From d0350b1c7df5073b8b7a5e82b2cde4e1c28a0bae Mon Sep 17 00:00:00 2001 From: Andrew James Steinmetz Date: Mon, 13 Nov 2023 15:38:06 -0700 Subject: [PATCH] Updates from Overleaf --- plasma-partition.tex | 81 ++++++++++++++++++++++---------------------- 1 file changed, 40 insertions(+), 41 deletions(-) diff --git a/plasma-partition.tex b/plasma-partition.tex index 795c850..1048cbf 100644 --- a/plasma-partition.tex +++ b/plasma-partition.tex @@ -71,7 +71,7 @@ \affiliation{Department of Physics, The University of Arizona, Tucson, AZ 85721, USA} %\date{\today} -\date{November 7, 2023} +\date{November 13, 2023} \begin{abstract} 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 magnetic dipole moment paramagnetic properties of the very dense primordial electron-positron $e^{+}e^{-}$-plasma, and obtain in quantitative terms the relatively small magnitude of the $e^{+}e^{-}$ magnetic moment polarization asymmetry required to produce a consistent self-magnetization in the universe. @@ -92,15 +92,15 @@ \section{Introduction} \end{align} 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}. -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 a primordial magnetic field (PMF) ${\cal B}_{\rm PMF}\simeq0.1{\rm\ nG}$ 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 arise from primordial sources predating the formation of stars. The presence of a primordial magnetic field (PMF) ${\cal B}_{\rm PMF}\simeq0.1{\rm\ nG}$ according to Ref.~\cite{Jedamzik:2020krr} could be sufficient to explain the Hubble tension. We investigate the novel hypothesis that the observed IGMF originates in the large scale non-Amp{\`e}rian (i.e non-current sourced in the `Gilbertian' sense~\cite{Rafelski:2017hce}) PMFs created in the dense cosmic $e^{+}e^{-}$-pair plasma by magnetic dipole moment paramagnetism competing with Landau's diamagnetism. {\xblue We evaluate the Gilbertian magnetic properties of the very dense $e^{+}e^{-}$ cosmic matter-antimatter plasma. The abundance of $e^{+}e^{-}$ is considered in \rsec{sec:abundance} and its thermal properties in \rsec{sec:thermal}. We establish the Gilbertian (non-Amp{\`e}rian = non-current) magnetism present in the plasma in \rsec{sec:magnetization} and demonstrate in \rsec{sec:ferro} that the non-interacting plasma is non-ferromagnetic. We characterize that such ferromagnetism can occur if a tiny polarization asymmetry arises from residual interactions.} -{\xblue Since the Gilbertian and Amp{\`e}rian mechanisms of magnetization are distinct physical phenomena the resulting spectral decomposition of magnetic space domains in a cosmological context cannot be assumed as being the same; we return to this question, albeit briefly, in \rsec{sec:lengthscale}.} In fact our study of pre-recombination Gilbertian dipole moment magnetization of the $e^{+}e^{-}$-plasma is in part motivated by the difficulty in generating Amp{\`e}rian 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}. +{\xblue Since the Gilbertian and Amp{\`e}rian mechanisms of magnetization are distinct physical phenomena, the resulting spectral decomposition of magnetic domains in a cosmological context cannot be assumed as identical. We return to this question, albeit briefly, in \rsec{sec:lengthscale}.} In fact our study of pre-recombination Gilbertian dipole moment magnetization of the $e^{+}e^{-}$-plasma is in part motivated by the difficulty in generating Amp{\`e}rian 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}. -In our 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. +In our 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] @@ -111,15 +111,15 @@ \section{Introduction} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -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{\`e}rian (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 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{\`e}rian (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 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 $T_\mathrm{split}=20.3\keV$. 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 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 $T_\mathrm{split}=20.3\keV$. Since electrons and positrons have opposite magnetic moments, the magnetized dense $e^{+}e^{-}$-plasma entails negligible net statistical average local spin density. 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}, following hadronization of the quark-gluon plasma (QGP) and below about $T\!=\!100\,000\keV$, in terms of energy density the 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 $2000\keV>T>20\keV$ and focus on the range $200\keV>T>20\keV$ 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 density, the 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 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 $2000\keV>T>20\keV$ and focus on the range $200\keV>T>20\keV$ where the most rapid antimatter abundance changes occur 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} 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, in the latter case the number of magnetic dipoles is fixed, while in the universe the (comoving) number can vary with $T$. +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} describes 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; in the latter case the number of magnetic dipoles is fixed, while in the universe the (comoving) number can vary with $T$. -\rsec{sec:ferro} covers the consequences of forced magnetization via a magnetic moment 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. +\rsec{sec:ferro} covers the consequences of forced magnetization via a magnetic moment polarization 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 particle spin in cosmic magnetic fields. