Z protons, N neutrons and Z electrons. For the elastic neutrino atom
scattering, the low-energy neutrinos are scattering off both atomic electrons
and the nucleus. Considering the effective low-energy CC and NC interactions,
the effective Hamiltonian in current-current interaction form can be written
as

$\displaystyle\mathcal{H}_{I}^{\mathrm{atom}}$
$\displaystyle=\mathcal{H}^{\mathrm{electron}}_{I}+\mathcal{H}^{\mathrm{nucleon}}_{I}=\frac{G_{F}}{\sqrt{2}}\,\left(j_{\mu}\,\mathcal{J}^{\mu}_{\mathrm{electron}}+j_{\mu}\,\mathcal{J}^{\mu}_{\mathrm{nucleon}}\right),$
(4.26)

where $\mathcal{J}^{\mu}_{\mathrm{nucleon}}$ denote the hadronic current for
nucleus, $j^{\mu}$ and $\mathcal{J}^{\mu}_{\mathrm{electron}}$ are the lepton
currents for neutrino and electron respectively. According to the weak
interaction theory, the lepton current for neutrino and electron can be
written as

$\displaystyle
j_{\mu}=\overline{\psi_{\nu}}\,\gamma_{\mu}\,\left(1-\gamma_{5}\right)\,\psi_{\nu},$
(4.27)
$\displaystyle\mathcal{J}^{\mu}_{\mathrm{electron}}=\overline{\psi_{e}}\,\gamma_{\mu}\,\left(1-\gamma_{5}\right)\,\psi_{e}\,\,\,\,\,(\mathrm{W^{\pm}\,exchange}),$
(4.28)
$\displaystyle\mathcal{J}^{\mu}_{\mathrm{electron}}=\overline{\psi_{e}}\,\gamma_{\mu}\,\left(c_{V}^{e}-c_{A}^{e}\gamma_{5}\right)\,\psi_{e}\,\,\,\,\,(\mathrm{Z^{0}\,exchange}),$
(4.29)

where $\psi_{\nu}$ and $\psi_{e}$ represent the spinor for the neutrino and
electron, respectively. From Eq. (4.11) the coupling coefficient for electrons
are $c^{e}_{V}=-1/2+2\sin^{2}\theta_{w}$ and $c^{e}_{A}=-1/2$. The hadronic
current for is given by the expression [Giunti and Kim (2007)]

$\displaystyle\mathcal{J}^{\mu}_{\mathrm{nucleon}}\equiv\overline{\psi_{t}}\,\gamma^{\mu}\left(c^{t}_{V}-c^{t}_{A}\gamma^{5}\right)\psi_{t},$
(4.30)

where subscript $t$ means the target constituents (protons and neutrons). From
Eq. (4.11) the coupling constants for proton(uud) and neutron(udd) are given
by

$\displaystyle
c^{p}_{V}=\frac{1}{2}-2\sin^{2}\theta_{w},\,\,\,\,c^{p}_{A}=\frac{1}{2},\,\,\,\,\,\mathrm{proton}$
(4.31) $\displaystyle
c^{n}_{V}=-\frac{1}{2}\,\,\,\,c^{n}_{A}=-\frac{1}{2},\,\,\,\,\,\mathrm{neutron}.$
(4.32)

To obtain the effective potential for atom, we need to average the effective
Hamiltonian over the electron and nucleon background. For the neutrino-nucleon
(proton,neutron) interaction, we only have the neutral current interaction via
$Z^{0}$ boson. However, for the neutrino-electron interaction, we can have
charged-current or neutral current interaction depending on the flavor or
neutrino. In following, we consider interaction between $\nu_{e}$ and
electrons first which includes both charged and neutral-currents interaction
for general discussion.

Considering atomic electrons as a gas of unpolarized electrons with a
statistical distribution function $f(E_{e})$, the effective potential for
neutrino-electron interaction can be obtained by averaging the effective
Hamiltonian over the electron background [Giunti and Kim (2007)]

$\displaystyle\langle{\mathcal{H}^{\mathrm{electron}}_{I}}\rangle$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}\int\,\frac{d^{3}p_{e}}{(2\pi)^{3}2E_{e}}\,f(E_{e},T)\left[\overline{\psi_{\nu}}(x)\,\gamma_{\mu}\left(1-\gamma_{5}\right)\,\psi_{\nu}(x)\right]$
$\displaystyle\times\frac{1}{2}\\!\sum_{h_{e}=\pm
1}\\!\\!\langle\,e^{-}(p_{e},h_{e})|\overline{\psi_{e}}\,\gamma^{\mu}\big{(}(1+c^{e}_{V})\\!-\\!(1+c^{e}_{A})\gamma_{5}\big{)}\,\psi_{e}|e^{-}(p_{e},h_{e})\rangle,$
(4.33)

where $h_{e}$ denotes the helicity of the electron. The average over helicity
of the electron matrix element can be calculated with Dirac spinor and gamma
matrix traces [Giunti and Kim (2007)]. Then the average effective Lagrangian
can be written as

$\displaystyle\langle{\mathcal{H}^{\mathrm{electron}}_{I}}\rangle$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}(1+c^{e}_{V})\int\,\frac{d^{3}p_{e}}{(2\pi)^{3}}f(E_{e})\left[\overline{\psi_{\nu}}(x)\,\frac{\gamma^{\mu}{p_{e}}_{\mu}}{E_{e}}\left(1-\gamma_{5}\right)\,\psi_{\nu}(x)\right]$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}\,(1+c^{e}_{V})\,\left[\int\,\frac{d^{3}p_{e}}{(2\pi)^{3}}f(E_{e})\left(\gamma^{0}-\frac{\vec{\gamma}\cdot\vec{{p}}_{e}}{E_{e}}\right)\right]\overline{\psi_{\nu}}(x)\left(1-\gamma_{5}\right)\psi_{\nu}(x)$
$\displaystyle=\left[\frac{G_{F}}{\sqrt{2}}\left(1+c^{e}_{V}\right)n_{e}\right]\,\overline{\psi_{\nu}}(x)\gamma^{0}\left(1-\gamma_{5}\right)\psi_{\nu}(x),$
(4.34)

where $n_{e}$ is the number density of the electron. In this case, the
effective potential for neutrino-atomic electron interaction can be written as

$\displaystyle
V^{\mathrm{electron}}_{I}=\frac{G_{F}}{\sqrt{2}}\left(1+c^{e}_{V}\right)n_{e}=\frac{G_{F}}{\sqrt{2}}\left(4\sin^{2}\theta_{w}+1\right)n_{e}.$
(4.35)

The same method can be applied to the neutrino-nuclear interactions. Following
the same approach and averaging the effective neutrino-nuclear Hamiltonian
over the nuclear background, the effective potential experienced by a neutrino
in a background of neutron/proton is given by [Giunti and Kim (2007)]

$\displaystyle
V_{I}^{\mathrm{proton}}=\frac{G_{F}}{\sqrt{2}}\left(1-4\sin^{2}\theta_{w}\right)n_{p},\qquad
V_{I}^{\mathrm{neutron}}=-\frac{G_{F}}{\sqrt{2}}\,n_{n},$ (4.36)

where $n_{p}$ and $n_{n}$ represent the number density of proton and neutron.
Combining the neutron and proton potential together, we define the effective
nucleon potential experienced by neutrino as

$\displaystyle
V_{I}^{\mathrm{nucleon}}\equiv-\frac{G_{F}}{\sqrt{2}}\bigg{[}1-\left(1-4\sin^{2}\theta_{w}\right)\xi\bigg{]}n_{n},\qquad\xi=n_{p}/n_{n},$
(4.37)

where $\xi$ is the ratio between proton and neutron number density.

In our study, we generalize the effective potential to the case of neutrino
coherent scattering with multi-atom system, we consider a neutrino coherent
forward scatters from a spherical symmetric system which is composed by atoms.
In this case, the neutrino scatters off every atom, and it is impossible to
identify which scatterer the neutrino interacts with and thus it is necessary
to sum over all possible contributions from each atom. In such circumstances,
it is appropriate to assume that the number density of electrons and neutrons
can be written as

$\displaystyle
n_{e}=Z_{e}\,\left(\frac{N_{\mathrm{atom}}}{V}\right),\,\,\,\,\mathrm{and}\,\,\,\,n_{n}=N\,\,\left(\frac{N_{\mathrm{atom}}}{V}\right),$
(4.38)

where $N_{\mathrm{atom}}$ is the number of atoms inside the system, $V$ is the
volume of system, $Z$ is the number of electrons, and $N$ is the number of
neutrons. Then the effective potential is given by

$\displaystyle V_{I}$
$\displaystyle=V_{I}^{\mathrm{electron}}+V_{I}^{\mathrm{nucleon}}$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}\left(\frac{N_{\mathrm{atom}}}{V}\right)\bigg{\\{}\left(4\sin^{2}\theta_{w}\pm
1\right)\,Z_{e}-\bigg{[}1-\left(1-4\sin^{2}\theta_{w}\right)\xi\bigg{]}\,N\bigg{\\}},$
(4.39)

where the $+$ sign is for electron neutrinos $\nu_{e}$ and the $-$ sign is for
muon(tau) neutrinos $\nu_{\mu,\tau}$, separately. From Eq. (4.1.1), it shows
that the effective potential depends on the number density of electrons and
nucleons contained within the wavelength. Thus by increasing the atoms
contained in the wavelength or selecting different atoms as targets, we can
enhance the effective potential and may be able to provide a sensitive way to
detect the cosmic neutrino background. Beside the detection of cosmic neutrino
background, the effective potential for multi-atom can also provide new
approaches for studying other aspects of neutrino physics in the future.

####  Matrix elements of incoherent neutrino scattering

To determine the freeze-out temperature (chemical/kinetic freeze-out) for a
given flavor of neutrinos, we need to know all the elastic and inelastic
interaction processes in the early Universe and compare their interaction rate
with Hubble expansion rate. In this section we summarize the matrix elements
for the neutrino annihilation/production processes and elastic scattering
processes which are relevant for investigating neutrino freezeout. These
matrix elements serve as one of the fundamental ingredients for solving the
Boltzmann equation [Birrell et al. (2014b)].

Considering the Universe with temperature $T\approx\mathcal{O}$(MeV), the
particle species in comisc plasma are given by:

$\displaystyle\mathrm{Particle\,\,species\,\,in\,\,plasma:}\left\\{\gamma,\,l^{-},\,l^{+},\,\nu_{e},\,\nu_{\mu},\,\nu_{\tau},\,\bar{\nu}_{e},\,\bar{\nu}_{\mu},\,\bar{\nu}_{\tau}\right\\},$
(4.40)

where $l^{\pm}$ represents the charged leptons. In this case, neutrinos can
interact with all these particles via weak interactions and remain in
equilibrium. In Table. 4.3 and Table. 4.4 we present the matrix elements
$|M|^{2}$ for different weak interaction processes in the early Universe.

In the calculation of transition amplitude, we use the low energy
approximation for $W^{\pm}$ and $Z^{0}$ massive propagators, i.e.

$\displaystyle\mbox{$Z^{0}$
boson}:\frac{-i\left[g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{M^{2}_{z}}\right]}{q^{2}-M^{2}_{z}}\approx\frac{ig_{\mu\nu}}{M^{2}_{z}},\quad$
$\displaystyle\mbox{$W^{\pm}$
boson}:\frac{-i\left[g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{M^{2}_{w}}\right]}{q^{2}-M^{2}_{w}}\approx\frac{ig_{\mu\nu}}{M^{2}_{w}},$
(4.41)

and consider the tree-level Feynman diagram contributions only. Then,
following the Feynman rules of weak interaction [Griffiths (2008)], we obtain
the matrix elements $|M|^{2}$ for different interaction processes.

Annihilation/Production |
---|---
Scattering Process |  Transition Amplitude $|M|^{2}$
$l^{-}+l^{+}\longrightarrow\nu_{l}+\bar{\nu}_{l}$ |  $32G^{2}_{F}\bigg{[}\left(1+2\sin^{2}\theta_{w}\right)^{2}\left(p_{1}\cdot p_{4}\right)\left(p_{2}\cdot p_{3}\right)$ $+\left(2\sin^{2}\theta_{w}\right)^{2}\left(p_{1}\cdot p_{3}\right)\left(p_{2}\cdot p_{4}\right)$ $+2\sin^{2}\theta_{w}\left(1+2\sin^{2}\theta_{w}\right)m^{2}_{l}\left(p_{3}\cdot p_{4}\right)\bigg{]}$
$l^{\prime-}+l^{\prime+}\longrightarrow\nu_{l}+\bar{\nu}_{l}$ |  $32G^{2}_{F}\bigg{[}\left(1-2\sin^{2}\theta_{w}\right)^{2}\left(p_{1}\cdot p_{4}\right)\left(p_{2}\cdot p_{3}\right)$ $+\left(2\sin^{2}\theta_{w}\right)^{2}\left(p_{1}\cdot p_{3}\right)\left(p_{2}\cdot p_{4}\right)$ $-2\sin^{2}\theta_{w}\left(1-2\sin^{2}\theta_{w}\right)m^{2}_{l^{\prime}}\left(p_{3}\cdot p_{4}\right)\bigg{]}$
$\nu_{l}+\bar{\nu}_{l}\longrightarrow\nu_{l}+\bar{\nu}_{l}$ |  $32G^{2}_{F}\bigg{[}\left(p_{1}\cdot p_{4}\right)\left(p_{2}\cdot p_{3}\right)\bigg{]}$
$\nu_{l^{\prime}}+\bar{\nu}_{l^{\prime}}\longrightarrow\nu_{l}+\bar{\nu}_{l}$ |  $32G^{2}_{F}\bigg{[}\left(p_{1}\cdot p_{4}\right)\left(p_{2}\cdot p_{3}\right)\bigg{]}$
Table 4.3: The transition amplitude for different annihilation and production processes. The definition of particle number is given by $1+2\leftrightarrow 3+4$, where $l,\,l^{\prime}=e,\,\mu,\,\tau\,(l\neq\,l^{\prime})$. Elastic Scattering Process ($\nu_{e}$) |
---|---
Scattering Process |  Transition Amplitude $|M|^{2}$
$\nu_{l}+l^{-}\longrightarrow\nu_{l}+l^{-}$ |  $32G^{2}_{F}\bigg{[}\left(1+2\sin^{2}\theta_{w}\right)^{2}\left(p_{1}\cdot p_{2}\right)\left(p_{3}\cdot p_{4}\right)$ $+\left(2\sin^{2}\theta_{w}\right)^{2}\left(p_{1}\cdot p_{4}\right)\left(p_{2}\cdot p_{3}\right)$ $-2\sin^{2}\theta_{w}\left(1+2\sin^{2}\theta_{w}\right)m^{2}_{l}\left(p_{1}\cdot p_{3}\right)\bigg{]}$
$\nu_{l}+l^{+}\longrightarrow\nu_{l}+l^{+}$ |  $32G^{2}_{F}\bigg{[}\left(1+2\sin^{2}\theta_{w}\right)^{2}\left(p_{1}\cdot p_{4}\right)\left(p_{2}\cdot p_{3}\right)$ $+\left(2\sin^{2}\theta_{w}\right)^{2}\left(p_{1}\cdot p_{2}\right)\left(p_{3}\cdot p_{4}\right)$ $-2\sin^{2}\theta_{w}\left(1+2\sin^{2}\theta_{w}\right)m^{2}_{l}\left(p_{1}\cdot p_{3}\right)\bigg{]}$
$\nu_{l}+l^{\prime-}\longrightarrow\nu_{l}+l^{\prime-}$ |  $32G^{2}_{F}\bigg{[}\left(1-2\sin^{2}\theta_{w}\right)^{2}\left(p_{1}\cdot p_{2}\right)\left(p_{3}\cdot p_{4}\right)$ $+\left(2\sin^{2}\theta_{w}\right)^{2}\left(p_{1}\cdot p_{4}\right)\left(p_{2}\cdot p_{3}\right)$ $+2\sin^{2}\theta_{w}\left(1-2\sin^{2}\theta_{w}\right)m^{2}_{l^{\prime}}\left(p_{1}\cdot p_{3}\right)\bigg{]}$
$\nu_{l}+l^{\prime+}\longrightarrow\nu_{l}+l^{\prime+}$ |  $32G^{2}_{F}\bigg{[}\left(1-2\sin^{2}\theta_{w}\right)^{2}\left(p_{1}\cdot p_{4}\right)\left(p_{2}\cdot p_{3}\right)$ $+\left(2\sin^{2}\theta_{w}\right)^{2}\left(p_{1}\cdot p_{2}\right)\left(p_{3}\cdot p_{4}\right)$ $+2\sin^{2}\theta_{w}\left(1-2\sin^{2}\theta_{w}\right)m^{2}_{l^{\prime}}\left(p_{1}\cdot p_{3}\right)\bigg{]}$
$\nu_{l}+\nu_{l}\longrightarrow\nu_{l}+\nu_{l}$ |  $\frac{1}{2!}\frac{1}{2!}\times 32G^{2}_{F}\bigg{[}4\left(p_{1}\cdot p_{2}\right)\left(p_{3}\cdot p_{4}\right)\bigg{]}$
$\nu_{l}+\bar{\nu}_{l}\longrightarrow\nu_{l}+\bar{\nu}_{l}$ |  $32G^{2}_{F}\bigg{[}4\left(p_{1}\cdot p_{4}\right)\left(p_{2}\cdot p_{3}\right)\bigg{]}$
$\nu_{l}+\nu_{l^{\prime}}\longrightarrow\nu_{l}+\nu_{l^{\prime}}$ |  $32G^{2}_{F}\bigg{[}\left(p_{1}\cdot p_{2}\right)\left(p_{3}\cdot p_{4}\right)\bigg{]}$
$\nu_{l}+\bar{\nu}_{l^{\prime}}\longrightarrow\nu_{l}+\bar{\nu}_{l^{\prime}}$ |  $32G^{2}_{F}\bigg{[}\left(p_{1}\cdot p_{4}\right)\left(p_{2}\cdot p_{3}\right)\bigg{]}$
Table 4.4: The transition amplitude for different elastic scattering
processes. The definition of particle number is given by $1+2\leftrightarrow
3+4$, where $l,\,l^{\prime}=e,\,\mu,\,\tau\,(l\neq\,l^{\prime})$.

###  Neutrinos in the early Universe

In the early Universe the neutrinos are kept in equilibrium with cosmic plasma
via the weak interaction processes. However, as the Universe expanded, these
weak interactions gradually became too slow to maintain equilibrium, then the
neutrinos ceased interacting and decoupled from the cosmic background. These
relic neutrino background carries great information about our early Universe
which can provide the information when our Universe is about $1$ sec old.

Today the Universe is believed to be filled with relic cosmic neutrinos.
Although the cosmic neutrino background is hard to detect, we can use entropy
conservation to infer their temperature and find that they should have the
temperature $T_{\nu}^{0}=1.9\,\mathrm{K}$ in the present epoch for massless
neutrinos. However, from the neutrino oscillation experiment, we know that the
the neutrinos are not massless particles.

