###### Abstract

We offer a survey of the matter-antimatter evolution within the primordial
Universe. While the origin of the tiny matter-antimatter asymmetry has
remained one of the big questions in modern cosmology, antimatter itself has
played a large role for much of the Universe’s early history. In our study of
the evolution of the Universe we adopt the position of the standard model
Lambda-CDM Universe implementing the known baryonic asymmetry. We present the
composition of the Universe across its temperature history while emphasizing
the epochs where antimatter content is essential to our understanding. Special
topics we address include the heavy quarks in quark-gluon plasma (QGP), the
creation of matter from QGP, the free-streaming of the neutrinos, the
vanishing of the muons, the magnetism in the electron-positron cosmos, and a
better understanding of the environment of the Big Bang Nucleosynthesis (BBN)
producing the light elements. We suggest but do not explore further that the
methods used in exploring the early Universe may also provide new insights in
the study of exotic stellar cores, magnetars, as well as gamma-ray burst (GRB)
events. We describe future investigations required in pushing known physics to
its extremes in the unique laboratory of the matter-antimatter early Universe.

###### keywords:

Particles, Plasmas and Electromagnetic Fields in Cosmology; Quarks to Cosmos;

xx 1 5 A SHORT SURVEY OF MATTER-ANTIMATTER EVOLUTION IN THE PRIMORDIAL
UNIVERSE Johann Rafelski, Jeremiah Birrell, Andrew Steinmetz, and Cheng Tao
Yang Johann Rafelski, Jeremiah Birrell, Andrew Steinmetz, and Cheng Tao Yang
Correspondence<EMAIL_ADDRESS>
Published: 27 June 2023 Universe 2023, 9(7), 309;
https://doi.org/10.3390/universe9070309

This article belongs to the Special Issue Remo Ruffini Festschrift, see
Acknowledgements

https://www.mdpi.com/journal/universe/special_issues/J0M337731D

###### Contents

  1. 1 Timeline of Particles and Plasmas in the Universe
    1. 1.1 Guide to $130\text{ GeV}>T>20\text{ keV}$
    2. 1.2 The five plasma epochs
    3. 1.3 The Lambda-CDM Universe
  2. 2 QGP Epoch
    1. 2.1 Conservation laws in QGP
    2. 2.2 Heavy flavor: Bottom and charm in QGP
  3. 3 Hadronic Epoch
    1. 3.1 The formation of matter
    2. 3.2 Strangeness abundance
    3. 3.3 Pion abundance
  4. 4 Leptonic Epoch
    1. 4.1 Thermal degrees of freedom
    2. 4.2 Muon abundance
    3. 4.3 Neutrino masses and oscillation
    4. 4.4 Neutrino freeze-out
    5. 4.5 Effective number of neutrinos
  5. 5 Electron-Positron Epoch
    1. 5.1 The last bastion of antimatter
    2. 5.2 Cosmic magnetism
    3. 5.3 Landau eigen-energies in cosmology
    4. 5.4 Electron-positron statistical physics
    5. 5.5 Charge neutrality and chemical potential
    6. 5.6 Magnetization of the electron-positron plasma
  6. 6 Looking in the Cosmic Rear-view Mirror

##  1 Timeline of Particles and Plasmas in the Universe

###  1.1 Guide to $130\text{ GeV}>T>20\text{ keV}$

This survey of the early Universe begins with quark-gluon plasma (QGP) at a
temperature of $T=130\text{ GeV}$. It then ends at a temperature of
$T=20\text{ keV}$ with the electron-positron epoch which was the final phase
of the Universe to contain significant quantities of antimatter. This defines
the “short” $t\approx 1/2$ hour time-span that will be covered. This work
presumes that the Universe is homogeneous and that in our casual domain, the
Universe’s baryon content is matter dominated. Our work is rooted in the
Universe as presented by Lizhi Fang and Remo Ruffini Fang, Lizhi and Ruffini
(1984, 1985, 1987). Within the realm of the Standard Model, we coherently
connect the differing matter-antimatter plasmas as each transforms from one
phase into another.

A more detailed description of particles and plasmas follows in Sect. 1.2. We
have adopted the standard $\Lambda$CDM model of a cosmological constant
($\Lambda$) and cold dark matter (CDM) where the Universe undergoes dynamical
expansion as described in the Friedmann-Lemaître-Robertson-Walker (FLRW)
metric. The contemporary history of the Universe in terms of energy density as
a function of time and temperature is shown in Fig. 1. The Universe’s past is
obtained from integrating backwards the proposed modern composition of the
Universe which contains $69\%$ dark energy, $26\%$ dark matter, $5\%$ baryons,
and $<1\%$ photons and neutrinos in terms of energy density. The method used
to obtain these results are found in Sect. 1.3.

After the general overview, we take the opportunity to enlarge in some detail
our more recent work in special topics. In Sect. 2, we describe the chemical
potentials of the QGP plasma species leading up to hadronization, Hubble
expansion of the QGP plasma, and the abundances of heavy quarks. In Sect. 3 we
discuss the formation of matter during hadronization, the role of strangeness,
and the unique circumstances which led to pions remaining abundant well after
all other hadrons were diluted or decayed. We review the roles of muons and
neutrinos in the leptonic epoch in Sect. 4. The $e^{\pm}$ plasma epoch is
described in Sect. 5 which is the final stage of the Universe where antimatter
played an important role. Here we introduce the statistical physics
description of electrons and positron gasses, their relation to the baryon
density, and the magnetization of the $e^{\pm}$ plasma prior to the
disappearance of the positrons shortly after Big Bang Nucleosynthesis (BBN). A
more careful look at the effect of the dense $e^{\pm}$ plasma on BBN is
underway. One interesting feature of having an abundant $e^{\pm}$ plasma is
the possibility of magnetization in the early Universe which we consider in
Sect. 5.2. We introduce in this work the spin magnetic moment polarization for
the first time in the context of cosmology. We address this using spin-
magnetization and mean-field theory where all the spins respond to the
collective bulk magnetism self generated by the plasma. We stop our survey at
a temperature of $T=20\text{ keV}$ with the disappearance of the positrons
signifying the end of antimatter dynamics at cosmological scales.

Figure 1: Contemporary and recent Universe composition: In this example we
assumed present day composition to be $69\%$ dark energy, $26\%$ dark matter,
$5\%$ baryons, $<1\%$ photons and neutrinos. The dashed line shows how
introduction of $2\times 0.1\text{ eV}$ mass in two of the three neutrinos
impacts the energy density evolution (Neutrino mass choice is just for
illustration. Other values are possible). The recombination temperature
$T_{r}\approx 0.25\text{ eV}$ delimits the era when the Universe was opaque
shown as the shaded region.

This primordial Universe is a plasma physics laboratory with unique properties
not found in terrestrial laboratories or stellar environments due to the high
amount of antimatter present. We suggest in Sect. 6 areas requiring further
exploration including astrophysical systems where positron content is
considerable and the possibility for novel compact objects with persistent
positron content is discussed. While the disappearance of baryonic matter is
well described in the literature, it has not always been appreciated how long
the leptonic ($\bar{\mu}=\mu^{+}$ and $\bar{e}=e^{+}$) antimatter remains a
significant presence in the Universe’s evolutionary history. We show that the
$e^{\pm}$ epoch is a prime candidate to resolve several related cosmic
mysteries such as early Universe matter in-homogeneity and the origin of
cosmic magnetic fields. While the plasma epochs of the early Universe are in
our long gone past, plasmas which share features with the primordial Universe
might possibly exist in the contemporary Universe today. Such extraordinary
stellar objects could poses properties dynamics relevant to gamma-ray burst
(GRB) Ruffini et al. (2001); Aksenov et al. (2008, 2010); Ruffini and
Vereshchagin (2013), black holes Ruffini et al. (2003, 2010, 2000) and neutron
stars (magnetars) Han et al. (2012); Belvedere et al. (2012).

###  1.2 The five plasma epochs

At an early time in the standard cosmological model, the Universe began as a
fireball, filling all space, with extremely high temperature and energy
density Rafelski (2015). Our domain of the present day Universe originated
from an ultra-relativistic plasma which contained almost a perfect symmetry
between matter and antimatter except for a small discrepancy of one part in
$10^{9}$ which remains a mystery today. There are two general solutions of
this problem both of which suppose that the Universe’s initial conditions were
baryon-antibaryon number symmetric in order to avoid ‘fine-tuning’ to a
specific value:

  * A

Case of baryonic number (charge) conservation: In order to separate space
domains in which either matter or antimatter is albeit very slightly dominant
we need a ‘force’ capable of dynamically creating this matter-antimatter
separation. This requires that two of the three Sakharov Sakharov (1967, 1991)
conditions be fulfilled:

    * 1.

Violation of CP-invariance allowing to distinguish matter from antimatter

    * 2.

Non-stationary conditions in absence of local thermodynamic equilibrium

Other than very distant antimatter domains Cohen et al. (1998) the missing
antimatter could be perhaps ‘stored’ in a compact structure Khlopov et al.
(2000); Blinnikov et al. (2015); Khlopov and Lecian (2023).

  * B

There is no known cause for baryon charge conservation. Therefore it is
possible to consider the full Sakharov model with

    * 3.

Absence of baryonic charge conservation

allowing the dynamical formation of the uniform matter-antimatter asymmetry
typically occurring prior to the epoch governed by physics confirmed by
current experiment to which environs we restrict this short survey. A well
studied example is the Affleck-Dine mechanism Affleck and Dine (1985).

Very early formation of baryon asymmetry is further supported by the finding
that the known CP-violation in the Standard Model’s weak sector is
insufficient to explain in quantitative terms the baryon asymmetry Rubakov and
Shaposhnikov (1996). However, baryon asymmetry could develop at a later stage
in Universe evolution. We show in this review that this remains a topic
deserving further investigation. In this work we take a homogeneous prescribed
baryon asymmetry obtained from observed baryon to photon ratio in the
Universe. Additional comments on the situation in the context of non-
equilibria processes are made in Sect. 2.2, at the end of Sect. 4.2, and in
Sect. 6.

The primordial hot Universe fireball underwent several practically adiabatic
phase changes which dramatically evolved its bulk properties as it expanded
and cooled. We present an overview Fig. 2 of particle families across all
epochs in the Universe, as a function of temperature and thus time. The comic
plasma, after the electroweak symmetry breaking epoch and presumably
inflation, occurred in the early Universe in the following sequence:

Figure 2: Normalized Universe constituent matter and radiation components
$\Omega_{i}$ are evolved over cosmological timescales (top scale, bottom scale
is temperature $T$) from contemporary observational cosmology to the QGP epoch
of the Universe. Vertical lines denote transitions between distinct epochs.
Solid neutrino (green) line shows contribution of massless neutrinos, while
the dashed line shows $1$ massless and $2\times 0.1$ eV neutrinos (Neutrino
mass choice is just for illustration. Other values are possible).

  1. 1.

Primordial quark-gluon plasma: At early times when the temperature was between
$130\text{ GeV}>T>150\text{ MeV}$ we have the building blocks of the Universe
as we know them today, including the leptons, vector bosons, and all three
families of deconfined quarks and gluons which propagated freely. As all
hadrons are dissolved into their constituents during this time, strongly
interacting particles $u,d,s,t,b,c,g$ controlled the fate of the Universe.
Here we will only look at the late-stage evolution at around $150\text{ MeV}$.

  2. 2.

Hadronic epoch: Around the hadronization temperature $T_{h}\approx 150\text{
MeV}$, a phase transformation occurred forcing the strongly interacting
particles such as quarks and gluons to condense into confined states Letessier
and Rafelski (2008). It is here where matter as we know it today forms and the
Universe becomes hadronic-matter dominated. In the temperature range
$150\text{ MeV}>T>20\text{ MeV}$ the Universe is rich in physics phenomena
involving strange mesons and (anti)baryons including (anti)hyperon abundances
Fromerth et al. (2012); Yang and Rafelski (2022).

  3. 3.

Lepton-photon epoch: For temperature $10\text{ MeV}>T>2\text{ MeV}$, the
Universe contained relativistic electrons, positrons, photons, and three
species of (anti)neutrinos. Muons vanish partway through this temperature
scale. In this range, neutrinos were still coupled to the charged leptons via
the weak interaction Birrell et al. (2014); Birrell (2014). During this time
the expansion of the Universe is controlled by leptons and photons almost on
equal footing.

  4. 4.

Final antimatter epoch: After neutrinos decoupled and become free-streaming,
referred to as neutrino freeze-out, from the cosmic plasma at $T=2\text{
MeV}$, the cosmic plasma was dominated by electrons, positrons, and photons.
We have shown in Grayson et al. (2023) that this plasma existed until
$T\approx 0.02\text{ MeV}$ such that BBN occurred within a rich electron-
positron plasma. This is the last time the Universe will contain a significant
fraction of its content in antimatter.

  5. 5.

Moving towards a matter dominated Universe: The final major plasma stage in
the Universe began after the annihilation of the majority of $e^{\pm}$ pairs
leaving behind a residual amount of electrons determined by the baryon
asymmetry in the Universe and charge conservation. The Universe was still
opaque to photons at this point and remained so until the recombination period
at $T\approx 0.25\text{ eV}$ starting the era of observational cosmology with
the CMB. This final epoch of the primordial Universe will not be described in
detail here, but is well covered in Aghanim et al. (2020).

Figure 3: The evolution of the photon reheating (black line) process in terms
of fractional temperature change in the Universe. Figure adapted from Rafelski
and Birrell (2014). The dashed portion is a qualitative description subject to
the exact model of QGP hadronization.

Each plasma outlined above contributes to the thermal behavior of the Universe
over time. This is illustrated in Fig. 3 where the fractional drop in
temperature during each plasma transformation is plotted. Each subsequent
plasma lowers the available degrees of freedom (as the particle inventory is
whittled away) as the Universe cools Wantz and Shellard (2010); Rafelski and
Birrell (2014). Each drop in degrees of freedom represents entropy being
pumped into the photons as entropy is conserved (up until local gravitational
processes become relevant) in an expanding Universe. As there are no longer
degrees of freedom to consume, thereby reheating the photon field further, the
fractional temperature remains constant today.

In Fig. 2 we begin on the right at the end of the QGP era. The first dotted
vertical line shows the QGP phase transition and hadronization, near
$T=150\text{ MeV}$. The hadron era proceeds with the disappearance of muons,
pions, and heavier hadrons. This constitutes a reheating period, with energy
and entropy from these particles being transferred to the remaining $e^{\pm}$,
photon, neutrino plasma. The black circle near $T=115\text{ MeV}$ denotes our
change from $2+1$-flavor lattice QCD Kronfeld (2013); D’Elia et al. (2013);
Bonati et al. (2014) data for the hadron energy density, taken from Borsanyi
et al. Borsanyi (2013); Borsanyi et al. (2014), to an ideal gas model
Bernstein (1988) at lower temperature. We note that the hadron ideal gas
energy density matches the lattice results to less than a percent at
$T=115\text{ MeV}$ Philipsen (2013).

To the right of the QGP transition region, the solid hadron line shows the
total energy density of quarks and gluons. From top to bottom, the dot-dashed
hadron lines to the right of the transition show the energy density fractions
of $2+1$-flavor (u,d,s) lattice QCD matter (almost indistinguishable from the
total energy density), charm, and bottom (both in the ideal gas
approximation). To the left of the transition the dot-dashed lines show the
pion, kaon, $\eta+f_{0}$, $\rho+\omega$, nucleon, $\Delta$, and Y
contributions to the energy fraction.

Continuing to the second vertical line at $T=\mathcal{O}(1\,\text{ MeV})$, we
come to the annihilation of $e^{\pm}$ and the photon reheating period. Notice
that only the photon energy density fraction increases, as we assume that
neutrinos are already decoupled at this time and hence do not share in the
reheating process, leading to a difference in photon and neutrino
temperatures. This is not strictly correct but it is a reasonable simplifying
assumption for the current purpose; see Mangano et al. (2005); Fornengo et al.
(1997); Mangano et al. (2002); Birrell et al. (2014). We next pass through a
long period, from $T=\mathcal{O}(1\,\text{ MeV})$ until
$T=\mathcal{O}(1\,\text{ eV})$, where the energy density is dominated by
photons and free-streaming neutrinos. BBN occurs in the approximate range
$T=40-70\text{ keV}$ and is indicated by the next two vertical lines in Fig.
2. It is interesting to note that, while the hadron fraction is insignificant
at this time, there is still a substantial background of $e^{\pm}$ pairs
during BBN (see Sect. 5.1).