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Cosmic electron-positron plasma abundance} @@ -127,9 +127,9 @@ \section{Cosmic electron-positron plasma abundance} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent As the universe cooled below temperature $T=m_{e}$ (the electron mass), the thermal electron and positron comoving density depleted by over eight orders of magnitude. At $T_{\rm split}=20.3\keV$, the charged lepton asymmetry (mirrored by baryon asymmetry and enforced by charge neutrality) became evident as the surviving excess electrons persisted while positrons vanished entirely from the particle inventory of the universe due to annihilation. -The electron-to-baryon density ratio $n_{e^{-}}/n_{B}$ is shown in~\rf{fig:densityratio} as the solid blue line while the positron-to-baryon ratio $n_{e^{+}}/n_{B}$ is represented by the dashed red line. These two lines overlap until the temperature drops below $T_{\rm split}=20.3\keV$ as positrons vanish from the universe marking the end of the $e^{+}e^{-}$-plasma and the dominance of the electron-proton $(e^{-}p)$-plasma. The two vertical dashed green lines denote temperatures $T=m_{e}\simeq511\keV$ and $T_{\rm split}=20.3\keV$. These results were obtained using charge neutrality and the baryon-to-photon content (entropy) of the universe; see details in~\cite{Rafelski:2023emw}. The two horizontal black dashed lines denote the relativistic $T\gg m_e$ abundance of $n_{e^{\pm}}/n_{B}=4.47\times10^{8}$ and post-annihilation abundance of $n_{e^{-}}/n_{B}=0.87$. Above temperature $T\simeq85\keV$, the $e^{+}e^{-}$ primordial plasma density exceeded that of the Sun's core density $n_{e}\simeq6\times10^{26}{\rm\ cm}^{-3}$~\cite{Bahcall:2000nu}. +The electron-to-baryon density ratio $n_{e^{-}}/n_{B}$ is shown in~\rf{fig:densityratio} as the solid blue line while the positron-to-baryon ratio $n_{e^{+}}/n_{B}$ is represented by the dashed red line. These two lines overlap until the temperature drops below $T_{\rm split}=20.3\keV$ as positrons vanish from the universe. This marks the end of the $e^{+}e^{-}$-plasma and begins the dominance of the electron-proton $(e^{-}p)$-plasma. The two vertical dashed green lines denote temperatures $T=m_{e}\simeq511\keV$ and $T_{\rm split}=20.3\keV$. These results were obtained using charge neutrality and the baryon-to-photon content (entropy) of the universe; see details in~\cite{Rafelski:2023emw}. The two horizontal black dashed lines denote the relativistic $T\gg m_e$ abundance of $n_{e^{\pm}}/n_{B}=4.47\times10^{8}$ and the post-annihilation abundance of $n_{e^{-}}/n_{B}=0.87$. Above temperature $T\simeq85\keV$, the $e^{+}e^{-}$ primordial plasma density exceeded that of the Sun's core density $n_{e}\simeq6\times10^{26}{\rm\ cm}^{-3}$~\cite{Bahcall:2000nu}. -Conversion of the dense $e^{+}e^{-}$-pair plasma into photons reheated the photon background~\cite{Birrell:2014uka} separating the photon and neutrino temperatures. The $e^{+}e^{-}$ annihilation and photon reheating period lasted no longer than an afternoon lunch break. Because of charge neutrality, the post-annihilation comoving ratio $n_{e^{-}}/n_{B}=0.87$~\cite{Rafelski:2023emw} is slightly offset from unity in~\rf{fig:densityratio} by the presence of bound neutrons in $\alpha$ particles and other neutron containing light elements produced during BBN epoch. +Conversion of the dense $e^{+}e^{-}$-pair plasma into photons reheated the photon background~\cite{Birrell:2014uka} separating the photon and neutrino temperatures. The $e^{+}e^{-}$ annihilation and photon reheating period lasted no longer than an afternoon lunch break. Because of charge neutrality, the post-annihilation comoving ratio $n_{e^{-}}/n_{B}=0.87$~\cite{Rafelski:2023emw} is slightly offset from unity in~\rf{fig:densityratio} by the presence of bound neutrons in $\alpha$ particles and other neutron containing light elements produced during the BBN epoch. To obtain a quantitative description of the above evolution, we study the bulk properties of the relativistic charged/magnetic gasses in a nearly homogeneous and isotropic primordial universe via the thermal Fermi-Dirac or Bose distributions. For matter $(e^{-};\ \sigma=+1)$ and antimatter $(e^{+};\ \sigma=-1)$ particles, a nonzero relativistic chemical potential $\mu_{\sigma}=\sigma\mu$ is caused by an imbalance of matter and antimatter. While the primordial electron-positron plasma era was overall charge neutral, there was a small asymmetry in the charged leptons from baryon asymmetry~\cite{Fromerth:2012fe,Canetti:2012zc} in the universe. Reactions such as $e^{+}e^{-}\leftrightarrow\gamma\gamma$ constrains the chemical potential of electrons and