From d99d63af12af7bbe5b1c4a1f65aa443379bafb88 Mon Sep 17 00:00:00 2001 From: Andrew James Steinmetz Date: Mon, 28 Aug 2023 05:38:43 -0700 Subject: [PATCH] final draft before arxiv --- plasma-partition.tex | 81 +++++++++++++++++++++----------------------- 1 file changed, 38 insertions(+), 43 deletions(-) diff --git a/plasma-partition.tex b/plasma-partition.tex index 855aaf9..623cd45 100644 --- a/plasma-partition.tex +++ b/plasma-partition.tex @@ -74,7 +74,7 @@ \date{August 22, 2023} \begin{abstract} -We explore the hypothesis that the abundant presence of relativistic antimatter (positrons) in the primordial universe is the source of the inter-galactic magnetic fields we observe in the universe today. We evaluate both Landau diamagnetic and spin paramagnetic properties of the very dense primordial electron-positron $e^{+}e^{-}$-plasma, and obtain in quantitative terms the magnitude of the relatively small $e^{+}e^{-}$ polarization asymmetry required to produce a consistent self-magnetization in the Universe. +We explore the hypothesis that the abundant presence of relativistic antimatter (positrons) in the primordial universe is the source of the intergalactic magnetic fields we observe in the universe today. We evaluate both Landau diamagnetic and spin paramagnetic properties of the very dense primordial electron-positron $e^{+}e^{-}$-plasma, and obtain in quantitative terms the magnitude of the relatively small $e^{+}e^{-}$ magnetic moment polarization asymmetry required to produce a consistent self-magnetization in the universe. \end{abstract} \keywords{Magnetization in primordial universe, magnetic properties of relativistic electron-positron plasma, intergalactic magnetic fields} @@ -85,39 +85,37 @@ \section{Introduction} \label{sec:introduction} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\noindent Macroscopic domains of magnetic fields have been found around compact objects (stars, planets, etc.); between stars; within galaxies; between galaxies in clusters; and in deep extra-galactic void spaces. Considering this ubiquity of magnetic fields in the universe~\cite{Giovannini:2017rbc,Giovannini:2003yn,Kronberg:1993vk}, we search for a common cosmic primordial mechanism considering the electron-positron $e^{+}e^{-}$-pair plasma~\cite{Rafelski:2023emw,Grayson:2023flr}.We investigate the novel hypothesis that the observed intergalactic magnetic fields (IGMF) originate in the large scale non-Amp\'erian (i.e non-current that is `Gilbertian'~\cite{Rafelski:2017hce}) primordial magnetic fields (PMF) created in the dense cosmic $e^+e^-$-pair-plasma by spin paramagnetism competing with Landau's diamag netism. -% Is this really needed?: In this work, IGMF will refer to experimentally observed intergalactic fields of any origin while primordial magnetic fields (PMF) refers to fields generated in primordial universe prior to recombination - -The bounds for intergalactic magnetic fields IGMF at a length scale of $1{\rm\ Mpc}$ are today~\cite{Neronov:2010gir,Taylor:2011bn,Pshirkov:2015tua,Jedamzik:2018itu,Vernstrom:2021hru} +\noindent Macroscopic domains of magnetic fields have been found around compact objects (stars, planets, etc.); between stars; within galaxies; between galaxies in clusters; and in deep extra-galactic void spaces. The bounds for intergalactic magnetic fields (IGMF) at a length scale of $1{\rm\ Mpc}$ are today~\cite{Neronov:2010gir,Taylor:2011bn,Pshirkov:2015tua,Jedamzik:2018itu,Vernstrom:2021hru} \begin{align} \label{igmf} 10^{-8}{\rm\ G}>\mathcal{B}_{\rm IGMF}>10^{-16}{\rm\ G}\,. \end{align} -Our study of pre-recombination Gilbertian magnetization of the $e^+e^-$-antimatter plasma is also motivated by the difficulty in generating Amp\'erian PMFs with the large coherent length scales implied by IGMF~\cite{Giovannini:2022rrl} though currently the length scale for PMFs are not well constrained either~\cite{AlvesBatista:2021sln}. The conventional elaboration of the origins for cosmic PMFs are detailed in~\cite{Gaensler:2004gk,Durrer:2013pga,AlvesBatista:2021sln}. +Considering the ubiquity of magnetic fields in the universe~\cite{Giovannini:2017rbc,Giovannini:2003yn,Kronberg:1993vk}, we search for a common cosmic primordial mechanism considering the electron-positron $e^{+}e^{-}$-pair plasma~\cite{Rafelski:2023emw,Grayson:2023flr}: We investigate the novel hypothesis that the observed IGMF originates in the large scale non-Amp\'erian (i.e non-current sourced in the `Gilbertian' sense~\cite{Rafelski:2017hce}) primordial magnetic fields (PMF) created in the dense cosmic $e^{+}e^{-}$-pair plasma by spin paramagnetism competing with Landau's diamagnetism. + +Our study of pre-recombination Gilbertian spin magnetization of the $e^{+}e^{-}$-plasma is also motivated by the difficulty in generating Amp\'erian PMFs with large coherent length scales implied by the IGMF~\cite{Giovannini:2022rrl}, though currently the length scale for PMFs are not well constrained either~\cite{AlvesBatista:2021sln}. The conventional elaboration of the origins for cosmic PMFs are detailed in~\cite{Gaensler:2004gk,Durrer:2013pga,AlvesBatista:2021sln}. -Faraday rotation from distant radio active galaxy nuclei (AGN)~\cite{Pomakov:2022cem} suggest that neither dynamo nor astrophysical processes would sufficiently account for the presence of magnetic fields in the universe today if the IGMF strength was around the upper bound of ${\cal B}_{\rm IGMF}\simeq30-60{\rm\ nG}$ as found in Ref.