We then come to the beginning of the matter dominated regime, where the energy
density is dominated by the combination of dark matter and baryonic matter.
This transition is the result of the redshifting of the photon and neutrino
energy, $\rho\propto a^{-4}\propto T^{4}$, whereas for non-relativistic matter
$\rho\propto a^{-3}\propto T^{3}$. Recombination and photon decoupling occurs
near the transition to the matter dominated regime, denoted by the (Fig. 2)
vertical line at $T=0.25\text{ eV}$.

Finally, as we move towards the present day CMB temperature of
$T_{\gamma,0}=0.235$ meV on the left hand side, we have entered the dark
energy dominated regime. For the present day values, we have used the energy
densities proscribed by the Planck parameters Ade et al. (2014) using Eq. (14)
and zero Universe spatial curvature. The photon energy density is fixed by the
CMB temperature $T_{\gamma,0}$ and the neutrino energy density is fixed by
$T_{\gamma,0}$ along with the photon to neutrino temperature ratio and
neutrino masses. Both constitute $<1\%$ of the current energy budget.

The Universe evolution and total energy densities were computed using massless
neutrinos, but for comparison we show the energy density of massive neutrinos
in the dashed green line. For the dashed line we used two neutrino flavors
with masses $m_{\nu}=0.1\text{ eV}$ and one massless flavor. Note that the
inclusion of neutrino mass causes the leveling out of the neutrino energy
density fraction during the matter dominated period, as compared to the
continued redshifting of the photon energy.

###  1.3 The Lambda-CDM Universe

Here we provide background on the standard $\Lambda$CDM cosmological (FLRW-
Universe) model that is used in the computation of the composition of the
Universe over time. We use the spacetime metric with metric signature
$(+1,-1,-1,-1)$ in spherical coordinates

$ds^{2}=c^{2}dt^{2}-a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+\sin^{2}(\theta)d\phi^{2})\right]$
(1)

characterized by the scale parameter $a(t)$ of a spatially homogeneous
Universe. The geometric parameter $k$ identifies the Gaussian geometry of the
spacial hyper-surfaces defined by co-moving observers. Space is a Euclidean
flat-sheet for the observationally preferred value $k=0$ Ade et al. (2014,
2016); Aghanim et al. (2020). In this case it can be more convenient to write
the metric in rectangular coordinates

$ds^{2}=c^{2}dt^{2}-a^{2}(t)\left[dx^{2}+dy^{2}+dz^{2}\right].$ (2)

We will work in units where $\hbar=1,\ c=1$.

The global Universe dynamics can be characterized by two quantities: the
Hubble parameter $H$, a strongly time dependent quantity on cosmological time
scales, and the deceleration parameter $q$:

$\displaystyle H(t)^{2}$
$\displaystyle\equiv\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi
G_{N}}{3}\rho_{tot}\,,$ (3)
$\displaystyle\frac{\ddot{a}}{a}=-qH^{2},\qquad\qquad q$
$\displaystyle\equiv-\frac{a\ddot{a}}{\dot{a}^{2}},\qquad\qquad\dot{H}=-H^{2}(1+q)\,,$
(4)

where $G_{N}$ is the Newtonian gravitational constant and $\rho_{tot}$ is the
energy density of the Universe and composed of the various energy densities in
the Universe. The deceleration parameter $q$ is defined in terms of the second
derivative of the scale parameter.

In Fig. 4 (left) we illustrate the late stage evolution of the parameters $H$
and $q$ given in Eq. (3) and Eq. (4) compared to temperature. This illustrates
how the Universe evolves according to the Friedmann equations Eq. (3) and Eq.
(4) above. The deceleration begins radiation dominated with $q=1$ and then
transitions to matter dominated $q=1/2$. Within the $\Lambda$CDM model the
contemporary Universe is undergoing a transition from matter dominated to dark
energy dominated, where the deceleration would settle on the asymptotic value
of $q=-1$ Rafelski and Birrell (2014). However, several alternate models:
phantom energy Caldwell et al. (2003), Chaplygin gas Bilic et al. (2002), or
more generally dynamic (spatially and/or time dependent) dark energy Benevento
et al. (2020) cannot be excluded in absence of strong evidence for the
constancy of dark energy.

Within the $\Lambda$CDM model only usual forms of energy are relevant before
recombination epoch, see Fig. 2. Any alternate model can be thus constrained
by understanding precisely the evolution of the Universe prior to this epoch.
Part of the program of this survey is to connect the late stage evolution to
the very early Universe during and prior to BBN accounting for the
unexpectedly considerable antimatter content.

The current tension in Hubble parameter measurements Perivolaropoulos and
Skara (2022); Di Valentino et al. (2021); Aluri et al. (2023) might benefit
from closer inspection of these earlier denser periods should these contribute
to modification of the conventional model of Universe expansion. We further
note that the JWST has recently discovered that galaxy formation began earlier
than predicted which requires reevaluation of early Universe matter
inhomogeneities Yan et al. (2023). Fig. 4 (right) shows the close relationship
between the redshift $z$ and the Hubble parameter. Deviations separating the
two occur from the transitions which changed the deceleration value.

Figure 4:  (left) The numerically solved later $t>10^{-1}\ \mathrm{yr}$
evolution of photon and neutrino background temperatures $T_{\gamma},\
T_{\nu}$ (black and black dashed lines) and the deceleration parameter $q$
(thin blue line) over the lifespan of the Universe. (right) The evolution of
the Hubble parameter $1/H$ (black line) and redshift $z$ (blue dashed line)
which is related to the scale parameter $a(t)$. Figure adapted from Rafelski
and Birrell (2014).

The Einstein equations with a cosmological constant $\Lambda$ corresponding to
dark energy are:

$G^{\mu\nu}=R^{\mu\nu}-\left(\frac{R}{2}+\Lambda\right)g^{\mu\nu}=8\pi
G_{N}T^{\mu\nu},\quad R=g_{\mu\nu}R^{\mu\nu}.$ (5)

The homogeneous and isotropic symmetry considerations imply that the stress
energy tensor is determined by an energy density and an isotropic pressure

$\displaystyle T^{\mu}_{\nu}=\mathrm{diag}(\rho,-P,-P,-P).$ (6)

It is common to absorb the Einstein cosmological constant $\Lambda$ into the
energy and pressure

$\rho_{\Lambda}=\frac{\Lambda}{8\pi G_{N}},\qquad
P_{\Lambda}=-\frac{\Lambda}{8\pi G_{N}}$ (7)

and we implicitly consider this done from now on.

Two dynamically independent Friedmann equations Weinberg (1972) arise using
the metric Eq. (1) in Eq. (5):

$\frac{8\pi
G_{N}}{3}\rho=\frac{\dot{a}^{2}+k}{a^{2}}=H^{2}\left(1+\frac{k}{\dot{a}^{2}}\right),\qquad\frac{4\pi
G_{N}}{3}(\rho+3P)=-\frac{\ddot{a}}{a}=qH^{2}.$ (8)

We can eliminate the strength of the interaction, $G_{N}$, solving both these
equations for ${8\pi G_{N}}/{3}$, and equating the result to find a relatively
simple constraint for the deceleration parameter:

$q=\frac{1}{2}\left(1+3\frac{P}{\rho}\right)\left(1+\frac{k}{\dot{a}^{2}}\right).$
(9)

For a spatially flat Universe, $k=0$, note that in a matter-dominated era
where $P/\rho<<1$ we have $q\simeq 1/2$; for a radiative Universe where
$3P=\rho$ we find $q=1$; and in a dark energy Universe in which $P=-\rho$ we
find $q=-1$. Spatial flatness is equivalent to the assertion that the energy
density of the Universe equals the critical density

$\rho=\rho_{\text{crit}}\equiv\frac{3H^{2}}{8\pi G_{N}}.$ (10)

The CMB power spectrum is sensitive to the deceleration parameter and the
presence of spatial curvature modifies $q$. The Planck results Ade et al.
(2014, 2016); Aghanim et al. (2020) constrain the effective curvature energy
density fraction,

$\Omega_{K}\equiv 1-\rho/\rho_{\text{crit}},$ (11)

to

$|\Omega_{K}|<0.005.$ (12)

This indicates a nearly flat Universe which is spatially Euclidean. We will
work within an exactly spatially flat cosmological model, $k=0$. As must be
the case for any solution of Einstein’s equations, Eq. (8) implies that the
energy momentum tensor of matter is divergence free:

$T^{\mu\nu};_{\nu}=0\Rightarrow-\frac{\dot{\rho}}{\rho+P}=3\frac{\dot{a}}{a}=3H.$
(13)

A dynamical evolution equation for $\rho(t)$ arises once we combine Eq. (13)
with Eq. (8), eliminating $H$. Given an equation of state $P(\rho)$, solutions
of this equation describes the dynamical evolution of matter in the Universe.
In practice, we evolve the system in both directions in time. On one side, we
start in the present era with the energy density fractions fit by the central
values found in Planck data Ade et al. (2014)

$H_{0}=67.4\,\text{km/s/Mpc},\hskip 5.69054pt\Omega_{b}=0.05,\hskip
5.69054pt\Omega_{c}=0.26,\hskip 5.69054pt\Omega_{\Lambda}=0.69,$ (14)

and integrate backward in time. On the other hand, we start in the QGP era
with an equation of state determined by an ideal gas of SM particles, combined
with a perturbative QCD equation of state for quarks and gluons Borsanyi et
al. (2014), and integrate forward in time. As the Universe continues to dilute
from dark energy in the future, the cosmic equation of state will become well
approximated by the de Sitter inflationary metric which is a special case of
FLRW.

##  2 QGP Epoch

###  2.1 Conservation laws in QGP

During the first $\Delta t\approx 30\ \mu$sec after the Big Bang, the early
Universe is a hot soup that containing the elementary primordial building
blocks of matter and antimatter Rafelski (2015). In particular it contained
the light quarks which are now hidden in protons and neutrons. Beyond this
there were also electrons, photons, neutrinos, and massive strange and charm
quarks. These interacting particle species were kept in chemical and thermal
equilibrium with one another. Gluons which mediated the color interaction are
very abundant as well. This primordial phase lasted as long as the temperature
of the Universe was more than 110,000 times than the expected temperature
$T_{\odot}=1.36\text{ keV}\ (1.58\times 10^{7}\ \mathrm{K})$ at the center of
the Sun Castellani et al. (1997).

Figure 5:  The evolution of the cosmic baryon chemical potential $\mu_{B}$
after hadronization (blue line). Curves for QGP (thin black line) created in
terrestrial accelerators for differing entropy-per-baryon $s/B$ values are
included Rafelski and Schnabel (1988). The boundary (red line) where QGP
condenses into hadrons is illustrated at an energy density of $0.5\text{
GeV}/\mathrm{fm}^{3}$ as determined through lattice computation Bazavov et al.
(2014).

The conditions in the early Universe and those created in relativistic
collisions of heavy atomic nuclei differ somewhat: whereas the primordial
quark-gluon plasma survives for about 25 $\mu$sec in the Big Bang, the
comparable extreme conditions created in ultra-relativistic nuclear collisions
are extremely short-lived Rafelski et al. (2001) on order of $10^{-23}$
seconds. As a consequence of the short lifespan of laboratory QGP in heavy-ion
collisions Ollitrault (1992); Petrán et al. (2013), they are not subject to
the same weak interaction dynamics Ryu et al. (2015) as the characteristic
times for weak processes are too lengthy Rafelski (1982). Therefore our
ability to recreate the conditions of the primordial QGP are limited due to
the relativistic explosive disintegration of the extremely hot dense
relativistic ‘fireballs’ created in modern accelerators. This disparity is
seen in Fig. 5 where the chemical potential of QGP $\mu_{q}=\mu_{B}/3$
Rafelski and Schnabel (1988) for various values of entropy-per-baryon $s/b$
relevant to relativistic particle accelerators are plotted alongside the
evolution of the cosmic hadronic plasma chemical potential. The confinement
transition boundary (red line in Fig. 5) was calculated using a parameters
obtained from Letessier and Rafelski (2023) in agreement with lattice results
Bazavov et al. (2014). The QGP precipitates hadrons in the cosmic fluid at a
far higher entropy ratio than those accessible by terrestrial means and the
two manifestations of QGP live far away from each other on the QCD phase
diagram Jacak and Muller (2012).

The work of Fromerth et. al. Fromerth et al. (2012) allows us to parameterize
the chemical potentials $\mu_{d}$, $\mu_{e}$, and $\mu_{\nu}$ during this
epoch as they are the lightest particles in each main thermal category:
quarks, charged leptons, and neutral leptons. The quark chemical potential is
determined by the following three constraints Fromerth et al. (2012):

Figure 6: Plot of the down quark chemical potential (black), electron chemical
potential (dotted red) and neutrino chemical potential (dashed green) as a
function of time. These are 2003 unpublished results of Fromerth & Rafelski
Fromerth and Rafelski (2002); also presented in Ref. Rafelski (2020)

  1. 1.

Electric charge neutrality $Q=0$, given by

$\displaystyle\frac{Q}{V}=n_{Q}\equiv\sum_{f}\,Q_{f}\,n_{f}(\mu_{f},T)=0$ (15)

where $Q_{f}$ is the charge and $n_{f}$ is the numerical density of each
species $f$. $Q$ is a conserved quantity in the Standard Model under global
$U(1)_{EM}$ symmetry. This is summed is over all particles present in the QGP
epoch.

  2. 2.

Baryon number and lepton number neutrality $B-L=0$, given by

$\displaystyle\frac{B-L}{V}=n_{B}-n_{L}\equiv\sum_{f}(B_{f}-L_{f})n_{f}(\mu_{f},T)=0$
(16)

where $L_{f}$ and $B_{f}$ are the lepton and baryon number for the given
species $f$. This condition is phenomenologically motivated by baryogenesis
and is exactly conserved in the Standard Model under global $U(1)_{B-L}$
symmetry. We note many Beyond-Standard-Model (BSM) models also retain this as
an exact symmetry though Majorana neutrinos do not.

  3. 3.

The entropy-per-baryon density ratio $s/n_{B}$ is a constant and can be
written as

$\displaystyle\frac{S}{B}=\frac{s}{n_{B}}=\frac{\sum_{f}s_{f}(\mu_{f},T)}{\sum_{f}B_{f}n_{f}(\mu_{f},T)}=\mathrm{const}$
(17)

where $s_{f}$ is the entropy density of given species $f$. As the expanding
Universe remains in thermal equilibrium, the entropy is conserved within a co-
moving volume. The baryon number within a co-moving volume is also conserved.
As both quantities dilute with $1/a(t)^{3}$ within a normal volume, the ratio
of the two is constant. This constraint does not become broken until spatial
inhomogeneities from gravitational attraction becomes significant, leading to
increases in local entropy.

At each temperature $T$, the above three conditions form a system of three
coupled, nonlinear equations of the three chosen unknowns (here we have
$\mu_{d}$, $\mu_{e}$, and $\mu_{\nu}$). In Fig. 6 we present numerical
solutions to the conditions Eq. (15)-Eq. (17) and plot the chemical potentials
as a function of time. As seen in the figure, the three potentials are in
alignment during the QGP phase until the hadronization epoch where the down
quark chemical potential diverges from the leptonic chemical potentials before
reaching an asymptotic value at late times. This asymptotic value is given as
approximately $\mu_{q}\approx m_{N}/3$ the mass of the nucleons and represents
the confinement of the quarks into the protons and neutrons at the end of
hadronization.

Figure 7: Plot of the down quark chemical potential $\mu_{d}$ as a function of
temperature for differing values of entropy-per-baryon $S/B$ ratios. (2003
unpublished, Fromerth & Rafelski Rafelski (2020))

This asymptotic limit is also shown in Fig. 7 where we present the down quark
chemical potential for different values of the entropy-to-baryon ratio. While
the $s/n_{B}$ ratio has large consequences for the plasma at high
temperatures, the chemical potential is insensitive to this parameter at low
temperatures the degrees of freedom are dominated by the remaining baryon
number rather than the thermal degrees of freedom of the individual quarks.
Therefore the entropy to baryon value today greatly controls the quark content
when the Universe was very hot. We note that the distribution of quarks in the
QGP plasma does not remain fixed to the Fermi-Dirac distribution for thermal
and entropic equilibrium. The quark partition function is instead

$\displaystyle\ln\mathcal{Z}_{\mathrm{quarks}}=\sum_{q}\ln\left(1+\Upsilon_{q}(t)e^{-\beta
E_{q}}\right)\,,\qquad\Upsilon_{q}(t)=\gamma_{q}(t)\lambda_{q}\,\qquad
q=u,d,c,s,t,b,$ (18)

which is summed over all quarks and their quantum numbers. In Eq. (18),
$\lambda_{q}$ is the quark fugacity while $\gamma_{q}(t)$ is the temporal
inhomogeneity of the population distribution Rafelski (2020). The product of
the two $\Upsilon_{q}(t)=\gamma_{q}(t)\lambda_{q}$ is then defined as the
generalized fugacity for the species. Because of nuclear reactions, these
distributions populate and depopulate over time which pulls the gas off
entropic equilibrium while retaining temperature $T$ with the rest of the
Universe Letessier and Rafelski (2023). When $\gamma\neq 1$, the entropy of
the quarks is no longer minimized. As entropy in the cosmic expansion is
conserved overall, this means the entropy gain or loss is then related to the
entropy moving between the quarks or its products.