positrons~\cite{Elze:1980er} as \begin{align} @@ -146,7 +146,7 @@ \section{Cosmic electron-positron plasma abundance} \end{align} In~\req{chargeneutrality}, $n_{p}$ is the observed total number density of protons in all baryon species. The parameter $V$ relays the proper volume under consideration and $\ln{\cal Z}_{e^{+}e^{-}}$ is the partition function for the electron-positron gas. The chemical potential defined in~\req{cpotential} is obtained from the requirement that the positive charge of baryons (protons, $\alpha$ particles, light nuclei produced after BBN) is exactly and locally compensated by a tiny net excess of electrons over positrons. -The abundance of baryons is itself fixed by the known abundance relative to photons~\cite{ParticleDataGroup:2022pth} and we employed the contemporary recommended value $n_B/n_\gamma=6.09\times 10^{-10}$. The resulting chemical potential needs to be evaluated carefully to obtain the behavior near to $T_{\rm split}=20.3\keV$ where the relatively small value of chemical potential $\mu$ rises rapidly so that positrons vanish from the particle inventory of the universe while nearly one electron per baryon remains. The detailed solution of this problem is found in Refs.\;\cite{Fromerth:2012fe,Rafelski:2023emw} leading to the results shown in~\rf{fig:densityratio}. These results are obtained allowing for Fermi-Dirac and Bose statistics, however it is often numerically sufficient to consider the Boltzmann distribution limit; see~\rsec{sec:paradia}. +The abundance of baryons is itself fixed by the known abundance relative to photons~\cite{ParticleDataGroup:2022pth} and we employed the contemporary recommended value $n_B/n_\gamma=6.09\times 10^{-10}$. The resulting chemical potential needs to be evaluated carefully to obtain the behavior near $T_{\rm split}=20.3\keV$, where the relatively small value of chemical potential $\mu$ rises rapidly so that positrons vanish from the particle inventory of the universe while nearly one electron per baryon remains. The detailed solution of this problem is found in Refs.\;\cite{Fromerth:2012fe,Rafelski:2023emw} leading to the results shown in~\rf{fig:densityratio}. These results are obtained allowing for Fermi-Dirac and Bose statistics, however it is often numerically sufficient to consider the Boltzmann distribution limit; see~\rsec{sec:paradia}. The partition function of the $e^{+}e^{-}$-plasma can be understood as the sum of four gaseous species \begin{align} @@ -165,20 +165,20 @@ \section{Cosmic electron-positron plasma abundance} \end{align} where $p_{z}$ is the momentum parallel to the field axis and electric charge is $e\equiv q_{e^{+}}=-q_{e^{-}}$. The index $\sigma$ in~\req{partition:1} is a sum over electron and positron states while $s$ is a sum over polarizations. The index $s$ refers to the spin along the field axis: parallel $(\uparrow;\ s=+1)$ or anti-parallel $(\downarrow;\ s=-1)$ for both particle and antiparticle species. -As the gas is electrically neutral, we will for the time being ignore charge-charge interactions. There is an additional deformation of the distribution from particle creation and destruction correlations; see Ch.~11 of~\cite{Letessier:2002ony} in the context of quark flavors. These will be not included as the considering volume is always large. The quantum numbers of the energy eigenstate $E$ will be elaborated on in~\rsec{sec:thermal}. +As the gas is electrically neutral, we will for the time being ignore charge-charge interactions. There is an additional deformation of the distribution from particle creation and destruction correlations; see Ch.~11 of~\cite{Letessier:2002ony} in the context of quark flavors. These will be not included as the considered volume is always large. The quantum numbers of the energy eigenstate $E$ will be elaborated in~\rsec{sec:thermal}. -We are explicitly interested in small asymmetries such as baryon excess over antibaryons, or one polarization over another. These are described by~\req{partition:2} as the following two fugacities: +We are explicitly interested in small asymmetries, such as baryon excess over antibaryons, or one polarization over another. These are described by~\req{partition:2} as the following two fugacities: \begin{itemize} \item[a.] Chemical fugacity $\lambda_{\sigma}$ \item[b.] Polarization fugacity $\xi_{\sigma,s}$ \end{itemize} -The chemical fugacity $\lambda_{\sigma}$ (defined in~\req{cpotential} above) describes deformation of the Fermi-Dirac distribution due to nonzero chemical potential $\mu$. An imbalance in electrons and positrons leads as discussed earlier to a nonzero particle chemical potential $\mu\neq0$. We then introduce a novel polarization fugacity $\xi_{\sigma,s}$ and polarization potential $\eta_{\sigma,s}=\sigma s\eta$. We propose the polarization potential follows analogous expressions as seen in~\req{cpotential} obeying +The chemical fugacity $\lambda_{\sigma}$ (defined in~\req{cpotential} above) describes deformation of the Fermi-Dirac distribution due to nonzero chemical potential $\mu$. An imbalance in electrons and positrons leads, as discussed earlier, to a nonzero particle chemical potential $\mu\neq0$. We then introduce a novel polarization