# Primordial electric fields before recombination in the early Universe

Xi-Bin Li
1College of Physics and Electronic Information, Inner Mongolia Normal
University,
81 Zhaowuda Road, Hohhot, 010022, Inner Mongolia, China
2Inner Mongolia Key Laboratory for Physics and Chemistry of Functional
Materials,
Inner Mongolia Normal University, 81 Zhaowuda Road, Hohhot, 010022, Inner
Mongolia, China
3School of Physics and Technology, Wuhan University, 430072, Wuhan, China
Corresponding author<EMAIL_ADDRESS>
###### Abstract

This work is a supplement on the previous research about primordial
electromagnetic fields. In this work, three important problems are discussed:
the evolution of primordial electric fields, the electric and particle
densities’ solitons in plasma before recombination and their influences on the
power spectra of cosmic microwave background. Detailed computations show that
the primordial electric fields dissipate by Landau damping effect on both
large scale and small scale and there is no impact on the spectrum. While,
before recombination, there exist solitary waves stably propagating in plasma
whose speed is significantly slower than that of baryonic acoustic
oscillations, working only at extremely small scale. On the other hand, the
amplitude of solitons is so weak that only a significantly small contribution
on the phase of baryon acoustic oscillations, so there merely exist the
messages about such electric solitary waves on the spectrum. In a word, as
relevant monographs on cosmology, neglecting the electromagnetic fields
(electric fields at least) is a reasonable treatment on the calculations of
cosmic microwave background. However, the protonic density fluctuations show a
form of KdV equation while its propagation as a stable solitary wave, leading
a probability to the origin of fluctuation promoting the generation and
evolution of galaxies.

##  1 Introduction

Primordial electromagnetic fields are often considered as the essential
components in early Universe [1, 2]. They relate to the electromagnetic
dynamical process in the primordial plasma, cosmological perturbation dynamics
and galaxies’ generation and evolution [2, 3, 4]. Precious studies focus
mainly on the primordial magnetic problems. The origin of the magnetic fields
have been widely studied and may generate in different ways, like inflation
[5, 6, 7], quark confinement [8, 9] and neutrino decoupling [10, 11, 12]. Such
a field plays an important role in the electrodynamic properties [1], baryon
acoustic oscillations [13, 14] and tensor perturbations [15, 16, 17]. It has
attracted a lot of interests on the study of the tensor perturbed mode induced
by the cosmic magnetic fields, since it they give rise to a anisotropy signal
on B-mode polarizations of cosmic microwave background (CMB) [18, 19, 20, 21,
22, 23, 24]. However, the average amplitude of magnetic field should be small
enough to be consistent with the observations on spatial large-scale isotropy.
This is the reason why linear nonequilibrium or instability method is applied
to study the perturbation dynamics [3, 25, 26, 27, 28, 29, 30].

But unfortunately, the importance of the study on primordial cosmological
electric fields has not been realized till now. It may be partly because the
previous work did not show any amazing results about electric fields or
because the attractive from electric effects in primordial plasma are not so
strong compared with it from magnetic effects. But it needs to point out, in
plasmonics, the electric properties have an equally important position as
magnetic ones [31, 32].

Vlasov equation, as the mathematical tool on the plasma statistical properties
in electromagnetic fields, is deeply studies during the last century and
widely used in the nuclear industry and other fields [33, 34, 35, 36, 37, 38].
Relevant theoretical results, like Landau damping effect, acoustic waves and
electric solitary wave, have been solidly tested by modern experiments.
Therefore it’s reasonable to deduce that the corresponding phenomena should
also occur in cosmic primordial plasma. Now the extension on relevant results
to the early Universe is attempted and several questions not discussed earlier
are going to propose in this paper. The first question is the generation of
magnetic field must be accompanied with the generation of electric field. So
it’s necessary to analyse the evolution of electric fields and discuss how to
propagate in the plasma. The second question is that whether there is a
special way to persist the nonlinear perturbation after recombination,
especially the protonic density fluctuation, since perturbations continue to
grow up in the absence of optical pressure during and after recombination. The
last question is whether primordial cosmological electric fields influence on
the CMB power spectra and how significantly they are.

In this paper, I mainly concentrate on answering these three problems. The
method, of course, is the Vlasov equation, which is a Boltzmann equation
describing a charged particle’s state in phase space moving in the
electromagnetic field. The optimal scenario to obtain the power spectra of CMB
is the Boltzmann equation, as illustrated in series of monographs [39, 40,
41]. So a potentially feasible method to study the effects on perturbed
dynamics, evolutions for large-scale structure and CMB from the
electromagnetic fields is the Vlasov equation. As the critical part of this
work, the first step is obtaining the Vlasov equation in perturbed FRW metric
which could be accomplished by applying the geodesic equation. Based on this
equation, the problems introduced previous could solved via the existing
mature solutions. Relevant conclusions are quite brief but persuasive. The
initial electric fields dissipate out via Landau damping, intrinsic electric
solitary waves propagate with a speed something like average thermal velocity
but not exactly and there is almost no influences on CMB power spectra except
at extremely small scale.

The following contents of this paper is organized as follows. In Sec. 2, the
Vlasov equation in perturbed FRW Universe is obtained via the geodesic
equation, which is the key step of this work. In next section, it’s proofed
the initial electric fields generated before recombination are gong to
dissipate, during which it is assumed the electrons state under a thermal
equilibrium and obey the Maxwell distribution. In Sec. 4, the existence of
electric solitary waves in primordial plasma is proofed via a series of
nonlinear equations, which could retain after recombination to help the
generations and evolutions of the galaxies and galaxy cluster. In Sec. 5, the
relation between the acoustic oscillations and cosmic electric fields is
analytically computed, and the influences on CMB are also discussed. In the
final section, brief conclusions and further research programmes are given.

##  2 Vlasov Equation in Perturbed FRW Universe

The scaler perturbation in perturbed FRW Universe is described by the metric

$\displaystyle g_{00}(\mathbf{x},t)$ $\displaystyle=-1-2\Psi(\mathbf{x},t),$
$\displaystyle g_{0i}(\mathbf{x},t)$ $\displaystyle=0,$ $\displaystyle
g_{ij}(\mathbf{x},t)$ $\displaystyle=a^{2}\delta_{ij}[1+2\Phi(\mathbf{x},t)],$
(1)

where $a$ is scale factor dependent only on cosmic time and $\Psi$, $\Phi$ are
two variables describing the scaler perturbation under conformal gauge. The
distribution function of a particle under equilibrium state depends on
position coordinate $\mathbf{x}$, momentum $\mathbf{p}$ and time $t$ whose
complete differential on $t$ reads

$\displaystyle\frac{\mathrm{d}f}{\mathrm{d}t}=\frac{\partial f}{\partial
t}+\frac{\partial f}{\partial x^{i}}\frac{\partial x^{i}}{\partial
t}+\frac{\partial f}{\partial p}\frac{\partial p}{\partial t}+\frac{\partial
f}{\partial\hat{p}^{i}}\frac{\partial\hat{p}^{i}}{\partial t}.$ (2)

The four-momentum is defined as

$\displaystyle P^{\mu}=mU^{\mu}$ (3)

which relats to the four-velocity

$\displaystyle U^{\mu}=\frac{\mathrm{d}x^{\mu}}{\mathrm{d}\lambda},$ (4)

where $m$ is the rest mass of a particle and $\lambda$ is the proper time. The
second term on the right-hand in Eq. (2) could be reexpress in terms of four-
momentum by applying the definitions in Eq. (3). Therefor,

$\displaystyle\frac{\mathrm{d}x^{i}}{\mathrm{d}t}=\frac{\mathrm{d}x^{i}}{\mathrm{d}\lambda}\frac{\mathrm{d}\lambda}{\mathrm{d}t}=\frac{P^{i}}{P^{0}}.$
(5)

The four-momentum of a massive particle is also derived in terms of particle’s
energy and momentum:

$\displaystyle P^{\mu}=\left[E(1-\Psi),p\hat{p}^{i}\frac{1-\Phi}{a}\right],$
(6)

where $p^{2}=g_{ij}P^{i}P^{j}$. Similarly, it’s also available to get the
complete differential of a massive particle’s, like electron, distribution
function

$\displaystyle\frac{\mathrm{d}f_{\mathrm{e}}}{\mathrm{d}t}=\frac{\partial
f_{\mathrm{e}}}{\partial t}+\frac{\partial f_{\mathrm{e}}}{\partial
x^{i}}\frac{\partial x^{i}}{\partial t}+\frac{\partial
f_{\mathrm{e}}}{\partial E}\frac{\partial E}{\partial t},$ (7)

where the term disappears in Eq. (2) with the derivative respect to
orientation vector of momentum $\hat{p}^{i}$. The electrodynamics equation in
curved space-time is written as

$\displaystyle\frac{\mathrm{D}P^{\mu}}{\mathrm{D}\lambda}=\frac{q}{c}F^{\mu\nu}U_{\nu},$
(8)

with $F^{\mu\nu}$ denoting the electromagnetic tensor. Setting $\mu=0$, Eq.
(8) describing a electron’s motion is rewritten in terms of Christoffel symbol
as

$\displaystyle m\frac{\mathrm{d}U^{0}}{\mathrm{d}\lambda}+m\Gamma^{0}_{\
\alpha\beta}P^{\alpha}P^{\beta}=-\mathrm{e}_{0}E^{i}U_{i}.$ (9)

Repeating the computations in Ref. [39], the derivative of energy to cosmic
time gives

$\displaystyle\frac{\mathrm{d}E}{\mathrm{d}t}=-\left(H\frac{p^{2}}{E}+\dot{\Phi}\frac{p^{2}}{E}+\frac{p\hat{p}^{i}}{a}\frac{\partial\Psi}{\partial
x^{i}}-\mathrm{e}_{0}\frac{\hat{p}^{i}}{a}\frac{p^{2}}{E}\frac{\partial\phi}{\partial
x^{i}}\right).$ (10)

In the equation above, the dot $\dot{\ }$ denotes the derivative with respect
to cosmic time $t$, $H=\dot{a}/a$ represents the Hubble parameter and $\phi$
means electric potential whose gradient relates to the electric field
$\mathbf{E}=-\nabla\phi$ appeared in Eq. (9). Instituting Eqs. (3), (5) and
(10) into Eq. (7), there is

$\displaystyle\frac{\partial f_{\mathrm{e}}}{\partial
t}+\frac{\hat{p}^{i}}{a}\frac{p}{E}\frac{\partial f_{\mathrm{e}}}{\partial
x^{i}}-\frac{\partial f_{\mathrm{e}}}{\partial
E}\left[H\frac{p^{2}}{E}+\frac{\partial\Phi}{\partial
t}\frac{p^{2}}{E}+\frac{p\hat{p}^{i}}{a}\frac{\partial\phi}{\partial
x^{i}}-\mathrm{e}_{0}\frac{1}{a}\frac{p^{2}\hat{p}^{i}}{E}\frac{\partial\phi}{\partial
x^{i}}\right]=0.$ (11)

This is the Vlasov equation that describes a electron’s statistical state in
phase space forced by a electric field moving in a perturbed FRW Unierse
(having neglected the collision term). This equation in flat spacetime has
been widely and deeply studied in the area of plasma physics and mathematics.

##  3 Damping of primordial electric field before decoupling

###  3.1 Linearized Vlasov equation

Eq. (11) describes an electron’s Boltzmann equation in electric field with
respect of cosmic time $t$, but in this section the derivative to proper time
$\lambda$ is quite convenience on the calculations. Define the three-velocity
with a new symbol

$\displaystyle v^{i}=\frac{\mathrm{d}x^{i}}{\mathrm{d}\lambda}.$ (12)

The complete differential of electron’s state in phase space reads

$\displaystyle\frac{\mathrm{d}f_{\mathrm{e}}}{\mathrm{d}\lambda}=\frac{\partial
f_{\mathrm{e}}}{\partial\lambda}+\frac{\partial
f_{\mathrm{e}}}{\partial\mathbf{x}}\cdot\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}\lambda}+\frac{\partial
f_{\mathrm{e}}}{\partial\mathbf{v}}\cdot\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}\lambda}=C.$
(13)

For convenience and easy understanding, from now on in this section, the
symbol of proper time $\lambda$ is replaced by the symbol $t$, and it means
proper time instead of cosmic time. Applying the geodesic equation (8) by
setting the component $\mu=i$ and inserting it into the Boltamann equation
(13), it arrives

$\displaystyle\dot{f}_{\mathrm{e}}+\mathbf{v}\cdot\frac{\partial
f_{\mathrm{e}}}{\partial\mathbf{x}}-\left[H\mathbf{v}+\dot{\Phi}\mathbf{v}+\frac{1}{a^{2}}\nabla\Psi+\frac{\mathrm{e}_{0}}{m}\mathbf{E}\right]\cdot\frac{\partial
f_{\mathrm{e}}}{\partial\mathbf{v}}=C,$ (14)

where the the dot $~{}\dot{~{}}~{}$ in this section represents the derivative
to proper time $\lambda$. In Eq. (14), the following equations or relations
have been used: $\Gamma^{i}_{\;00}=\partial_{i}\Psi/a^{2}$,
$\Gamma^{i}_{\;j0}=\delta_{ij}(H+\partial\Phi/\partial t)$ and
$\mathrm{d}t/\mathrm{d}\lambda\simeq 1$ since nonrelativistic limit. The
collision term in Eq. (14) is decomposed into two parts $C=-\varepsilon
f_{\mathrm{e}}+n_{\mathrm{e}}\int\mathrm{d}\Omega\,\partial
f_{\mathrm{e}}/\partial\Omega$. The part including the angular integral in
collision is independent on state function $f_{\mathrm{e}}$ and it is only a
nonhomogeneous term, so in the following computation it is not taken into
account.