In practice, the generalized fugacity is $\Upsilon=1$ during the QGP epoch as
the quarks in early Universe remained in both thermal and entropic
equilibrium. This is because the Universe’s expansion was many orders of
magnitude slower than the process reaction and decay timescales Letessier and
Rafelski (2023). However near the hadronization temperature, heavy quarks
abundance and deviations from chemical equilibrium have not yet been studied
in great detail. We show in Sect. 2.2 and Yang and Rafelski (2020) that the
bottom quarks can deviate from chemical equilibrium $\gamma\neq 1$ by breaking
the detailed balance between reactions of the quarks.

###  2.2 Heavy flavor: Bottom and charm in QGP

In the QGP epoch, up and down $(u,d)$ (anti)quarks are effectively massless
and remain in equilibrium via quark-gluon fusion. Strange $(s)$ (anti)quarks
are in equilibrium via weak, electromagnetic, and strong interactions until
$T\sim 12\text{ MeV}$ Yang and Rafelski (2022). In this section, we focus on
the heavier charm and bottom $(c,b)$ (anti)quarks. In primordial QGP, the
bottom and charm quarks can be produced from strong interactions via quark-
gluon pair fusion processes and disappear via weak interaction decays. For
production, we have the following processes

$\displaystyle q+q$ $\displaystyle\longrightarrow b+\bar{b},\qquad
q+q\longrightarrow c+\bar{c},$ (19) $\displaystyle g+g$
$\displaystyle\longrightarrow b+\bar{b},\qquad g+g\longrightarrow c+\bar{c},$
(20)

for bottom and charm and

$\displaystyle b\longrightarrow c+l+\nu_{l},\qquad b\longrightarrow
c+q+\bar{q}$ (21) $\displaystyle c\longrightarrow s+l+\nu_{l},\qquad
c\longrightarrow s+q+\bar{q}$ (22)

for their decay. A detailed calculation of production and decay rate can be
found in Yang and Rafelski (2020).

Figure 8: Comparison of Hubble time $1/H$, quark lifespan $\tau_{q}$, and
characteristic time for production via quark-gluon pair fusion for (top
figure) charm and (bottom figure) bottom quarks as a function of temperature.
Both figures end at approximately the hadronization temperature of
$T_{h}\approx 150\text{ MeV}$. Three different masses $m_{b}=4.2\text{ GeV}$
(blue short dashes), $4.7\text{ GeV}$, (solid black), $5.2\text{ GeV}$ (red
long dashes) for bottom quarks are plotted to account for its decay width.

In the early Universe within the temperature range
$130\,\mathrm{GeV}>T>150\,\mathrm{MeV}$ we have the following particles:
photons, $8_{c}$-gluons, $W^{\pm}$, $Z^{0}$, three generations of
$3_{c}$-quarks and leptons in the primordial QGP. The Hubble parameter can be
written as the sum of particle energy densities $\rho_{i}$ for each species

$\displaystyle H^{2}=\frac{8\pi
G_{N}}{3}\left(\rho_{\gamma}+\rho_{\mathrm{lepton}}+\rho_{\mathrm{quark}}+\rho_{g,{W^{\pm}},{Z^{0}}}\right),$
(23)

where $G_{N}$ is Newton’s constant of gravitation. Ultra-relativistic
particles (which are effectively massless) and radiation dominate the speed of
expansion.

The Universe’s characteristic expansion time constant $1/H$ is seen in Fig. 8
(both top and bottom figures). The (top) figure plots the relaxation time for
the production and decay of charm quarks as a function of temperature. For the
entire duration of QGP, the Hubble time is larger than the decay lifespan and
production times of the charm quark. Therefore, the heavy charm quark remains
in equilibrium as its processes occur faster than the expansion of the
Universe. Additionally, the charm quark production time is faster than the
charm quark decay. The faster quark-gluon pair fusion keeps the charm in
chemical equilibrium up until hadronization. After hadronization, charm quarks
form heavy mesons that decay into multi-particles quickly. Charm content then
disappears from the Universe’s particle inventory.

In Fig. 8 (bottom) we plot the relaxation time for production and decay of the
bottom quark with different masses as a function of temperature. It shows that
both production and decay are faster than the Hubble time $1/H$ for the
duration of QGP. Unlike charm quarks however, the relaxation time for bottom
quark production intersects with bottom quark decay at a temperatures
dependant on the mass of the bottom. This means that the bottom quark
decouples from the primordial plasma before hadronization as the production
process slows down at low temperatures. The speed of weak interaction decays
then dilutes bottom quark content of the QGP plasma pulling the distribution
off equilibrium with $\Upsilon\neq 1$ (see Eq. (18)) in the temperature domain
below the crossing point, but before hadronization. All of this occurs with
rates faster than Hubble expansion and thus as the Universe expands, the
system departs from a detailed chemical balance rather than thermal freezeout.

Figure 9: The generalized fugacity $\Upsilon_{b}$ of free unconfined bottom
quark as a function of temperature in QGP up to the hadronization temperature
of $T_{h}\approx 150\text{ MeV}$ for three different bottom masses
$m_{b}=4.2\text{ GeV}$ (solid blue), $4.7\text{ GeV}$, (solid black),
$5.2\text{ GeV}$ (solid red).

Let us describe the dynamical non-equilibrium of bottom quark abundance in QGP
in more detail. The competition between decay and production reaction rates
for bottom quarks in the early Universe can be written as

$\displaystyle\frac{1}{V}\frac{dN_{b}}{dt}=\big{(}\,1-\Upsilon^{2}_{b}\,\big{)}\,R^{\mathrm{Source}}_{b}-\Upsilon_{b}\,R^{\mathrm{Decay}}_{b}\;,$
(24)

where $N_{b}$ is the bottom quark abundance, $\Upsilon_{b}$ is the general
fugacity of bottom quarks, and $R^{\mathrm{Source}}_{b}$ and
$R^{\mathrm{Decay}}_{b}$ are the thermal reaction rates per volume of
production and decay of bottom quark, respectively Yang and Rafelski (2020).
The bottom source rate is controlled by quark-gluon pair fusion rate which
vanishes upon hadronization. The decay rate depends on whether the bottom
quarks are unconfined and free or bound within B-mesons which is controlled by
the plasma temperature. Under the adiabatic approximation , we solve for the
generalized bottom fugacity $\Upsilon_{b}$ in Eq. (24) yielding

$\displaystyle\Upsilon_{b}=\frac{R^{\mathrm{Decay}}_{b}}{2R^{\mathrm{Source}}_{b}}\left[\sqrt{1+\left(2R^{\mathrm{Source}}_{b}/R^{\mathrm{Decay}}_{b}\right)^{2}}-1\right]\,.$
(25)

In Fig. 9 we show the fugacity of the bottom quarks as a function of
temperature $T=0.3\sim 0.15\text{ GeV}$ for different masses of bottom quarks.
In all cases, we have prolonged non-equilibrium $\Upsilon_{b}\neq 1$ because
the decay and production rates of bottom quarks are of comparable temporal
size to one another. The bottom content of QGP is exhausted as
$\Upsilon_{b}\rightarrow 0$ as the Universe cools in temperature. For smaller
masses, some bottom quark content is preserved up until hadronization as the
strong interaction formation rate slows the depletion from weak decay near the
QGP to HG phase transformation.

As demonstrated above, the bottom quark flavor is capable to imprint arrow in
time on physical processes being out of chemical equilibrium during the epoch
$T=0.3\sim 0.15\text{ GeV}$. This is one of the required Sakharov condition
(see Sect. 1.2) for baryogenesis. Our results provide a strong motivation to
explore the physics of baryon non-conservation involving the bottom quarks and
bound $b\bar{b}$ bottonium states in a thermal environment. Given that the
non-equilibrium of bottom flavor arises at a relatively low QGP temperature
allows for the baryogenesis to occur across primordial QGP hadronization epoch
Yang and Rafelski (2020). This result establishes the temperature era for the
non-equilibrium abundance of bottom quarks.

##  3 Hadronic Epoch

###  3.1 The formation of matter

It is in this epoch that the matter of the Universe, including all the baryons
which make up visible matter today, was created Fromerth and Rafelski (2002);
Rafelski (2020). Unlike the fundamental particles, such as the quarks or W and
Z, the mass of these hadrons is not due to the Higgs mechanism, but rather
from the condensation of the QCD vacuum Rafelski (2015); Roberts (2021, 2023).
The quarks from which protons and neutrons are made have a mass more than 100
times smaller than these nucleons. The dominant matter mass-giving mechanism
arises from quark confinement Hagedorn (1985). Light quarks are compressed by
the quantum vacuum structure into a small space domain a hundred times smaller
than their natural ‘size’. A heuristic argument can be made by considering the
variance in valance quark momentum $\Delta p$ required by the Heisenberg
uncertainty principle by confining them to a space of order $\Delta x\approx
1\ \mathrm{fm}$ and the energy density of the attractive gluon field required
to balance that outward pressure. That energy cost then manifests as the
majority of the nucleon mass. The remaining few percent of mass is then due to
the fact that quarks also have inertial mass provided by the Higgs mechanism
as well as the electromagnetic mass for particles with charge.

The QGP-hadronization transformation is not instantaneous and involves a
transitory period containing both hadrons and QGP Rafelski (2020). Therefore
the conservation laws outlined in Eq. (15) - Eq. (17) can be violated in one
phase as long as it is equally compensated in the other phase. This means the
partition function during hadronization, and thus the formation of matter,
should be parameterized between the hadron gas (HG) component and QGP
component as

$\displaystyle\ln\mathcal{Z}_{tot}=f_{HG}(T)\ln\mathcal{Z}_{HG}+\left[1-f_{HG}(T)\right]\ln\mathcal{Z}_{QGP}\,,$
(26)

where $f_{HG}(T)$ is the proportion of the phase space occupied by the hadron
gas with values between $0<f_{HG}<1$. The charge neutrality condition Eq. (15)
is then modified to be

$\displaystyle
n_{Q,HG+QGP}=f_{HG}(T)n_{HG,Q}+\left[1-f_{HG}(T)\right]n_{QGP,Q}=0\,.$ (27)

At a temperature of $T_{h}\approx 150\text{ MeV}$, the quarks and gluons
become confined and condense into hadrons (both baryons and mesons). During
this period, the number of baryon-antibaryon pairs is sufficiently high that
the asymmetry (of $\sim 1$ in $10^{9}$) would be essentially invisible until a
temperature of between $40-50\text{ MeV}$. We note that CPT symmetry is
protected by the lack of asymmetry in normal Standard Model reactions to some
large factor by the accumulation of scattering events through the majority of
the Universe’s evolution. CPT-violation is similarly restricted by possible
mass difference in the Kaons Fromerth and Rafelski (2003) via the hypothetical
difference in strange-antistrange quark masses which are expected to be small
if not identically zero.

Figure 10: The fractional energy density of the luminous Universe (photons and
leptons (white), mesons (blue), and hadrons (red)) as a function of the
temperature of the Universe from hadronization to the contemporary era. This
figure is a companion figure to Fig. 2. (2003 unpublished, Fromerth & Rafelski
Rafelski (2020))

In Fig. 10, we present the fraction of visible radiation and matter split
between the baryons, mesons, and photons and leptons. For a brief early
Universe period after QGP hadronization when the large amount of antimatter
found in antiquarks converted into the dense gas of hadrons, their
contribution to the energy density of the Universe competed with that of
radiation and leptons Rafelski (2020). Mass of matter will not emerge again
until the late Universe after recombination though by that point dark matter
would become the dominant form of matter in the cosmos.

The chemical potential of baryons after hadronization can be determined by the
conserved baryon-per-entropy ratio under adiabatic expansion. Considering the
net baryon density in the early Universe with temperature range
$150\,\mathrm{MeV}>T>5\text{ MeV}$ Yang and Rafelski (2022) we write

$\displaystyle\frac{\left(n_{B}-n_{\overline{B}}\right)}{s}$
$\displaystyle=\frac{1}{s}\left[\left(n_{p}-n_{\overline{p}}\right)+\left(n_{n}-n_{\overline{n}}\right)+\left(n_{Y}-n_{\overline{Y}}\right)\right]$
$\displaystyle=\frac{45}{2\pi^{4}g^{s}_{\ast}}\sinh\left[\frac{\mu_{B}}{T}\right]F_{N}\left[1+\frac{F_{Y}}{F_{N}}\sqrt{\frac{1+e^{-\mu_{B}/T}\,F_{Y}/F_{K}}{1+e^{\mu_{B}/T}\,F_{Y}/F_{K}}}\right].$
(28)

where $\mu_{B}$ is the baryon chemical potential, $g_{*}^{s}$ represents the
effective entropic degrees of freedom, and we employ phase-space functions
$F_{i}$ for the set of nucleon $N$, kaon $K$, and hyperon $Y$ particles. These
functions are defined in Section 11.4 of Letessier and Rafelski (2023) and
given by

$\displaystyle F_{N}=\sum_{N_{i}}\,g_{N_{i}}W(m_{N_{i}}/T)\;,\quad
N_{i}=n,p,\Delta(1232),$ (29) $\displaystyle
F_{K}=\sum_{K_{i}}\,g_{K_{i}}W(m_{K_{i}}/T)\;,\quad
K_{i}=K^{0},\overline{K^{0}},K^{\pm},K^{\ast}(892),$ (30) $\displaystyle
F_{Y}=\sum_{Y_{i}}\,g_{Y_{i}}W(m_{Y_{i}}/T)\;,\quad
Y_{i}=\Lambda,\Sigma^{0},\Sigma^{\pm},\Sigma(1385),$ (31)

where $g_{N_{i},K_{i},Y_{i}}$ is the degeneracy of each baryonic species. We
define the function $W(x)=x^{2}K^{\mathrm{B}}_{2}(x)$ where
$K^{\mathrm{B}}_{2}$ is the modified Bessel functions of integer order “$2$”.

The net baryon-per-entropy-ratio can be obtained from the present-day
measurement of the net baryon-per-photon ratio
$\left(n_{B}-n_{\overline{B}}\right)/n_{\gamma}$, where $n_{\gamma}$ is the
contemporary photon number density from the CMB Yang and Rafelski (2022). This
value is determined to be

$\displaystyle\frac{n_{B}-n_{\overline{B}}}{s}=\left.\frac{n_{B}-n_{\overline{B}}}{s}\right|_{t_{0}}=(0.865\pm
0.008)\times 10^{-10}\;.$ (32)

We arrive at this ratio from considering the observed baryon-per-photon ratio
Tanabashi et al. (2018) of

$\displaystyle\frac{n_{B}-n_{\overline{B}}}{n_{\gamma}}=(0.609\pm 0.006)\times
10^{-9}\,,$ (33)

as well as the entropy-per-particle Fromerth et al. (2012) for massless bosons
and fermions

$\displaystyle s/n|_{\mathrm{boson}}\approx 3.60\,,\qquad
s/n|_{\mathrm{fermion}}\approx 4.20\,.$ (34)

Figure 11: The baryon (blue solid line) and antibaryon (red solid line) number
density as a function of temperature in the range $150\text{ MeV}>T>5\text{
MeV}$. The green dashed line is the extrapolated value for baryon density. The
temperature $T=38.2\text{ MeV}$ (black dashed vertical line) is denoted when
the ratio $n_{\overline{B}}/(n_{B}-n_{\overline{B}})=1$ which define the
condition where antibaryons disappear from the Universe.