fugacity $\xi_{\sigma,s}$ and polarization potential $\eta_{\sigma,s}=\sigma s\eta$. We propose the polarization potential follows analogous expressions as seen in~\req{cpotential} obeying \begin{align} \label{spotential} \eta\equiv\eta_{+,+}=\eta_{-,-}\,,\quad\eta=-\eta_{\pm,\mp}\,,\quad\xi_{\sigma,s}\equiv\exp{\frac{\eta_{\sigma,s}}{T}}\,. \end{align} -An imbalance in polarization within a region of volume $V$ results in a nonzero magnetic moment potential $\eta\neq0$. Conveniently since antiparticles have opposite sign of charge and magnetic moment, the same magnetic moment is associated with opposite spin orientation for particles and antiparticles independent of degree of spin-magnetization. A completely particle-antiparticle symmetric magnetized plasma will have therefore zero total angular momentum. This is of course very different from the situation today of a matter dominated universe. +An imbalance in polarization within a region of volume $V$ results in a nonzero magnetic moment potential $\eta\neq0$. Conveniently, since antiparticles have opposite sign of charge and magnetic moment, the same magnetic moment is associated with opposite spin orientation for particles and antiparticles independent of degree of spin-magnetization. A completely particle-antiparticle symmetric magnetized plasma will have therefore zero total angular momentum. This is of course very different from the plasma situations of today in our matter dominated universe. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Theory of magnetized matter-antimatter plasmas} @@ -251,7 +251,7 @@ \section{Theory of magnetized matter-antimatter plasmas} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -Since we address the temperature interval $200\keV>T>20\keV$ where the effects of quantum Fermi statistics on the $e^{+}e^{-}$-pair plasma are relatively small, but the gas is still considered relativistic, we will employ the Boltzmann approximation to the partition function in~\req{partition:1}. However, we extrapolate our results for presentation completeness up to $T\simeq 4m_{e}$. +Since we address the temperature interval $200\keV>T>20\keV$ where the effects of quantum Fermi statistics on the $e^{+}e^{-}$-pair plasma are relatively small, but the gas is still considered relativistic, we employ the Boltzmann approximation to the partition function in~\req{partition:1}. However, we extrapolate our results for presentation completeness up to $T\simeq 4m_{e}$. In general, modifications due to quantum statistical phase-space reduction for fermions are expected to suppress results by about 20\% in the extrapolated regions. We will continue to search for semi-analytical solutions for Fermi statistics in relativistic $e^{+}e^{-}$-pair gasses to compliment the Boltzmann solution offered here. @@ -334,9 +334,9 @@ \subsection{Unified treatment of para and diamagnetism} \label{xfunc} x_{s'}&=\frac{{\tilde m}_{s'}}{T}=\sqrt{\frac{m_{e}^{2}}{T^{2}}+b_{0}\left(1-\frac{g}{2}s'\right)}\,. \end{align} -The latter two terms in~\req{boltzmann} proportional to $b_{0}K_{1}$ and $b_{0}^{2}K_{0}$ are the uniquely magnetic terms present containing both spin and Landau orbital influences in the partition function. The $K_{2}$ term is analogous to the textbook-case of free Fermi gas~\cite{greiner2012thermodynamics}, being modified only by spin effects. +The latter two terms in~\req{boltzmann} proportional to $b_{0}K_{1}$ and $b_{0}^{2}K_{0}$ are the uniquely magnetic terms present containing both spin and Landau orbital influences in the partition function. The $K_{2}$ term is analogous to the textbook free Fermi gas~\cite{greiner2012thermodynamics} modified only by spin effects. -This \lq separation of concerns\rq\ can be rewritten as +This `separation of concerns' can be rewritten as \begin{align} \label{spin} \ln{\cal Z}_{\rm S}&=\frac{T^{3}V}{\pi^{2}}\sum_{s'}^{\pm1}\left[\xi_{s'}\cosh{\frac{\mu}{T}}\right]\left(x_{s'}^{2}K_{2}(x_{s'})\right)\,,\\ @@ -349,7 +349,7 @@ \subsection{Unified treatment of para and diamagnetism} where the spin (S) and spin-orbit (SO) partition functions can be considered independently. When the magnetic scale $b_{0}$ is small, the spin-orbit term~\req{spinorbit} becomes negligible leaving only paramagnetic effects in~\req{spin} due to spin. In the non-relativistic limit,~\req{spin} reproduces a quantum gas whose Hamiltonian is defined as the free particle (FP) Hamiltonian plus the magnetic dipole (MD) Hamiltonian which span two independent Hilbert spaces ${\cal H}_{\rm FP}\otimes{\cal H}_{\rm MD}$. -Writing the partition function as~\req{boltzmann} instead of~\req{partitionpower:1} has the additional benefit that the partition function remains finite in the free gas $({\cal B}\rightarrow0)$ limit. This is because the free Fermi gas and~\req{spin} are mathematically analogous to one another. As the Bessel $K_{\nu}$ functions are evaluated as functions of $x_{\pm}$ in~\req{xfunc}, the \lq free\rq\ part of the partition $K_{2}$ is still subject to dipole magnetization effects. In the limit where ${\cal B}\rightarrow0$, the free Fermi gas is recovered in both the Boltzmann approximation $k=1$ and the general case $\sum_{k=1}^{\infty}$. +Writing