The square mass difference $\Delta m^{2}_{ij}$ has been experimentally
measured, from the neutrino oscillation experiment [Workman et al. (2022)]:

$\displaystyle\Delta{m}_{21}^{2}=7.39^{+0.21}_{-0.20}\times
10^{-5}\,\mathrm{eV}^{2},$ (4.42)
$\displaystyle\Delta{m}_{32}^{2}=2.45^{+0.03}_{-0.03}\times
10^{-3}\,\mathrm{eV}^{2}.$ (4.43)

and neutrino mass eigenvalue can be ordered in the normal mass hierarchy
($m_{1}\ll m_{2}<m_{3}$) or inverted mass hierarchy ($m_{3}\ll m_{1}<m_{2}$).
All three mass states remained relativistic until the temperature dropped
below their rest mass. These results allow for the possibility that one mass
eigenstate or two mass eigenstates of neutrinos may become non-relativistic
today.

####  Overview of neutrino freeze-out in the early Universe

The properties of the neutrino background are influenced by the details of the
freeze-out or decoupling process at a temperature
$T=\mathcal{O}(2\mathrm{MeV})$. In the literature one finds estimates of
freeze-out temperatures based on a comparison of Hubble expansion with
neutrino scattering length and considering only number changing (i.e.
chemical) processes. In the paper [Birrell et al. (2014b)], we employ a
similar definition of freeze-out temperature in the context of the Boltzmann
equation and refine the results by noting that there are three different
freeze-out processes for neutrinos:

1\. Neutrino chemical freeze-out: the temperature at which neutrino number
changing processes such as $e^{-}e^{+}\to\nu\overline{\nu}$ effectively cease.
After chemical freeze-out, there are no reactions that, in a noteworthy
fashion, can change the neutrino abundance and so particle number is
conserved.

2\. Neutrino kinetic freeze-out: the temperature at which the neutrino
momentum exchanging interactions such as $e^{\pm}\nu\to e^{\pm}\nu$ are no
longer occurring rapidly enough to maintain an equilibrium momentum
distribution.

3\. Collisions between neutrinos: Those collisions $\nu\nu\to\nu\nu$ are
capable of reequilibrating energy within and between neutrino flavor families.
These processes end at a yet lower temperature and the neutrinos will be truly
free-streaming from that point on.

To estimate the freeze-out temperature, we need to solve the Boltzmann
equation with different types of collision terms with the transition matrices
from Table. 4.3 and Table. 4.4. The paper [Birrell et al. (2014b) ] developed
a new method for analytically simplifying the collision integrals and showing
that the neutrino freeze-out temperature is controlled by standard model (SM)
parameters. The freeze-out temperature depends only on the magnitude of the
Weinberg angle in the form $\sin^{2}\theta_{W}$ , and a dimensionless relative
interaction strength parameter $\eta$,

$\displaystyle\eta\equiv M_{p}m_{e}^{3}G_{F}^{2},\qquad
M_{p}^{2}\equiv\frac{1}{8\pi G_{N}},$ (4.44)

a combination of the electron mass $m_{e}$, Newton constant $G_{N}$ (expressed
above in terms of Planck mass $M_{p}$), and the Fermi constant $G_{F}$. The
dimensionless interaction strength parameter $\eta$ in the present-day vacuum
has the value

$\displaystyle\eta_{0}\equiv\left.M_{p}m_{e}^{3}G_{F}^{2}\right|_{0}=0.04421.$
(4.45)

The magnitude of $\sin^{2}\theta_{W}$ is not fixed within the SM and could be
subject to variation as a function of time or temperature. In Fig. 4.1 we show
the dependence of neutrino freeze-out temperatures for $\nu_{e}$ and
$\nu_{\mu,\tau}$ on SM model parameters $\sin^{2}\theta_{W}$ and $\eta$ in
detail. The impact of SM parameter values on neutrino freeze-out and the
discussion of the implications and connections of this work to other areas of
physics, namely Big Bang Nucleosynthesis and dark radiation can be found in
detail in [Dreiner et al. (2012); Boehm et al. (2012); Blennow et al. (2012);
Birrell et al. (2014b)]. A comprehensive investigation of neutrino freezeout,
and a novel approach to analytically simplify the collision integrals for the
Boltzmann equation can be found in Dr. Jeremiah Birrell‘s PhD thesis [Birrell
(2014)].

Figure 4.1: Freeze-out temperatures for electron neutrinos (left) and $\mu$,
$\tau$ neutrinos (right) for the three types of freeze-out processes adapted
from paper [Birrell et al. (2014b)]. Top panels print temperature curves as a
function of $\sin^{2}\theta_{W}$ for $\eta=\eta_{0}$, the vertical dashed line
is $\sin^{2}\theta_{W}=0.23$; bottom panels are printed as a function of
relative change in interaction strength $\eta/\eta_{0}$ obtained for
$\sin^{2}\theta_{W}=0.23$.

####  Lepton number and effective number of neutrinos

Neutrinos decoupled from the cosmic plasma in the early Universe at a
temperature of $T=\mathcal{O}(2\mathrm{MeV})$ and became free-streaming.
However, after freezeout neutrinos still continue to play a significant role
in the evolution of the Universe and have a huge impact on cosmological
observations such as Big Bang Nucleosynthesis (BBN), the Cosmic Microwave
Background (CMB), and the matter spectrum for large scale structure. This is
due to the sensitivity of the Hubble parameter to the total energy density in
the Universe. Besides photons, neutrinos are the most abundant species and
contribute significantly to the relativistic energy density throughout the
early Universe, affecting the Hubble expansion rate significantly.

The contribution of energy density from the neutrino sector can be described
by the effective number of neutrinos $N_{\nu}^{\mathrm{eff}}$, which captures
the number of relativistic degrees of freedom for neutrinos as well as any
reheating that occurred in the sector after freeze-out. The effective number
of neutrino is defined as

$\displaystyle
N_{\nu}^{\mathrm{eff}}\equiv\frac{\rho^{\mathrm{tot}}_{\nu}}{\frac{7\pi^{2}}{120}\left(\frac{4}{11}\right)^{4/3}T_{\gamma}^{4}}\;,$
(4.46)

where $\rho_{\nu}^{\mathrm{tot}}$ is the total energy density in neutrinos and
$T_{\gamma}$ is the photon temperature. $N_{\nu}^{\mathrm{eff}}$ is defined
such that three neutrino flavors with zero participation of neutrinos in
reheating during $e^{\pm}$ annihilation results in $N_{\nu}^{\mathrm{eff}}=3$.
The factor of $\left(4/11\right)^{1/3}$ relates the photon temperature to the
free-streaming neutrinos temperature, under the assumption of zero neutrino
reheating after $e^{\pm}$ annihilation. The currently accepted theoretical
value is $N_{\nu}^{\mathrm{eff}}=3.046$, after including the slight effect of
neutrino reheating [Mangano et al. (2005); Birrell et al. (2014b)]. The
favored value of $N_{\nu}^{\mathrm{eff}}$ can be found by fitting to CMB data.
In 2013 the Planck collaboration found $N_{\nu}^{\mathrm{eff}}=3.36\pm 0.34$
(CMB only) and $N_{\nu}^{\mathrm{eff}}=3.62\pm 0.25$ (CMB and $H_{0}$) [Ade et
al. (2014)].

To explain the experimental value of $N_{\nu}^{\mathrm{eff}}$, many studies
aim to improve the calculation of neutrino decoupling in the early Universe,
including exploring the dependence of freeze-out on natural constants [Birrell
et al. (2014b)], the entropy transfer from $e^{\pm}$ annihilation and finite
temperature correction [Dicus et al. (1982); Heckler (1994); Fornengo et al.
(1997)], neutrino decoupling with flavor oscillations [Mangano et al. (2002,
2005)], and investigating nonstandard neutrino interactions [Morgan (1981);
Fukugita and Yazaki (1987); Elmfors et al. (1997); Vogel and Engel (1989);
Mangano et al. (2006); Giunti and Studenikin (2009); Mangano et al. (2006)].

The standard cosmological model assumes that the lepton asymmetry
$L\equiv[N_{\mathrm{L}}-N_{\overline{\mathrm{L}}}]/N_{\gamma}$ (normalized
with the photon number) between leptons and anti-leptons is small, similar to
the baryon asymmetry
$B=[N_{\mathrm{B}}-N_{\overline{\mathrm{B}}}]/N_{\gamma}$; most often it is
assumed $L=B$. Barenboim, Kinney, and Park [Barenboim et al. (2017); Barenboim
and Park (2017)] noted that the lepton asymmetry of the Universe is one of the
most weakly constrained parameters is cosmology and they propose that models
with leptogenesis are able to accommodate a large lepton number asymmetry
surviving up to today. Moreover, the discrepancy between $H_{\mathrm{CMB}}$
and $H_{0}$ has increased [Riess et al. (2018a, b); Aghanim et al. (2020a)].
The Hubble tension and the possibility that leptogenesis in the early Universe
resulted in neutrino asymmetry motivate our study of the dependence of
$N_{\nu}^{\mathrm{eff}}$ on lepton asymmetry, $L$. In our work [Yang et al.
(2018a)] we consider $L\simeq 1$ and explore how this large cosmological
lepton yield relates to the effective number of (Dirac) neutrinos
$N^{\mathrm{eff}}_{\nu}$.

##### Relation between $N_{\nu}^{\mathrm{eff}}$ and neutrino chemical
potential

We consider now neutrinos decouple [Birrell et al. (2015)] at a temperature of
$T_{f}\simeq 2\,\mathrm{MeV}$ and are subsequently free-streaming. Assuming
exact thermal equilibrium at the time of decoupling, the neutrino distribution
can be subsequently written as (see [Birrell et al. (2014a)] and references
therein)

$\displaystyle
f_{\nu}=\frac{1}{\exp{\left(\sqrt{\frac{E^{2}-m_{\nu}^{2}}{T_{\nu}^{2}}+\frac{m^{2}_{\nu}}{T^{2}_{f}}}-\sigma\frac{\mu_{\nu}}{T_{f}}\right)+1}}\;,\qquad
T_{\nu}\equiv\frac{a(t_{f})}{a(t)}T_{f},$ (4.47)

where $\sigma=+1(-1)$ denotes particles (antiparticles) and we define the
effective neutrino temperature $T_{\nu}$ by the red-shifting of momentum in
the comoving volume element of the Universe.

Since the freeze-out temperature $T_{f}\gg m_{\nu}$ and also neutrino
temperature $T_{\nu}\gg m_{\nu}$ in the domain of our analysis, we consider
the massless limit in Eq. (4.47). Under this approximation, the total neutrino
energy density can be written as

$\displaystyle\rho_{\nu}^{\mathrm{tot}}$
$\displaystyle=\frac{g_{\nu}\,T_{\nu}^{4}}{2\pi^{2}}\left[\frac{7\pi^{4}}{60}+\frac{\pi^{2}}{2}\left(\frac{\mu_{\nu}}{T_{f}}\right)^{\\!\\!2}+\frac{1}{4}\left(\frac{\mu_{\nu}}{T_{f}}\right)^{\\!\\!4}\right].$
(4.48)

Substituting Eq. (4.48) into the definition of the effective number of
neutrinos Eq. (4.46), we obtain

$\displaystyle
N_{\nu}^{\mathrm{eff}}\\!\\!=\\!3\\!\left(\frac{11}{4}\right)^{\\!\\!\frac{4}{3}}\\!\\!\left(\frac{T_{\nu}}{T_{\gamma}}\right)^{\\!\\!4}\\!\left[1\\!+\\!\frac{30}{7\pi^{2}}\\!\\!\left(\frac{\mu_{\nu}}{T_{f}}\right)^{\\!\\!2}\\!\\!+\frac{15}{7\pi^{4}}\\!\\!\left(\frac{\mu_{\nu}}{T_{f}}\right)^{\\!\\!4}\right].$
(4.49)

From Eq. (4.49) we have for the standard photon reheating ratio
$T_{\nu}/T_{\gamma}=(4/11)^{1/3}$ [Kolb and Turner (1990b)] and degeneracy
$g_{\nu}=3$ (flavor), the relation between the effective number of neutrinos
and the chemical potential at freezeout

$\displaystyle
N_{\nu}^{\mathrm{eff}}=3\left[1+\frac{30}{7\pi^{2}}\left(\frac{\mu_{\nu}}{T_{f}}\right)^{\\!\\!2}+\frac{15}{7\pi^{4}}\left(\frac{\mu_{\nu}}{T_{f}}\right)^{\\!\\!4}\right].$
(4.50)

To solve the neutrino chemical potential $\mu_{\nu}/T_{f}$ as a function of
the effective number of neutrinos, we can neglect the $(\mu_{\nu}/T_{f})^{4}$
term in Eq. (4.50) because $m_{\nu}\ll T_{f}$ and obtain

$\displaystyle\frac{\mu_{\nu}}{T_{f}}=\pm\sqrt{\frac{7\pi^{2}}{30}\left(\frac{N_{\nu}^{\mathrm{eff}}}{3}-1\right)}.$
(4.51)

In Fig. 4.2 we plot the free-streaming neutrino chemical potential
$|\mu_{\nu}|/T_{f}$ as a function of the effective number of neutrinos
$N_{\nu}^{\mathrm{eff}}$. For comparison, the solid (blue) line is the exact
solution of $|\mu_{\nu}|/T_{f}$ by solving Eq. (4.50) numerically, and the
(red) dashed line is the approximate solution Eq. (4.51) by neglecting the
$(\mu_{\nu}/T_{f})^{4}$ in calculation. In the parameter range of interest, we
show that the term $(\mu_{\nu}/T_{f})^{4}$ only contributes $\approx 2\%$ to
the calculation and henceforth we neglect it, and use the approximation Eq.
(4.51).

The SM value of the effective number of neutrinos, $N_{\nu}^{\mathrm{eff}}=3$,
is obtained under the assumption that the neutrino chemical potentials are not
essential, i.e., $\mu_{\nu}\ll T_{f}$. From Fig. 4.2, to interpret the
literature values $N_{\nu}^{\mathrm{eff}}=3.36\pm 0.34$ (CMB only) and
$N_{\nu}^{\mathrm{eff}}=3.62\pm 0.25$ (CMB and $H_{0}$), we require
$0.52\leqslant\mu_{\nu}/T_{f}\leqslant 0.69$. These values suggest a possible
neutrino-antineutrino asymmetry at freezeout, i.e. a difference between the
number densities of neutrinos and antineutrinos.

Figure 4.2: The free-streaming neutrino chemical potential $|\mu_{\nu}|/T_{f}$
as a function of the effective number of neutrinos $N_{\nu}^{\mathrm{eff}}$.
The solid (blue) line is the exact solution and the (red) dashed line is the
approximate solution neglecting the $(\mu_{\nu}/T_{f})^{4}$ term; the maximum
difference in the domain shown is about $2\%$.

##### Dependence of $N_{\nu}^{\mathrm{eff}}$ on lepton asymmetry

We now obtain the relation between neutrino chemical potential and the baryon
to lepton ratio. Let us consider the neutrino freezeout temperature
$T_{f}\simeq 2.0$ MeV; here we treat neutrino freezeout as occurring
instantaneously and prior to $e^{\pm}$ annihilation (implying zero neutrino
reheating). Comoving lepton (and baryon) number is conserved after the epoch
of leptogenesis (baryogenesis, respectively) which precedes the epoch under
consideration in this work ($T\lesssim 2$ MeV).

The lepton-density asymmetry $\ell$ at neutrino freeze-out can be written as

$\displaystyle\ell_{f}\equiv\big{(}n_{e}-n_{\overline{e}}\big{)}_{f}+\sum_{i=e,\mu,\tau}\big{(}n_{\nu_{i}}-n_{\overline{\nu}_{i}}\big{)}_{f},$
(4.52)

where we use the subscript $f$ to indicate that the quantities should be
evaluated at the neutrino freeze-out temperature. As a first approximation,
here we assume that all neutrinos freeze-out at the same temperature and their
chemical potentials are the same; i.e.,

$\displaystyle\mu_{\nu}=\mu_{\nu_{e}}=\mu_{\nu_{\mu}}=\mu_{\nu_{\tau}}.$
(4.53)

Furthermore, neutrino oscillation implies that neutrino number is freely
exchanged between flavors; i.e.,
$\nu_{e}\rightleftharpoons\nu_{\mu}\rightleftharpoons\nu_{\tau}$, and we can
assume that all neutrino flavors share the same population. Under these
assumptions, the lepton-density asymmetry can be written as

$\displaystyle\ell_{f}=\big{(}n_{e}-n_{\overline{e}}\big{)}_{f}+\big{(}n_{\nu}-n_{\overline{\nu}}\big{)}_{f},$
(4.54)

where the three flavors are accounted for by taking the degeneracy $g_{\nu}=3$
in the last term. The difference in yield of neutrinos and antineutrinos can
be written as

$\displaystyle\left(n_{\nu}-n_{\overline{\nu}}\right)_{f}=\frac{g_{\nu}}{6\pi^{2}}T^{3}_{f}\bigg{[}\pi^{2}\left(\frac{\mu_{\nu}}{T_{f}}\right)+\left(\frac{\mu_{\nu}}{T_{f}}\right)^{\\!\\!3}\bigg{]}.$
(4.55)

On the other hand, the baryon-density asymmetry $b$ at neutrino freezeout is
given by

$\displaystyle
b_{f}\equiv\big{(}n_{p}-n_{\overline{p}}\big{)}_{f}+\big{(}n_{n}-n_{\overline{n}}\big{)}_{f}\approx\big{(}n_{p}+n_{n}\big{)}_{f},$
(4.56)

where $n_{\overline{n}}$ and $n_{\overline{p}}$ are negligible in the
temperature range we consider here. Taking the ratio $\ell_{f}/b_{f}$, using
charge neutrality, and introducing the entropy density we obtain

$\displaystyle\left(\frac{\ell_{f}}{b_{f}}\right)\approx\left(\frac{n_{p}}{n_{B}}\right)_{f}+\left(n_{\nu}-n_{\overline{\nu}}\right)_{f}\left(\frac{s}{n_{B}}\right)_{f}\frac{1}{s_{f}},\qquad
n_{B}=(n_{p}+n_{n}),$ (4.57)

where we introduce the notation $n_{B}$ for the baryon number density. The
proton concentration at neutrino freeze-out is given by

$\displaystyle\left(\frac{n_{p}}{n_{B}}\right)_{f}$
$\displaystyle=\frac{1}{1+(n_{n}/n_{p})_{f}}=\frac{1}{1+\exp{\big{[}-\left(Q+\mu_{\nu}\right)/T_{f}\big{]}}},$
(4.58)

with $Q=m_{n}-m_{p}=1.293\,\mathrm{MeV}$. We neglect the electron chemical
potential in the last step because the $e^{\pm}$ asymmetry is determined by
the proton density, and at energies of order a few MeV, the proton density is
small, i.e., $\mu_{e}\ll T_{f}$.

