of two parameters $A_{1}$ and $A_{2}$.
Substituting the coefficients (A.2) in Eq. (2.1.2) leads to the following
generic complex scalar Lagrangian $\mathcal{L}_{4}$;

$\displaystyle{\mathcal{L}}_{4}$
$\displaystyle{}=A_{1}[\,\,(\,\,(\Box\pi)^{2}\partial_{\alpha}\pi^{*}\partial^{\alpha}\pi^{*}+2\Box\pi\partial_{\nu}\pi\partial^{\nu}\partial^{\alpha}\pi^{*}\partial_{\alpha}\pi^{*}-\partial_{\mu}\partial_{\nu}\pi\partial^{\mu}\partial^{\nu}\pi\partial^{\alpha}\pi^{*}\partial_{\alpha}\pi^{*}+\rm{c.c.}\,\,)$
$\displaystyle{}-2\partial_{\mu}\pi^{*}\partial^{\mu}\partial^{\nu}\pi\partial_{\nu}\partial_{\alpha}\pi^{*}\partial^{\alpha}\pi]+A_{2}[\Box\pi^{*}\Box\pi\partial_{\alpha}\pi^{*}\partial^{\alpha}\pi-\partial_{\mu}\partial_{\nu}\pi^{*}\partial^{\mu}\partial^{\nu}\pi\partial^{\alpha}\pi^{*}\partial_{\alpha}\pi]$
$\displaystyle{}+E[\partial_{\mu}\pi^{*}\partial^{\mu}\partial^{\nu}\pi\partial_{\nu}\partial_{\alpha}\pi\partial^{\alpha}\pi^{*}-\Box\pi\partial_{\nu}\pi^{*}\partial^{\nu}\partial^{\alpha}\pi\partial_{\alpha}\pi^{*}+\rm{c.c.}]$
(A.12)

Now, there are only two free parameters $A_{1}$ and $A_{2}$ in the Lagrangain.
The action for the above generic Lagrangian (A.2) is given by,

${S}_{4}=\int d^{4}x~{}\mathcal{L}_{4}$ (A.13)

We can identify the generic action (A.13) as the summation of the following
actions:

$\begin{split}S^{(1)}_{4}&=A_{1}\int
d^{4}x\;(\,\partial_{\mu}\partial^{\mu}\pi\partial_{\nu}\partial^{\nu}\pi\partial_{\alpha}\pi^{*}\partial^{\alpha}\pi^{*}+\rm{c.c.}\,\,)\\\
{S}^{(2)}_{4}&=A_{2}\int
d^{4}x\;\partial_{\mu}\partial^{\mu}\pi^{*}\partial_{\nu}\partial^{\nu}\pi\partial_{\alpha}\pi^{*}\partial^{\alpha}\pi\\\
{S}^{(3)}_{4}&=2A_{1}\int
d^{4}x\;(\,\partial_{\mu}\partial^{\mu}\pi\partial_{\nu}\pi\partial^{\nu}\partial^{\alpha}\pi^{*}\partial_{\alpha}\pi^{*}+\rm{c.c.}\,\,)\\\
{S}^{(4)}_{4}&=-E\int
d^{4}x\;(\,\partial_{\mu}\partial^{\mu}\pi\partial_{\nu}\pi^{*}\partial^{\nu}\partial^{\alpha}\pi\partial_{\alpha}\pi^{*}+\rm{c.c.}\,\,)\\\
{S}^{(5)}_{4}&=-A_{1}\int
d^{4}x\;(\,\partial_{\mu}\partial_{\nu}\pi\partial^{\mu}\partial^{\nu}\pi\partial^{\alpha}\pi^{*}\partial_{\alpha}\pi^{*}+\rm{c.c.}\,\,)\\\
{S}^{(6)}_{4}&=-A_{2}\int
d^{4}x\;\partial_{\mu}\partial_{\nu}\pi^{*}\partial^{\mu}\partial^{\nu}\pi\partial^{\alpha}\pi^{*}\partial_{\alpha}\pi\\\
{S}^{(7)}_{4}&=E\int
d^{4}x\;(\,\partial_{\mu}\pi\partial^{\mu}\partial^{\nu}\pi^{*}\partial_{\nu}\partial_{\alpha}\pi^{*}\partial^{\alpha}\pi+\rm{c.c.}\,\,)\\\
{S}^{(8)}_{4}&=-2A_{1}\int
d^{4}x\;\partial_{\mu}\pi^{*}\partial^{\mu}\partial^{\nu}\pi^{*}\partial_{\nu}\partial_{\alpha}\pi\partial^{\alpha}\pi\\\
{S}^{(9)}_{4}&=-2A_{1}\int
d^{4}x\;\partial_{\mu}\pi^{*}\partial^{\mu}\partial^{\nu}\pi\partial_{\nu}\partial_{\alpha}\pi^{*}\partial^{\alpha}\pi\end{split}$
(A.14)

where

${S}_{4}=\sum_{i=1}^{9}\;{S}^{(i)}_{4}$ (A.15)

Having obtained the generic 4th order Galileon action (A.14), our next step is
to calculate the equations of motion of $\pi$ ( for $\pi^{*}$ is straight
forward). We define the quantity $\mathcal{E}_{i}$ as the variation of the
${S^{(i)}}_{CGG}$ with respect to $\pi$ (with respect to $\pi^{*}$ for
${\mathcal{E}_{i}}^{*}$) as,

$\mathcal{E}^{i}_{4}=\frac{\delta{S}^{(i)}_{4}}{\delta\pi}$ (A.16)

Hence, equation of motion for the generic action (A.15) is given by,

$\displaystyle\mathcal{E}_{4}=\sum_{i=1}^{9}\;\mathcal{E}_{4}^{i}$ (A.17)