the partition function as~\req{boltzmann} instead of~\req{partitionpower:1} has the additional benefit that the partition function remains finite in the free gas $({\cal B}\rightarrow0)$ limit. This is because the free Fermi gas and~\req{spin} are mathematically analogous to one another. As the Bessel $K_{\nu}$ functions are evaluated as functions of $x_{\pm}$ in~\req{xfunc}, the `free' part of the partition $K_{2}$ is still subject to dipole magnetization effects. In the limit where ${\cal B}\rightarrow0$, the free Fermi gas is recovered in both the Boltzmann approximation $k=1$ and the general case $\sum_{k=1}^{\infty}$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Charge chemical potential response} @@ -361,9 +361,9 @@ \subsection{Charge chemical potential response} \sinh\frac{\mu}{T}=n_{p}\frac{\pi^{2}}{T^{3}}\\ \left[\sum_{s'}^{\pm1}\xi_{s'}\!\left(\!x_{s'}^{2}K_{2}(x_{s'})\!+\!\frac{b_{0}}{2}x_{s'}K_{1}(x_{s'})\!+\!\frac{b_{0}^{2}}{12}K_{0}(x_{s'}\!)\!\right)\!\right]^{-1}\!. \end{multline} -\req{chem} is fully determined by the right-hand-side expression if the magnetic moment fugacity is set to unity $\eta=0$ implying no external bias to the number of polarizations except as a consequence of the difference in energy eigenvalues. In practice, the latter two terms in~\req{chem} are negligible to chemical potential in the bounds of the primordial $e^{+}e^{-}$-plasma considered and only becomes relevant for extreme (see~\rf{fig:chemicalpotential}) magnetic field strengths well outside our scope. +\req{chem} is fully determined by the right-hand-side expression if the magnetic moment fugacity is set to unity $\eta=0$. This implies no external bias to the number of polarizations except as a consequence of the difference in energy eigenvalues. In practice, the latter two terms in~\req{chem} are negligible to chemical potential in the bounds of the primordial $e^{+}e^{-}$-plasma considered and only becomes relevant for extreme (see~\rf{fig:chemicalpotential}) magnetic field strengths well outside our scope. -\req{chem} simplifies if there is no external magnetic field $b_{0}=0$ into +\req{chem} simplifies in the limit of zero external magnetic field $b_{0}\rightarrow0$ into \begin{align} \label{simpchem:1} \sinh\frac{\mu}{T}=n_{p}\frac{\pi^{2}}{T^{3}}\left[2\cosh\frac{\eta}{T}\left(\frac{m_{e}}{T}\right)^{2}K_{2}\left(\frac{m_{e}}{T}\right)\right]^{-1}\,. @@ -378,7 +378,7 @@ \subsection{Charge chemical potential response} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -In~\rf{fig:chemicalpotential} we plot the chemical potential $\mu/T$ in~\req{chem} and~\req{simpchem:1} which characterizes the importance of the charged lepton asymmetry as a function of temperature. Since the baryon (and thus charged lepton) asymmetry remains fixed, the suppression of $\mu/T$ at high temperatures indicates a large pair density which is seen explicitly in~\rf{fig:densityratio}. The black line corresponds to the $b_{0}=0$ and $\eta=0$ case. +In~\rf{fig:chemicalpotential} we plot the chemical potential $\mu/T$ in~\req{chem} and~\req{simpchem:1} which characterizes the importance of the charged lepton asymmetry as a function of temperature. Since the baryon (and thus charged lepton) asymmetry remains fixed, the suppression of $\mu/T$ at high temperatures indicates a large pair density which is seen explicitly in~\rf{fig:densityratio}. The black line corresponds to the $b_{0}=0$ and $\eta=0$ cases. 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 magnetic moment potential until $\eta$ reaches a significant fraction of the electron mass $m_{e}$ in size. The chemical potential for large values of magnetic moment potential $\eta=\{100,\ 200,\ 300,\ 400,\ 500\}\,\keV$ are also plotted as dashed black lines with $b_{0}=0$. @@ -414,23 +414,22 @@ \section{Gilbertian magnetization of electron-positron plasma} \label{defmagetization} {\cal M}\equiv\frac{T}{V}\frac{\partial}{\partial{\cal B}}\ln{{\cal Z}_{e^{+}e^{-}}} = \frac{T}{V}\left(\frac{\partial b_{0}}{\partial{\cal B}}\right)\frac{\partial}{\partial b_{0}}\ln{{\cal Z}_{e^{+}e^{-}}}\,, \end{align} -Magnetization arising from other components in the cosmic gas (protons, neutrinos, etc.) could in principle also be included. Localized inhomogeneities of matter evolution are often non-trivial and generally be solved numerically using magneto-hydrodynamics (MHD)~\cite{melrose2008quantum,Vazza:2017qge,Vachaspati:2020blt,Stoneking:2020egj}. In the context of MHD, primordial magnetogenesis from fluid flows in the electron-positron epoch was considered in~\cite{Gopal:2004ut,Perrone:2021srr}. {\xblue We note in passing that the possible conservation of magnetic helicity~\cite{Boyarsky:2011uy} +Magnetization arising from other components in the cosmic gas (protons, neutrinos, etc.) could in principle also be included. Localized inhomogeneities and matter evolution are often non-trivial and generally are solved numerically using magneto-hydrodynamics (MHD)~\cite{melrose2008quantum,Vazza:2017qge,Vachaspati:2020blt,Stoneking:2020egj}. In the context of MHD, primordial magnetogenesis from fluid flows in the electron-positron epoch was considered in~\cite{Gopal:2004ut,Perrone:2021srr}. {\xblue