diff --git a/a-short-survey-mdpi.pdf b/a-short-survey-mdpi.pdf index f165a5a..8a20faf 100644 Binary files a/a-short-survey-mdpi.pdf and b/a-short-survey-mdpi.pdf differ diff --git a/a-short-survey-mdpi.tex b/a-short-survey-mdpi.tex index 92e7031..8d176fa 100644 --- a/a-short-survey-mdpi.tex +++ b/a-short-survey-mdpi.tex @@ -87,7 +87,7 @@ %\conference{} % An extended version of a conference paper % Abstract (Do not insert blank lines, i.e. \\) -\abstract{In celebration of Professor Remo Ruffini's birthday, his contributions to astrophysics and cosmology and the large number of students and young scientists he mentored, (see \rf{RemoAtWork}) we offer here 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.} +\abstract{In celebration of Professor Remo Ruffini's birthday, his contributions to astrophysics and cosmology and the large number of students and young scientists he mentored, (see \rf{RemoAtWork}) 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.} \keyword{Particles, Plasmas and Electromagnetic Fields in Cosmology; Quarks to Cosmos;} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -118,7 +118,7 @@ \subsection{Guide to 130 GeV > T > 20 keV}\label{sec:Guide} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -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 \rsec{Summary} 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)~\cite{Ruffini:2001fe,Aksenov:2008ze,Aksenov:2010vi,Ruffini:2012it}, black holes~\cite{Ruffini:2003yt,Ruffini:2009hg,Ruffini:2000yu} and neutron stars (magnetars)~\cite{Han:2011er,Belvedere:2012uc}. +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 \rsec{sec:Summary} 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)~\cite{Ruffini:2001fe,Aksenov:2008ze,Aksenov:2010vi,Ruffini:2012it}, black holes~\cite{Ruffini:2003yt,Ruffini:2009hg,Ruffini:2000yu} and neutron stars (magnetars)~\cite{Han:2011er,Belvedere:2012uc}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The five plasma epochs}\label{sec:Timeline} \noindent 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~\cite{Rafelski:2015cxa}. The ultra-relativistic plasma produced in the early Universe 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. We repeat the standard wisdom that the known CP-violation in the Standard Model's weak sector is insufficient to explain the baryon asymmetry we see today. Additionally, three conditions are required in cosmology to explain the asymmetry outlined by Sakharov~\cite{Sakharov:1967dj,Sakharov:1988vdp}: @@ -127,23 +127,25 @@ \subsection{The five plasma epochs}\label{sec:Timeline} \item Violation of CP-invariance \item Non-stationary conditions in absence of local thermodynamic equilibrium \end{itemize} -In this work we take the baryon asymmetry as a given parameter (though additional comments on the situation in the context of non-equilibria processes are made in \rsec{BottomCharm} and \rsec{Summary}). This fireball then underwent several phases changes which dramatically evolved the gross properties of the Universe as it expanded and cooled. Evolutionary processes in the primordial Universe are taken to be adiabatic. We present an overview \rf{CosmicFraction} 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: +In this work we take the baryon asymmetry as a given parameter (though additional comments on the situation in the context of non-equilibria processes are made in \rsec{sec:BottomCharm} and \rsec{sec:Summary}). This fireball then underwent several phases changes which dramatically evolved the gross properties of the Universe as it expanded and cooled. Evolutionary processes in the primordial Universe are taken to be adiabatic. We present an overview \rf{CosmicFraction} 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: + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure} \centerline{\includegraphics[trim=70 120 60 100,clip,width=\textwidth,width=\linewidth]{./plots/energy_fractions.pdf}} \caption{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). \label{CosmicFraction}} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{enumerate} - \item \textbf{Primordial quark-gluon plasma}: At early times when the temperature was between $130\ \mathrm{GeV}>T>150\ \mathrm{MeV}$ 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 $150\MeV$. - \item \textbf{Hadronic epoch}: Around the hadronization temperature $T_h\approx150\ \mathrm{MeV}$, a phase transformation occurred forcing the strongly interacting particles such as quarks and gluons to condense into confined states~\cite{Letessier:2005qe}. It is here where matter as we know it today forms and the Universe becomes hadronic-matter dominated. In the temperature range $ 150\ \mathrm{MeV}>T>20\ \mathrm{MeV}$ the Universe is rich in physics phenomena involving strange mesons and (anti)baryons including (anti)hyperon abundances~\cite{Fromerth:2012fe,Yang:2021bko}. - \item \textbf{Lepton-photon epoch}: For temperature $10\ \mathrm{MeV}>T>2\ \mathrm{MeV}$, 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.