Using Eq. (A.16), the equations of motion corresponding to nine actions are

$\displaystyle\mathcal{E}^{(1)}_{4}$
$\displaystyle=2A_{1}\,[\partial_{\mu}\partial^{\mu}\Box\pi\,\partial_{\alpha}\pi^{*}\partial^{\alpha}\pi^{*}+4\,\partial_{\mu}\Box\pi\,\partial_{\alpha}\pi^{*}\partial^{\mu}\partial^{\alpha}\pi^{*}+2\,\Box\pi\,\partial_{\alpha}\Box\pi^{*}\,\partial^{\alpha}\pi^{*}+2\,\Box\pi\,\partial_{\mu}\partial_{\alpha}\pi^{*}\,\partial^{\mu}\partial^{\alpha}\pi^{*}$
$\displaystyle\hskip
28.45274pt-2\,\partial^{\alpha}\Box\pi^{*}\,\Box\pi^{*}\,\partial_{\alpha}\pi-(\Box\pi^{*})^{2}\,\Box\pi]$
(A.18) $\displaystyle\mathcal{E}^{(2)}_{4}$
$\displaystyle=A_{2}\,[\partial_{\mu}\partial^{\mu}\Box\pi^{*}\,\partial_{\alpha}\pi\partial^{\alpha}\pi^{*}+2\,\partial_{\mu}\Box\pi^{*}\,\partial^{\mu}\partial^{\alpha}\pi\partial_{\alpha}\pi^{*}+2\,\partial_{\nu}\Box\pi^{*}\,\partial_{\alpha}\pi\partial^{\nu}\partial^{\alpha}\pi^{*}+\Box\pi^{*}\,\partial_{\alpha}\Box\pi\,\partial^{\alpha}\pi^{*}$
$\displaystyle{}+\Box\pi^{*}\,\partial^{\alpha}\Box\pi^{*}\,\partial_{\alpha}\pi+2\,\Box\pi^{*}\,\partial_{\nu}\partial_{\alpha}\pi\partial^{\nu}\partial^{\alpha}\pi^{*}-\partial_{\alpha}\Box\pi^{*}\,\Box\pi\,\partial^{\alpha}\pi^{*}-\Box\pi^{*}\,\partial_{\alpha}\Box\pi\,\partial^{\alpha}\pi^{*}-(\Box\pi^{*})^{2}\,\Box\pi]$
(A.19) $\displaystyle\mathcal{E}^{(3)}_{4}$
$\displaystyle=2A_{1}\,[\,2\,\partial^{\alpha}\partial^{\nu}\Box\pi^{*}\,\partial_{\nu}\pi^{*}\,\partial_{\alpha}\pi+2\,\partial^{\nu}\Box\pi^{*}\,\partial^{\alpha}\pi\,\partial_{\nu}\partial_{\alpha}\pi^{*}+2\partial^{\alpha}\Box\pi^{*}\Box\pi^{*}\,\partial_{\alpha}\pi+(\Box\pi^{*})^{2}\,\Box\pi$
$\displaystyle{}+2\,\partial^{\mu}\partial^{\nu}\partial^{\alpha}\pi^{*}\,\partial_{\mu}\partial_{\nu}\pi\,\partial_{\alpha}\pi^{*}+2\,\partial^{\mu}\partial^{\nu}\partial^{\alpha}\pi^{*}\,\partial_{\mu}\partial_{\alpha}\pi^{*}\,\partial_{\nu}\pi+2\,\partial^{\nu}\partial^{\alpha}\pi^{*}\,\partial_{\mu}\partial_{\nu}\pi\,\partial^{\mu}\partial_{\alpha}\pi^{*}$
$\displaystyle{}-\Box\pi\,\partial^{\nu}\partial^{\alpha}\pi^{*}\,\partial_{\nu}\partial_{\alpha}\pi^{*}]$
(A.20) $\displaystyle\mathcal{E}^{(4)}_{4}$
$\displaystyle=-E\,[\,2\,\partial^{\nu}\Box\pi^{*}\,\partial_{\nu}\partial_{\alpha}\pi\,\partial^{\alpha}\pi^{*}+4\,\partial^{\mu}\partial^{\nu}\partial^{\alpha}\pi\,\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial_{\alpha}\pi^{*}+2\,\partial^{\mu}\partial^{\nu}\pi^{*}\,\partial_{\nu}\partial_{\alpha}\pi\,\partial_{\mu}\partial^{\alpha}\pi^{*}$
$\displaystyle{}+2\,\partial_{\alpha}\partial_{\nu}\Box\pi\,\partial^{\nu}\pi^{*}\,\partial^{\alpha}\pi^{*}\,+2\,\partial^{\nu}\Box\pi\,\partial_{\alpha}\partial_{\nu}\pi^{*}\,\partial^{\alpha}\pi^{*}+2\,\partial^{\nu}\Box\pi\,\Box\pi^{*}\,\partial_{\nu}\pi^{*}+2\,\partial^{\alpha}\Box\pi^{*}\,\Box\pi\,\partial_{\alpha}\pi^{*}$
$\displaystyle{}+\Box\pi\,\partial_{\nu}\partial_{\alpha}\pi^{*}\,\partial^{\nu}\partial^{\alpha}\pi^{*}+(\Box\pi^{*})^{2}\,\Box\pi-\partial_{\nu}\Box\pi^{*}\,\partial^{\nu}\partial^{\alpha}\pi^{*}\,\partial_{\alpha}\pi-2\,\Box\pi^{*}\,\partial^{\alpha}\Box\pi^{*}\,\partial_{\alpha}\pi$
$\displaystyle{}-2\,\Box\pi^{*}\,\partial^{\nu}\partial^{\alpha}\pi^{*}\,\partial_{\nu}\partial_{\alpha}\pi-\partial_{\alpha}\Box\pi^{*}\,\partial^{\nu}\partial^{\alpha}\pi^{*}\,\partial_{\nu}\pi]$
(A.21) $\displaystyle\mathcal{E}^{(5)}_{4}$
$\displaystyle=-2A_{1}\,[\,\partial_{\nu}\partial^{\nu}\Box\pi\,\partial_{\alpha}\pi^{*}\,\partial^{\alpha}\pi^{*}+2\,\partial^{\nu}\Box\pi\,\partial^{\alpha}\pi^{*}\,\partial_{\nu}\partial_{\alpha}\pi^{*}+2\,\partial^{\mu}\Box\pi\,\partial^{\alpha}\pi^{*}\,\partial_{\mu}\partial_{\alpha}\pi^{*}$
$\displaystyle{}+2\,\partial^{\mu}\partial^{\nu}\pi\,\partial_{\nu}\partial_{\alpha}\pi^{*}\,\partial_{\mu}\partial^{\alpha}\pi^{*}+2\,\partial^{\mu}\partial^{\nu}\pi\,\partial^{\alpha}\pi^{*}\,\partial_{\nu}\partial_{\mu}\partial_{\alpha}\pi^{*}\,-2\,\partial_{\nu}\partial_{\mu}\partial_{\alpha}\pi^{*}\,\partial^{\mu}\partial^{\nu}\pi^{*}\,\partial^{\alpha}\pi$
$\displaystyle{}-\Box\pi\,\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\mu}\partial^{\nu}\pi^{*}]$
(A.22) $\displaystyle\mathcal{E}^{(6)}_{4}$
$\displaystyle=-A_{2}\,[\,\partial_{\nu}\partial^{\nu}\Box\pi^{*}\,\partial_{\alpha}\pi\,\partial^{\alpha}\pi^{*}+\partial_{\nu}\Box\pi^{*}\,\partial^{\nu}\partial^{\alpha}\pi\,\partial_{\alpha}\pi^{*}+\partial_{\nu}\Box\pi^{*}\,\partial_{\alpha}\pi\,\partial^{\nu}\partial^{\alpha}\pi^{*}+\partial_{\mu}\Box\pi^{*}\,\partial^{\mu}\partial^{\alpha}\pi\,\partial_{\alpha}\pi^{*}$
$\displaystyle{}+\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\mu}\partial^{\alpha}\pi\,\partial^{\nu}\partial_{\alpha}\pi^{*}+\partial_{\mu}\Box\pi^{*}\,\partial_{\alpha}\pi\,\partial^{\mu}\partial^{\alpha}\pi^{*}+\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\nu}\partial^{\alpha}\pi\,\partial^{\mu}\partial_{\alpha}\pi^{*}$
$\displaystyle{}+\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial_{\alpha}\pi\,\partial^{\mu}\partial^{\nu}\partial^{\alpha}\pi^{*}-\partial^{\mu}\partial^{\nu}\pi\,\partial^{\alpha}\pi^{*}\,\partial_{\alpha}\partial_{\mu}\partial_{\nu}\pi^{*}-\Box\pi^{*}\,\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\mu}\partial^{\nu}\pi]$
(A.23) $\displaystyle\vspace{1cm}\mathcal{E}^{(7)}_{4}$
$\displaystyle=E\,[\,2\,\partial^{\mu}\partial^{\alpha}\Box\pi\,\partial_{\mu}\pi^{*}\,\partial_{\alpha}\pi^{*}+4\,\partial^{\mu}\partial^{\nu}\partial^{\alpha}\pi\,\partial_{\nu}\partial_{\mu}\pi^{*}\,\partial_{\alpha}\pi^{*}+2\,\partial_{\mu}\Box\pi\,\Box\pi^{*}\,\partial^{\mu}\pi^{*}$
$\displaystyle{}+2\,\partial_{\alpha}\Box\pi\,\partial_{\mu}\pi^{*}\,\partial^{\mu}\partial^{\alpha}\pi^{*}+2\,\Box\pi^{*}\,\partial_{\nu}\partial_{\alpha}\pi\,\partial^{\nu}\partial^{\alpha}\pi^{*}+2\,\partial^{\nu}\Box\pi^{*}\,\partial_{\nu}\partial_{\alpha}\pi\,\partial^{\alpha}\pi^{*}$
$\displaystyle{}+2\,\partial_{\nu}\partial_{\alpha}\pi\,\partial^{\nu}\partial^{\mu}\pi^{*}\,\partial_{\mu}\partial^{\alpha}\pi^{*}+2\,\partial_{\nu}\partial_{\alpha}\pi\,\partial_{\mu}\pi^{*}\,\partial^{\nu}\partial^{\mu}\partial^{\alpha}\pi^{*}-2\,\partial^{\nu}\Box\pi^{*}\,\partial_{\nu}\partial_{\alpha}\pi^{*}\,\partial^{\alpha}\pi$
$\displaystyle{}-2\,\partial^{\mu}\partial^{\nu}\pi^{*}\,\partial_{\mu}\partial_{\nu}\partial_{\alpha}\pi^{*}\,\partial^{\alpha}\pi-2\,\partial^{\mu}\partial^{\nu}\pi^{*}\,\partial_{\nu}\partial_{\alpha}\pi^{*}\,\partial_{\mu}\partial^{\alpha}\pi]$
(A.24) $\displaystyle\mathcal{E}^{(8)}_{4}$
$\displaystyle=-2\,A_{1}\,[\,\partial^{\mu}\partial^{\alpha}\Box\pi^{*}\,\partial_{\mu}\pi^{*}\,\partial_{\alpha}\pi+\partial^{\mu}\Box\pi^{*}\,\partial_{\alpha}\partial_{\mu}\pi^{*}\,\partial^{\alpha}\pi+\partial^{\mu}\Box\pi^{*}\,\Box\pi\,\partial_{\mu}\pi^{*}$
$\displaystyle{}\hskip
28.45274pt+2\,\partial_{\alpha}\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\nu}\partial^{\mu}\pi^{*}\,\partial^{\alpha}\pi+\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\nu}\partial^{\mu}\pi^{*}\,\Box\pi+\partial^{\alpha}\partial^{\mu}\partial^{\nu}\pi^{*}\,\partial_{\mu}\pi^{*}\,\partial_{\nu}\partial_{\alpha}\pi$
$\displaystyle{}\hskip
28.45274pt+\partial^{\mu}\partial^{\nu}\pi^{*}\,\partial_{\mu}\partial_{\alpha}\pi^{*}\,\partial_{\nu}\partial^{\alpha}\pi+\partial^{\mu}\partial^{\nu}\pi^{*}\,\partial_{\mu}\pi^{*}\,\partial_{\nu}\Box\pi-\partial^{\mu}\partial^{\nu}\partial^{\alpha}\pi^{*}\,\partial_{\nu}\partial_{\alpha}\pi\,\partial_{\mu}\pi^{*}$
$\displaystyle{}\hskip
28.45274pt-\partial^{\mu}\partial^{\nu}\pi^{*}\partial_{\nu}\Box\pi\partial_{\mu}\pi^{*}-\partial^{\mu}\partial^{\nu}\pi^{*}\partial_{\nu}\partial_{\alpha}\pi\partial^{\alpha}\partial_{\mu}\pi^{*}]$
(A.25) $\displaystyle\mathcal{E}^{(9)}_{4}$
$\displaystyle=-2\,A_{1}\,[\,\partial^{\mu}\partial^{\alpha}\Box\pi^{*}\,\partial_{\mu}\pi^{*}\,\partial_{\alpha}\pi+\partial^{\mu}\partial^{\nu}\partial^{\alpha}\pi^{*}\,\partial_{\nu}\partial_{\mu}\pi^{*}\,\partial_{\alpha}\pi+\partial^{\mu}\partial^{\nu}\partial^{\alpha}\pi^{*}\,\partial_{\mu}\pi^{*}\,\partial^{\nu}\partial^{\alpha}\pi$
$\displaystyle{}\hskip
28.45274pt+\partial_{\alpha}\Box\pi^{*}\,\Box\pi^{*}\,\partial^{\alpha}\pi+\partial_{\alpha}\partial_{\nu}\pi^{*}\,\partial^{\nu}\Box\pi^{*}\,\partial^{\alpha}\pi+\partial_{\nu}\partial_{\alpha}\pi^{*}\,\Box\pi^{*}\,\partial^{\nu}\partial^{\alpha}\pi$
$\displaystyle{}\hskip
28.45274pt+\partial_{\alpha}\Box\pi^{*}\,\partial_{\mu}\pi^{*}\,\partial^{\mu}\partial^{\alpha}\pi+-\partial^{\mu}\partial^{\nu}\pi\,\partial_{\nu}\Box\pi^{*}\,\partial_{\mu}\pi^{*}]$
(A.26)