The amplitude of electric field is so weak that the deviation to the
equilibrium about electron’s distribution function is quite slight:

$\displaystyle
f_{\mathrm{e}}(\mathbf{x},\mathbf{v},t)=n_{\mathrm{e}0}(\mathbf{v},t)+\delta
n_{\mathrm{e}}(\mathbf{x},\mathbf{v},t),$ (15)

where $n_{\mathrm{e}0}$ is the probability density function under equilibrium
state in absence of scaler perturbations in Eq. (1). instituting Eq. (15) into
Eq. (14) and consider only the first order, together with the Maxwell
equation, it shows the equations

$\displaystyle\frac{\partial\delta n}{\partial
t}+\mathbf{v}\cdot\frac{\partial\delta
n}{\partial\mathbf{x}}-\left[H\mathbf{v}+\dot{\Phi}\mathbf{v}+\frac{1}{a^{2}}\nabla\Psi+\frac{\mathrm{e}_{0}}{m}\mathbf{E}\right]\cdot\frac{\partial
n_{\mathrm{e}0}}{\partial\mathbf{v}}=-\varepsilon\delta n,$ (16)
$\displaystyle\quad\quad\quad\quad\rho(\mathbf{x},t)=-\mathrm{e}_{0}\int\delta
n(\mathbf{x},\mathbf{v},t)\mathrm{d}^{3}v,$ (17)
$\displaystyle\quad\quad\quad\quad\mathbf{j}(\mathbf{x},t)=-\mathrm{e}_{0}\int\delta
n(\mathbf{x},\mathbf{v},t)\mathbf{v}\mathrm{d}^{3}v,$ (18)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\nabla\cdot\mathbf{E}=4\pi\rho(\mathbf{x},t),$
(19) $\displaystyle\quad\quad\quad\quad\frac{\partial\mathbf{E}}{\partial
t}=-4\pi\mathbf{j}(\mathbf{x},t),\ \ \ \ \nabla\times\mathbf{E}=0.$ (20)

$\rho$ in Eq. (17) is the charge density while $\mathbf{j}$ in Eq. (18) is the
current density.

Set the perturbed probability density function shows an oscillating form

$\displaystyle\delta n=\delta
n(\mathbf{v})\mathrm{e}^{\mathrm{i}\left[\mathbf{k}\cdot\mathbf{x}-\int^{t}\omega(t^{\prime})\mathrm{d}t^{\prime}\right]}.$
(21)

Therefor, Eq. (16) becomes by neglecting $\Phi$ and $\Psi$

$\displaystyle-\mathrm{i}\omega\delta
n+\mathrm{i}\mathbf{v}\cdot\mathbf{k}\delta n+\varepsilon\delta
n=\left[H\mathbf{v}+\frac{\mathrm{e}_{0}}{m}\mathbf{E}\right]\cdot\frac{\partial
n_{0}}{\partial\mathbf{v}}.$ (22)

Then, Eq. (22) gives the fractional solution to perturbed probability density
function:

$\displaystyle\delta
n=\frac{1}{\mathrm{i}[\mathbf{k}\cdot\mathbf{v}-\omega(t)-\mathrm{i}\varepsilon]}\left[H\mathbf{v}+\frac{\mathrm{e}_{0}}{m}\mathbf{E}\right]\cdot\frac{\partial
n_{0}}{\partial\mathbf{v}}.$ (23)

###  3.2 Polarization of plasma gas

Polarization vector indicates the deviation from electric neutrality, and it
defines

$\displaystyle\mathbf{D}_{0}=\mathbf{E}_{0}+4\pi\mathbf{P},$ (24)

where $\mathbf{D}$ is the electric displacement vector which connects the
electric field $\mathbf{E}$ with permittivity tensor $\varepsilon_{ij}$:

$\displaystyle D_{0i}\exp$
$\displaystyle\left[\mathrm{i}\left(\mathbf{k}\cdot\mathbf{x}-\int^{t}\omega(t^{\prime})\mathrm{d}t^{\prime}\right)\right]=$
$\displaystyle\varepsilon_{ij}(\omega,\mathbf{k},t)E_{0j}\exp\left[\mathrm{i}\left(\mathbf{k}\cdot\mathbf{x}-\int^{t}\omega(t^{\prime})\mathrm{d}t^{\prime}\right)\right].$
(25)

The permittivity tensor could be decomposed into two components, the
transverse one and longitudinal one. In this paper, the only component,
transverse permittivity $\varepsilon_{1}$, is used which means the direction
of electric field $\mathbf{E}_{0}$ is parallel to the direction of waves
$\mathbf{k}$, or $\mathbf{E}_{0}\varparallel\mathbf{k}$. So, polarization
vector is

$\displaystyle 4\pi\mathbf{P}_{0}=(\varepsilon_{1}-1)\mathbf{E}_{0}.$ (26)

According to Eq. (80), polarization vector is able to define in such a way:

$\displaystyle\nabla\cdot\mathbf{P}$
$\displaystyle=-a^{3}n_{\mathrm{e}}-\int^{t}a^{3}\dot{\Phi}\mathrm{d}t^{\prime},$
(27) $\displaystyle\frac{\partial\mathbf{P}}{\partial t}$
$\displaystyle=a^{2}n_{\mathrm{e}}\mathbf{v}.$ (28)

Of course, the definitions above indicate $\mathbf{P}$ propagates as a form of
oscillating wave

$\displaystyle\mathbf{P}=\mathbf{P}_{0}\exp\left[\mathrm{i}\left(\mathbf{k}\cdot\mathbf{x}-\int^{t}\omega(t^{\prime})\mathrm{d}t^{\prime}\right)\right].$
(29)

Thus, inserting Eq. (23) and (29) into Eq. (27) gives

$\displaystyle\mathrm{i}\mathbf{k}\cdot\mathbf{P}$
$\displaystyle=\mathrm{e}_{0}a^{3}\int\frac{\mathrm{d}^{3}v}{\mathrm{i}(\mathbf{k}\cdot\mathbf{v}-\omega)}\frac{\partial
n_{0}}{\partial\mathbf{v}}\cdot\left(H\mathbf{v}+\frac{\mathrm{e}_{0}}{m}\mathbf{E}\right)$
$\displaystyle=\frac{\mathrm{e}_{0}^{2}}{m}a^{3}\mathbf{E}\cdot\int\frac{\mathrm{d}^{3}v}{\mathrm{i}(\mathbf{k}\cdot\mathbf{v}-\omega-\mathrm{i}\varepsilon)}\frac{\partial
n_{0}}{\partial\mathbf{v}}+\mathrm{e}_{0}Ha^{3}\int\frac{\mathrm{d}^{3}v}{\mathrm{i}(\mathbf{k}\cdot\mathbf{v}-\omega-\mathrm{i}\varepsilon)}\frac{\partial
n_{0}}{\partial\mathbf{v}}\cdot\mathbf{v}.$ (30)

The equation above is rewritten in a short form
$\mathrm{i}\mathbf{k}\cdot\mathbf{P}=\mathbf{E}\cdot\mathbf{K}+\mathbf{K}_{0}$,
compared which with Eq. (26) gives the expression to the transverse
permittivity tensor

$\displaystyle\varepsilon_{1}(\omega,\mathbf{k},t)=1-\frac{4\pi\mathrm{e}_{0}^{2}}{mk^{2}}a^{3}\int\frac{\mathrm{d}^{3}v}{\mathrm{i}(\mathbf{k}\cdot\mathbf{v}-\omega-\mathrm{i}\varepsilon)}\frac{\partial
n_{0}}{\partial\mathbf{v}}\cdot\mathbf{k}.$ (31)

By applying Eq. (28), it gives the expression of the longitudinal permittivity
tensor

$\displaystyle\varepsilon_{t}(\omega,\mathbf{k},t)=1+\frac{4\pi\mathrm{e}_{0}^{2}}{m\omega}a^{3}\int\frac{n_{0}(\mathbf{v})\mathrm{d}^{3}v}{\mathrm{i}(\mathbf{k}\cdot\mathbf{v}-\omega-\mathrm{i}\varepsilon)},$
(32)

but this formula will not be used any longer, so there’s no need to show any
more interpretations.

###  3.3 Expression of electric fields

As discussed above, density function of electron could contain a slight
perturbation as a function of $t$

$\displaystyle n(\mathbf{x},\mathbf{k},t)=n_{0}(\textbf{v},t)+\delta
n(\textbf{r},\textbf{v},t),$ (33)

whose initial condition is

$\displaystyle
n(\mathbf{x},\mathbf{k},0)=n_{0}(\textbf{v},0)+g(\textbf{r},\textbf{v},0).$
(34)

Start with Eq. (16), (17) and (20), perturbed density function $\delta n$ and
electric potential $\varphi$ satisfy the equations

$\displaystyle\frac{\partial\delta n}{\partial
t}+\mathbf{v}\cdot\frac{\partial\delta
n}{\partial\mathbf{x}}-\left[H\mathbf{v}+\dot{\Phi}\mathbf{v}+\frac{1}{a^{2}}\nabla\Psi+\frac{\mathrm{e}_{0}}{m}\mathbf{E}\right]\cdot\frac{\partial
n_{\mathrm{e}0}}{\partial\mathbf{v}}=-\varepsilon\delta n,$ (35)
$\displaystyle\nabla^{2}\varphi=4\pi\mathrm{e}_{0}\int\delta
n(\mathbf{x},\mathbf{v},t)\mathrm{d}^{3}v.$ (36)

$\delta n$ and $\varphi$ are transformed into wave number vector space by
Fourier transformation

$\displaystyle\delta
n_{\mathbf{k}}(\mathbf{v},t)=\int_{-\infty}^{+\infty}\delta
n(\mathbf{x},\mathbf{v},t)\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{x}}\mathrm{d}^{3}x,$
(37)

and

$\displaystyle\varphi_{\mathbf{k}}(t)=\int_{-\infty}^{+\infty}\varphi(\mathbf{x},t)\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{x}}\mathrm{d}^{3}x.$
(38)

Switching Eq. (35) and (36) into Fourier space by neglecting the perturbed
scaler variables $\Phi$ and $\Psi$

$\displaystyle\frac{\partial\delta n_{\mathbf{k}}(\mathbf{v},t)}{\partial t}$
$\displaystyle+\mathrm{i}\mathbf{k}\cdot\mathbf{v}\delta
n_{\mathbf{k}}(\mathbf{v},t)+\varepsilon\delta
n_{\mathbf{k}}(\mathbf{v},t)-\left[H\mathbf{v}-\frac{\mathrm{e}_{0}}{m}\varphi_{\mathbf{k}}(t)\mathrm{i}\mathbf{k}\right]\cdot\frac{\partial
n_{0}(\mathbf{v})}{\partial\mathbf{v}}=0$ (39)

and

$\displaystyle a^{-2}k^{2}\varphi_{\mathbf{k}}(t)=4\pi\mathrm{e}_{0}\int\delta
n_{\mathbf{k}}(\mathbf{v},t)\mathrm{d}^{3}v$ (40)

Then, Laplace transforms on $\delta n_{\mathbf{k}}(\mathbf{v},t)$ and
$\varphi_{\mathbf{k}}(t)$ about proper time $t$ give

$\displaystyle\delta_{\mathbf{k},s}(\mathbf{v})=\int_{0}^{\infty}\mathrm{e}^{-st}n_{\mathbf{k}}(\mathbf{v},t)\mathrm{d}t$
(41)

and

$\displaystyle\varphi_{\mathbf{k},s}=\int_{0}^{\infty}\mathrm{e}^{-st}\varphi_{\mathbf{k}}(t)\mathrm{d}t,$
(42)

whose inverse transformation is

$\displaystyle\varphi_{\mathbf{k}}(t)=\frac{1}{2\pi\mathrm{i}}\int_{\sigma-\mathrm{i}\infty}^{\sigma+\mathrm{i}\infty}\varphi_{\mathbf{k},s}\mathrm{e}^{st}\mathrm{d}s=\frac{1}{2\pi\mathrm{i}}\sum_{i}\textrm{Res}\left(\varphi_{\mathbf{k},s_{i}}\mathrm{e}^{s_{i}t}\right)$
(43)

with $\textrm{Res}(\cdots)$ denoting the $i$’th residual of integrand
$\varphi_{\mathbf{k},s}\mathrm{e}^{st}$. While the Laplace transform on the
first term $\partial\delta n_{\mathbf{k}}(\mathbf{v},t)/\partial t$ including
a time derivative in Eq. (35) reads

$\displaystyle\int_{0}^{\infty}\frac{\partial\delta
n_{\mathbf{k}}(\mathbf{v},t)}{\partial t}\mathrm{e}^{-st}\mathrm{d}t=s\delta
n_{\mathbf{k},s}(\mathbf{v})+g_{\mathbf{k}}(\mathbf{v}),$ (44)

where $g_{\mathbf{k}}(\mathbf{v})$ represents the Fourier transformation on
$g(\mathbf{x},\mathbf{v},0)$.