~\cite{Birrell:2012gg,Birrell:2014ona}. During this time the expansion of the Universe is controlled by leptons and photons almost on equal footing. - \item \textbf{Final antimatter epoch}: After neutrinos decoupled and become free-streaming, referred to as neutrino freeze-out, from the cosmic plasma at $T=2\ \mathrm{MeV}$, the cosmic plasma was dominated by electrons, positrons, and photons. We have shown in~\cite{Chris:2023abc} that this plasma existed until $T\approx0.02\ \mathrm{MeV}$ 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. - \item \textbf{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 $T\approx0.25\ \mathrm{eV}$ 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~\cite{Planck:2018vyg}. + \item \textbf{Primordial quark-gluon plasma}: At early times when the temperature was between $130\GeV>T>150\MeV$ 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 $150\MeV$. + \item \textbf{Hadronic epoch}: Around the hadronization temperature $T_h\approx150\MeV$, a phase transformation occurred forcing the strongly interacting particles such as quarks and gluons to condense into confined states~\cite{Letessier:2005qe}. It is here where matter as we know it today forms and the Universe becomes hadronic-matter dominated. In the temperature range $ 150\MeV>T>20\MeV$ the Universe is rich in physics phenomena involving strange mesons and (anti)baryons including (anti)hyperon abundances~\cite{Fromerth:2012fe,Yang:2021bko}. + \item \textbf{Lepton-photon epoch}: For temperature $10\MeV>T>2\MeV$, 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.~\cite{Birrell:2012gg,Birrell:2014ona}. During this time the expansion of the Universe is controlled by leptons and photons almost on equal footing. + \item \textbf{Final antimatter epoch}: After neutrinos decoupled and become free-streaming, referred to as neutrino freeze-out, from the cosmic plasma at $T=2\MeV$, the cosmic plasma was dominated by electrons, positrons, and photons. We have shown in~\cite{Chris:2023abc} that this plasma existed until $T\approx0.02\MeV$ 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. + \item \textbf{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 $T\approx0.25\eV$ 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~\cite{Planck:2018vyg}. \end{enumerate} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] - %\begin{center} \centering \includegraphics[width=\textwidth]{./plots/degrees_of_freedom.PNG} \caption{The evolution of the photon reheating (black line) process in terms of fractional temperature change in the Universe. Figure adapted from~\cite{Rafelski:2013yka}. The dashed portion is a qualitative description subject to the exact model of QGP hadronization.} @@ -157,7 +159,7 @@ \subsection{The five plasma epochs}\label{sec:Timeline} 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 $T=\mathcal{O}(1\, \MeV)$, we come to the annihilation of $e^\pm$ and the photon reheating period. Notice that only the photon energy density fraction increases here, as we assume here 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~\cite{Mangano:2005cc,Fornengo:1997wa,Mangano:2001iu,Birrell:2012gg}. We next pass through a long period, from $T=\mathcal{O}(1\, \MeV)$ until $T=\mathcal{O}(1\, \eV)$, where the energy density is dominated by photons and free-streaming neutrinos. BBN occurs in the approximate range $T=40-70\keV$ and is indicated by the next two vertical lines in \rf{CosmicFraction}. 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 \rsec{sec:ElectronPositronDensity}). +Continuing to the second vertical line at $T=\mathcal{O}(1\, \MeV)$, 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~\cite{Mangano:2005cc,Fornengo:1997wa,Mangano:2001iu,Birrell:2012gg}. We next pass through a long period, from $T=\mathcal{O}(1\, \MeV)$ until $T=\mathcal{O}(1\, \eV)$, where the energy density is dominated by photons and free-streaming neutrinos. BBN occurs in the approximate range $T=40-70\keV$ and is indicated by the next two vertical lines in \rf{CosmicFraction}. 