Considering the inventory of strange mesons and baryons in the cosmos after
hadronization, we evaluated the temperature of the net baryon disappearance in
Fig. 11. In solving Eq. (3.1) numerically, we plot the baryon and antibaryon
number density as a function of temperature in the range
$150\,\mathrm{MeV}>T>5\,\mathrm{MeV}$. The temperature where antibaryons
disappear from the Universe inventory can be defined when the ratio
$n_{\overline{B}}/(n_{B}-n_{\overline{B}})=1$. This condition was reached at
temperature $T=38.2\text{ MeV}$ which is in agreement with the qualitative
result in Kolb and Turner Kolb and Turner (1990). After this temperature, the
net baryon density dilutes with a residual co-moving conserved quantity
determined by the baryon asymmetry.

The antibaryon disappearance temperature does not depend on baryon and lepton
number neutrality $L=B$. Rather, it depends only on the baryon-per-entropy
ratio which is assumed to be constant during the Universe’s evolution, a
condition which is maintained well after the plasmas discussed here vanish.
The assumption of co-moving baryon number conservation is justified by the
wealth of particle physics experiments, and the co-moving entropy conservation
in an adiabatic evolving Universe is a common assumption.

###  3.2 Strangeness abundance

As the energy contained in QGP is used up to create mesons, that is massive
particles containing matter and antimatter, the high abundance of
(anti)strange $(s,\bar{s})$ quark pairs present in the plasma is preserved. A
smaller abundance of (anti)charm $(c,\bar{c})$ can combine with abundant
strange quarks to form ‘exotic’ heavy mesons. With time, charmness and later
strangeness decay away as these flavors are heavier than the light $(u,d)$
quarks and antiquarks. Unlike charm, which disappears from the particle
inventory relatively quickly, strangeness can still persist Yang and Rafelski
(2022) in the Universe until $T\approx\mathcal{O}(10\text{ MeV})$. As already
noted, the meson sector is of particular interest in our work since mesons
carry antimatter in form of their antiquark component. After the loss of
antibaryons at $T=38.2\text{ MeV}$, Fig. 11, the remaining light mesons then
act as a proxy for the hadronic antimatter evolution.

We illustrate this by considering an unstable strange particle $S$ decaying
into two particles $1$ and $2$ which themselves have no strangeness content.
In a dense and high-temperature plasma with particles $1$ and $2$ in thermal
equilibrium, the inverse reaction populates the system with particle $S$. This
is written schematically as

$\displaystyle S\Longleftrightarrow
1+2,\qquad\mathrm{Example}:K^{0}\Longleftrightarrow\pi+\pi\,.$ (35)

The natural decay of the daughter particles provides the intrinsic strength of
the inverse strangeness production reaction rate. As long as both decay and
production reactions are possible, particle $S$ abundance remains in thermal
equilibrium. This balance between production and decay rates is called a
detailed balance. The thermal reaction rate per time and volume for two-to-one
particle reactions $1+2\rightarrow 3$ has been presented before Kuznetsova et
al. (2008); Kuznetsova and Rafelski (2010). In full kinetic and chemical
equilibrium, the reaction rate per time per volume is given by Kuznetsova and
Rafelski (2010) :

$\displaystyle R_{12\to
3}=\frac{g_{3}}{(2\pi)^{2}}\,\frac{m_{3}}{\tau^{0}_{3}}\,\int^{\infty}_{0}\frac{p^{2}_{3}dp_{3}}{E_{3}}\frac{e^{E_{3}/T}}{e^{E_{3}/T}\pm
1}\Phi(p_{3})\;,$ (36)

where $\tau^{0}_{3}$ is the vacuum lifetime of particle $3$. The positive sign
$``+"$ is for the case when particle $3$ is a boson, while it is negative
$``-"$ for fermions. The function $\Phi(p_{3})$ in the non-relativistic limit
$m_{3}\gg p_{3},T$ can be written as

$\displaystyle\Phi(p_{3}\to 0)=2\frac{1}{(e^{E_{1}/T}\pm 1)(e^{E_{2}/T}\pm
1)}.$ (37)

When back-reactions are faster than the Universe expansion, a condition we
characterize in the following, we can explore the Universe composition
assuming both kinetic and particle abundance equilibrium (chemical
equilibrium). In Fig. 12 we numerically solve for the chemical potential of
strangeness and show the chemical equilibrium particle abundance ratios Yang
and Rafelski (2022) for various mesons, the baryons, and their antiparticles.
In the temperature range $150\text{ MeV}>T>40\text{ MeV}$ the Universe is rich
in physics phenomena involving strange mesons and (anti)baryons including
(anti)hyperon abundances. While antibaryons vanish after temperature $T\approx
40\text{ MeV}$, kaons persist compared to baryons until $T=20\text{ MeV}$. For
temperatures $T<20\text{ MeV}$, the Universe becomes light-quark baryons
dominant. Pions $\pi(q\bar{q})$ persist the longest of the mesons (a feature
explored in Sect. 3.3) until $T=5.6\text{ MeV}$. Pions are the most abundant
hadrons in this period because of their low mass and the inverse decay
reaction $\gamma+\gamma\rightarrow\pi^{0}$ which assures chemical equilibrium
Kuznetsova et al. (2008).

Below $T=5.6\text{ MeV}$, we have $n_{\pi}/n_{B}<1$ and the number density of
pion become sub-dominate compared to the remaining baryons. It is important to
realize that hadrons always are a part of the evolving Universe, a point we
wish to see emphasized more in literature. For temperatures $150\text{
MeV}>T>20\text{ MeV}$ the Universe is meson-dominant with (anti)strangeness
well represented in the meson sector with $s=\bar{s}$. Below temperature
$T<13\text{ MeV}$, strangeness inventory is mostly found in the hyperons as we
have $(s-\bar{s})\neq 0$. We note that hyperons never exceed baryon content
throughout the hadron epoch. This period of meson physics ends the stage of
the Universe where antimatter was dominant in the quark sector.

Figure 12: Ratios of hadronic particle number densities as a function of
temperature $150\text{ MeV}>T>5\text{ MeV}$ in the early Universe with baryon
$B$ yields: Pions $\pi(q\bar{q})$ (brown line), kaons $K(q\bar{s})$ (blue
line), antibaryon $\overline{B}$ (black line), hyperon $Y$ (red line) and
antihyperons $\overline{Y}$ (dashed red line). Also shown is the
$\overline{K}/Y$ ratio (purple line) and the $\bar{B}$ to asymmetry
$B-\bar{B}$ ratio (green line). Temperature crossings are included (as
vertical dashed black lines) at $T=40\text{ MeV},\ 20\text{ MeV},\ 13\text{
MeV},\ 5.6\text{ MeV}$ as different abundances become sub-dominate compared to
other species. The dashed brown line represents the drop in overall pion $\pi$
abundance when the vanishing of the charged pions $\pi^{\pm}$ from the
particle inventory is taken into account.

In Fig. 13 we schematically show important source reactions for strange quark
abundance in baryons and mesons considering both open and hidden strangeness
($s\bar{s}$-content). The important strangeness processes (involving both the
quark and lepton sectors) are

$\displaystyle l^{-}+l^{+}$
$\displaystyle\leftrightarrow\phi\,,\qquad\rho+\pi\leftrightarrow\phi\,,$
$\displaystyle\pi+\pi$ $\displaystyle\leftrightarrow
K\,,\qquad\Lambda\leftrightarrow\pi+N\,,\qquad\mu^{\pm}+\nu\leftrightarrow
K^{\pm}\,.$ (38)

Muons and pions are coupled through electromagnetic reactions

$\displaystyle\mu^{+}+\mu^{-}\leftrightarrow\gamma+\gamma\,,\qquad\pi^{0}\leftrightarrow\gamma+\gamma\,,$
(39)

to the photon background and retain their chemical equilibrium respectively
Rafelski and Yang (2021); Kuznetsova et al. (2008). The large
$\phi\leftrightarrow K+K$ rate assures $\phi$ and $K$ are in relative chemical
equilibrium.

Figure 13: The strangeness abundance changing reactions in the primordial
Universe. Red circles show strangeness carrying hadronic particles and thick
red lines denote effectively instantaneous reactions. Thick black lines show
relatively strong hadronic reactions. Figure 14: The hadronic reaction
relaxation times $\tau_{i}$ in the meson sector as a function of temperature
compared to Hubble time $1/H$ (black solid line). The following processes are
presented: The leptonic (solid blue line) and strong (dashed blue line) kaon
$K$ processes, the electronic (solid dark red line) and muonic (dashed dark
red line) phi meson $\phi$ processes, the forward and backward (thick black
lines) electromagnetic pion $\pi$ processes, and the strong (red lines) rho
meson $\rho$ processes.

Once the primordial Universe expansion rate (given as the inverse of the
Hubble parameter $1/H$) overwhelms the strongly temperature-dependent back-
reaction, the decay $S\rightarrow 1+2$ occurs out of balance and particle $S$
disappears from the Universe. In order to determine where exactly strangeness
disappears from the Universe inventory we explore the magnitudes of a
relatively large number of different rates of production and decay processes
and compare these with the Hubble time constant Yang and Rafelski (2022).
Strangeness then primarily resides in two domains:

  * •

Strangeness in the mesons

  * •

Strangeness in the (anti)hyperons

In the meson domain, the relevant interaction rates competing with Hubble time
are the reactions

$\displaystyle\pi+\pi$ $\displaystyle\leftrightarrow
K\,,\qquad\mu^{\pm}+\nu\leftrightarrow K^{\pm}\,,$ $\displaystyle l^{+}+l^{-}$
$\displaystyle\leftrightarrow\phi\,,\qquad\rho+\pi\leftrightarrow\phi\,,\qquad\pi+\pi\leftrightarrow\rho\,.$
(40)

The relaxation times $\tau_{i}$ for these processes are compared with Hubble
time in Fig. 14. The criteria for a detailed reaction balance is broken once a
process crosses above the Hubble time $1/H$ and thus can no longer be
considered as subject to adiabatic evolution. As the Universe cools, these
various processes freeze out as they cross this threshold. In Table 1 we show
the characteristic strangeness reactions and their freeze-out temperatures in
the hadronic epoch.

Table 1: The characteristic strangeness reaction, their freeze-out temperature, and temperature width in the hadronic epoch. Reactions | freeze-out Temperature (MeV) |  $\Delta T_{f}$ (MeV)
---|---|---
$\mu^{\pm}\nu\rightarrow K^{\pm}$ |  $T_{f}=33.8$ MeV |  $3.5$ MeV
$e^{+}e^{-}\rightarrow\phi$ |  $T_{f}=24.9$ MeV |  $0.6$ MeV
$\mu^{+}\mu^{-}\rightarrow\phi$ |  $T_{f}=23.5$ MeV |  $0.6$ MeV
$\pi\pi\rightarrow K$ |  $T_{f}=19.8$ MeV |  $1.2$ MeV
$\pi\pi\rightarrow\rho$ |  $T_{f}=12.3$ MeV |  $0.2$ MeV

Once freeze-out occurs and the corresponding detailed balance is broken, the
inverse decay reactions act like a “hole” in the strangeness abundance
siphoning strangeness out of the Universe’s particle inventory. The first
freeze-out reaction is the weak interaction kaon production process

$\displaystyle\mu^{\pm}+\nu_{\mu}\rightarrow K^{\pm}\,,\qquad
T_{f}^{K^{\pm}}=33.8\text{ MeV}\,,$ (41)

which is followed by the electromagnetic $\phi$ meson production process

$\displaystyle l^{-}+l^{+}\rightarrow\phi\,,\qquad T_{f}^{\phi}=23\sim
25\text{ MeV}\,.$ (42)

Hadronic kaon production via pions follows next in the freeze-out process

$\displaystyle\pi+\pi\rightarrow K\,,\qquad T_{f}^{K}=19.8\text{ MeV}\,.$ (43)

as it becomes slower than the Hubble expansion. The reactions

$\displaystyle\gamma+\gamma\leftrightarrow\pi\,,\qquad\rho+\pi\leftrightarrow\phi$
(44)

remain faster compared to $1/H$ for the duration of the hadronic plasma epoch.
Most $\rho$ meson decays are faster Tanabashi et al. (2018) than $\rho$ meson
producing processes and cannot contribute to the strangeness creation in the
meson sector. Below the temperature $T<20\text{ MeV}$, all the detail balances
in the strange meson sector are broken by freeze-out and the strangeness
inventory in meson sector disappears rapidly.

Figure 15: Thermal reaction rate $R$ per volume and time for important
hadronic strangeness production, exchange and decay processes as a function of
temperature $150\,\mathrm{MeV}>T>10\,\mathrm{MeV}$. The following processes
are presented: $\Lambda\leftrightarrow N\pi$ (solid black line),
$K\leftrightarrow\pi\pi$ (solid green line), $\pi N\leftrightarrow\Lambda K$
(solid blue line), $\bar{K}N\leftrightarrow\Lambda\pi$ (solid red line). Two
temperature crossings are denoted at $T=40\text{ MeV},\ 12.9\text{ MeV}$.

Were it not for the small number of baryons present, strangeness would
entirely vanish with the loss of the mesons. In order to understand
strangeness in hyperons in the baryonic domain, we evaluated the reactions

$\displaystyle\pi+N\leftrightarrow
K+\Lambda\,,\qquad\overline{K}+N\leftrightarrow\Lambda+\pi\,,\qquad\Lambda\leftrightarrow
N+\pi\,,$ (45)

for strangeness production, exchange, and decay respectively in detail. The
general form for thermal reaction rate per volume is discussed in Ch. 17 of
Letessier and Rafelski (2023). In Fig. 15 we show that for $T<20\text{ MeV}$,
the reactions for the hyperon $\Lambda$ production is dominated by
$\overline{K}+N\leftrightarrow\Lambda+\pi$. Both strangeness and
antistrangeness disappear from the Universe via the reactions

$\displaystyle\Lambda\rightarrow N+\pi\,,\qquad K\to\pi+\pi\,,$ (46)

which conserves $s=\bar{s}$. Beginning with $T=12.9\text{ MeV}$, the dominant
reaction is $\Lambda\leftrightarrow N+\pi$, which shows that at lower
temperatures strangeness content resides in the $\Lambda$ baryon. This
behavior is seen explicitly in Fig. 12 where the hyperon abundance (of which
the $\Lambda$ baryon is a member) exceeds the rapidly diminishing kaon
abundance as the Universe cools. While hyperons never form a dominant
component of the hadronic content of the Universe, it is an important life-
boat for strangeness persisting after the more transitory mesons. In this
case, the strangeness abundance becomes asymmetric and we have $s\gg\bar{s}$
at temperatures $T<12.9\text{ MeV}$. Hence, strange hyperons and antihyperons
could enter into dynamic non-equilibrium condition including $\langle
s-\bar{s}\rangle\neq 0$. The primary conclusion of the study of strangeness
production and content in the early Universe, following on QGP hadronization,
is that the relevant temperature domains indicate a complex interplay between
baryon and meson (strange and non-strange) abundances and non-trivial
decoupling from equilibrium for strange and non-strange mesons.

###  3.3 Pion abundance

Pions ($q\bar{q},q\in u,d$), the lightest hadrons, are the dominant hadrons in
the hadronic era and the most abundant hadron family well into the leptonic
epoch (see Sect. 4). The neutral pion $\pi^{0}$ vacuum lifespan of
$\tau_{\pi^{0}}^{0}=(8.52\pm 0.18)\times 10^{-17}$ seconds Tanabashi et al.
(2018) is far shorter compared to the Hubble expansion time of
$1/H=(10^{-3}\sim 10^{-4})$ seconds within this epoch as depicted in Fig. 14.

