# Seminal Electromagnetic fields from preinflation

1 Daniela Magos111<EMAIL_ADDRESS>and 1,2 Mauricio Bellini
<EMAIL_ADDRESS>1 Instituto de Investigaciones Físicas de Mar del Plata
- IFIMAR,
Consejo Nacional de Investigaciones Científicas y Técnicas \- CONICET, Mar del
Plata, Argentina.
2 Departamento de Física, Facultad de Ciencias Exactas y Naturales,
Universidad Nacional de Mar del Plata, Funes 3350, C.P. 7600, Mar del Plata,
Argentina.

###### Abstract

We investigate the geometric dynamics of the primordial electric and magnetic
fields during the early stages of the universe by extending a recently
introduced quantum algebra BMM ; BMAS . We work on an extended model of
gravity that considers the boundary terms from the Einstein-Hilbert action as
geometric quantum fluctuations of the spacetime. We propose that the extended
Riemann manifold is generated by a new connection
${\hat{\delta\Gamma}}^{\mu}_{\alpha\beta}$. This connection contains geometric
information about the fluctuations of gravitational and electromagnetic fields
in the vacuum, which could have been crucial during the primordial stages of
the universe’s evolution. We revisit a preinflationary cosmological model mb
with a variable time scale and negative spatial curvature, such that the
universe begins with a null initial background energy density. We observed the
emergence of large scale magnetic fields starting from small values during the
early phases of the universe’s evolution. Subsequently, these fields decrease
to reach present day values on the order of $\left<\hat{\delta B}\right>\simeq
10^{-12}\,{\rm G}$ on cosmological scales (between $10^{24}$ and $10^{26}$
meters). This significant deviation from inflationary models eliminates the
need to impose excessively large initial values on these fields.

##  I Introduction and motivation

One of the open questions in cosmology is to explain how the primordial
electromagnetic fields and gravitational waves (GW) were generated and evolved
in the primordial universe. The study of these fundamental phenomena is
essential to gain insight into the physical processes that took place during
the early expansion of the universe. A particularly intriguing possibility is
to provide a geometric explanation for the origin of electromagnetic field
fluctuations. Their source is to be understood in the realm of quantum
mechanics, and the expectation value of the flux associated with these
fluctuations will locally modify the curvature of spacetime. Furthermore, it
is possible to detect the primordial electromagnetic fields through the
radiation of the cosmic microwave background, for instance, detecting traces
of $B$-mode polarization in the electromagnetic waves would indirectly provide
evidence for the existence of primordial GW. Concerning to the generation
mechanisms of cosmological magnetic fields, it has been suggested that seed
magnetic fields could originate from phase transitions occurring in the early
universe 6 ; 7 ; thorne ; rebhan ; giova ; rab ; fammb . However, it is
natural to look for the possibility of generating large-scales magnetic fields
during preinflation with strength according with observational data on
cosmological scales: $\ll 10^{-9}$ G (Gauss). According to mbs ; prd0 ; prd ,
the origin of the magnetic fields could have been caused by certain primordial
magnetic monopoles during preinflation. On intergalactic scales, there is
compelling evidence supporting the existence of a pre-galactic seed magnetic
field on the order of $10^{-16}\,{\rm G}$. This is based on observations of
high-energy TeV photons emitted by distant blazars 26 . However, more recent
studies suggest a present-day lower magnetic field strength, around
$10^{-18}\,{\rm G}$ 27 ; 66 .

Inflation in0 ; in1 ; in11 ; in2 ; in3 ; in33 ; in333 ; in4 ; b1 ; bcms ; npb
provides an optimal environment for the creation of large-scale primordial
fields with significant coherence TW ; Ra . Meanwhile, preinflation Kur1 ;
Kur2 ; Kur3 ; GR  describes an exponential MB , (or super exponential dark23 )
expansion with a null Hubble parameter initially, and later increases during
the expansion. During preinflation the spatial curvature is negative and the
universe is created without matter with initially null energy density. It is
expected that during preinflation the large-scale coherence of primordial
fields can be maintained and the rapid expansion during the era of super
exponential expansion facilitates the generation of fields that are correlated
on exceedingly large scales through the stretching of wave modes.
Additionally, vacuum fluctuations of the electromagnetic field can be
triggered while a mode is within the Hubble radius, and these can subsequently
transform into classical fluctuations when modes exceed the Hubble radius
mebe0 . Furthermore, it is expected that during preinflation, any existing
densities of charged particles are greatly diluted by the expansion, resulting
in the universe becoming a poor conductor on sub-Hubble scales. As a result,
field generation from a zero field is not excluded by magnetic flux
conservation. However, the Friedman-Robertson-Walker (FRW) universe is
conformally flat and the Maxwell theory exhibits conformal invariance, so that
magnetic fields generated at preinflation would diminish significantly due to
the accelerated expansion of the universe. A possibility to solve this problem
relies in produce non-trivial magnetic fields in which conformal invariance is
broken.

In this work, we shall introduce a new connection,

$\hat{\delta\Gamma}^{\mu}_{\alpha\beta}=b\,\hat{\Omega}^{\mu}\,g_{\alpha\beta},$
(1)

with parameter $b$, metric tensor of the Riemannian manifold $g_{\alpha\beta}$
and the flux of the quantum field $\hat{\Omega}^{\mu}$ through the closed
$3d$-hypersurface when the varied action results null. The connection given by
the Eq. (1) encodes geometric information pertaining to gravitational and
electromagnetic fluctuations in relation to the curved background represented
by a Riemann manifold. It can be regarded as a Weyl-type connection weyl ,
incorporating both gravitational and electromagnetic aspects.

The manuscript is organized as follows: in Sect. II, we revisit an extend
general relativity by using the connections proposed in the Eq. (1) to
describe the boundary terms in the relativistic dynamics. Furthermore, we
obtain the dynamics for the gravito-electromagnetic field and we obtain the
dynamics for a particular gauge where both contributions, gravitational and
electromagnetic are decoupled. In Sect. III we revisit the preinflationary
model with variable time-scale and negative spatial curvature mb , which we
will use to study the evolution of electromagnetic fluctuations in the
universe. In Sect. IV we develop the formalism to the quantum electromagnetic
fluctuations in preinflation. We solve the dymamic equations and in Sec. V we
obtain the primordial electric and magnetic fields along with their
corresponding expectation values during preinflation, using an extension of a
new quantum algebra proposed in BMM . We also estimate the present-day values
of magnetic fields on cosmological scales. Finally, in Sect. VI we develop
some final comments and conclusions.

##  II Extended General Relativity with boundary contributions

Let us consider the Einstein-Hilbert action ${\cal I}$, which describes
gravity and matter,

${\cal I}=\int_{V}d^{4}x\,\sqrt{-g}\left[\frac{R}{2\kappa}+{\cal
L}_{m}\right],$ (2)

where $g$ is the determinant of the background metric tensor
$g_{\alpha\beta}$, $R$ is the background scalar curvature, $\kappa=8\pi
G/c^{4}$ is the Einstein constant ($G$ is the gravitational constant and $c$
is the light velocity in vacuum), and ${{\cal L}_{m}}$ is the Lagrangian
density that describes the background physical dynamics. The variation of the
Einstein-Hilbert action with boundary terms included is

$\delta{\cal I}=\int d^{4}x\sqrt{-g}\left[\delta
g^{\alpha\beta}\left(G_{\alpha\beta}+\kappa
T_{\alpha\beta}\right)+g^{\alpha\beta}\delta R_{\alpha\beta}\right]=0,$ (3)

where ${{T}}_{\alpha\beta}$ is the background stress tensor:
${{T}}_{\alpha\beta}=2\frac{\delta{{\cal L}_{m}}}{\delta
g^{\alpha\beta}}-g_{\alpha\beta}{{\cal L}_{m}}$. The Einstein tensor, denoted
by $G_{\alpha\beta}=R_{\alpha\beta}-\frac{1}{2}\,R\,g_{\alpha\beta}$, is
defined using the background (and symmetric) Ricci tensor $R_{\alpha\beta}$
which describes the geometry of the background using the Levi-Civita
connections. The scalar curvature $R$ is given by
$R=g^{\alpha\beta}\,R_{\alpha\beta}$.

To include the boundary terms in the global dynamics, we shall consider cases
where the curvature variations $\delta R_{\alpha\beta}$ are originated by
variations of the metric tensor RiBe ; mobe . In other words, we shall
consider that $\delta g_{\alpha\beta}$ are the geometrical sources of $\delta
R_{\alpha\beta}$,

$\delta R_{\alpha\beta}=\lambda\left(x\right)\,\delta g_{\alpha\beta},$ (4)

where $\delta R_{\alpha\beta}$ and $\delta g_{\alpha\beta}$ will be considered
on an extended manifold. Therefore the varied action, Eq. (3), takes the form

$\delta{\cal I}=\int d^{4}x\sqrt{-g}\left[\delta
g^{\alpha\beta}\left(G_{\alpha\beta}-\lambda\left(x\right)\,g_{\alpha\beta}+\kappa
T_{\alpha\beta}\right)\right]=0.$ (5)

where we have used that $\delta g^{\alpha\beta}\,g_{\alpha\beta}=-\delta
g_{\alpha\beta}\,g^{\alpha\beta}$. The Eq. (5) implies that

$G_{\alpha\beta}-\lambda\left(x\right)\,{g}_{\alpha\beta}=-\kappa\,T_{\alpha\beta}.$
(6)

To obtain the field equations, the boundary terms with $\lambda(x^{\mu})$ in
the Eq. (6) can be assimilated to the redefined stress tensor:
$\bar{T}_{\alpha\beta}={T}_{\alpha\beta}-\frac{1}{\kappa}\lambda(x^{\mu})\,g_{\alpha\beta}$,
then we obtain that
$\nabla_{\beta}\,G^{\alpha\beta}=\nabla_{\beta}\,\bar{T}^{\alpha\beta}=0$, and
the background dynamics for the physical fields comes from the equations

$\nabla_{\beta}\,T^{\alpha\beta}=\frac{1}{\kappa}\,g^{\alpha\beta}\,\frac{\partial\lambda(x^{\mu})}{\hskip-8.5359pt\partial
x^{\beta}}.$ (7)

Hence, the flux due to the boundary terms ($g^{\alpha\beta}\delta
R_{\alpha\beta}$) in the minimized action, acts as the source for the physical
fields.