Multiply $\mathrm{e}^{-st}$ on both sides of Eq. (39) and then integrate from
$0$ to $\infty$. Using Eq. (44), then Eq. (39) becomes an equation as a
function of parameter $s$:

$\displaystyle s\delta n_{\textbf{k},s}(\textbf{v})-$ $\displaystyle
g_{\textbf{k}}(\textbf{v})+\mathrm{i}\textbf{k}\cdot\textbf{v}\delta
n_{\textbf{k},s}+\varepsilon\delta
n_{\textbf{k},s}(\textbf{v})-H\textbf{v}\cdot\frac{\partial
n_{0}}{\partial\textbf{v}}$
$\displaystyle+\frac{4\pi\mathrm{e}_{0}^{2}}{mk^{2}}\mathrm{i}a^{3}\textbf{k}\cdot\frac{\partial
n_{0}}{\partial\textbf{v}}\int\delta
n_{\textbf{k},s}(\textbf{v}^{\prime})\mathrm{d}^{3}v^{\prime}=0.$ (45)

Collate the equation above, and integrate over all $\mathbf{v}$ to find the
solution

$\displaystyle\left(1-\frac{4\pi
e_{0}^{2}a^{3}}{mk^{2}}\int\frac{1}{\textbf{k}\cdot\textbf{v}-is-i\varepsilon}\textbf{k}\cdot\frac{\partial
n_{0}}{\partial\textbf{v}}\mathrm{d}^{3}v\right)\int\delta
n_{\textbf{k},s}(\textbf{v}^{\prime})\mathrm{d}^{3}v^{\prime}$
$\displaystyle=$
$\displaystyle\int\frac{g_{\textbf{k}}(\textbf{v})}{s+\mathrm{i}\textbf{k}\cdot\textbf{v}+\varepsilon}\mathrm{d}^{3}v+H\int\frac{1}{s+\mathrm{i}\textbf{k}\cdot\textbf{v}+\varepsilon}\textbf{v}\cdot\frac{\partial
n_{0}}{\partial\textbf{v}}\mathrm{d}^{3}v.$ (46)

Multiplying $4\pi\mathrm{e}_{0}/k^{2}$ on both sides of Eq. (46) and applying
the definitions on transverse permittivity
$\varepsilon_{1}(\omega,\mathbf{k},t)$ in Eq. (31) and electric potential
$\varphi_{\mathbf{k},s}$ in Eq. (42), the electric potential is obtained

$\displaystyle\varphi_{\mathbf{k},s}=\frac{\mathrm{i}4\pi\mathrm{e}_{0}a^{2}}{k^{2}\varepsilon_{1}(\mathrm{i}s,\textbf{k})}\left(\int\frac{g_{\textbf{k}}(\textbf{v})\mathrm{d}^{3}v}{\textbf{k}\cdot\textbf{v}-\mathrm{i}s-\mathrm{i}\varepsilon}+\int\frac{H\textbf{v}\cdot\frac{\partial
n_{0}}{\partial\textbf{v}}\mathrm{d}^{3}v}{\textbf{k}\cdot\textbf{v}-\mathrm{i}s-\mathrm{i}\varepsilon}\right).$
(47)

It notes the Laplace transformations above play only on the oscillating
variables, like $\delta n$, but avoid the scalar factor $a$. This is because
the intrinsic oscillating frequency $\Omega_{\mathrm{e}}$ is extremely larger
than the Hubble expansion rate $H$ as discussed at the of this section.
Besides, the exactly analytic expression of $a$ is not going to be involved at
all in this section.

###  3.4 Plasma with Maxwell distribution

The electron’s mass is much lighter than the background particles, like
proton, so it’s reasonable to assume electrons is on the equilibrium which
follow the Maxwell distribution [42]

$\displaystyle
n_{\mathrm{e}}(\mathbf{v})={N_{\mathrm{e}}}{\sqrt{\mathrm{det}g_{ij}}}\left(\frac{m}{2\pi
k_{\mathrm{B}}T}\right)^{\frac{3}{2}}\exp\left(-\frac{g_{ij}P^{i}v^{j}}{2k_{\mathrm{B}}T}\right),$
(48)

here $g_{ij}=a^{2}\delta_{ij}$ denoting the background metric and
$N_{\mathrm{e}}\propto a^{-3}$ denoting the particle number density. Define a
new variable about velocity $\tilde{\mathbf{v}}=a\mathbf{v}$, then the
distribution functions follows

$\displaystyle\tilde{n}_{0}(\tilde{\mathbf{v}})=N_{\mathrm{e}0}\left(\frac{m}{2\pi
k_{\mathrm{B}}T}\right)^{\frac{3}{2}}\exp\left(-\frac{m\tilde{\mathbf{v}}^{2}}{2k_{\mathrm{B}}T}\right),$
(49)

with $N_{\mathrm{e}0}\equiv N_{\mathrm{e}}a^{3}$ representing electron’s
number density at present time. Integrating Eq. (49) over $\tilde{v}_{y}$ and
$\tilde{v}_{z}$ generates the Maxwell distribution along $x$ direction

$\displaystyle\tilde{n}_{0}(\tilde{v}_{x})=N_{\mathrm{e}0}\left(\frac{m}{2\pi
k_{\mathrm{B}}T}\right)^{\frac{1}{2}}\exp\left(-\frac{m\tilde{v}_{x}^{2}}{2k_{\mathrm{B}}T}\right).$
(50)

With the discussions above and setting $\mathbf{k}$ parallel to $x$-axis, the
transverse permittivity could be reexpressed as follow:

$\displaystyle\varepsilon_{1}=1-\frac{4\pi e_{0}^{2}a}{k^{2}m}\int
k\frac{\partial\tilde{n}_{0}}{\partial\tilde{v}_{x}}\frac{\mathrm{d}\tilde{v}_{x}}{k_{\mathrm{p}}\tilde{v}_{x}-\omega-\mathrm{i}\varepsilon},$
(51)

with definition of physical wave number $k_{\mathrm{p}}=k/a$. Generally, the
ratio $\varepsilon/k_{\mathrm{p}}\ll 1$, so it is labeled as $0^{+}$ below for
convenience. Instituting Eq. (50) into Eq. (51), it arrives

$\displaystyle\varepsilon_{1}=1-$ $\displaystyle\frac{4\pi
e_{0}^{2}}{k_{\mathrm{p}}^{2}m}\int\frac{\mathrm{d}\tilde{v}_{x}}{\tilde{v}_{x}-\frac{\omega}{k_{\mathrm{p}}}-\mathrm{i}0}N_{\mathrm{e}0}\left(\frac{m}{2\pi
k_{\mathrm{B}}T}\right)^{\frac{1}{2}}\left(-\frac{m\tilde{v}_{x}}{k_{\mathrm{B}}T}\right)\exp\left(-\frac{m\tilde{v}_{x}^{2}}{2k_{\mathrm{B}}T}\right).$
(52)

It’s convenient to define the following parameters for the subsequent
computations

$\displaystyle\begin{cases}\lambda_{\mathrm{e}}^{-1}=\sqrt{\dfrac{4\pi
e_{0}^{2}N_{\mathrm{e}0}}{k_{\mathrm{B}}T}},\quad
v_{\mathrm{e}}=\sqrt{\dfrac{k_{\mathrm{B}}T}{m}},\\\
z=\dfrac{\tilde{v}_{x}}{\sqrt{2}v_{\mathrm{e}}},\quad\quad
x=\dfrac{\omega}{\sqrt{2}k_{\mathrm{p}}v_{\mathrm{e}}}.\end{cases}$ (53)

The parameters in the equations above used to reflect the plasma
characteristics are $\lambda_{\mathrm{e}}$, which means the Debye radius at
the moment of recombination, and $v_{\mathrm{e}}$, which corresponds to the
average thermal velocity. Thus Eq. (52) reexpresses

$\displaystyle\varepsilon_{1}$
$\displaystyle=1+\frac{1}{k_{\mathrm{p}}^{2}\lambda_{\mathrm{e}}^{2}}\int\frac{\mathrm{d}\tilde{v}_{x}}{\tilde{v}_{x}-\frac{\omega}{k_{\mathrm{p}}}-\mathrm{i}0}\frac{1}{\sqrt{2\pi}}\frac{\tilde{v}_{x}}{v_{\mathrm{e}}}\exp\left(-\frac{\tilde{v}_{x}^{2}}{2v_{\mathrm{e}}^{2}}\right)$
$\displaystyle=1+\frac{1}{k_{\mathrm{p}}^{2}\lambda_{\mathrm{e}}^{2}}\frac{1}{\sqrt{\pi}}\int^{\infty}_{0}\frac{z}{z-x-\mathrm{i}0}e^{-z^{2}}\mathrm{d}z$
$\displaystyle=1+\frac{1}{k_{\mathrm{p}}^{2}\lambda_{\mathrm{e}}^{2}}\frac{1}{\sqrt{\pi}}\int^{\infty}_{0}\frac{(z-x)+x}{z-x-\mathrm{i}0}e^{-z^{2}}\mathrm{d}z$
$\displaystyle=1+\frac{1}{k_{\mathrm{p}}^{2}\lambda_{\mathrm{e}}^{2}}\left[1+\frac{x}{\sqrt{\pi}}\int^{\infty}_{0}\frac{\mathrm{e}^{-z^{2}}}{z-x-\mathrm{i}0}\mathrm{d}z\right]$
$\displaystyle=1+\frac{1}{k_{\mathrm{p}}^{2}\lambda_{\mathrm{e}}^{2}}[1+F(x)].$
(54)

In the last equality, integral of Gaussian type (153) has been used and $F(x)$
is a function as a form of Cauchy’s type

$\displaystyle
F(x)\equiv\frac{x}{\sqrt{\pi}}\int^{\infty}_{0}\frac{\mathrm{e}^{-z^{2}}}{z-x-\mathrm{i}0}\mathrm{d}z.$
(55)

According to the introductions of Cauchy’s type in Appendix A, the
permittivity at high frequency limit $\omega\gg k_{\mathrm{p}}v_{\mathrm{e}}$,
or $x\gg 1$, approximates

$\displaystyle\varepsilon_{1}\approx
1-\frac{\Omega_{\mathrm{e}}^{2}}{\omega^{2}}\left(1+\frac{3k_{\mathrm{p}}^{2}v_{\mathrm{e}}^{2}}{\omega^{2}}\right)+\mathrm{i}\sqrt{\frac{\pi}{2}}\frac{\omega\Omega_{\mathrm{e}}^{2}}{(k_{\mathrm{p}}v_{\mathrm{e}})^{3}}\exp\left(-\frac{\omega^{2}}{2k_{\mathrm{p}}^{2}v_{\mathrm{e}}^{2}}\right),$
(56)

where

$\displaystyle\Omega_{\mathrm{e}}\equiv\frac{v_{\mathrm{e}}}{\lambda_{\mathrm{e}}}=\sqrt{\frac{4\pi
N_{\mathrm{e}0}e_{0}^{2}}{m}}$ (57)

is defined as the Langmuir frequency. In the low frequency limit $x\ll 1$, it
evaluates

$\displaystyle\varepsilon_{1}\approx
1+\left(\frac{\Omega_{\mathrm{e}}}{k_{\mathrm{p}}v_{\mathrm{e}}}\right)^{2}\left[1-\left(\frac{\omega}{k_{\mathrm{p}}v_{\mathrm{e}}}\right)^{2}-\mathrm{i}\sqrt{\frac{\pi}{2}}\frac{\omega}{k_{\mathrm{p}}v_{\mathrm{e}}}\right],\
\ (x\ll 1).$ (58)

Applying parameters in Eq. (53), the second part of electric potential in Eq.
(47) arrives