However, as we will see, for our study of $N_{\nu}^{\mathrm{eff}}$ we will be
interested in the case of a large lepton-to-baryon ratio. From Eq. (4.58) it
is apparent that this can only be achieved through the second term in Eq.
(4.57), with the first term then being negligible, as it is smaller than $1$.
So we further approximate

$\displaystyle\left(\frac{\ell_{f}}{b_{f}}\right)\approx\left(n_{\nu}-n_{\overline{\nu}}\right)_{f}\left(\frac{s}{n_{B}}\right)_{f}\frac{1}{s_{f}}.$
(4.59)

We retained the full expression Eq. (4.58) in our above discussion to show
that the presence of a chemical potential $\mu_{\nu}\simeq 0.2\,Q$ could lead
to small, perhaps noticeable, effects on pre-BBN proton and neutron abundance.
We defer this unrelated discussion to a separate future work. Note that for
large $|\mu_{\nu}|$, Eq. (4.59) implies that the signs of $\mu_{\nu}$ and
$\ell_{f}$ are the same. However, for very small $\mu_{\nu}$ the sign of
$\ell_{f}$ is determined by the interplay between (anti)electrons and
(anti)neutrinos; i.e., there is competition between the two terms in Eq.
(4.54).

In general, the total entropy density at freeze-out can be written

$\displaystyle s_{f}=\frac{2\pi^{2}}{45}g^{s}_{\ast}(T_{f})\,T_{f}^{3},$
(4.60)

where the $g^{s}_{\ast}$ counts the degree of freedom for relativistic
particles [Kolb and Turner (1990b)]. At $T_{f}\simeq 2\mathrm{MeV}$, the
relativistic species in the early Universe are photons, electron/positrons,
and $3$ neutrino species. We have

$\displaystyle g^{s}_{\ast}$
$\displaystyle=g_{\gamma}+\frac{7}{8}\,g_{e^{\pm}}+\frac{7}{8}\,g_{\nu\bar{\nu}}\left(\frac{T_{\nu}}{T_{\gamma}}\right)^{\\!\\!3}\bigg{[}1+\frac{15}{7\pi^{2}}\left(\frac{\mu_{\nu}}{T_{f}}\right)^{\\!\\!2}\bigg{]}=10.75+\frac{45}{4\pi^{2}}\left(\frac{\mu_{\nu}}{T_{f}}\right)^{\\!\\!2}\;,$
(4.61)

where the degrees of freedom are given by $g_{\gamma}=2$, $g_{e^{\pm}}=4$, and
$g_{\nu\bar{\nu}}=6$, and we have $T_{\nu}=T_{\gamma}=T_{f}$ at neutrino
freeze-out.

Figure 4.3: The ratio $B/|L|$ between the net baryon number and the net lepton
number as a function of $N^{\mathrm{eff}}_{\nu}$: The solid blue line shows
$B/|L|$. The vertical (red) dotted lines represent the values $3.36\leqslant
N_{\nu}^{\mathrm{eff}}\leqslant 3.62$, which correspond to $1.16\times
10^{-9}\leqslant B/|L|\leqslant 1.51\times 10^{-9}$ (horizontal dashed lines).

Finally, since the entropy-per-baryon from neutrino freeze-out up to the
present epoch is constant, we can obtain this value by considering the
Universe’s entropy content today [Fromerth et al. (2012)]. For $T\ll
1\,\mathrm{MeV}$, the entropy content today is carried by photons and
neutrinos, yielding

$\displaystyle\left(\frac{s}{n_{B}}\right)_{t_{0}}$
$\displaystyle=\frac{\sum_{i}\,s_{i}}{n_{B}}=\frac{n_{\gamma}}{n_{B}}\,\bigg{(}\frac{s_{\gamma}}{n_{\gamma}}+\frac{s_{\nu}}{n_{\gamma}}+\frac{s_{\bar{\nu}}}{n_{\gamma}}\bigg{)}\;$
(4.62)
$\displaystyle=\left(\frac{1}{B}\right)_{\\!\\!t_{0}}\\!\\!\left[\frac{s_{\gamma}}{n_{\gamma}}+\frac{4}{3T_{\nu}}\frac{\rho_{\nu}^{\mathrm{tot}}}{n_{\gamma}}-\frac{\mu_{\nu}}{T_{f}}\left(\frac{n_{\nu}-n_{\bar{\nu}}}{n_{\gamma}}\right)\right]_{t_{0}}\;,$
(4.63)

where $t_{0}$ denotes the present day values, we have
$B=n_{B}/n_{\gamma}=0.605\times 10^{-9}$ (CMB) [Patrignani et al. (2016)] from
today’s observation. The entropy per particle for a massless boson at zero
chemical potential is $(s/n)_{\mathrm{boson}}\approx 3.602$.

Substituting Eq. (4.55) and Eq. (4.60) into Eq. (4.59) yields the lepton-to-
baryon ratio

$\displaystyle\frac{L}{B}=\frac{45}{4\pi^{4}}\frac{\pi^{2}(\mu_{\nu}/T_{f})+(\mu_{\nu}/T_{f})^{3}}{10.75+{45}(\mu_{\nu}/T_{f})^{2}/{4\pi^{2}}}\left(\frac{s}{n_{B}}\right)_{\\!\\!t_{0}}\;,$
(4.64)

in terms of $\mu_{\nu}/T_{f}$ which is given by Eq.(4.51) and the present day
entropy-per-baryon ratio. In Fig. 4.3 we show the ratio between the net baryon
number and the net lepton number as a function of the effective number of
neutrino species $N^{\mathrm{eff}}_{\nu}$ with the parameter
$B|_{t_{0}}=0.605\times 10^{-9}$(CMB). We find that the values
$N_{\nu}^{\mathrm{eff}}=3.36\pm 0.34$ and $N_{\nu}^{\mathrm{eff}}=3.62\pm
0.25$ require the ratio between baryon number and lepton number to be
$1.16\times 10^{-9}\leqslant\,B/|L|\leqslant 1.51\times 10^{-9}$. These values
are close to the baryon-to-photon ratio $0.57\times 10^{-9}\leqslant
B\leqslant 0.67\times 10^{-9}$.

In summary, motivated by the necessity to explain a slightly faster Universe
expansion, we believe that there is need for additional unobserved particles,
leading to an increase in the Universe expansion rate. Considerable effort has
been made in this direction, e.g., by introducing exotic and new ‘dark’
particles, see [Birrell and Rafelski (2015)] and references therein. In this
work a similar effect is achieved by introducing lepton asymmetry in the
Universe. We connected the lepton asymmetry in the Universe with the chemical
neutrino potential $\mu_{\nu}$, and further evaluated the consequences for the
Universe expansion. We have explored the other natural scenario regarding the
baryon number-to-lepton number ratio. Instead of $B\simeq|L|$, we found that
$0.4\leqslant|L|\leqslant 0.52$ and $B\simeq 1.33\times 10^{-9}|L|$ reconciles
the CMB and current epoch results for the Hubble expansion parameter.

The large lepton asymmetry from cosmic neutrino can also affect the neutron
lifespan in cosmic plasma which is one of the important parameter controlling
BBN element abundances. In general the neutron lifespan dependence on
temperature of the cosmic medium. When temperature
$T=\mathcal{O}(\mathrm{MeV})$, neutron decay occurs in the plasma of
electron/positron and neutrino/antineutrino. Electrons and neutrinos in the
background plasma can reduce the neutron decay rate by Fermi suppression to
the neutron decay rate. Furthermore, the neutrino background can still provide
the suppression after electron/positron pair annihilation becomes nearly
complete. In this case,the large neutrino chemical potential from lepton
asymmetry would play an important role and needs to be accounted for in the
precision study of the neutron lifespan in the cosmic plasma.

##  Chapter 5 Charged leptons in cosmic plasma

Charged leptons played significant roles in the dynamics and evolution of the
early Universe. They were kept in equilibrium via electromagnetic and weak
interactions. In this chapter, I examine a dynamical model of the abundance of
charged leptons $\mu^{\pm}$ and $e^{\pm}$ in the early Universe obtaining
their disappearance temperature, the condition when they disappear from the
particle inventory. Of particular interest is the dense electron-positron
plasma present during the early Universe evolution. I study the damping rate
and the magnetization process in this dense $e^{\pm}$ plasma in the early
Universe.

###  Overview of charge leptons in early Universe

The $\tau^{\pm}$ leptons can undergo various decay processes via the weak
interaction in the early Universe, and is the only charged lepton that can
decay into hadrons because of its heavy mass ($m_{\tau}=1776.86$ MeV). The
principle decay channels of $\tau^{\pm}$ are given by

$\displaystyle\tau^{-}\rightarrow\nu_{\tau}+e^{-}+\bar{\nu}_{e},\qquad\tau^{-}\rightarrow\nu_{\tau}+\mu^{-}+\bar{\nu}_{\mu},$
(5.1)
$\displaystyle\tau^{-}\rightarrow\nu_{\tau}+\pi^{-},\qquad\qquad\,\tau^{+}\rightarrow\bar{\nu}_{\tau}+\pi^{+},$
(5.2)

where the vacuum lifespan for $\tau^{\pm}$ is given by [Workman et al. (2022)]

$\displaystyle\tau_{\tau}=(290.3\pm 0.5)\times 10^{-15}\,\mathrm{sec}.$ (5.3)

Moreover, following the decay of $\tau^{\pm}$ into pions, these pions
subsequently decay into a muon and a neutrino through the reaction

$\displaystyle\pi^{-}\rightarrow\nu_{\mu}+\mu^{-},\qquad\qquad\,\pi^{+}\rightarrow\bar{\nu}_{\mu}+\mu^{+},$
(5.4)

with pion vacuum lifespan $\tau_{\pi}=2.6033\times 10^{-8}$ sec [Workman et
al. (2022)]. In this scenario, $\tau^{\pm}$ disappears from the Universe via
multiparticle decay processes. These decay processes can contribute as one of
the sources for the production of neutrinos and muons in the early Universe.

The $\mu^{\pm}$ lepton abundance is an important quantity required for the
understanding of several fundamental questions regarding properties of the
primordial Universe, particularly in relation to the freeze-out of strangeness
flavor in the early Universe. We recall that the strangeness decay often
proceeds into muons, energy thresholds permitting, as the charged kaons K±
have a 63% branching into $\mu+\bar{\nu}_{\mu}$. Should muons fall out of
thermal abundance equilibrium this would directly impact the detailed balance
back-reaction processes. Another, indirect influence on strangeness in early
Universe arises through the nearly exclusive decay of charged pions into
$\mu+\bar{\nu}_{\mu}$. Without chemical abundance equilibrium this back
reaction stops too impacting pions and thus all other hadronic particles in
the Universe.

On the other hand, we will show that the lightest charged leptons $e^{\pm}$
can persist via the reaction $\gamma\gamma\to e^{-}e^{+}$ until the
temperature $T=20$ keV in the early Universe. After $T=20$ keV, the positron
rapidly disappears through annihilation, leaving only residual electrons to
maintain the Universe’s charge neutrality. The existence of an electron-
positron plasma plays a pivotal role in several aspects of the early Universe
as follows:

1\. The role of electron-positron plasma has not received the appropriate
attention in the days of precision Big-Bang nucleosynthesis studies. The
standard BBN model indicates that the synthesis of light elements typically
takes place at temperatures around $86\,\mathrm{keV}>T_{BBN}>50\,\mathrm{keV}$
[Pitrou et al. (2018)]. Within this temperature range there are millions of
electron-positron pairs per charged nucleon, providing an electron-positron-
rich plasma environment for nucleosynthesis which leads to modifications in
the Coulomb potential due to the screening effect. Furthermore, the electron-
positron densities can reach millions of times normal atomic densities. The
presence of these $e\bar{e}$-pairs before and during BBN has been acknowledged
by Wang, Bertulani and Balantekin [Wang et al. (2011)] nearly a decade ago.

2\. The Universe today is filled with magnetic fields at various scales and
strengths both within galaxies and in deep extra-galactic space. The origin of
these magnetic fields is currently unknown. In the early Universe, when
temperature $T>20$ keV, we have dense $e^{\pm}$ plasma. The significant
magnetic moments of electrons and positrons also provide opportunities to
investigate spin magnetization process.

Understanding the abundances of muons and electrons/positrons provides
essential insights into the evolution of the primordial Universe. In the
following we discuss the muon density at persistence temperature in section
5.2, and explore the electron/positron plasma properties, including the damped
rate and magnetization in section 5.3.

###  Muon–antimuon in the early Universe

Our interest in strangeness flavor freeze-out in the early Universe requires
the understanding of the abundance of muons in the early Universe. The
specific question needing an answer is at which temperature muons remain in
abundance (chemical) equilibrium established predominantly by electromagnetic
and weak interaction processes, allowing detailed-balance back-reactions to
influence strangeness abundance.

In the early Universe in the the cosmic plasma muons of mass $m_{\mu}=105.66$
MeV can be produced by the following interaction processes

$\displaystyle\gamma+\gamma\longrightarrow\mu^{+}+\mu^{-},\qquad$
$\displaystyle e^{+}+e^{-}\longrightarrow\mu^{+}+\mu^{-}\;,$ (5.5)
$\displaystyle\pi^{-}\longrightarrow\mu^{-}+\bar{\nu}_{\mu},\qquad$
$\displaystyle\pi^{+}\longrightarrow\mu^{+}+\nu_{\mu}\;.$ (5.6)

The back reactions for all above processes are in detailed balance, provided
all particles shown on the right hand side (RHS) exist in chemical abundance
equilibrium in the Universe. We recall the empty space (no plasma) at rest
lifetime of pions $\tau_{\pi}=2.6033\times 10^{-8}$ sec.

However, all produced muons can also decay via the reactions

$\mu^{-}\rightarrow\nu_{\mu}+e^{-}+\bar{\nu}_{e},\qquad\mu^{+}\rightarrow\bar{\nu}_{\mu}+e^{+}+\nu_{e}\,,$
(5.7)

with the empty space (no plasma) at rest lifetime $\tau_{\mu}=2.197\times
10^{-6}\,\mathrm{sec}$. We thus must establish the range of temperature in
which production processes exceed in speed the decay process.

The temperature range of our interests is the Universe when $m_{\mu}\gg T$. In
this case the the Boltzmann approximation is appropriate for studying massive
particles muons and pions. The thermal decay rate per volume and time for
muons $\mu^{\pm}$ (and pions $\pi^{\pm}$) in the Boltzmann limit are given by
[Kuznetsova and Rafelski (2010b)]:

$\displaystyle
R_{\mu}=\frac{g_{\mu}}{2\pi^{2}}\left(\frac{T^{3}}{\tau_{\mu}}\right)\left(\frac{m_{\mu}}{T}\right)^{2}K_{1}(m_{\mu}/T)\;,$
(5.8) $\displaystyle
R_{\pi}=\frac{g_{\pi}}{2\pi^{2}}\left(\frac{T^{3}}{\tau_{\pi}}\right)\left(\frac{m_{\pi}}{T}\right)^{2}K_{1}(m_{\pi}/T)\;,$
(5.9)

where the lifespan of $\mu^{\pm}$ and $\pi^{\pm}$ in the vacuum were given
above. This rate accounts for both the density of particles in chemical
abundance equilibrium and the effect of time dilation present when particles
are in thermal motion with respect to observer at rest in the local reference
frame. The effects of Fermi blocking or boson stimulated emission have been
neglected.

The thermal averaged reaction rate per volume for the reaction
$a\overline{a}\rightarrow b\overline{b}$ in Boltzmann approximation is given
by [Letessier and Rafelski (2002)]

$\displaystyle R_{a\overline{a}\rightarrow
b\overline{b}}=\frac{g_{a}g_{\overline{a}}}{1+I}\frac{T}{32\pi^{4}}\int_{s_{th}}^{\infty}ds\frac{s(s-4m^{2}_{a})}{\sqrt{s}}\sigma_{a\overline{a}\rightarrow
b\overline{b}}~{}K_{1}(\sqrt{s}/T),$ (5.10)

where $s_{th}$ is the threshold energy for the reaction,
$\sigma_{a\overline{a}\rightarrow b\overline{b}}$ is the cross section for the
given reaction, and $K_{1}$ is the modified Bessel function of integer order
$”1”$. We introduce the factor $1/1+I$ to avoid the double counting of
indistinguishable pairs of particles; we have $I=1$ for an identical pair and
$I=0$ for a distinguishable pair.

The leading order invariant matrix elements for the reactions
$e^{+}+e^{-}\to\mu^{+}+\mu^{-}$ and $\gamma+\gamma\to\mu^{+}+\mu^{-}$, are
introduced in this work by [Kuznetsova et al. (2008)]

$\displaystyle|M_{e\bar{e}\to\mu\bar{\mu}}|^{2}=$ $\displaystyle
32\pi^{2}\alpha^{2}\frac{(m_{\mu}^{2}-t)^{2}+(m_{\mu}^{2}-u)^{2}+2m_{\mu}^{2}s}{s^{2}},\quad
m_{\mu}\gg m_{e}\;,$ (5.11)
$\displaystyle|M_{\gamma\gamma\to\mu\bar{\mu}}|^{2}=$ $\displaystyle
32\pi^{2}\alpha^{2}\bigg{[}\left(\frac{m_{\mu}^{2}-u}{m_{\mu}^{2}-t}+\frac{m_{\mu}^{2}-t}{m_{\mu}^{2}-u}\right)+4\left(\frac{m_{\mu}^{2}}{m_{\mu}^{2}-t}+\frac{m_{\mu}^{2}}{m^{2}_{\mu}-u}\right)$
(5.12) $\displaystyle\hskip
28.45274pt-4\left(\frac{m_{\mu}^{2}}{m^{2}_{\mu}-t}+\frac{m^{2}_{\mu}}{m^{2}_{\mu}-u}\right)^{2}\bigg{]}\;,$

where $s,t,u$ are the Mandelstam variables. The cross section required in Eq.
(5.10) can be obtained by integrating the matrix elements Eq. (5.11) and Eq.
(5.12) over the Mandelstam variable $t$ [Kuznetsova and Rafelski (2010b)]. We
have

$\displaystyle\sigma_{e\bar{e}\to\mu\bar{\mu}}=\frac{64\pi\alpha^{2}}{48\pi}\left(\frac{1+2m^{2}_{\mu}/s}{s-4m_{e}^{2}}\right)\sqrt{1-\frac{4m^{2}_{\mu}}{s}},$
(5.13)
$\displaystyle\sigma_{\gamma\gamma\to\mu\bar{\mu}}=\frac{\pi}{2}\left(\frac{\alpha}{m_{\mu}}\right)^{2}(1-\beta^{2})\left[(3-\beta^{4})\ln\frac{1+\beta}{1-\beta}-2\beta(2-\beta^{2})\right],$
(5.14) $\displaystyle\beta=\sqrt{1-4m^{2}_{\mu}/s}$ (5.15)

Substituting the cross sections into Eq. (5.10) we obtain the production rates
for $e\bar{e}\to\mu\bar{\mu}$ and $\gamma\gamma\to\mu\bar{\mu}$ respectively.

In Fig. 5.1 we show the invariant thermal reaction rates per volume and time
for rates of relevance, as a function of temperature $T$. As the temperature
decreases in the expanding Universe, the initially dominant production rates
($e\bar{e},\gamma\gamma\to\mu\bar{\mu}$) decrease with decreasing temperature,
and eventually cross the $\mu^{\pm}$ decay rates. Muon abundance disappears as
soon as any decay rate is faster than the fastest production rate.
Specifically after the Universe cools below the temperature
$T_{\mathrm{disappear}}=4.195$ MeV, the dominant reaction is the muon decay.
Due to the relatively slow expansion of the Universe, the disappearance of
muons is sudden, and the abundance of muons vanishes as soon as a decay rate
surpasses the dominant production rate.