Thus, the equation of motion for the generic 4th order action (A.15) is:

$\displaystyle\mathcal{E}_{4}$
$\displaystyle=\left(A_{1}+\frac{A_{2}}{2}\right)[\,\,\Box\pi\,\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\mu}\partial^{\nu}\pi^{*}+2\,\Box\pi^{*}\,\partial_{\mu}\partial_{\nu}\pi\,\partial^{\mu}\partial^{\nu}\pi^{*}$
$\displaystyle{}\hskip
28.45274pt-2\,\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\nu}\partial^{\alpha}\pi\,\partial^{\mu}\partial_{\alpha}\pi^{*}-(\,\Box\pi^{*})^{2}\,\Box\pi\,]$
(A.27)

To compare with the earlier results [148], we discuss special cases by taking
some specific values of the coefficients $A_{1}$ and $A_{2}$.

Case 1: $A_{1}=0$ Equations of motion for $\pi$ is given by

$\displaystyle\mathcal{E}_{4}^{A_{1}=0}$
$\displaystyle=\frac{A_{2}}{2}[\,\,\Box\pi\,\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\mu}\partial^{\nu}\pi^{*}+2\,\Box\pi^{*}\,\partial_{\mu}\partial_{\nu}\pi\,\partial^{\mu}\partial^{\nu}\pi^{*}-2\,\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\nu}\partial^{\alpha}\pi\,\partial^{\mu}\partial_{\alpha}\pi^{*}-(\,\Box\pi^{*})^{2}\,\Box\pi\,]$
(A.28)

which in the limit $\pi=\pi^{*}$ gives

$\displaystyle\mathcal{E}_{4}^{A_{1}=0}$
$\displaystyle=\frac{A_{2}}{2}\,[\,3\,\Box\pi\,\partial_{\mu}\partial_{\nu}\pi\,\partial^{\mu}\partial^{\nu}\pi-2\,\partial_{\mu}\partial_{\nu}\pi\,\partial^{\nu}\partial^{\alpha}\pi\,\partial^{\mu}\partial_{\alpha}\pi-(\Box\pi)^{3}\,\,]$
(A.29)

Case 2: $A_{2}=0$ Equations of motion for $\pi$ is given by

$\displaystyle\mathcal{E}_{4}^{A_{2}=0}$
$\displaystyle=A_{1}[\,\,\Box\pi\,\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\mu}\partial^{\nu}\pi^{*}+2\,\Box\pi^{*}\,\partial_{\mu}\partial_{\nu}\pi\,\partial^{\mu}\partial^{\nu}\pi^{*}-2\,\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\nu}\partial^{\alpha}\pi\,\partial^{\mu}\partial_{\alpha}\pi^{*}-(\,\Box\pi^{*})^{2}\,\Box\pi\,]$
(A.30)

which in the limit $\pi=\pi^{*}$ gives

$\displaystyle\mathcal{E}_{4}^{A_{2}=0}$
$\displaystyle=A_{1}\,\left[\,3\,\Box\pi\,\partial_{\mu}\partial_{\nu}\pi\,\partial^{\mu}\partial^{\nu}\pi-2\,\partial_{\mu}\partial_{\nu}\pi\,\partial^{\nu}\partial^{\alpha}\pi\,\partial^{\mu}\partial_{\alpha}\pi-(\Box\pi)^{3}\,\,\right]$
(A.31)

Case 3: $A_{1}=A_{2}$ Equations of motion for $\pi$ is given by

$\displaystyle\mathcal{E}_{4}^{A_{1}\neq A_{2}}$
$\displaystyle=A_{1}\,\left[\,\frac{3}{2}\,\Box\pi\,\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\mu}\partial^{\nu}\pi^{*}+3\,\Box\pi^{*}\,\partial_{\mu}\partial_{\nu}\pi\,\partial^{\mu}\partial^{\nu}\pi^{*}\right.$
$\displaystyle{}\hskip
28.45274pt\left.-3\,\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\nu}\partial^{\alpha}\pi\,\partial^{\mu}\partial_{\alpha}\pi^{*}-\frac{3}{2}\,(\Box\pi^{*})^{2}\,\Box\pi\,\,\right]$
(A.32)

which in the limit $\pi=\pi^{*}$ gives

$\displaystyle\mathcal{E}_{4}^{A_{1}=A_{2}}$
$\displaystyle=\frac{3A_{1}}{2}\,\,[\,3\,\Box\pi\,\partial_{\mu}\partial_{\nu}\pi\,\partial^{\mu}\partial^{\nu}\pi-2\,\partial_{\mu}\partial_{\nu}\pi\,\partial^{\nu}\partial^{\alpha}\pi\,\partial^{\mu}\partial_{\alpha}\pi-(\Box\pi)^{3}\,\,]$
(A.33)

Case 4: $A_{1}\neq A_{2}$ Equations of motion for $\pi$ is given by

$\displaystyle\mathcal{E}_{4}^{A_{1}\neq A_{2}}$
$\displaystyle=\left(A_{1}+\frac{A_{2}}{2}\right)[\,\,\Box\pi\,\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\mu}\partial^{\nu}\pi^{*}+2\,\Box\pi^{*}\,\partial_{\mu}\partial_{\nu}\pi\,\partial^{\mu}\partial^{\nu}\pi^{*}-2\,\partial_{\mu}\partial_{\nu}\pi^{*}\,\partial^{\nu}\partial^{\alpha}\pi\,\partial^{\mu}\partial_{\alpha}\pi^{*}$
$\displaystyle{}-(\,\Box\pi^{*})^{2}\,\Box\pi\,]$ (A.34)

which in the limit $\pi=\pi^{*}$ gives

$\displaystyle\mathcal{E}_{4}^{A_{1}\neq A_{2}}$
$\displaystyle=\left(A_{1}+\frac{A_{2}}{2}\right)[\,3\,\Box\pi\,\partial_{\mu}\partial_{\nu}\pi\,\partial^{\mu}\partial^{\nu}\pi-2\,\partial_{\mu}\partial_{\nu}\pi\,\partial^{\nu}\partial^{\alpha}\pi\,\partial^{\mu}\partial_{\alpha}\pi-(\Box\pi)^{3}\,\,]$
(A.35)

From the above cases, it is interesting to see that equations of motion are
independent of the values of $A_{1}$ and $A_{2}$ and the equations of motion
of all the four cases are identical to Ref. [148] in flat space-time. Hence,
for simplicity, we set $A_{1}=0$ in the Lagrangian for the fourth order
complex scalar Galileon Lagrangian.

###  A.3 Gauge fixing

In this Appendix, we show that the Galileon complex scalar action (2.1.3) is
indeed invariant under the Gauge transformation. Replacing,
$\partial_{\mu}\rightarrow{D}_{\mu}=\partial_{\mu}+\alpha A_{\mu}$ (where we
fix $\alpha$ at the end of the calculation) implies

$\displaystyle\partial_{\mu}\partial^{\nu}\pi\rightarrow{D}_{\mu}{D}^{\nu}\pi=(\partial_{\mu}+\alpha
A_{\mu})(\partial^{\nu}+\alpha
A^{\nu})\pi={\partial_{\mu}\,^{\nu}{\pi}+\alpha(\partial_{\mu}{A^{\nu}}\pi+A^{\nu}\partial_{\mu}{\pi}+A_{\mu}\partial^{\nu}{\pi})+\alpha\alpha
A_{\mu}A^{\nu}\pi}$ (A.36)

Under the local U(1) gauge transformation: $\pi\rightarrow\pi
e^{-ie\theta(x)}$, above expression becomes