\,\cite{Vernstrom:2021hru}. Presence of magnetic fields of this magniture would then require that at least some portion of the IGMF to arise from primordial sources predating the formation of stars. The presence of ${\cal B}_{\rm PMF}\simeq0.1{\rm\ nG}$ according to Ref.\,\cite{Jedamzik:2020krr} could be sufficient to explain the Hubble tension. +Faraday rotation from distant radio active galaxy nuclei (AGN)~\cite{Pomakov:2022cem} suggest that neither dynamo nor astrophysical processes would sufficiently account for the presence of magnetic fields in the universe today if the IGMF strength was around the upper bound of ${\cal B}_{\rm IGMF}\simeq30-60{\rm\ nG}$ as found in Ref.~\cite{Vernstrom:2021hru}. The presence of magnetic fields of this magnitude would then require that at least some portion of IGMFs to arise from primordial sources predating the formation of stars. The presence of ${\cal B}_{\rm PMF}\simeq0.1{\rm\ nG}$ according to Ref.~\cite{Jedamzik:2020krr} could be sufficient to explain the Hubble tension. -In this work our focus is to establish the Gilbertian (non-Amp\'erian=non-current) magnetic properties of the very dense $e^+e^-$ cosmic matter-antimatter plasma: The magnetization of the early universe requires a large density of strong magnetic dipoles. Due to their large magnetic moment ($\propto 1/m_e$) electrons and positrons magnetically dominate the Universe. The dense electron-positron $e^{+}e^{-}$-plasma characterized in~\rf{fig:densityratio}: We show the antimatter (positron) abundance as ratio to the prevailing baryon density as a function of cosmic photon temperature $T$. In this work we measure $T$ in units of energy (keV) thus we set the Boltzmann constant to $k_{B}=1$. We consider all results in temporal sequence in the expanding universe, thus we begin with high $T$ and early times on the left in~\rf{fig:densityratio} and end at lower $T$ and later times on the right. +In this work our focus is to establish the Gilbertian (non-Amp\'erian = non-current) magnetic properties of the very dense $e^{+}e^{-}$ cosmic matter-antimatter plasma. In this framework, the magnetization of the early universe requires a large density of strong magnetic dipoles. Due to their large magnetic moment ($\propto e/m_e$) electrons and positrons magnetically dominate the universe. The dense $e^{+}e^{-}$-plasma is characterized in~\rf{fig:densityratio}: We show the antimatter (positron) abundance as a ratio to the prevailing baryon density as a function of cosmic photon temperature $T$. In this work we measure $T$ in units of energy (keV) thus we set the Boltzmann constant to $k_{B}=1$. We consider all results in temporal sequence in the expanding universe, thus we begin with high $T$ and early times on the left in~\rf{fig:densityratio} and end at lower $T$ and later times on the right. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] \centering -\includegraphics[width=0.48\textwidth]{plots/EEPlasmaDensityRatio_new01.jpg} +\includegraphics[width=0.45\textwidth]{plots/EEPlasmaDensityRatio_new01.jpg} \caption{Number density of electron $e^{-}$ and positron $e^{+}$ to baryon ratio $n_{e^{\pm}}/n_{B}$ as a function of photon temperature in the universe. See text in~\rsec{sec:abundance} for further details.} \label{fig:densityratio} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -We evaluate the polarization required for PMF-magnitude of the spontaneous Gilbertian magnetization. Magnetic flux persistence implies that once $e^+e^-$-pair-plasma fades out, the ambient large scale Gilbertian magnetic field is maintained by the induced Amp\'erian (current) sources arising in the residual $e^-p^+\alpha^{++}$-plasma ultimately leading to the observed large scale structure IGMF. +We evaluate the magnetic moment polarization required for PMF magnitude of the spontaneous Gilbertian magnetization. Magnetic flux persistence implies that once the $e^{+}e^{-}$-pair plasma fades out, the ambient large scale Gilbertian magnetic field is maintained by the induced Amp\'erian (current) sources arising in the residual $e^{-}p^{+}\alpha^{++}$-plasma ultimately leading to the observed large scale structure IGMF. -As we see in~\rf{fig:densityratio} at $T>m_ec^2=511\keV$ the $e^{+}e^{-}$-pair abundance was nearly 450 million pairs per baryon, dropping to about 100 million pairs per baryon at the pre-BBN temperature $T=100\keV$. Pairs persist down to $T=21$\,keV, above the number of $e+e^-$-pairs is large compared to the residual `unpaired' electrons neutralizing the baryon charge locally. Since