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 \rsec{sec:ElectronPositronDensity}). 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 (\rf{CosmicFraction}) vertical line at $T=0.25\eV$. @@ -170,7 +172,6 @@ \subsection{The Lambda-CDM Universe}\label{sec:Cosmo} \noindent 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 \beqn\label{metric} ds^2=c^2dt^2-a^2(t)\left[ \frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2(\theta)d\phi^2)\right] -%g_{00}=1, \quad g_{rr}=-\frac{a^2}{1-kr^2}, \quad g_{\theta\theta}=-a^2r^2, \quad g_{\phi\phi}=-a^2 r^2\sin^2\theta \eeqn 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 $k=0$~\cite{Planck:2013pxb,Planck:2015fie,Planck:2018vyg}. In this case it can be more convenient to write the metric in rectangular coordinates \beqn\label{metric2} @@ -188,7 +189,6 @@ \subsection{The Lambda-CDM Universe}\label{sec:Cosmo} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] - %\begin{center} \centering \includegraphics[width=\textwidth]{./plots/deceleration_evolution.PNG} \caption{ (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~\cite{Rafelski:2013yka}.} @@ -235,16 +235,16 @@ \subsection{The Lambda-CDM Universe}\label{sec:Cosmo} \begin{equation} |\Omega_K|<0.005. \end{equation} -This indicates a nearly flat Universe which is spatially Euclidean. We will work here within an exactly spatially flat cosmological model, $k=0$. +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, \req{hubble} implies that the energy momentum tensor of matter is divergence free: \beqn\label{divTmn} T^{\mu\nu};_\nu =0 \Rightarrow -\frac{\dot\rho}{\rho+P}=3\frac{\dot a}{a}=3H. \eeqn -A dynamical evolution equation for $\rho(t)$ arises once we combine \req{divTmn} with \req{hubble}, 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 Planck data~\cite{Planck:2013pxb}, +A dynamical evolution equation for $\rho(t)$ arises once we combine \req{divTmn} with \req{hubble}, 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~\cite{Planck:2013pxb} \begin{equation}\label{Planck_params} -H_0=67.74\text{km/s/Mpc},\hspace{2mm} \Omega_b=0.05,\hspace{2mm} \Omega_c=0.26, \hspace{2mm}\Omega_\Lambda=0.69, +H_0=67.4\,\text{km/s/Mpc},\hspace{2mm} \Omega_b=0.05,\hspace{2mm} \Omega_c=0.26, \hspace{2mm}\Omega_\Lambda=0.69, \end{equation} -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~\cite{Borsanyi:2013bia}, and integrate forward in time. As the Universe continues to dilute from dark energy in the future, the cosmic equation of state is becoming well approximated by the de Sitter inflationary metric which is a special case of FLRW. +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~\cite{Borsanyi:2013bia}, 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{QGP Epoch}\label{sec:QGP} @@ -253,9 +253,7 @@ \subsection{Conservation laws in QGP}\label{sec:Conservation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] - %\begin{center} \centering - %\includegraphics[width=\textwidth]{./plots/phaseQGP} \includegraphics[width=\textwidth]{./plots/QGPDiagram_A.jpg} \caption{ 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~\cite{Rafelski:1987nv}. The boundary (red line) where QGP condenses into hadrons is illustrated at an energy density of $0.5\GeV/\mathrm{fm}^{3}$ as determined through lattice computation~\cite{HotQCD:2014kol}.} \label{phaseQGP} @@ -265,15 +263,16 @@ \subsection{Conservation laws in QGP}\label{sec:Conservation} 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~\cite{Rafelski:2001hp} on order of $10^{-23}$ seconds. As a consequence of the short lifespan of laboratory QGP in heavy-ion collisions~\cite{Ollitrault:1992bk,Petran:2013lja}, they are not subject to the same weak interaction dynamics~\cite{Ryu:2015vwa} as the characteristic times for weak processes are too lengthy~\cite{Rafelski:1982ii}. 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 \rf{phaseQGP} where the chemical potential of QGP $\mu_{q}=\mu_{B}/3$~\cite{Rafelski:1987nv} 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 \rf{phaseQGP}) was calculated using a parameters obtained from~\cite{Letessier:2002ony} in agreement with lattice results~\cite{HotQCD:2014kol}. 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~\cite{Jacak:2012dx}. The work of Fromerth et. al.