###  II.1 Quantum boundary terms

In this work, we are interested in the flux originated from quantum
gravitational and electromagnetic fields. This means that the varied
connection, Eq. (1) will be quantum operators. In this case, the boundary
terms in the Eq. (3), ${g}^{\alpha\beta}\hat{\delta
R}_{\alpha\beta}\equiv\hat{\delta\Theta}$, describe the flux of the 4-vector
$\hat{\delta
W}^{\alpha}=b^{-1}\left(\hat{\delta\Gamma}^{\epsilon}_{\beta\epsilon}{g}^{\beta\alpha}-\hat{\delta\Gamma}^{\alpha}_{\beta\gamma}{g}^{\beta\gamma}\right)$,
through the 3D closed hypersurface $\partial M$;
$\hat{\delta\Theta}=\nabla_{\alpha}\hat{\delta W}^{\alpha}$. Explicitly, the
flux takes the form

$\hat{\delta\Theta}=\lambda(x^{\mu})g^{\alpha\beta}\hat{\delta
g}_{\alpha\beta}=-3\,\nabla_{\alpha}\,\hat{\Omega}^{\alpha},$ (8)

where $\hat{\Omega}^{\alpha}$ will be a 4-vector field that takes into account
gravitational and electromagnetic contributions, that describes the geometric
perturbations with respect to the background manifold. The differential
operator $\nabla_{\alpha}$ denotes the covariant derivative on the Riemann
manifold. To depict the variation of the Ricci tensor on the extended
manifold, $\hat{\delta R}_{\alpha\beta}$, we shall use an extension of the
Palatini expression pal

$\hat{\delta
R}_{\alpha\beta}=b^{-1}\,\left[\left(\hat{\delta\Gamma}^{\mu}_{\alpha\mu}\right)_{\|\beta}-\left(\hat{\delta\Gamma}^{\mu}_{\alpha\beta}\right)_{\|\mu}\right].$
(9)

Here, $\left(...\right)_{\|\mu}$ denotes the covariant derivative of
$\left(...\right)$ on the extended manifold. The covariant derivative of the
background metric tensor on the extended manifold with self-interactions
included mb , is

$\hat{g}_{\alpha\beta\|\mu}=\nabla_{\mu}g_{\alpha\beta}-\hat{\delta\Gamma}^{\nu}_{\alpha\mu}\,g_{\nu\beta}-\hat{\delta\Gamma}^{\nu}_{\beta\mu}\,g_{\alpha\nu}+2\eta\,g_{\alpha\beta}\,\hat{\Omega}_{\mu},$
(10)

where $\nabla_{\mu}g_{\alpha\beta}=0$ is the covariant derivative of the
metric tensor on the Riemann manifold and $\eta$ is a constant that takes into
account the magnitude of self-interactions. In this framework, the variation
of the metric tensor on the extended manifold, can be defined as

$\hat{\delta g}_{\alpha\beta}=\hat{g}_{\alpha\beta\|\mu}\,U^{\mu},$ (11)

where $U^{\alpha}=\frac{dx^{\alpha}}{\hskip-4.26773ptdS}$ are the components
of the relativistic observers that moves on the Riemann manifold.

By requiring the integrand in the Eq. (3) to vanish, we obtain the following
equation

$\left[b\,\left(\hat{\Omega}^{\alpha}U^{\beta}+\hat{\Omega}^{\beta}\,U^{\alpha}\right)-2\eta
g^{\alpha\beta}\hat{\Omega}_{\mu}\,U^{\mu}\right]\,\left(G_{\alpha\beta}+\kappa\,T_{\alpha\beta}\right)=3\left[\nabla_{\mu}\hat{\Omega}^{\mu}+\left(2b+\eta\right)\hat{\Omega}_{\mu}\,\hat{\Omega}^{\mu}\right].$
(12)

Here, we have used the fact that

$\hat{\delta g}^{\alpha\beta}=g^{\alpha\beta}_{\hskip
11.38092pt\|\mu}\,U^{\mu}=b\,\left(\hat{\Omega}^{\alpha}\,U^{\beta}+\hat{\Omega}^{\beta}\,U^{\alpha}\right)-2\,\eta\,g^{\alpha\beta}\,\hat{\Omega}_{\mu}\,U^{\mu}.$
(13)

We are aimed to describe a linear equation for $\hat{\Omega}^{\alpha}$, so
that we use the gauge $\eta=-2b$ into the Eq. (12), and using the Eq. (6), we
obtain

$\nabla_{\alpha}\hat{\Omega}^{\alpha}=6\,b\,\lambda(x^{\mu})\,\hat{\Omega}_{\alpha}U^{\alpha}.$
(14)

Keep in mind that the way observers perceive the source varies based on their
relative motion with respect to the source. This is reflected in the right
side of the Eq. (14) by the components of the relativistic velocity,
$U^{\mu}$. Additionally, we assume that the minimum spatial scale is the
Planck length: $L_{p}=\left(\frac{\hbar G}{c^{3}}\right)^{1/2}$, which can be
related directly with the parameter $b$, $L_{p}=\varepsilon\,b$,
($\varepsilon\ll 1$), such that, we can impose the following renormalization

$\left<V\left|g^{\alpha\beta}\,\hat{\delta
g}_{\alpha\beta}\right|V\right>=-\frac{1}{b},$ (15)

where $\left|V\right>$ is a quantum state on the Riemann manifold in the
Heisenberg representation. In this representation, the quantum operators
evolve in spacetime while the quantum states are squeezed in time. Since our
focus in this work is on cosmological applications, we will assume that the
expectation value of a certain quantum operator $\hat{O}(x^{\mu})$ can be
expressed as the space $3d$-volumetric expectation value
$\left<V\right|\hat{O}(x^{\mu})\left|V\right>=O(t)$. Therefore the
cosmological parameter will be given by

$\lambda(t)=-b\,\left<V\left|\hat{\delta\Theta}\right|V\right>=3\,b\,\left<V\left|\nabla_{\mu}\hat{\Omega}^{\mu}\right|V\right>,$
(16)

where $b=\frac{L_{p}}{\varepsilon}$ has units of length. When the metric
tensor changes, it can cause fluctuations in the curvature of spacetime.
Therefore, the curvature fluctuations can be expressed as the Eq. (4), where
$\lambda(t)$ is proportional to the flux of $\hat{\Omega}^{\mu}$.

###  II.2 Extended manifold for gravito-electromagnetic fields

To jointly describe gravitational and electromagnetic field fluctuations, we
propose a formulation for the field $\hat{\Omega}^{\alpha}$ that incorporates
both gravitational and electromagnetic fluctuations

$\hat{\Omega}^{\alpha}\equiv\hat{\sigma}^{\alpha}+a\,\hat{\delta A}^{\alpha},$
(17)

where

$\hat{\sigma}\equiv g^{\alpha\beta}\hat{\delta\Psi}_{\alpha\beta}.$ (18)

The gravitational perturbations are encoded in
$\hat{\delta\Psi}_{\alpha\beta}$, $a$ is a coupling constant with units
$\left[a\right]={\rm\frac{s\,Coul}{kg\,m^{3}}}$. Furthermore
$\hat{\sigma}_{\alpha}\equiv\hat{\sigma}_{,\alpha}$ denotes the partial
derivative of $\hat{\sigma}$ with respect to the coordinate $x^{\alpha}$ and
$\hat{\delta\hat{A}}^{\alpha}$ are the fluctuations of the electromagnetic
$4$-potential.

Such that, using the Eqs. (17) and (18) into the Eq. (12), the following
equations of motion for the quantum fields are obtained

$\displaystyle\lambda(t)\,\left(2b-8\eta\right)\left(g^{\epsilon\theta}U^{\alpha}\nabla_{\alpha}\hat{\delta\Psi}_{\epsilon\theta}\,+a\,U_{\alpha}\,\hat{\delta
A}^{\alpha}\,\right)-3(2b+\eta)g^{\epsilon\theta}g^{\alpha\beta}\nabla_{\mu}\hat{\delta\Psi}_{\epsilon\theta}\nabla^{\mu}\hat{\delta\Psi}_{\alpha\beta}$
$\displaystyle-3\,g^{\epsilon\theta}\Box\hat{\delta\Psi}_{\epsilon\theta}-3a\nabla_{\mu}\hat{\delta
A}^{\mu}-6(2b+\eta)\,a\,g^{\epsilon\theta}\hat{\delta
A}^{\mu}\,\nabla_{\mu}\hat{\delta\Psi}_{\epsilon\theta}\,-3\left(2b+\eta\right)\,a^{2}\hat{\delta
A}^{\mu}\hat{\delta A}_{\mu}=0.$ (19)

This equation can be split in the following independent equations

$\Box\hat{\delta\Psi}_{\epsilon\theta}+(2b+\eta)\nabla_{\mu}\hat{\delta\Psi}_{\epsilon\theta}\,\nabla^{\mu}\hat{\sigma}+2(2b+\eta)\,a\,\hat{\delta
A}^{\mu}\nabla_{\mu}\hat{\delta\Psi}_{\epsilon\theta}=\frac{2}{3}(b-4\eta)\,\lambda(t)\,U^{\mu}\,\nabla_{\mu}\hat{\delta\Psi}_{\epsilon\theta},$
(20a) $\nabla_{\mu}\hat{\delta A}^{\mu}+(2b+\eta)\,a\,\hat{\delta
A}^{\mu}\hat{\delta
A}_{\mu}=\frac{1}{3}(2b-8\eta)\,\lambda(t)\,U_{\mu}\hat{\delta A}^{\mu}.$
(20b)

Given that these equations are nonlinear and coupled, our next step will be to
investigate a specific gauge that decouples the fields and linearizes the
dynamical equations.

For $\eta=-2b$, the Eqs. (20) can be written as two decoupled and linear
differential equations,

$\Box\hat{\delta\Psi}_{\epsilon\theta}=6\,b\,\lambda(t)\,U^{\mu}\,\nabla_{\mu}\hat{\delta\Psi}_{\epsilon\theta},$
(21a) $\nabla_{\mu}\hat{\delta A}^{\mu}=6\,b\,\lambda(t)\,U_{\mu}\hat{\delta
A}^{\mu}.$ (21b)

The Eq. (21a) describes the wave dynamics for the components of GW,
$\hat{\delta\Psi}_{\epsilon\theta}$ with sources. In section IV, we study on
detail the dynamics of the electromagnetic fluctuations $\hat{\delta A}^{\mu}$
during preinflation, which are described by the Eq. (21b). On the other hand,
the dynamics for the trace of the GW components
$\hat{\sigma}=g^{\alpha\beta}\,\hat{\delta\Psi}_{\alpha\beta}$, can be written
as

$\Box\,\hat{\sigma}=6\,b\,\lambda(t)\,U_{\mu}\,\hat{\sigma}^{\mu}.$ (22)

Notice that, when the system is isolated and the boundary terms in the Eq. (5)
are null and the standard Einstein equations with $\lambda(t)=0$ are
recovered. The dynamics of GW in preinflation was studied in detail in a
previous work mb .

##  III Preinflation with negative spatial curvature

In this section, we will explore a model of the universe driven by a phantom
scalar field, and electromagnetic fields minimally coupled to gravity.
Assuming an electromagnetically neutral universe, the expectation value of
these fields over the Riemannian background will be considered to be zero,
only during the expansion, the fluctuations of the electromagnetic fields will
be relevant. However, as we shall see in (III.2), the gravitational
contributions of the geometric fluctuations will be applicable in the dynamics
of the universe’s expansion.

Let us study a Lagrangian density given by a scalar field $\hat{\varphi}$ and
the electromagnetic fields encoded in the Faraday tensor $\hat{F}_{\mu\nu}$,
to describe a preinflationary universe

${{\cal
L}}_{m}=\left[\frac{1}{2}g^{\alpha\beta}\hat{\varphi}_{,\alpha}\hat{\varphi}_{,\beta}-V(\hat{\varphi})\right]-\frac{1}{4}\,\hat{F}^{\mu\nu}\,\hat{F}_{\mu\nu}.$
(23)

These fields are minimally coupled to gravity. The scalar field is responsible
for the expansion of the universe and the electromagnetic tensor
$\hat{F}_{\mu\nu}$ is defined by
$\hat{F}_{\mu\nu}=\nabla_{\mu}\hat{A}_{\nu}-\nabla_{\nu}\hat{A}_{\mu},$, where
$\hat{A}^{\mu}\equiv\left(\hat{\Phi}(\vec{x},t)/c,\,\hat{A}^{i}(\vec{x},t)\right)$
is 4-electromagnetic potential such that $\hat{\Phi}(\vec{x},t)$ is the
electric potential and $\hat{A}^{i}(\vec{x},t)$ is the vector potential. We
assume an electromagnetically neutral universe on the background spacetime,
such that

$\left<V\right|\hat{A}^{\mu}(\vec{x},t)\left|V\right>=0.$ (24)

Therefore, the contributions of the electromagnetic potential to the dynamics
will be given by the fluctuations $\hat{\delta A}^{\mu}(\vec{x},t)$, with
respect to $\left<V\right|\hat{A}^{\mu}(\vec{x},t)\left|V\right>$.