electrons and positrons have opposite magnetic moments, the magnetized dense $e^+e^-$-plasma entails negligible net local spin density in statistical average, the residual very, very small spin polarization of unpaired electrons complements the magnetic field induced polarization of the proton component. +As we see in~\rf{fig:densityratio} at $T>m_ec^2=511\keV$ the $e^{+}e^{-}$-pair abundance was nearly 450 million pairs per baryon, dropping to about 100 million pairs per baryon at the pre-BBN temperature of $T=100\keV$. The number of $e^{+}e^{-}$-pairs is large compared to the residual `unpaired' electrons neutralizing the baryon charge locally down to $T=21\keV$. Since electrons and positrons have opposite magnetic moments, the magnetized dense $e^{+}e^{-}$-plasma entails negligible net local spin density in statistical average. The residual very small polarization of unpaired electrons complements the magnetic field induced polarization of the proton component. -As shown in Fig.\,2 in Ref.\,\cite{Rafelski:2023emw} below about $T=100\,000$\,keV in terms of energy density following hadronization of QGP the Universe in its first hour consists of photons, neutrinos and the $e^+e^-$-pair plasma. Massive dark matter and dark energy are negligible. While we study the polarization of $e^{+}e^{-}$-plasma we do not address here its origin. However, we recall that the pair plasma decouples from the neutrino background near to $T=2000\keV$~\cite{Birrell:2014uka}. Therefore we consider the magnetic properties of the $e^{+}e^{-}$-pair plasma in the temperature range $2000\keV>T>21\keV$ and focus on the range $200\keV>T>21\keV$ where the most rapid antimatter abundance changes occurs and where the Boltzmann approximation is valid. This is notably the final epoch where antimatter exists in large quantities in the cosmos~\cite{Rafelski:2023emw}. +As shown in Fig.\,2 in Ref.~\cite{Rafelski:2023emw}, following hadronization of the quark-gluon plasma (QGP) and below about $T\!=\!100\,000\keV$, in terms of energy densitythe early universe's first hour consists of photons, neutrinos and the $e^{+}e^{-}$-pair plasma. Massive dark matter and dark energy are negligible during this era. While we study the magnetic moment polarization of $e^{+}e^{-}$-plasma we do not address here its origin. However, we recall that the pair plasma decouples from the neutrino background near to $T=2000\keV$~\cite{Birrell:2014uka}. Therefore we consider the magnetic properties of the $e^{+}e^{-}$-pair plasma in the temperature range $2000\keV>T>21\keV$ and focus on the range $200\keV>T>21\keV$ where the most rapid antimatter abundance changes occurs and where the Boltzmann approximation is valid. This is notably the final epoch where antimatter exists in large quantities in the cosmos~\cite{Rafelski:2023emw}. -The abundance of antimatter shown in~\rf{fig:densityratio} is obtained and discussed in more detail in~\rsec{sec:abundance}. Our analysis in~\rsec{sec:thermal} of the relativistic Fermion partition function focuses on the Gilbertian spin and Landau paramagnetic contribution to magnetization. We show in~\rsec{sec:magnetization} accounting for the matter-antimatter asymmetry present in the universe that magnetization is nonzero. +The abundance of antimatter shown in~\rf{fig:densityratio} is obtained and discussed in more detail in~\rsec{sec:abundance}. Our analysis in~\rsec{sec:thermal} the four relativistic fermion gases (particle and antiparticle and both polarizations) where the spin and spin-orbit contributions are evaluated in~\rsec{sec:paradia}. The influence of magnetization on the charge chemical potential is determined in~\rsec{sec:chem}. We show in~\rsec{sec:magnetization}, accounting for the matter-antimatter asymmetry present in the universe, that magnetization is nonzero. Our description of relativistic paramagnetism is covered in~\rsec{sec:paramagnetism}. The balance between paramagnetic and diamagnetic response is evaluated as a function of particle gyromagnetic ratio in~\rsec{sec:gfac}. The per-lepton magnetization is examined in~\rsec{sec:perlepton} distinguishing between cosmic and laboratory cases. -Our description of relativistic paramagnetism is covered in~\rsec{sec:paramagnetism}. We further demonstrate in~\rsec{sec:ferro} that magnetization can be spontaneously increased in strength near the IGMF upper limit seen in~\req{igmf} given sufficient spin polarization. {\color{red} Andrew - maybe a few more phrases about content here? Sections IVC (g-factor), IVD (counterintuitve plasma per particle magnetization) are not addressed and Section V could be broken into A (chem spin pot) and B (self-magnetization). } Our findings are summarized in~\rsec{sec:conclusions} and a wealth of future follow-up projects described, mostly depending on introduction of transport theory that accounts for spin of particles in presence of a magnetic field. -{\color{red} Andrew - Note I moved the following paragraphs to end of manuscript, they break the flow.