~\cite{Fromerth:2012fe} 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~\cite{Fromerth:2012fe}: + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] - %\begin{center} \centering \includegraphics[width=\textwidth]{./plots/mu_combo.pdf} \caption{Plot of the down quark chemical potential (black), electron chemical potential (dotted red) and neutrino chemical potential (dashed green) as a function of time. (2003 unpublished, Fromerth \& Rafelski~\cite{Rafelski:2019twp})} \label{QGPchem1} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{enumerate} \item Electric charge neutrality $Q=0$, given by \begin{align}\label{QGP_Q} @@ -295,7 +294,6 @@ \subsection{Conservation laws in QGP}\label{sec:Conservation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] - %\begin{center} \centering \includegraphics[width=\textwidth]{./plots/Tmud1.pdf} \caption{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~\cite{Rafelski:2019twp})} @@ -308,10 +306,10 @@ \subsection{Conservation laws in QGP}\label{sec:Conservation} \end{align} which is summed over all quarks and their quantum numbers. In \req{QuarkDistribution}, $\lambda_{q}$ is the quark fugacity while $\gamma_{q}(t)$ is the temporal inhomogeneity of the population distribution~\cite{Rafelski:2019twp}. 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~\cite{Letessier:2002ony}. When $\gamma\neq1$, 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~\cite{Letessier:2002ony}. However near the hadronization temperature, heavy quarks abundance and deviations from chemical equilibrium have not yet been studied in great detail. We show in \rsec{BottomCharm} and~\cite{Yang:2020nne,Yang:2023bot} that the bottom quarks can deviate from chemical equilibrium $\gamma\neq1$ by breaking the detailed balance between reactions of the quarks. +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~\cite{Letessier:2002ony}. However near the hadronization temperature, heavy quarks abundance and deviations from chemical equilibrium have not yet been studied in great detail. We show in \rsec{sec:BottomCharm} and~\cite{Yang:2020nne,Yang:2023bot} that the bottom quarks can deviate from chemical equilibrium $\gamma\neq1$ by breaking the detailed balance between reactions of the quarks. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Heavy flavor: Bottom and charm in QGP}\label{BottomCharm} +\subsection{Heavy flavor: Bottom and charm in QGP}\label{sec:BottomCharm} \noindent In the QGP epoch, up and down $(u,d)$ quarks are effectively massless and remain in equilibrium via quark-gluon fusion. Strange $(s)$ quarks are in equilibrium via weak, electromagnetic, and strong interactions until $T\sim12\MeV$~\cite{Yang:2021bko}. In this section, we focus on the heavier charm and bottom $(c,b)$ 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 \begin{align} q+q&\longrightarrow b+\bar b,\qquad q+q\longrightarrow c+\bar c,\\ @@ -326,12 +324,11 @@ \subsection{Heavy flavor: Bottom and charm in QGP}\label{BottomCharm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure} %[ht] -\begin{center} -\includegraphics[width=\textwidth]{./plots/CharmQuark_QGP.jpg}\\ -\includegraphics[width=\textwidth]{./plots/BQuarkReactionTime_bottom.jpg} -\caption{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 $T_{h}\approx150\MeV$. Three different masses $m_{b}={4.2\GeV\ \mathrm{(blue\ short\ dashes)},\ 4.7\GeV\ \mathrm{(solid\ black)},\ 5.2\GeV\ \mathrm{(red\ long\ dashes)}}$ for bottom quarks are plotted to account for its decay width.} + \centering + \includegraphics[width=0.9\textwidth]{./plots/CharmQuark_QGP.jpg} + \includegraphics[width=0.9\textwidth]{./plots/BQuarkReactionTime_bottom.jpg} + \caption{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 $T_{h}\approx150\MeV$. Three different masses $m_{b}=4.2\GeV$ (blue short dashes), $4.7\GeV$, (solid black), $5.2\GeV$ (red long dashes) for bottom quarks are plotted to account for its decay width.} \label{BCreaction_fig} -\end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -344,14 +341,13 @@ \subsection{Heavy flavor: Bottom and charm in QGP}\label{BottomCharm} The Universe's characteristic expansion time constant $1/H$ is seen in \rf{BCreaction_fig} (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 \rf{BCreaction_fig} (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 $\Upsilon\neq1$ (see \req{QuarkDistribution}) 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. + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[t] -%\begin{center} \centering \includegraphics[width=\textwidth]{./plots/BquarkFugacity.jpg} -\caption{The generalized fugacity $\Upsilon_{b}$ of free unconfined bottom quark as a function of temperature in QGP up to the hadronization temperature of $T_{h}\approx150\MeV$ for three different bottom masses $m_{b}={4.2\GeV\ \mathrm{(solid\ blue)},\ 4.7\GeV\ \mathrm{(solid\ black)},\ 5.2\GeV\ \mathrm{(solid\ red)}}$.