At seeing such a large discrepancy in characteristic times, one is tempted to
presume that the decay process dominates and that $\pi^{0}$ disappears quickly
in the hadronic gas. However, in the high temperature $T=\mathcal{O}(100\text{
MeV})\sim\mathcal{O}(10\text{ MeV})$ thermal bath of this era, the inverse
decay reaction forms neutral pions $\pi^{0}$ at rate corresponding to the
decay process maintaining the abundance of the species (see Fig. 12). In
general, $\pi^{0}$ is produced in the QED plasma predominantly by thermal two-
photon fusion:

$\displaystyle\gamma+\gamma\rightarrow\pi^{0}.$ (47)

This formation process is simply the inverse of the dominant decay process.
While we do not address it in detail here, the $\pi^{\pm}$ charged pions are
also in thermal equilibrium with the other pions species via hadronic and
electromagnetic reactions

$\displaystyle\pi^{0}+\pi^{0}\leftrightarrow\pi^{+}+\pi^{-}\,\qquad
l^{+}+l^{-}\leftrightarrow\pi^{+}+\pi^{-}\,,\qquad\gamma+\gamma\leftrightarrow\pi^{+}+\pi^{-}\,.$
(48)

Of these, the hadronic interaction is the fastest and controls the charged
pion abundance most directly Kuznetsova (2009); Fromerth et al. (2012) such
that the condition

$\displaystyle\rho_{\pi^{0}}\sim\rho_{\pi^{\pm}}\,,$ (49)

where $\rho$ is the energy density of the species and is maintained for most
of the hadronic era. We point out that the in the late (colder) hadronic era,
the charged pions will scatter off the remaining baryons with asymmetric
reactions due to the lack of antibaryons. The smallness of the electronic
$e^{+}e^{-}$ formation of $\pi^{0}$ is characterized by its small branching
ratio in $\pi^{0}$ decay $B=\Gamma_{ee}/\Gamma_{\gamma\gamma}=6.2\pm 0.5\times
10^{-8}$ Tanabashi et al. (2018) which can be neglected compared to photon
fusion. The general form for invariant production rates and relaxation time is
discussed in Kuznetsova et al. (2008) where we have for the photon fusion
process

$\displaystyle R_{\gamma\gamma\to\pi^{0}}=$
$\displaystyle\int\frac{d^{3}{p_{\pi}}}{(2\pi\
)^{3}2E_{\pi}}\int\frac{d^{3}{p_{2\,\gamma}}}{(2\pi\
)^{3}2E_{2\,\gamma}}\int\frac{d^{3}{p_{1\,\gamma}}}{(2\pi\
)^{3}2E_{1\,\gamma}}\left(2\pi\right)^{4}\delta^{4}\left(p_{1\,\gamma}+p_{2\,\gamma}-p_{\pi}\right)\times$
$\displaystyle\sum_{spin}\left|\langle
p_{1\,\gamma}p_{2\,\gamma}\left|M\right|p_{\pi}\rangle\right|^{2}f_{\pi}(p_{\pi})f_{\gamma}(p_{1\,\gamma})f_{\gamma}(p_{2\,\gamma})\Upsilon^{-2}_{\gamma}\Upsilon_{\pi^{0}}^{-1}e^{u\cdot
p_{\pi}/T}\,,$ (50)

where $\Upsilon_{i}$ is the fugacity and $f_{i}$ is the Bose-Einstein
distribution of particle $i$, and $M$ is the matrix element for the process.
Since the $\gamma+\gamma\to\pi^{0}$ is the dominant mechanism of pion
production, we can omit all sub-dominant processes, and the dynamic equation
of $\pi^{0}$ abundance can be written as Fromerth et al. (2012):

$\displaystyle\frac{d}{dt}\Upsilon_{\pi^{0}}=\frac{1}{\tau_{T}}\Upsilon_{\pi^{0}}+\frac{1}{\tau_{S}}\Upsilon_{\pi^{0}}+\frac{1}{\tau_{\pi^{0}}}\left(\Upsilon^{2}_{\gamma}-\Upsilon_{\pi^{0}}\right)\,,$
(51)

where $\tau_{T}$ and $\tau_{S}$ are the kinematic relaxation times for
temperature and entropy evolution and $\tau_{\pi^{0}}$ is the chemical
relaxation time for $\pi^{0}$. We have

$\displaystyle\frac{1}{\tau_{T}}$
$\displaystyle\equiv-T^{3}g^{*}\frac{d(n_{\pi}/(\Upsilon_{3}g^{*}T^{3}))/dT}{dn_{\pi}/d{\Upsilon_{3}}}{\dot{T}},$
$\displaystyle\frac{1}{\tau_{S}}$
$\displaystyle\equiv-\frac{n_{\pi}/\Upsilon_{3}}{dn_{\pi}/d{\Upsilon_{3}}}\frac{d\ln(g^{*}VT^{3})}{dT}\dot{T},$
$\displaystyle\tau_{\pi^{0}}$
$\displaystyle\equiv\frac{dn_{\pi^{0}}/d\Upsilon_{\pi^{0}}}{R_{\pi^{0}}}\,,$
(52)

Where $n_{\pi^{0}}$ is the number density of pions. A minus sign is introduced
in the above expressions to maintain $\tau_{T}$, $\tau_{S}>0$. Since entropy
is conserved within the radiation-dominated epoch, we have $T^{3}V={\rm
constant}$ thus $d(T^{3}V(T))/dT=0$. This implies the entropic relaxation time
is infinite yielding $1/\tau_{S}=0$. The effect of Universe expansion and
dilution of number density is described by $1/\tau_{T}$. Comparing $\tau_{T}$
to the chemical relaxation time $\tau_{\pi^{0}}$ can provide the quantitative
condition for freeze-out from chemical equilibrium. In the case of pion mass
being much larger than the temperature, $m_{\pi}\gg T$, we have Kuznetsova
(2009)

$\displaystyle\tau_{T}\approx\frac{T}{m_{\pi}H}.$ (53)

In Fig. 14 we compare the relaxation time of $\tau_{\pi^{0}}$ to the Hubble
time $1/H$ which shows that $\tau_{\pi^{0}}\ll 1/H$. In such a case, the yield
of $\pi^{0}$ is expected to remain in chemical equilibrium (even as its
thermal number density gradually decreases) with no freeze-out temperature
occurring. This makes pions distinct from all other meson species. This
phenomenon can be attributed to the high population of photons as in such an
environment, it remains sufficiently probable to find high-energy photons to
fuse back into neutral pions $\pi^{0}$ Fromerth et al. (2012) for the duration
of large pion abundance. As shown in Fig. 12, pions remained as proxy for
hadronic matter and antimatter down to $T=5.6\text{ MeV}$.

##  4 Leptonic Epoch

###  4.1 Thermal degrees of freedom

The leptonic epoch, dominated by photons and both charged and neutral leptons,
is notable for being the last time where neutrinos played an active role in
the Universe’s thermal dynamics before decoupling and becoming free-streaming.
In the early stage of this plasma after the hadronization era ended
$T\approx\mathcal{O}(10\text{ MeV})$, neutrinos represented the highest energy
density followed by the light charged leptons and then finally the photons.
The differing relativistic limit energy densities can be related by

$\displaystyle\rho_{e^{\pm}}\approx\left(2\times\frac{7}{8}\right)\rho_{\gamma}\,,\qquad\rho_{\nu}\approx\left(3\times\frac{7}{8}\right)\rho_{\gamma}\,.$
(54)

The reason for this hierarchy is because of the degrees of freedom Letessier
and Rafelski (2023); Rafelski and Birrell (2014) available in each species in
thermal equilibrium; the factor $7/8$ arises from the difference in pressure
contribution between bosons and fermions.

While photons only exhibit two polarization degrees of freedom, the charged
light leptons could manifest as both matter (electrons), antimatter
(positrons) and as well as two polarizations yielding $2\times 2=4$. The
neutral leptons made up of the neutrinos however had three thermally active
species $3\times 2=6$ boosting their energy density in that period to more
than any other contribution. The muon-antimuon energy density was also
controlled by its degrees of freedom matching that of $e^{\pm}$ until
$T\approx\mathcal{O}(100\text{ MeV})$, still well within the hadronic epoch,
when the heavier lepton no longer satisfied the ultra-relativistic (and thus
massless) limit. This separation of the two lighter charge lepton dynamics is
seen in Fig. 2 after hadronization.

The known cosmic degrees of freedom require that if and when neutrinos are
Dirac-like and have chiral right-handed (matter) components, then these right
handed components must not drive the neutrino effective degrees of freedom
$N^{\nu}_{\mathrm{eff}}$ away from three. In a more general context the non-
interacting sterile neutrinos could also inflate $N_{\mathrm{eff}}^{\nu}$
during this epoch for the same reasoning Kopp et al. (2011); Hamann et al.
(2011); Kopp et al. (2013); Lello and Boyanovsky (2015); Birrell and Rafelski
(2015) or have a connection to dark matter Weinberg (2013); Giusarma et al.
(2014). The neutrino degrees of freedom will be more fully discussed in Sect.
4.5.

###  4.2 Muon abundance

As seen in Sect. 3.2, muon abundance and their associated reactions are
integral to the understanding of the strangeness and antistrangeness content
of the primordial Universe Yang and Rafelski (2022). Therefore we determine to
what extent and temperature (anti)muons remained in chemical abundance
equilibrium. Without a clear boundary separating the hadronic epoch from the
leptonic epoch, there is complete overlap in the hadronic and leptonic species
dynamics in the period $T=\mathcal{O}(10\text{ MeV})\sim\mathcal{O}(1\text{
MeV})$.

In the cosmic plasma, muons can be produced by predominately electromagnetic
and weak interaction processes

$\displaystyle\gamma+\gamma\longrightarrow\mu^{+}+\mu^{-},\qquad$
$\displaystyle e^{+}+e^{-}\longrightarrow\mu^{+}+\mu^{-}\;,$ (55)
$\displaystyle\pi^{-}\longrightarrow\mu^{-}+\bar{\nu}_{\mu},\qquad$
$\displaystyle\pi^{+}\longrightarrow\mu^{+}+\nu_{\mu}\;.$ (56)

Provided that all particles shown on the left-hand side of each reaction
(namely the photons, electrons(positrons) and charged pions) exist in chemical
equilibrium, the back-reaction for each of the above processes occurs in
detailed balance.

The scattering angle averaged thermal reaction rate per volume for the
reaction $a\overline{a}\rightarrow b\overline{b}$ in Boltzmann approximation
is given by Letessier and Rafelski (2023)

$\displaystyle R_{a\overline{a}\rightarrow
b\overline{b}}=\frac{g_{a}g_{\overline{a}}}{1+I}\frac{T}{32\pi^{4}}\int_{s_{th}}^{\infty}ds\frac{s(s-4m^{2}_{a})}{\sqrt{s}}\sigma_{a\overline{a}\rightarrow
b\overline{b}}K_{1}(\sqrt{s}/T),$ (57)

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. We introduce the factor $1/(1+I)$ to avoid the double counting
of indistinguishable pairs of particles where $I=1$ for an identical pair and
$I=0$ for a distinguishable pair.

The muon weak decay processes are

$\mu^{-}\rightarrow\nu_{\mu}+e^{-}+\bar{\nu}_{e},\qquad\mu^{+}\rightarrow\bar{\nu}_{\mu}+e^{+}+\nu_{e}\,,$
(58)

with the vacuum life time $\tau_{\mu}=2.197\times 10^{-6}$ seconds producing
(anti)neutrino pairs of differing flavor and electrons(positrons). We recall
the considerable shorter vacuum lifetime of pions
$\tau_{\pi^{\pm}}=2.6033\times 10^{-8}$ seconds. The thermal decay rate per
volume in the Boltzmann limit is Kuznetsova et al. (2008)

$\displaystyle
R_{i}=\frac{g_{i}}{2\pi^{2}}\left(\frac{T^{3}}{\tau_{i}}\right)\left(\frac{m_{i}}{T}\right)^{2}K_{1}(m_{i}/T)$
(59)

where $\tau_{i}$ is the vacuum lifespan of a given particle $i$.

Figure 16: The thermal reaction rate per volume for muon related reactions as
a function of temperature adapted from Rafelski and Yang (2021). The dominant
reaction rates for $\mu^{\pm}$ production are: The $\gamma\gamma$ channel
(blue dashed line), $e^{\pm}$ (red dashed line), these two combined as the
total electromagnetic rate (pink solid line), and the charged pion decay feed
channel (black solid line). The muon decay rate is also shown (green solid
line). The crossing point between the electromagnetic production processes and
the muonic decay rate is seen as the dashed vertical black line at $T_{\rm
dis}=4.2\text{ MeV}$.

These production and decay rates for muonic processes are evaluated in
Rafelski and Yang (2021). From this, we can determine the temperature when
muons rather suddenly disappear from the particle inventory of the Universe
which occurs when their decay rate exceeds their production rate. In Fig. 16
we show the invariant thermal reaction rates per volume and time for the
relevant muon reactions. As the temperature decreases in the expanding
Universe, the initially dominant production rates become rapidly smaller due
to the mass threshold effect. This is allowing the production and decay rates
to become equal. The characteristic times are much faster than the Hubble time
(not shown in Fig. 16). Muon abundance therefore disappears just when the
decay rate overwhelms production at the temperature
$T_{\mathrm{dis}}=4.20\text{ MeV}$.

In Fig. 17 we show that the number density ratio of muons to baryons
$n_{\mu^{\pm}}/n_{B}$ at the muon disappearance temperature
$T_{\mathrm{dis}}=4.20\text{ MeV}$ is $n_{\mu^{\pm}}/n_{\mathrm{B}}\approx
0.91$ Yang and Rafelski (2022). Interestingly, this means that the muon
abundance may still be able to influence baryon evolution up to this point
because their number density is comparable to that of baryons (there are no
antibaryons). This coincidence of abundance offers a novel and tantalizing
model-building opportunity for both baryon-antibaryon separation models and/or
strangelet formation models.

Figure 17: The density ratio between $\mu^{\pm}$ and baryons
$n_{\mu^{\pm}}/n_{B}$ (blue solid line) is plotted as a function of
temperature. The red dashed line indicates a density ratio value of
$n_{\mu^{\pm}}/n_{B}=1$. The density ratio at the muon disappearance
temperature (vertical black dashed line) is about
$n_{\mu^{\pm}}/n_{\mathrm{B}}(T_{\mathrm{dis}})\approx 0.911$.

###  4.3 Neutrino masses and oscillation

Neutrinos are believed to have a small, but nonzero mass due to the phenomenon
of flavor oscillation Fukuda et al. (1998); Eguchi et al. (2003); Fogli et al.
(2006). This is seen in the flux of neutrinos from the Sun, and also in
terrestrial reactor experiments. In the Standard Model neutrinos are produced
via weak charged current (mediated by the W boson) as flavor eigenstates. If
the neutrino was truly massless, then whatever flavor was produced would be
immutable as the propagating state. However, if neutrinos have mass, then they
propagate through space as their mass-momentum eigenstates. Neutrino masses
can be written in terms of an effective theory where the mass term contains
various couplings between neutrino states determined by some BSM theory. The
exact form of such a BSM theory is outside the scope of this work, we refer
the reader to some standard references Giunti and Studenikin (2015); Fritzsch
and Xing (2000); Giunti and Kim (2007); Fritzsch (2015).

Within the Standard Model keeping two degrees of freedom for each neutrino
flavor the Majorana fermion mass term is given by

$\displaystyle\mathcal{L}_{m}^{Maj.}=-\frac{1}{2}\bar{\nu}^{\alpha}_{L}M^{M}_{\alpha\beta}(\nu^{\beta}_{L})^{c}+\mathrm{h.c.}\,,$
(60)

where $\nu^{c}=\hat{C}(\bar{\nu})^{T}$ is the charge conjugate of the neutrino
field. The operator $\hat{C}=i\gamma^{2}\gamma^{0}$ is the charge conjugation
operator. An interesting consequence of neutrinos being Majorana particles is
that they would be their own antiparticles like photons allowing for
violations of total lepton number. Neutrinoless double beta decay is an
important, yet undetected, evidence for Majorana nature of neutrinos Dolinski
et al. (2019). Majorana neutrinos with small masses can be generated from some
high scale via the See-Saw mechanism Arkani-Hamed et al. (2001); Ellis and
Lola (1999); Casas and Ibarra (2001) which ensures that the degrees of freedom
separate into heavy neutrinos and light nearly massless Majorana neutrinos.
The See-Saw mechanism then provides an explanation for the smallness of the
neutrino masses as has been experimentally observed.

A flavor eigenstate $\nu^{\alpha}$ can be described as a superposition of mass
eigenstates $\nu^{k}$ with coefficients given by the Pontecorvo-Maki-Nakagawa-
Sakata (PMNS) mixing matrix King and Luhn (2013); Fernandez-Martinez et al.
(2016) which are both in general complex and unitary. This is given by

$\displaystyle\nu^{\alpha}=\sum_{k}^{n}U^{\ast}_{\alpha
k}\nu^{k},\qquad\alpha=e,\mu,\tau,\qquad k=1,2,3,\dots,n$ (61)

where $U$ is the PMNS mixing matrix. The PMNS matrix is the lepton equivalent
to the CKM mixing matrix which describes the misalignment between the quark
flavors and their masses. For Majorana neutrinos, there can be up to three
complex phases $(\delta,\rho,\gamma)$ which are CP-violating Pascoli et al.
(2007) which are present when the number of generations is $n\geq 3$. For
Dirac-like neutrinos, only the $\delta$ complex phase is required. In
principle, the number of mass eigenstates can exceed three, but is restricted
to three generations in most models. By standard convention Schwartz (2014)
found in the literature we parameterize the rotation matrix $U$ as

$\displaystyle
U=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{13}s_{23}e^{i\delta}&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}&c_{13}s_{23}\\\
s_{12}s_{23}-c_{12}s_{13}c_{23}e^{i\delta}&-c_{12}s_{23}-s_{12}s_{13}c_{23}e^{i\delta}&c_{13}c_{23}\end{pmatrix}\times\begin{pmatrix}1&&\\\
&e^{i\rho}&\\\ &&e^{i\gamma}\end{pmatrix}\,,$ (62)

where $c_{ij}=\mathrm{cos}(\theta_{ij})$ and
$s_{ij}=\mathrm{sin}(\theta_{ij})$. In this convention, the three mixing
angles $(\theta_{12},\theta_{13},\theta_{23})$, are understood to be the Euler
angles for generalized rotations.