} +\rsec{sec:ferro} covers the consequences of forced magnetization via a spin polarization chemical potential. We find in~\rsec{sec:spinpot} that magnetization can be spontaneously increased in strength near the IGMF upper limit seen in~\req{igmf} given sufficient magnetic moment polarization. A model of self-magnetization is explored in~\rsec{sec:self} which indicates the need for flux conserving currents at low temperatures. Our findings are summarized in~\rsec{sec:conclusions}. We also suggest and a wealth of future follow-up projects mostly depending on introduction of transport theory that accounts for spin of particles in presence of a magnetic field. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Cosmic electron-positron plasma abundance} @@ -322,7 +320,7 @@ \subsection{Unified treatment of para and diamagnetism} \sum_{n=0}^{\infty}W_{1}(n)=\int_{0}^{\infty}{\rm d}n\,W_{1}(n)+\frac{1}{2}\left[W_{1}(\infty)+W_{1}(0)\right]\\ +\frac{1}{12}\left[\frac{\partial W_{1}}{\partial n}\bigg\vert_{\infty}-\frac{\partial W_{1}}{\partial n}\bigg\vert_{0}\right]+{\cal R} \end{multline} -where ${\cal R}$ is the resulting power series and error remainder of the integration defined in terms of Bernoulli polynomials. Euler-Maclaurin integration is rarely convergent, and in this case serves only as an approximation within the domain where the error remainder is small and bounded; see Ref.\,\cite{greiner2012thermodynamics} for the non-relativistic case. In this analysis, we keep the zeroth and first order terms in the Euler-Maclaurin formula. We note that regularization of the excess terms in~\req{eulermaclaurin} is done in the context of strong field QED~\cite{greiner2008quantum} though that is outside our scope. +where ${\cal R}$ is the resulting power series and error remainder of the integration defined in terms of Bernoulli polynomials. Euler-Maclaurin integration is rarely convergent, and in this case serves only as an approximation within the domain where the error remainder is small and bounded; see Ref.~\cite{greiner2012thermodynamics} for the non-relativistic case. In this analysis, we keep the zeroth and first order terms in the Euler-Maclaurin formula. We note that regularization of the excess terms in~\req{eulermaclaurin} is done in the context of strong field QED~\cite{greiner2008quantum} though that is outside our scope. After truncation of the series and error remainder and combining~\req{partitionpower:1} through~\req{eulermaclaurin}, the partition function can then be written in terms of modified Bessel $K_{\nu}$ functions of the second kind, yielding \begin{align} @@ -382,14 +380,12 @@ \subsection{Charge chemical potential response} The para-diamagnetic contribution from~\req{spinorbit} does not appreciably influence $\mu/T$ until the magnetic scales involved become incredibly large well outside the observational bounds defined in~\req{igmf} and~\req{tbscale} as seen by the dotted blue curves of various large values $b_{0}=\{25,\ 50,\ 100,\ 300\}$. The chemical potential is also insensitive to forcing by the spin potential until $\eta$ reaches a significant fraction of the electron mass $m_{e}$ in size. The chemical potential for large values of spin potential $\eta=\{100,\ 200,\ 300,\ 400,\ 500\}\,\keV$ are also plotted as dashed black lines with $b_{0}=0$. -It is interesting to note that there are crossing points where a given chemical potential can be described as either an imbalance in spin-polarization or presence of external magnetic field. While spin potential suppresses the chemical potential at low temperatures, external magnetic fields only suppress the chemical potential at high temperatures. +It is interesting to note that there are crossing points where a given chemical potential can be described as either an imbalance in magnetic moment polarization or presence of external magnetic field. While spin potential suppresses the chemical potential at low temperatures, external magnetic fields only suppress the chemical potential at high temperatures. The profound insensitivity of the chemical potential to these parameters justifies the use of the free particle chemical potential (black line) in the ranges of magnetic field strength considered for cosmology. Mathematically this can be understood as $\xi$ and $b_{0}$ act as small corrections in the denominator of~\req{chem} if expanded in powers of these two parameters. -\section{Gilbertian magnetic properties of electron-positron plasma} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Gilbertian magnetization of electron-positron plasma} +\section{Gilbertian magnetization of electron-positron plasma} \label{sec:magnetization} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent The total magnetic flux within a region of space can be written as the sum of external fields and the magnetization of the medium via @@ -446,7 +442,7 @@ \subsection{Gilbertian magnetization of electron-positron plasma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] \centering - \includegraphics[width=0.48\textwidth]{plots/Magnetization_Hc_new004.png} + \includegraphics[width=0.45\textwidth]{plots/Magnetization_Hc_new004.png} \caption{The magnetization ${\mathfrak M}$, with $g=2$, of the primordial $e^{+}e^{-}$-plasma is plotted as a function of temperature.