} + \caption{The generalized fugacity $\Upsilon_{b}$ of free unconfined bottom quark as a function of temperature in QGP up to the hadronization temperature of $T_{h}\approx150\MeV$ for three different bottom masses $m_{b}=4.2\GeV$ (solid blue), $4.7\GeV$, (solid black), $5.2\GeV$ (solid red).} \label{UpsilonBottom_fig} -%\end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -382,7 +378,6 @@ \subsection{The formation of matter}\label{sec:Creation} \begin{align}\label{QGP_transQ} n_{Q,HG+QGP}=f_{HG}(T)n_{HG,Q}+\left[1-f_{HG}(T)\right]n_{QGP,Q}=0\,. \end{align} -%n_{Q,B}=f_{HG}(T)A_{HG}(T)n_{HG,Q}+\left[1-f_{HG}(T)\right]A_{QGP}(T)n_{QGP,Q}-A_{L}(T)n_{Q,L}\, At a temperature of $T_{h}\approx150\MeV$, 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 $\sim1$ in $10^{9}$) would be essentially invisible until a temperature of between $40-50\MeV$. 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 via the difference in strange-antistrange masses which are expected to be small if not identically zero. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -391,7 +386,6 @@ \subsection{The formation of matter}\label{sec:Creation} \includegraphics[width=\textwidth]{./plots/hadron_content.png} \caption{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 \rf{CosmicFraction}. (2003 unpublished, Fromerth \& Rafelski~\cite{Rafelski:2019twp})} \label{hadron_content} -%\end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -425,12 +419,10 @@ \subsection{The formation of matter}\label{sec:Creation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] -%\begin{center} \centering \includegraphics[width=\textwidth]{./plots/Baryon_Antibaryon_cm.jpg} \caption{The baryon (blue solid line) and antibaryon (red solid line) number density as a function of temperature in the range $150\MeV>T>5\MeV$. The green dashed line is the extrapolated value for baryon density. The temperature $T=38.2\MeV$ (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.} \label{Baryon_fig} -%\end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -438,8 +430,7 @@ \subsection{The formation of matter}\label{sec:Creation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Strangeness abundance}\label{sec:Strangeness} -\noindent -As the energy contained in QGP is used up to create matter and antimatter particles, the high abundance of strange $(s,\bar{s})$ quark pairs present in the plasma is preserved as exotic hadronic condensates containing strange quarks. After hadronization, both charm $(c,\bar{c})$ and strange quarks can form heavy mesons. With time, strangeness and charmness decay away as they are heavier than the light $(u,d)$ quarks and antiquarks. However, unlike charm which disappears from the particle inventory quickly, strangeness can still persist~\cite{Yang:2021bko} in the Universe until $T\approx\mathcal{O}(10\MeV)$. +\noindent As the energy contained in QGP is used up to create matter and antimatter particles, the high abundance of strange $(s,\bar{s})$ quark pairs present in the plasma is preserved as exotic hadronic condensates containing strange quarks. After hadronization, both charm $(c,\bar{c})$ and strange quarks can form heavy mesons. With time, strangeness and charmness decay away as they are heavier than the light $(u,d)$ quarks and antiquarks. However, unlike charm which disappears from the particle inventory quickly, strangeness can still persist~\cite{Yang:2021bko} in the Universe until $T\approx\mathcal{O}(10\MeV)$. 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 \begin{align} @@ -459,12 +450,10 @@ \subsection{Strangeness abundance}\label{sec:Strangeness} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[bt] -%\begin{center} \centering \includegraphics[width=\textwidth]{./plots/Meson_Baryon_density_ratio_CTYang.jpg} \caption{Ratios of hadronic particle number densities as a function of temperature $150\MeV>T>5\MeV$ 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 $T=40\MeV,\ 20\MeV,\ 13\MeV,\ 5.6\MeV$ 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.} \label{EquilibPartRatiosFig} -%\end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -490,12 +479,10 @@ \subsection{Strangeness abundance}\label{sec:Strangeness} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] -%\begin{center} \centering \includegraphics[width=1.0\linewidth]{./plots/Strangeness_Hubble_CTYang_V2.jpg} \caption{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.