Figure 5.1: We plot the thermal reaction rate per volume for different
reactions as a function of temperature. We found that dominant reactions for
$\mu^{\pm}$ production are ${\gamma+\gamma\to\mu^{+}+\mu^{-}}$ and
$e^{+}+e^{-}\to\mu^{+}+\mu^{-}$, and the total production rate crosses the
decay rate of $\mu^{\pm}$ at temperature $T_{dissapear}\approx 4.195$ MeV.

On the other hand, considering the number density for nonrelativistic
$\mu^{\pm}$ in the Boltzmann approximation, we have

$\displaystyle
n_{\mu^{\pm}}=\frac{g_{\mu^{\pm}}}{2\pi^{2}}T^{3}\left(\frac{m_{\mu}}{T}\right)^{2}K_{2}(m_{\mu}/T)=g_{\mu^{\pm}}\left(\frac{m_{\mu}T}{2\pi}\right)^{3/2}e^{-{m_{\mu}}/{T}}\;.$
(5.16)

then the number density between $n_{\mu^{\pm}}$ and baryon $n_{B}$ can be
written as

$\displaystyle\frac{n_{\mu^{\pm}}}{n_{\mathrm{B}}}=\frac{n_{\mu^{\pm}}}{s}\frac{s}{n_{\mathrm{B}}}=\frac{n_{\mu^{\pm}}}{s}\left(\frac{s}{n_{\mathrm{B}}}\right)_{\\!t_{0}},$
(5.17)

where we used that $s/n_{\mathrm{B}}$ remains constant and $t_{0}$ represent
present day value. The present value is given by $(n_{B}/s)_{t_{0}}\approx
8.69\times 10^{-11}$ (detail please see Chapter 1). The entropy density $s$
can be characterized introducing $g^{s}_{\ast}$, the total number of
‘entropic’ degrees of freedom

$\displaystyle s=\frac{2\pi^{2}}{45}g^{s}_{\ast}T^{3}\;.$ (5.18)

For temperature $10\,\mathrm{MeV}>T>3$ MeV, the massless photons, nearly
relativistic electron/positrons, and practically massless neutrinos contribute
to the degree of freedom $g^{s}_{\ast}$. In this case, the number density
between $n_{\mu^{\pm}}$ and baryon $n_{B}$ in the temperature interval we
consider $10\,\mathrm{MeV}>T>3$ MeV is given by

$\displaystyle\frac{n_{\mu^{\pm}}}{n_{\mathrm{B}}}=\frac{45}{2\pi^{2}}\frac{g_{\mu^{\pm}}}{g^{s}_{\ast}}\left(\frac{m_{\mu}}{2\pi
T}\right)^{3/2}e^{-{m_{\mu}}/{T}}\;\left(\frac{s}{n_{\mathrm{B}}}\right)_{\\!t_{0}}.$
(5.19)

Figure 5.2:  The density ratio between $\mu^{\pm}$ and baryons as a function
of temperature. The density ratio at muon disappearance temperature is about
$n_{\mu^{\pm}}/n_{\mathrm{B}}(T_{\mathrm{disappear}})\approx 0.911$, and
around the temperature $T\approx 4.212$ MeV the density ratio
$n_{\mu^{\pm}}/n_{\mathrm{B}}\approx 1$.

In Fig. 5.2 we show the muon to baryon density ratio Eq. (5.19) as a function
of $T$. We see that the muon abundance $T=10$ MeV exceeds that of baryons by a
factor 500,000 while at muon disappearance temperature
$n_{\mu^{\pm}}/n_{\mathrm{B}}(T_{\mathrm{disappear}})\approx 0.911$. The
number density $n_{\mu^{\pm}}$ and $n_{\mathrm{B}}$ abundances are equal at
around the temperature $T_{\mathrm{equal}}\approx
4.212\,\mathrm{MeV}>T_{\mathrm{disappear}}$. This means that the muon
abundance may still be able to influence baryon evolution because their number
density is comparable to the baryon density.

The primary insight of this work is that aside of protons, neutrons and other
nonrelativistic particles, both positively and negatively charged muons
$\mu^{\pm}$ are present in thermal equilibrium and in non-negligible abundance
for $T>T_{\mathrm{dissapear}}\approx 4.195$ MeV. This offers a new and
tantalizing model building opportunity for anyone interested in baryon-
antibaryon separation in the primordial Universe, strangelet formation, and
perhaps other exotic primordial structure formation mechanisms.

###  Electron-positron plasma in the early Universe

In the early Universe, after the neutrino freeze-out at $T\approx 2$ MeV, the
Universe is controlled by the electron-positron-photon plasma. In this
section, we demonstrate the rich electron-positron plasma in the early
Universe by examining the chemical potential $\mu_{e}$ in the charge-neutral
and entropy-conserving Universe. We study the microscope collision property of
electron-positron plasma and explore the spin response of the electron-
positron plasma to external and self-magnetization fields, thus developing
methods for future detailed study.

####  Electron chemical potential in the early Universe

In this section, we derive the dependence of electron chemical potential, and
hence $e^{\pm}$ density, on the photon background temperature by employing the
following physical principles:

  1. 1.

Charge neutrality of the Universe:

$\displaystyle n_{e}-n_{\overline{e}}=n_{p}-n_{\overline{p}}\approx\,n_{p},$
(5.20)

where $n_{e}$ and $n_{\overline{e}}$ denotes the number density of electron
and positron.

  2. 2.

Neutrinos decouple (freeze-out) at a temperature $T_{f}\simeq 2$ MeV, after
which they free stream through the Universe with an effective temperature
[Birrell et al. (2014a)]

$\displaystyle T_{\nu}(t)=T_{f}a(t_{f})/a(t),$ (5.21)

where $a(t)$ is the FLRW Universe scale factor.

  3. 3.

Total comoving entropy is conserved. At $T\leq T_{f}$ the dominant
contributors to entropy are photons, $e^{\pm}$, and neutrinos. In addition,
after neutrino freeze-out, neutrino comoving entropy is independently
conserved [Birrell et al. (2014a)]. This implies that the combined comoving
entropy in $\gamma$, $e^{\pm}$ is also conserved for $T_{\gamma}\leq T_{f}$.

Motivated by the fact that comoving entropy in $\gamma$, $e^{\pm}$ is
conserved after neutrino freeze-out, we rewrite the charge neutrality
condition, Eq.(5.20) in the form

$\displaystyle
n_{e}-n_{\overline{e}}=X_{p}\frac{n_{B}}{s_{\gamma,e,\overline{e}}}s_{\gamma,e,\overline{e}},\qquad
X_{p}\equiv\frac{n_{p}}{n_{B}},$ (5.22)

where $n_{B}$ is the number density of baryons, and
$s_{\gamma,e,\overline{e}}$ is the combined entropy density in photons,
electrons, and positrons. During the Universe expansion, the comoving entropy
and baryon number are conserved quantities, hence the ratio
$n_{B}/s_{\gamma,e,\overline{e}}$ is conserved. We have

$\displaystyle\frac{n_{B}}{s_{\gamma,e,\overline{e}}}=\left(\frac{n_{B}}{s_{\gamma,e,\overline{e}}}\right)_{t_{0}}\\!\\!\\!\\!=\left(\frac{n_{B}}{s_{\gamma}}\right)_{t_{0}}\\!\\!\\!\\!=\left(\frac{n_{B}}{n_{\gamma}}\right)_{t_{0}}\left(\frac{n_{\gamma}}{s_{\gamma}}\right)_{t_{0}},$
(5.23)

where the subscript $t_{0}$ denotes the present day value, and the second
equality is obtained by observing that the present day $e^{\pm}$-entropy
density is negligible compared to the photon entropy density. We can evaluate
the ratio by giving the present day baryon-to-photon ratio:
$n_{B}/n_{\gamma}=6.05\times 10^{-10}$(CMB) [Workman et al. (2022)] and the
entropy per particle for a massless boson:$(s/n)_{\mathrm{boson}}\approx
3.602$ [Letessier and Rafelski (2002)].

The total entropy density of photons and electron/positron can be written as

$\displaystyle
s_{\gamma,e,\overline{e}}=\frac{2\pi^{2}}{45}g_{\gamma}\,T_{\gamma}^{3}+\frac{\rho_{e,\overline{e}}+P_{e,\overline{e}}}{T_{\gamma}}-\frac{\mu_{e}}{T_{\gamma}}(n_{e}-n_{\overline{e}}),$
(5.24)

where $\rho_{e,\overline{e}}=\rho_{e}+\rho_{\overline{e}}$ and
$P_{e,\overline{e}}=P_{e}+P_{\overline{e}}$ are the total energy density and
pressure of electrons/positron respectively. The energy density and pressure
in electrons and positrons are given by

$\displaystyle\frac{\rho_{e,\overline{e}}}{T_{\gamma}^{4}}=\frac{g_{e}}{2\pi^{2}}M_{e}^{4}\bigg{[}$
$\displaystyle\int_{1}^{\infty}\frac{u^{2}\sqrt{u^{2}-1}du}{\exp(M_{e}u-b_{e})+1}+\int_{1}^{\infty}\frac{u^{2}\sqrt{u^{2}-1}du}{\exp(M_{e}u+b_{e})+1}\bigg{]}\,,$
(5.25)

and

$\displaystyle\frac{P_{e,\overline{e}}}{T_{\gamma}^{4}}=\frac{g_{e}}{6\pi^{2}}M_{e}^{4}\bigg{[}$
$\displaystyle\int_{1}^{\infty}\frac{(u^{2}-1)^{3/2}du}{\exp(M_{e}u-b_{e})+1}+\int_{1}^{\infty}\frac{(u^{2}-1)^{3/2}du}{\exp(M_{e}u+b_{e})+1}\bigg{]},$
(5.26)

where we introduce the dimensionless variables as follows:

$\displaystyle u=\frac{E}{m_{e}},\qquad M_{e}=\frac{m_{e}}{T_{\gamma}},\qquad
b_{e}=\frac{\mu_{e}}{T_{\gamma}}.$ (5.27)

By incorporating Eq.(5.22) and Eq.(5.24), the charge neutrality condition can
be expressed as

$\displaystyle\left[1+X_{p}\left(\frac{n_{B}}{n_{\gamma}}\right)_{t_{0}}\left(\frac{n_{\gamma}}{s_{\gamma}}\right)_{t_{0}}\frac{\mu_{e}}{T_{\gamma}}\right]\frac{n_{e}-n_{\overline{e}}}{T_{\gamma}^{3}}$
$\displaystyle\qquad\qquad\qquad=X_{p}\left(\frac{n_{B}}{n_{\gamma}}\right)_{t_{0}}\left(\frac{n_{\gamma}}{s_{\gamma}}\right)_{t_{0}}\left(\frac{2\pi^{2}}{45}g_{\gamma}+\frac{\rho_{e,\overline{e}}+P_{e,\overline{e}}}{T_{\gamma}^{4}}\right).$
(5.28)

Using the Fermi distribution, the number density of electrons over positrons
in the early Universe is given by

$\displaystyle n_{e}-n_{\overline{e}}$
$\displaystyle=\frac{g_{e}}{2\pi^{2}}\left[\int_{0}^{\infty}\frac{p^{2}dp}{\exp{\left((E-\mu_{e})\right)/T_{\gamma}}+1}\right.\left.-\int_{0}^{\infty}\frac{p^{2}dp}{\exp{\left((E+\mu_{e})/T_{\gamma}\right)}+1}\right]$
$\displaystyle=\frac{g_{e}}{2\pi^{2}}{T_{\gamma}^{3}}\tanh(b_{e})M_{e}^{3}\int_{1}^{\infty}\\!\\!\\!\\!\frac{u\sqrt{u^{2}-1}du}{1+\cosh(M_{e}u)/\cosh(b_{e})}.$
(5.29)

Substituting Eq.(5.3.1) into Eq.(5.3.1) and giving the value of $X_{p}$, the
charge neutrality condition can be solved to determine $\mu_{e}/T_{\gamma}$ as
a function of $M_{e}$ and $T_{\gamma}$.

Figure 5.3: Left axis: The chemical potential of an electron as a function of
photon temperature $T=T_{\gamma}$ with $X_{p}=0.878$ and
$n_{B}/n_{\gamma}=6.05\times 10^{-10}$. Right axis: the ratio of
electron(positron) number density to baryon density as a function of
temperature. The blue solid line is the electron density, the red dashed line
is the positron density, and the green dotted line is the number density with
$\mu_{e}=0$. We found that when electron chemical potential $\mu_{e}\approx
T=0.02\,\mathrm{MeV}$ the positron density decreases because of the
annihilation.

In Fig. 5.3 (left axis) we solve Eq.(5.3.1) numerically and plot the electron
chemical potential as a function of temperature with the following parameters:
proton concentration $X_{p}=0.878$ from observation [Workman et al. (2022)]
and $n_{B}/n_{\gamma}=6.05\times 10^{-10}$ from CMB. We can see the value of
chemical potential is comparatively small $\mu_{e}/T\approx 10^{-6}\sim
10^{-7}$ during the BBN temperature range, implying an equal number of
electrons and positrons in plasma. From the ratio of electron (positron)
number density to baryon density in Fig. 5.3 (right axis) we can see that
during the accepted BBN temperature range the Universe was filled with an
electron-positron rich plasma. For example when the temperature is around
$T=70\,\mathrm{keV}$ the density of electrons and positrons is comparatively
large in the early Universe $n_{e^{\pm}}\approx 10^{7}\,n_{B}$. Later when the
temperature is around $T=20.3\,\mathrm{keV}$, the positron density decreases,
leading to the transformation of the pair plasma to an electron-proton plasma.

####  Microscope damping rate of electron-positron plasma

In electron-positron plasma, the major reactions between photons and
$e^{+}e^{-}$ pairs are inverse Compton scattering, Møller scattering, and
Bhabha scattering:

$\displaystyle e^{\pm}+\gamma\longrightarrow e^{\pm}+\gamma,\qquad
e^{\pm}+e^{\pm}\longrightarrow e^{\pm}+e^{\pm},\qquad
e^{\pm}+e^{\mp}\longrightarrow e^{\pm}+e^{\mp}.$ (5.30)

The general formula for thermal reaction rate per volume is discussed in
[Letessier and Rafelski (2002)] (Eq.(17.16), Chapter 17). For inverse Compton
scattering we have

$\displaystyle
R_{e^{\pm}\gamma}=\frac{g_{e}g_{\gamma}}{16\left(2\pi\right)^{5}}T\int_{m_{e}^{2}}^{\infty}\\!\\!\\!\\!ds\frac{K_{1}(\sqrt{s}/T)}{\sqrt{s}}\int^{0}_{-(s-m_{e}^{2})^{2}/s}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!dt\,|M_{e^{\pm}\gamma}|^{2},$
(5.31)

and for Møller and Bhabha reactions we have

$\displaystyle
R_{e^{\pm}e^{\pm}}=\frac{g_{e}g_{e}}{16\left(2\pi\right)^{5}}T\\!\\!\int_{4m_{e}^{2}}^{\infty}\\!\\!\\!\\!ds\frac{K_{1}(\sqrt{s}/T)}{\sqrt{s}}\int^{0}_{-(s-4m_{e}^{2})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!dt\,|M_{e^{\pm}e^{\pm}}|^{2},$
(5.32) $\displaystyle
R_{e^{\pm}e^{\mp}}=\frac{g_{e}g_{e}}{16\left(2\pi\right)^{5}}T\\!\\!\int_{4m_{e}^{2}}^{\infty}\\!\\!\\!\\!ds\frac{K_{1}(\sqrt{s}/T)}{\sqrt{s}}\int^{0}_{-(s-4m_{e}^{2})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!dt\,|M_{e^{\pm}e^{\mp}}|^{2},$
(5.33)

where $g_{i}$ is the degeneracy of particle $i$, $|M|^{2}$ is the matrix
element for a given reaction, $K_{1}$ is the Bessel function of order $1$, and
$s,t,u$ are Mandelstam variables. The leading order matrix element associated
with inverse Compton scattering can be expressed in the Mandelstam variables
[Kuznetsova and Rafelski (2012); Kuznetsova et al. (2010)] we have

$\displaystyle|M_{e^{\pm}\gamma}|^{2}\\!=32\pi^{2}\alpha^{2}\bigg{[}$
$\displaystyle
4\left(\frac{m_{e}^{2}}{m_{e}^{2}-s}+\frac{m_{e}^{2}}{m_{e}^{2}-u}\right)^{2}$
$\displaystyle\qquad\qquad-\frac{4m_{e}^{2}}{m_{e}^{2}-s}-\frac{4m_{e}^{2}}{m_{e}^{2}-u}-\frac{m_{e}^{2}-u}{m_{e}^{2}-s}-\frac{m_{e}^{2}-s}{m_{e}^{2}-u}\bigg{]},$
(5.34)

and for Møller and Bhabha scattering we have

$\displaystyle|M_{e^{\pm}e^{\pm}}|^{2}\\!=64\pi^{2}\alpha^{2}\bigg{[}$
$\displaystyle\frac{s^{2}+u^{2}+8m_{e}^{2}(t-m_{e}^{2})}{2(t-m^{2}_{\gamma})^{2}}$
$\displaystyle\quad+\frac{{s^{2}+t^{2}}+8m_{e}^{2}(u-m_{e}^{2})}{2(u-m_{\gamma}^{2})^{2}}+\frac{\left({s}-2m_{e}^{2}\right)\left({s}-6m_{e}^{2}\right)}{(t-m_{\gamma}^{2})(u-m_{\gamma}^{2})}\bigg{]},$
(5.35)

and

$\displaystyle|M_{e^{\pm}e^{\mp}}|^{2}=64\pi^{2}\alpha^{2}\bigg{[}$
$\displaystyle\frac{s^{2}+u^{2}+8m_{e}^{2}(t-m_{e}^{2})}{2(t-m^{2}_{\gamma})^{2}}$
$\displaystyle\quad+\frac{u^{2}+t^{2}+8m_{e}^{2}(s-m_{e}^{2})}{2(s-m^{2}_{\gamma})^{2}}+\frac{\left({u}-2m_{e}^{2}\right)\left({u}-6m_{e}^{2}\right)}{(t-m^{2}_{\gamma})(s-m^{2}_{\gamma})}\bigg{]},$
(5.36)

where we introduce the photon mass $m_{\gamma}$ to account the plasma effect
and avoid singularity in reaction matrix elements.

The photon mass $m_{\gamma}$ in plasma is equal to the plasma frequency
$\omega_{p}$, where we have [Kislinger and Morley (1976)]

$\displaystyle
m^{2}_{\gamma}=\omega^{2}_{p}=8\pi\alpha\int\frac{d^{3}p_{e}}{(2\pi)^{3}}\left(1-\frac{p_{e}^{2}}{3E_{e}^{2}}\right)\frac{f_{e}+f_{\bar{e}}}{E_{e}},$
(5.37)

where $E_{e}=\sqrt{p_{e}^{2}+m^{2}_{e}}$. In the BBN temperature range
$86\,\mathrm{keV}>T_{BBN}>50\,\mathrm{keV}$ we have $m_{e}\gg T$ and
considering the nonrelativistic limit for electron-positron plasma, we obtain

$\displaystyle
m^{2}_{\gamma}=\frac{4\pi\alpha}{2m_{e}}\left(\frac{2m_{e}T}{\pi}\right)^{3/2}e^{-m_{e}/T}\cosh\left(\frac{\mu_{e}}{T}\right).$
(5.38)

In the BBN temperature range, we have $\mu_{e}/T\ll 1$, which implies the
equal number of electrons and positrons in plasma.