We note in passing that the possible conservation of magnetic helicity~\cite{Boyarsky:2011uy} relates to current induced magnetic fields. We do not expect this conservation law to hold for our Gilbertian spin based magnetization. } - We introduce dimensionless units for magnetization ${\mathfrak M}$ by defining the critical field strength \begin{align} {\cal B}_{C}\equiv\frac{m_{e}^{2}}{e}\,,\qquad{\mathfrak M}\equiv\frac{\cal M}{{\cal B}_{C}}\,. \end{align} The scale ${\cal B}_{C}$ is where electromagnetism is expected to become subject to non-linear effects, though luckily in our regime of interest, electrodynamics should be linear. We note however that the upper bounds of IGMFs in~\req{igmf} (with $b_{0}=10^{-3}$; see~\req{tbscale}) brings us to within $1\%$ of that limit for the external field strength in the temperature range considered. -The total magnetization ${\mathfrak M}$ can be broken into the sum of magnetic moment parallel ${\mathfrak M}_{+}$ and magnetic moment anti-parallel ${\mathfrak M}_{-}$ contributions +The total magnetization ${\mathfrak M}$ can be broken into the sum of magnetic moment parallel ${\mathfrak M}_{+}$ and anti-parallel ${\mathfrak M}_{-}$ contributions \begin{align} \label{g2mag} {\mathfrak M}={\mathfrak M}_{+}+{\mathfrak M}_{-}\,. \end{align} -We note that the expression for the magnetization simplifies significantly for $g=2$ which is the \lq natural\rq\ gyro-magnetic factor~\cite{Evans:2022fsu,Rafelski:2022bsv} for Dirac particles. For illustration, the $g=2$ magnetization from~\req{defmagetization} is then +We note that the expression for the magnetization simplifies significantly for $g=2$ which is the `natural' gyro-magnetic factor~\cite{Evans:2022fsu,Rafelski:2022bsv} for Dirac particles. For illustration, the $g=2$ magnetization from~\req{defmagetization} is then \begin{align} \label{g2magplus} {\mathfrak M}_{+}&=\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\xi\cosh{\frac{\mu}{T}}\left[\frac{1}{2}x_{+}K_{1}(x_{+})+\frac{b_{0}}{6}K_{0}(x_{+})\right]\,,\\ @@ -458,13 +457,13 @@ \subsection{Magnetic response of electron-positron plasma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent In~\rf{fig:magnet}, we plot the magnetization as given by~\req{g2magplus} and~\req{g2magminus} with the magnetic moment potential set to unity $\xi=1$. The lower (solid red) and upper (solid blue) bounds for cosmic magnetic scale $b_{0}$ are included. The external magnetic field strength ${\cal B}/{\cal B}_{C}$ is also plotted for lower (dotted red) and upper (dotted blue) bounds. Since the derivative of the partition function governing magnetization may manifest differences between Fermi-Dirac and the here used Boltzmann limit more acutely, out of abundance of caution, we indicate extrapolation outside the domain of validity of the Boltzmann limit with dashes. -We see in~\rf{fig:magnet} that the $e^{+}e^{-}$-plasma is overall paramagnetic and yields a positive overall magnetization which is contrary to the traditional assumption that matter-antimatter plasma lack significant magnetic responses of their own in the bulk. With that said, the magnetization never exceeds the external field under the parameters considered which shows a lack of ferromagnetic behavior. +We see in~\rf{fig:magnet} that the $e^{+}e^{-}$-plasma is overall paramagnetic and yields a positive magnetization which is contrary to the traditional assumption that matter-antimatter plasma lack significant magnetic responses of their own in the bulk. With that said, the magnetization never exceeds the external field under the parameters considered which shows a lack of ferromagnetic behavior. -The large abundance of pairs causes the smallness of the chemical potential seen in~\rf{fig:chemicalpotential} at high temperatures. As the universe expands and temperature decreases, there is a rapid decrease of the density $n_{e^{\pm}}$ of $e^{+}e^{-}$-pairs. This is the primary the cause of the rapid paramagnetic decrease seen in~\rf{fig:magnet} above $T_\mathrm{split}=20.3\keV$. At lower temperatures $T<20.3\keV$ there remains a small electron excess (see~\rf{fig:densityratio}) needed to neutralize proton charge. These excess electrons then govern the residual magnetization and dilutes with cosmic expansion. +The large abundance of pairs causes the smallness of the chemical potential seen in~\rf{fig:chemicalpotential} at high temperatures. As the universe expands and temperature decreases and there is a rapid decrease of the density $n_{e^{\pm}}$ of $e^{+}e^{-}$-pairs. This is the primary cause of the rapid paramagnetic decrease seen in~\rf{fig:magnet} above $T_\mathrm{split}=20.3\keV$. At lower temperatures $T<20.3\keV$ there remains a small electron excess (see~\rf{fig:densityratio}) needed to neutralize proton charge. These excess electrons then govern the residual magnetization and dilutes with cosmic expansion. -An interesting feature of~\rf{fig:magnet} is that the magnetization in the full temperature range increases as a function of temperature. This is contrary to Curie's law~\cite{greiner2012thermodynamics} which stipulates that paramagnetic susceptibility of a laboratory material is inversely proportional