$\displaystyle\varphi_{\mathbf{k},s}^{(2)}=$
$\displaystyle\frac{H}{k_{\mathrm{p}}}\frac{\mathrm{i}4\pi
e_{0}}{k^{2}a\varepsilon_{1}}\int\frac{\mathrm{d}\tilde{v}}{\tilde{v}-\frac{\mathrm{i}s}{k_{\mathrm{p}}}-\mathrm{i}0}\cdot\frac{\partial\tilde{n}_{0}(\tilde{v}_{x})}{\partial\tilde{v}_{x}}$
$\displaystyle+2\frac{H}{k_{\mathrm{p}}}\frac{\mathrm{i}4\pi
e_{0}}{k^{2}a\varepsilon_{1}}\int\frac{\mathrm{d}^{2}\tilde{v}}{\tilde{v}_{x}-\frac{\mathrm{i}s}{k_{\mathrm{p}}}-\mathrm{i}0}N_{\mathrm{e}0}\left(\frac{m}{2\pi
k_{\mathrm{B}}T}\right)^{\frac{3}{2}}$ $\displaystyle\ \ \ \ \
\times\exp\left(-\frac{m(\tilde{v}_{x}^{2}+\tilde{v}_{y}^{2}+\tilde{v}_{z}^{2})}{2k_{\mathrm{B}}T}\right)\cdot\left(-\frac{m\tilde{v}_{y}^{2}}{k_{\mathrm{B}}T}\right)$
$\displaystyle=$
$\displaystyle\frac{H}{k_{\mathrm{p}}}\frac{v_{\mathrm{e}}m}{\mathrm{i}a^{3}e_{0}\varepsilon_{1}}\frac{4\pi
e_{0}^{2}N_{\mathrm{e}0}}{k_{\mathrm{p}}^{2}k_{\mathrm{B}}T}\left[\sqrt{\frac{2}{\pi}}\int_{0}^{+\infty}\frac{z^{2}\mathrm{e}^{-z^{2}}}{z-x-\mathrm{i}0}\mathrm{d}z\right.$
$\displaystyle\left.\quad+2\int_{0}^{+\infty}\frac{\mathrm{e}^{-z^{2}}}{z-x-\mathrm{i}0}\frac{\mathrm{d}z}{\sqrt{2\pi}}\cdot\int_{0}^{+\infty}z^{\prime
2}\mathrm{e}^{-z^{\prime 2}}\mathrm{d}z^{\prime}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{\mathrm{i}\varepsilon_{1}(\omega,\mathbf{k})}\frac{\sqrt{2}Hv_{\mathrm{e}}m}{k_{\mathrm{p}}a^{3}e_{0}}\frac{1}{k_{\mathrm{p}}^{2}\lambda_{\mathrm{e}}^{2}}\left[\frac{1}{\sqrt{\pi}}\int_{0}^{+\infty}\frac{z^{2}\mathrm{e}^{-z^{2}}}{z-x-\mathrm{i}0}\mathrm{d}z\right.$
$\displaystyle\left.\quad\quad\quad\quad\quad\quad\quad\quad+\frac{1}{\sqrt{\pi}}\int\frac{\mathrm{e}^{-z^{2}}}{z-x-\mathrm{i}0}\mathrm{d}z\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\mathrm{i}\varepsilon_{1}(\omega,\mathbf{k})}\frac{1}{k_{\mathrm{p}}^{3}\lambda_{\mathrm{e}}^{2}\lambda_{H}}\left[1+xF(x)+2F(x)+\frac{F(x)}{x}\right],$
(59)

where

$\displaystyle\lambda_{H}^{-1}\equiv\frac{\sqrt{2}Hv_{\mathrm{e}}m}{k_{\mathrm{p}}a^{3}e_{0}}$
(60)

is a parameter relates to Hubble expansion rate, and Eq. (153) has been used
during the calculations above.

Finally, we get the exact expression of electric potential under FRW metric in
absence of $\Phi$ and $\Psi$

$\displaystyle\varphi_{\mathbf{k},s}=\frac{\mathrm{i}4\pi
e_{0}}{k_{\mathrm{p}}^{2}}\frac{\chi(\mathrm{i}s,\mathbf{k})}{\varepsilon_{1}(\mathrm{i}s,\mathbf{k})}-\frac{\mathrm{i}}{k_{\mathrm{p}}^{3}\lambda_{\mathrm{e}}^{2}\lambda_{H}}\frac{G({\mathrm{i}s}/{\sqrt{2}v_{\mathrm{e}}k_{\mathrm{p}}})}{\varepsilon_{1}(\mathrm{i}s,\mathbf{k})},$
(61)

where

$\displaystyle\chi(\mathrm{i}s,\mathbf{k})=\int\frac{\tilde{g}_{\mathbf{k}}(v_{x})\mathrm{d}v_{x}}{kv_{x}-\mathrm{i}s-\mathrm{i}0}$
(62)

and

$\displaystyle G(x)=1+xF(x)+2F(x)+{F(x)}/{x}.$ (63)

###  3.5 Diffusion damping of primordial electric fields

During the epoch before decoupling, electrons state on a thermal equilibrium
with photons, but the protons have already exited the thermal equilibrium
state with photons for along period, so $T_{\mathrm{e}}\gg T_{\textrm{p}}$.
Thus the high frequency limit means $\omega\gg k_{\mathrm{B}}T_{\mathrm{e}}\gg
k_{\mathrm{B}}T_{\textrm{p}}$. Eq. (56) rewrites by setting $\beta\equiv
k_{\mathrm{p}}v_{\mathrm{e}}/\omega\ll 1$

$\displaystyle\frac{\omega^{2}}{\Omega_{\mathrm{e}}^{2}}=1+3\beta^{2}-\mathrm{i}\sqrt{\frac{2}{\pi}}\beta^{-3}\mathrm{e}^{-1/2\beta^{2}}.$
(64)

The first iteration gives the first order zero point

$\displaystyle\omega=\Omega_{\mathrm{e}}$ (65)

Then the second time gives the solution where the term $\beta^{2}$ has taken
into account:

$\displaystyle\omega=\Omega_{\mathrm{e}}(1+3k_{\mathrm{p}}^{2}\lambda_{\mathrm{e}}^{2})^{1/2}.$
(66)

Finally, inserting Eq. (66) into Eq. (64), we get a complex solution
$\omega=\omega^{\prime}+\mathrm{i}\omega^{\prime\prime}$, with

$\displaystyle\omega^{\prime}=\Omega_{\mathrm{e}}\left(1+\frac{3}{2}k_{\mathrm{p}}^{2}\lambda_{\mathrm{e}}^{2}\right),$
(67)
$\displaystyle\omega^{\prime\prime}=-\sqrt{\frac{\pi}{8}}\frac{\Omega_{\mathrm{e}}}{(k_{\mathrm{p}}\lambda_{\mathrm{e}})^{3}}\exp\left[-\frac{1}{2(k_{\mathrm{p}}\lambda_{\mathrm{e}})^{2}}-\frac{3}{2}\right].$
(68)

Similarly, the zero point at low frequency is

$\displaystyle\omega^{\prime}=k_{\mathrm{p}}v_{\mathrm{e}},\ \ \text{and}\
\omega^{\prime\prime}=-\sqrt{\frac{\pi}{8}}\omega^{\prime}.$ (69)

Thus the root of equation $\varphi_{\mathrm{i}s,\mathbf{k}}=0$ reads
$s=\mathrm{i}\omega^{\prime}-\gamma$ with $\gamma=-\omega^{\prime\prime}$.
Base on Eq. (43), it leads to the expression of electric potential

$\displaystyle\varphi_{\mathbf{k}}(t)=\frac{2e_{0}}{k_{\mathrm{p}}^{2}}\sum_{i}R_{i}\mathrm{e}^{s_{i}t}-\frac{1}{2\pi
k_{\mathrm{p}}^{3}\lambda_{\mathrm{e}}^{2}\lambda_{H}}\sum_{i}R^{\prime}_{i}\mathrm{e}^{s_{i}t}.$
(70)

Obviously, the potential is going to damp out by Landau damping, so the
initial condition before electron’s recombination, like neutrino’s decoupling,
affects almost nothing on cosmic microwave background. Although the potential
may be a large amplitude on large scale, it still damps away as well with time
increasing.

On the other hand, although the electric potential damps out with time
increasing, the particle density $\delta n$ has an oscillating mode. Inserting
Eq. (47) into Eq. (45), we have

$\displaystyle\delta_{\mathbf{k},s}=\frac{1}{s+\mathrm{i}\mathbf{k}\cdot\mathbf{v}}$
$\displaystyle\left\\{g_{\mathbf{k}}(v)+\left[H\mathbf{v}+\frac{4\pi
e_{0}^{2}}{mk^{2}}\left(\frac{\chi(\mathrm{i}s,\mathbf{k})}{\varepsilon_{1}(\mathrm{i}s,\mathbf{k})}+\frac{\varsigma(\mathrm{i}s,\mathbf{k})}{\varepsilon_{1}(\mathrm{i}s,\mathbf{k})}+\frac{C(\mathrm{i}s,\mathbf{k})}{\varepsilon_{1}(\mathrm{i}s,\mathbf{k})}\right)\mathbf{k}\right]\cdot\frac{\partial
n_{0}}{\partial\mathbf{v}}\right\\},$ (71)

where $C(\mathrm{i}s,\mathbf{k})$ is the Laplace and Fourier transform on the
nonhomogeneous part in collision term. Except the singularities at
$\varepsilon=0$, the point $s=-\mathrm{i}\mathbf{k}\cdot\mathbf{v}$ is also a
singularity. So the inverse Laplace transformation on Eq. (71) gives an
oscillating mode without damping

$\displaystyle\delta
n_{\mathbf{k}}(\mathbf{v},t)\propto\mathrm{e}^{-\mathrm{i}\mathbf{k}\cdot\mathbf{v}}.$
(72)

Furthermore, it’s necessary to characterize the attenuating scale of electric
fields by estimating the order of $a^{3/2}\lambda_{\mathrm{e}}H$ or
$a^{-3/2}\Omega_{\mathrm{e}}/H$:

$\displaystyle\frac{H^{2}}{(\Omega_{\mathrm{e}}a^{-3/2})^{2}}=\frac{\varepsilon_{0}mc^{2}a^{3}}{4\pi
e_{0}^{2}N_{\mathrm{e}0}}\frac{H^{2}_{0}}{c^{2}}\frac{H^{2}}{H_{0}^{2}}.$ (73)

This number is covenient to calculate under natural unit system:
$\varepsilon_{0}/4\pi e_{0}^{2}=2.22\times
10^{45}\;\mathrm{kg}^{-1}\cdot\mathrm{m}^{-3}\cdot\mathrm{s}^{2}$,
$mc^{2}=0.51\;\mathrm{MeV}=8.16\times
10^{-14}\;\mathrm{kg}\cdot\mathrm{m}^{2}\cdot\mathrm{s}^{-2}$,
$N_{\mathrm{e}0}=11.244X_{\mathrm{e}}\Omega_{\text{m}}h^{2}\;\mathrm{m}^{-3}$
and $H_{0}/c=3.33\times 10^{-4}h\;\text{Mpc}^{-1}$. At early epoch of the
Universe, the main component is either radiation or matter, so
$H^{2}/H_{0}^{2}=\Omega_{\text{m}}a^{-3}[1+a_{\text{eq}}/a]$. Then Eq. (73)
becomes

$\displaystyle\frac{a^{3}H^{2}}{\Omega_{\mathrm{e}}^{2}}=2.11\times
10^{-20}\frac{\Omega_{\text{m}}h^{2}}{X_{\mathrm{e}}\Omega_{\text{b}}h^{2}}\left(1+\frac{a_{\text{eq}}}{a}\right).$
(74)

Recent Planck data shows $\Omega_{\text{m}}h^{2}=0.1424$ and
$\Omega_{\text{b}}h^{2}=0.022447$ [43, 44]. Saha equation predicts
$X_{\mathrm{e}}\sim 10^{-2}$ at present time. With the discussions above, Eq.
(73) estimates

$\displaystyle{a^{3}H^{2}}/{\Omega_{\mathrm{e}}^{2}}=1.91\times
10^{-19}\left(1+\frac{a_{\text{eq}}}{a}\right)\sim 10^{-18},$ (75)

thus ${a^{3/2}H}/{\Omega_{\mathrm{e}}}\sim 10^{-9}$. While, it also estimates
$a^{3/2}\lambda_{\mathrm{e}}H\sim 1$. This estimation indicates that the
electric field entering Hubble horizon damps out immediately by Landau damping
effect. So the primordial electric fields give not any effects on the power
spectra of cosmic microwave background.

##  4 Soliton of electric fields inside plasma before decoupling

In this section, let’s discuss the nonlinear effect in the primordial plasma.

###  4.1 Basic equations via moment method

Back to the Vlasov equation appeared in Eq. (11) in terms of derivative to
cosmic time $t$ instead of proper time in Sec. 3. Define two variables

$\displaystyle
n_{\mathrm{e}}\equiv\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}f_{\mathrm{e}},$
(76)

and

$\displaystyle
n_{\mathrm{e}}v^{i}\equiv\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{p\hat{p}^{i}}{E}f_{\mathrm{e}}.$
(77)

Multiply $\mathrm{d}^{3}p/(2\pi)^{3}$ on both sides of Eq. (11) and integrate
over all momentum space,

$\displaystyle\quad\frac{\partial}{\partial
t}\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}f_{\mathrm{e}}+\frac{1}{a}\frac{\partial}{\partial
x^{i}}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{p\hat{p}^{i}}{E}f_{\mathrm{e}}$
$\displaystyle-\left[H+\dot{\Phi}\right]\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{\partial
f_{\mathrm{e}}}{\partial
E}\frac{p^{2}}{E}-\frac{1}{a}\frac{\partial\Psi}{\partial
x^{i}}\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{\partial
f_{\mathrm{dm}}}{\partial E}\hat{p}^{i}p$
$\displaystyle+\frac{e_{0}}{a}\partial_{i}\phi\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{p\hat{p}^{i}}{E}f_{\mathrm{e}}=0,$
(78)

The third term on the left hand integrates by part

$\displaystyle\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{p^{2}}{E}\frac{\partial
f_{\mathrm{e}}}{\partial E}$
$\displaystyle=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{\partial
f_{\mathrm{e}}}{\partial p}p$
$\displaystyle={-3}\frac{4\pi}{(2\pi)^{3}}\int_{0}^{+\infty}p^{2}f_{\mathrm{e}}\mathrm{d}p$
$\displaystyle={-3}n_{\mathrm{e}},$ (79)

where the relation $(p/E)(\partial f_{\mathrm{e}}/\partial E)=\partial
f_{\mathrm{e}}/\partial p$ has been used in first equality. Because electrons
during the matter dominated epoch are nonrelativistic particles, $v^{i}$ is
regarded as a first order variable. Inserting Eqs. (76), (77) and (79) into
Eq. (78) and neglecting the terms higher than first order, it reduces to

$\displaystyle\frac{\partial n_{\mathrm{e}}}{\partial
t}+3[H+\dot{\Phi}]n_{\mathrm{e}}+\frac{1}{a}\partial_{i}(n_{\mathrm{e}}v^{i})=0.$
(80)

This is called the zero order moment equation.

