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Expand Up @@ -100,7 +100,7 @@ \section{Timeline of Particles and Plasmas in the Universe}\label{sec:Intro}
\subsection{Guide to \texorpdfstring{$130\GeV>T>20\keV$}{130\GeV>T>20\keV}}\label{sec:Guide}
\noindent This survey of the early Universe begins with quark-gluon plasma (QGP) at a temperature of $T=130\GeV$. It then ends at a temperature of $T=20\keV$ with the electron-positron epoch which was the final phase of the Universe to contain significant quantities of antimatter. This defines the \lq\lq short\rq\rq\ $t\approx1/2$ 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~\cite{fang1984cosmology,fang1985galaxies,fang1987quantum}. 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 \rsec{sec:Timeline}. 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–RobertsonWalker (FLRW) metric. The contemporary history of the Universe in terms of energy density as a function of time and temperature is shown in \rf{CosmicDensity}. 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 \rsec{sec:Cosmo}.
A more detailed description of particles and plasmas follows in \rsec{sec:Timeline}. 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-Lema\^itre-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 \rf{CosmicDensity}. 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 \rsec{sec:Cosmo}.
After the general overview, we take the opportunity to enlarge in some detail our more recent work in special topics. In \rsec{sec:QGP}, 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 \rsec{sec:Hadrons} 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 \rsec{sec:Leptonic}. The $e^{\pm}$ plasma epoch is described in \rsec{sec:ElectronPositron} 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. We begin to 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 $T=20\keV$ with the disappearance of the positrons signifying the end of antimatter dynamics at cosmological scales.
Expand Down Expand Up @@ -298,7 +298,7 @@ \subsection{Conservation laws in QGP}\label{sec:Conservation}
\begin{align}\label{QGP_sB}
\frac{S}{B}=\frac{s}{n_B}=\frac{\sum_fs_f(\mu_f,T)}{\sum_fB_fn_f(\mu_f,T)}=\mathrm{const}
\end{align}
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 inhomogeneitiess from gravitational attraction becomes significant, leading to increases in local entropy.
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.
\end{enumerate}
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 \rf{QGPchem1} we present numerical solutions to the conditions \req{QGP_Q}-\req{QGP_sB} 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.
Expand Down Expand Up @@ -688,7 +688,7 @@ \subsection{Neutrino masses and oscillation} \label{sec:Neutrinos}
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. {\xblue 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~\cite{Dolinski:2019nrj}.} Majorana neutrinos with small masses can be generated from some high scale via the See-Saw mechanism~\cite{Arkani-Hamed:1998wuz,Ellis:1999my,Casas:2001sr} 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 PontecorvoMakiNakagawaSakata (PMNS) mixing matrix~\cite{King:2013eh,FernandezMartinez:2016lgt} which are both in general complex and unitary. This is given by
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~\cite{King:2013eh,FernandezMartinez:2016lgt} which are both in general complex and unitary. This is given by
\begin{align}\label{NuFlavors}
\nu^{\alpha}=\sum_k^nU^\ast_{\alpha k}\nu^{k}, \qquad\alpha=e,\mu,\tau,\qquad k=1,2,3,\dots,n
\end{align}
Expand All @@ -708,14 +708,14 @@ \subsection{Neutrino masses and oscillation} \label{sec:Neutrinos}
\end{alignat}
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 eigenmasses are generally considered to be small with values no more than $0.1\eV$. 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~\cite{ParticleDataGroup:2022pth}
The neutrino proper masses are generally considered to be small with values no more than $0.1\eV$. 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~\cite{ParticleDataGroup:2022pth}
\begin{align}\label{NuOscillation}
P_{\alpha\rightarrow\beta}
=&\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)\notag\\
&+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}
\end{align}
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~\cite{ParticleDataGroup:2022pth}. 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)~\cite{Avignone:2007fu,Esteban:2020cvm} the masses are organized as two of the three neutrino eigenmasses are close together in value.
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~\cite{ParticleDataGroup:2022pth}. 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)~\cite{Avignone:2007fu,Esteban:2020cvm} 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~\cite{NuSTEC:2017hzk,DUNE:2020ypp}) 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.
Expand Down Expand Up @@ -777,7 +777,7 @@ \subsection{Neutrino freeze-out}\label{sec:Freezeout}
After neutrinos freeze-out, the neutrino co-moving entropy is independently conserved. However, the presence of electron-positron rich plasma until $T=20\keV$ 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 $T=20\keV$ 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.~\cite{Mangano:2005cc}, neutrino freeze-out including flavour oscillations is shown to be a negligible effect.
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.~\cite{Mangano:2005cc}, neutrino freeze-out including flavor oscillations is shown to be a negligible effect.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Effective number of neutrinos}\label{sec:EffectiveNeutrino}
Expand Down Expand Up @@ -949,7 +949,7 @@ \subsection{Electron-positron statistical physics}\label{sec:Partition}
\end{align}
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 $\gamma\neq1$ 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 inequilibria could spontaneously develop. For the remainder of this work, we will retain $\gamma(x)=1$. The energy $E_{n}^\pm$ can be written as
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
\begin{align}
E_{n}^\pm&=\sqrt{p^2_z+\tilde m^2_\pm+2eBn},\qquad\tilde{m}^2_\pm=m^2_e+eB\left(1\mp\frac{g}{2}\right)\,,
\end{align}
Expand Down Expand Up @@ -1052,7 +1052,7 @@ \section{Looking in the Cosmic Rear-view Mirror}\label{sec:Summary}
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~\cite{Borsanyi:2016ksw,Rafelski:2013qeu,Petran:2013lja,Philipsen:2012nu}, 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 \rf{phaseQGP}, and longevity of the plasma. We described (see \rsec{sec:BottomCharm}) 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~\cite{Yang:2020nne}.
Bottom nonequilibrium is one among a few interesting results presented bridging the temperature gap between QGP hadronization at temperature $T\simeq150\MeV$ and neutrino freeze-out. Specifically we shown {\bf persistence of:}
Bottom non-equilibrium is one among a few interesting results presented bridging the temperature gap between QGP hadronization at temperature $T\simeq150\MeV$ and neutrino freeze-out. Specifically we shown {\bf persistence of:}
\begin{itemize}
\item Strangeness abundance, present beyond the loss of the antibaryons at $T=38.2\MeV$.
\item 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 $T=5.6\MeV$.
Expand Down
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