diff --git a/a-short-survey-mdpi.tex b/a-short-survey-mdpi.tex index 0ef2e23..e2ba869 100644 --- a/a-short-survey-mdpi.tex +++ b/a-short-survey-mdpi.tex @@ -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–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}. +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. @@ -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. @@ -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 Pontecorvo–Maki–Nakagawa–Sakata (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} @@ -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