The neutrino proper masses are generally considered to be small with values no
more than $0.1\text{ eV}$. Because of this, neutrinos produced during fusion
within the Sun or radioactive fission in terrestrial reactors on Earth
propagate relativistically. Evaluating freely propagating plane waves in the
relativistic limit yields the vacuum oscillation probability between flavors
$\nu_{\alpha}$ and $\nu_{\beta}$ written as Workman et al. (2022)

$\displaystyle P_{\alpha\rightarrow\beta}=$
$\displaystyle\delta_{\alpha\beta}-4\sum_{i<j}^{n}\mathrm{Re}\left[U_{\alpha
i}U^{\ast}_{\beta i}U^{\ast}_{\alpha j}U_{\beta
j}\right]\sin^{2}\\!\\!\left(\frac{\Delta m^{2}_{ij}L}{4E}\right)$
$\displaystyle+2\sum_{i<j}^{n}\mathrm{Im}\left[U_{\alpha i}U^{\ast}_{\beta
i}U^{\ast}_{\alpha j}U_{\beta j}\right]\sin\\!\\!\left(\frac{\Delta
m^{2}_{ij}L}{2E}\right),\qquad\Delta m^{2}_{ij}\equiv{m^{2}_{i}-m^{2}_{j}}$
(63)

where $L$ is the distance traveled by the neutrino between production and
detection. The square mass difference $\Delta m^{2}_{ij}$ has been
experimentally measured Workman et al. (2022). As oscillation only restricts
the differences in mass squares, the precise values of the masses cannot be
determined from oscillation experiments alone. It is also unknown under what
hierarchical scheme (normal or inverted) Avignone et al. (2008); Esteban et
al. (2020) the masses are organized as two of the three neutrino proper masses
are close together in value.

It is important to point out that oscillation does not represent any physical
interaction (except when neutrinos must travel through matter which modulates
the $\nu_{e}$ flavor Alvarez-Ruso et al. (2018); Abi et al. (2020)) or change
in the neutrino during propagation. Rather, for a given production energy, the
superposition of mass eigenstates each have unique momentum and thus unique
group velocities. This mismatch in the wave propagation leads to the
oscillatory probability of flavor detection as a function of distance.

We further note that non-interacting BSM so called sterile neutrinos of any
mass have not yet been observed despite extensive searching. The existence of
such neutrinos, if they were ever thermally active in the early cosmos would
leave fingerprints on the Cosmic Neutrino Background (CNB) spectrum Birrell
and Rafelski (2015). The presence of an abnormally large anomalous magnetic
moment Morgan (1981); Fukugita and Yazaki (1987); Vogel and Engel (1989);
Elmfors et al. (1997); Giunti and Studenikin (2009, 2015); Canas et al. (2016)
for the neutrino would also possibly leave traces in the evolution of the
early Universe.

###  4.4 Neutrino freeze-out

The relic neutrino background (or CNB) is believed to be a well-preserved
probe of a Universe only a second old which at some future time may become
experimentally accessible. The properties of the neutrino background are
influenced by the details of the freeze-out or decoupling process at a
temperature $T=\mathcal{O}(2\text{ MeV})$. The freeze-out process, whereby a
particle species stops interacting and decouples from the photon background,
involves several steps that lead to the species being described by the free-
streaming momentum distribution. We outline freeze-out properties, including
what distinguishes it from the equilibrium distributions Birrell et al.
(2014).

Chemical freeze-out of a particle species occurs at the temperature, $T_{ch}$,
when particle number changing processes slow down and the particle abundance
can no longer be maintained at an equilibrium level. Prior to the chemical
freeze-out temperature, number changing processes are significant and keep the
particle in chemical (and thermal) equilibrium, implying that the distribution
function has the Fermi-Dirac form, obtained by maximizing entropy at fixed
energy (parameter $1/T$) and particle number (parameter $\lambda$)

$f_{c}(t,E)=\frac{1}{\lambda\exp(E/T)+1},\text{ for }T(t)>T_{ch}.$ (64)

Kinetic freeze-out occurs at the temperature, $T_{f}$, when momentum
exchanging interactions no longer occur rapidly enough to maintain an
equilibrium momentum distribution. When $T_{f}<T(t)<T_{ch}$, the number-
changing process no longer occurs rapidly enough to keep the distribution in
chemical equilibrium but there is still sufficient momentum exchange to keep
the distribution in thermal equilibrium. The distribution function is
therefore obtained by maximizing entropy, with fixed energy, particle number,
and antiparticle number separately. This implies that the distribution
function has the form

$f_{k}(t,E)=\frac{1}{\Upsilon^{-1}\exp(E/T)+1},\text{ for }T_{f}<T(t)<T_{ch}.$
(65)

The time dependent generalized fugacity $\Upsilon(t)$ controls the occupancy
of phase space and is necessary once $T(t)<T_{ch}$ in order to conserve
particle number.

For $T(t)<T_{f}$ there are no longer any significant interactions that couple
the particle species of interest and so they begin to free-stream through the
Universe, i.e. travel on geodesics without scattering. The Einstein-Vlasov
equation can be solved, see Choquet-Bruhat (2008), to yield the free-streaming
momentum distribution

$f(t,E)=\frac{1}{\Upsilon^{-1}e^{\sqrt{p^{2}/T^{2}+m^{2}/T_{f}^{2}}}+1}$ (66)

where the free-streaming effective temperature

$T(t)=\frac{T_{f}a(t_{k})}{a(t)}$ (67)

is obtained by redshifting the temperature at kinetic freeze-out. The
corresponding free-streaming energy density, pressure, and number densities
are given by

$\displaystyle\rho$
$\displaystyle=\frac{d}{2\pi^{2}}\\!\int_{0}^{\infty}\\!\\!\\!\frac{\left(m^{2}+p^{2}\right)^{1/2}p^{2}dp}{\Upsilon^{-1}e^{\sqrt{p^{2}/T^{2}+m^{2}/T_{f}^{2}}}+1},$
(68) $\displaystyle P$
$\displaystyle=\frac{d}{6\pi^{2}}\\!\int_{0}^{\infty}\\!\\!\\!\frac{\left(m^{2}+p^{2}\right)^{-1/2}p^{4}dp}{\Upsilon^{-1}e^{\sqrt{p^{2}/T^{2}+m^{2}/T_{f}^{2}}}+1},$
(69) $\displaystyle n$
$\displaystyle=\frac{d}{2\pi^{2}}\\!\int_{0}^{\infty}\\!\\!\\!\frac{p^{2}dp}{\Upsilon^{-1}e^{\sqrt{p^{2}/T^{2}+m^{2}/T_{f}^{2}}}+1},$
(70)

where $d$ is the degeneracy of the particle species. These differ from the
corresponding expressions for an equilibrium distribution in Minkowski space
by the replacement $m\rightarrow mT(t)/T_{f}$ only in the exponential.

The separation of the freeze-out process into these three regimes is of course
only an approximation. In principle, there is a smooth transition between
them. However, it is a very useful approximation in cosmology. See Mangano et
al. (2005); Birrell et al. (2015) for methods capable of resolving these
smooth transitions.

Figure 18: 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. (2014). 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
$\sin^{2}\theta_{W}=0.23$.

To estimate the freeze-out temperature we need to solve the Boltzmann equation
with different types of collision terms. In Birrell et al. (2014) we detail a
new method for analytically simplifying the collision integrals and show 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 $\eta$,

$\displaystyle\eta\equiv M_{p}m_{e}^{3}G_{F}^{2},\qquad
M_{p}^{2}\equiv\frac{1}{8\pi G_{N}},$ (71)

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

$\displaystyle\eta_{0}\equiv\left.M_{p}m_{e}^{3}G_{F}^{2}\right|_{0}=0.04421.$
(72)

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. 18 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. (2014).

After neutrinos freeze-out, the neutrino co-moving entropy is independently
conserved. However, the presence of electron-positron rich plasma until
$T=20\text{ keV}$ provides 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 $e^{\pm}$ entropy can still
transfer to neutrinos until temperature $T=20\text{ keV}$ and can modify free
streaming distribution and the effective number of neutrinos.

We expect that incorporating oscillations into the freeze-out calculation
would yield a smaller freeze-out temperature difference between neutrino
flavors as oscillation provides a mechanism in which the heavier flavors
remain thermally active despite their direct production becoming suppressed.
In work by Mangano et. al. Mangano et al. (2005), neutrino freeze-out
including flavor oscillations is shown to be a negligible effect.

###  4.5 Effective number of neutrinos

The population of each flavor of neutrino is not a fixed quantity throughout
the evolution of the Universe. In the earlier hot Universe, the population of
neutrinos is controlled thermally and to maximize entropy, each flavor is
equally filled. As the expansion factor $a(t)$ is radiation dominated for much
of this period (see Fig. 2), the CMB is ultimately sensitive to the total
energy density within the neutrino sector (which is sometimes referred to as
the dark radiation contribution). This is 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. This quantity is related to the total energy density
in the neutrino sector as well as the photon background temperature of the
Universe via

$\displaystyle
N_{\nu}^{\mathrm{eff}}\equiv\frac{\rho^{\mathrm{tot}}_{\nu}}{\frac{7\pi^{2}}{120}\left(\frac{4}{11}\right)^{4/3}T_{\gamma}^{4}}\;,$
(73)

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 (effective) temperature of the free-streaming
neutrinos after $e^{\pm}$ annihilation, under the assumption of zero neutrino
reheating. Strictly speaking, the number of true degrees of freedom is exactly
determined by the number of neutrino families and available quantum numbers,
therefore deviations of $N_{\nu}^{{\rm eff}}>3$ are to be understood as
reheating which goes into the neutrino energy density $\rho_{\nu}^{{\rm
tot}}$.

Experimentally, $N_{\mathrm{eff}}$ has been determined from CMB data by the
Planck collaboration Aghanim et al. (2020) in their 2018 analysis yielding
$N_{\nu,exp}^{\mathrm{eff}}=2.99\pm 0.17$ though this value has evolved
substantially since their 2013 and 2015 analyses Ade et al. (2014, 2016).
Precise study of neutrino decoupling (as outlined in Sect. 4.4) and thus
freeze-out can improve the predictions for the value of
$N_{\nu}^{\mathrm{eff}}$. Many studies focus on improving the calculation of
decoupling through various means such as

  1. 1.

Determining the dependence of freeze-out on the natural constants found in the
Standard Model of particle physics Birrell et al. (2014); Birrell (2014).

  2. 2.

The entropy transfer from electron-positron annihilation and finite
temperature correction at neutrino decoupling Dicus et al. (1982); Heckler
(1994); Fornengo et al. (1997).

  3. 3.

Neutrino decoupling with flavor oscillations Mangano et al. (2002, 2005).
Nonstandard neutrino interactions have been investigated, including neutrino
electromagnetic Morgan (1981); Fukugita and Yazaki (1987); Elmfors et al.
(1997); Vogel and Engel (1989); Mangano et al. (2006); Giunti and Studenikin
(2009) and nonstandard neutrino electron coupling Mangano et al. (2006).

As $N_{\nu}^{\mathrm{eff}}$ is only a measure of the relativistic energy
density leading up to photon decoupling, a natural alternative mechanism for
obtaining $N_{\nu}^{\mathrm{eff}}>3$ is the introduction of additional,
presently not discovered, weakly interacting massless particles Anchordoqui
and Goldberg (2012); Abazajian et al. (2012); Anchordoqui et al. (2013);
Steigman (2013); Giusarma et al. (2014). Alternatively, theories outside
conventional freeze-out considerations have been proposed to explain the
tension in $N_{\mathrm{eff}}$ including: QGP as the possible source of
$N_{\mathrm{eff}}$ or connection between lepton asymmetry $L$ and
$N_{\nu}^{\mathrm{eff}}$.

The natural consistency of the reported CMB range of $N_{\nu}^{\mathrm{eff}}$
with the range of QGP hadronization temperatures, motivates the exploration of
a connection between $N_{\nu}^{\mathrm{eff}}$ and the decoupling of sterile
particles at and below the QGP phase transition Birrell and Rafelski (2015).
This demonstrates that that $N_{\nu}^{\mathrm{eff}}>3.05$ can be associated
with the appearance of several light particles at QGP hadronization in the
early Universe that either are weakly interacting in the entire space or is
only allowed to interact within the deconfined domain, in which case their
coupling would be strong. Such particles could leave a clear dark radiation
experimental signature in relativistic heavy-ion experiments that produce the
deconfined QGP phase.

In standard $\Lambda$CDM, the asymmetry between leptons and antileptons
$L\equiv[N_{\mathrm{L}}-N_{\overline{\mathrm{L}}}]/N_{\gamma}$ (normalized
with the photon number) is generally assumed to be small (nano-scale) such
that the net normalized lepton number equals the net baryon number $L=B$ where
$B=[N_{\mathrm{B}}-N_{\overline{\mathrm{B}}}]/N_{\gamma}$. Barenboim, Kinney,
and Park Barenboim et al. (2017); Barenboim and Park (2017) note 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.

If lepton number is grossly broken, this could provide a connection between
cosmic neutrino properties and the baryon-antibaryon asymmetry present in the
Universe today Barenboim and Park (2017). We quantify in Yang et al. (2018)
the impact of large lepton asymmetry on Universe expansion and show that there
is another ‘natural’ choice $L\simeq 1$, making the net lepton number and net
photon number in the Universe similar. Thus because $N_{\nu}^{{\rm eff}}$ can
be understood as a characterization of the relativistic dark radiation energy
content in the early Universe, independent of its source, there still remains
ambiguity in regard to measurements of $N_{\nu}^{\rm eff}$.

##  5 Electron-Positron Epoch

###  5.1 The last bastion of antimatter

The electron-positron epoch of the early Universe was home to Big Bang
Nucleosynthesis (BBN), the annihilation of most electrons and positrons
reheating both the photon and neutrino fields, as well as setting the stage
for the eventual recombination period which would generate the cosmic
microwave background (CMB). The properties of the electron-positron $e^{\pm}$
plasma in the early Universe has not received appropriate attention in an era
of precision BBN studies Pitrou et al. (2018). The presence of $e^{\pm}$ pairs
before and during BBN has been acknowledged by Wang, Bertulani and Balantekin
Wang et al. (2011); Hwang et al. (2021) over a decade ago. This however was
before necessary tools were developed to explore the connection between
electron and neutrino plasmas Mangano et al. (2005); Birrell et al. (2014,
2014).

Figure 19: The $e^{\pm}$ number densities as a function of temperature in the
range $2\,\mathrm{MeV}>T>10\,\mathrm{keV}$. The blue solid line is the
electron density $n_{e^{-}}$, the red solid line is the positron density
$n_{e^{+}}$, and the brown solid line is the baryon density $n_{B}$. For
comparison, we also show the green dotted line as the solar electron density
within the solar core Bahcall et al. (2001).

During the late stages of the $e^{\pm}$ epoch where BBN occurred, the matter
content of the Universe was still mostly dominated by the light charged
leptons by many orders of magnitude even though the Hubble parameter was still
mostly governed by the radiation behavior of the neutrinos and photons. In
Fig. 19 we show that the dense $e^{\pm}$ plasma in the early Universe under
the hypothesis charge neutrality and entropy conservation as a function of
temperature $2\,\mathrm{MeV}>T>10\,\mathrm{keV}$ Grayson et al. (2023). The
plasma is electron-positron rich, i.e, $n_{e^{\pm}}\gg n_{B}$ in the early
Universe until leptonic annihilation at $T_{\mathrm{split}}=20.36\
\mathrm{keV}$. For $T<T_{\mathrm{split}}$ the positron density $n_{e^{+}}$
quickly vanishes because of annihilation leaving only a residual electron
density as required by charge conservation.