To discuss the collisional plasma by the linear response theory, it is
convenient to define the average relaxation rate for the electron-positron
plasma as follows:

$\displaystyle\kappa=\frac{R_{e^{\pm}e^{\pm}}+R_{e^{\pm}e^{\mp}}+R_{e^{\pm}\gamma}}{\sqrt{n_{e^{-}}n_{e^{+}}}}\approx\frac{R_{e^{\pm}e^{\pm}}+R_{e^{\pm}e^{\mp}}}{\sqrt{n_{e^{-}}n_{e^{+}}}},$
(5.39)

where the density function ${\sqrt{n_{e^{-}}n_{e^{+}}}}$ in the Boltzmann
limit is given by

$\displaystyle{\sqrt{n_{e^{-}}n_{e^{+}}}}=\frac{g_{e}}{2\pi^{3}}T^{3}\left(\frac{m_{e}}{T}\right)^{2}K_{2}(m_{e}/T).$
(5.40)

In Fig. 5.4, we show the reaction rates for Møller reaction, Bhabha reaction,
and inverse Compton scattering as a function of temperature. For temperatures
$T>12.0$ keV, the dominant reactions in plasma are Møller and Bhabha
scatterings between electrons and positrons. Thus in the BBN temperature
range, we can neglect the inverse Compton scattering. The total relaxation
rate is approximately constant $\kappa=10\sim 12$ keV during the BBN. For
$T<20.3$ keV the relaxation rate $\kappa$ decreases rapidly because of
positron annihilation. At this temperature, the composition of plasma begins
to change from an electron-positron plasma to an electron-baryon plasma.

Figure 5.4: The relaxation rate $\kappa$ as a function of temperature in
nonrelativistic electron-positron plasma. For comparison, we show reaction
rates for Møller reaction $e^{-}+e^{-}\to e^{-}+e^{-}$ (blue line), Bhabha
reaction $e^{-}+e^{+}\to e^{-}+e^{+}$ (red line), and inverse Compton
scattering $e^{-}+\gamma\to e^{-}+\gamma$ (green line) respectively. It shows
that the dominant reactions during BBN are the Møller and Bhabha scatterings
between electrons and positrons. The total relaxation rate Eq.(5.39) is shown
in the black line. It shows that we have $\kappa=10\sim 12$ keV during the BBN
temperature range. For comparison, the Debye mass
$m_{D}=\omega_{p}\sqrt{m_{e}/T}$(purple line) is shown as a function of
temperature.

##### From static to damped dynamic screening

At present, the observation of light element (e.g. D, 3He, 4He, and 7Li)
abundances produced in Big-Bang nucleosynthesis (BBN) offers a reliable probe
of the early Universe before the recombination. Much effort of the BBN study
is currently being made to reconcile the discrepancies and tensions between
theoretical predictions and observations of light element abundances, e.g. 7Li
problem [Pitrou et al. (2018); Fields (2011)]. Current models assume that the
Universe was essentially void of anything but reacting light nucleons and
electrons needed to keep the local baryon density charge-neutral, a situation
similar to the experimental environment where empirical nuclear reaction rates
are obtained.

The electron-positron plasma influences light element abundances through
electromagnetic screening of the nuclear potential. The electron cloud
surrounding the charge of an ion screens other nuclear charges far from its
own radius and reduces the Coulomb barrier. In nuclear reactions, the
reduction of Coulomb barrier makes the penetration probability easier and
enhance the thermonuclear reaction rates. In this case, the modification of
the nuclei interaction due to the plasma screening effect may plays a key role
in the formation of light element in the BBN.

The enhancement factor of thermonuclear reaction rates and screening potential
are calculated by Salpeter in 1954 [Salpeter (1954)], which describes the
static screening effects for the thermonuclear reactions. In an isotropic and
homogeneous plasma the Coulomb potential of a point-like particle with charge
$Ze$ at rest is modified into [Salpeter (1954)]

$\displaystyle\phi_{\text{stat}}(r)=\frac{Ze}{4\pi\epsilon_{0}r}e^{-m_{D}r},$
(5.41)

where $m_{D}$ is the Debye mass. After that it has been exploited widely in
BBN for static screening [Salpeter and van Horn (1969); Famiano et al.
(2016)].

Subsequently, the study of dynamical screening for moving ions has been taken
into account [Carraro et al. (1988); Gruzinov (1998); Hwang et al. (2021)].
When a test charge moves with a velocity that is enough to react with the
background charge in plasma, the Coulomb potential is modified by the
dynamical effect. However, the applications focus on the weakly interacting
electron-positron plasma only.

In our separate work [Grayson et al. (2023)] we use the linear response theory
adapted by C.Grayson to to describe the inter nuclear potential in electron-
positron plasma during BBN. We improve the prior efforts by evaluation and
inclusion of the collision damping rate due to scattering in the dense plasma
medium and provide an approximate analytic formula that can be readily used to
estimate the effect of screening on internuclear potential. For comprehensive
discussion and the application of the damped dynamic screening see [Grayson et
al. (2023)].

####  Magnetization of the electron-positron plasma

In the present-day Universe, we have magnetic fields [Giovannini (2004);
Kronberg (1994a, b)] at various scales and strengths both within galaxies and
in deep extra-galactic space far away from matter sources. Current
observations suggest the upper and lower bounds for the Extra-Galactic
Magnetic Field (EGMF) are given by [Neronov and Vovk (2010); Taylor et al.
(2011); Pshirkov et al. (2016); Jedamzik and Saveliev (2019); Vernstrom et al.
(2021)]

$\displaystyle 10^{-8}{\mathrm{G}}>B_{\mathrm{EGFM}}>10^{-16}{\mathrm{G}}\,.$
(5.42)

The origin for EGMF today is currently unknown; different models are
considered in lectures [Widrow et al. (2012); Vazza et al. (2021)]. In our
work [Rafelski et al. (2023)], we investigate the hypothesis that the observed
EGMF are primordial in nature, predating even the recombination epoch. Under
this hypothesis, the first best candidate is the electron-positron plasma.
This is because for the temperature range $200\,\mathrm{keV}>T>20$ keV, we
still have relatively large quantity of both $e^{\pm}$ in the the early
Universe plasma. In addition, electrons and positrons have the largest
magnetic moments in nature, are likely to have been magnetized in the early
Universe due to spin orientation. These provide the possibility origins for a
primordial magnetic field.

As the Universe undergoes the isentropic expansion, the temperature gradually
decreases as $T\propto 1/a(t)$, where $a(t)$ represents the scale factor. The
assumption is made that the magnetic flux is conserved over comoving surfaces,
implying that the primordial relic field is expected to dilute as $B\propto
1/a(t)^{2}$ [Rafelski et al. (2023)]. Combining these cosmological redshift
relations, we can introduce a dimensionless cosmic magnetic scale that remains
unchanged during the evolution of the Universe

$\displaystyle
b\equiv\frac{e{B}}{T^{2}}=\left(\frac{e{B}}{T^{2}}\right)_{t_{0}}=b_{0}={\rm\
const.}\qquad 10^{-3}>b_{0}>10^{-11}\,.$ (5.43)

The upper and lower bounds for $b_{0}$ are estimated by using the present day
EGMF observations Eq. (5.42) and the present CMB temperature
$T_{0}=2.7\,\mathrm{K}\approx 2.3\times 10^{-4}$ eV [Aghanim et al. (2020b)].
As $b_{0}$ is a constant of expansion, this means the contemporary small
bounded values of may have once represented large magnetic fields in the early
Universe and require detailed study in a different epoch of the Universe.
Therefore, correctly describing the dynamics of this $e^{\pm}$ plasma is of
interest when considering the origin of extra-galactic magnetic fields (EGMF).

In the following, we will demonstrate that fundamental quantum statistical
analysis can lead to further insights on the behavior of magnetized plasma,
and show that the $e^{\pm}$ plasma is overall paramagnetic and yields a
positive overall magnetization, which is contrary to the traditional
assumption that matter-antimatter plasma lack significant magnetic responses.
For more detailed discussion of electron-positron plasma magnetization, please
see [Steinmetz et al. (2023a)].

##### Electron-positron partition function

To study the statistical behavior of the $e^{\pm}$ system in a magnetic field,
we utilize the general Fermion partition function [Elze et al. (1980)]

$\displaystyle\ln\mathcal{Z}=\sum_{\alpha}\ln\left(1+e^{-\beta(E-\eta)}\right)\,,$
(5.44)

where $\beta=1/T$, $\alpha$ is the set of all quantum numbers in the system,
and $\eta$ is the generalized chemical potential. In the case of a magnetized
$e^{\pm}$ system, we consider it as a system of four quantum species:
Particles and antiparticles, and spin aligned and anti-aligned. Taken
together, we consider a system where electrons and positrons can be spin
aligned or anti-aligned with the magnetic field $B$ and the partition function
of the system can be written as

$\displaystyle\ln\mathcal{Z}_{tot}=\frac{2eBV}{(2\pi)^{2}}\sum_{\sigma}^{\pm
1}\sum_{s}^{\pm
1/2}\sum_{n=0}^{\infty}\int^{\infty}_{0}dp_{z}\left[\ln\left(1+\Upsilon(x)e^{(\sigma\eta_{e}+s\eta_{s})/T}e^{-\beta
E_{n}^{s}}\right)\right]\,,$ (5.45)

where $n$ is the principle quantum number for the Landau levels. The parameter
$\eta_{e}$ is the electron chemical potential and $\eta_{s}$ is the spin
chemical potential [Steinmetz et al. (2023a)]. The parameter $\Upsilon(x)$ is
the fugacity of the Fermi gas. In this thesis we will focus on the case
$\Upsilon(x)=1$ and $\eta_{s}=0$ , we leave the general case $\Upsilon(x)\neq
1$ and $\eta_{s}\neq 0$ for future work.

In the following, we will retain $\Upsilon(x)=1$ and consider the case
$\eta_{s}/T\ll 1$ for the first approximation. Then the partition function
becomes

$\displaystyle\ln\mathcal{Z}_{tot}=\frac{2eBV}{(2\pi)^{2}}\sum_{s}^{\pm
1/2}\sum_{n=0}^{\infty}\int^{\infty}_{0}\\!\\!dp_{z}\bigg{[}\ln\left(1+e^{-\beta(E_{n}^{s}-\eta_{e})}\right)+\ln\left(1+e^{-\beta(E_{n}^{s}+\eta_{e})}\right)\bigg{]}.$
(5.46)

Considering the $e^{\pm}$ plasma in a uniform magnetic field $B$ pointing
along the $z$-axis, the energy $E_{n}^{\pm}$ of electron/positron system can
be written as [Rafelski et al. (2023)]

$\displaystyle
E_{n}^{\pm}=\sqrt{p^{2}_{z}+\tilde{m}^{2}_{\pm}+2eBn},\qquad\tilde{m}^{2}_{\pm}=m^{2}_{e}+eB\left(1\mp\frac{g}{2}\right)\,,$
(5.47)

where the $\pm$ script refers to spin aligned and anti-aligned eigenvalues.
The parameter $g$ is the gyro-magnetic ($g$-factor) of the particle.

To simplify the partition function, we consider the expansion of the
logarithmic function as follows:

$\displaystyle\ln\left(1+x\right)=\sum^{\infty}_{k=1}\frac{(-1)^{k+1}}{k}x^{k},\,\,\,\,\,\,\,\mathrm{for}\,|x|<1.$
(5.48)

Then the partition function of electron/positron system can be written as

$\displaystyle\ln\mathcal{Z}_{tot}=$
$\displaystyle\frac{2eBV}{(2\pi)^{2}}\sum_{n=0}^{\infty}\int^{\infty}_{0}\\!\\!dp_{z}\sum^{\infty}_{k=1}\frac{(-1)^{k+1}}{k}\bigg{[}e^{k\beta\mu_{e}}+e^{-k\beta\mu_{e}}\bigg{]}e^{-k\beta
E_{n}^{\pm}}$
$\displaystyle=\frac{2eBV}{(2\pi)^{2}}\sum_{n=0}^{\infty}\sum^{\infty}_{k=1}\frac{(-1)^{k+1}}{k}\bigg{[}2\cosh{(k\beta\mu_{e})}\bigg{]}\int_{0}^{\infty}dp_{z}\,e^{-k\beta
E_{n}^{\pm}}.$ (5.49)

Using the general definition of Bessel function:

$\displaystyle K_{\nu}(\beta
m)=\frac{\sqrt{\pi}}{\Gamma({\nu-1/2})}\frac{1}{m}\left(\frac{\beta}{2m}\right)^{\nu-1}\int_{0}^{\infty}\,dp\,p^{2\nu-2}e^{-\beta
E}\,\,\,\,\,\,\,\mathrm{for}\,\nu>1/2,$ (5.50)

the integral over $dp_{z}$ can be written as

$\displaystyle\int_{0}^{\infty}dp_{z}\,e^{-k\beta E_{n}^{\pm}}$
$\displaystyle=\frac{\Gamma{(1/2)}}{\sqrt{\pi}}\sqrt{\tilde{m}^{2}_{\pm}+2eBn}\,\,K_{1}\\!\\!\left({k\sqrt{\tilde{m}^{2}_{\pm}+2eBn}}/{T}\right)$
$\displaystyle=\sqrt{\tilde{m}^{2}_{\pm}+2eBn}\,\,K_{1}\\!\\!\left({k\sqrt{\tilde{m}^{2}_{\pm}+2eBn}}/{T}\right).$
(5.51)

In this case, the partition function becomes

$\displaystyle\ln\mathcal{Z}_{tot}$
$\displaystyle=\frac{2eBV}{(2\pi)^{2}}\sum_{n=0}^{\infty}\sum^{\infty}_{k=1}\frac{(-1)^{k+1}}{k}\bigg{[}2\cosh{(k\beta\mu_{e})}\bigg{]}\sqrt{\tilde{m}^{2}_{\pm}+2eBn}\,\,K_{1}({k\sqrt{\tilde{m}^{2}_{\pm}+2eBn}}/{T})$
$\displaystyle=\frac{2eBTV}{(2\pi)^{2}}\sum^{\infty}_{k=1}\frac{(-1)^{k+1}}{k^{2}}\bigg{[}2\cosh{(k\beta\mu_{e})}\bigg{]}\sum_{n=0}^{\infty}W^{\pm}_{1}(n),$
(5.52)

where we introduce the function $W^{\pm}_{1}(n)$ as follows

$\displaystyle
W^{\pm}_{1}(n)\equiv\frac{k\sqrt{\tilde{m}^{2}_{\pm}+2eBn}}{T}\,\,K_{1}\\!\\!\left({k\sqrt{\tilde{m}^{2}_{\pm}+2eBn}}/{T}\right).$
(5.53)

Considering the Euler-Maclaurin formula to replace the sum over Landau levels,
we have

$\displaystyle\sum^{\infty}_{n=0}W^{\pm}_{1}(n)=\int^{\infty}_{0}\\!\\!dn\,W^{\pm}_{1}(n)$
$\displaystyle+\frac{1}{2}\bigg{[}W^{\pm}_{1}(\infty)+W^{\pm}_{1}(0)\bigg{]}$
$\displaystyle\qquad+\frac{1}{12}\bigg{[}\left.\frac{\partial
W^{\pm}_{1}}{\partial n}\right|_{\infty}-\left.\frac{\partial
W^{\pm}_{1}}{\partial n}\right|_{0}\bigg{]}+R,$ (5.54)

where $R$ is the error remainder which is defined by integrals over Bernoulli
polynomials which is small and can be neglected [Elze et al. (1980)]. Using
the properties of Bessel function we have

$\displaystyle\frac{\partial W^{\pm}_{1}}{\partial
n}=-\frac{k^{2}eB}{T^{2}}K_{0}\left({k\sqrt{\tilde{m}^{2}_{\pm}+2eBn}}/{T}\right),\qquad
W^{\pm}_{1}(\infty)=0,$ (5.55)
$\displaystyle\int^{\infty}_{a}\\!\\!dx\,x^{2}K_{1}(x)=a^{2}K_{2}(a),$ (5.56)

then we obtain

$\displaystyle\sum^{\infty}_{n=0}W^{\pm}_{1}(n)$
$\displaystyle=\left(\frac{T^{2}}{k^{2}eB}\right)\left[\left(\frac{k\tilde{m}_{\pm}}{T}\right)^{2}K_{2}(k\tilde{m}_{\pm}/T)\right]+\frac{1}{2}\left[\left(\frac{k\tilde{m}_{\pm}}{T}\right)K_{1}(k\tilde{m}_{\pm}/T)\right]$
$\displaystyle\qquad+\frac{1}{12}\left[\left(\frac{k^{2}eB}{T^{2}}\right)K_{0}(k\tilde{m}_{\pm}/T)\right].$
(5.57)

Replacing the sum over Landau levels by the integral, the partition function
becomes

$\displaystyle\ln\mathcal{Z}_{tot}=\ln\mathcal{Z}_{free}+\ln\mathcal{Z}_{B}\,,$
(5.58)

where we define the partition functions as

$\displaystyle\ln\mathcal{Z}_{free}=\frac{T^{3}V}{2\pi^{2}}\left[2\cosh{\left(\frac{\eta_{e}}{T}\right)}\right]\sum_{i=\pm}x_{i}^{2}K_{2}\left(x_{i}\right)\,,\qquad
x_{i}=\frac{\tilde{m}_{i}}{T}$ (5.59)
$\displaystyle\ln\mathcal{Z}_{B}=\frac{eBTV}{2\pi^{2}}\left[2\cosh{\left(\frac{\eta_{e}}{T}\right)}\right]\sum_{i=\pm}\bigg{[}\frac{x_{i}}{2}K_{1}\left(x_{i}\right)+\frac{b_{0}}{12}K_{0}\left(x_{i}\right)\bigg{]}\,.$
(5.60)

The partition function $\ln(\mathcal{Z}_{free})$ in Eq. (5.59) represents the
general form of the Fermi partition function for $e^{\pm}$ with ”effective
mass” $\tilde{m}_{\pm}$ in our system. When the magnetic field $B=0$ the
function $\ln(\mathcal{Z}_{free})$ will go back to the general form of the
Fermi partition function without the external field. The partition function
$\ln\mathcal{Z}_{B}$ gives us the partition with magnetic field effect to the
order $\mathcal{O}(eB)$ and $\mathcal{O}(eB)^{2}$.

In the temperature domain $200\,\mathrm{keV}>T>20$ keV, we have $m_{e}\gg T$,
and it suffices to consider the Boltzmann limit of the quantum distributions.
Considering the Boltzmann approximation for non-relativistic electrons and
positrons we can rewrite Eq. (5.59) - Eq. (5.60) and obtain

$\displaystyle\ln\mathcal{Z}_{tot}\\!=\\!\frac{T^{3}V}{2\pi^{2}}\left[2\cosh\left(\frac{\eta_{e}}{T}\right)\right]\sum_{i=\pm}\left\\{x_{i}^{2}K_{2}\left(x_{i}\right)+\frac{b_{0}}{2}x_{i}K_{1}\left(x_{i}\right)+\frac{b^{2}_{0}}{12}K_{0}\left(x_{i}\right)\right\\}.$
(5.61)

Given the partition function Eq. (5.61), we can explore the chemical potential
and magnetization of $e^{\pm}$ plasma in the early Universe under the
hypothesis of charge neutrality and entropy conservation.