To describe the background expanding universe with a variable time scale, we
examine the line element where the spatial contributions are written in
spherical coordinates

$ds^{2}=e^{-2\int\gamma(t)\,dt}\,c^{2}\,dt^{2}-e^{2\int
H(t)\,dt}\,\left[\frac{dr^{2}}{1-K\,r^{2}}+r^{2}\left(d\theta^{2}+\sin^{2}(\theta)d\vartheta^{2}\right)\right].$
(25)

Here, $c$ is the light velocity in the vacuum, $H(t)=\dot{a}/a(t)$ is the
Hubble parameter with a scale factor $a(t)$, $K=-(H_{0}/c)^{2}$ is the spatial
curvature, with $H_{0}$ is the Hubble parameter at certain time $t_{0}$, and
$\gamma(t)$ is a function that considers the variable timescale along the
expansion of the universe rnbh . Throughout the paper, we will be using

$\gamma\equiv-\frac{\dot{H}}{H},$ (26)

where the dot denotes derivative with respect to the time variable. The scalar
field $\hat{\varphi}$, drives the expansion of the universe, and can be
written as a semiclassical expansion bcms

$\hat{\varphi}\left(x^{\alpha}\right)=\left<V\right|\hat{\varphi}\left(x^{\alpha}\right)\left|V\right>+\hat{\delta\varphi}\left(x^{\alpha}\right).$
(27)

The expectation value of the scalar field
$\hat{\varphi}\left(x^{\alpha}\right)$, is only a function of time,
$\left<V\right|\hat{\varphi}\left(x^{\alpha}\right)\left|V\right>=\phi_{c}(t)$.
The scalar field fluctuations are given by the differential equation

$\ddot{\hat{\delta\varphi}}+\left(3H-\frac{\dot{H}}{H}\right)\,\dot{\hat{\delta\varphi}}-\left(\frac{H}{H_{0}}\right)^{2}\,e^{-2\int
H(t)\,dt}\,\nabla^{2}\hat{\delta\varphi}+{V}^{\prime\prime}(\phi_{c})\,\hat{\delta\varphi}=0,$
(28)

where

$\nabla^{2}\hat{\delta\varphi}\equiv\left(1-K\,r^{2}\right)\,\frac{\partial^{2}\hat{\delta\varphi}}{\partial
r^{2}}+\frac{1}{r^{2}}\frac{\partial^{2}\hat{\delta\varphi}}{\partial\theta^{2}}+\frac{1}{r^{2}\,\sin^{2}(\theta)}\frac{\partial^{2}\hat{\delta\varphi}}{\partial\vartheta^{2}}+\left(\frac{2}{r}-Kr\right)\frac{\partial\hat{\delta\varphi}}{\partial
r}+\frac{\cot{(\theta)}}{r^{2}}\,\frac{\partial\hat{\delta\varphi}}{\partial\theta}.$
(29)

The physical origin of the scalar field fluctuations $\hat{\delta\varphi}$ are
responsible for the back-reaction effects that alter the background dynamics
of the primordial universe dark231 , and must be consider to fully describe
the dynamics of the preinflationary universe.

###  III.1 Background dynamics and phantom field

The dynamics of the expectation value for the scalar field, $\phi_{c}(t)$ is

$\ddot{\phi}_{c}+\left[3\,H-\frac{\dot{H}}{H}\right]\dot{\phi}_{c}+\left.\frac{\delta\bar{\Upsilon}(\varphi)}{\delta\varphi}\right|_{\varphi\equiv\phi_{c}}=0,$
(30)

where the effective potential $\bar{\Upsilon}(\phi_{c})$, with back-reaction
contribution is

$\bar{\Upsilon}(\phi_{c})=\left[V(\phi_{c})\,\left(\frac{H}{H_{0}}\right)^{2}+b\,\frac{\left<V\right|\hat{\delta\Theta}\left|V\right>}{8\pi\,G}\right]=\left(\frac{H}{H_{0}}\right)^{2}\,\left[V(\phi_{c})-\frac{\lambda(t)}{8\pi\,G}\right].$
(31)

The extended Einstein equations with the boundary terms included and
$\gamma=-\dot{H}/H$, are given by

$3H(t)^{2}\left[1+3\,\alpha(t)\right]=8\pi\,G\,\left[\frac{\dot{\phi}^{2}_{c}}{2}+\bar{\Upsilon}(\phi_{c})\right],$
(32a)
$-3H(t)^{2}\left[3+\alpha(t)\right]=8\pi\,G\,\left[\frac{\dot{\phi}^{2}_{c}}{2}-\bar{\Upsilon}(\phi_{c})\right],$
(32b)

where the contribution of the spatial curvature is given by
$\alpha(t)=(c^{2}\,K/H^{2}_{0})\,e^{-2\int H(t)\,dt}$ mb . Additionally, the
total pressure with boundary terms included takes the form
$\bar{P}_{T}=\left(\frac{H}{H_{0}}\right)^{2}\,\left[P+\frac{\lambda(t)}{8\pi
G}\right]$, and the total energy density is
$\bar{\rho}_{T}=\left(\frac{H}{H_{0}}\right)^{2}\,\left[\rho-\frac{\lambda(t)}{8\pi
G}\right]$.

From the Eqs. (32), we obtain

$\displaystyle\dot{\phi_{c}}^{2}=\frac{\alpha(t)}{4\pi G}\,H^{2}(t),$ (33)
$\displaystyle\bar{\Upsilon}[\phi_{c}(t)]=\frac{[3+2\alpha(t)]}{8\pi
G}\,H^{2}(t).$ (34)

The equation of state is given by
$\bar{\omega}=\frac{\bar{P}_{T}}{\bar{\rho}_{T}}=-1+\frac{2\alpha}{3(1+\alpha)}$,
which corresponds to

$\phi_{c}(t)=\sqrt{\frac{\alpha(t)}{4\pi G}}.$ (35)

For $K<0$, we obtain that $\alpha(t)<0$ and $\phi_{c}(t)$ is an imaginary
function. In this instance, the kinetic component of the energy density has a
negative value, $\dot{\phi}_{c}^{2}/2<0$. This aligns with cosmological models
featuring phantom fields that satisfy an equation of state: $\bar{\omega}<-1$.
Similar to a previous study dark23 , we can adopt a Hubble parameter which
becomes zero at $t=0$, and increases throughout the initial expansion. If we
take $n\gg 1$, we can regard the Hubble parameter during preinflation as

$H(t)=H_{0}\,\left(\frac{t}{t_{0}}\right)^{1/n},$ (36)

where we must remember that $H_{0}$ is the value of the Hubble parameter in
some particular moment of the history of the universe: $t_{0}$. We shall
choose $t_{0}$ at the moment when the Cosmic Microwave Background Radiation
(CMBR) is emitted, $i.e$, at $t_{0}=3.8\times 10^{5}$ years. Furthermore, we
require that $K=-(H_{0}/c)^{2}$ in order to the initial energy density value
be null $\left.\rho\right|_{t=0}=0$. This implies the hypothesis that
initially, the universe is absent of matter in a background sense. Notice that
the value for spacial curvature is the inverse of the squared Hubble distance
at the moment when CMBR is emitted.

We are using $\gamma$ given by the Eq. (26), such that, the metric of the Eq.
(25) takes the form

$ds^{2}=\left(\frac{H(t)}{H_{0}}\right)^{2}\,c^{2}\,dt^{2}-e^{2\int
H(t)\,dt}\,\left[\frac{dr^{2}}{1+(H_{0}/c)^{2}\,r^{2}}+r^{2}\left(d\theta^{2}+\sin^{2}(\theta)\,d\vartheta^{2}\right)\right],$
(37)

where $H(t)$ is given by the Eq. (36). As in a previous work dark231 , the
cosmological parameter yields

$\lambda_{0}=-\frac{\mu}{3}\,H^{2}_{0}=3\,b\,\left<V\right|\nabla_{\mu}\hat{\sigma}^{\mu}\left|V\right>,$
(38)

which alters the background dynamics of the universe [see Eq. (16)]. This
alteration is originated by fluctuations in the spacetime curvature through
the effective contyributions of the boundary terms
$\left<V\left|\hat{\delta\Theta}\right|V\right>$ in the varied action, Eq.
(5). These terms are geometrically explained by the flux of $\Omega^{\mu}$
through the closed $3d$-hypersurface. In a recent work dark231 , corrections
due to CMB lensing contributions on the Hubble parameter ref  were calculated:
$\delta H_{0}=\sqrt{\mu/3}\,H_{0}=0.03\,{\rm\frac{km}{s\,Mpc}}$, in such a way
that $\mu\approx 5.8\times 10^{-7}$.

###  III.2 New algebra for the geometric fluctuations

To compute $\lambda_{0}$ from the expected value of the geometric flux of
$\hat{\Omega}^{\mu}$, it will be necessary to introduce a nonlinear quantum
algebra associated with a differential operator $\hat{\frac{\partial}{\partial
x^{\mu}}}$ BMM , in such a way that the Eq. (16) holds, providing us with a
geometric invariant for the renormalization of our theory.

To find the expectation value for the field, $\hat{\sigma}^{\mu}$, we can use
the Eqs. (16) and (14), to obtain

$\left<V\right|\hat{\Omega}^{\mu}\,U_{\mu}\left|V\right>=\frac{1}{18\,b^{2}},$
(39)

using the Eq. (17), $\left<V\right|\hat{\Omega}^{\mu}\,U_{\mu}\left|V\right>$
can be decomposed as

$\left<V\right|\hat{\Omega}^{\mu}\,U_{\mu}\left|V\right>=U_{\mu}\left<V\right|\hat{\sigma}^{\mu}\left|V\right>+a\,U_{\mu}\left<V\right|\hat{\delta
A}^{\mu}\left|V\right>.$ (40)

The last term in the Eq. (40) vanishes because we are considering
$\left<V\right|\hat{\delta A}^{\mu}\left|V\right>=0$. Therefore, we obtain
that

$\left<V\right|\hat{\Omega}^{\mu}\,U_{\mu}\left|V\right>=U_{\mu}\left<V\right|\hat{\sigma}^{\mu}\left|V\right>=\frac{1}{18\,b^{2}}.$
(41)

This means that the expectation value for the constant $\lambda_{0}$ will be
purely of gravitational nature, because $\lambda_{0}$ is given the expectation
value of the curvature contributions
$U_{\mu}\left<V\right|\hat{\sigma}^{\mu}\left|V\right>$. Furthermore, from the
Eq. (21b), we obtain that the expectation value

$\left<V\right|\nabla_{\mu}\hat{\delta A}^{\mu}\left|V\right>=0.$ (42)

This result guarantees that the classical effects of the electromagnetic
potential components on the background metric have a vanishing flux.

To calculate the Eq. (41), we must consider that the field $\hat{\sigma}$ is a
quantum scalar field which can be written as a Fourier expansion in terms of
their modes

$\hat{\sigma}\left(x^{\mu}\right)=\frac{1}{(2\pi)^{3/2}}\,\int
d^{3}\,p\,\left[\hat{C}_{p}\,{\sigma}(p,x^{\mu})+\hat{C}^{\dagger}_{p}\,{\sigma}^{*}(p,x^{\mu})\right],$
(43)

where, $\hat{C}^{\dagger}_{p}$ and $\hat{C}_{p}$ represent respectively the
creation and annihilation operators. Furthermore, ${\sigma}(p,x^{\mu})$
corresponds to the modes of the field with momentum $p=\frac{k}{\hbar}$, where
$k$ denotes the wavenumber.