} \label{fig:magnet} \end{figure} @@ -471,7 +467,7 @@ \subsection{Magnetic response of electron-positron plasma} \begin{figure}[ht] \centering \includegraphics[width=0.45\textwidth]{plots/GFactor_05.png} - \caption{The magnetization $\mathfrak M$ as a function of $g$-factor plotted for several temperatures with magnetic scale $b_{0}=10^{-3}$ and spin polarization fugacity $\xi=1$.} + \caption{The magnetization $\mathfrak M$ as a function of $g$-factor plotted for several temperatures with magnetic scale $b_{0}=10^{-3}$ and polarization fugacity $\xi=1$.} \label{fig:gfac} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -540,10 +536,12 @@ \subsection{Laboratory versus the relativistic electron-positron-universe} While not shown, if~\rf{fig:momentperlepton} was extended to lower temperatures, the magnetization per lepton of the constant field case would be greater than the non-constant case which agrees with our intuition that magnetization is easier to achieve at lower temperatures. This feature again highlights the importance of flux conservation in the system and the uniqueness of the primordial cosmic environment. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Spin polarization and ferromagnetism} +\section{Magnetic moment polarization and ferromagnetism} \label{sec:ferro} - +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Spin chemical potential} +\label{sec:spinpot} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent Up to this point, we have neglected the impact that a nonzero spin potential $\eta\neq0$ (and thus $\xi\neq1$) would have on the primordial $e^{+}e^{-}$-plasma magnetization. In the limit that $(m_{e}/T)^2\gg b_0$ the magnetization given in~\req{arbg:1} and~\req{arbg:2} is entirely controlled by the spin fugacity $\xi$ asymmetry generated by the spin potential $\eta$ yielding up to first order ${\cal O}(b_{0})$ in magnetic scale \begin{multline} \label{ferro} @@ -571,12 +569,12 @@ \subsection{Self-magnetization} \label{sec:self} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\noindent One exploratory model we propose is to fix the spin polarization asymmetry, described in~\req{spotential}, to generate a homogeneous magnetic field which dissipates as the universe cools down. In this model, there is no pre-existing external primordial magnetic field generated by some unrelated physics, but rather the $e^{+}e^{-}$-plasma itself is responsible for the creation of $({\cal B}_{\rm PMF}\ne 0)$ field by virtue of spin polarization. +\noindent One exploratory model we propose is to fix the magnetic moment polarization asymmetry, described in~\req{spotential}, to generate a homogeneous magnetic field which dissipates as the universe cools down. In this model, there is no pre-existing external primordial magnetic field generated by some unrelated physics, but rather the $e^{+}e^{-}$-plasma itself is responsible for the creation of $({\cal B}_{\rm PMF}\ne 0)$ field by virtue of spin polarization. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] \centering \includegraphics[width=0.45\textwidth]{plots/Spinchemical_03.png} - \caption{The spin potential $\eta$ and chemical potential $\mu$ are plotted under the assumption of self-magnetization through a nonzero spin polarization in bulk of the plasma.} + \caption{The spin potential $\eta$ and chemical potential $\mu$ are plotted under the assumption of self-magnetization through a nonzero magnetic moment polarization in bulk of the plasma.} \label{fig:self} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -586,11 +584,11 @@ \subsection{Self-magnetization} \label{selfmag} {\mathfrak M}(b_{0})=\frac{{\cal M}(b_0)}{{\cal B}_{C}}\longleftrightarrow\frac{\cal B}{{\cal B}_{C}}=b_{0}\frac{T^{2}}{m_{e}^{2}}\,, \end{align} -which sets the total magnetization as a function of itself. The spin polarization described by $\eta\rightarrow\eta(b_{0},T)$ then becomes a fixed function of the temperature and magnetic scale. The underlying assumption would be the preservation of the homogeneous field would be maintained by scattering within the gas (as it is still in thermal equilibrium) modulating the polarization to conserve total magnetic flux. +which sets the total magnetization as a function of itself. The magnetic moment polarization described by $\eta\rightarrow\eta(b_{0},T)$ then becomes a fixed function of the temperature and magnetic scale. The underlying assumption would be the preservation of the homogeneous field would be maintained by scattering within the gas (as it is still in thermal equilibrium) modulating the polarization to conserve total magnetic flux. The result of the self-magnetization assumption in~\req{selfmag} for the potentials is plotted in~\rf{fig:self}. The solid lines indicate the curves for $\eta/T$ for differing values of $b_{0}=\{10^{-11},\ 10^{-7},\ 10^{-5},\ 10^{-3}\}$ which become dashed above $T=300\keV$ to indicate that the Boltzmann approximation is no longer appropriate though the general trend should remain unchanged. -The dotted lines are the curves for the chemical potential $\mu/T$. At high temperatures we see that a relatively small $\eta/T$ is needed to produce magnetization owing to the large densities present.