Multiplying $\dfrac{\mathrm{d}^{3}p}{(2\pi)^{3}}\dfrac{p\hat{p}^{i}}{E}$ on
both sides of Eq. (11) and integrating over $\mathbf{p}$, this gives the first
order moment equation

$\displaystyle\quad\quad\frac{\partial}{\partial
t}\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{p\hat{p}^{j}}{E}f_{\mathrm{e}}+\frac{1}{a}\frac{\partial}{\partial
x^{i}}\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{p^{2}\hat{p}^{i}\hat{p}^{j}}{E^{2}}f_{\mathrm{e}}$
$\displaystyle-[H+\dot{\Phi}]\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{\partial
f_{\mathrm{e}}}{\partial
E}\frac{p^{3}\hat{p}^{j}}{E}-\frac{1}{a}\frac{\partial\Psi}{\partial
x_{i}}\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{\partial
f_{\mathrm{e}}}{\partial E}\frac{\hat{p}^{i}\hat{p}^{j}p^{2}}{E}$
$\displaystyle\quad\quad\quad+e_{0}\frac{1}{a}\frac{\partial\phi}{\partial
x_{i}}\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{\partial
f_{\mathrm{e}}}{\partial E}\frac{p^{2}\hat{p}^{i}\hat{p}^{j}}{E^{2}}=0.$ (81)

The second term in Eq. (81) reexpresses as

$\displaystyle\int\frac{\mathrm{d}^{3}}{(2\pi)^{3}}\frac{p^{2}\hat{p}^{i}\hat{p}^{j}}{E^{2}}f_{\mathrm{e}}=$
$\displaystyle\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}f_{\mathrm{e}}\left(\frac{p\hat{p}^{i}}{E}-v^{i}\right)\left(\frac{p\hat{p}^{j}}{E}-v^{j}\right)$
$\displaystyle+\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}f_{\mathrm{e}}v^{i}v^{j}=P^{ij}+n_{\mathrm{e}}v^{i}v^{j},$
(82)

where $P^{ij}$ is called pressure tensor which corresponds to the anisotropy.
The third term equals approximately to $-4\partial_{i}(n_{\mathrm{e}}v^{j})/a$
by making use of the relation $(p/E)(\partial f_{\mathrm{e}}/\partial
E)=\partial f_{\mathrm{e}}/\partial p$ and the fourth term equals to
$n_{\mathrm{e}}\partial_{j}\Psi/a$. The last integration is calculated by part
explicitly as previous integrations. This integral is

$\displaystyle\int\frac{\mathrm{d}^{3}}{(2\pi)^{3}}\frac{p^{2}\hat{p}^{i}\hat{p}^{j}}{E^{2}}\frac{\partial
f_{\mathrm{e}}}{\partial E}$
$\displaystyle=\int\mathrm{d}\Omega\hat{p}^{i}\hat{p}^{j}\int_{0}^{\infty}\frac{\partial
f_{\mathrm{e}}}{\partial p}\frac{p}{E}$
$\displaystyle=-\int\mathrm{d}\Omega\hat{p}^{i}\hat{p}^{j}\int_{0}^{\infty}\mathrm{d}p\left(\frac{3p^{2}}{E}-\frac{p4}{E^{3}}\right)$
$\displaystyle=-3\delta^{ij}\frac{n_{\mathrm{e}}}{m},$ (83)

where the additional fact is applied

$\displaystyle\int\mathrm{d}\Omega\hat{p}^{i}\hat{p}^{j}=\delta^{ij}\frac{4\pi}{3}.$
(84)

Therefore, with the discussions above, Eq. (81) becomes

$\displaystyle\frac{\partial(n_{\mathrm{e}}v^{j})}{\partial t}+\frac{1}{a}$
$\displaystyle\partial_{i}(n_{e}v^{i}v^{j})+\frac{1}{a}\partial_{i}P^{ij}+4Hn_{\mathrm{e}}v^{j}$
$\displaystyle+\frac{n_{\mathrm{e}}}{a}\frac{\partial\Psi}{\partial
x^{j}}-\frac{1}{a}\frac{e_{0}}{m}\frac{\partial\phi}{\partial x^{j}}=0.$ (85)

Regroup the first two terms in the equation above

$\displaystyle\quad\frac{\partial(n_{\mathrm{e}}v^{j})}{\partial
t}+\frac{1}{a}\frac{\partial(n_{\mathrm{e}}v^{i}v^{j})}{\partial
x^{i}}=\left[\frac{\partial n_{\mathrm{e}}}{\partial
t}+\frac{1}{a}\frac{\partial(n_{\mathrm{e}}v^{i})}{\partial
t}\right]v^{j}+n_{\mathrm{e}}\left[\frac{\partial v^{j}}{\partial
t}+\frac{1}{a}v^{i}\partial_{i}v^{j}\right].$ (86)

Instituting Eq. (80) and (86) into Eq. (85) and neglecting the terms higher
than first order but retaining the term $v^{j}\partial_{j}v^{i}$, it obtains
the evolutionary equation of velocity

$\displaystyle\dot{v}^{i}+Hv^{i}+\frac{1}{a}v^{j}\partial_{j}v^{i}+\frac{1}{a}\partial_{i}\Psi-\frac{1}{a}\frac{e_{0}}{m}\partial_{i}\phi=0.$
(87)

Note that the terms containing $v^{j}\partial_{j}v^{i}$ is retained since the
following computations will make use of it. Eq. (87) is also able to describe
a proton’s velocity equation just by exchanging $-e_{0}/m$ to $e_{0}/M$, with,
of course, $M$ denoting the mass of photon.

###  4.2 Nonlinear equations for electric fields

Electrons stay on a equilibrium state with photons and they move evidently
much faster than protons, so electrons obey the Maxwell distribution

$\displaystyle
N_{\mathrm{e}}=N_{\mathrm{e}0}\exp\left(\frac{e_{0}\phi}{k_{\mathrm{B}}T_{\mathrm{e}}}\right)$
(88)

with

$\displaystyle N_{\mathrm{e}0}=\int
n_{\mathrm{e}}(\mathbf{x},\mathbf{v},t)\mathrm{d}^{3}v.$ (89)

Thus we have the following equations

$\displaystyle\nabla^{2}\phi=4\pi
e_{0}\left[N_{e0}\exp\left(\dfrac{e_{0}\phi}{k_{\mathrm{B}}T_{\mathrm{e}}}\right)-N_{\mathrm{p}}\right],$
(90a) $\displaystyle\dfrac{\partial N_{\mathrm{p}}}{\partial
t}+\dfrac{1}{a}\dfrac{\partial(N_{\mathrm{p}}v_{\mathrm{p}}^{i})}{\partial
x^{i}}+3[H+\dot{\Phi}]N_{\mathrm{p}}=0,$ (90b) $\displaystyle\dfrac{\partial
v_{\mathrm{p}}^{2}}{\partial
t}+Hv_{\mathrm{p}}^{i}+\dfrac{1}{a}\partial_{i}\Psi+v_{\mathrm{p}}^{i}\partial_{i}v_{\mathrm{p}}^{j}=-\frac{1}{a}\frac{e_{0}}{M}\partial_{i}\phi.$
(90c)

To nondimensionalize the equations, it’s necessary to define such parameters
as below:

$\displaystyle\tilde{\phi}=\frac{e_{0}\phi}{k_{\mathrm{B}}T_{\mathrm{e}}},\
\tilde{N}_{\mathrm{e}0}=a^{3}N_{\mathrm{e}0},\
\tilde{N}_{\mathrm{p}}=a^{3}N_{p},\ \tilde{T}=aT,\
\tilde{N}=\frac{\tilde{N}_{\mathrm{p}}}{\tilde{N}_{\mathrm{e}}},$
$\displaystyle\ \ \tilde{\lambda}_{\mathrm{e}}^{-1}=\sqrt{\frac{4\pi
e_{0}^{2}\tilde{N}_{\mathrm{e}0}}{k_{\mathrm{B}}\tilde{T}_{\mathrm{e}}}},\
\tilde{x}=\frac{x}{\tilde{\lambda}_{\mathrm{e}}},\
\tilde{\Omega}_{\text{p}}=\sqrt{\frac{4\pi
e_{0}^{2}\tilde{N}_{\mathrm{e}0}}{M}},\ \tilde{t}=\tilde{\Omega}_{\text{p}}t,$
$\displaystyle\quad\quad\quad\quad\tilde{u}=\frac{v_{\mathrm{p}}}{\sqrt{k_{\mathrm{B}}\tilde{T}_{\mathrm{e}}/M}}\
\ \text{and}\ \ \mathrm{d}\tilde{\tau}=\mathrm{d}\tilde{t}/a.$ (91)

Multiply $a^{2}/k_{\mathrm{B}}T_{\mathrm{e}}$, Eq. (90a) reduces to

$\displaystyle\frac{\partial^{2}}{\partial
x^{2}}\frac{e_{0}\phi}{k_{\mathrm{B}}T_{\mathrm{e}}}=\frac{4\pi
e_{0}^{2}\tilde{N}_{\mathrm{e}0}}{k_{\mathrm{B}}\tilde{T}_{\mathrm{e}}}\left[\exp\left(\frac{e_{0}\phi}{k_{\mathrm{B}}T_{\mathrm{e}}}\right)-\frac{\tilde{N}_{\mathrm{p}}}{\tilde{N}_{\mathrm{e}0}}\right].$
(92)

Inserting relevant parameters in Eq. (91), it leads to a nondimensional
equation

$\displaystyle\frac{\partial^{2}\tilde{\phi}}{\partial\tilde{x}^{2}}=e^{\tilde{\phi}}-\tilde{N}.$
(93)

Multiply
$a\tilde{\lambda}_{\mathrm{e}}/(\tilde{N}_{\mathrm{e}0}\sqrt{k_{\mathrm{B}}\tilde{T}_{\mathrm{e}}})$
on both sides of Eq. (90b):

$\displaystyle\frac{\tilde{\lambda}_{\mathrm{e}}a}{\sqrt{k_{\mathrm{B}}\tilde{T}_{\mathrm{e}}/M}}\cdot\frac{\partial}{\partial
t}\left(\frac{\tilde{N}_{\mathrm{p}}}{\tilde{N}_{\mathrm{e}0}}\right)+\frac{\tilde{\lambda}_{\mathrm{e}}}{a}\frac{\partial}{\partial
x}\left(\frac{\tilde{N}_{\mathrm{p}}}{\tilde{N}_{\mathrm{e}0}}\cdot\frac{\tilde{v}_{\mathrm{p}}}{\sqrt{{k_{\mathrm{B}}\tilde{T}_{\mathrm{e}}}/{M}}}\right)=0,$
(94)

which leads to

$\displaystyle\frac{\partial\tilde{N}}{\partial\tilde{t}}+\frac{1}{a}\frac{\partial}{\partial\tilde{x}}(\tilde{N}\tilde{u})=0.$
(95)

Similarly, Eq. (90c) becomes

$\displaystyle\frac{\partial\tilde{u}}{\partial\tilde{\tau}}+\frac{\tilde{u}}{a}\frac{\partial\tilde{u}}{\partial\tilde{x}}=-\frac{\partial\tilde{\phi}}{\partial\tilde{x}}.$
(96)

###  4.3 Nonlinear solution

The electrical neutrality condition provides $\tilde{N}=1+\tilde{n}$ with
$\tilde{n}\ll 1$. The linearized equations to Eq. (93), (95) and (96) read

$\displaystyle\frac{\partial\tilde{\phi}^{2}}{\partial\tilde{x}^{2}}=\tilde{\phi}-\tilde{n},$
(97a)
$\displaystyle\frac{\partial\tilde{n}}{\partial\tilde{\tau}}+\frac{1}{a}\frac{\partial\tilde{u}}{\partial\tilde{x}}=0,$
(97b)
$\displaystyle\frac{\partial\tilde{u}}{\partial\tilde{\tau}}+\frac{\partial\tilde{\phi}}{\partial\tilde{x}}=0.$
(97c)

Take derivative of Eq. (97a) to $\tilde{\tau}$ and insert Eq. (97b) into it,

$\displaystyle\frac{\partial^{3}\tilde{\phi}}{\partial\tilde{x}^{2}\partial\tilde{\tau}}=\frac{\partial\tilde{\phi}}{\partial\tilde{\tau}}-\frac{\partial\tilde{n}}{\partial\tilde{\tau}}=\frac{\partial\tilde{\phi}}{\partial\tilde{\tau}}+\frac{1}{a}\frac{\partial\tilde{u}}{\partial\tilde{x}}.$
(98)