To obtain the quantum representation of
$\hat{\sigma}_{\alpha}\equiv\hat{\frac{\partial}{\partial
x^{\alpha}}}\hat{\sigma}(x^{\mu})$, we use the nonlinear quantum algebra
introduced in BMM  to describe quantum differential operators:

$\left[\hat{\frac{\partial}{\partial
x^{\prime\alpha}}},\hat{\sigma}^{\dagger}_{p}(x^{\mu})\right]=-\frac{i\,p^{\prime}_{\alpha}}{\,\hbar}\frac{1}{b^{2}}\,\left[\hat{\sigma}_{p^{\prime}}(x^{\prime\mu}),\hat{\sigma}^{\dagger}_{p}(x^{\mu})\right],$
(44)

where
$\hat{\sigma}^{\dagger}_{p}(x^{\mu})=\hat{C}^{\dagger}_{p}\,\sigma^{*}(p,x^{\mu})$
and $\hat{\sigma}_{p}(x^{\mu})=\hat{C}_{p}\,\sigma(p,x^{\mu})$. Furthermore,
$p_{\alpha}$ are the $\alpha$th-components of the $4$-vector momentum ${\bf
p}$. When the background spacetime is represented by orthogonal coordinates it
is possible to develop the expansions

$\sigma(p,x^{\mu})=e^{\frac{i}{\hbar}p_{\alpha}x^{\alpha}}\,\xi(p,\tau),\qquad\sigma^{*}(p,x^{\mu})=e^{-\frac{i}{\hbar}p_{\alpha}x^{\alpha}}\,\xi^{*}(p,\tau).$
(45)

Therefore, applying the algebra introduced in the Eq. (44) to the expansion
(43), we derive the expectation value of $\hat{\sigma}_{\alpha}$ on the
background Riemann manifold

$\left<V\right|\hat{\sigma}_{\alpha}\left|V\right>=\frac{-i}{(2\pi)^{3/2}\,\hbar^{3}b^{2}}\int
d^{3}p\left(\frac{p^{\prime}_{\alpha}}{\hbar}\right)\left<V\left|\left[\hat{C}_{p},\hat{C}^{\dagger}_{p^{\prime}}\right]\right|V\right>\,\|\sigma(p,x^{\mu})\|^{2},$
(46)

where

$\left<V\left|\left[\hat{C}_{p},\hat{C}^{\dagger}_{p^{\prime}}\right]\right|V\right>=i\,\left(2\pi\right)^{3/2}\,\hbar^{3}b^{3}\,\delta^{(3)}(\vec{p}-\vec{p}\,^{\prime}).$
(47)

Such that, the expectation value of $\hat{\sigma}_{\alpha}$, is

$\left<V\right|\hat{\sigma}_{\alpha}\left|V\right>=b\,\frac{p_{\alpha}}{\hbar}\,\left\|\sigma(p,x^{\mu})\right\|^{2},$
(48)

which allows obtaining the following invariant.

$\left<V\right|\hat{\sigma}_{\alpha}\left|V\right>\,U^{\alpha}=b\,\frac{p_{\alpha}\,U^{\alpha}}{\hbar}\,\left\|\sigma(p,x^{\mu})\right\|^{2}=\frac{1}{18\,b^{2}},$
(49)

where we have used the Eq. (41).

Thus, we have introduced the algebra of the Eq. (44), enabling the existence
of a consistent theory of quantum gravity without divergences.

##  IV Primordial electromagnetic fluctuations

To describe the electromagnetic fields we study a Lagrangian density ${\cal
L}_{EM}$

${\cal L}_{EM}=-\frac{1}{4}\,\hat{F}^{\mu\nu}\,\hat{F}_{\mu\nu},$ (50)

The electromagnetic equations of motion, comes from the field equation
$\nabla_{\mu}\hat{F}^{\mu\nu}=0$, that takes the form

$\Box\hat{\delta A}^{\alpha}+g^{\alpha\nu}\,R_{\nu\mu}\,\hat{\delta
A}^{\mu}=\nabla^{\alpha}\left(\nabla_{\mu}\hat{\delta
A}^{\mu}\right)=6\,\nabla^{\alpha}\left(\lambda(t)\,U_{0}\,\hat{\delta
A}^{0}\right).$ (51)

The last equality is because we are considering a co-moving frame a co-moving
frame with $U_{0}=\sqrt{g_{00}}$ and $U_{i}=0$.

The Eq. (51) describes the dynamics for the fluctuations of the
electromagnetic field components, which fluctuate with respect to a null mean
value, because we are assuming that the universe is electromagnetically
neutral with respect to the background spacetime.

The four dynamical equations for $\hat{\delta A}^{\alpha}$, are explicitly

$\displaystyle\ddot{\hat{\Phi}}+\Big{(}3H-\gamma\Big{)}\dot{\hat{\Phi}}-\left(3H^{2}+3H\gamma+\dot{\gamma}\right)\hat{\Phi}-e^{-2\int(H+\gamma)dt}\bigg{\\{}\nabla^{2}\hat{\Phi}\bigg{.}$
$\displaystyle\bigg{.}-2H\left[\partial_{1}\hat{\delta
A}^{1}+\partial_{2}\,\hat{\delta A}^{2}+\partial_{3}\hat{\delta
A}^{3}+\cot{(\theta)}\,\hat{\delta
A}^{2}+\left(\frac{2}{r}+\frac{K\,r}{1-K\,r^{2}}\right)\hat{\delta
A}^{1}\right]\bigg{\\}}\,$ $\displaystyle=$ $\displaystyle 0,$ (52a)
$\displaystyle\ddot{\hat{\delta
A}}^{1}+\Big{(}5H+\gamma\Big{)}\dot{\hat{\delta A}}^{1}+6H^{2}\,\hat{\delta
A}^{1}-e^{-2\int(H+\gamma)dt}\left\\{\nabla^{2}\hat{\delta
A}^{1}+\frac{1}{1-Kr^{2}}\left(3K-\frac{1}{r^{2}}\right)\,\hat{\delta
A}^{1}\right.$ $\displaystyle\left.+2K\,r\,\partial_{1}\hat{\delta
A}^{1}-2(1-K\,r^{2})\left[h(t)\,\partial_{1}\Phi-\frac{1}{r}\left(\partial_{2}\hat{\delta
A}^{2}+\partial_{3}\hat{\delta A}^{3}\right)\right]\right\\}$ $\displaystyle=$
$\displaystyle 0,$ (52b) $\displaystyle\ddot{\hat{\delta
A}}^{2}+\Big{(}5H+\gamma\Big{)}\dot{\hat{\delta A}}^{2}+6H^{2}\,\hat{\delta
A}^{2}-e^{-2\int(H+\gamma)dt}\left\\{\nabla^{2}\hat{\delta
A}^{2}-\left(4K-\frac{1}{r^{2}}+\frac{\cot^{2}{(\theta)}}{r^{2}}\right)\,\hat{\delta
A}^{2}\right.$
$\displaystyle\left.+2\left(K\,r-\frac{1}{r}\right)\partial_{1}\hat{\delta
A}^{2}-\frac{2}{r^{2}}\,\left[h(t)\,\partial_{2}\Phi+\frac{1}{r}\partial_{2}\hat{\delta
A}^{1}-\cot{(\theta)}\partial_{3}\hat{\delta A}^{3}\right]\right\\}$
$\displaystyle=$ $\displaystyle 0,$ (52c) $\displaystyle\ddot{\hat{\delta
A}}^{3}+\Big{(}5H+\gamma\Big{)}\dot{\hat{\delta A}}^{3}+6H^{2}\,\hat{\delta
A}^{3}-e^{-2\int(H+\gamma)dt}\left\\{\nabla^{2}\hat{\delta
A}^{3}+2\frac{\cot^{2}{(\theta)}}{r^{2}}\,\partial_{2}\hat{\delta
A}^{3}-4K\,\hat{\delta A}^{3}\right.$
$\displaystyle\left.-2\left(K\,r-\frac{1}{r}\right)\partial_{1}\hat{\delta
A}^{3}-\frac{2}{r^{2}\sin^{2}{\theta}}\,\left[h(t)\,\partial_{3}\Phi+\frac{1}{r}\partial_{3}\hat{\delta
A}^{1}+\cot{(\theta)}\partial_{3}\hat{\delta A}^{2}\right]\right\\}$
$\displaystyle=$ $\displaystyle 0,$ (52d)

where we have introduced the function

$h(t)=H(t)-3\,b\,\lambda_{0}\,\left(\frac{H}{H_{0}}\right)^{3},$ (53)

with $\lambda_{0}$ a constant given by the Eq. (38), and the Laplacian
$\nabla^{2}\hat{\delta A}^{\alpha}$, written in spherical coordinates, is

$\nabla^{2}\hat{\delta
A}^{\alpha}\equiv\left(1+(H_{0}/c)^{2}\,r^{2}\right)\,\frac{\partial^{2}\hat{\delta
A}^{\alpha}}{\partial r^{2}}+\frac{1}{r^{2}}\frac{\partial^{2}\hat{\delta
A}^{\alpha}}{\partial\theta^{2}}+\frac{1}{r^{2}\,\sin^{2}(\theta)}\frac{\partial^{2}\hat{\delta
A}^{\alpha}}{\partial\vartheta^{2}}+\left(\frac{2}{r}+3\,(H_{0}/c)^{2}r\right)\frac{\partial\hat{\delta
A}^{\alpha}}{\partial
r}+\frac{\cot{(\theta)}}{r^{2}}\,\frac{\partial\hat{\delta
A}^{\alpha}}{\partial\theta}.$ (54)

The spatial curvature $K=-(H_{0}/c)^{2}$ was chosen to obtain an initial null
energy density, and we must remember that $H\gamma+\dot{H}=0$, because of the
Eq. (26).

###  IV.1 Fluctuations of the Electric potential

The dynamics for the fluctuations of the electric potential
$\hat{\Phi}(x^{\alpha})$, are governed by the Eq. (52a), which, according to
the conditions given in the Eq. (21b), takes the form of an homogeneous
differential equation for $\hat{\Phi}$

$\ddot{\hat{\Phi}}+\big{(}3H-\gamma+2h(t)\big{)}\dot{\hat{\Phi}}+\big{(}2H^{2}+2h(t)\,(2H-3\gamma)+H\gamma-\dot{\gamma}\big{)}\hat{\Phi}-e^{-2\int(H+\gamma)dt}\nabla^{2}\hat{\Phi}=0,$
(55)

where $h(t)$ is given in the Eq. (53).