$\displaystyle{D}_{\mu}{D}^{\nu}(\pi e^{-ie\theta})$
$\displaystyle=e^{-ie\theta}(\partial_{\mu}\,^{\nu}{\pi}+\alpha\partial_{\mu}{A^{\nu}}\pi+A^{\nu}\alpha\partial_{\mu}{\pi}+A_{\mu}\alpha\partial^{\nu}{\pi}+A_{\mu}A^{\nu}\alpha\alpha\pi$
$\displaystyle{}-ie(\partial^{\nu}{\pi}\partial_{\mu}{\theta}+\partial_{\mu}{\pi}\partial^{\nu}{\theta}+\partial_{\mu}\,^{\nu}{\theta}\pi+A^{\nu}\alpha\partial_{\mu}{\theta}\pi+A_{\mu}\alpha\partial^{\nu}{\theta}\pi)+ieie\partial_{\mu}{\theta}\partial^{\nu}{\theta}\pi$
(A.37)

Under the gauge transformation $A_{\mu}\rightarrow
A_{\mu}+\partial_{\mu}\theta$, Eq. (A.3) becomes,

$\displaystyle{D}_{\mu}{D}^{\nu}(\pi e^{-ie\theta})$ $\displaystyle\rightarrow
e^{-ie\theta}[\partial_{\mu}\,^{\nu}{\pi}+\alpha(\partial_{\mu}{A^{\nu}}\pi+\partial_{\mu}{\partial^{\nu}\theta}\pi+A^{\nu}\partial_{\mu}{\pi}+\partial^{\nu}\theta\partial_{\mu}{\pi}+A_{\mu}\partial^{\nu}{\pi}+\partial_{\mu}\theta\partial^{\nu}{\pi}$
$\displaystyle{}+\alpha A_{\mu}A^{\nu}\pi+\alpha
A_{\mu}\partial^{\nu}\theta\pi+\alpha
A^{\nu}\partial_{\mu}\theta\pi+\alpha\partial_{\mu}\theta\partial^{\nu}\theta\pi)-ie(\partial^{\nu}{\pi}\partial_{\mu}{\theta}+\partial_{\mu}{\pi}\partial^{\nu}{\theta}+\partial_{\mu}\,^{\nu}{\theta}\pi$
$\displaystyle{}+\alpha
A^{\nu}\partial_{\mu}{\theta}\pi+\alpha\partial^{\nu}\theta\partial_{\mu}{\theta}\pi+\alpha
A_{\mu}\partial^{\nu}{\theta}\pi+\alpha\partial_{\mu}\partial^{\nu}{\theta}\pi)+ieie\partial_{\mu}{\theta}\partial^{\nu}{\theta}\pi]$
(A.38)

For the ${D}_{\mu}{D}^{\nu}\pi$ to be invariant under the local gauge
transformations, we set $\alpha=ie$ in Eq. (A.3). Hence, the action (2.9) is
invariant under the following simultaneous transformations:

$\displaystyle A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\theta\,\,;\hskip
14.22636pt\pi\rightarrow\pi e^{-ie\theta(x)}$ (A.39)

and the quantity ${D}_{\mu}{D}^{\nu}\pi$ will transform as

$\displaystyle{D}_{\mu}{D}^{\nu}(\pi
e^{-ie\theta})=(\partial_{\mu}\,^{\nu}{\pi}+\alpha\partial_{\mu}{A^{\nu}}\pi+A^{\nu}\alpha\partial_{\mu}{\pi}+A_{\mu}\alpha\partial^{\nu}{\pi}+A_{\mu}A^{\nu}\alpha\alpha\pi)e^{-ie\theta}$
(A.40)

Similar analysis can be done for $\pi^{*}\rightarrow\pi^{*}e^{ie\theta(x)}$
also.

###  A.4 Galileon scalar electrodynamics in curved space-time

In this Appendix, we show that the action (2.2.1) leads to higher derivative
equations of motion. We also show addition of the non-minimal term leads to
second order equations of motion. The variation of the action (2.2.1) with
respect to $\pi$ yield the EOM:

$\displaystyle\mathcal{E}^{\rm{min}}_{4}$
$\displaystyle=\frac{\omega}{2\lambda_{\pi}^{6}}[\,\,2\,\nabla_{\mu}\nabla^{\mu}\pi^{*}\,\nabla_{\nu}\nabla_{\alpha}\pi\,\nabla^{\nu}\nabla^{\alpha}\pi^{*}-\nabla_{\mu}\nabla^{\mu}\pi^{*}\,\nabla_{\nu}\nabla^{\nu}\pi^{*}\,\nabla_{\alpha}\nabla^{\alpha}\pi-2\,\nabla_{\mu}\nabla_{\nu}\pi^{*}\,\nabla^{\mu}\nabla^{\alpha}\pi^{*}\,\nabla^{\nu}\nabla_{\alpha}\pi$
$\displaystyle{}+\nabla_{\mu}\nabla^{\mu}\pi\,\nabla^{\mu}\nabla^{\nu}\pi^{*}\,\nabla_{\mu}\nabla_{\nu}\pi^{*}+2\,\nabla_{\alpha}\pi\,\nabla^{\alpha}\pi^{*}\,(\,\nabla^{\nu}\nabla_{\nu}\nabla_{\mu}\nabla^{\mu}\pi^{*}-\nabla^{\nu}\nabla^{\mu}\nabla_{\mu}\nabla_{\nu}\pi^{*}\,)$
$\displaystyle{}+\nabla^{\alpha}\pi^{*}\,\nabla_{\nu}\pi^{*}\,(\,\nabla^{\mu}\nabla_{\mu}\nabla^{\nu}\nabla_{\alpha}\pi+\nabla^{\alpha}\nabla^{\nu}\nabla_{\mu}\nabla^{\mu}\pi\,)-\nabla^{\alpha}\pi^{*}\,\nabla_{\mu}\pi^{*}\,(\,\nabla^{\nu}\nabla^{\mu}\nabla_{\nu}\nabla_{\alpha}\pi+\nabla_{\alpha}\nabla_{\nu}\nabla^{\mu}\nabla^{\nu}\pi\,)$
$\displaystyle{}+3\,\nabla^{\nu}\nabla^{\alpha}\pi\,\nabla_{\alpha}\pi^{*}\,(\,\nabla_{\nu}\nabla_{\mu}\nabla^{\mu}\pi^{*}-\nabla_{\mu}\nabla^{\mu}\nabla_{\nu}\pi^{*}\,)+2\,\nabla^{\nu}\nabla^{\alpha}\pi^{*}\,\nabla_{\alpha}\pi\,(\,\nabla_{\nu}\nabla_{\mu}\nabla^{\mu}\pi^{*}-\nabla_{\mu}\nabla^{\mu}\nabla_{\nu}\pi^{*}\,)$
$\displaystyle{}+\nabla^{\mu}\nabla_{\mu}\pi\,\nabla^{\alpha}\pi^{*}\,(\,\nabla_{\nu}\nabla^{\nu}\nabla_{\alpha}\pi^{*}-\nabla_{\alpha}\nabla^{\nu}\nabla_{\nu}\pi^{*}\,)+2\,\nabla^{\mu}\nabla^{\nu}\pi^{*}\,\nabla^{\alpha}\pi^{*}\,(\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}\pi-\nabla_{\nu}\nabla_{\mu}\nabla_{\alpha}\pi\,)$
$\displaystyle{}+\nabla^{\nu}\nabla^{\alpha}\pi^{*}\,\nabla_{\alpha}\pi^{*}\,(\,\nabla_{\nu}\nabla_{\mu}\nabla^{\mu}\pi-\nabla_{\mu}\nabla^{\mu}\nabla_{\nu}\pi\,)+2\,\nabla^{\mu}\nabla^{\nu}\pi\,\nabla^{\alpha}\pi^{*}\,(\,\nabla_{\alpha}\nabla_{\mu}\nabla_{\nu}\pi^{*}-\nabla_{\nu}\nabla_{\alpha}\nabla_{\mu}\pi^{*}\,)$
$\displaystyle{}+2\,\nabla^{\mu}\nabla^{\nu}\pi^{*}\,\nabla^{\alpha}\pi\,(\,\nabla_{\mu}\nabla_{\alpha}\nabla_{\nu}\pi^{*}-\nabla_{\nu}\nabla_{\mu}\nabla_{\alpha}\pi^{*}\,)\,\,]$
(A.41)

Using the following commutation properties of covariant derivatives:

$\displaystyle[\nabla_{\mu},\nabla_{\nu}]\nabla^{\alpha}\pi$
$\displaystyle=R^{\alpha}_{\rho\mu\nu}\,\nabla^{\rho}\pi;\;\nabla_{\nu}\nabla_{\mu}\nabla_{\alpha}\pi-\nabla_{\alpha}\nabla_{\mu}\nabla_{\nu}\pi=R^{\rho}_{\mu\alpha\nu}\nabla_{\rho}\pi;\;$
$\displaystyle{}\nabla_{\alpha}\nabla_{\nu}\nabla^{\nu}\pi-\nabla_{\nu}\nabla^{\nu}\nabla_{\alpha}\pi=-R_{\nu\alpha}\nabla^{\nu}\pi,$
(A.42)