~\rf{fig:self} also shows that the chemical potential does not deviate from the free particle case until the spin polarization becomes sufficiently high which indicates that this form of self-magnetization would require the annihilation of positrons to be incomplete even at lower temperatures. +The dotted lines are the curves for the chemical potential $\mu/T$. At high temperatures we see that a relatively small $\eta/T$ is needed to produce magnetization owing to the large densities present.~\rf{fig:self} also shows that the chemical potential does not deviate from the free particle case until the magnetic moment polarization becomes sufficiently high which indicates that this form of self-magnetization would require the annihilation of positrons to be incomplete even at lower temperatures. This is seen explicitly in~\rf{fig:polarswap} where we plot the numerical density of particles as a function of temperature for spin aligned $(+\eta)$ and spin anti-aligned $(-\eta)$ species for both positrons $(-\mu)$ and electrons $(+\mu)$. Various self-magnetization strengths are also plotted to match those seen in~\rf{fig:self}. The nature of $T_{\rm split}$ changes under this model since antimatter and polarization states can be extinguished separately. Positrons persist where there is insufficient electron density to maintain the magnetic flux. Polarization asymmetry therefore appears physical only in the domain where there is a large number of matter-antimatter pairs. @@ -607,38 +605,35 @@ \subsection{Self-magnetization} The low $T$-behavior of~\rf{fig:polarswap} will need further corroboration after Amp{\'e}rian currents as a source of magnetic field are incorporated: The Gilbertian sources, here spin paramagnetism and Landau diamagnetism, may not be dominant magnetic sources when $e^{+}e^{-}$-pairs are of comparable number to the residual electron and proton abundance. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Summary and Outlook} +\section{Summary and discussion} \label{sec:conclusions} -\noindent This work is an effort to interpret the intergalactic magnetic fields of today as originating in primordial fields generated in the first hour of universe existence. In~\rsec{sec:thermal} we reminded of the comoving in expanding universe magnitude, $b_{0}$, of the magnetic field expressed in dimensionless units estimated between $10^{-3}>b_{0}>10^{-11}$. We believe considering conservation of magnetic flux that $b_{0}$ once created is most likely a conserved property of the expanding universe. - -We have demonstrated that the $e^{+}e^{-}$-pair plasma is an appropriate non-electrical current candidate source for the primordial field: it is very dense, made of particles with highest magnetic moment, and displays a strong paramagnetic response. Therefore expanding on our work in~\cite{Rafelski:2023emw}, we explored its paramagnetic magnetization in the early universe temperature range between $2000\keV>T>20\keV$. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\noindent This work is an effort to interpret the intergalactic magnetic fields of today as originating in primordial fields generated in the first hour of the universe's existence. In~\rsec{sec:abundance} have demonstrated that the $e^{+}e^{-}$-pair plasma is an appropriate non-electrical current candidate source for the primordial field: It is (a) very dense, (b) made of particles with highest magnetic moment in nature, (c) and displays a strong paramagnetic response. Therefore expanding on our work in~\cite{Rafelski:2023emw}, we explored its paramagnetic magnetization in the early universe temperature range between $2000\keV>T>20\keV$. -Antiparticles $(e^{+})$ have the opposite sign of charge, and thus magnetic moment, compared to particles $(e^{-})$. Therefore in an $e^{+}e^{-}$-pair plasma, net magnetization can be associated with opposite spin orientations for particles and antiparticles without the accompaniment of a net angular momentum in the volume considered. This is of course very different from the the matter dominated universe arising below $T\simeq20\keV$ which includes the current epoch. +In~\rsec{sec:thermal} we define the comoving scale, $b_{0}$, of the magnetic field expressed in dimensionless units estimated between $10^{-3}>b_{0}>10^{-11}$. We believe considering conservation of magnetic flux that $b_{0}$ once created is most likely a conserved property of the expanding universe. Antiparticles $(e^{+})$ have the opposite sign of charge, and thus