Since we are interested in describing the electric potential during the
preinflationary epoch, we can write it as a Fourier expansion in terms of
their modes,

$\displaystyle\hat{\Phi}\left(x^{\alpha}\right)=\int\frac{d^{3}k}{(2\pi)^{3/2}}\sum_{\iota=1}^{2}\,{}_{\iota}{\cal\varepsilon}^{0}(k)\,\,\sum_{l,m}\,\left[\varsigma^{(0)}(k,t){}_{\iota}\hat{\sigma}_{klm}^{(0)}(x^{j})+\left(\varsigma^{(0)}(k,t)\right)^{*}{}_{\iota}\hat{\sigma}_{klm}^{(0)\dagger}(x^{j})\right],$
(56)

where $-l\leq m\leq l$,
${}_{\iota}\hat{\sigma}_{klm}^{(0)}(x^{j})={}_{\iota}\hat{A}_{klm}\,Y_{lm}(\theta,\vartheta)\,R^{(0)}_{l}\big{(}k\,(\vec{r}\cdot\hat{e}_{r})\big{)}$
and
${}_{\iota}\hat{\sigma}_{klm}^{(0)\dagger}(x^{j})={}_{\iota}\hat{A}^{\dagger}_{klm}\,Y^{*}_{lm}(\theta,\vartheta)\,\left(R^{(0)}_{l}\big{(}k\,(\vec{r}\cdot\hat{e}_{r})\big{)}\right)^{*}$.
Here, $Y_{lm}(\theta,\vartheta)$ are the spherical harmonic and
$R^{(0)}_{l}[k\,(\vec{r}\cdot\hat{e}_{r})]$ are the solutions for the radial
differential equation. The operators ${}_{\iota}\hat{A}^{\dagger}_{k}$ and
${}_{\iota}\hat{A}_{k}$ are the creation and destruction operators for a given
$\iota$–polarization, that comply with the algebra

$\displaystyle\left[{}_{\iota}\hat{A}_{klm},\;_{\varsigma}\hat{A}_{k^{\prime}l^{\prime}m^{\prime}}^{\dagger}\right]=\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}\;\delta({k}-{k^{\prime}})\,\delta_{ll^{\prime}}\,\delta_{mm^{\prime}}\,\delta_{\iota\varsigma},$
(57)
$\displaystyle\left[{}_{\iota}\hat{A}_{klm},\;_{\iota}\hat{A}_{k^{\prime}l^{\prime}m^{\prime}}\right]=\left[{}_{\iota}\hat{A}_{klm}^{\dagger},\;_{\iota}\hat{A}_{k^{\prime}l^{\prime}m^{\prime}}^{\dagger}\right]=0,$
(58)

for ${}_{\iota}\hat{A}_{klm}=\,_{\iota}\hat{A}_{k}\,A_{lm}$, such that
$A_{lm}=\sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}}$. The indices $\iota$
denote the $2$-possible polarization modes and $\varsigma(k,t)$ are the time
dependent modes for a given wave number $k$.

The canonical quantization requires that

$\left[\hat{\delta
A}_{0}(t,\vec{r}),\,\hat{\Pi}^{\mu}(t,\vec{r^{\prime}})\right]=i\delta^{\mu}_{0}\,\delta^{(3)}(\vec{r}-\vec{r^{\prime}}),$
(59)

where $\hat{\Pi}^{\beta}=\frac{\delta{\cal L}_{EM}}{\hat{\delta
A}_{\beta;\alpha}}\left.\right|_{\alpha=0}=g^{\mu\beta}\nabla_{\mu}\hat{\delta
A}^{\alpha}-g^{\alpha\mu}\nabla_{\mu}\hat{\delta
A}^{\beta}\left.\right|_{\alpha=0}$. To write the electric potential
$\hat{\Phi}\left(x^{\alpha}\right)$ as real field, we can define the real
operators
${}_{\iota}\hat{a}_{klm}=\frac{1}{2}\left({}_{\iota}\hat{A}_{klm}+_{\iota}\hat{A}^{\dagger}_{klm}\right)$
and
$\bar{R}^{(0)}_{l}[k\,(\vec{r}\cdot\hat{e}_{r})]=\frac{1}{2}\left\\{{R}^{(0)}_{l}[k\,(\vec{r}\cdot\hat{e}_{r})]+[{R}^{(0)}]^{*}_{l}[k\,(\vec{r}\cdot\hat{e}_{r})]\right\\}$

$\displaystyle\hat{\Phi}\left(x^{\alpha}\right)=\int\frac{dk}{2\pi^{2}}\sum_{\iota=1}^{2}\,{}_{\iota}{\cal
P}^{0}({k},t)\,\,\sum_{l,m}\,k^{2}\,\left[{}_{\iota}\hat{a}_{klm}\,\bar{Y}_{lm}(\theta,\vartheta)\,\bar{R}^{(0)}_{l}[k\,(\vec{r}\cdot\hat{e}_{r})]\right],$
(60)

where
$\bar{Y}_{lm}(\theta,\vartheta)=\frac{1}{2}\left[{Y}_{lm}(\theta,\vartheta)+{Y}^{*}_{lm}(\theta,\vartheta)\right]$
are the real spherical harmonic. Furthermore, ${}_{\iota}{\cal P}^{0}({k},t)$
are the time-dependent polarization components of each $k$-mode, for the
electric potential $\hat{\Phi}\left(x^{\alpha}\right)\equiv
c\;\hat{A}^{0}\left(x^{\alpha}\right)$.

${}_{1}{\cal P}^{0}=Re[\varsigma^{(0)}(k,t)],\quad_{2}{\cal
P}^{0}=Im[\varsigma^{(0)}(k,t)].$ (61)

The time dependent modes $\varsigma^{(0)}(k,t)$ obey the differential equation

$\ddot{\varsigma}^{(0)}+\big{[}3H-\gamma+2h(t)\big{]}\dot{\varsigma}^{(0)}+\Big{[}2H^{2}+2h(t)\,\big{(}2H-3\gamma\big{)}+H\gamma-\dot{\gamma}+\kappa_{0}^{2}\,e^{-2\int(H+\gamma)dt}\Big{]}\varsigma^{(0)}=0,$
(62)

where $\kappa_{0}$ is a constant to be determined.

In the Fig. 1 we have plotted the sum of the polarization components for the
electric potential $\hat{\Phi}$ (solid red line) with $l=200$. It is worth
noting that in the early stages of the universe’s evolution, the sum decreases
from initially high values then begins to increase again until reaching its
maximum shortly after the emission of the CMBR. Subsequently, it decreases
monotonically until the present day.

Figure 1: The polarization modes $\sum_{\iota}{}_{\iota}{\cal P}^{0}(k,t)$ of
the electric potential $\hat{\Phi}$, (solid red line) and the polarization
modes $\sum_{\iota}{}_{\iota}{\cal P}^{i}(k,t)$ of $\hat{\delta A}^{i}$ for
$i=1,2,3$, (dashed green lines) are shown. All curves are depicted for
$l=200$.

The solution for the radial differential equation

$\displaystyle\big{(}1+(H_{0}/c)^{2}\,r^{2}\big{)}\frac{d^{2}}{dr^{2}}\bar{R}^{(0)}_{l}(k\,r)+\left(\frac{2}{r}+3(H_{0}/c)^{2}\,r\right)\frac{d}{dr}\bar{R}^{(0)}_{l}(k\,r)+\left(\kappa_{0}^{2}-\frac{l(l+1)}{r^{2}}\right)\bar{R}^{(0)}_{l}(k\,r)=0,$
(63)

is

$\displaystyle\bar{R}^{(0)}_{l}(k\,r)=\frac{1}{\sqrt{r}}$
$\displaystyle\left\\{{\rm C_{1}}\,{\it
LP}\left[\sqrt{1-(c\,\kappa_{0}/H_{0})^{2}}-1/2,\;\sqrt{l(l+1)+1/4},\sqrt{1+(H_{0}/c)^{2}r^{2}}\right]\right.$
$\displaystyle+$ $\displaystyle\;\;\left.{\rm C_{2}}\,{\it
LQ}\left[\sqrt{1-(c\,\kappa_{0}/H_{0})^{2}}-1/2,\;\sqrt{l(l+1)+1/4},\sqrt{1+(H_{0}/c)^{2}r^{2}}\right]\right\\},$
(64)

where ${\rm C_{1},C_{2}}$ are constants. Furthermore ${\it
LP}\left[a_{1},a_{2},z\right]$ and ${\it LQ}\left[a_{1},a_{2},z\right]$ are
the second kind associated Legendre functions, with parameters $a_{1},a_{2}$
and variable $z$.

###  IV.2 Dynamics of $\hat{\delta A}^{i}$ components

Similarly to the electric potential $\hat{\Phi}$, we propose a Fourier
expansion for $\hat{\delta A}^{i}$, ($i=1,2,3$) as a complex field

$\displaystyle\hat{\delta
A}^{i}\left(x^{\alpha}\right)=\int\frac{d^{3}k}{(2\pi)^{3/2}}\sum_{\iota=1}^{2}\,{}_{\iota}{\cal\varepsilon}^{i}(k)\,\sum_{l,m}\,\left[\xi^{(i)}(k,t){}_{\iota}\hat{\sigma}_{klm}(x^{j})+\,\left(\xi^{(i)}(k,t)\right)^{*}{}_{\iota}\hat{\sigma}_{klm}^{(i)\dagger}(x^{j})\right],$

where
${}_{\iota}\hat{\sigma}_{klm}^{(i)}(x^{j})={}_{\iota}\hat{B}_{klm}\,Y_{lm}(\theta,\vartheta)\,R^{(i)}_{l}\big{(}k\,(\vec{r}\cdot\hat{e}_{r})\big{)}$
and
${}_{\iota}\hat{\sigma}_{klm}^{(i)\dagger}(x^{j})={}_{\iota}B^{\dagger}_{klm}\,Y^{*}_{lm}(\theta,\vartheta)\,\left(R^{(i)}_{l}\big{(}k\,(\vec{r}\cdot\hat{e}_{r})\big{)}\right)^{*}\,$,
such that, the Eq.(IV.2) can be written in terms of their modes as a real
field,

$\displaystyle\hat{\delta
A}^{i}\left(x^{\alpha}\right)=\int\frac{dk}{2\pi^{2}}\sum_{\iota=1}^{2}\,{}_{\iota}{\cal
P}^{i}({k},t)\,\,\sum_{l,m}\,k^{2}\,\left({}_{\iota}\hat{b}_{klm}^{(i)}\,\bar{Y}_{lm}(\theta,\vartheta)\,\bar{R}^{(i)}_{l}\left[k\,(\vec{r}\cdot\hat{e}_{r})\right]\right),$
(66)

where the functions $\bar{Y}_{lm}(\theta,\vartheta)$ are the spherical
harmonics and
${}_{\iota}\hat{b}_{klm}^{(i)}=\frac{1}{2}\left({}_{\iota}\hat{B}_{klm}+\,{}_{\iota}{\hat{B}_{klm}^{\dagger}}\right)$,
with ${}_{\iota}\hat{B}_{klm}$ and ${}_{\iota}\hat{B}^{\dagger}_{klm}$ the
destruction and creation operators, respectively, and they obey the algebra

$\displaystyle\left[{}_{\iota}\hat{B}_{klm},\;_{\varsigma}\hat{B}_{k^{\prime}l^{\prime}m^{\prime}}^{\dagger}\right]=\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}\;\delta({k}-{k^{\prime}})\,\delta_{ll^{\prime}}\,\delta_{mm^{\prime}}\,\delta_{\iota\varsigma},$
(67)
$\displaystyle\left[{}_{\iota}\hat{B}_{klm},\;_{\iota}\hat{B}_{k^{\prime}l^{\prime}m^{\prime}}\right]=\left[{}_{\iota}\hat{B}_{klm}^{\dagger},\;_{\iota}\hat{B}_{k^{\prime}l^{\prime}m^{\prime}}^{\dagger}\right]=0.$
(68)

The polarization modes of $\hat{\delta A}^{i}$, are given by

${}_{1}{\cal P}^{i}(k,t)=Re[\xi^{(i)}(\kappa_{i},t)],\quad_{2}{\cal
P}^{i}(k,t)=Im[\xi^{(i)}(k,t)]$ (69)

which define a plane which is orthogonal to the wave propagation.