to temperature. However, Curie's law applies to systems with fixed number of particles which is not true in our situation; see~\rsec{sec:perlepton}. +An interesting feature of~\rf{fig:magnet} is that the magnetization in the full temperature range increases as a function of temperature. This is contrary to Curie's law~\cite{greiner2012thermodynamics} which stipulates that paramagnetic susceptibility of a laboratory material is inversely proportional to temperature. However, Curie's law applies to systems with a fixed number of particles which is not true in our situation; see~\rsec{sec:perlepton}. -A further consideration is possible hysteresis as the $e^{+}e^{-}$ density drops with temperature. It is not immediately obvious the gas's magnetization should simply \lq degauss\rq\ so rapidly without further consequence. If the very large paramagnetic susceptibility present for $T\simeq m_{e}$ is the origin of an overall magnetization of the plasma, the conservation of magnetic flux through the comoving surface ensures that the initial residual magnetization is preserved at a lower temperature by Faraday induced kinetic flow processes however our model presented here cannot account for such effects. Some consequences of enforced magnetization are considered in~\rsec{sec:ferro}. +A further consideration is possible hysteresis as the $e^{+}e^{-}$ density drops with temperature. It is not immediately obvious the gas's magnetization should simply `degauss' so rapidly without further consequence. If the very large paramagnetic susceptibility present for $T\simeq m_{e}$ is the origin of an overall magnetization of the plasma, the conservation of magnetic flux through the comoving surface ensures that the initial residual magnetization is preserved at a lower temperature by Faraday induced kinetic flow processes. However our model presented here cannot account for such effects. Some consequences of enforced magnetization are considered in~\rsec{sec:ferro}. Early universe conditions may also apply to some extreme stellar objects with rapid change in $n_{e^{\pm}}$ with temperatures above $T_\mathrm{split}=20.3\keV$. Production and annihilation of $e^{+}e^{-}$-plasmas is also predicted around compact stellar objects~\cite{Ruffini:2009hg,Ruffini:2012it} potentially as a source of gamma-ray bursts (GRB). @@ -513,12 +512,12 @@ \subsection{g-factor balance between para and diamagnetism} \subsection{Laboratory versus the relativistic electron-positron-universe} \label{sec:perlepton} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\noindent Despite the relatively large magnetization seen in~\rf{fig:magnet}, the average contribution per lepton is only a small fraction of its overall magnetic moment indicating the magnetization is only loosely organized. Specifically, the magnetization regime we are in is described by +\noindent Despite the relatively large magnetization seen in~\rf{fig:magnet}, the average contribution per lepton is only a small fraction of its overall magnetic moment indicating the magnetization is loosely organized. Specifically, the magnetization regime we are in is described by \begin{align} \label{fractionalmagnetization} {\cal M}\ll\mu_{B}\frac{N_{e^{+}}+N_{e^{-}}}{V}\,,\qquad\mu_{B}\equiv\frac{e}{2m_{e}}\,, \end{align} -where $\mu_{B}$ is the Bohr magneton and $N=nV$ is the total particle number in the proper volume V. To better demonstrate that the plasma is only weakly magnetized, we define the average magnetic moment per lepton given by along the field ($z$-direction) axis as +where $\mu_{B}$ is the Bohr magneton and $N=nV$ is the total particle number in the proper volume V. To better demonstrate that the plasma is only weakly magnetized, we define the average magnetic moment per lepton along the field ($z$-direction) axis as \begin{align} \label{momentperlepton} \vert\vec{m}\vert_{z}\equiv\frac{{\cal M}}{n_{e^{-}}+n_{e^{+}}}\,,\qquad\vert\vec{m}\vert_{x}=\vert\vec{m}\vert_{y}=0\,. @@ -534,7 +533,7 @@ \subsection{Laboratory versus the relativistic electron-positron-universe} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -The average magnetic moment $\vert\vec{m}\vert_{z}$ defined in~\req{momentperlepton} is plotted in~\rf{fig:momentperlepton} which displays how essential the external field is on the \lq per lepton\rq\ magnetization. Both the $b_{0}=10^{-11}$ (lower plot, red curve) and $b_{0}=10^{-3}$ (upper plot, blue curve) cosmic magnetic scale bounds are plotted in the Boltzmann approximation. The dashed lines indicate where this approximation is only qualitatively correct. For illustration, a constant magnetic field case (solid green line) with a comoving reference value chosen at temperature $T_{0}=10\keV$ is also plotted. +The average magnetic moment $\vert\vec{m}\vert_{z}$ defined in~\req{momentperlepton} is plotted in~\rf{fig:momentperlepton} which displays how essential the external field is for `per lepton' magnetization. Both the $b_{0}=10^{-11}$ (lower plot, red curve) and $b_{0}=10^{-3}$ (upper plot, blue curve) cosmic magnetic scale bounds are plotted in the Boltzmann approximation. The dashed lines indicate where this approximation is only qualitatively correct. For illustration, a