##### Electron chemical potential under magnetic field

We explore the chemical potential of electron-positron plasma in a uniform
magnetic field $B$ in the early Universe under the hypothesis of charge
neutrality and entropy conservation. Considering the temperature after
neutrino freeze-out, the charge neutrality condition can be written as

$\displaystyle\left(n_{e}-n_{\bar{e}}\right)=n_{p}=X_{p}\,\left(\frac{n_{B}}{s_{\gamma,e}}\right)\,s_{\gamma,e},\qquad
X_{p}\equiv\frac{n_{p}}{n_{B}}\,,$ (5.62)

where $n_{p}$ and $n_{B}$ is the number density of protons and baryons
respectively. Using the partition function Eq. (5.61), the net number density
of electrons in Boltzmann approximation can be written as

$\displaystyle\left(n_{e}-n_{\bar{e}}\right)$
$\displaystyle=\frac{T}{V}\frac{\partial}{\partial\eta_{e}}\ln\mathcal{Z}_{tot}$
$\displaystyle=\frac{T^{3}}{2\pi^{2}}\left[2\sinh{(\eta_{e}/T)}\right]\sum_{i=\pm}\left[x_{i}^{2}K_{2}(x_{i})+\frac{b_{0}}{2}x_{i}K_{1}(x_{i})+\frac{b^{2}_{0}}{12}K_{0}(x_{i})\right]\,.$
(5.63)

Substituting Eq. (5.3.3) into the charge neutrality condition Eq. (5.62), we
can solve the chemical potential of electron $\eta_{e}/T$ numerically. We have

$\displaystyle\sinh{(\eta_{e}/T)}$
$\displaystyle=\frac{2\pi^{2}}{2T^{3}}\,\frac{X_{p}(n_{B}/s_{\gamma,e})s_{\gamma,e}}{\sum_{i=\pm}\left[x_{i}^{2}K_{2}(x_{i})+\frac{b_{0}}{2}x_{i}K_{1}(x_{i})+\frac{b^{2}_{0}}{12}K_{0}(x_{i})\right]}\,,$
(5.64)
$\displaystyle\longrightarrow\frac{2\pi^{2}n_{p}}{2T^{3}}\,\frac{X_{p}(n_{B}/s_{\gamma,e})s_{\gamma,e}}{2x^{2}K_{2}(x)},\qquad
x=m_{e}/T,\qquad\mathrm{for}\,\,b_{0}=0.$ (5.65)

We see in Eq. (5.65) that for the case $b_{0}=0$, the chemical potential
agrees with the free particle result in [Grayson et al. (2023)]. In Fig. 5.5
we plot the chemical potential of electron as a function of temperature with
different value of $b_{0}$. It shows that the chemical potential is not
sensitive to the magnetic field because the small value of
$10^{-3}>b_{0}>10^{-11}$ can be neglected in Eq. (5.64).

Figure 5.5: The chemical potential of electron as a function of temperature in
the magnetic field $b_{0}$ with $X_{p}=0.878$ and $n_{B}/n_{\gamma}=6.05\times
10^{-10}$. The red dashed line represents the magnetic field $b_{0}=1.1\times
10^{-11}$ and blue line labels the magnetic field $b_{0}=5.5\times 10^{-3}$

##### Electron-positron magnetization

Considering the magnetized electron-positron partition function Eq. (5.61), it
is convenient to introduce the dimensionless magnetization
$\overline{\mathcal{M}}$ and the critical field $B_{c}$ as follows

$\displaystyle\overline{\mathcal{M}}\equiv\frac{M}{\mathcal{B}_{c}}=\frac{1}{\mathcal{B}_{c}}\left(\frac{T}{V}\frac{\partial\ln\mathcal{Z}_{tot}}{\partial
B}\right)\,\qquad\mathcal{B}_{c}=\frac{m_{e}^{2}}{e}\,.$ (5.66)

Substituting the partition function Eq. (5.61) into Eq. (5.66), the total
magnetization ${\overline{\mathcal{M}}}$ can be broken into the sum of spin
parallel $\overline{\mathcal{M}}_{+}$ and spin anti-parallel
$\overline{\mathcal{M}}_{-}$ magnetization. We have

$\displaystyle{\overline{\mathcal{M}}}={\overline{\mathcal{M}}_{+}}+{\overline{\mathcal{M}}_{-}},$
(5.67)
$\displaystyle\overline{\mathcal{M}}_{\pm}=\frac{e^{2}T^{2}}{2\pi^{2}m_{e}^{2}}\left[2\cosh\left(\frac{\eta_{e}}{T}\right)\right]\left\\{c_{1}(x_{\pm})K_{1}(x_{i})+c_{0}K_{0}(x_{\pm})\right\\}\,,$
(5.68)

where the coefficients are given by

$\displaystyle c_{1}(x_{\pm})$
$\displaystyle=\left[\frac{1}{2}-\left(\frac{1}{2}\pm\frac{g}{4}\right)\left(1+\frac{b^{2}_{0}}{12x^{2}_{\pm}}\right)\right]x_{\pm}\,,\qquad
c_{0}=\left[\frac{1}{6}-\left(\frac{1}{4}\pm\frac{g}{8}\right)\right]b_{0}\,.$
(5.69)

Substituting the chemical potential Eq. (5.64) into Eq. (5.67), we can solve
the magnetization numerically.

Figure 5.6: The magnetization $\overline{\cal M}=\mathcal{M}/\mathcal{B}_{C}$,
with $g=2$, of the primordial $e^{+}e^{-}$ plasma is plotted as a function of
temperature. The lower (solid red) and upper (solid blue) bounds for cosmic
magnetic scale $b_{0}$ are included. The external magnetic field strength
${\cal B}/{\cal B}_{C}$ is also plotted in for lower (dashed red) and upper
(dashed blue) bounds. The spin fugacity is set to unity.

In this thesis we focus on considering the case for $g=2$. In this case, the
electron-positron magnetization can be written as

$\displaystyle{\overline{\mathcal{M}}}={\overline{\mathcal{M}}_{+}}+{\overline{\mathcal{M}}_{-}}$
(5.70) $\displaystyle{\overline{\cal
M}}_{+}=\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\cosh{\frac{\eta_{e}}{T}}\left[\frac{1}{2}x_{+}K_{1}(x_{+})+\frac{b_{0}}{6}K_{0}(x_{+})\right]\,,$
(5.71) $\displaystyle{\overline{\cal
M}}_{-}=-\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\cosh{\frac{\eta_{e}}{T}}\left[\left(\frac{1}{2}+\frac{b_{0}^{2}}{12x_{-}^{2}}\right)x_{-}K_{1}(x_{-})+\frac{b_{0}}{3}K_{0}(x_{-})\right]\,,$
(5.72)

where $x_{\pm}$ are given by

$\displaystyle x_{+}=\frac{m_{e}}{T},\qquad
x_{-}=\sqrt{\frac{m_{e}^{2}}{T^{2}}+2b_{0}}$ (5.73)

The discussion for the case $g\neq 2$ can be found in [Steinmetz et al.
(2023a)].

In Fig. 5.6, we present the magnetization Eq. (5.70) for the case $g=2$ as a
function of temperature. It shows that the magnetization depends on the
magnetic scale $b_{0}$ and the $e^{+}e^{-}$ plasma possesses an overall
paramagnetic property, resulting in a positive magnetization
$\overline{\mathcal{M}}$. This paramagnetic property is contrary to the
conventional assumption that matter-antimatter plasmas lack significant
inherent magnetic responses. However, the magnetization never exceeds the
external field under the parameters considered, which shows a lack of
ferromagnetic behavior. As the Universe cooled, the dropping magnetization
slowed at $T_{\mathrm{split}}=20.3$ keV, where positrons vanished. Thereafter
the remaining electron density diluted with cosmic expansion.

In this section, we have explored the electron-positron plasma considering
external and self-magnetization fields without spin potential $\eta_{s}/T\ll
1$. However the nonzero spin potential $\eta_{s}\neq 0$ would have an impact
on the primordial $e^{+}e^{-}$ plasma. In general, the magnetization is also a
function of the spin potential $\eta_{s}$, and would be one important
parameter that control the spin direction of primordial gas which allows for
magnetization even in the absence of external magnetic fields. For further
discussion see [Steinmetz et al. (2023a)].

##  Chapter 6 Outlook: research publications underway

In this chapter, I address the ongoing research projects I hope to complete by
the end of year $2023$, which include: The low temperature baryongenesis
originating in nonequilibrium of bottom flavor; Population of Higgs in the
early Universe; Extra neutrino from microscopic processes after freeze-out;
and Self-consistent relaxation rate for electron-positron plasma in the early
Universe..

###  Possibility of bottom-catalyzed low temperature baryongenesis

From the PDG, the observed baryon-to-photon density ratio $\eta$ today is
given by $5.8\times 10^{-10}\leqslant\eta\leqslant 6.5\times 10^{-10}$
[Tanabashi et al. (2018)] where the $\eta=(6.12\pm 0.04)\times 10^{-10}$
[Workman et al. (2022)] is used in our calculation. This observed value is the
evidence of baryon asymmetry and quantifies the matter-antimatter asymmetry in
the Universe. In the past, the small value of the baryon asymmetry could be
interpreted as simply due to the initial conditions in the Universe. However,
in the current standard cosmological model, the inflation can erase any pre-
existing asymmetry between baryons and anti-baryons. In this case, we need
baryogenesis to generate of excess of baryon number compared to anti-baryon
number in order to create the observed baryon number today.

The precise epoch responsible for the observed matter genesis $\eta$ in the
early Universe has not been established yet. Several mechanisms have been
proposed to explain baryogenesis with investigations typically focusing on the
temperature range between GUT phase transition $T_{\mathrm{G}}\simeq
10^{16}\,\mathrm{GeV}$ and the electroweak phase transition near
$T_{\mathrm{W}}\simeq 130\,\mathrm{GeV}$ [Kuzmin et al. (1985, 1987); Arnold
and McLerran (1987); Kolb et al. (1996); Riotto and Trodden (1999); Nielsen
and Takanishi (2001); Giudice et al. (2004); Davidson et al. (2008); Morrissey
and Ramsey-Musolf (2012)].

In following section we present arguments that the Sakharov conditions
[Sakharov (1967)] for matter asymmetry to form also appear during QGP
hadronization era near to $T_{\mathrm{H}}\simeq 150\,\mathrm{MeV}$, and show
the possibility that bottom catalyzed the baryongenesis.

##### Overview of Sakhraov conditions

In 1967, the Soviet physicist Sakharov first formulated the three conditions
necessary to permit baryongenesis in the early Universe [Sakharov (1967)] and
in 1991 he refined the three conditions as follows [Sakharov (1990)]:

  * •

Absence of baryonic charge conservation

  * •

Violation of CP-invariance

  * •

Non-stationary conditions in absence of local thermodynamic equilibrium

Since the inflation erases any initial asymmetry between baryons and anti-
baryons, the first condition, a violation of baryon number $B$, must exist to
have the present baryon asymmetry. The second Sakharov condition requiring
$CP$ violation assures us that we can recognize a universal difference between
matter and antimatter, thus one abundance can be enhanced compared to the
other.

The third condition, departure from thermal equilibrium, is one of the
important prerequisites for matter genesis, this is because it creates the
arrow in time for the Universe. In general, the thermal equilibrium implies
both chemical (abundance) equilibrium and kinetic (equipartition of energy)
equilibrium. The observed baryon and lepton numbers cannot be generated in a
full thermal (chemical and kinetic) equilibrium, because even if the required
processes are occurring, the net effect is cancelled out by the equal number
of back-reactions. We believe that the presence of chemical non-equilibrium is
more relevant – kinetic equilibrium is usually established much more quickly
and has less impact on the actual particle abundances [Koch et al. (1986);
Birrell et al. (2015)].

Chemical non-equilibrium can be achieved by breaking the detailed balance
between particle production reaction and annihilation/decay. When the Universe
expands and temperature cools down, the production process slows down and is
not able to keep up with decay reactions. Then the detailed balance is broken
and creates the arrow in time for the Universe. In this case, the third
condition of Sakharov can be interpreted as:

  * •

Non-stationary conditions in absence of detailed balance,

where the ideal thermodynamic equilibrium is generalized to the concept of
detailed balance.

##### Is there enough bottom flavor to matter?

Considering that FLRW-Universe evolves conserving entropy, and that baryon and
lepton number following on the era of matter genesis is conserved, the current
day baryon $B$ to entropy $S$, $B/S$-ratio must be achieved during matter
genesis. The estimates of present day baryon-to-photon density ratio $\eta$
allows the determination of the present value of baryon per entropy ratio
[Rafelski (2020); Letessier and Rafelski (2002); Fromerth and Rafelski (2002);
Fromerth et al. (2012)]:

$\displaystyle\left(\frac{B}{S}\right)_{t_{0}}\\!\\!\\!\\!=\eta\left(\frac{n_{\gamma}}{\sigma_{\gamma}+\sigma_{\nu}}\right)_{\\!t_{0}}\\!\\!\\!\\!=(8.69\pm
0.05)\\!\\!\times\\!\\!10^{-11},$ (6.1)

where the subscript $t_{0}$ denotes the present day value, where
$\eta=(6.12\pm 0.04)\times 10^{-10}$ [Tanabashi et al. (2018)] is used in
calculation. Here we consider that the Universe today is dominated by photons
and free-streaming low mass neutrinos [Birrell et al. (2014a)], and
$\sigma_{\gamma}$ and $\sigma_{\nu}$ are the entropy density for photons and
neutrinos, respectively.

In chemical equilibrium the ratio of bottom quark (pair) density $n_{b}^{th}$
to entropy density $\sigma=S/V$ just above quark-gluon hadronization
temperature $T_{\mathrm{H}}=150\sim 160\,\mathrm{MeV}$ is
$n_{b}^{th}/\sigma=10^{-10}\sim 10^{-13}$ (see Fig. 2.1). By studying the
bottom density per entropy near to the hadronization temperature and comparing
it to the baryon-per-entropy ratio $B/S$ we found there is sufficient
abundance of bottom quarks for the proposed matter genesis mechanism to be
relevant.

##### Non-stationary conditions in absence of detailed balance

We have demonstrated that the bottom quark nonequliibrium occurs near to QGP
phase transition around the temperature $T=0.3\sim 0.15$ GeV in Fig. 2.4 and
Fig. 2.6. We demonstrate that the competition between weak interaction decay
and the strong interaction fusion processes is responsible for driving the
bottom quark departure from the equilibrium in the early Universe. In all
cases we see prolonged non-equilibrium which provides the arrow of time for
baryongenesis.

##### Violation of $CP$-invariance

In general, violation of $CP$ asymmetry can occur in the amplitudes of hadron
decay. The weak interaction $CP$ violation arises from the components of
Cabibbo-Kobayashi-Maskawa (CKM) matrix associated with quark-level transition
amplitude and $CP$-violating phase.

Given that the non-equilibrium of bottom flavor arises at relatively low QGP
temperature, the bottom quark decay occurs from preformed $\mathrm{B}_{x}$
meson states, $x=u,d,s,c$ [Karsch et al. (1988); Brambilla et al. (2010);
Aarts et al. (2011); Brambilla et al. (2018); Bazavov et al. (2018); Offler et
al. (2019)]. These decays violate aside of the $CP$ symmetry, see for example
[Aaij et al. (2020b, a)]. The exploration of the here interesting $CP$
symmetry breaking in B${}_{c}(b\bar{c})$ decay is in progress [Tully (2019);
Amhis et al. (2021); Tanabashi et al. (2018)]. Present measurements of
$CP$-violation suggest that the CP asymmetry parameter is around
$\delta_{CP}\approx 10^{-3}$ [Tanabashi et al. (2018)].

##### Nonequilibrium bottom and baryon asymmetry

The off equilibrium phenomenon of bottom quark around the temperature range
$T=0.3\sim 0.15$ GeV can provide the non-chemical equilibrium condition for
baryogenesis to occur in the primordial-QGP hadronization era. Furthermore,
let us consider the scenario where all bottom quarks are confined within
$B_{c}^{\pm}$ meson. In this case, the decay of charged mesons in the
primordial-QGP can be a source of $CP$ violation. However, it remains
uncertain whether the decay of $B_{c}^{\pm}$ mesons contributes to baryon
violation. Our postulation is as follows: the baryon asymmetry is produced by
the bottom quark disappearance via the irreversible decay of $B^{\pm}_{c}$
meson during the off-equilibrium process. Once a baryon symmetry exists in
universe, it will also produce the asymmetry between leptons and anti-leptons
which is similar to the baryon asymmetry by the $L=B$.

The heavy $B_{c}^{\pm}$ meson decay into multi-particles in plasma is
associated with the irreversible process. This is because after decay the
daughter particles can interact with plasma and distribute their energy to
other particles and reach equilibrium with the plasma quickly. In this case
the energy required for the inverse reaction to produce $B_{c}^{\pm}$ meson is
difficult to overcome and therefore we have an irreversible process for multi-
particle decay in plasma.

The rapid $B_{c}^{\pm}$ decay and bottom reformation speed at picosecond scale
assures that there are millions of individual microscopic processes involving
bottom quark production and decay before and during the hadronization epoch of
QGP. In this case, we have an Urca process for the bottom quark, i.e. a
cycling reaction that produces the bottom quark which subsequently disappears
via the $B_{c}^{\pm}$ meson decay. The Urca process is a fundamental physical
process and has been stuying the realms of in astrophysics and nuclear
physics. In our case, for bottom quark as a example: at low temperature, the
number of bottom quark cycling can be estimated as

$\displaystyle\left.\mathrm{C_{cycle}}\right|_{T=0.2\mathrm{GeV}}=\frac{\tau_{H}}{\tau_{B_{c}}}\approx
2\times 10^{7},$ (6.2)

where the lifespan of $B_{c}^{\pm}$ is $\tau_{\mathrm{B}_{c}}\approx
0.51\times 10^{-12}\,\mathrm{sec}$ and at temperature $T=0.2$ GeV the Hubble
time is $\tau_{H}=1/H=1.272\times 10^{-5}$ sec. The Urca process plays a
significant role by potentially amplifying any small and currently unobserved
violation of baryon number associated with the bottom quark. The small baryon
asymmetry is enhanced by the Urca-like process with cycling
${\tau^{\ast}_{H}}/{\tau_{\ast}}$ in the early Universe. This amplification
would be crucial for achieving the required strength for today’s observation.

Full understanding of the possibility for baryogenesis to occur in primordial-
QGP hadronization still requires a detailed study of the relation between
baryon asymmetry and irreversible process; we are working on it now. Here, our
interest is to show that our results provide a strong motivation to explore
the physics of baryon nonconservation involving the bottomnium mesons or/and
bottom quarks in thermal environment.

###  Population of Higgs in the early Universe

In our earlier study, the bottom $(b)$ quark abundance depends on the
competition between the strong interaction fusion and weak interaction decay
rates. This lead to the off equilibrium phenomenon of the bottom quark near
the hadronization temperature. The same idea can be applied to other heavy
particles in QGP. In following we focus on the Higgs abundance first and
develop methods for future detailed study.