####  IV.2.1 $r$-component $\hat{\delta A^{1}}$

To solve the system of Eqs. (52b-52d), we can essay a particular solution of
the Eq. (52b) as two differential equations

$\displaystyle e^{2\int(H+\gamma)dt}\left[\ddot{\hat{\delta
A}}^{1}+\left(5H+\gamma\right)\dot{\hat{\delta A}}^{1}+6H^{2}\,\hat{\delta
A}^{1}\right]$ $\displaystyle=$ $\displaystyle\nabla^{2}\hat{\delta
A}^{1}+\frac{2}{r}\partial_{r}\hat{\delta
A}^{1}+\frac{3+3(H_{0}/c)^{2}\,r^{2}+2(H_{0}/c)^{4}\,r^{4}}{r^{2}\,\big{(}1+(H_{0}/c)^{2}\,r^{2}\big{)}}\,\hat{\delta
A}^{1},$ (70a) $\displaystyle-\cot{(\theta)}\,{\hat{\delta A}}^{2}$
$\displaystyle=$
$\displaystyle\dot{\hat{\Phi}}+r\,h(t)\partial_{r}\hat{\Phi}+\left[2h(t)+H-\gamma\right]\hat{\Phi},$
(70b)

where $h(t)$ is given in the Eq. (53) and we have used the condition provided
by the Eq. (21b).

The time dependent modes $\xi^{(i)}(k,t)$ are solutions of the differential
equation

$\ddot{\xi}^{(i)}+\left[5H+\gamma\right]\,\dot{\xi}^{(i)}+\left[6H^{2}+{\kappa_{i}}^{2}\,e^{-2\int\left(H+\gamma\right)dt}\right]\,{\xi^{(i)}}=0,$
(71)

for some constant $\kappa_{i}$ and $i=1$. The polarization components for
fixed $k$ and a numeric solution for the Eq. (71) is depicted in the Fig, (1)
by the dashed green curve. Furthermore,
$\bar{R}^{(1)}_{l}\left[k\,(\vec{r}\cdot\hat{e}_{r})\right]$ obeys the
differential equation:

$\displaystyle\left(1+(H_{0}/c)^{2}\,r^{2}\right)\frac{d^{2}}{dr^{2}}\bar{R}^{(1)}_{l}(k\,r)$
$\displaystyle+$
$\displaystyle\left(\frac{4}{r}+3(H_{0}/c)^{2}\,r\right)\frac{d}{dr}\bar{R}^{(1)}_{l}(k\,r)$
(72) $\displaystyle+$
$\displaystyle\left[{\kappa_{1}}^{2}+\frac{3+3(H_{0}/c)^{2}\,r^{2}+2(H_{0}/c)^{4}\,r^{4}}{r^{2}\,\big{(}1+(H_{0}/c)^{2}\,r^{2}\big{)}}-\frac{l(l+1)}{r^{2}}\right]\bar{R}^{(1)}_{l}(k\,r)=0.$

The general solution of the Eq. (72) can be written as follows

$\displaystyle\bar{R}^{(1)}_{l}(k\,r)=\frac{\sqrt{1+(H_{0}\,r/c)^{2}}}{r^{3/2}}\,$
$\displaystyle\left\\{{\rm C_{1}^{(1)}}{\it
LP}\left[i\,\sqrt{1+(c\,\kappa_{1}/H_{0})^{2}}-1/2,\;\sqrt{l(l+1)-3/4},\sqrt{1+(H_{0}/c)^{2}r^{2}}\right]\right.$
$\displaystyle+$ $\displaystyle\;\;\left.{\rm C_{2}^{(1)}}\,{\it
LQ}\left[i\,\sqrt{1+(c\,\kappa_{1}/H_{0})^{2}}-1/2,\;\sqrt{l(l+1)+3/4},\sqrt{1+(H_{0}/c)^{2}r^{2}}\right]\right\\}$
(73)

where ${\it LP}$ and $\it LQ$ are the associated Legendre functions, and
$C_{1}^{(1)}$ and $C_{2}^{(1)}$ are constants.

On the other hand, from the Eq. (70b) we find that the angle $\theta$ must be
constant, $\theta_{*}\equiv\operatorname{arcCot}(\alpha_{1})$ (for some
$\alpha_{1}$), and in consequence the spherical harmonics only depends on
$\vartheta$, and we can write

$\big{(}\partial_{\theta}\bar{Y}_{lm}\big{)}\left.\right|_{\theta_{*}}=\alpha_{2}\bar{Y}_{lm}(\theta_{*},\vartheta).$
(74)

Besides, the following relations are satisfied

$\displaystyle\alpha_{1}\,{}_{\iota}\hat{b}_{klm}^{(2)}$ $\displaystyle=$
$\displaystyle-_{\iota}\hat{a}_{klm}$ (75a) ${}_{\iota}{\cal
P}^{2}(k,t)\bar{R}^{(2)}_{l}(k\,r)$ $\displaystyle=$
${}_{\iota}{\cal\dot{P}}^{0}(k,t)\bar{R}^{(0)}_{l}(k\,r)+\big{(}2h(t)+H-\gamma\big{)}\,_{\iota}{\cal
P}^{0}(k,t)\bar{R}^{(0)}_{l}(k\,r)+h(t)_{\iota}{\cal
P}^{0}(k,t)r\frac{d\bar{R}^{(0)}_{l}}{dr}.$ (75b)

####  IV.2.2 $\theta$-component $\hat{\delta A^{2}}$

To solve the Eq. (52c), we proceed similarly to the previous case, and split
it into

$\displaystyle e^{2\int(H+\gamma)dt}\left[\ddot{\hat{\delta
A}}^{2}+\Big{(}5H+\gamma\Big{)}\dot{\hat{\delta A}}^{2}+6H^{2}\,\hat{\delta
A}^{2}\right]$ $\displaystyle=$ $\displaystyle\nabla^{2}\hat{\delta
A}^{2}-\left(4K-\frac{1}{r^{2}}+\frac{\alpha_{1}^{2}}{r^{2}}\right)\,\hat{\delta
A}^{2}+2\left(K\,r-\frac{1}{r}\right)\partial_{1}\hat{\delta A}^{2}$ (76a)
$\displaystyle\alpha_{1}\alpha_{3}\hat{\delta A}^{3}$ $\displaystyle=$
$\displaystyle\alpha_{2}\left(h(t)\,\hat{\Phi}+\frac{1}{r}\hat{\delta
A}^{1}\right),$ (76b)

from the Eq. (76b) we found that the angle $\vartheta$ is also a constant
$\vartheta_{*}=\frac{1}{m}\arctan\left(\frac{\alpha_{3}}{m}\right)$. The
equation for the time dependent modes $\xi^{(2)}(k,t)$ has the same form as in
the Eq. (71), with $i=2$. Such behaviour is showed in the Fig. 1.

The radial component
$\bar{R}^{(2)}_{l}\left[k\,(\vec{r}\cdot\hat{e}_{r})\right]$ satisfies the
differential equation

$\displaystyle\left(1+\frac{H_{0}^{2}}{c^{2}}\,r^{2}\right)\frac{d^{2}}{dr^{2}}\bar{R}^{(2)}_{l}(k\,r)+\frac{H_{0}^{2}}{c^{2}}\,r\frac{d}{dr}\bar{R}^{(2)}_{l}(k\,r)+\frac{1}{r^{2}}\left(1-4\frac{H_{0}^{2}}{c^{2}}r^{2}-\alpha_{1}^{2}-l(l+1)+{\kappa_{2}}^{2}r^{2}\right)\bar{R}^{(2)}_{l}(k\,r)=0.$
(77)

The general solution for the Eq. (77), is

$\displaystyle\bar{R}^{(2)}_{l}(k\,r)=\sqrt{r}$ $\displaystyle\left\\{{\rm
C_{1}^{(2)}}\,{\it
LP}\left[i\sqrt{4+(c\,\kappa_{2}/H_{0})^{2}}-1/2,\;\sqrt{l(l+1)-3/4+\alpha_{1}^{2}},\sqrt{1+(H_{0}/c)^{2}r^{2}}\right]\right.$
$\displaystyle+$ $\displaystyle\;\;\left.{\rm C_{2}^{(2)}}\,{\it
LQ}\left[i\sqrt{4+(c\,\kappa_{2}/H_{0})^{2}}-1/2,\;\sqrt{l(l+1)-3/4+\alpha_{1}^{2}},\sqrt{1+(H_{0}/c)^{2}r^{2}}\right]\right\\},$
(78)

where $C_{1}^{(2)}$ and $C_{2}^{(2)}$ are constants and $\it LP,\;LQ$ are the
associated Legendre functions. Furthermore, regarding to the Eq.(76b), we can
deduce the following relationships

$\displaystyle\alpha_{1}\,\alpha_{3}{}_{\iota}\hat{b}_{klm}^{(3)}$
$\displaystyle=$ $\displaystyle-_{\iota}\hat{a}_{klm},$ (79a) ${}_{\iota}{\cal
P}^{3}(k,t)\bar{R}^{(3)}_{l}(k\,r)$ $\displaystyle=$ $\displaystyle
h(t)\,{}_{\iota}{\cal
P}^{0}(k,t)\bar{R}^{(0)}_{l}(k\,r)+\alpha_{2}{}_{\iota}{\cal
P}^{1}(k,t)\frac{\bar{R}^{(1)}_{l}(k\,r)}{r},$ (79b) $\displaystyle=$
$\displaystyle\alpha_{2}{}_{\iota}{\cal
P}^{2}(k,t)\bar{R}^{(2)}_{l}(k\,r)+\big{(}1-\alpha_{2}\big{)}h(t)\,{}_{\iota}{\cal
P}^{0}(k,t)\bar{R}^{(0)}_{l}(k\,r).$ (79c)

Notice that from the Eqs. (75b) and (79c) we can obtain ${}_{\iota}{\cal
P}^{3}(k,t)\bar{R}^{(3)}_{l}(k\,r)$, in terms of $\hat{\Phi}$.

####  IV.2.3 $\vartheta$-component $\hat{\delta A^{3}}$

Following the same line of thought, we now proceed to break down the Eq. (52d)
into the following

$\displaystyle e^{2\int(H+\gamma)dt}\left[\ddot{\hat{\delta
A}}^{3}+\Big{(}5H+\gamma\Big{)}\dot{\hat{\delta A}}^{3}+6H^{2}\,\hat{\delta
A}^{3}\right]$ $\displaystyle=$ $\displaystyle\nabla^{2}\hat{\delta
A}^{3}+2\frac{\alpha_{1}^{2}\,\alpha_{2}}{r^{2}}\,\hat{\delta
A}^{3}-4K\,\hat{\delta
A}^{3}-2\left(K\,r-\frac{1}{r}\right)\partial_{1}\hat{\delta A}^{3},$ (80a)
$\displaystyle-\frac{1}{r}\hat{\delta A}^{1},$ $\displaystyle=$ $\displaystyle
h(t)\,\hat{\Phi}+\alpha_{1}\hat{\delta A}^{2}.$ (80b)

The time dependent modes $\xi^{(3)}(k,t)$ obeys the Eq. (71), with $i=3$, and
the corresponding polarization modes are depicted in Fig. 1. Notice that the
evolution of the polarization modes, ${}_{\iota}{\cal P}^{i}(k,t)\|$, for
$\hat{\delta A}^{i}$, shown as dashed green lines in Fig. 1, are identical for
$i=1,2,3$. However, they are smaller than the one for the electric potential,
i.e., $[{}_{1}{\cal P}^{0}(k,t)+{}_{2}{\cal P}^{0}(k,t)]\;<\;[{}_{1}{\cal
P}^{0}(k,t)+{}_{2}{\cal P}^{0}(k,t)]$.