magnetic moment, compared to particles $(e^{-})$. Therefore in an $e^{+}e^{-}$-pair plasma, net magnetization can be associated with opposite spin orientations for particles and antiparticles without the accompaniment of a net angular momentum in the volume considered. This is of course very different from the the matter dominated universe arising below $T\simeq20\keV$ which includes the current epoch. The $e^{+}e^{-}$-pair plasma environment is well beyond the reach of all present day laboratory and known astrophysical environments. As seen in~\rf{fig:densityratio} the lower temperature limit, where the last $e^{+}e^{-}$-pair disappeared, is 15 times the Sun's core temperature~\cite{Bahcall:2000nu} $T_{\odot}=1.37\keV$. Laboratory conditions to explore our results depend on presence of $e^{+}e^{-}$-pair abundance which in turn depends on sufficiently stable thermal photon content. -Both Landau diamagnetism and spin paramagnetism are relevant in the analysis of dense $e^{+}e^{-}$-plasma Gilbertian (non-current) magnetization. The high temperature relevance of spin paramagnetism relies on the high abundance of pairs. In the theoretical treatment of~\rsec{sec:thermal}, this is accounted for by introducing effective polarization mass $\tilde{m}$ in~\req{effmass:2}. This allows for the separation of the spin portion of the relativistic partition function from the spin-orbital portion (Landau diamagnetism) in~\rsec{sec:paradia}. An expression valid in the Boltzmann limit of the gas is presented. In~\rsec{sec:chem} we determined the effect of magnetism on the chemical potential; +Both Landau diamagnetism and spin paramagnetism are relevant in the analysis of dense $e^{+}e^{-}$-plasma Gilbertian (non-current) magnetization. The high temperature relevance of spin paramagnetism relies on the high abundance of pairs. In the theoretical treatment of~\rsec{sec:thermal}, this is accounted for by introducing effective polarization mass $\tilde{m}$ in~\req{effmass:2}. This allows for the separation of the spin portion of the relativistic partition function from the spin-orbital portion (Landau diamagnetism) in~\rsec{sec:paradia}. In~\rsec{sec:chem} we determined the effect of magnetism on the chemical potential; see~\rf{fig:chemicalpotential}. -This novel approach to high temperature magnetization allows using~\rsec{sec:magnetization} to show that the $e^{+}e^{-}$-plasma paramagnetic response (see~\req{g2magplus} and~\req{g2magminus}) is dominated by the varying abundance of electron-positron pairs, exponentially decreasing with decreasing $T$ for $T150\,000$\,keV, the up-quark has the largest natural charge-to-mass $e/m$ ratio among elementary particles. A connection from the quark-gluon plasma to the $e^{+}e^{-}$-plasma then requires understanding of the impact of the hadronization process on magnetization, and vice-verse, consideration of hadronization as a source of the magnetizatin mechanism. We note that the contribution of $\mu^{+}\mu^{-}$-plasma to magnetization is reduced by a factor $\simeq 200$ compared to $e^{+}e^{-}$-plasma due to the $\propto e/m_\mu$ behavior of magnetic moments. We also note that the complex neutrino decoupling process near to $T=2\,000$\,keV should be explored as a source of magnetization mechanism. +Beyond $e^{+}e^{-}$-plasma, the quark-gluon plasma at $T>150\,000\keV$ is also of great interest. The up-quark has the largest natural charge-to-mass $e/m$ ratio among elementary particles besides the electron. A connection from the quark-gluon plasma to the $e^{+}e^{-}$-plasma then requires understanding of the impact of the hadronization process on magnetization, and vice-verse, a consideration of hadronization as a magnetization mechanism. We note that the contribution of $\mu^{+}\mu^{-}$-plasma to magnetization is reduced by a factor $\simeq 200$ compared to $e^{+}e^{-}$-plasma due to the $\propto e/m_{\mu}$ behavior of magnetic moments. We also note that the complex neutrino decoupling process near to $T=2\,000\keV$ should be explored as a source of magnetization mechanism. -Near to $T=80\keV$ just prior to BBN we have $4.47\times10^{8}$ $e^{+}e^{-}$-pairs per baryon and a primordial magnetic field in range of $10^{9}-10^{1}$ Gauss. BBN thus occurs in an environment as different as can be imagined from the empty space network of nuclear reactions explored. Our work creates the question in what way the presence of a primordial magnetic field could have impacted BBN and vice-verse, if BBN could provide the mechanism for spontaneous magnetization. The $e^{+}e^{-}$-pair impact is already being considered~\cite{Grayson:2023flr}. - -Below $T=20\keV$ the universe's particle inventory is dominated by electrons, protons, and $\alpha$-particles. In order to conserve the magnetic flux originating in the polarized homogeneous $e^{+}e^{-}$-pair plasma a very different model will need to be developed allowing for fragmentation of the homogeneous plasma universe into polarization domains along with the substitution