Considering the temperature range $10\,\mathrm{GeV}>T>1\,\mathrm{GeV}$ in the
early Universe, the number density of the Higgs can be written as

$\displaystyle
n_{H}=\frac{\Upsilon_{H}}{2\pi^{2}}T^{3}\left(\frac{m_{H}}{T}\right)^{2}K_{2}(m_{H}/T),$
(6.3)

where $\Upsilon_{H}$ is the Higgs fugacity parameter, and $m_{H}=125$ GeV is
the mass of Higgs. In the temperature range we consider here, we have
$m_{H}\gg T$ which allows us to consider the Boltzmann limit for the
calculation of Higgs number density.

Figure 6.1: The density ratio between Higgs and baryon asymmetry as a function
of temperature with condition $\Upsilon_{H}=1$. It shows that the
$n_{H}=(n_{B}-n_{\bar{B}})$ at temperature $T=5.7$ GeV.

Using constant baryon-per-entropy ratio, the density between Higgs and baryon
asymmetry ($u,d$ quark-antiquark asymmetry) can be written as

$\displaystyle\frac{n_{H}}{(n_{B}-n_{\bar{B}})}=\frac{n_{H}}{s_{tot}}\,\left(\frac{s_{tot}}{n_{B}-n_{\bar{B}}}\right)=\frac{n_{H}}{s_{tot}}\left(\frac{s_{\gamma,\nu}}{n_{B}-n_{\bar{B}}}\right)_{\\!t_{0}},$
(6.4)

where the present day value of baryon per entropy ratio is given by Eq.
(1.34). The entropy density in QGP can be written as

$\displaystyle s_{tot}=\frac{2\pi^{2}}{45}g^{s}_{\ast}T_{\gamma}^{3},\qquad
g^{s}_{\ast}=\sum_{i=\mathrm{g},\gamma}g_{i}\left({\frac{T_{i}}{T_{\gamma}}}\right)^{3}+\frac{7}{8}\sum_{i=l^{\pm},\nu,u,d}g_{i}\left({\frac{T_{i}}{T_{\gamma}}}\right)^{3},$
(6.5)

where we consider the massless particles in QGP only as a first estimation. In
Fig. 6.1, we plot the density ratio between Higgs and baryon asymmetry for the
case $\Upsilon_{H}=1$. It shows that at temperature $T=5.7$ GeV, the ratio is
equal to one, i.e., $n_{H}=(n_{B}-n_{\bar{B}})$ which means that the Higgs
abundance could influence baryon evolution because the Higgs number density is
comparable to the baryon number density. This result motivated us to examine
the Higgs boson and to study its dynamic abundance in detail during the QGP
epoch.

In the QGP epoch, the dominant production of the Higgs boson is the bottom
fusion reaction:

$\displaystyle b+\overline{b}\longrightarrow H,$ (6.6)

which is the inverse decay process of $H\to b+\overline{b}$. On the other
hand, Higgs abundance disappears via the $W,Z$ decay channel as follows:

$\displaystyle H\longrightarrow
WW^{\ast},ZZ^{\ast}\longrightarrow\mathrm{anything}.$ (6.7)

where $W^{\ast},Z^{\ast}$ represent the virtual bosons. Once Higgs decays into
the $W$ and $Z$ bosons, the short lifespan of $W,Z$ mean they decay into
multi-particles and reequilibrate with the plasma quickly. In this case the
energy required for the inverse decay reaction to produce the Higgs boson is
difficult to overcome in the temperature we are interested in. Similar to the
case of bottom quark we studied, the competition between the production and
decay of Higgs require the dynamic study of the particle abundance in the
early Universe. We aim to apply the knowledge from our study of bottom quark
to the Higgs boson, and to examine all other possible sources for Higgs
production in QGP and developing methods for future study before the end of
year 2023.

###  After neutrino freeze-out: Extra neutrinos from microscope processes

After neutrinos chemical freeze-out, the number of neutrinos is independently
conserved. However, the presence of electron-positron rich plasma until $T=20$
keV permits the reaction $\gamma\gamma\to e^{-}e^{+}\to\nu\bar{\nu}$ to occur
even after neutrinos decouple from the cosmic plasma. This suggests the small
amount of extra neutrinos can be produced until temperature $T=20$ keV and can
modify the free streaming distribution and the effective number of neutrinos.
In this section, we examine the possible source of extra neutrino from
electron-positron plasma and develop methods for future detailed study.

Considering that neutrinos decouple at $T_{f}=2$ MeV and become free streaming
after freeze-out. The presence of electron-positron plasma environment from
$2\,\mathrm{MeV}>T>0.02$ MeV can allow the following weak reaction to occur:

$\displaystyle\gamma+\gamma\longrightarrow
e^{-}+e^{+}\longrightarrow\nu+\bar{\nu}.$ (6.8)

Given the thermal reaction rate per volume $R_{\gamma\gamma\to e\overline{e}}$
for reaction $\gamma\gamma\to e\overline{e}$ and
$R_{e\overline{e}\to\nu\overline{\nu}}$ for reaction
$e\overline{e}\to\nu\overline{\nu}$, then the thermal reaction rate per volume
for $\gamma\gamma\to e^{-}e^{+}\to\nu\bar{\nu}$ can be written as

$\displaystyle R_{\gamma\to e\to\nu}=R_{\gamma\gamma\to
e\overline{e}}\left(\frac{R_{e\overline{e}\to\nu\overline{\nu}}}{R_{\gamma\gamma\to
e\overline{e}}+R_{e\overline{e}\to\nu\overline{\nu}}}\right)\approx
R_{e\overline{e}\to\nu\overline{\nu}}$ (6.9)

In Fig. 6.2 we plot the thermal reaction rate per volume for relevant
reactions as a function of temperature $2\,\mathrm{MeV}>T>0.05\,\mathrm{MeV}$.
It shows that the dominant reaction for the process $\gamma\gamma\to
e^{-}e^{+}\to\nu\bar{\nu}$ is the $e\overline{e}\to\nu\overline{\nu}$ and can
be approximated $R_{\gamma\to e\to\nu}=R_{e\overline{e}\to\nu\overline{\nu}}$
in the temperature we are interested in.

Figure 6.2: The thermal reaction rate per volume as a function of temperature
$2\,\mathrm{MeV}>T>0.05\,\mathrm{MeV}$. The dominant reaction for the process
$\gamma\gamma\to e^{-}e^{+}\to\nu\bar{\nu}$ is the
$e\overline{e}\to\nu\overline{\nu}$ and we have $R_{\gamma\to
e\to\nu}=R_{e\overline{e}\to\nu\overline{\nu}}$.

Given the thermal reaction rate, the dynamic equation describing the relic
neutrino abundance after freeze-out can be expressed as:

$\displaystyle\frac{dn_{\nu}}{dt}+3Hn_{\nu}=R_{e\overline{e}\to\nu\overline{\nu}}(T_{\gamma,e^{\pm}})-R_{\nu\overline{\nu}\to
e\overline{e}}(T_{\nu}),$ (6.10)

where $n_{\nu}$ is the number density of neutrinos and $H$ is the Hubble
parameter. The parameter $T_{\gamma,e^{\pm}}$ is the equilibrium temperature
between photons and $e^{\pm}$ and $T_{\nu}$ is the temperature for free-
streaming neutrinos:

$\displaystyle T_{\nu}=\frac{a(t_{f})}{a(t)}T_{f},$ (6.11)

where $T_{f}$ is the neutrino freezeout temperature. After neutrinos decoupled
from the cosmic plasma, we have $T_{\nu}\neq T_{\gamma,e^{\pm}}$. This is
because the conservation of entropy, after freezeout, the relic neutrino
entropy is conserved independently and the entropy from $e^{\pm}$ annihilation
flows solely into photons and reheats the photons’ temperature. However, after
neutrino freezeout, extra entropy from electron-positron plasma can still flow
into the free-streaming neutrino sector via the reaction $\gamma\gamma\to
e^{-}e^{+}\to\nu\bar{\nu}$. To describe this novel situation, kinetic theory
for entropy production needs to be adapted, a topic we will address in the
future. Here we neglect this extra entropy and consider the standard scenario
for first approximation.

In Fig. 6.3 we plot the temperature ratio $T_{\nu}/T_{\gamma,e^{\pm}}$, the
rate ratio $R_{\nu\overline{\nu}\rightarrow
e\overline{e}}/R_{e\overline{e}\rightarrow\nu\overline{\nu}}$ and
$(R_{e\overline{e}\rightarrow\nu\overline{\nu}}-R_{\nu\overline{\nu}\rightarrow
e\overline{e}})/R_{e\overline{e}\rightarrow\nu\overline{\nu}}$ as a function
of temperature. It shows that after neutrino freezeout, the back reaction
$\nu\overline{\nu}\rightarrow e\overline{e}$ becomes smaller compared to the
reaction $e\overline{e}\rightarrow\nu\overline{\nu}$ as the temperature cools
down. This is because as $T_{\nu}$ cools down, the density of relic neutrinos
becomes so low and their energy becomes too small to interact. However, the
hot and rich electron-positron plasma can still annihilate into neutrino pairs
without any difficulties.

Figure 6.3: The temperature ratio $T_{\nu}/T_{\gamma,e^{\pm}}$ (blue line),
the rate ratio $R_{\nu\overline{\nu}\rightarrow
e\overline{e}}/R_{e\overline{e}\rightarrow\nu\overline{\nu}}$ (red line) and
$(R_{e\overline{e}\rightarrow\nu\overline{\nu}}-R_{\nu\overline{\nu}\rightarrow
e\overline{e}})/R_{e\overline{e}\rightarrow\nu\overline{\nu}}$ (green line) as
a function of temperature. It shows that the reaction
$\nu\overline{\nu}\rightarrow e\overline{e}$ is small compare to the reaction
$e\overline{e}\rightarrow\nu\overline{\nu}$ as temperature cooling down.

Solving the dynamic equation of neutrino abundance Eq.(6.10), the general
solution can be written as

$\displaystyle n_{\nu}(T)=n_{\mathrm{relic}}(T)+n_{\mathrm{extra}}(T),\qquad
T=T_{\gamma,e^{\pm}},$ (6.12)

where $n_{\mathrm{relic}}$ represents the relic neutrino number density and
$n_{\mathrm{extra}}$ is the extra number density from the $e^{\pm}$
annihilation. The relic neutrino density is given by

$\displaystyle
n_{\mathrm{relic}}=n_{\nu}^{0}\exp\left(-3\int_{t_{i}}^{t}{dt^{\prime}}H(t^{\prime})\right)=n_{\nu}^{0}\exp\left(3\int_{T_{i}}^{T}\frac{dT^{\prime}}{T^{\prime}}(1+\mathcal{F})\right),$
(6.13) $\displaystyle
n^{0}_{\nu}=g_{\nu}\frac{3\zeta(3)}{4\pi^{2}}T^{3}_{i},\qquad
F=\frac{T}{3g^{\ast}_{s}}\frac{dg^{\ast}_{s}}{dT},$ (6.14)

where $T_{i}$ is the initial temperature and $g^{\ast}_{s}$ is the entropy
degrees of freedom. The extra neutrino density can be written as

$\displaystyle n_{\mathrm{extra}}=-$
$\displaystyle\exp\left(3\int_{T_{i}}^{T}\frac{dT^{\prime}}{T^{\prime}}(1+\mathcal{F})\right)$
$\displaystyle\times$
$\displaystyle\int_{T_{i}}^{T}\frac{dT^{\prime}}{T^{\prime}}\frac{R_{e\overline{e}}(T^{\prime})-R_{\nu\overline{\nu}}(T^{\prime}_{\nu})}{H(T^{\prime})}\left(1+\mathcal{F}\right)\exp\left(-3\int_{T_{i}}^{T^{\prime}}\frac{dT^{\prime\prime}}{T^{\prime\prime}}(1+\mathcal{F})\right).$
(6.15)

Figure 6.4: the ratio between $n_{extra}/n_{relic}$ as a function of
temperature with different neutrino freezeout temperature $T_{f}$. It shows
that the higher freezeout temperature $T_{f}$ the higher number of extra
neutrinos can be produced.

In Fig. 6.4 we plot the ratio between $n_{\mathrm{extra}}/n_{\mathrm{relic}}$
as a function of temperature with different neutrino freeze-out temperature
$T_{f}$. It shows that the number of extra neutrinos depends strongly on the
parameter $T_{f}$. This is because the freeze-out temperature determines the
timing of the entropy transfer between $e^{\pm}$ and photon, which
subsequently affects the evolution of temperature ratio between neutrinos and
photons in the early Universe. The temperature ratio affects the rate ratio
between $\nu\overline{\nu}\to e\overline{e}$ and
$e\overline{e}\to\nu\overline{\nu}$, because once the neutrino is too cold and
the back reaction $\nu\overline{\nu}\to e\overline{e}$ can not maintain the
balance, the $e^{\pm}$ annihilation starts to feed the extra neutrinos to the
relic neutrino background.

In addition to the annihilation of electron-positron pairs, there are other
sources that can contribute to the presence of extra neutrinos in the early
Universe. These additional sources include particle physics phenomena and
plasma effects: neutrinos from charged leptons $\mu^{\pm},\tau^{\pm}$ decay,
neutrinos from the $\pi^{\pm}$ decay, and neutrino radiation from massive
photon decay in electron-positron rich plasma. All of these potential sources
of extra neutrinos can impact the distribution of freely streaming neutrinos
and the effective number of neutrinos. Understanding these effects is crucial
to comprehending how the neutrino component influences the expansion of the
Universe, as well as the potential implications for large-scale structure
formation and the spectrum of relic neutrinos.

###  The $e^{\pm}$ plasma relaxation rate: Self-consistence approach

In electron-positron plasma, the photon mass appears as $m_{\gamma}^{2}$ in
the transition matrices for Møller and Bhabha reactions, which is one of
important parameters in the calculation of the relaxation rate in $e^{\pm}$
plasma. When evaluating Møller and Bhabha scattering, we include the
temperature-dependent mass of the photon obtained in plasma theory without
damping. In general, the effective mass of the photon depends on the property
of the plasma. Considering the linear response theory, the dispersion relation
for the photon in nonrelativistic $e^{\pm}$ plasma is given by [Formanek et
al. (2021)]

$\displaystyle w^{2}=|k|^{2}+\frac{w}{w+i\kappa}w_{pl}^{2},$ (6.16)

where $w_{pl}$ is the plasma frequency and $\kappa$ is the average collision
rate of $e^{\pm}$ plasma. The effective plasma frequency in damped plasma can
be solved by considering the case $|k|^{2}=0$ [Formanek et al. (2021)]

$\displaystyle
w_{\pm}=-i\frac{\kappa}{2}\pm\sqrt{w^{2}_{pl}-\frac{\kappa^{2}}{4}}.$ (6.17)

It shows that the plasma frequency in damped plasma $w_{\pm}$ is a function of
$\kappa.$ In this case, the effective photon mass in damped plasma is also a
function of the scattering rate. We have

$\displaystyle m_{\gamma}=w_{\pm}(w_{pl},\kappa)=m_{\gamma}(w_{pl},\kappa),$
(6.18)

where the photon mass $m_{\gamma}=w_{+}$ for the underdamped plasma
$w_{pl}>\kappa/2$, and $m_{\gamma}=w_{-}$ for overdamped plasma
$w_{pl}<\kappa/2$. Eq. (6.18) shows that computed damping strength $\kappa$ is
the dominant scale for collisional plasma and it is also the main parameter
determining the photon mass in plasma.

Substituting the effective photon mass Eq. (6.18) into the definition of the
average relaxation rate Eq. (5.39), we obtain the self-consistent equation for
damping rate $\kappa$ as

$\displaystyle\kappa\,$
$\displaystyle\left[\frac{g_{e}}{2\pi^{3}}T^{3}\left(\frac{m_{e}}{T}\right)^{2}K_{2}(m_{e}/T)\right]$
$\displaystyle=\frac{g_{e}g_{e}}{32\pi^{4}}T\\!\\!\int_{4m_{e}^{2}}^{\infty}\\!\\!\\!\\!ds\frac{s(s-4m^{2}_{e})}{\sqrt{s}}K_{1}(\sqrt{s}/T)\bigg{[}\sigma_{e^{\pm}e^{\pm}}(s,w_{pl},\kappa)+\sigma_{e^{\pm}e^{\mp}}(s,w_{pl},\kappa)\bigg{]},$
(6.19)

where the cross sections depend on the parameter $w_{pl}$ and $\kappa$, and
the variable $\kappa$ appears on both sides of the equation so we need solve
the equation numerically to determine the $\kappa$ value that satisfies this
condition.

Depending on the nature of the plasma (overdamped or underdamped plasma), we
can establish the photon mass in collision plasma based on two distinct
conditions as follows:

  * •

Case 1. The plasma frequency is larger than the collision rate
$w_{pl}>\kappa/2$, we have

$\displaystyle
m_{\gamma}=w_{+}=-i\frac{\kappa}{2}+\sqrt{w^{2}_{pl}-\frac{\kappa^{2}}{4}}.$
(6.20)

  * •

Case 2. The plasma frequency is smaller than the collision rate
$w_{pl}<\kappa/2$, we have

$\displaystyle
m_{\gamma}=w_{-}=-i\left(\frac{\kappa}{2}+\sqrt{\frac{\kappa^{2}}{4}-w^{2}_{pl}}\right).$
(6.21)

In Fig. 6.5 it shows that during the BBN temperature range $50\leqslant
T\leqslant 86$ keV, the plasma frequency is smaller than the collision rate
$w_{pl}<\kappa/2$. In this case, the effective photon mass in collision plasma
during BBN epoch is given by Eq.(6.21).

Figure 6.5: The relaxation rate $\kappa/2$ (blue line) as a function of
temperature in nonrelativistic electron-positron plasma. For comparison, we
show the plasma frequency $\omega_{pl}$ in the red line. It shows that for
$T>145.5$ MeV, the plasma frequency is larger than the collision rate
$w_{pl}>\kappa/2$; for temperature $T<145.5$ MeV, we have $\kappa/2>w_{pl}$.
For temperature $T<20.3$ keV, the composition of plasma is changed to electron
and proton, which is beyond our current study because of unequal numbers of
electrons and positrons.

To calculate the cross section for Møller and Bhabha scattering we need to
include the imaginary photon mass in the calculation of transition matrix
elements. In general, the real part of photon mass in the calculation includes
the effective photon-electron/positron scattering in plasma, and the imaginary
part of photon mass contributes to the decay width of massive photon in
plasma. To estimate the effect of photon mass on the damping rate $\kappa$, we
first consider the effective mass corresponding to photon-electron/positron
scattering in plasma, and leave the photon decay for future study.