While the radial equation for
$\bar{R}^{(3)}_{l}\left[k\,(\vec{r}\cdot\hat{e}_{r})\right]$ satisfies the
differential equation

$\displaystyle\left(1+\frac{H_{0}^{2}}{c^{2}}\,r^{2}\right)\frac{d^{2}\bar{R}^{(3)}_{l}(k\,r)}{\hskip-28.45274ptdr^{2}}+\left(\frac{4}{r}+5\frac{H_{0}^{2}}{c^{2}}\,r\right)\frac{d\bar{R}^{(3)}_{l}(k\,r)}{\hskip-31.2982ptdr}-\frac{1}{r^{2}}\left[4\frac{H_{0}^{2}}{c^{2}}-2\alpha_{1}\alpha_{2}-l(l+1)-{\kappa_{3}}^{2}r^{2}\right]\bar{R}^{(3)}_{l}(k\,r)=0.$
(81)

The solution can be written in terms of associated Legendre functions $\it LP$
and $\it LQ$ as follows:

$\displaystyle\bar{R}^{(3)}_{l}(k\,r)=\frac{1}{r^{3/2}}$
$\displaystyle\left\\{{\rm C_{1}^{(3)}}\,{\it
LP}\left[i(c\,\kappa_{3}/H_{0})-1/2,\;\sqrt{l(l+1)+9/4-2\alpha_{1}\alpha_{2}},\sqrt{1+(H_{0}/c)^{2}r^{2}}\right]\right.$
$\displaystyle+$ $\displaystyle\;\;\left.{\rm C_{2}^{(3)}}\,{\it
LQ}\left[i(c\,\kappa_{3}/H_{0})-1/2,\;\sqrt{l(l+1)+9/4-2\alpha_{1}\alpha_{2}},\sqrt{1+(H_{0}/c)^{2}r^{2}}\right]\right\\},$
(82)

where $C_{1}^{(3)}$ and $C_{2}^{(3)}$ are constants. Using the Eq. (80b) and
the Fourier expansion proposed in the Eq. (66), we obtain that

$\displaystyle{}_{\iota}\hat{b}_{klm}^{(1)}$ $\displaystyle=$
${}_{\iota}\hat{a}_{klm},$ (83a) ${}_{\iota}{\cal
P}^{1}(k,t)\bar{R}^{(1)}_{l}(k\,r)$ $\displaystyle=$ $\displaystyle
r\,{}_{\iota}{\cal P}^{2}(k,t)\bar{R}^{(2)}_{l}(k\,r)-r\,h(t){}_{\iota}{\cal
P}^{0}(k,t)\bar{R}^{(0)}_{l}(k\,r).$ (83b)

Summarizing, we have found that, for each pair of angles

$\big{(}\theta_{*},\vartheta_{*}\big{)}=\left(\operatorname{arcCot}{\left(\alpha_{1}\right)},\frac{1}{m}\arctan\left({\frac{\alpha_{3}}{m}}\right)\right),$
(84)

the spacial components of the electromagnetic 4-potential can be expressed in
terms of the electric potential $\hat{\Phi}$, as follows

$\displaystyle\hat{\delta A}^{1}$ $\displaystyle=$ $\displaystyle
r\,\dot{\hat{\Phi}}+r^{2}\,h(t)\partial_{r}\hat{\Phi}+r\,\big{[}h(t)+H-\gamma\big{]}\hat{\Phi},$
(85a) $\displaystyle\hat{\delta A}^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{\alpha_{1}}\bigg{[}\dot{\hat{\Phi}}+r\,h(t)\partial_{r}\hat{\Phi}+\big{(}2h(t)+H-\gamma\big{)}\hat{\Phi}\bigg{]},$
(85b) $\displaystyle\hat{\delta A}^{3}$ $\displaystyle=$
$\displaystyle-\frac{\alpha_{2}}{\alpha_{1}\alpha_{3}}\bigg{[}\dot{\hat{\Phi}}+r\,h(t)\partial_{r}\hat{\Phi}+\big{[}h(t)+H-\gamma+h(t)/\alpha_{2}\big{]}\hat{\Phi}\bigg{]}.$
(85c)

Once obtained the $\hat{\delta A}^{i}$ fields, we can compute the primordial
electric and magnetic fields in the early universe.

##  V Primordial electric and magnetic fields

Let us consider the electromagnetic fields orthogonal to the relativistic
velocity $U^{\mu}=\frac{dx^{\mu}}{dS}$, so that $\hat{\delta
E}_{\mu}U^{\mu}=0$ and $\hat{\delta B}_{\mu}U^{\mu}=0$. Thus, we can express
the electric and magnetic fields in a following compact form, review

$\displaystyle\hat{\delta E}^{\mu}$ $\displaystyle=$
$\displaystyle\hat{F}^{\mu\nu}\,U_{\nu},$ (86) $\displaystyle\hat{\delta
B}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\epsilon^{\mu\nu\rho\lambda}\,U_{\nu}\,\hat{F}_{\rho\lambda}.$
(87)

Here, $\epsilon^{\mu\nu\rho\lambda}$ is the totally anti symmetric Levy-Civita
symbol, such that $\epsilon^{0123}=1,\;(-1)$ for any even (odd) permutation of
(0,1,2,3) and 0 if any index is repeated. Furthermore, choosing a co-moving
gauge, i.e., $U^{0}=\sqrt{g^{00}}$ and $U^{i}=0$, the nonzero components of
the electric and magnetic fields consist only of the following spatial
components

$\displaystyle\hat{\delta E}^{i}$ $\displaystyle=$
$\displaystyle\hat{F}^{i0}\,U_{0}=U_{0}\left(g^{ij}\hat{\nabla}_{j}\hat{\Phi}-g^{0j}\hat{\nabla}_{j}\hat{\delta
A}^{i}\right),$ (88a) $\displaystyle\hat{\delta B}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\epsilon^{i0jk}\,U_{0}\,\hat{F}_{jk}=-\frac{1}{2}\,\epsilon^{0ijk}U_{0}\left(\hat{\nabla}_{j}(\hat{\delta
A}_{k})-\hat{\nabla}_{k}(\hat{\delta A}_{j})\right),$ (88b)

for $i=1,2,3$. With the metric given by the Eq. (37), the components of the
electric field, Eq. (88a), take the form

$\displaystyle\hat{\delta E}^{1}$ $\displaystyle=$
$\displaystyle-(1+(H_{0}/c)^{2}\,r^{2})\,\left(\frac{H(t)}{H_{0}}\right)\,e^{-\int
2Hdt}\,\hat{\partial}_{1}\left(\hat{\Phi}/c\right)-\,\left(\frac{H(t)}{H_{0}}\right)^{3}\,\dot{\hat{\delta
A}}^{1},$ (89a) $\displaystyle\hat{\delta E}^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{r^{2}}\,\left(\frac{H(t)}{H_{0}}\right)\,e^{-\int
2Hdt}\,\hat{\partial}_{2}\left(\hat{\Phi}/c\right)-\,\left(\frac{H(t)}{H_{0}}\right)^{3}\,\dot{\hat{\delta
A}}^{2},$ (89b) $\displaystyle\hat{\delta E}^{3}$ $\displaystyle=$
$\displaystyle-\frac{1}{r^{2}{\sin^{2}\theta}}\,\left(\frac{H(t)}{H_{0}}\right)\,e^{-\int
2Hdt}\,\hat{\partial}_{3}\left(\hat{\Phi}/c\right)-\,\left(\frac{H(t)}{H_{0}}\right)^{3}\,\dot{\hat{\delta
A}}^{3}.$ (89c)

On the other hand, the spatial magnetic field components, Eq. (88b), are

$\displaystyle\hat{\delta B}^{1}$ $\displaystyle=$ $\displaystyle
r^{2}\,\left(\frac{H(t)}{H_{0}}\right)\,e^{\int
2Hdt}\left[\tilde{\alpha_{1}}\,\hat{\partial}_{2}\,\hat{\delta
A}^{3}-\hat{\partial}_{3}\,\hat{\delta
A}^{2}+2\alpha_{1}\tilde{\alpha_{1}}\,\hat{\delta A}^{3}\right],$ (90a)
$\displaystyle\hat{\delta B}^{2}$ $\displaystyle=$
$\displaystyle\,\left(\frac{H(t)}{H_{0}}\right)\,e^{\int
2Hdt}\left[r^{2}\,\tilde{\alpha_{1}}\,\hat{\partial}_{1}\,\hat{\delta
A}^{3}-\frac{1}{1+(H_{0}/c)^{2}\,r^{2}}\,\hat{\partial}_{3}\,\hat{\delta
A}^{1}+\frac{2}{r}\,\hat{\delta A}^{3}\right],$ (90b)
$\displaystyle\hat{\delta B}^{3}$ $\displaystyle=$
$\displaystyle\,\left(\frac{H(t)}{H_{0}}\right)\,e^{\int
2Hdt}\,\left[r^{2}\,\hat{\partial}_{1}\,\hat{\delta
A}^{2}-\frac{1}{1+(H_{0}/c)^{2}\,r^{2}}\,\hat{\partial}_{2}\,\hat{\delta
A}^{1}+\frac{2}{r}\,\hat{\delta A}^{2}\right],$ (90c)

where
$\tilde{\alpha}_{1}\equiv\sin^{2}{\left[\operatorname{arcCot}\left(\alpha_{1}\right)\right]}$,
and the vector potential $\hat{\delta A}^{i}$ can be written as in the Eq.
(IV.2).

Figure 2: Temporal evolution for the expectation value of magnetic field for
$n=200$, with different wavelengths sub-Hubble values
$\lambda=\\{0.01,\;0.1,\;0.5,\;0.9,\;1\\}\times c/H_{0}$, and $l=500$. In all
cases, the magnetic field reaches its maximum at $t_{\text{max}}\sim
29.45\times 10^{16}$ s and then decreases, reaching present day values on the
order of $\left<V|B(t_{f})|V\right>\simeq 10^{-12}$ G. The largest values
correspond to the biggest value of $\lambda$.