we get

$\displaystyle\mathcal{E}_{4}^{\rm{min}}=\frac{\omega}{2\lambda_{\pi}^{6}}[\,-2\,\nabla_{\alpha}\pi\,\nabla^{\alpha}\pi^{*}\,\nabla^{\nu}\nabla^{\mu}\pi^{*}\,R_{\mu\nu}-\nabla_{\alpha}\pi\,\nabla^{\alpha}\pi^{*}\,\nabla^{\mu}\pi^{*}\,\nabla_{\mu}R+2\,\nabla^{\mu}\pi^{*}\,\nabla^{\alpha}\pi^{*}\,\nabla^{\nu}\nabla^{\rho}\pi\,R_{\rho\mu\alpha\nu}$
$\displaystyle-\nabla^{\mu}\pi^{*}\,\nabla^{\alpha}\pi^{*}\,\nabla_{\mu}\nabla^{\nu}\pi\,R_{\nu\alpha}-\nabla^{\mu}\pi^{*}\,\nabla^{\alpha}\pi^{*}\,\nabla^{\rho}\pi\,\nabla_{\rho}R_{\alpha\mu}-3\,\nabla^{\alpha}\nabla^{\nu}\pi\,\nabla_{\alpha}\pi^{*}\,\nabla^{\mu}\pi^{*}\,R_{\mu\nu}$
$\displaystyle-2\,\nabla^{\mu}\pi^{*}\,\nabla_{\alpha}\pi\,\nabla^{\nu}\nabla^{\alpha}\pi^{*}\,R_{\mu\nu}+\nabla^{\nu}\pi^{*}\,\nabla^{\alpha}\pi^{*}\,\nabla^{\mu}\nabla_{\mu}\pi\,R_{\nu\alpha}-2\,\nabla^{\nu}\nabla^{\alpha}\pi^{*}\,\nabla_{\alpha}\pi^{*}\,\nabla^{\mu}\pi\,R_{\mu\nu}$
$\displaystyle+2\,\nabla_{\mu}\nabla^{\mu}\pi^{*}\,\nabla_{\nu}\nabla_{\alpha}\pi\,\nabla^{\nu}\nabla^{\alpha}\pi^{*}-\nabla_{\mu}\nabla^{\mu}\pi^{*}\,\nabla_{\nu}\nabla^{\nu}\pi^{*}\,\nabla_{\alpha}\nabla^{\alpha}\pi-2\,\nabla_{\mu}\nabla_{\nu}\pi^{*}\,\nabla^{\mu}\nabla^{\alpha}\pi^{*}\nabla^{\nu}\nabla_{\alpha}\pi$
$\displaystyle+\nabla_{\mu}\nabla^{\mu}\pi\,\nabla^{\mu}\nabla^{\nu}\pi^{*}\,\nabla_{\mu}\nabla_{\nu}\pi^{*}\,\,]$
(A.43)

Note that the second and fifth terms in the above equation are higher
derivative (of metric) terms. In order to remove these terms we need to add
some non-minimal terms in the action in such a way that on varying the total
action, all the derivatives of the Ricci tensor and Ricci scalar vanish. The
equation of motion for the non-minimal action (2.2.1) is given by:

$\displaystyle\mathcal{E}^{\rm{nm}}_{4}$
$\displaystyle=\frac{\omega}{4\lambda_{\pi}^{6}}[\,2\,\nabla_{\mu}\nabla^{\mu}\pi^{*}\,\nabla_{\nu}\pi^{*}\,\nabla^{\nu}\pi\,R+2\,\nabla_{\mu}\nabla_{\nu}\pi\,\nabla^{\mu}\pi^{*}\,\nabla^{\nu}\pi^{*}\,R+2\,\nabla_{\mu}\pi^{*}\,\nabla^{\mu}\pi\,\nabla^{\alpha}\pi^{*}\,\nabla_{\alpha}R$
$\displaystyle{}-\nabla_{\mu}\nabla^{\mu}\pi\,\nabla_{\nu}\pi^{*}\,\nabla^{\nu}\pi^{*}\,R+2\,\nabla_{\mu}\nabla^{\mu}\pi\,\nabla^{\nu}\pi^{*}\,\nabla^{\alpha}\pi^{*}\,R_{\nu\alpha}+4\,\nabla^{\alpha}\nabla^{\mu}\pi^{*}\,\nabla_{\alpha}\pi\,\nabla^{\nu}\pi^{*}\,R_{\mu\nu}$
$\displaystyle{}+2\,\nabla^{\alpha}\pi\,\nabla^{\mu}\pi^{*}\,\nabla^{\nu}\pi^{*}\,\nabla_{\alpha}R_{\mu\nu}+4\,\nabla^{\mu}\nabla_{\alpha}\pi^{*}\,\nabla^{\alpha}\pi^{*}\,\nabla^{\nu}\pi
R_{\mu\nu}++2\nabla_{\alpha}\pi^{*}\nabla^{\alpha}\pi^{*}\nabla^{\mu}\nabla^{\nu}\pi\,R_{\mu\nu}\,\,]$
(A.44)

If we add the above non minimal action in the action (2.2.1) then it will
cancel all the higher order derivative terms in the EOM. So varying the action
${S}_{4}^{\rm{min}}+{S}^{\rm{nm}}_{4}$ with respect to $\pi$, we get

$\displaystyle\mathcal{E}^{\prime}$
$\displaystyle=\frac{\omega}{2\lambda_{\pi}^{6}}\,\,[\,\,2\,\Box\pi^{*}\,\nabla_{\nu}\nabla_{\alpha}\pi\,\nabla^{\nu}\nabla^{\alpha}\pi^{*}-(\,\Box\pi^{*})^{2}\,\Box\pi+\Box\pi\,\nabla^{\mu}\nabla^{\nu}\pi^{*}\,\nabla_{\mu}\nabla_{\nu}\pi^{*}$
$\displaystyle{}-2\,\nabla_{\mu}\nabla_{\nu}\pi^{*}\,\nabla^{\mu}\nabla^{\alpha}\pi^{*}\,\nabla^{\nu}\nabla_{\alpha}\pi+\Box\pi^{*}\,\nabla_{\mu}\pi^{*}\,\nabla^{\mu}\pi\,R-\frac{1}{2}\Box\pi\,\nabla_{\alpha}\pi^{*}\,\nabla^{\alpha}\pi^{*}\,R$
$\displaystyle{}+\nabla_{\alpha}\pi^{*}\,\nabla_{\mu}\pi^{*}\,\nabla^{\mu}\nabla^{\alpha}\pi\,R+2\,\Box\pi\,\nabla^{\nu}\pi^{*}\,\nabla^{\mu}\pi^{*}\,R_{\nu\mu}+\nabla_{\alpha}\pi^{*}\,\nabla^{\alpha}\pi^{*}\,\nabla^{\nu}\nabla^{\mu}\pi\,R_{\mu\nu}$
$\displaystyle{}-2\,\nabla_{\alpha}\pi\,\nabla^{\alpha}\pi^{*}\,\nabla^{\nu}\nabla^{\mu}\pi^{*}\,R_{\mu\nu}-\nabla^{\mu}\pi^{*}\,\nabla^{\alpha}\pi^{*}\,\nabla_{\mu}\nabla^{\nu}\pi\,R_{\nu\alpha}$
$\displaystyle{}+2\,\nabla^{\mu}\pi^{*}\,\nabla^{\alpha}\pi^{*}\,\nabla^{\nu}\nabla^{\rho}\pi\,R_{\rho\mu\alpha\nu}-3\,\nabla^{\alpha}\nabla^{\nu}\pi\,\nabla_{\alpha}\pi^{*}\,\nabla^{\mu}\pi^{*}\,R_{\mu\nu}]$
(A.45)

###  A.5 Consistency check with real scalar galileon

In this section, we show systematically that our fourth order action
($S^{\rm{min}}_{4}+S^{\rm{nm}}_{4}$ ) exactly matches with the results of
Deffayet et al [149] in the $\pi=\pi^{*}$ limit. Eq. (2.18) can be written as,

$\displaystyle\left.{S}_{4}^{\rm{min}}+{S}_{4}^{\rm{nm}}\right|_{\pi=\pi^{*}}$
$\displaystyle=A_{2}\,\int
d^{4}x\,\,\sqrt{-g}\,\,\left[\,(\Box\pi)^{2}\,\nabla_{\alpha}\pi\,\nabla^{\alpha}\pi+\Box\pi\,\nabla_{\nu}\pi\,\nabla^{\nu}\nabla^{\alpha}\pi\,\nabla_{\alpha}\pi\right.$
$\displaystyle~{}~{}-\left.\,\nabla_{\mu}\nabla_{\nu}\pi\,\nabla^{\mu}\nabla^{\nu}\pi\,\nabla_{\alpha}\pi\,\nabla^{\alpha}\pi-\nabla^{\mu}\nabla^{\nu}\pi\,\nabla_{\nu}\nabla_{\alpha}\pi\,\nabla_{\mu}\pi\,\nabla^{\alpha}\pi\,\right]$
(A.46) $\displaystyle{}+\frac{A_{2}}{4}\int
d^{4}x\,\,\sqrt{-g}\,\,\nabla_{\alpha}\pi\nabla^{\alpha}\pi\nabla^{\mu}\pi\nabla^{\nu}\pi\,\,G_{\mu\nu}$
$\displaystyle{}-\frac{3A_{2}}{4}\int
d^{4}x\,\,\sqrt{-g}\,\,\nabla_{\alpha}\pi\nabla^{\alpha}\pi\nabla^{\mu}\pi\nabla^{\nu}\pi
R_{\mu\nu}$

where $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}\,g_{\mu\nu}\,R$. Concentrating on the
last term in the above equation (A.5) and using the relation
$(\nabla_{\mu}\nabla_{\nu}\nabla^{\nu}\pi-\nabla_{\nu}\nabla^{\nu}\nabla_{\mu}\pi)\,\nabla^{\mu}\pi=-R_{\mu\nu}\nabla^{\mu}\pi\nabla^{\nu}\pi$,
ignoring the total derivative terms, we get:

$\displaystyle-\frac{3A_{2}}{4}\nabla_{\alpha}\pi\nabla^{\alpha}\pi\nabla^{\mu}\pi\nabla^{\nu}\pi
R_{\mu\nu}$
$\displaystyle=-\frac{3A_{2}}{4}\nabla_{\alpha}\pi\nabla^{\alpha}\pi\nabla_{\mu}\nabla^{\mu}\nabla_{\nu}\pi\nabla^{\nu}\pi+\frac{3A_{2}}{4}\nabla_{\alpha}\pi\nabla^{\alpha}\pi\nabla_{\nu}\nabla^{\mu}\nabla_{\mu}\pi\nabla^{\nu}\pi$
$\displaystyle{}=\frac{3A_{2}}{2}\nabla_{\mu}\nabla_{\alpha}\pi\nabla^{\mu}\nabla^{\nu}\pi\nabla_{\nu}\pi\nabla^{\alpha}\pi+\frac{3A_{2}}{4}\nabla_{\mu}\nabla_{\nu}\pi\nabla^{\mu}\nabla^{\mu}\pi\nabla^{\nu}\pi\nabla_{\alpha}\pi\nabla^{\alpha}\pi$
$\displaystyle{}\hskip
28.45274pt-\frac{3A_{2}}{4}(\Box\pi)^{2}\nabla_{\alpha}\pi\nabla^{\alpha}\pi-\frac{3A_{2}}{2}\Box\pi\nabla^{\mu}\nabla^{\nu}\pi\nabla_{\mu}\pi\nabla_{\nu}\pi\,.$
(A.47)

Using Eqs. (A.5) and (A.5), we get:

$\displaystyle\left.{S}_{4}^{\rm{min}}+{S}_{4}^{\rm{nm}}\right|_{\pi=\pi^{*}}$
$\displaystyle=\frac{A_{2}}{4}\,\int
d^{4}x\,\,\sqrt{-g}\,\,\left[\,(\Box\pi)^{2}\,\nabla_{\alpha}\pi\,\nabla^{\alpha}\pi-2\Box\pi\,\nabla_{\nu}\pi\,\nabla^{\nu}\nabla^{\alpha}\pi\,\nabla_{\alpha}\pi\right.$
$\displaystyle{}\hskip
42.67912pt-\left.\,\nabla_{\mu}\nabla_{\nu}\pi\,\nabla^{\mu}\nabla^{\nu}\pi\,\nabla_{\alpha}\pi\,\nabla^{\alpha}\pi+2\nabla^{\mu}\nabla^{\nu}\pi\,\nabla_{\nu}\nabla_{\alpha}\pi\,\nabla_{\mu}\pi\,\nabla^{\alpha}\pi\,\right]$
$\displaystyle{}+\frac{A_{2}}{4}\int
d^{4}x\,\,\sqrt{-g}\,\,\nabla_{\alpha}\pi\nabla^{\alpha}\pi\nabla^{\mu}\pi\nabla^{\nu}\pi\,\,G_{\mu\nu}$
(A.48)

Thus, the above action (A.5) matches with the action in Ref. [149]. It is
important to note that there is an overall sign difference between our result
and obtained by Deffayet et al [149]. This is due to the signature convention
of the metric and hence our definition of energy-momentum tensor i.e.
$T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g^{\mu\nu}}$, differs by
an overall negative sing compared to Deffayet et al [149].

####  A.5.1 Galileon scalar electrodynamics in curved space time for $A_{2}=0$

Before discussing the other special case $A_{2}=0$, we obtain the non-minimal
coupling terms for the full action. Consider the action corresponding to the
full Lagrangian (A.2) and varying the action with respect to $\pi$ gives the
equation of motion for the generic action in curved spacetime. As our main
focus is to obtain the non-minimal action, we collect only the fourth order
derivative terms in the field as we have done in Appendix (A.4). After
collecting the fourth order terms we get,

$\displaystyle\mathcal{E}^{\rm{min}}_{4}$
$\displaystyle=\frac{A_{2}}{2}\left[\,\,2\nabla_{\alpha}\pi\,\nabla^{\alpha}\pi^{*}\,(\,\nabla^{\nu}\nabla_{\nu}\nabla_{\mu}\nabla^{\mu}\pi^{*}-\nabla^{\nu}\nabla^{\mu}\nabla_{\mu}\nabla_{\nu}\pi^{*}\,)\right.$
$\displaystyle{}\left.+\nabla_{\alpha}\pi^{*}\,\nabla^{\mu}\pi^{*}\,(\,\nabla_{\nu}\nabla^{\nu}\nabla^{\alpha}\nabla_{\mu}\pi+\nabla_{\mu}\nabla^{\alpha}\nabla_{\nu}\nabla^{\nu}\pi-\nabla_{\nu}\nabla^{\alpha}\nabla^{\nu}\nabla_{\mu}\pi-\nabla_{\mu}\nabla_{\nu}\nabla^{\alpha}\nabla^{\nu}\pi\,)\right]$
$\displaystyle{}+A_{1}\left[\,\,2\nabla_{\alpha}\pi^{*}\,\nabla^{\alpha}\pi^{*}\,(\,\nabla^{\nu}\nabla_{\nu}\nabla_{\mu}\nabla^{\mu}\pi-\nabla^{\nu}\nabla^{\mu}\nabla_{\mu}\nabla_{\nu}\pi\,)\right.$
(A.49)
$\displaystyle{}\left.+\nabla^{\alpha}\pi^{*}\,\nabla^{\mu}\pi^{*}\,(\,\nabla_{\nu}\nabla^{\alpha}\nabla^{\nu}\nabla_{\mu}\pi+\nabla_{\mu}\nabla_{\nu}\nabla^{\alpha}\nabla^{\nu}\pi-\nabla^{\nu}\nabla_{\nu}\nabla^{\alpha}\nabla_{\mu}\pi-\nabla_{\mu}\nabla^{\alpha}\nabla_{\nu}\nabla^{\nu}\pi\,)\right.$
$\displaystyle{}\left.+2\nabla_{\alpha}\pi^{*}\,\nabla_{\nu}\pi\,(\,\nabla^{\mu}\nabla_{\mu}\nabla^{\nu}\nabla^{\alpha}\pi^{*}+\nabla^{\nu}\nabla^{\alpha}\nabla^{\mu}\nabla_{\mu}\pi^{*}-\nabla^{\nu}\nabla^{\mu}\nabla^{\alpha}\nabla_{\mu}\pi^{*}-\nabla_{\mu}\nabla_{\alpha}\nabla^{\mu}\nabla_{\nu}\pi^{*}\,)\right]$

using the commutation properties of the covariant derivatives and using (A.4)
we get

$\displaystyle\mathcal{E}^{\rm{min}}_{4}$
$\displaystyle=-\frac{A_{2}}{2}\nabla_{\alpha}\pi^{*}\nabla^{\alpha}\pi\nabla^{\mu}\pi^{*}\nabla_{\mu}R-A_{1}\nabla_{\alpha}\pi^{*}\nabla^{\alpha}\pi^{*}\nabla^{\mu}\pi\nabla_{\mu}R-\frac{A_{2}}{2}\nabla^{\alpha}\pi^{*}\nabla^{\mu}\pi^{*}\nabla^{\nu}\pi\nabla_{\nu}R_{\alpha\mu}$
$\displaystyle{}+A_{1}\nabla^{\alpha}\pi^{*}\nabla^{\mu}\pi^{*}\nabla^{\nu}\pi\nabla_{\nu}R_{\alpha\mu}-2A_{1}\nabla^{\alpha}\pi^{*}\nabla^{\mu}\pi^{*}\nabla^{\nu}\pi\nabla_{\mu}R_{\alpha\nu}\,.$
(A.50)

We now consider the following general non-minimal action

$\displaystyle S^{\rm{nm}}_{4}=\int
d^{4}x\sqrt{-g}\,\,\mathcal{L}^{\rm{nm}}_{4}$ (A.51)

where

$\displaystyle\mathcal{L}^{\rm{nm}}_{4}$
$\displaystyle=-\left[\nabla_{\alpha}\pi^{*}\nabla^{\alpha}\pi^{*}\nabla^{\mu}\pi\nabla^{\nu}\pi+\nabla_{\alpha}\pi\nabla^{\alpha}\pi\nabla^{\mu}\pi^{*}\nabla^{\nu}\pi^{*}\right]\left(g_{1}\,R_{\mu\nu}+g_{2}\,g_{\mu\nu}R\right)$
$\displaystyle{}-\nabla_{\alpha}\pi^{*}\nabla^{\alpha}\pi\nabla^{\mu}\pi^{*}\nabla^{\nu}\pi]\left(h_{1}\,R_{\mu\nu}+h_{2}\,g_{\mu\nu}R\right)$
(A.52)

where $g_{1},g_{2},h_{1}$ and $h_{2}$ are arbitray constants which we will fix
later. Variation of the action (A.51) with respect to $\pi$ and collecting
only derivatives of the Ricci tensor and Ricci scalar terms because we are
interested in fixing the coefficients $g_{1},g_{2},h_{1},h_{2}$, we obtain