For BBN temperature $50\leqslant T\leqslant 86$ keV, we have $w_{pl}<\kappa$
and the effective photon mass can be approximated as

$\displaystyle m^{2}_{\gamma}=w_{-}w_{-}^{\ast}$
$\displaystyle=\left(\frac{\kappa}{2}+\sqrt{\frac{\kappa^{2}}{4}-w^{2}_{pl}}\right)^{2}=\frac{\kappa^{2}}{2}\left[\left(1-\frac{2w^{2}_{pl}}{\kappa^{2}}\right)+\sqrt{1-\frac{4w^{2}_{pl}}{\kappa^{2}}}\right]$
$\displaystyle=\frac{\kappa^{2}}{2}\left[\left(1-\frac{2w^{2}_{pl}}{\kappa^{2}}\right)+\left(1-\frac{2w^{2}_{pl}}{\kappa^{2}}+\cdots\right)\right]\approx\kappa^{2}.$
(6.22)

where we consider the limit $w^{2}_{pl}/\kappa^{2}\ll 1$ and effective photon
mass is equal to the average collision rate in plasma
$m^{2}_{\gamma}\approx\kappa$.

Substituting the photon mass $m^{2}_{\gamma}=\kappa^{2}$ for overdamping
plasma into the relaxation rate of electron-positron Eq. (6.4), and
introducing the following dimensionless variables

$\displaystyle x=\sqrt{s}/T,\qquad a=m_{\gamma}/T=\kappa/T,\qquad b=m_{e}/T,$
(6.23)

Figure 6.6: The relaxation rate $\kappa$ that satisfies Eq.(6.4) as a function
of temperature $50\leqslant T\leqslant 86$ keV. It shows that for overdamping
plasma, we have $m^{2}_{\gamma}=\kappa^{2}$, and $\kappa=1.832$ keV when
$T=86$ keV and $\kappa=0.350$ keV when $T=50$ keV. The minor fluctuations are
a result of the restricted numerical precision.

then the relaxation rate of electron-positron can be written as

$\displaystyle\left[\frac{g_{e}}{2\pi^{2}}T^{4}\left(\frac{m_{e}}{T}\right)^{\\!2}\\!K_{2}(m_{e}/T)\right]\,\left(\frac{\kappa}{T}\right)$
$\displaystyle\qquad\qquad\qquad=\frac{g^{2}_{e}\alpha^{2}}{8\pi^{3}}T^{4}\\!\\!\int_{2b}^{\infty}\\!dxK_{1}(x)\left[\mathcal{F}_{e^{\pm}e^{\pm}}(x,\kappa/T)+\mathcal{F}_{e^{\pm}e^{\mp}}(x,\kappa/T)\right],$
(6.24)

where the functions $\mathcal{F}_{e^{\pm}e^{\pm}}$ and
$\mathcal{F}_{e^{\pm}e^{\mp}}$ are given by

$\displaystyle\mathcal{F}_{e^{\pm}e^{\pm}}(x,a=\kappa/T)$
$\displaystyle=\left\\{2\left[3a^{2}+4b^{2}+\frac{4(b^{4}-a^{4})}{x^{2}-4b^{2}+2a^{2}}\right]\ln\left(\frac{a^{2}}{x^{2}-4b^{2}+a^{2}}\right)\right.$
$\displaystyle\left.+\frac{(x^{2}-4b^{2})(8b^{4}+2a^{4}+3a^{2}x^{2}+2x^{4}-4b^{2}(2x^{2}+a^{2}))}{a^{2}(x^{2}-4b^{2}+a^{2})}\right\\}$
(6.25)

and

$\displaystyle\mathcal{F}_{e^{\pm}e^{\mp}}(x,a=\kappa/T)$
$\displaystyle=\left\\{\frac{2x^{2}(a^{2}+x^{2})-4b^{4}}{x^{2}-a^{2}}\ln\left(\frac{a^{2}}{x^{2}-4b^{2}+a^{2}}\right)\right.$
$\displaystyle+\frac{(x^{2}-4b^{2})(3x^{2}+4b^{2}+2a^{2})}{(x^{2}-a^{2})}+\frac{x^{6}-12b^{4}x^{2}-16b^{6}}{3(x^{2}-a^{2})^{2}}$
$\displaystyle\left.+\frac{(x^{2}-4b^{2})(8b^{4}+2a^{4}+3a^{2}x^{2}+2x^{4}-4b^{2}(2x^{2}+a^{2}))}{a^{2}(x^{2}-4b^{2}+a^{2})}\\!\right\\}.$
(6.26)

To determine the $\kappa$ that satisfies Eq. (6.4), we can solve it
numerically. In Fig. 6.6, we plot the relaxation rate $\kappa$ that satisfies
Eq. (6.4) as a function of temperature $50\,\mathrm{keV}\leqslant T\leqslant
86$ keV. It shows that during the BBN temperature range, we have overdamping
electron/positron plasma $w_{pl}<\kappa$, and the effective photon mass
$m^{2}_{\gamma}=\kappa^{2}$. The relaxation rate $\kappa=1.832\sim 0.350$ keV
during the BBN temperature range, which is smaller than the relaxation rate
without damping photon mass (See Fig. 5.4, where the relaxation rate
$\kappa=10\sim 12$ keV during the BBN temperature).

Our first estimation implies that the relaxation rate is sensitive to the
photon mass in damped plasma. To address the self-consistent evaluation of
damping rate in plasma requires the development of a well-defined, self-
consistent approach, where both damping and photon properties in plasma are
determined in a mutually consistent manner. We aim to include the full photon
mass effect (both real and imaginary parts) in the study and improve our
calculation for the next step to complete the project.

##  Chapter 7 Summary and conclusion

We have studied the evolution of the early Universe from the QGP epoch to the
$e^{\pm}$ plasma $300\,\mathrm{MeV}>T>0.02\,\mathrm{MeV}$. Our effort focuses
on understanding how the plasma state impacts and modifies elementary
interactions between particles and thus the properties of the early Universe.
In the QGP epoch, we investigate the heavy-quark bottom and charm near the QGP
hadronization temperature $300\,\mathrm{MeV}>T>150$ MeV. We show that the
faster quark-gluon pair fusion keeps the charm quarks in chemical equilibrium
up until hadronization. After hadronization, charm quarks form heavy mesons
that decay quickly into multi-particles and causes charm quarks to vanish from
inventory of particles in the Universe. For the bottom quarks, the quark
production rate competes with bottom decay rate as a function of temperature.
When the Universe’s temperature is near to the QGP phase transition
$300\,\mathrm{MeV}>T>150$ MeV, the bottom quark breaks the detail balance and
disappearance from particle inventory provides the arrow in time and which can
be the ‘sweet-spot’ for Sakharov conditions.

After hadronization, the free quarks and gluons become confined within
baryon/mesons and the Universe becomes hadronic-matter dominated. In the
temperature range $150\,\mathrm{MeV}>T>20\,\mathrm{MeV}$, the Universe is rich
in physics phenomena involving strange mesons and (anti)baryons including
(anti)hyperon abundances. Considering the inventory of the Universe with
strange mesons and baryons, the antibaryons disappear from the Universe at
temperature $T=38.2$ MeV. Strangeness can be produced by the inverse decay
reactions via weak, electromagnetic, and strong interactions in the early
Universe until $T\approx 13$ MeV. For $T>20$ MeV, the strangeness is
dominantly present in the meson. For $20>T>13$ MeV, the strangeness can
present in the hyperons and anti-strangeness in kaon keeping symmetry
$s=\overline{s}$. Below the temperature $T<13$ MeV a new regime opens up in
which the tiny residual strangeness abundance in hyperons and is governed by
weak decays with no reequilibration with mesons.

When the temperature in the Universe is about $T\approx 10$ MeV, the main
ingredients that control the Universe evolution are: photons, neutrinos,
electrons and positrons. The massive meson, baryon, and $\mu^{\pm}$ and
$\tau^{\pm}$ are also present, but their small number density can be neglected
when considering the energy density of the Universe expansion. For temperature
$10\,\mathrm{MeV}>T>2$ MeV the Universe is controlled by the weak interaction
between neutrinos and matter. We explore the neutrino coherent and incoherent
scatterings with matter and apply them to study the neutrino freeze-out
(incoherent scattering) in the early Universe and long wavelength cosmic
neutrino-atom scattering (coherent scattering). After neutrino freeze-out
$T_{f}\approx 2\,\mathrm{MeV}$, it becomes free-streaming and the number of
neutrinos is independently conserved. We investigate the relation between the
lepton number $L$ and the effective number of neutrinos
$N^{\mathrm{eff}}_{\nu}$, and explore the impact of a large cosmological
lepton asymmetry on the Universe evolution. Instead of $B\simeq|L|$, we found
that $0.4\leqslant|L|\leqslant 0.52$ and $B\simeq 1.33\times 10^{-9}|L|$
reconciles the CMB and current epoch results for the Hubble expansion
parameter.

Considering the temperature after neutrino freeze-out:
$2\,\mathrm{MeV}>T>0.02$ MeV, the cosmic plasma is dominated by the photon,
electrons and positrons. The massive $\mu^{\pm}$ abundance disappears at
$T_{\mathrm{disappear}}=4.195$ MeV as soon as the muon decay rate becomes
faster than muon production rate. The muon-baryon density ratio at muon
persistence temperature is equal to
$n_{\mu}^{\pm}/n_{B}(T_{\mathrm{disappear}})\approx 0.911$, which implies that
the muon abundance could influence baryon evolution because muon number
density is comparable to the baryon number density.

We demonstrate that the presence of rich electron-positron plasma can last
until temperature $T=20.3$ keV by calculating the chemical potential of
electrons in the universe that maintains charge neutrality and entropy
conservation. We evaluate the microscope damping rate with temperature
dependent photon mass in electron-positron plasma which is one important
variable to study the inter-nuclear electromagnetic potentials in damped
plasma. Finally we examine the magnetization process within dense electron-
positron plasma and show that it has paramagnetic properties when subjected to
an external field. Our study of magnetization can provide insights to
understand the magnetized plasma and the possible origin of primordial
magnetic field and developing methods for future detailed study.

Looking forward, we will refine our understanding of the evolution of the
early Universe by studying the topics that follow. For electron-positron
plasma: we will improve our calculation on the damping rate $\kappa$ and
develop a self-consistent approach where both damping and photon properties in
plasma are determined in a mutually consistent manner. For the neutrino
sector: we will examine the sources of extra neutrino and entropy transfer
from microscope process after neutrino freezeout in details. By studying these
phenomena, we aim to enhance our understanding of the role of neutrinos in the
evolution of universe. For the heavy particle in QGP: we will focus on
understanding the behavior of heavy quarks in extreme environments, and refine
our study of nonequilibrium bottom and its potential applications on
baryongenesis at low temperature. We will also study the dynamic Higgs
abundance under the competition between production and decay to understand the
departure from equilibrium of Higgs in the early Universe.

In summary, our research offers an initial glimpse into the first hour of the
Universe’s history, exploring the intricate relationship between fundamental
particles and plasma in the early Universe. We hope that the output of our
study will be beneficial to all parties concerned while at the same time
contribute to the knowledge enhancement in the understanding of the early
Universe.

##  Appendix A Decomposition of Fermi gas into zero and finite temperature
components

The most interesting physics of finite temperature Fermi gasses occurs around
the Fermi surface, which needs a mathematical tool that can capture the finite
temperature behavior of the Fermi-Dirac distribution in an analytic fashion.
We provide a novel form of the Fermi distribution that can separate the Fermi
gas into zero and finite temperature components analytically which is useful
for addressing physics beyond the zero temperature approximation. This is an
ongoing research project and will be submitted to International Journal of
Theoretical Physics.

##### A novel form of Fermi distribution

We have empirically identified the following novel way to state the form of
the Fermi distribution. We obtained this form while seeking to carry out
cosmological computations involving shifts in behavior from high to very low-
temperature physics.

$\displaystyle f$ $\displaystyle\equiv\frac{1}{e^{(E-\widetilde{\mu})/T}+1}$
$\displaystyle=\Theta\left(\frac{\widetilde{\mu}-E}{T}\right)+e^{-|E-\widetilde{\mu}|/T}\;\left[\frac{1}{2}\mathrm{sgn}\left(\frac{E-\widetilde{\mu}}{T}\right)+\frac{1}{2}\tanh\left(\frac{E-\widetilde{\mu}}{2T}\right)\right],$
(A.1)

where the Heaviside step function $\Theta(x)$ and sign function
$\mathrm{sgn}(x)$ are given by

$\displaystyle\Theta(x)=\left\\{\begin{array}[]{r}1,\quad\mathrm{for}\quad{x}>0\\\
1/2,\quad\mathrm{for}\quad{x}=0\\\
0,\quad\mathrm{for}\quad{x}<0\end{array}\right.\,,\qquad\mathrm{sgn}(x)=\left\\{\begin{array}[]{r}+1,\quad\mathrm{for}\quad{x}>0\\\
0,\quad\mathrm{for}\quad{x}=0\\\ -1,\quad\mathrm{for}\quad{x}<0\\\
\end{array}\right.\,.$ (A.8)

The first term in Eq. (A) represents the zero temperature limit of the Fermi
distribution, and the finite temperature contributions in square brackets are
weighted by a decaying exponential function. One of the benefits of weighting
the finite temperature contributions with decaying exponential function is
that it can provide a good asymptotic limit numerically and the numerical
evaluations will naturally center around the Fermi surface of the system.

##### Mathematical proof

We note that the right hand side (RHS) of Eq. (A) comprises several non-
analytical functions, also called distributions. On first sight it is hard to
believe that these cancel to create the analytical Fermi function format seen
on the left hand side (LHS). In the following, we will demonstrate that the
two sides equal each other. To demonstrate the novel form of the Fermi
distribution, it is convenient to introduce the dimensionless variable as
follows:

$\displaystyle x=\frac{E-\widetilde{\mu}}{T}.$ (A.9)

With this dimensionless variable, the Fermi function can be written as

$\displaystyle
f(x)=\Theta(-x)+e^{-|x|}\left[\frac{1}{2}\mathrm{sgn}\left(x\right)+\frac{1}{2}\tanh\left(\frac{x}{2}\right)\right].$
(A.10)

We use following properties of the sign function to help simplify the
expression:

$\displaystyle\mathrm{sgn}(x)\equiv\frac{|x|}{x}\equiv\frac{x}{|x|}\;,\quad\mathrm{sgn}(0)=0,\qquad\mathrm{sgn}(x)=2\Theta(x)-1\;,$
(A.11) $\displaystyle\mathrm{sgn}^{2}(x)\sinh(x)=\sinh(x)\;.$ (A.12)

Since $\sinh(x)$ vanishes at $x=0$, we don’t need to worry about what value to
assign to $\mathrm{sgn}^{2}(x)$ at $x=0$. Using the properties above, we can
replace the step function and exponential function as follows:

$\displaystyle\Theta(-x)=\frac{1}{2}(1-\mathrm{sgn}(x)\;,$ (A.13)
$\displaystyle e^{-|x|}=\cosh|x|-\sinh|x|=\cosh x-\mathrm{sgn}(x)\sinh x\;.$
(A.14)

Then the Fermi distribution function can be written as

$\displaystyle f=\frac{1}{2}$
$\displaystyle+\left[\cosh{x}-\mathrm{sgn}(x)\sinh{x}-1\right]\frac{1}{2}\mathrm{sgn}(x)$
$\displaystyle\qquad\qquad\qquad+\left[\cosh{x}-\mathrm{sgn}(x)\sinh{x}\right]\frac{1}{2}\tanh{(x/2)}\;.$
(A.15)

Using the properties of the singular function Eq. (A.12), the distribution
function can be written as

$\displaystyle f=f_{R}+f_{I}$ (A.16) $\displaystyle
f_{R}=\frac{1}{2}\left(1-\sinh x+\cosh x\tanh(x/2)\right)$ (A.17)
$\displaystyle f_{I}=\mathrm{sgn}(x)\frac{1}{2}\left(\cosh x-1-\sinh
x\tanh(x/2)\right),$ (A.18)

where $f_{R}$ and $f_{I}$ represent the regular and irregular part of the
distribution, respectively.

For the irregular part $f_{I}$, we use the properties of hyperbolic functions

$\displaystyle\cosh x-1=2\sinh^{2}(x/2),\qquad\sinh x=2\sinh(x/2)\cosh(x/2).$
(A.19)

Then it can be written as

$\displaystyle f_{I}$
$\displaystyle=\mathrm{sgn}(x)\frac{1}{2}\left[2\sinh^{2}(x/2)-2\sinh(x/2)\cosh(x/2)\tanh(x/2)\right]$
(A.20)
$\displaystyle=\mathrm{sgn}(x)\frac{1}{2}\left[2\sinh^{2}(x/2)-2\sinh^{2}(x/2)\right]$
(A.21) $\displaystyle=0.$ (A.22)

We show that the irregular part $f_{I}$ vanishes as an identity. On the other
hand, the regular part $f_{R}$ can be simplified, as shown below:

$\displaystyle f_{R}$
$\displaystyle=\frac{1}{2}\left[1-\frac{1}{2}\left(e^{x}-e^{-x}\right)+\frac{1}{2}\left(e^{x}+e^{-x}\right)\frac{e^{x}-1}{e^{x}+1}\right]$
(A.23)
$\displaystyle=\frac{1}{2(e^{x}+1)}\left[\left(e^{x}+1\right)-\frac{1}{2}\left[\left(e^{x}+1\right)\left(e^{x}-e^{-x}\right)-\left(e^{x}-1\right)\left(e^{x}+e^{-x}\right)\right]\right]$
(A.24)
$\displaystyle=\frac{1}{2(e^{x}+1)}\left[\left(e^{x}+1\right)-\left(e^{x}-1\right)\right]$
(A.25) $\displaystyle=f.$ (A.26)

Finally, we consider the LHS and RHS of Eq. (A) at $x=0$. With the singular
function properties as given we see that at $x=0$ both LHS and RHS of Eq. (A)
are equal to $1/2$, and in the first derivative of the RHS at $x=0$ the two
$\delta(x)$-terms

$\displaystyle\frac{d\Theta((\widetilde{\mu}-E)/T)}{dE}=-\frac{1}{T}\delta\left(\frac{\widetilde{\mu}-E}{T}\right)\;,\,\,\frac{d\,\mathrm{sgn}((E-\widetilde{\mu})/T)}{dE}=\frac{2}{T}\delta\left(\frac{E-\widetilde{\mu}}{T}\right)\;,$
(A.27)

cancel exactly as required, since there is no $\delta(x)$ on LHS. This
encourages us to believe that all of the singular expressions cancel. This
completes the demonstration of the exact validity of Eq. (A).

##### Numerical illustration

In Fig. A.1 we plot the exact Fermi-distribution (LHS of Eq. (A) with solid
lines and novel form of Fermi-distribution (RHS of Eq. (A)) with dashed lines
as a function of energy with different parameters. It demonstrates that the
LHS and RHS of Eq. (A) are equivalent to each other numerically.

Figure A.1: The exact Fermi-distribution (LHS of Eq. (A) with solid lines and
novel form of Fermi-distribution (RHS of Eq. (A)) with dashed lines as a
function of energy with different parameters. Top: we compare the RHS and LHS
of Eq. (A) with different chemical potential: $\widetilde{\mu}=0,\pm 0.1,\pm
0.4\,\pm 0.6\,\mathrm{MeV}$ at temperature $T=0.012\,\mathrm{MeV}$. Bottom:
the Fermi distribution with different temperatures
$T=0.511,0.0511,0.012\,\mathrm{MeV}$ for chemical potential
$\widetilde{\mu}=\pm 0.461\,\mathrm{MeV}$.

To illustrate the zero and finite temperature contribution to the Fermi