###  V.1 Expectation value of the electromagnetic fields

To compute the expectation value of the electric and magnetic fields on the
curved background spacetime governed by the Riemann manifold, we need to
define an extension of the nonlinear quantum algebra introduced in BMM , which
includes the quantum differential operator $\hat{\frac{\partial}{\partial
x^{\prime j}}}$ that acts on the operator
${}_{\iota}\hat{\sigma}^{(\alpha)\dagger}_{klm}(x^{i})$, such that, the Eq.
(44) transforms into

$\left[\hat{\frac{\partial}{\partial x^{\prime
j}}},{}_{\iota}\hat{\sigma}^{(\alpha)\dagger}_{klm}(x^{i})\right]=-\frac{i\,k^{\prime}_{j}}{b^{2}}\,\left[{}_{\iota}\hat{\sigma}_{k^{\prime}l^{\prime}m^{\prime}}^{(\alpha)}(x^{\prime
i}),{}_{\iota}\hat{\sigma}^{(\alpha)\dagger}_{klm}(x^{i})\right],$ (91)

where
${}_{\iota}\hat{\sigma}_{klm}^{(\alpha)}(x^{i})={}_{\iota}\hat{B}_{klm}\,{}_{\iota}\sigma^{(\alpha)}(k,x^{i})$
and
${}_{\iota}{\hat{\sigma}^{(\alpha)\dagger}_{klm}}(x^{i})={}_{\iota}\hat{B}^{\dagger}_{klm}\,\left(\sigma^{(\alpha)}(k,x^{i})\right)^{*}$.
Therefore, we obtain

$\left<V\left|\hat{\frac{\partial}{\partial x^{\prime j}}}\hat{\delta
A}^{\nu}\right|V\right>=\frac{1}{b}\sum_{\iota=1}^{2}\sum_{l,m}\frac{(2l+1)(l-m)!}{4\pi(l+m)!}\,k_{j}\,{}_{\iota}P^{\nu}(k,t)\|{}_{\iota}\sigma_{klm}^{(\nu)}\|^{2},$
(92)

where we have used the fact that

$\left<V\left|\left[{}_{\iota}\hat{B}_{klm},\,{}_{\varsigma}\hat{B}_{k^{\prime}l^{\prime}m^{\prime}}^{\dagger}\right]\right|V\right>=ib\,\sqrt{2\pi}\,\frac{(2l+1)(l-m)!}{2(l+m)!}\,\delta^{(3)}(\vec{k}-\vec{k}\,^{\prime})\,\delta_{\iota\varsigma}\,\delta_{ll^{\prime}}\,\delta_{mm^{\prime}}.$
(93)

Therefore, the expectation value of the electric field is

$\left<V\left|\hat{\delta E}^{i}\right|V\right>=-\frac{U_{0}}{4\pi
b}g^{ij}\sum_{\iota=1}^{2}\sum_{klm}\frac{(2l+1)(l-m)!}{(l+m)!}k_{j}\,{}_{\iota}P^{0}(k,t)\|{}_{\iota}\sigma_{klm}^{(0)}\|^{2},\hskip
28.45274pti,j=1,2,3,$ (94)

where the wave numbers related to spherical coordinates
$(x^{1},x^{2},x^{3})=(r,\theta,\vartheta)$, are: $k_{1}=k$, $k_{2}=l\,k$ and
$k_{3}=m\,k$. The expectation values of the magnetic field components, Eq.
(90), for $(\theta_{*},\vartheta_{*})$, given in the Eq. (84), are

$\displaystyle\left<V\left|\hat{\delta
B}^{r}(t,r,\theta_{*},\vartheta_{*})\right|V\right>$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{4\pi b}\,\left(\frac{H(t)}{H_{0}}\right)\,e^{\int
2Hdt}\sum_{\iota=1}^{2}\sum_{l,m}\frac{(2l+1)(l-m)!}{(l+m)!}$ (95a)
$\displaystyle\times$
$\displaystyle\left[\tilde{\alpha_{1}}\,l\,{}_{\iota}P^{\vartheta}(k,t)\|{}_{\iota}\sigma_{klm}^{(3)}\|^{2}-m\,{}_{\iota}P^{\theta}(k,t)\,\|{}_{\iota}\sigma_{klm}^{(2)}\|^{2}\right],$
$\displaystyle\left<V\left|\hat{\delta
B}^{\theta}(t,r,\theta_{*},\vartheta_{*})\right|V\right>$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{4\pi b}\,\left(\frac{H(t)}{H_{0}}\right)\,e^{\int
2Hdt}\sum_{\iota=1}^{2}\sum_{l,m}\frac{(2l+1)(l-m)!}{(l+m)!}$ (95b)
$\displaystyle\times$
$\displaystyle\left[\tilde{\alpha_{1}}\,k\,{}_{\iota}P^{\vartheta}(k,t)\|{}_{\iota}\sigma_{klm}^{(3)}\|^{2}-\frac{m}{r^{2}\big{(}1+(H_{0}/c)^{2}\,r^{2}\big{)}}\,{}_{\iota}P^{r}(k,t)\,\|{}_{\iota}\sigma_{klm}^{(1)}\|^{2}\right],$
$\displaystyle\left<V\left|\hat{\delta
B}^{\vartheta}(t,r,\theta_{*},\vartheta_{*})\right|V\right>$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{4\pi b}\,\left(\frac{H(t)}{H_{0}}\right)\,e^{\int
2Hdt}\sum_{\iota=1}^{2}\sum_{l,m}\frac{(2l+1)(l-m)!}{(l+m)!}$ (95c)
$\displaystyle\times$
$\displaystyle\left[k\,{}_{\iota}P^{\theta}(k,t)\,\|{}_{\iota}\sigma_{klm}^{(2)}\|^{2}-\frac{l}{r^{2}\big{(}1+(H_{0}/c)^{2}\,r^{2}\big{)}}\,{}_{\iota}P^{r}(k,t)\,\|{}_{\iota}\sigma_{klm}^{(1)}\|^{2}\right],$

where we have used that $\left<V\left|\hat{\delta A}^{\nu}\right|V\right>=0$.
Finally, the squared strength of the magnetic field can be computed as follows

$\left<V\left|\hat{\delta B}\right|V\right>^{2}=\left<V\left|\hat{\delta
B}^{i}\right|V\right>\left<V\left|\hat{\delta B}_{i}\right|V\right>.$ (96)

The time evolution of the expectation value of the magnetic field is depicted
in the Fig, (2), for different wavelength values: $\lambda$ and $l=500$. Note
that, the magnetic fields reach their maximum value at $t=29.45\times 10^{1}6$
s and then, smoothly decrease.

In Table (1), we present the initial and final values of the magnetic field,
$\left<V|B(t_{0})|V\right>$ and $\left<V|B(t_{f})|V\right>$ respectively, for
$n=200$ and various present-day wavelength values relevant for cosmological
scales. Here, $t_{0}=4.35\times 10^{13}$ s corresponds to the moment when CMBR
was emitted, and $t_{f}=4.35\times 10^{17}$ s, represents the present-day age
of the universe.

$\lambda\,/10^{25}$ $[m]$ |  $k\,/10^{-25}\;\;[m^{-1}]$ |  $\left<B_{0}\right>\,/10^{-16}$ [G] |  $\left<B_{f}\right>\,/10^{-12}$ [G] |  $\Delta B\,/10^{-12}$ [G]
---|---|---|---|---
16.26744329 | 0.386242951 | 2.716352592 | 3.304304126 | 3.304032491
13.55620274 | 0.463491541 | 4.280950842 | 5.207557948 | 5.207129853
12.20058247 | 0.514990601 | 4.090227643 | 4.975552921 | 4.975143898
6.778101370 | 0.926983083 | 3.467601289 | 4.218159778 | 4.217813018
1.135562027 | 4.634915418 | 3.163467821 | 3.848196954 | 3.847880607
0.135562027 | 46.34915418 | 3.150291818 | 3.832169020 | 3.831853990
Table 1: Expectation values of the magnetic field for different values of
$\lambda$, $k$ and $l=500$.
$\left<B_{0,f}\right>\equiv\left<V|B(t_{0,f})|V\right>$ where
$t_{0}=4.35\times 10^{13}$ s corresponds to the CMB emission, and
$t_{f}=4.35\times 10^{17}$ s is the age of the universe. $\Delta
B\equiv|\left<B_{0}\right>-\left<B_{f}\right>|$.

For completeness of the work, we also compare the magnitude of the magnetic
field for different values of $l$ and fixed $\lambda$ as is indicated in the
plots of Fig. 3, it is observed that the magnitude of $\left<V\right>$ is
bigger according to $l$.

Figure 3: Temporal evolution for the expectation values for magnetic fields
for $n=200$, with different wavelengths sub-Hubble values
$\lambda=\\{0.01,\;0.1,\;0.5,\;1\\}\times c/H_{0}$, and
$l=\\{200,\;300,\;400,\;600\\}$, and their comparative values with $l=500$.
Notice that in all cases the magnetic fields reach their maximum at
$t_{*}=29.45\times 10^{16}$ s, and then decrease, taking present day values of
the order of $\left<V|B(t_{f})|V\right>\simeq 10^{-12}$ G.

##  VI Conclusions

In this work, we have explored a model of an extended general relativity,
where the boundary terms are related with the pertubations of the spacetime.
The extending Riemann manifold is described by the varied connection proposed
into the Eq. (1). In this frame we studied the dynamics of the inflaton field
and compute the expectation value of the flux of
$\hat{\Omega}^{\mu}=\hat{\sigma}^{\mu}+a\hat{\delta A}^{\mu}$ through a closed
$3d$-hypersurface, which alters the Einstein equations that describes the
background dynamics of the universe. This 4-vector field $\hat{\Omega}^{\mu}$
contains geometric information related to primordial gravitational and
electromagnetic fluctuations. The expectation value of its flux is solely due
to gravitational contributions, because we have required an electromagnetic
neutral background, $\left<V\right|\nabla_{\mu}\hat{\delta
A}^{\mu}\left|V\right>=0$. Therefore, the cosmological constant $\lambda_{0}$
is exclusively originated by the flux of
$\hat{\sigma}^{\mu}=\hat{\partial}_{\mu}\hat{\sigma}$, as can be seen in the
Eq. (38), which means that $\lambda_{0}$ alters the background dynamics
described by the Einstein equations
$G_{\alpha\beta}-\lambda_{0}\,{g}_{\alpha\beta}=-\kappa\,T_{\alpha\beta}$, as
a consequence of the rugosity of spacetime with respect to the background
curvature.

We studied the dynamics of the gravito-electromagnetic field during
preinflation. Choosing the gauge $\eta=-2b$, we obtained a decoupled dynamics
for GW and electromagnetic fields in a model with variable time-scale and
negative spatial curvature. In this work we have focused in the study of the
electromagnetic fields. We proposed a Fourier expansion for each component of
the 4-electromagnetic potential, and we found that for each pair of fixed
angles $(\theta_{*},\vartheta_{*})$, the vector potential fluctuations
$\hat{\delta A}^{i}$ can be written in terms of the electric potential
$\hat{\Phi}$, as is reported in the Eq. (85). Te polarization modes of the
electric potential are dominant with respect to those of the vector potential,
as can be appreciated in the Fig. (1).

In the context of quantum electromagnetic fluctuations in preinflation, and
using an extension of the quantum algebra proposed in BMM ; BMAS , we have
calculated the strength of primordial magnetic fields, which were depicted in
Figs. 2 and 3. In all cases, the magnetic field reaches its peak at
$t_{\text{max}}=29.45\times 10^{16}$ s, and then gradually decreases. They
take present-day values of the order of $\left<V|B(t_{f})|V\right>\simeq
10^{-12}$ G. It is important to note the distinction between our results and
those obtained by other models, as seen, for instance MemBell . In our case
magnetic fields grow from small initial values during the early stages of the
universe’s evolution, and thereafter decrease. This significant divergence
eliminates the necessity of imposing excessive large initial values upon these
fields, to recover the present day observed values.

Observe that the choice of the gauge $\eta=-2b$ greatly simplifies our
calculations, making it possible to make an analytical treatment of the
dynamics of $\hat{\delta A}^{\mu}$. As a prospective avenue for future
research, it would be highly intriguing to explore an alternate gauge in which
the coupling between $\hat{\delta A}^{\mu}$ and
$\hat{\delta\Psi}_{\alpha\beta}$ can be examined. Such an investigation,
particularly the exploration of the transfer of B-modes from GW to
electromagnetic waves observable in the CMBR spectrum, holds notable
importance. This task appears to necessitate numerical calculations for a
comprehensive analysis.

## CRediT authorship contribution statement

Daniela Magos: Ideas, Conceptualization, Calculation, Writing. Mauricio
Bellini: Ideas, Conceptualization, Calculation, Writing.

## Declaration of competing interest

The authors declare the following financial interests/personal relationships
which may be considered as potential competing interests: The authors reports
financial support was provided by National Scientific and Technical Research
Council. The authors reports financial support was provided by National
University of Mar del Plata.

## Data availability

No data was used for the research described in the article.

## Acknowledgements

The authors acknowledge CONICET, Argentina (PIP 11220200100110CO) and UNMdP
(EXA1055/22) for financial support. Daniela Magos thanks CONICET for the 2020
postdoctoral fellowship with Latin American countries.

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