$\displaystyle\mathcal{E}^{\rm{nm}}_{4}$
$\displaystyle=g_{1}\nabla_{\alpha}\pi^{*}\nabla^{\alpha}\pi^{*}\nabla^{\mu}\pi\nabla_{\mu}R+4g_{2}\nabla_{\alpha}\pi^{*}\nabla^{\alpha}\pi^{*}\nabla^{\mu}\pi\nabla_{\mu}R+\frac{h_{1}}{2}\nabla_{\alpha}\pi^{*}\nabla^{\alpha}\pi\nabla^{\mu}\pi^{*}\nabla_{\mu}R$
(A.53)
$\displaystyle{}+2h_{2}\nabla_{\alpha}\pi^{*}\nabla^{\alpha}\pi\nabla^{\mu}\pi^{*}\nabla_{\mu}R+2g_{1}\nabla^{\mu}\pi^{*}\nabla^{\nu}\pi^{*}\nabla^{\alpha}\pi\nabla_{\alpha}R_{\mu\nu}+h_{1}\nabla^{\mu}\pi^{*}\nabla^{\nu}\pi\nabla^{\alpha}\pi^{*}\nabla_{\alpha}R_{\mu\nu}\,.$

Adding Eqs. (A.5.1) and (A.53) such that all the derivative terms of Ricci
tensor and scalar vanish, we obtain the following relations :

$\displaystyle-A_{1}+g_{1}+4g_{2}$ $\displaystyle=0$ (A.54)
$\displaystyle-\frac{A_{2}}{2}+A_{1}+2g_{1}$ $\displaystyle=0$ (A.55)
$\displaystyle-\frac{A_{2}}{2}+2h_{2}+\frac{h_{1}}{2}$ $\displaystyle=0$
(A.56) $\displaystyle-2A_{1}+h_{1}$ $\displaystyle=0$ (A.57)

which implies

$\displaystyle g_{1}$ $\displaystyle=\frac{A_{2}}{4}-\frac{A_{1}}{2}$ (A.58)
$\displaystyle g_{2}$ $\displaystyle=\frac{3A_{1}}{8}-\frac{A_{2}}{16}$ (A.59)
$\displaystyle h_{1}$ $\displaystyle=2A_{1}$ (A.60) $\displaystyle h_{2}$
$\displaystyle=-\frac{A_{1}}{2}+\frac{A_{2}}{4}\,.$ (A.61)

Substituting the above values in Eq. (A.5.1), we get

$\displaystyle\mathcal{L}^{\rm{nm}}_{4}$
$\displaystyle=\frac{A_{1}}{2}\left[\nabla_{\alpha}\pi^{*}\nabla^{\alpha}\pi^{*}\nabla^{\mu}\pi\nabla^{\nu}\pi+\nabla_{\alpha}\pi\nabla^{\alpha}\pi\nabla^{\mu}\pi^{*}\nabla^{\nu}\pi^{*}\right]\left(R_{\mu\nu}-\frac{3}{4}\,g_{\mu\nu}R\right)$
$\displaystyle{}-\frac{A_{2}}{4}\left[\nabla_{\alpha}\pi^{*}\nabla^{\alpha}\pi^{*}\nabla^{\mu}\pi\nabla^{\nu}\pi+\nabla_{\alpha}\pi\nabla^{\alpha}\pi\nabla^{\mu}\pi^{*}\nabla^{\nu}\pi^{*}\right]\left(R_{\mu\nu}-\frac{1}{4}\,g_{\mu\nu}R\right)$
$\displaystyle{}-2A_{1}\nabla_{\alpha}\pi^{*}\nabla^{\alpha}\pi\nabla^{\mu}\pi^{*}\nabla^{\nu}\pi]\left(R_{\mu\nu}-\frac{1}{4}\,g_{\mu\nu}R\right)-\frac{A_{2}}{4}\nabla_{\alpha}\pi^{*}\nabla^{\alpha}\pi\nabla^{\mu}\pi^{*}\nabla^{\nu}\pi\,g_{\mu\nu}R$
(A.62)

We note that the above non-minimal action identical with $A_{1}=0$ case
considered in Appendix (A.4). In $\pi=\pi^{*}$ limit, this case also leads to
identical equations of motion as in Ref. [149].

###  A.6 Vector Galileon action $S_{VEG}$

In Ref. [2017-Nandi.Shankaranarayanan-JCAP], the authors obtained a vector
Galileon model that leads to second-order equations. For completeness, in this
appendix, we have listed below the vector Galileon action. The complete vector
Galileon action can be written as

${S}_{VEC}={S}_{\rm VG}+\lambda_{\rm VG}\sum_{i=1}^{12}{S}_{Vi}.$ (A.63)

where, $\lambda_{VG}$ is coupling constant

$\displaystyle{S}_{VG}$ $\displaystyle=\lambda_{\rm
VG}\,\int\,d^{4}x\,\sqrt{-g}\,\epsilon^{\alpha\gamma\nu}\epsilon^{\mu\eta\beta}\,\nabla_{\alpha\beta}A_{\gamma}\,\nabla_{\mu\nu}A_{\eta}$
(A.64) $\displaystyle{S}_{V1}$ $\displaystyle=E_{1}\,\int
d^{4}x\,\sqrt{-g}\,{g}^{\mu\nu}{g}^{\alpha\beta}{g}^{\gamma\delta}\,R_{\mu\nu}\,{\nabla}_{\alpha}{A}_{\gamma}\,{\nabla}_{\beta}{A}_{\delta}$
(A.65) $\displaystyle{S}_{V2}$ $\displaystyle=E_{2}\,\int
d^{4}x\,\sqrt{-g}\,{g}^{\mu\alpha}{g}^{\nu\beta}{g}^{\gamma\delta}\,{R}_{\mu\nu}\,{\nabla}_{\alpha}{A}_{\gamma}\,{\nabla}_{\beta}{A}_{\delta}$
(A.66) $\displaystyle{S}_{V3}$ $\displaystyle=E_{3}\,\int
d^{4}x\,\sqrt{-g}\,{g}^{\mu\nu}{g}^{\alpha\beta}{g}^{\gamma\delta}\,{R}_{\mu\nu}\,{\nabla}_{\alpha}{A}_{\beta}\,{\nabla}_{\gamma}{A}_{\delta}$
(A.67) $\displaystyle{S}_{V4}$ $\displaystyle=E_{4}\,\int
d^{4}x\,\sqrt{-g}\,{g}^{\mu\nu}{g}^{\alpha\delta}{g}^{\gamma\beta}\,{R}_{\mu\nu}\,{\nabla}_{\alpha}{A}_{\beta}\,{\nabla}_{\gamma}{A}_{\delta}$
(A.68) $\displaystyle{S}_{V5}$ $\displaystyle=E_{5}\,\int
d^{4}x\,\sqrt{-g}\,{g}^{\mu\gamma}{g}^{\alpha\beta}{g}^{\nu\delta}\,{R}_{\mu\nu}\,{\nabla}_{\alpha}{A}_{\beta}\,{\nabla}_{\gamma}{A}_{\delta}$
(A.69) $\displaystyle{S}_{V6}$ $\displaystyle=E_{6}\,\int
d^{4}x\,\sqrt{-g}\,{g}^{\mu\alpha}{g}^{\nu\delta}{g}^{\gamma\beta}\,{R}_{\mu\nu}\,{\nabla}_{\alpha}{A}_{\beta}\,{\nabla}_{\gamma}{A}_{\delta}$
(A.70) $\displaystyle{S}_{V7}$ $\displaystyle=E_{7}\,\int
d^{4}x\,\sqrt{-g}\,{g}^{\mu\alpha}{g}^{\nu\beta}{g}^{\gamma\zeta}{g}^{\delta\eta}\,{R}_{\alpha\beta\gamma\delta}\,{\nabla}_{\mu}{A}_{\nu}\,{\nabla}_{\zeta}{A}_{\eta}$
(A.71) $\displaystyle{S}_{V8}$ $\displaystyle=E_{8}\,\int
d^{4}x\,\sqrt{-g}\,{g}^{\mu\alpha}{g}^{\eta\beta}{g}^{\gamma\zeta}{g}^{\delta\nu}\,{R}_{\alpha\beta\gamma\delta}\,{\nabla}_{\mu}{A}_{\nu}\,{\nabla}_{\zeta}{A}_{\eta}~{}~{}~{}~{}~{}~{}$
(A.72) $\displaystyle{S}_{V9}$ $\displaystyle=E_{9}\,\int
d^{4}x\,\sqrt{-g}\,{g}^{\alpha\beta}{g}^{\gamma\delta}{g}^{\mu\nu}\,{R}_{\alpha\beta}\,{R}_{\gamma\delta}\,{A}_{\mu}{A}_{\nu}$
(A.73) $\displaystyle{S}_{V10}$ $\displaystyle=E_{10}\,\int
d^{4}x\,\sqrt{-g}\,{g}^{\alpha\beta}{g}^{\gamma\mu}{g}^{\delta\nu}\,{R}_{\alpha\beta}\,{R}_{\gamma\delta}\,{A}_{\mu}{A}_{\nu}$
(A.74) $\displaystyle{S}_{V11}$ $\displaystyle=E_{11}\,\int
d^{4}x\,\sqrt{-g}\,{g}^{\alpha\gamma}{g}^{\beta\delta}{g}^{\mu\nu}\,{R}_{\alpha\beta}\,{R}_{\gamma\delta}\,{A}_{\mu}{A}_{\nu}~{}~{}~{}~{}~{}~{}$
(A.75) $\displaystyle{S}_{V12}$ $\displaystyle=E_{12}\,\int
d^{4}x\,\sqrt{-g}\,{g}^{\alpha\gamma}{g}^{\beta\mu}{g}^{\delta\nu}\,{R}_{\alpha\beta}\,{R}_{\gamma\delta}\,{A}_{\mu}{A}_{\nu}$
(A.76)

where $E_{i}$’s are dimensionless coefficients. Note that $E_{i}$ are related
to $\lambda_{\rm VG}$.