} \label{reaction_time_tot} -%\end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -552,12 +539,10 @@ \subsection{Strangeness abundance}\label{sec:Strangeness} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] -\begin{center} \centering \includegraphics[width=\linewidth]{./plots/NewHyperonRate_CTYang.jpg} \caption{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 $T=40\MeV,\ 12.9\MeV$.} \label{Lambda_Rate_volume.fig} -\end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -632,7 +617,6 @@ \subsection{Muon abundance} \label{sec:Muons} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] -%\begin{center} \centering \includegraphics[width=0.9\columnwidth]{./plots/MuonRate_new2.jpg} \caption{The thermal reaction rate per volume for muon related reactions as a function of temperature adapted from~\cite{Rafelski:2021aey}. The dominant reaction rates for $\mu^\pm$ production are printed as follows: The $\gamma\gamma$ channel (blue dashed line), $e^{\pm}$ (red dashed line), the combined electromagnetic rate (pink solid line), and the charged pion decay 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 denoted by the dashed vertical black line at $T_{\rm dis}=4.2\MeV$.} @@ -661,7 +645,6 @@ \subsection{Muon abundance} \label{sec:Muons} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] -%\begin{center} \centering \includegraphics[width=0.9\columnwidth]{./plots/DensityRatio_new2.jpg} \caption{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 $n_{\mu^\pm}/n_\mathrm{B}(T_\mathrm{dis})\approx0.911$.} @@ -700,8 +683,6 @@ \subsection{Neutrino masses and oscillation} \label{sec:Neutrinos} 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} \begin{align}\label{NuOscillation} - %P_{\alpha\rightarrow\beta} - %=\left|\sum_{j}U^{\ast}_{\alpha j}U_{\beta j}\exp{\left(-i\frac{\Delta m^2_{kj}}{2E}L\right)}\right|^{2},\qquad\Delta m^2_{kj}\equiv{m^2_k-m^2_j} P_{\alpha\rightarrow\beta} =&\delta_{\alpha\beta}-4\sum_{iB_{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~\cite{Neronov:2010gir}. There are generally considered two possible origins~\cite{Widrow:2011hs,Vazza:2021vwy} 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 still require primordial seed fields at some point to act as catalyst. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -903,7 +880,7 @@ \subsection{Landau eigen-energies in cosmology}\label{sec:Landau} \label{XScale} 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)}\,. \end{alignat} 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. -%We note here one important difference between KGP and DP eigen-energies in the context of cosmology: The anomalous magnetic moment portion of the DP statistics is suppressed by $1/a(t)$ over cosmological time while the AMM contribution is preserved in the KGP model. That the Universe's expansion makes a distinction between $g=2$ magnetic moment and AMM for DP fermions appears as a rather artificial and nonphysical trait. While the suppression of AMM may often be small for particles such as electrons, this suppression is non-trivial for particles with large AMM values such as the proton. That cosmological redshift would push DP protons to be described by $g=2$ eign-energies in the non-relativistic limit counts as a malaise for the model and further strengthens our thinking that the KGP model is more appropriate for cosmological studies. + Following reasoning outlined in~\cite{rafelski2023study} and~\cite{Steinmetz:2018ryf} we will proceed using the KGP eigen-energies. Motivated by \req{XScale}, we can introduce a dimensionless cosmic magnetic scale which is frozen in the homogeneous case as \begin{alignat}{1} \label{Bo} b_{0}\equiv\frac{eB}{T^{2}}=\frac{eB\hbar c^{2}}{(k_{B}T)^{2}}\ \mathrm{(S.I)}\,, @@ -935,7 +912,7 @@ \subsection{Electron-positron statistical physics}\label{sec:Partition} \label{PartFuncB}\ln\mathcal{Z}_{tot}=\frac{2eBV}{(2\pi)^2}\sum_{\sigma}^{\pm1}\sum_{s}^{\pm1/2}\sum_{n=0}^\infty\int^\infty_{0}dp_z\left[\ln\left(1+\Upsilon_{\sigma}^{s}(x)e^{-\beta E_{n}^{s}}\right)\right]\,,\\ \label{Fugacity}\Upsilon_{\sigma}^{s}(x)=\gamma(x)\lambda_{\sigma}^{s}\,,\qquad\lambda_{\sigma}^{s}=e^{(\sigma\eta_{e}+s\eta_{s})/T}\,, \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 here 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 +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 \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} @@ -959,8 +936,9 @@ \subsection{Electron-positron statistical physics}\label{sec:Partition} \end{align} \req{lnZ} 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. -\subsection{Charge neutrality and chemical potential} -\noindent Here we explore the chemical potential of dense magnetized electron-positron plasma in the early Universe under the hypothesis of charge neutrality and entropy conservation. We focus on the temperature interval at the post-BBN temperature range $20\keV1$ in early Universe.