Another derivative of Eq. (98) to $\tilde{\tau}$ gives

$\displaystyle\frac{\partial^{4}\tilde{\phi}}{\partial\tilde{x}^{2}\partial\tilde{\tau}^{2}}$
$\displaystyle=\frac{\partial^{2}\tilde{\phi}}{\partial\tilde{\tau}^{2}}+\frac{\partial^{2}\tilde{u}}{\partial\tilde{x}\partial\tilde{\tau}}$
$\displaystyle=\frac{\partial^{2}\tilde{\phi}}{\partial\tilde{\tau}^{2}}-\frac{\partial^{2}\tilde{\phi}}{\partial\tilde{x}^{2}}-\frac{\mathrm{d}a^{-1}}{\mathrm{d}\tilde{\tau}}\frac{\partial\tilde{u}}{\partial\tilde{x}}.$
(99)

Since the Langmuir frequency $\tilde{\Omega}_{\text{p}}$ is much lager than
the Hubble expansion rate $H$, i. e. $\tilde{\Omega}_{\text{p}}\gg H$, as
discussed in Sec. 3.5

$\displaystyle\frac{\mathrm{d}a^{-1}}{\mathrm{d}\tilde{\tau}}=-\frac{1}{a^{2}}\frac{\mathrm{d}a}{\mathrm{d}t}\frac{\mathrm{d}t}{\mathrm{d}\tilde{\tau}}=-\frac{H}{\tilde{\Omega}_{\text{p}}}\ll
1.$ (100)

Thus we finally obtain the nonlinear partial differential equation

$\displaystyle\frac{\partial\tilde{\phi}^{2}}{\partial\tilde{\tau}^{2}}-\frac{1}{a}\frac{\partial\tilde{\phi}^{2}}{\partial\tilde{x}^{2}}=\frac{\partial^{4}\tilde{\phi}}{\partial\tilde{x}^{2}\partial\tilde{\tau}^{2}}.$
(101)

This equation describes a solitary wave meaning its profile would not change
as it propagates. Setting

$\displaystyle\tilde{\phi}\propto\mathrm{e}^{\mathrm{i}\left[\tilde{k}\tilde{x}-\int^{\tilde{\tau}}\tilde{\omega}(\tilde{\tau})\mathrm{d}\tilde{\tau}\right]}$
(102)

with $\tilde{k}=k\tilde{\lambda}_{\mathrm{e}}$ and
$\tilde{\omega}=\omega/\tilde{\Omega}_{\text{p}}$, inserting which into Eq.
(101), we get the dispersion relation

$\displaystyle\omega^{2}=\frac{\tilde{\Omega}_{\text{p}}^{2}}{a}\frac{k^{2}\tilde{\lambda}_{\mathrm{e}}^{2}}{1+k^{2}\tilde{\lambda}_{\mathrm{e}}^{2}}$
(103)

by applying the limit condition
$H\ll(\mathrm{d}\omega/\mathrm{d}t)/\omega\ll\tilde{\Omega}_{\text{p}}$. If
consider only the linear part of Eq. (101) and return to natural unit, we have

$\displaystyle\frac{\partial^{2}\tilde{\phi}}{\partial\tau^{2}}-\frac{k_{\mathrm{B}}\tilde{T}_{\mathrm{e}}}{aM}\frac{\partial^{2}\tilde{\phi}}{\partial
x^{2}}=0.$ (104)

Define the average thermal velocity

$\displaystyle c_{\textrm{s}}=\sqrt{k_{\mathrm{B}}T_{\mathrm{e}}/M},$ (105)

which also stands for the phase velocity
$c_{\textrm{s}}=a^{-1/2}\tilde{\lambda}_{\mathrm{e}}\tilde{\Omega}_{\text{p}}$.
$c_{\textrm{s}}$ is nether the thermal velocity of electrons nor protons,
instead it’s their collective group velocity since the tight coupling with
each other before recombination. So solitary equation (101) propagates with
velocity $c_{\textrm{s}}$ approximately.

Now give further analysis on Eq. (101). Introduce the Mach number which
represents the ratio of actual velocity to phase velocity

$\displaystyle\mathrm{M}_{\mathrm{a}}\equiv\frac{\mathrm{d}x/\mathrm{d}\tau}{a^{1/2}c_{\textrm{s}}}=\frac{\mathrm{d}x/\mathrm{d}\tau}{\tilde{\lambda}_{\mathrm{e}}\tilde{\Omega}_{\text{p}}}\propto
a^{-\frac{1}{2}}.$ (106)

$\tilde{\phi}$, $\tilde{N}$ and $\tilde{u}$ only dependent on a combination
variable

$\displaystyle\tilde{\eta}=\tilde{x}-\int^{\tilde{\tau}}\mathrm{M}_{\mathrm{a}}\mathrm{d}\tilde{\tau}^{\prime}.$
(107)

Then equations in (90) becomes a new set of dynamic equations in terms of
$\eta$:

$\displaystyle\frac{\mathrm{d}^{2}\tilde{\phi}}{\mathrm{d}\tilde{\eta}^{2}}=\mathrm{e}^{\tilde{\phi}}-\tilde{N},$
(108a)
$\displaystyle-\mathrm{M}_{\mathrm{a}}\frac{\mathrm{d}\tilde{N}}{\mathrm{d}\tilde{\eta}}+\frac{1}{a}\frac{\mathrm{d}(\tilde{N}\tilde{u})}{\mathrm{d}\tilde{\eta}}=0,$
(108b)
$\displaystyle-\mathrm{M}_{\mathrm{a}}\frac{\mathrm{d}\tilde{u}}{\mathrm{d}\tilde{\eta}}+\frac{\tilde{u}}{a}\frac{\mathrm{d}\tilde{u}}{\mathrm{d}\tilde{\eta}}=-\frac{\mathrm{d}\tilde{\phi}}{\mathrm{d}\tilde{\eta}},$
(108c)

with boundary condition $\tilde{\phi}\to 0$,
$\dfrac{\mathrm{d}\tilde{\phi}}{\mathrm{d}\tilde{\eta}}\to 0$, $\tilde{u}\to
0$, $\tilde{N}\to 1$ and
$\dfrac{\mathrm{d}\tilde{N}}{\mathrm{d}\tilde{\eta}}\to 0$ when
$|x|\to\infty$, $|\tilde{\eta}|\to\infty$.

Integrate Eq. (108b) over $\tilde{\eta}$ from $\tilde{\eta}$ to $\infty$

$\displaystyle-\mathrm{M}_{\mathrm{a}}\int_{\eta}^{+\infty}\frac{\mathrm{d}\tilde{N}}{\mathrm{d}\tilde{\eta}^{\prime}}\mathrm{d}\tilde{\eta}^{\prime}=\frac{1}{a}\int_{\tilde{\eta}}^{+\infty}\frac{\mathrm{d}(\tilde{N}\tilde{u})}{\mathrm{d}\tilde{\eta}^{\prime}}\mathrm{d}\tilde{\eta}^{\prime}$
(109)

to get the first equation between $\tilde{n}$ and $\tilde{N}$

$\displaystyle\tilde{u}=\mathrm{M}_{\mathrm{a}}a(1-\frac{1}{\tilde{N}}).$
(110)

Integrating on Eq. (108c), we have

$\displaystyle\left(\mathrm{M}_{\mathrm{a}}a^{\frac{1}{2}}-a^{-\frac{1}{2}}\tilde{u}\right)^{2}=\mathrm{M}_{\mathrm{a}}^{2}a-2\tilde{\phi}.$
(111)

Instituting Eq. (110) into the equation above, it arrives

$\displaystyle\tilde{N}=\frac{\mathrm{M}_{\mathrm{a}}a^{\frac{1}{2}}}{\sqrt{(\mathrm{M}_{\mathrm{a}}a^{\frac{1}{2}})^{2}-2\tilde{\phi}}}.$
(112)

Then instituting this equation into Eq. (108a), there is

$\displaystyle\frac{\mathrm{d}^{2}\tilde{\phi}}{\mathrm{d}\tilde{\eta}^{2}}=\mathrm{e}^{\tilde{\phi}}-\frac{\mathrm{M}_{\mathrm{a}}a^{\frac{1}{2}}}{\sqrt{\left(\mathrm{M}_{\mathrm{a}}a^{\frac{1}{2}}\right)^{2}-2\tilde{\phi}}}$
(113)

Next multiply $\dfrac{\mathrm{d}\tilde{\phi}}{\mathrm{d}\tilde{\eta}}$ and
integral over $\tilde{\eta}$

$\displaystyle\frac{1}{2}\left(\frac{\mathrm{d}\tilde{\phi}}{\mathrm{d}\tilde{\eta}}\right)^{2}=\mathrm{e}^{\tilde{\phi}}+\mathrm{M}_{\mathrm{a}}a^{\frac{1}{2}}\sqrt{\left(\mathrm{M}_{\mathrm{a}}a^{\frac{1}{2}}\right)^{2}-2\tilde{\phi}}+\left(\mathrm{M}_{\mathrm{a}}a^{\frac{1}{2}}+1\right).$
(114)

As discussed previous, $\mathrm{M}_{\mathrm{a}}a^{\frac{1}{2}}$ is number or
variable that deviates slightly from the unit, so define
$\varepsilon\equiv\mathrm{M}_{\mathrm{a}}a^{\frac{1}{2}}-1\ll 1$. Thus Eq.
(114) rewrites

$\displaystyle\left(\frac{\mathrm{d}\tilde{\phi}}{\mathrm{d}\tilde{\eta}}\right)^{2}=\frac{2}{3}\tilde{\phi}^{2}(3\varepsilon-\tilde{\phi}).$
(115)

The solution to Eq. (115) is finally expressed as

$\displaystyle\tilde{\phi}$
$\displaystyle=3\varepsilon\;\mathrm{sech}^{2}\left[\left(\frac{\varepsilon}{2}\right)^{\frac{1}{2}}\left(\tilde{x}-\int^{\tilde{\tau}}\mathrm{M}_{\mathrm{a}}\mathrm{d}\tilde{\tau}^{\prime}\right)\right]$
$\displaystyle\simeq
3\varepsilon\;\mathrm{sech}^{2}\left[\left(\frac{\varepsilon}{2}\right)^{\frac{1}{2}}\left(\tilde{x}-\int^{\tilde{\tau}}a^{-1/2}\mathrm{d}\tilde{\tau}^{\prime}\right)\right],$
(116)

where $\mathrm{sech}(x)$ is the hyperbolic secant function.

###  4.4 Korteweg-de Vires equation of density perturbation

Since $\varepsilon\equiv\mathrm{M}_{\mathrm{a}}a^{\frac{1}{2}}-1\ll 1$, expand
density ratio, electric potential and velocity as series of $\varepsilon$:

$\displaystyle\tilde{N}=1+\varepsilon N^{(1)}+\varepsilon^{2}N^{(2)}+\cdots,$
(117a)
$\displaystyle\tilde{\phi}=\varepsilon\phi^{(1)}+\varepsilon^{2}\phi^{(2)}+\cdots,$
(117b)
$\displaystyle\tilde{u}=\varepsilon\tilde{u}^{(1)}+\varepsilon^{2}\tilde{u}^{(2)}+\cdots.$
(117c)

Define new quantities appeared in Eq. (116)

$\displaystyle\xi\equiv\varepsilon^{1/2}\left(\tilde{x}-\int^{\tilde{\tau}}\mathrm{M}_{\mathrm{a}}\mathrm{d}\tilde{\tau}^{\prime}\right)=\varepsilon^{1/2}\left(\tilde{x}-\int^{\tilde{\tau}}a^{-1/2}\mathrm{d}\tilde{\tau}^{\prime}\right)$
(118)

and

$\displaystyle\eta=\varepsilon^{3/2}\int^{\tilde{\tau}}a^{-1/2}\mathrm{d}\tilde{\tau}^{\prime}.$
(119)

Change the coordinate from $(\tilde{x},\tilde{\tau})$ to $(\xi,\eta)$

$\displaystyle\frac{\partial}{\partial\tilde{x}}=\varepsilon^{1/2}\frac{\partial}{\partial\xi},\
\frac{\partial}{\partial\tilde{\tau}}=a^{-1/2}\varepsilon^{3/2}\frac{\partial}{\partial\eta}-a^{-1/2}\varepsilon^{1/2}\frac{\partial}{\partial\xi}.$
(120)

Thus Eq. (108) becomes

$\displaystyle\left\\{\begin{array}[]{l}\varepsilon\dfrac{\partial^{2}\tilde{\phi}^{\prime}}{\partial\xi^{2}}=\mathrm{e}^{\tilde{\phi}}-\tilde{N},\\\
a^{-1/2}\varepsilon\dfrac{\partial\tilde{N}}{\partial\eta}-a^{-1/2}\dfrac{\partial\tilde{N}}{\partial\xi}+\dfrac{1}{a}\dfrac{\partial(\tilde{N}\tilde{u})}{\partial\xi}=0,\\\
a^{-1/2}\varepsilon\dfrac{\partial\tilde{u}}{\partial\eta}-a^{-1/2}\dfrac{\partial\tilde{u}}{\partial\xi}+\dfrac{\tilde{u}}{a}\dfrac{\partial\tilde{u}}{\partial
xi}=-\dfrac{\partial\tilde{\phi}}{\partial\xi}.\end{array}\right.$ (124)