##  Appendix B Helical magnetic field from Riemann coupling: Details

In this Appendix, we provide more details on the results from chapter 3.

###  B.1 Series expansion of the Bessel function

For completeness, in this appendix, we obtain the series expansion of the
Bessel functions to the leading order terms [294]. It is useful to define the
following quantity:

$\displaystyle
F(\tau)=\exp\left[\frac{i\pi}{\alpha}\left\lfloor\frac{\pi-\arg(\tau)-\arg\left(\frac{\sqrt{k}\,\varsigma}{\alpha}\right)}{2\pi}\right\rfloor\right]$
(B.1)

where $\left\lfloor\cdots\right\rfloor$ represents the floor function. Up to
leading order, the series expansion for the Bessel functions are:

$\displaystyle
J_{\frac{1}{2\alpha}}\left(\frac{\varsigma\,\sqrt{k}}{\alpha}\tau\,\right)=\frac{F(\tau)}{\Gamma\left(1+\frac{1}{2\alpha}\right)}\left(\frac{\varsigma\,\sqrt{k}}{2\alpha}\right)^{\frac{1}{2\alpha}}\tau^{\frac{1}{2\alpha}}\left(1-\frac{\varsigma^{2}k\tau^{2}}{2\alpha(1+2\alpha)}+O\left(\tau^{3}\right)\right),$
(B.2) $\displaystyle
Y_{\frac{1}{2\alpha}}\left(\frac{\varsigma\,\sqrt{k}}{\alpha}\tau\,\right)=\frac{F(\tau)}{\pi}\left(\frac{\varsigma\,\sqrt{k}}{2\alpha}\right)^{\frac{1}{2\alpha}}\Gamma\left(-\frac{1}{2\alpha}\right)\cos\left(\frac{\pi}{2\alpha}\right)\left(-\tau^{\frac{1}{2\alpha}}+\frac{k\,\varsigma^{2}\,\tau^{2+\frac{1}{2\alpha}}}{2\alpha(1+2\alpha)}+O\left(\tau^{3}\right)\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{F(\tau)^{-1}}{\pi}\left(\frac{2\alpha}{\varsigma\,\sqrt{k}}\right)^{\frac{1}{2\alpha}}\Gamma\left(\frac{1}{2\alpha}\right)\left(-\tau^{-\frac{1}{2\alpha}}+\frac{k\,\varsigma^{2}\,\tau^{2-\frac{1}{2\alpha}}}{2\alpha(-1+2\alpha)}+O\left(\tau^{3}\right)\right),$
(B.3) $\displaystyle
J_{\frac{1}{2\alpha}}\left(-\frac{i\varsigma\,\sqrt{k}}{\alpha}\tau\,\right)=\frac{\tilde{F}(\tau)}{\Gamma\left(1+\frac{1}{2\alpha}\right)}\left(-\frac{i\,\varsigma\,\sqrt{k}}{2\alpha}\right)^{\frac{1}{2\alpha}}\tau^{\frac{1}{2\alpha}}\left(1+\frac{\varsigma^{2}k\tau^{2}}{2\alpha(1+2\alpha)}+O\left(\tau^{3}\right)\right)$
(B.4) $\displaystyle
Y_{\frac{1}{2\alpha}}\left(-i\frac{\varsigma\,\sqrt{k}}{\alpha}\tau\,\right)=-\frac{\tilde{F}(\tau)}{\pi}\left(-\frac{i\,\varsigma\,\sqrt{k}}{2\alpha}\right)^{\frac{1}{2\alpha}}\Gamma\left(-\frac{1}{2\alpha}\right)\cos\left(\frac{\pi}{2\alpha}\right)\tau^{\frac{1}{2\alpha}}\left(1+\frac{k\varsigma^{2}\tau^{2}}{2\alpha(1+2\alpha)}+O\left(\tau^{3}\right)\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{}-\frac{\tilde{F}(\tau)^{-1}}{\pi}\left(-\frac{2\alpha}{\,i\,\varsigma\,\sqrt{k}}\right)^{\frac{1}{2\alpha}}\Gamma\left(\frac{1}{2\alpha}\right)\tau^{-\frac{1}{2\alpha}}\left(1+\frac{k\,\varsigma^{2}\,\tau^{2}}{2\alpha(-1+2\alpha)}+O\left(\tau^{3}\right)\right).$
(B.5)

At leading order, the helicity modes are:

$\displaystyle A_{+}(\tau,k)$
$\displaystyle=F(\tau)\,\left(\frac{\varsigma\,\sqrt{k}}{2\alpha}\right)^{\frac{1}{2\alpha}}\left[\frac{C_{1}}{\Gamma\left(1+\frac{1}{2\alpha}\right)}-\frac{C_{2}}{\pi}\Gamma\left(-\frac{1}{2\alpha}\right)\cos\left(\frac{\pi}{2\alpha}\right)\right]$
$\displaystyle-~{}~{}C_{2}\frac{F(\tau)^{-1}}{\pi}\left(\frac{2\alpha}{\varsigma\,\sqrt{k}}\right)^{\frac{1}{2\alpha}}\Gamma\left(\frac{1}{2\alpha}\right)\tau^{-\frac{1}{\alpha}}$
(B.6) $\displaystyle A_{-}(\tau,k)$
$\displaystyle=\tilde{F}(\tau)\left(-i\frac{\varsigma\,\sqrt{k}}{2\alpha}\right)^{\frac{1}{2\alpha}}\left[\,\frac{C_{3}}{\Gamma\left(1+\frac{1}{2\alpha}\right)}-\frac{C_{4}}{\pi}\Gamma\left(-\frac{1}{2\alpha}\right)\cos\left(\frac{\pi}{2\alpha}\right)\right]$
$\displaystyle{}-~{}~{}C_{4}\frac{\tilde{F}(\tau)^{-1}}{\pi}\left(-\frac{2\alpha}{i\varsigma\,\sqrt{k}}\right)^{\frac{1}{2\alpha}}\Gamma\left(\frac{1}{2\alpha}\right)\tau^{-\frac{1}{\alpha}}$
(B.7)

Figure B.1: Plot of the Floor function for the range of $-10^{8}<\eta<10^{8}$
and $-50<\beta<-2$.

It is convenient to define the following quantities:

$\displaystyle\mathcal{F}(\tau)$
$\displaystyle=F(\tau)\,\left(\frac{\varsigma}{2\alpha}\right)^{\frac{1}{2\alpha}},$
(B.8) $\displaystyle\tilde{\mathcal{F}}(\tau)$
$\displaystyle=\tilde{F}(\tau)\,\left(-i\frac{\varsigma}{2\alpha}\right)^{\frac{1}{2\alpha}},$
(B.9) $\displaystyle C(\tau)$
$\displaystyle=F(\tau)\,\left(\frac{\varsigma}{2\alpha}\right)^{\frac{1}{2\alpha}}\left[\frac{C_{1}}{\Gamma\left(1+\frac{1}{2\alpha}\right)}-\frac{C_{2}}{\pi}\Gamma\left(-\frac{1}{2\alpha}\right)\cos\left(\frac{\pi}{2\alpha}\right)\right],$
(B.10) $\displaystyle\tilde{C}(\tau)$
$\displaystyle=\tilde{F}(\tau)\left(-i\frac{\varsigma}{2\alpha}\right)^{\frac{1}{2\alpha}}\left[\,\frac{C_{3}}{\Gamma\left(1+\frac{1}{2\alpha}\right)}-\frac{C_{4}}{\pi}\Gamma\left(-\frac{1}{2\alpha}\right)\cos\left(\frac{\pi}{2\alpha}\right)\right]$
(B.11)

B.1 is the 3-D plot of the floor function for the range of values of $\eta$
and $\beta$ that are consistent with generic models of inflation. As can be
seen from the plot, the floor function is zero in the range of interest. Due
to this reason, we have suppressed the time dependence in above quantities
i.e.,
$\mathcal{F}(\tau)=\mathcal{F},\tilde{\mathcal{F}}(\tau)=\tilde{\mathcal{F}}$
and $C(\tau)=C,\tilde{C}(\tau)=\tilde{C}$. We have used this in computing the
energy densities in Sec. (3.2).

###  B.2 Power spectrum for slow-roll inflation

In Sec. (3.2), we obtained the power-spectrum for the helical fields in the
power-law inflation. In this section, we obtain the power-spectrum for slow-
roll inflation.

To do that, we first obtain the power-spectrum for the power-law inflation by
assuming that $M$ is slowly varying and is related to the Hubble parameter
$H$, i. e., $M\sim H\sim\mathcal{H}/{a}$.

####  B.2.1 Power law inflation

The equation of motion (4.18) for power inflation (4.16) is:

$\displaystyle
A_{h}^{\prime\prime}+\left[k^{2}+\frac{8hk\xi}{\eta}\right]\,A_{h}=0.$ (B.12)

where $\xi=\frac{\beta(\beta+2)}{(\beta+1)}$, solution of the above equation
for super horizon modes are given by

$\displaystyle A_{+}(\eta,k)$