} \label{Case2_fig} @@ -1032,14 +1008,13 @@ \subsection{Magnetization of the electron-positron plasma}\label{sec:Magnetizati \end{itemize} Using the cosmic magnetic scale parameter $b_0$ and chemical potential $\eta_e/T$ we solve the magnetization numerically. In \rf{Case2_fig}, we plot the magnetization $M/B$ as a function of temperature $T$ showing that the magnetization depends on the magnetic field $b_0$ strongly. This is because for a small magnetic field $b_0$ the dominant term in \req{Magnetization_001} and \req{Magnetization_002} is $xK_1(x)/b_0$. For a given $b_0$, the value of magnetization can be larger than the externally proscribed magnetic field, i.e. $M/B>1$ which shows the possibility that magnetic domains can be formed in the early Universe. - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Looking in the Cosmic Rear-view Mirror}\label{Summary} +\section{Looking in the Cosmic Rear-view Mirror}\label{sec:Summary} \noindent 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 was created along the path of the Universe's evolution, as most think, the previously discussed Sakharov conditions (see \rsec{sec:Timeline}) 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~\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{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,Yang:2023bot}. +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,Yang:2023bot}. -Bottom nonequilibrium is one among a few interesting results presented here bridging the temperature gap between QGP hadronization at temperature $T\simeq150\MeV$ and neutrino freeze-out. Specifically we shown {\bf persistence of:} +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:} \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 $T=5.6\MeV$. @@ -1048,9 +1023,7 @@ \section{Looking in the Cosmic Rear-view Mirror}\label{Summary} 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 $T\in 3.5-1\MeV$ (range spanning differing flavor freeze-out, chemical equilibria, and even variation in standard natural constants; see \rf{fig:freezeoutT}), the neutrinos freeze-out and decouple from the rest of the thermally active matter in the Universe. We consider neutrino decoupling condition as a function of elementary constants: If these constants were not all ``constant'' or significantly temperature dependent, a noticeable entropy flow of annihilating $e^{\pm}$ plasma into neutrinos could be present, generating additional so-called neutrino degrees of freedom. -We presented a detailed study of the evolving disappearance of the lightest antimatter, the positrons; we quantify the magnitude of the large positron abundance during and after Big Bang Nucleosynthesis (BBN), see \rf{Density_fig}. In fact the energy density of electron-positron plasma exceeds greatly that of baryonic matter during and following the BBN period with the last positrons vanishing from the Universe near temperature $T=20\keV$, see \rf{ratio_fig}. - -Looking forward, we note that some of the topics we explored deserve a more intense followup work: \\[-0.7cm] +We presented a detailed study of the evolving disappearance of the lightest antimatter, the positrons; we quantify the magnitude of the large positron abundance during and after Big Bang Nucleosynthesis (BBN), see \rf{Density_fig}. In fact the energy density of electron-positron plasma exceeds greatly that of baryonic matter during and following the BBN period with the last positrons vanishing from the Universe near temperature $T=20\keV$, see \rf{ratio_fig}. Looking forward, we note that some of the topics we explored deserve a more intense followup work: \begin{itemize} \item The study of matter baryogenesis in the context of bottom quarks chemical non-equilibrium persistence near to QGP hadronization; \item The impact of relatively dense $e^{\pm}$ plasma on BBN processes; @@ -1087,7 +1060,6 @@ \section{Looking in the Cosmic Rear-view Mirror}\label{Summary} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \reftitle{References} -%\bibliographystyle{ieeetr} \bibliography{refs-mdpi-final.bib} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document}