Then the solution to the first order of Eq. (124) reads

$\displaystyle\phi^{(1)}=n^{(1)}=a^{-\frac{1}{2}}n^{(1)}.$ (125)

The equations to the order of $\varepsilon^{2}$ reads

$\displaystyle\frac{\partial^{2}\phi^{(1)}}{\partial\xi^{2}}=\phi^{(2)}+\frac{1}{2}\left[\phi^{(1)}\right]^{2}-N^{(2)},$
(126a)
$\displaystyle-a^{-1/2}\frac{n^{(2)}}{\partial\xi}+a^{-1/2}\frac{n^{(1)}}{\partial\eta}+\dfrac{1}{a}\dfrac{\partial(N^{(1)}u^{(1)})}{\partial\xi}+\dfrac{1}{a}\dfrac{u^{(2)}}{\partial\xi}=0,$
(126b)
$\displaystyle-a^{-1/2}\frac{u^{(2)}}{\partial\xi}+a^{-1/2}\frac{u^{(1)}}{\partial\eta}+\dfrac{u^{(1)}}{a}\dfrac{\partial
u^{(1)}}{\partial\xi}=-\frac{\partial\phi^{(2)}}{\partial\xi}.$ (126c)

Eliminate $\phi^{(2)}$ by inserting Eq. (126a) to Eq. (126c) and replace
$u^{(2)}$ and $n^{(2)}$ by applying Eq. (126b), then we have the first order
of density perturbation $n^{(1)}$

$\displaystyle\frac{\partial n^{(1)}}{\partial\eta}+n^{(1)}\frac{\partial
n^{(1)}}{\partial\xi}+\frac{a^{1/2}}{2}\frac{\partial^{3}n^{(1)}}{\partial\xi^{3}}=0.$
(127)

For convenience, it’s necessary to introduce variables
$n^{(1)}=(a^{1/2}/2)^{1/3}\theta$ and $\xi=(a^{1/2}/2)^{1/3}\rho$, inserting
which in Eq. (127),

$\displaystyle\frac{\partial\theta}{\partial\eta}+\theta\frac{\partial\theta}{\partial\rho}+\frac{\partial\theta}{\partial\rho^{3}}=0.$
(128)

This is the famous Korteweg-de Vries (KdV) equation.

The profile of unimodal wave is the same as Eq. (116) plotted in Figs. 1 and
2, implying a stationary density fluctuation propagates without dissipation.
Figs. 1 and 2 illustrate both the electric and energy density fluctuations
since they comes from the pertubation of particles’ number density. This
perturbation could be considered as the initial condition of the generation
and evolution of galaxies since it would not disappear during and after photon
recombination due to the gravitational attraction in absence of optic
dissipation. Besides the amplitude of such a fluctuation should not be
sufficiently large because of propotion to $\varepsilon\ll 1$ but enough to
regard it as an appropriate starting collapsing condition [3]. It also
illustrates the width of solitary wave’s profile has a close value of
galaxies’ diameter.

###  4.5 Numerical Simulation

As the calculations in Sec. 3, the electric characteristic length evaluates
$\tilde{\lambda}_{\mathrm{e}}^{-1}=7.637\times 10^{-5}\;\text{Mpc}^{-1}$. If
the major contributions are radiation and matters, the conformal time is
briefly expressed

$\displaystyle\tau=\frac{2}{\sqrt{\Omega_{\text{m}}H_{0}^{2}}}\Big{[}\sqrt{a+a_{\text{eq}}}-\sqrt{a_{\text{eq}}}\Big{]}.$
(129)

Applying the recent Planck data, relevant cosmic data are best fitted as
$a_{*}=1/1090$, $a_{\text{eq}}=1/3400$, where $a_{\text{eq}}$ and $a_{*}$
denote the scale factor at the present of radiation and matter equivalent and
recombination. We have $c\tau_{*}=280.8\;\text{Mpc}$ and
$c\tau_{\text{eq}}=112.9\;\text{Mpc}$ and $\tau_{\text{eq}}/\tau_{*}=0.4021$.
Then get the expression for scale factor as a function of conformal time via
the inverse transform on Eq. (129)

$\displaystyle a$
$\displaystyle=\left[\dfrac{\tau\sqrt{\Omega_{\text{m}}H_{0}^{2}}}{2}+\sqrt{a_{\text{eq}}}\right]^{2}-a_{\text{eq}}$
$\displaystyle=\left[\left(\sqrt{a+a_{\text{eq}}}-\sqrt{a_{\text{eq}}}\right)\eta+\sqrt{a_{\text{eq}}}\right]-a_{\text{eq}}$
(130)

with $\eta=\tau/\tau_{*}$. Thus, the formula in hyperbolic secant function
becomes

$\displaystyle\tilde{x}-\int_{\tilde{\tau}_{\text{eq}}}^{\tilde{\tau}}\mathrm{M}_{\mathrm{a}}\mathrm{d}\tilde{\tau}=\tilde{x}-\int_{\tilde{\tau}_{\text{eq}}}^{\tilde{\tau}}(1+\varepsilon)c_{\textrm{s}}\mathrm{d}\tilde{\tau}$
$\displaystyle=$
$\displaystyle\tilde{\lambda}_{\mathrm{e}}^{-1}\left[x-(1+\varepsilon)c\tau_{*}\sqrt{\frac{k_{\mathrm{B}}T_{0}}{Mc^{2}}}\int_{\tau_{\text{eq}}}^{\tau}a^{-1/2}\mathrm{d}(\tau/\tau_{*})\right]$
$\displaystyle=$
$\displaystyle\tilde{\lambda}_{\mathrm{e}}^{-1}\left[x-(1+\varepsilon)c\tau_{*}\sqrt{\frac{k_{\mathrm{B}}T_{0}}{Mc^{2}}}\int_{\tau_{\text{eq}}/\tau_{*}}^{\eta}a^{-1/2}(\eta)\mathrm{d}\eta\right].$
(131)

Fig. 1 and Fig. 2 plot the solitary evolutions with $\varepsilon=0.01$ and
$\varepsilon=0.005$. The differences between them appear only the amplitude
and width of the profile. It’s notable such electric solitary waves vanish
gradually while the electrons coupled with protons, but the density
perturbation persists still during the process because gravitation still works
but optical pressure dissipates. The width of solitons’ profile are around
$D\sim 0.01\;\text{Mpc}$, which exactly approximate to the galaxies’
diameters. So this may indicate the primordial nonlinear density perturbations
after recombination promote the formation of galaxies.

Figure 1: The profiles of solitary waves in Eq. (116) with $\varepsilon=0.01$
at the moments $\tau/\tau_{*}=$0.5, 0.6, 0.7, 0.8, 0.9 and 1.0 from left to
right respectively. Figure 2: The profiles of solitary waves with
$\varepsilon=0.005$.

Before the ending of this section, point needs to be explained. In Sec. 3 the
primordial electric fields damp away by Landau damping, but in this section,
the fields propagate as solitons instead of attenuation. Are they paradoxical?
Of course not. The former propagating as oscillating waves, or transverse wave
precisely, these waves absorb by plasma during propagating. While the latter
being a solitary wave, or longitudinal wave precisely, these waves remain the
stability during their propagations, which is an intrinsic property inside the
plasma matters.

##  5 Acoustic Oscillations

Before the recombination epoch, the electrons and photons coupled tight with
each other caused by Compton scattering, so the free path is much shorter than
the Hubble horizon.

The unintegrated equations for proton and electron read

$\displaystyle\frac{\mathrm{d}f_{\mathrm{e}}(\textbf{x},\textbf{q},t)}{\mathrm{d}t}=\langle
C_{\mathrm{e}\mathrm{p}}\rangle_{QQ^{\prime}q^{\prime}}+\langle
C_{\mathrm{e}\gamma}\rangle_{pp^{\prime}q^{\prime}},$ (132)
$\displaystyle\frac{\mathrm{d}f_{\mathrm{p}}(\textbf{x},\textbf{Q},t)}{\mathrm{d}t}=\langle
C_{\mathrm{e}\mathrm{p}}\rangle_{qq^{\prime}Q^{\prime}}+\langle
C_{\mathrm{p}\gamma}\rangle_{pp^{\prime}Q^{\prime}},$ (133)

where $C_{\mathrm{e}\mathrm{p}}$ denotes the Coulomb scattering,
$C_{\mathrm{e}\gamma}$ represents the Compton scattering between electron and
photon which is inversely proportional to $m_{\mathrm{e}}^{2}$, and
$\langle\cdots\rangle_{QQ^{\prime}q^{\prime}}$ means the integral over
momentum ${\bf Q}$, ${\bf Q},$ and ${\bf q}^{\prime}$. Multiplying
$\dfrac{\mathrm{d}^{3}p}{(2\pi)^{2}}\dfrac{m_{\mathrm{e}}p\hat{p}^{i}}{E}$ on
both sides of Eq. (132) and
$\dfrac{\mathrm{d}^{3}p}{(2\pi)^{2}}\dfrac{m_{\mathrm{p}}p\hat{p}^{i}}{E}$ on
both sides of Eq. (133), and integrating on momentum ${\bf q}$, it finally
arrives at the baryonic velocity equation in Fourier space:

$\displaystyle{v}_{\mathrm{b}}^{\prime}+Hv_{b}^{\prime}+\mathrm{i}k\Psi-\mathrm{i}\frac{e_{0}}{m_{\mathrm{e}}}k\phi=\eta^{\prime}\frac{4\rho_{\mathrm{r}}}{3\rho_{\mathrm{b}}}\big{[}3\mathrm{i}{\Theta}_{1}+v_{\mathrm{b}}\big{]},$
(134)

where $v_{\mathrm{b}}$ denotes the baryonic velocity, $\tilde{\Theta}_{1}$
denotes the dipole moment of perturbed photon’s temperature, $\eta$ means the
optic depth and prime ′ is the derivative to conformal time $\tau$. The
Boltamann equations of photon for first two momentum give

$\displaystyle\Theta_{0}^{\prime}+k\Theta_{1}$ $\displaystyle=-\Phi^{\prime},$
(135) $\displaystyle\Theta_{1}^{\prime}-\frac{k}{3}\Theta_{0}$
$\displaystyle=\frac{k\Psi}{3}+\mathrm{i}\left[\Theta_{1}-\frac{\mathrm{i}v_{\mathrm{b}}}{3}\right].$
(136)

Insert Eq. (134) into itself and remain until the first order about baryonic
velocity,

$\displaystyle
v_{\mathrm{b}}=-3\mathrm{i}\Theta_{1}+\frac{R}{\eta^{\prime}}\left[v_{\mathrm{b}}^{\prime}+\frac{a^{\prime}}{a}\nu_{b}+\mathrm{i}k\Psi-\mathrm{i}\frac{e_{0}}{m_{e}}k\phi\right]$
(137)

with $R\equiv 3\rho_{\mathrm{b}}^{(0)}/4\rho_{\mathrm{r}}^{(0)}$. Then Eq.
(136) becomes by inserting Eq. (137)

$\displaystyle\Theta_{1}-\frac{k}{3}=\frac{k\Psi}{3}+R\left(-\Theta_{1}^{\prime}-\frac{a^{\prime}}{a}\Theta_{1}+\frac{1}{3}k\Psi-\frac{1}{3}\frac{e_{0}}{m_{\mathrm{e}}}k\phi\right).$
(138)

It could be rewritten as

$\displaystyle\Theta_{1}^{\prime}+\frac{a^{\prime}}{a}\frac{R}{1+R}\Theta_{1}-\frac{k}{3(1+R)}\Theta_{0}=\frac{k}{3}\Psi-\frac{k}{3}\frac{e_{0}}{m_{\mathrm{e}}}\frac{\phi}{1+R}.$
(139)

Eliminating $\Theta_{1}$ by inserting Eq. (135) into Eq. (139), we finally get
the photons coupled with gravity containing the electric fields

$\displaystyle\left[\frac{\mathrm{d}^{2}}{\mathrm{d}\tau^{2}}+\frac{R^{\prime}}{1+R}\frac{\mathrm{d}}{\mathrm{d}\tau}+k^{2}c_{\mathrm{s}}^{2}\right]\left(\Theta_{0}+\Phi\right)=-\frac{k^{2}}{3}\Psi+\frac{k^{2}}{3(1+R)}(\Phi+\bar{\phi}),$
(140)

with

$\displaystyle c_{\mathrm{s}}=\sqrt{\frac{1}{3(1+R)}}$ (141)

and

$\displaystyle\bar{\phi}=\frac{e_{0}\phi}{m_{\mathrm{e}}c^{2}}.$ (142)

According to the Green’s function theory, the photons’ temperature coupled
with gravity can be constructed if neglecting the damping term and considering
only the oscillating term