of Gilbertian magnetic dipole moment sources of magnetic fields by Amp{\`e}rian kinetic current curl response. The effort to connect the bulk magnetization due to discrete dipoles by Amp{\`e}rian magnetization generated through currents and inhomogeneous flows will require study of transport equations allowing for spin polarization. +Near to $T=80\keV$ just prior to BBN we have $4.47\times10^{8}$ $e^{+}e^{-}$-pairs per baryon and a primordial magnetic field in range of $10^{9}-10^{1}$ Gauss. BBN thus occurs in an environment as different as can be imagined from the empty space network of nuclear reactions explored. Our work creates the question in what way the presence of a primordial magnetic field could have impacted BBN and vice-verse, if BBN could provide the mechanism for spontaneous magnetization. The $e^{+}e^{-}$-pair impact is already being considered~\cite{Grayson:2023flr}. -% TO FAR REACHING: A mechanism could be found in the kinetic coupling between the two (spin and particle) chemical potentials of the $e^{+}e^{-}$-plasma. Local charge neutrality then creates a response of the baryon density to magnetization. This model could provide a mechanism necessary to generate collapsing (and rotating) cosmic structure and galaxies. +Below $T=20\keV$ the universe's particle inventory is dominated by electrons, protons, and $\alpha$-particles. In order to conserve the magnetic flux originating in the polarized homogeneous $e^{+}e^{-}$-pair plasma a very different model will need to be developed allowing for fragmentation of the homogeneous plasma universe into polarization domains and evolving Amp{\`e}rian kinetic current curl responses. The effort to connect the bulk magnetization due to discrete dipoles by Amp{\`e}rian magnetization generated through currents and inhomogeneous flows will require study of transport equations allowing for magnetic moment polarization. -Recent measurements by the James Webb Space Telescope (JWST)~\cite{Yan:2022sxd,adams2023discovery,arrabal2023spectroscopic} indicate that galaxy formation began with maturing galaxies already present at a large redshift value of $z\gtrsim10$ within the first 500 million years of the universe. This requires gravitational collapse to begin in a hotter environment. Additionally the observation of supermassive (with millions of solar masses) black holes already present~\cite{CEERSTeam:2023qgy} in this same high redshift era are indicating the need for exceptionally small-scale high mass density regions in the early universe. There is a natural mechanism present in our consideration needed to create the above condition: As the Universe evolved the rapid $10^{8}$ drop in $e^{+}e^{-}$ abundance within the temperature range $100\keV>T>21\keV$ shown in~\rf{fig:densityratio} could be inducing dynamical currents preserving (comoving) magnetic flux in the emerging $p^+\alpha^{++}e^{-}$-plasma and in turn generate vortex seeds for small scale baryonic matter localization which could support anisotropies in the cosmic microwave background (CMB)~\cite{Jedamzik:2013gua,Abdalla:2022yfr}. +Recent measurements by the James Webb Space Telescope (JWST)~\cite{Yan:2022sxd,adams2023discovery,arrabal2023spectroscopic} indicate that maturing galaxies already present at a large redshift value of $z\gtrsim10$ within the first 500 million years of the universe. This requires gravitational collapse to begin earlier in a hotter environment. Additionally the observation of supermassive (with millions of solar masses) black holes already present~\cite{CEERSTeam:2023qgy} in this same high redshift era indicate the need for exceptionally small-scale high mass density regions in the early universe. There is a natural mechanism present in our work needed to create the above condition: As the universe evolved, the rapid $10^{8}$ drop in $e^{+}e^{-}$ abundance within the temperature range $100\keV>T>21\keV$ shown in~\rf{fig:densityratio} could be inducing dynamical currents preserving (comoving) magnetic flux in the emerging $p^{+}\alpha^{++}e^{-}$-plasma and in turn generate vortex seeds for small scale baryonic matter localization which could support anisotropies in the cosmic microwave background (CMB)~\cite{Jedamzik:2013gua,Abdalla:2022yfr}. -We believe that this exploration of the magnetization of $e^+e^{-}$-plasma provides a novel and credible proposal for interpretation and exploration of magnetic fields penetrating in ubiquitous manner the Universe. This work shows that the feromagnetic and diamagnetic $e^+e^{-}$-plasma properties may play a pivotal r\^ole in understanding the primordial universe. \\ +To conclude: This work shows that the paramagnetic and diamagnetic $e^{+}e^{-}$-plasma properties may play a pivotal role in understanding the primordial universe. In particular we have shown that the possible self-magnetization of the cosmic $e^{+}e^{-}$-plasma provides a novel and credible proposal for interpretation and exploration of magnetic fields in the universe. \acknowledgments