From 5ee67b901b3b0dc6874a978b35c919fca2e42ce3 Mon Sep 17 00:00:00 2001 From: Andrew James Steinmetz Date: Sun, 27 Aug 2023 10:26:54 -0700 Subject: [PATCH] revise round three --- plasma-partition.tex | 84 +++++++++++++++++++++++++------------------- 1 file changed, 48 insertions(+), 36 deletions(-) diff --git a/plasma-partition.tex b/plasma-partition.tex index 60f4c39..a1e231a 100644 --- a/plasma-partition.tex +++ b/plasma-partition.tex @@ -64,7 +64,7 @@ \begin{document} -\title{Matter-antimatter origin of cosmic magnetism: A first look} +\title{Matter-antimatter origin of cosmic magnetism} \author{Andrew Steinmetz\orc{\orcC}} \author{Cheng Tao Yang\orc{\orcB}} \author{Johann Rafelski\orc{\orcA}} @@ -74,10 +74,10 @@ \date{August 22, 2023} \begin{abstract} -We explore the hypothesis that the abundant presence of relativistic antimatter (positrons) in the primordial universe is responsible for the inter-galactic magnetic fields we observe in the universe today. We evaluate the magnetic properties of the 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. +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. \end{abstract} -\keywords{early universe cosmology, magnetization, electron-positron plasma, intergalactic magnetic fields} +\keywords{Magnetization in primordial universe, magnetic properties of relativistic electron-positron plasma, intergalactic magnetic fields} \maketitle @@ -85,9 +85,9 @@ \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 surprisingly 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 primordial mechanism. +\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. -We investigate the hypothesis that the observed intergalactic magnetic fields (IGMF) predate the recombination epoch. In this work, IGMF will refer to experimentally observed intergalactic fields of any origin while primordial magnetic fields (PMF) refers to fields generated via early universe processes possibly as far back as inflation. +We investigate the novel hypothesis that the observed intergalactic magnetic fields (IGMF) originate in the cosmic dense $e^+e^-$-plasma which predates considerably the observable recombination epoch. 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 by processes acting possibly as far back as inflation: in this work we evaluate the magnituspontanous magnetization in the dense relativistic $e^+e^-$-pair-plasma providing source of the PMF originating . In general a spontanous magnetization which are persistent and thus can cause the observed ubiquity of IGMF magnetic fields. We consider in this work a novel hypothesis that the magnetization of the early universe is driven by relativistic spin paramagnetism originating in the dense electron-positron $e^{+}e^{-}$ plasma shown in \rf{fig:densityratio} in terms of the antimatter (positron) abundance ratio to the prevailing baryon density. Our hypothesis coincidentally occurs around the era $(T\sim80\keV)$ of Big Bang Nucleosynthesis (BBN). Electrons and positrons (having the largest magnetic moments in nature) in the pre-BBN early universe could induce a magnetic field in presence of a tiny spin orientation polarization. Since electrons and positrons have opposite magnetic moments, the induced magnetization does not entail a net local spin density statistical average. @@ -125,7 +125,7 @@ \subsection{Short survey of observational cosmic magnetism} 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}. Such strong magnetic fields would then require that at least some portion of the IGMF arise from primordial sources that predate the formation of stars. Ref.~\cite{Jedamzik:2020krr} propose further that the presence of a magnetic field of ${\cal B}_{\rm PMF}\simeq0.1{\rm\ nG}$ could be sufficient to explain the Hubble tension. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Electron-positron abundance} +\section{Cosmic electron-positron plasma abundance} \label{sec: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. @@ -184,7 +184,7 @@ \section{Electron-positron abundance} An imbalance in polarization within a region of volume $V$ results in a nonzero spin 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Theory of thermal matter-antimatter plasmas} +\section{Theory of magnetized matter-antimatter plasmas} \label{sec:thermal} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent As the universe undergoes isotropic expansion, the temperature decreases adiabatically~\cite{Abdalla:2022yfr} and conserves entropy as @@ -261,7 +261,7 @@ \section{Theory of thermal matter-antimatter plasmas} 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Spin and spin-orbit partition functions} +\subsection{Unified treatment of para and diamagnetism} \label{sec:paradia} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent We will proceed in this section with the Boltzmann approximation for the limit where $T\lesssim m_e$. The partition function shown in equation \req{partition:1} can be rewritten removing the logarithm as @@ -357,7 +357,7 @@ \subsection{Spin and spin-orbit partition functions} 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 spin 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{Chemical potential} +\subsection{Charge chemical potential response} \label{sec:chem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent In presence of a magnetic field in the Boltzmann approximation, the charge neutrality condition \req{chargeneutrality} becomes @@ -391,8 +391,10 @@ \subsection{Chemical potential} 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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Electron-positron plasma magnetization} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{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 @@ -455,8 +457,7 @@ \section{Electron-positron plasma magnetization} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Relativistic paramagnetism} +\subsection{Magnetic response of electron-positron plasma} \label{sec:paramagnetism} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent In \rf{fig:magnet}, we plot the magnetization as given by \req{g2magplus} and \req{g2magminus} with the spin 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. @@ -481,7 +482,7 @@ \section{Relativistic paramagnetism} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Dependency on g-factor} +\subsection{g-factor balance between para and diamagnetism} \label{sec:gfac} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -513,7 +514,7 @@ \subsection{Dependency on g-factor} These integer and half-integer points represent when the two Landau towers of orbital levels match up exactly. Therefore, we propose a more natural transition between the spinless diamagnetic gas of $g=0$ and a paramagnetic gas is $g=1$. A more careful analysis is required to confirm this, but that our numerical value is close to unity is suggestive. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Magnetization per lepton} +\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 @@ -544,8 +545,10 @@ \subsection{Magnetization per lepton} 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 potential and ferromagnetism} +\section{Spin polarization and ferromagnetism} \label{sec:ferro} + +\subsection{Spin chemical potential} \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} @@ -573,8 +576,7 @@ \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 external primordial magnetic field $({\cal B}_{\rm PMF}=0)$ generated by some unrelated physics, but rather the $e^{+}e^{-}$ plasma itself is responsible for the field by virtue of spin polarization. - +\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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] \centering @@ -595,49 +597,59 @@ \subsection{Self-magnetization} 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. -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 where polarization states are extinguished rather than antimatter states. +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. In~\rf{fig:self} the bottom frame shows how positrons can persist longer in primordial plasma except for the case of the lower cosmic bound of $b_{0}=10^{-11}$ where there is sufficient electron density to maintain the magnetic flux. The ratio thereafter climbs back up towards unity as the electrons become too diluted requiring a resurgence of positrons. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{figure}[ht] +\begin{figure}%[ht] \centering - \includegraphics[width=0.45\textwidth]{plots/NumeberDensitySpin.png} - \caption{The number density $n_{e^{\pm}}$ of polarized electrons (top) and positrons (bottom) under the self-magnetization model for differing values of $b_{0}$} +\includegraphics[width=0.45\textwidth]{plots/ElectronDenisty_SpinChemicalPotential004.jpg}\\ +\includegraphics[width=0.45\textwidth]{plots/DensityRatioTest.jpg} + \caption{(top) The number density $n_{e^{\pm}}$ of polarized electrons and positrons under the self-magnetization model for differing values of $b_{0}$. (bottom) The ratio of electron states to positron states of opposite polarizations.} \label{fig:polarswap} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Matter inhomogeneities in the cosmic plasma} -\label{sec:inhomogeneous} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\noindent In general, an additional physical constraint is required to fully determine $\mu$ and $\eta$ simultaneously as both potentials have mutual dependency (see \rsec{sec:ferro}). We note that spin polarizations are not required to be in balanced within a single species to preserve angular momentum. For magnetized domains or finite volumes, boundary/surface conditions would also need to be considered. - -The CMB~\cite{Planck:2018vyg} indicates that the early universe was home to domains of slightly higher and lower baryon densities which resulted in the presence of galactic super-clusters, cosmic filaments, and great voids seen today. However, the CMB, as measured today, is blind to the localized inhomogeneities required for gravity to begin galaxy and supermassive black hole formation. - -Such acute inhomogeneities distributed like a dust~\cite{Grayson:2023flr} in the plasma would make the proton density sharply and spatially dependant $n_{p}\rightarrow n_{p}(x)$ which would directly affect the potentials $\mu(x)$ and $\eta(x)$ and thus the density of electrons and positrons locally. This suggests that $e^{+}e^{-}$ may play a role in the initial seeding of gravitationally collapsing systems. If the primordial plasma were home to such small localized magnetic domains, the associated nonzero local angular momentum within these domains would provide a natural mechanism for the formation of rotating galaxies today. - -Recent measurements by the James Webb Space Telescope (JWST)~\cite{Yan:2022sxd,adams2023discovery,arrabal2023spectroscopic} indicate that galaxy formation began surprisingly early at large redshift values