$\displaystyle\Theta_{0}+\Phi=c_{1}s_{1}(\tau)+c_{2}s_{2}(\tau)$
$\displaystyle\quad+\frac{k^{2}}{3}\int_{0}^{\tau}[\Phi+\bar{\phi}-\Psi](\tau^{\prime})\frac{s_{1}(\tau^{\prime})s_{2}(\tau)-s_{1}(\tau)s_{2}(\tau^{\prime})}{s_{1}(\tau^{\prime})s^{\prime}_{2}(\tau^{\prime})-s^{\prime}_{1}(\tau^{\prime})s_{2}(\tau^{\prime})}\mathrm{d}\tau^{\prime}$
$\displaystyle=[\Theta_{0}(0)+\Phi(0)]\cos[kr_{\text{s}}(\tau)]$
$\displaystyle\quad+\frac{k}{\sqrt{3}}\int_{0}^{\tau}[\Phi+\bar{\phi}-\Psi](\tau^{\prime})\sin[k(r_{\text{s}}(\tau)-r_{\text{s}}(\tau^{\prime}))]\mathrm{d}\tau^{\prime}.$
(143)

where $s_{1}(k,\tau)=\sin[kr_{\text{s}}(\tau)]\ \text{and}\
s_{2}(k,\tau)=\cos[kr_{\text{s}}(\tau)]$ are the two homogeneous solutions,
and the sound horizon is defined as

$\displaystyle
r_{\text{s}}(\tau)=\int_{0}^{\tau}\mathrm{d}\tau^{\prime}c_{\text{s}}(\tau^{\prime}).$
(144)

Compared with the condition in absence of electric field, Eq. (143) does not
show any differences except a phase in terms of a sine integral

$\displaystyle\Delta\varphi=\frac{k}{\sqrt{3}}\int_{0}^{\tau}\bar{\phi}(\tau^{\prime})\sin[k(r_{\text{s}}(\tau)-r_{\text{s}}(\tau^{\prime}))]\mathrm{d}\tau^{\prime}.$
(145)

As discussed in Sec. 3 and Sec. 4, the primordial electric fields damp out
since Landau damping, so no need to calculate the influence on the acoustic
oscillations. However the intrinsic electric fields in plasma propagates as
solitary waves, so it needs to evaluate the amplitude of this contribution.
$\tilde{\phi}$ has been achieved in Eq. (116), the temperature before
recombination approximates $k_{\mathrm{B}}T\sim 0.1\text{eV}$, the energy of
rest electron $m_{\mathrm{e}}c^{2}\sim 0.5\text{MeV}$ and the amplitude of
soliton $\varepsilon\sim 0.01$. Thus the additional term’s amplitude
approximates $\bar{\phi}\sim 10^{-8}$. While the gravitational perturbation is
observed at order $\Phi,\ \Psi\sim 10^{-5}$. So the electric fields merely
increase the amplitude of acoustic oscillations by a order of $10^{-3}$, which
means the CMB’s power spectra almost change nothing at acoustic peak. Besides,
the electrical acoustic wave, appeared in Eq. (105), propagates much slower
than the gravitational acoustic wave, appeared in Eq. (141), the ratio between
electrical acoustic horizon and gravitational acoustic horizon is

$\displaystyle
r=\frac{r_{\mathrm{s}}^{(\mathrm{e})}}{r_{\mathrm{s}}^{(\text{g})}}=\frac{\int_{0}^{\tau}c_{\mathrm{s}}^{(\mathrm{e})}\mathrm{d}\tau^{\prime}\;}{\int_{0}^{\tau}c_{\mathrm{s}}^{(\text{g})}\mathrm{d}\tau^{\prime}\;}\sim\sqrt{\frac{k_{\mathrm{B}}T_{0}}{Mc^{2}}\cdot\frac{\Omega_{\mathrm{b}}h^{2}}{\Omega_{\mathrm{r}}h^{2}}}\sim
10^{-4}.$ (146)

This means the electrical acoustic peaks in CMB’s power spectra locate at
extremely large $\ell$’s, which roughly corresponds to the scale of galaxies
or galaxy cluster, consistent to the discussions in Sec. 4.

Figure 3: The monopole at recombination in a standard CDM model with
$\varepsilon=0.01$ and $\varepsilon=100$.

The simulation on Eq. (140) needs a further treatment on solitary solution in
(116). Multiplying $\tau_{*}$ on both sides of Eq. (140) and defining
$\eta=\tau/\tau_{*}$, then it rewrites

$\displaystyle\left[\frac{\mathrm{d}^{2}}{\mathrm{d}\eta^{2}}+\frac{\mathrm{d}R/\mathrm{d}\eta}{1+R}\frac{\mathrm{d}}{\mathrm{d}\eta}+(\tau_{*}/\tau_{0})^{2}(k\eta_{0})^{2}c_{\mathrm{s}}^{2}\right]\left(\Theta_{0}+\Phi\right)=$
$\displaystyle\quad\quad-\left(\frac{\tau_{*}}{\tau_{0}}\right)^{2}\frac{(k\tau_{0})^{2}}{3}\Psi+\left(\frac{\tau_{*}}{\tau_{0}}\right)^{2}\frac{(k\tau_{0})^{2}}{3(1+R)}(\Phi+\bar{\phi}),$
(147)

here $\tau_{0}=14161.5\,\text{Mpc}$ denoting the conformal time at present
time. Based on the Fourier transform

$\displaystyle\mathcal{F}[\mathrm{sech}(x)]=k\pi\,\mathrm{scsh}\left(\dfrac{k\pi}{2}\right),$
(148)

the transformation on $e_{0}\tilde{\phi}/mc^{2}$ reads

$\displaystyle\mathcal{F}[\bar{\phi}]=$
$\displaystyle\frac{k_{\mathrm{B}}T_{0}}{mc^{2}a}\frac{3(2\varepsilon)^{1/2}\tilde{\lambda}_{\mathrm{e}}}{c\tau_{0}}$
$\displaystyle\times\exp\left(\mathrm{i}\sqrt{\frac{\varepsilon}{2}}\frac{c\tau_{*}}{\tilde{\lambda}_{\mathrm{e}}}\sqrt{\frac{k_{\mathrm{B}}T_{0}}{Mc^{2}}}\int_{0}^{\eta}a^{-\frac{1}{2}}\mathrm{d}\eta^{\prime}\right)$
$\displaystyle\times\pi
kc\tau_{0}\,\mathrm{scsh}\left(\sqrt{\frac{2}{\varepsilon}}\frac{\pi\tilde{\lambda}_{\mathrm{e}}}{2c\tau_{0}}kc\tau_{0}\right),$
(149)

where $\mathrm{scsh}(x)$ is the hyperbolic cosecant function. The Bardeen
potentials follow the convenient fits [45]:

$\displaystyle\Phi(k,y)=\bar{\Phi}\left\\{[1-T(k)]\exp[-0.11(ky/k_{\text{eq}})^{1.6}]+T(k)\right\\},$
(150)
$\displaystyle\Psi(k,y)=\bar{\Psi}\left\\{[1-T(k)]\exp[-0.097(ky/k_{\text{eq}})^{1.6}]+T(k)\right\\},$
(151)

where $T(k)$ is the BBKS transfer function and $y\equiv a/a_{\text{eq}}$.
Inserting the relevant equations into Eq. (147), then the numerical results
are simulated in Fig. 3. The black solid line represents the exact solution
with $\varepsilon=0.01$ which almost totally coincides with the line of
$\varepsilon=0$. While, the red dashed line represents the condition with
$\varepsilon=100$. This condition goes against the assumption $\varepsilon\ll
1$, but it has already illustrated that the electric potential contributes a
phase on the CMB’s power spectrum.

This section could be included in a brief statement: the CMB’s power spectra
could merely distinguish the contributions from primordial electric fields.

##  6 Conclusions and further discussions

In this paper, three problems are discussed: the damping of primordial
electric fields, electric solitons and their effects on acoustic oscillations.
As calculated in relevant cosmological monographs [39], the effects on CMB
from electromagnetic fields are often ignored, although, in plasma, such
effects always play extremely significant roles.

First, the evolutions of primordial electric fields are semi-precisely
calculated, but the results are similarly with the ones in plasma physics,
which show the primordial fields dissipate out no matter at sub-horizon scale
or at super-horizon scale by Landau damping effect, especially the fields
having entered the electric acoustic horizon. It means the effects of
neutrino’s recombination don’t need take into account since the initial
perturbations have already damped out till the electron’s recombination. In a
word, there is no need to consider the electromagnetic initial conditions but
just only consider the gravitational initial conditions from inflation.
Second, I proof that the electric fields propagate as solitary waves instead
of oscillating waves and their speed is much slower than baryonic acoustic
oscillations. And third, it illustrates the electric solitons merely affect
the shape of CMB’s power spectra. However, the protonic density fluctuation
also shows a form of solitary wave described by a KdV equation. Since proton
has a much large mass than electron dose, such a nonlinear perturbation grows
continuously in the presence only of the gravitation, so it’s regarded as the
initial evolutionary condition of galaxies.

This paper illustrates the origin of electromagnetic fields after
recombination, and offers an effective method to compute the electric
influences on CMB’s power spectra. In Eq. (8), the mechanic equation
containing only the electric fields is obtained by setting the component
$\mu=0$. Similarly, the equation including the electric and magnetic fields
could be obtained by setting $\mu=i$. It’s not hard to imagine the Vlasov
equation of the second moment will introduce the higher order tensors, like
magnetic fields, anisotropic tensor (or pressure tensor) and tensor
perturbations. Proper cutoff on higher moment will simplify the calculations.
But the computations will be quite complex and relevant results must be
attractive.

## Acknowledgment

This project is supported by the National Natural Science Foundation of China
(Grants No. 11864030) and Scientific Research Funding Project for Introduced
High Level Talents of IMNU (Grants No. 2020YJRC001). I also thank my
undergraduate students Jian-Kia Yin, Jia-Xin Ma, Jia-Xian Liu and Lu-Jie Shi
help me input the formulas.

##  Appendix A Integral of Cauchy’s type

$F(x)$ defined in Eq. (55) could analytically calculated under two special
conditions. The first is high frequency limit, i. e. $x\gg 1$, then it
approximates

$\displaystyle F(x)$
$\displaystyle=\frac{x}{\sqrt{\pi}}\int^{+\infty}_{-\infty}\frac{\mathrm{e}^{-z^{2}}}{z-x-\mathrm{i}0}\mathrm{d}z$
$\displaystyle=\mathrm{P.\;V.}\frac{x}{\sqrt{\pi}}\int^{+\infty}_{-\infty}\frac{e^{-z^{2}}}{z-x}\mathrm{d}z+\frac{x}{\sqrt{\pi}}\mathrm{i}\pi\mathrm{e}^{-x^{2}}$
$\displaystyle=\mathrm{P.\;V.}\frac{1}{\sqrt{\pi}}\int^{+\infty}_{-\infty}\frac{e^{-z^{2}}}{\frac{z}{x}-1}\mathrm{d}z+\mathrm{i}\sqrt{\pi}x\mathrm{e}^{-x^{2}}$
$\displaystyle\simeq-\mathrm{P.\;V.}\frac{1}{\sqrt{\pi}}\int^{+\infty}_{-\infty}\mathrm{d}z\;\mathrm{e}^{-z^{2}}\sum_{n=0}^{\infty}\left(\frac{z}{x}\right)^{n}+\mathrm{i}\sqrt{\pi}x\mathrm{e}^{-x^{2}}$
$\displaystyle\simeq-1-\frac{1}{2x^{2}}-\frac{3}{4x^{4}}+\mathrm{i}\sqrt{\pi}x\mathrm{e}^{-x^{2}},$
(152)

where $\mathrm{P.\;V.}$ means the Cauchy’s principal value [46], and it has
been employed two Gaussian integrals [47]:

$\displaystyle\int_{-\infty}^{+\infty}x^{2n}\mathrm{e}^{-px^{2}}$
$\displaystyle\mathrm{d}x=\frac{(2n-1)!!}{(2p)^{n}}\sqrt{\frac{\pi}{p}},$
$\displaystyle(p>0,\quad n=0,\;1,\;\cdots)$ (153)

and

$\displaystyle\int_{-\infty}^{+\infty}x^{2n+1}\mathrm{e}^{-px^{2}}\mathrm{d}x=0,\
(p>1,\quad n=\cdots,\;-1,\;0,\;1,\;\cdots).$ (154)

The low frequency limit condition is evaluated as

$\displaystyle F(x)$
$\displaystyle=\frac{x}{\sqrt{\pi}}\int_{-\infty}^{+\infty}\frac{\mathrm{e}^{-(z-x)^{2}}}{z-\mathrm{i}0}\mathrm{d}z$
$\displaystyle=\mathrm{P.\;V.}\frac{x}{\sqrt{\pi}}\int_{-\infty}^{+\infty}\mathrm{d}z\;\frac{\mathrm{e}^{-z^{2}}}{z}\left[1-2zx+\frac{1}{2!}(4z^{2}-2)x^{2}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\left.-\frac{1}{3!}(8z^{3}+12z)x^{3}+\cdots\right]+\mathrm{i}\sqrt{\pi}x$
$\displaystyle\simeq-2x^{2}+\frac{4}{3}x^{4}+\mathrm{i}\sqrt{\pi}x,$ (155)

where, of course, Eq. (153) and (154) have also been applied.

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