Figure 20: The energy density ratio $\chi$ (solid blue line) between $e^{\pm}$
and baryons as a function of temperature from
$10\,\mathrm{keV}<T<200\,\mathrm{keV}$. The dashed red line crossing point
represents where the baryon density exceeds that of the electron-positron
pairs.

The temperatures during this epoch were also cool enough that the electrons
and positrons could be described as partially non-relativistic to fairly good
approximation while also still being as energy dense as the Solar core making
it a relatively unique plasma environment not present elsewhere in cosmology.
Considering the energy density between non-relativistic $e^{\pm}$ and baryons,
we can write the ratio of energy densities as

$\displaystyle\chi$
$\displaystyle\equiv\frac{\rho_{e}+\rho_{\bar{e}}}{\rho_{p}+\rho_{n}}$
$\displaystyle=\frac{m_{e}(n_{e}+n_{\bar{e}})}{m_{p}n_{p}+m_{n}n_{n}}=\frac{m_{e}(n_{e}+n_{\bar{e}})}{n_{B}(m_{p}X_{p}+m_{n}X_{n})}=\left(\frac{n_{e}+n_{\bar{e}}}{n_{B}}\right)\,\left(\frac{m_{e}}{m_{p}X_{p}+{m_{n}X_{\alpha}}/2}\right)\,,$
(74)

where we consider all neutrons as bound in ${}^{4}H_{e}$ after BBN. Species
ratios $X_{p}=n_{p}/n_{B}$ and $X_{\alpha}=n_{\alpha}/n_{B}$ are given by the
PDG Workman et al. (2022) as

$\displaystyle X_{p}=0.878,\qquad X_{\alpha}=0.245\,,$ (75)

with masses

$\displaystyle m_{e}=0.511\,\mathrm{MeV},\qquad
m_{p}=938.272\,\mathrm{MeV},\qquad m_{n}=939.565\,\mathrm{MeV}\,.$ (76)

In Fig. 20 we plot the energy density ratio Eq. (5.1) as a function of
temperature $10\,\mathrm{keV}<T<200\,\mathrm{keV}$. This figure shows that the
energy density of electron and positron is dominant until $T=28.2\text{ keV}$,
i.e., at higher temperatures we have $\rho_{e}\gg\rho_{B}$. Until around
$T\approx 85\text{ keV}$, the $e^{\pm}$ number density remained higher than
that of the solar core, though notably at a much higher temperature than the
Sun’s core of $T_{\odot}=1.36\text{ keV}$ Castellani et al. (1997). After
$T=28.2\text{ keV}$, where $\rho_{e}\ll\rho_{B}$, the ratio becomes constant
around $T=20\text{ keV}$ because of positron annihilation and charge
neutrality.

###  5.2 Cosmic magnetism

The Universe today filled with magnetic fields Kronberg (1994) at various
scales and strengths both within galaxies and in deep extra-galactic space far
and away from matter sources. Extra-galactic magnetic fields (EGMF) are not
well constrained today, but are required by observation to be non-zero
Anchordoqui and Goldberg (2002); Widrow (2002) with a magnitude between
$10^{-12}\ \mathrm{T}>B_{EGMF}>10^{-20}\ \mathrm{T}$ over Mpc coherent length
scales. The upper bound is constrained from the characteristics of the CMB
while the lower bound is constrained by non-observation of ultra-energetic
photons from blazars Neronov and Vovk (2010). There are generally considered
two possible origins Widrow et al. (2012); Vazza et al. (2021) for extra-
galactic magnetic fields: (a) matter-induced dynamo processes involving
Amperian currents and (b) primordial (or relic) seed magnetic fields whose
origins may go as far back as the Big Bang itself. It is currently unknown
which origin accounts for extra-galactic magnetic fields today or if it some
combination of the two models. Even if magnetic fields in the Universe today
are primarily driven via amplification through Amperian matter currents, such
models could still benefit from the presence of primordial fields to act as
catalyst. The purpose of this section is then to consider the magnetization
properties of the $e^{\pm}$ plasma period due to spin which has not yet been
considered.

While matter (and thus electrons) are relatively dilute today, the early
Universe plasmas contained relatively large quantity of both matter $(e^{-})$
and antimatter $(e^{+})$. We explore here the spin response of the electron-
positron plasma to external and self-magnetization fields thus developing
methods for future detailed study.

Figure 21: Qualitative value of the primordial magnetic field over the
evolutionary lifespan of the Universe. The upper and lower black lines
represent extrapolation of the EGMF bounds into the past. The major phases of
the Universe are indicated with shaded regions. The values of the Schwinger
critical field (purple line) and the upper bound of surface magnetar field
strength (blue line) are included for scale.

As magnetic flux is conserved over co-moving surfaces, we see in Fig. 21 that
the primordial relic field is expected to dilute as $B\propto 1/a(t)^{2}$.
This means the contemporary small bounded values of $5\times 10^{-12}\
\mathrm{T}>B_{relic}>10^{-20}\ \mathrm{T}$ (coherent over $\mathcal{O}(1\
\mathrm{Mpc})$ distances) may have once represented large magnetic fields in
the early Universe. Therefore, correctly describing the dynamics of this
$e^{\pm}$ plasma is of interest when considering modern cosmic mysteries such
as the origin of extra-galactic magnetic fields Anchordoqui and Goldberg
(2002); Neronov and Vovk (2010). While most approaches tackle magnetized
plasmas from the perspective of classical or semi-classical magneto-
hydrodynamics (MHD) Berezhiani et al. (1992); Berezhiani and Mahajan (1995);
Schlickeiser et al. (2018); Melrose (2013), our perspective is to demonstrate
that fundamental quantum statistical analysis can lead to further insights on
the behavior of magnetized plasmas.

As a starting point, we consider the energy eigenvalues of charged fermions
within a homogeneous magnetic field. Here, we have several choices: We could
assume the typical Dirac energy eigenvalues with gyro-magnetic g-factor set to
$g=2$. But as electrons, positrons and most plasma species have anomalous
magnetic moments (AMM), we require a more complete model. Particle dynamics of
classical particles with AMM are explored in Rafelski et al. (2018); Formanek
et al. (2018, 2020, 2021). Another option would be to modify the Dirac
equation with a Pauli term Thaller (2013), often called the Dirac-Pauli (DP)
approach, via

$\displaystyle\hat{H}_{\mathrm{AMM}}=-a\frac{e}{2m_{e}}\frac{\sigma_{\mu\nu}F^{\mu\nu}}{2}\,,$
(77)

where $\sigma_{\mu\nu}$ is the spin tensor proportional to the commutator of
the gamma matrices and $F^{\mu\nu}$ is the EM field tensor. For the duration
of this section, we will remain in natural units $(\hbar=c=k_{B}=1)$ unless
explicitly stated otherwise. The AMM is defined via g-factor as

$\displaystyle\frac{g}{2}=1+a\,.$ (78)

This approach, while straightforward, would complicate the energies making
analytic understanding and clarity difficult without a clear benefit.
Modifying the Dirac equation with Eq. (77) yields the following eigen-energies

$\displaystyle
E_{n}^{s}|_{DP}=\sqrt{\left(\sqrt{m_{e}^{2}+2eB\left(n+\frac{1}{2}-s\right)}-\frac{eB}{2m}(g-2)s\right)^{2}+p_{z}^{2}}$
(79)

This model for the electron-positron plasma of the early Universe has been
used in work such as Strickland et. al. Strickland et al. (2012). Our work in
this section is then in part a companion piece which compares and contrasts
the DP model of fermions to our preferred model for the AMM via the Klein-
Gordon-Pauli (KGP) equation given by

$\displaystyle\left(\left(i\partial_{\mu}-eA_{\mu}\right)^{2}-m_{e}^{2}-e\frac{g}{2}\frac{\sigma_{\mu\nu}F^{\mu\mu}}{2}\right)\Psi=0\,.$
(80)

We wish to emphasize, that each of the three above models (Dirac, DP, KGP) are
distinct and have differing physical consequences and are not interchangeable
which we explored in the context of hydrogen-like atoms in Steinmetz et al.
(2019). Recent work done in Rafelski et al. (2023) discuss the benefits of KGP
over other approaches for $g\neq 2$ from a quantum field theory perspective.
Exploring the statistical behavior of KGP in a cosmological context can lead
to new insights in magnetization which may be distinguished from pure $g=2$
behavior of the Dirac equation or the _ad hoc_ modification imposed by the
Pauli term in DP. One major improvement of the KGP approach over the more
standard DP approach is that the energies take eigenvalues which are
mathematically similar to the Dirac energies. Considering the $e^{\pm}$ plasma
in a uniform magnetic field $B$ pointing along the $z$-axis, the energy of
$e^{\pm}$ fermions can be written as

$\displaystyle E_{n}^{s}$ $\displaystyle=\sqrt{p^{2}_{z}+\tilde{m}^{2}+2eBn},$
(81) $\displaystyle\tilde{m}^{2}$
$\displaystyle=m^{2}_{e}+eB\left(1-gs\right),\qquad s=\pm\frac{1}{2},\qquad
n=0,1,2,3,\dots$

where $n$ is the principle quantum number for the Landau levels and $s$ is the
spin quantum number. Here we introduce a notion of effective mass $\tilde{m}$
which inherits the spin-specific part of the energy adding them to the mass.
This convention is also generalizable to further non-minimal electromagnetic
models with more exotic energy contributions such that we write a general
replacement as

$\displaystyle m_{e}^{2}\rightarrow\tilde{m}^{2}(B)\,.$ (82)

This definition also pulls out the ground state Landau energy separating it
from the remainder of the Landau tower of states. One restriction is that the
effective mass must remain positive definite in our analysis thus we require

$\displaystyle\tilde{m}^{2}(B)=m^{2}_{e}+eB\left(1-gs\right)>0\,.$ (83)

This condition fails under ultra-strong magnetic fields of order

$\displaystyle
B_{\mathrm{crit}}=\frac{m_{e}^{2}}{ea}=\frac{\mathcal{B}_{S}}{a}\approx
3.8\times 10^{12}\ \mathrm{T}\,,$ (84)

where $\mathcal{B}_{S}$ is the Schwinger critical field strength. For
electrons, this field strength is well above the window of magnetic field
strengths of interest during the late $e^{\pm}$ epoch.

###  5.3 Landau eigen-energies in cosmology

There is another natural scale for the magnetic field besides Eq. (84) when
considering the consequences of FLRW expansion on the $e^{\pm}$ gas. As the
Universe expands, different terms in the energies and thus partition function
evolve as a function of the scale factor $a(t)$ which arises in the FLRW
metric. We can consider the expansion to be an adiabatic process which results
in a smooth shifting of the relevant dynamical quantities. From the
conservation of magnetic flux through a co-moving surface, the magnetic field
under expansion starting at some initial time $t_{0}$ is given by

$\displaystyle B(t)=B(t_{0})\frac{a(t_{0})^{2}}{a(t)^{2}}\,.$ (85)

As the Universe expands, the temperature also cools as the cosmological
redshift reduces the momenta of particles in the Universe lowering their
contribution to the energy content of the Universe. This cosmological redshift
is written as

$\displaystyle p_{i}(t)=p_{i}(t_{0})\frac{a(t_{0})}{a(t)}\,,\qquad
T(t)=T(t_{0})\frac{a(t_{0})}{a(t)}\,.$ (86)

The momenta scale with the same factor as temperature as it is the origin of
cosmological redshift. The energy of massive free particles in the Universe
scales differently based on their momentum (and thus temperature). When hot
and relativistic, particle energy scales with inverse scale factors like
radiation. However as particles transition to non-relativistic momenta, their
energies scale with the inverse square of the scale factor like magnetic flux.

$\displaystyle E(t)=E(t_{0})\frac{a(t_{0})}{a(t)}\xrightarrow{\mathrm{NR}}\
E(t_{0})\frac{a(t_{0})^{2}}{a(t)^{2}}\,.$ (87)

This occurs because of the functional dependence of energy on momentum in the
relativistic versus non-relativistic cases. The argument in the Boltzmann
statistical factor is given by

$\displaystyle X_{n}^{s}\equiv\frac{E_{n}^{s}}{T}\,.$ (88)

We can explore this relationship for the magnetized system explicitly by
writing out Eq. (88) using the KGP eigen-energies as

$\displaystyle
X_{n}^{s}=\sqrt{\frac{m_{e}^{2}}{T^{2}}+\frac{p_{z}^{2}}{T^{2}}+\frac{2eB}{T^{2}}\left(n+\frac{1}{2}-\frac{gs}{2}\right)}\,,$
(89)

where we now introduce the expansion scale factor via Eq. (85) - Eq. (86). The
Boltzmann factor can then be written as

$\displaystyle
X_{n}^{s}[a(t)]=\sqrt{\frac{m_{e}^{2}}{T^{2}(t_{0})}\frac{a(t)^{2}}{a(t_{0})^{2}}+\frac{p_{z}^{2}(t_{0})}{T^{2}(t_{0})}+\frac{2eB(t_{0})}{T^{2}(t_{0})}\left(n+\frac{1}{2}-\frac{gs}{2}\right)}\,.$
(90)

This reveals that only the mass contribution is dynamic over cosmological
time. For any given eigen-state, the mass term increases driving the state
into the non-relativistic limit while the momenta and magnetic contributions
are frozen by initial conditions.

Following reasoning outlined in Rafelski et al. (2023) and Steinmetz et al.
(2019) we will proceed using the KGP eigen-energies. Motivated by Eq. (90), we
can introduce a dimensionless cosmic magnetic scale which is frozen in the
homogeneous case as

$\displaystyle b_{0}\equiv\frac{eB}{T^{2}}=\frac{eB\hbar c^{2}}{(k_{B}T)^{2}}\
\mathrm{(S.I)}\,,$ (91)

where we’ve included the expression explicitly in full SI units. We can
estimate the value of $b_{0}$ from the bounds of the extra-galactic magnetic
field strength and the temperature of the Universe today. If the origin of
deep space extra-galactic magnetic fields are relic fields from the early
Universe, which today are expected to exist between $5\times 10^{-12}\
\mathrm{T}>B_{relic}>10^{-20}\ \mathrm{T}$, then at temperature $T=2.7\
\mathrm{K}$, the value of the cosmic magnetic scale is between

$\displaystyle 5.5\times 10^{-3}>b_{0}>1.1\times 10^{-11}\,.$ (92)

This should remain constant in the Universe at-large up to the last epoch the
Universe was sufficiently magnetized to disturb this value. As the electron-
proton $(e^{-}p)$ plasma which generated the CMB was relatively dilute over
its duration, it was unlikely sufficiently magnetized to significantly alter
this value over extra-galactic scales. Rather, the first candidate plasma,
going backwards in time, to have been sufficiently magnetized and dense to
have set the relic field magnetic scale would have been the electron-positron
plasma which existed during the duration of Big Bang Nucleosynthesis (BBN) and
beforehand.

Higher order non-minimal magnetic contributions which can be introduced via
Eq. (82) to the eigen-energies like $\approx\mu_{B}^{2}B^{2}/T^{2}$ are even
more suppressed over cosmological time which drives the system into minimal
electromagnetic coupling with the exception of the anomalous magnetic moment
in the KGP eigenenergies. It is interesting to note that cosmological
expansion serves to “smooth out” the characteristics of more complex BSM
electrodynamics erasing them from a statistical perspective in favor of the
minimal or minimal-like dynamics. As $b_{0}$ is a constant of expansion,
assuming the electron-proton plasma between the CMB and electron-positron
annihilation did not greatly disturbed it, we can calculate the remnant values
at the temperature $T=50\ \mathrm{keV}$ (which takes place in the middle of
BBN) with the expression

$\displaystyle B(T)=\frac{b_{0}}{e}T^{2}\,,$ (93)

yielding a range of field strengths

$\displaystyle 2.3\times 10^{5}\ \mathrm{T}>B(T=50\ \mathrm{keV})>4.6\times
10^{-4}\ \mathrm{T}\,,$ (94)

during which the electron-positron plasma in the Universe had a number density
comparable to that of the Solar core Bahcall et al. (2001). We note that while
the density of leptons is comparable to that of the solar core during this
period, the temperature is not. The $e^{\pm}$ plasma during BBN was far hotter
than the solar core’s comparatively cool temperature of $T_{\odot}=1.37\text{
keV}$ Castellani et al. (1997).