constant magnetic field case (solid green line) with a comoving reference value chosen at temperature $T_{0}=10\keV$ is also plotted. If the field strength is held constant, then the average magnetic moment per lepton is suppressed at higher temperatures as expected for magnetization satisfying Curie's law. The difference in~\rf{fig:momentperlepton} between the non-constant (red and blue solid curves) case and the constant field (solid green curve) case demonstrates the importance of the conservation of primordial magnetic flux in the plasma, required by~\req{bscale}. @@ -547,7 +546,7 @@ \section{Magnetic moment polarization and ferromagnetism} \subsection{Magnetic moment chemical potential} \label{sec:spinpot} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\noindent Up to this point, we have neglected the impact that a nonzero magnetic moment 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 magnetic moment fugacity $\xi$ asymmetry generated by the magnetic moment potential $\eta$ yielding up to first order ${\cal O}(b_{0})$ in magnetic scale +\noindent Up to this point, we have neglected the impact that a nonzero polarization 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 polarization fugacity $\xi$ generated by the polarization potential $\eta$ yielding up to first order ${\cal O}(b_{0})$ in magnetic scale \begin{multline} \label{ferro} \lim_{m_{e}^{2}/T^{2}\gg b_0}{\mathfrak M}=\frac{g}{2}\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\sinh{\frac{\eta}{T}}\cosh{\frac{\mu}{T}}\left[\frac{m_{e}}{T}K_{1}\left(\frac{m_{e}}{T}\right)\right]\\ @@ -555,17 +554,17 @@ \subsection{Magnetic moment chemical potential} +{\cal O}\left(b_{0}^{2}\right) \end{multline} -Given~\req{ferro}, we can understand the magnetic moment potential as a kind of \lq ferromagnetic\rq\ influence on the primordial gas which allows for magnetization even in the absence of external magnetic fields. This interpretation is reinforced by the fact the leading coefficient is $g/2$. +Given~\req{ferro}, we can understand the polarization potential as a kind of `ferromagnetic' influence on the primordial gas which allows for magnetization even in the absence of external magnetic fields. This interpretation is reinforced by the fact that the leading coefficient is $g/2$. We suggest that a variety of physics could produce a small nonzero $\eta$ within a domain of the gas. Such asymmetries could also originate statistically as while the expectation value of free gas polarization is zero, the variance is likely not. -As $\sinh{\eta/T}$ is an odd function, the sign of $\eta$ also controls the alignment of the magnetization. In the high temperature limit~\req{ferro} with strictly $b_{0}=0$ assumes a form of to lowest order for brevity +As $\sinh{\eta/T}$ is an odd function, the sign of $\eta$ also controls the alignment of the magnetization. In the high temperature limit~\req{ferro}, with strictly $b_{0}=0$, is to lowest order for brevity \begin{align} \label{hiTferro} \lim_{m_{e}/T\rightarrow0}{\mathfrak M}\vert_{b_{0}=0}=\frac{g}{2}\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\frac{\eta}{T}\,, \end{align} -While the limit in~\req{hiTferro} was calculated in only the Boltzmann limit, it is noteworthy that the high temperature (and $m\rightarrow0$) limit of Fermi-Dirac distributions only differs from the Boltzmann result by a proportionality factor. +While~\req{hiTferro} was calculated in only the Boltzmann limit, it is noteworthy that the high temperature (and $m\rightarrow0$) limit of Fermi-Dirac distributions only differs from the Boltzmann result by a proportionality factor. The natural scale of the $e^{+}e^{-}$ magnetization with only a small magnetic moment fugacity ($\eta<1\eV$) fits easily within the bounds of the predicted magnetization during this era if the IGMF measured today was of primordial origin. The reason for this is that the magnetization seen in~\req{g2magplus},~\req{g2magminus} and~\req{ferro} are scaled by $\alpha{\cal B}_{C}$ where $\alpha$ is the fine structure constant. @@ -573,8 +572,8 @@ \subsection{Magnetic moment chemical potential} \subsection{Self-magnetization} \label{sec:self} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\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 the $({\cal B}_{\rm PMF}\ne 0)$ field by virtue of magnetic moment 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 magnetic moment polarization. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] \centering @@ -589,7 +588,7 @@ \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 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. +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 maintained by scattering within the gas (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. @@ -607,7 +606,7 @@ \subsection{Self-magnetization} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -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 magnetic dipole moment 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. +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 magnetic dipole moment 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. {\xblue %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%