###  5.4 Electron-positron statistical physics

We now turn our attention now to the statistical behavior of the $e^{\pm}$
system. We can utilize the general fermion partition function given by Elze et
al. (1980)

$\displaystyle\ln\mathcal{Z}=\sum_{\alpha}\ln\left(1+e^{-\beta(E-\eta)}\right)\,,$
(95)

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

$\displaystyle\ln\mathcal{Z}_{tot}=\frac{2eBV}{(2\pi)^{2}}\sum_{\sigma}^{\pm
1}\sum_{s}^{\pm
1/2}\sum_{n=0}^{\infty}\int^{\infty}_{0}dp_{z}\left[\ln\left(1+\Upsilon_{\sigma}^{s}(x)e^{-\beta
E_{n}^{s}}\right)\right]\,,$ (96)
$\displaystyle\Upsilon_{\sigma}^{s}(x)=\gamma(x)\lambda_{\sigma}^{s}\,,\qquad\lambda_{\sigma}^{s}=e^{(\sigma\eta_{e}+s\eta_{s})/T}\,,$
(97)

where $\eta_{e}$ is the electron chemical potential and $\eta_{s}$ is the spin
chemical potential for the generalized fugacity $\lambda_{\sigma}^{s}$. The
parameter $\gamma(x)$ is a spatial field which controls the distribution
inhomogeneity of the Fermi gas. Inhomogeneities can arise from the influence
of other forces on the gas such as gravitational forces. Deviations of
$\gamma\neq 1$ represent configurations of reduced entropy (maximum entropy
yields the normal Fermi distribution itself with $\gamma=1$) without pulling
the system off a thermal temperature.

This situation is similar to that of the quarks during QGP, but instead the
deviation is spatial rather than in time. This is precisely the kind of
behavior that may arise in the $e^{\pm}$ epoch as the dominant photon thermal
bath keeps the Fermi gas in thermal equilibrium while spatial inhomogeneity
could spontaneously develop. For the remainder of this work, we will retain
$\gamma(x)=1$. The energy $E_{n}^{\pm}$ can be written as

$\displaystyle E_{n}^{\pm}$
$\displaystyle=\sqrt{p^{2}_{z}+\tilde{m}^{2}_{\pm}+2eBn},\qquad\tilde{m}^{2}_{\pm}=m^{2}_{e}+eB\left(1\mp\frac{g}{2}\right)\,,$
(98)

where the $\pm$ script refers to spin aligned and anti-aligned eigenvalues. As
we are interested in the temperature domain $T=50\ \mathrm{keV}$, we can
consider a semi-relativistic approach obtained by the Boltzmann approximation.
Taking the limit $m_{e}/T\ll 1$, we obtain the first order Boltzmann
approximation for semi-relativistic electrons and positrons. The Euler-
Maclaurin formula is used to replace the sum over Landau levels with an
integration which lets us split the partition function into three segments

$\displaystyle\ln\mathcal{Z}_{tot}=\ln\mathcal{Z}_{free}+\ln\mathcal{Z}_{B}+\ln\mathcal{Z}_{R}\,,$
(99)

where we define

$\displaystyle\ln\mathcal{Z}_{free}=\frac{T^{3}V}{2\pi^{2}}\sum_{i=\pm}\left[2\cosh{\left(\frac{\eta_{e}^{i}}{T}\right)}\right]x_{i}^{2}K_{2}\left(x_{i}\right)\,,\qquad
x_{i}=\frac{\tilde{m}_{i}}{T}$ (100)
$\displaystyle\ln\mathcal{Z}_{B}=\frac{eBTV}{2\pi^{2}}\sum_{i=\pm}\left[2\cosh{\left(\frac{\eta_{e}^{i}}{T}\right)}\right]\bigg{[}\frac{x_{i}}{2}K_{1}\left(x_{i}\right)+\frac{b_{0}}{12}K_{0}\left(x_{i}\right)\bigg{]}\,,$
(101)
$\displaystyle\ln\mathcal{Z}_{R}=\frac{eBTV}{\pi^{2}}\sum_{i=\pm}\left[2\cosh{\left(\frac{\eta_{e}^{i}}{T}\right)}\right]R.$
(102)

The parameter $R$ is the error remainder which is defined by integrals over
Bernoulli polynomials. The parameter $\eta_{e}^{\pm}$ indicates that the
chemical potential may be modified by the spin chemical potential and is in
general non-zero as defined in Eq. (97).

While this would require further derivation to demonstrate explicitly, the
benefit of the Euler-Maclaurin approach is if the error contribution remains
finite or bound for the magnetized partition function, then a correspondence
between the free Fermi partition function (with noticeably modified effective
mass $\tilde{m}_{\pm}$) and the magnetized Fermi partition function can be
established. The mismatch between the summation and integral in the Euler-
Maclaurin formula would then encapsulate the immediate magnetic response and
deviation from the free particle phase space.

While we label $\ln\mathcal{Z}_{free}$ in Eq. (100) as the “free” partition
function, this is not strictly true as this contribution to the overall
partition function is a function of the effective mass we defined earlier in
Eq. (82). When determining the magnetization of the quantum Fermi gas,
derivatives of the magnetic field $B$ will not fully vanish on this first term
which will resulting in an intrinsic magnetization which is distinct from the
contribution from the ground state and mismatch between the quantized Landau
levels and the continuum of the free momentum. Specifically, this free Fermi
contribution represents the magnetization that uniquely arises from the spin
magnetic energy rather than orbital contributions.

Assuming the error remainder $R$ is small and can be neglected, we can rewrite
Eq. (100) - Eq. (101) obtaining

$\displaystyle\ln\mathcal{Z}_{tot}=\frac{T^{3}V}{2\pi^{2}}\sum_{i=\pm}\left[2\cosh\left(\frac{\eta_{e}^{i}}{T}\right)\right]\left\\{x_{i}^{2}K_{2}\left(x_{i}\right)+\frac{b_{0}}{2}x_{i}K_{1}\left(x_{i}\right)+\frac{b^{2}_{0}}{12}K_{0}\left(x_{i}\right)\right\\}\,.$
(103)

Eq. (103) is a surprisingly compact expression containing only tractable
functions and will be our working model for the remainder of the work. Note
that the above does not take into consideration density inhomogeneities and is
restricted to the domain where the plasma is well described as a Maxwell-
Boltzmann distribution. With that said, we have not taken the non-relativistic
expansion of the eigen-energies.

###  5.5 Charge neutrality and chemical potential

We explore the chemical potential of dense magnetized electron-positron plasma
in the early Universe under the hypothesis of charge neutrality and entropy
conservation. To learn about orders of magnitude we set in the following
$\eta_{e}=\eta_{e}^{+}=\eta_{e}^{-}$ and focus on the interval in the post-BBN
temperature range $50\text{ keV}>T>20\text{ keV}$. We return to the full
problem under separate cover. The charge neutrality condition can be written
as

$\displaystyle\left(n_{e}-n_{\bar{e}}\right)=n_{p}=\left(\frac{n_{p}}{n_{B}}\right)\,\left(\frac{n_{B}}{s_{\gamma,\nu,e}}\right)\,s_{\gamma,\nu,e}=X_{p}\left(\frac{n_{B}}{s_{\gamma,\nu}}\right)\,s_{\gamma,\nu},\qquad
X_{p}\equiv\frac{n_{p}}{n_{B}}\,,$ (104)

where $n_{p}$ and $n_{B}$ is the number density of protons and baryons
respectively.

The radiation entropy component is given by $s_{\gamma,\nu}$. The entropy
density contribution of $e^{\pm}$ is negligible compared to the photon and
neutrino entropy density at post-BBN temperatures $50\text{ keV}>T>20\text{
keV}$ because the low densities of $n_{e}\ll n_{\gamma,\nu}$ relative to the
photon and neutrino gasses. The entropy density can be written as Kolb and
Turner (1990)

$\displaystyle s=\frac{2\pi^{2}}{45}g_{s}T_{\gamma}^{3},\qquad
g_{s}=\sum_{i=boson}g_{i}\left(\frac{T_{i}}{T_{\gamma}}\right)^{3}+\frac{7}{8}\sum_{i=fermion}g_{i}\left(\frac{T_{i}}{T_{\gamma}}\right)^{3}\,,$
(105)

where $g_{s}$ is the effective degree of freedom that contribute from boson
and fermion species. The parameters $X_{p}$ and $(n_{B}/s)$ (see Eq. (32)) can
be determined by the observation, yielding $X_{p}=0.878\pm 0.015$ Workman et
al. (2022). The net number density of electrons can be obtained by using the
partition function of electron-positron plasma in the Boltzmann limit Eq.
(103) (with $g=2$) as follows:

$\displaystyle\left(n_{e}-n_{\bar{e}}\right)$
$\displaystyle=\frac{T}{V}\frac{\partial}{\partial\eta_{e}}\ln\mathcal{Z}_{tot}$
(106)
$\displaystyle=\frac{T^{3}}{2\pi^{2}}\left[2\sinh{(\eta_{e}/T)}\right]\sum_{i=\pm}\left[x_{i}^{2}K_{2}(x_{i})+\frac{b_{0}}{2}x_{i}K_{1}(x_{i})+\frac{b^{2}_{0}}{12}K_{0}(x_{i})\right]\,.$

Substituting Eq. (106) into the charge neutrality condition Eq. (104) we can
solve the chemical potential of electron $\eta_{e}/T$ yielding

$\displaystyle\sinh{(\eta_{e}/T)}$
$\displaystyle=\frac{2\pi^{2}}{2T^{3}}\,\frac{X_{p}(n_{B}/s_{\gamma,\nu})s_{\gamma,\nu}}{\sum_{i=\pm}\left[x_{i}^{2}K_{2}(x_{i})+\frac{b_{0}}{2}x_{i}K_{1}(x_{i})+\frac{b^{2}_{0}}{12}K_{0}(x_{i})\right]}\,,$
(107)
$\displaystyle\longrightarrow\frac{2\pi^{2}n_{p}}{2T^{3}}\,\frac{X_{p}(n_{B}/s_{\gamma,\nu})s_{\gamma,\nu}}{2x^{2}K_{2}(x)},\qquad
x=m_{e}/T,\qquad\mathrm{for}\,\,b_{0}=0\,.$ (108)

We see in Eq. (108) that for the case $b_{0}=0$, the chemical potential agrees
with the free particle result in Grayson et al. (2023).

###  5.6 Magnetization of the electron-positron plasma

We consider the electron-positron plasma in the mean field approximation where
the external field is representative of the “bulk” internal magnetization of
the gas. Each particle is therefore responding to the averaged magnetic flux
generated by its neighbors as well as any global external field contribution.
Considering the magnetized electron-positron partition function Eq. (103) we
introduce dimensionless magnetization in S.I units and the critical field as
follows

$\displaystyle\frac{M}{H_{c}}=\frac{1}{H_{c}}\frac{k_{B}T}{V}\frac{\partial\ln\mathcal{Z}_{tot}}{\partial
B}\,,\qquad H_{c}$ $\displaystyle=\frac{B_{c}}{\mu_{0}}\qquad
B_{c}=\frac{m_{e}^{2}c^{4}}{e\hbar c^{2}}\,.$ (109)

Applying Eq. (109) to Eq. (103) we arrive at the expression

$\displaystyle M_{\pm}$
$\displaystyle=\frac{eT^{2}}{2\pi^{2}}\left[2\cosh\left(\frac{\eta_{e}}{T}\right)\right]\left\\{c_{1}(x_{\pm})K_{1}(x_{i})+c_{0}K_{0}(x_{\pm})\right\\}\,,$
(110) $\displaystyle c_{1}(x_{\pm})$
$\displaystyle=\left[\frac{1}{2}-\left(\frac{1}{2}\pm\frac{g}{4}\right)\left(1+\frac{b^{2}_{0}}{12x^{2}_{\pm}}\right)\right]x_{\pm}\,,\qquad
c_{0}=\left[\frac{1}{6}-\left(\frac{1}{4}\pm\frac{g}{8}\right)\right]b_{0}\,.$
(111)

Figure 22: Estimate for the spin magnetization as a function of temperature in
the range $10^{3}\text{ keV}>T>10\text{ keV}$, see text for detail.

Substituting the chemical potential Eq. (107) into Eq. (110) we can solve the
magnetization $M$ numerically. Considering the case $g=2$ the magnetization
can be written as the sum of the aligned and anti-aligned polarizations

$\displaystyle M=M_{+}+M_{-}\,,$ (112)

where the functions $M_{\pm}$ are defined as

  * A.

The aligned polarized gas is described by $\tilde{m}_{+}=m_{e}$ and
$x=\tilde{m}_{+}/T$. The magnetization of this contribution is therefore

$\displaystyle M_{+}$
$\displaystyle=\frac{eT^{2}}{\pi^{2}}\sqrt{1+\sinh^{2}(\eta_{e}/T)}\left(\frac{1}{2}x_{+}K_{1}(x_{+})+\frac{b_{0}}{6}K_{0}(x_{+})\right)$
(113)

  * B.

The spin anti-aligned gas has effective masses
$\tilde{m}_{-}=\sqrt{m^{2}_{e}+2eB}$, and $x_{-}=\tilde{m}_{-}/T$. This yields
a magnetization contribution of

$\displaystyle M_{-}$
$\displaystyle=-\frac{eT^{2}}{\pi^{2}}\sqrt{1+\sinh^{2}(\eta_{e}/T)}\left[\left(\frac{1}{2}+\frac{b_{0}^{2}}{12x_{-}^{2}}\right)x_{-}K_{1}(x_{-})+\frac{b_{0}}{3}K_{0}(x_{-})\right]$
(114)

Using the cosmic magnetic scale parameter $b_{0}$ and chemical potential
$\eta_{e}/T$ we solve the magnetization numerically. In Fig. 22, we present
the outcome of this estimate. The solid lines (red for the lower bound of
$b_{0}$ and blue for the higher bound of $b_{0}$) showing that the
magnetization depends on the magnetic scale $b_{0}$.

##  6 Looking in the Cosmic Rear-view Mirror

The present day Universe seems devoid of antimatter but the primordial
Universe was nearly matter-antimatter symmetric. There was only a fractional
nano-scale excess of matter which today makes up the visible matter we see
around us. All that remains of the tremendous initial amounts of matter-
antimatter from the Big Bang is now seen as background thermal entropy. The
origin of this nano-matter excess remains to this day an unresolved puzzle. If
matter asymmetry emerged along the path of the Universe’s evolution, as most
think, the previously discussed Sakharov conditions (see Sect. 1.2) must be
fulfilled.

We explored several major epochs in the Universe evolution where antimatter,
in all its diverse forms, played a large roll. Emphasis was placed on
understanding the thermal and chemical equilibria arising within the context
of the Standard Model of particle physics. We highlighted that primordial
quark-gluon plasma (QGP, which existed for $\approx 25\;\mu$sec) is an
important antimatter laboratory with its gargantuan antimatter content. Study
of the QGP fireballs created in heavy-ion collisions performed today informs
our understanding of the early Universe and vice versa Borsanyi et al. (2016);
Rafelski (2013); Petrán et al. (2013); Philipsen (2013), even though the
primordial quark-gluon plasma under cosmic expansion explores a location in
the phase diagram of QCD inaccessible to relativistic collider experiments
considering both net baryon density, see Fig. 5, and longevity of the plasma.
We described (see Sect. 2.2) that the QGP epoch near to hadronization
condition possessed bottom quarks in a non-equilibrium abundance: This novel
QGP-Universe feature may be of interest in consideration of the QGP epoch as
possible source for baryon asymmetry Yang and Rafelski (2020).

Bottom non-equilibrium is one among a few interesting results presented
bridging the temperature gap between QGP hadronization at temperature $T\simeq
150\text{ MeV}$ and neutrino freeze-out. Specifically we shown persistence of:

  * •

Strangeness abundance, present beyond the loss of the antibaryons at
$T=38.2\text{ MeV}$.

  * •

Pions, which are equilibrated via photon production long after the other
hadrons disappear; these lightest hadrons are also dominating the Universe
baryon abundance down to $T=5.6\text{ MeV}$.

  * •

Muons, disappearing at around $T=4.2\text{ MeV}$, the condition when their
decay rate outpaces their production rate.

At yet lower temperatures neutrinos make up the largest energy fraction in the
Universe driving the radiation dominated cosmic expansion. Partway through
this neutrino dominated Universe, in temperature range $T\in 3.5-1\text{ MeV}$
(range spanning differing flavor freeze-out, chemical equilibria, and even
variation in standard natural constants; see Fig. 18), the neutrinos freeze-