of $z\gtrsim10$ within the first 500 million years of the universe requiring gravitational collapse to begin in a hotter environment than expected. Additionally the observation of supermassive black holes already present~\cite{CEERSTeam:2023qgy} in this same high redshift period (with millions of solar masses) indicates the need for exceptionally local high density regions in the early universe whose generation is not yet explained and likely need to exist long before the recombination epoch. +However, this low $T$-behavior will need further corroboration after Amp\'erian 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} \label{sec:conclusions} -\noindent Expanding on our work in~\cite{Rafelski:2023emw}, we explored the paramagnetic magnetization of the $e^{+}e^{-}$ thermal Fermi gas in the temperature range of the early universe between $2000\keV>T>20\keV$: At the lower limit far higher than the Sun's core temperature~\cite{Bahcall:2000nu} of $T_{\odot}=1.37\keV$ the last $e^{+}e^{-}$ pair disappeared. This era of the universe combines strong magnetic fields, high matter-antimatter density, and relatively high temperature well beyond reach of all laboratory and astrophysics environments. +\noindent This is an effort to interpret the IGMF of today as primordial fields where the $e^{+}e^{-}$ of the era displayed a paramagnetic response. Expanding on our work in~\cite{Rafelski:2023emw}, we explored the paramagnetic magnetization of the $e^{+}e^{-}$ thermal Fermi gas in the temperature range of the early universe between $2000\keV>T>20\keV$: At the lower limit (far higher than the Sun's core temperature~\cite{Bahcall:2000nu} of $T_{\odot}=1.37\keV$) the last $e^{+}e^{-}$ pair disappeared as seen in \rf{fig:densityratio}. This era of the universe as described in \rsec{sec:abundance} combines strong magnetic fields, high matter-antimatter density, and relatively high temperature. This domain is well beyond the reach of all present day laboratory and known astrophysics environments. -In this work we studied paramagnetic properties of the primordial universe $e^{+}e^{-}$ plasma. Our analysis shows that $e^{+}e^{-}$ plasma paramagnetic response is dominated by the varying abundance of electron-positron pairs and decreases with decreasing $T$ unlike laboratory cases where number of magnetic particles is constant. +In \rsec{sec:thermal} we consider $b_{0}$, the magnitude of the comoving magnetic field in the universe expressed in dimensionless units. The actual value is estimated between $10^{-3}>b_{0}>10^{-11}$ and once created is most likely a characteristic conserved property of expending universe. We recognize that both Landau diamagnetism and spin paramagnetism are relevant in the consideration of magnetization of a dense $e^+e^-$-plasma. The recognition of high temperature relevance of spin paramagnetism relies on high abundance of pairs. In theoretical treatment this accounted for introducing effective polarization mass $\tilde{m}$ in \req{effmass:2}. This allows separation of the spin portion of the relativistic partition function from the spin-orbital portion (Landau diamagnetism) part 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; see \rf{fig:chemicalpotential}. -Because $b_0$, the scaled magnetic field in the universe is unchanged, this implies conservation of magnetic flux. We showed how this can be realized introducing spin polarization. Spin polarization within the $e^{+}e^{-}$ pair plasma is characterized introducing a chemical potential. We obtained the required primordial degree of spin polarization necessary to understand today's IGMF. Our study demonstrates that the early universe only required a small asymmetry in spin polarization to produce required magnetic fields. +This novel approach to high temperature magnetization allows us in \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 and unlike laboratory cases where number of magnetic particles is constant decreases with decreasing $T$. However, in the domain of interest we determine in \rsec{sec:gfac} that magnetization is not sensitive to the anomalous magnetic moment of the electron. Considering magnetization as a function of $g$-factor we further find a transition seen in \rf{fig:gfac} between paramagnetic and diamagnetic gasses. + +%Because $b_0$, the scaled magnetic field in the universe is unchanged, this implies conservation of magnetic flux. The enhancement this has on + +REVISE In \rsec{sec:self} we explored spin asymmetry introducing asociated chemical potential. The per-lepton magnetization is shown in \rsec{sec:perlepton} and \rf{fig:momentperlepton}. We showed in \rsec{sec:ferro} how magnetization can be induced by introducing spin polarization via a novel spin fugacity. Forced spin polarization within the $e^{+}e^{-}$ pair plasma is characterized introducing a potential into the partition function. We obtained in \rsec{sec:self} the required primordial degree of spin polarization necessary to understand today's IGMF. Our study demonstrates that the early universe only required a small asymmetry in spin polarization to produce required magnetic fields. %Forward -Our results can be both improved and lead to extension of present day paradigms. Addressing first high temperature domain, an improvement on the here presented results would be to introduce a full inventory of particles including neutrinos before these disappear or decouple at lower temperatures and to do this using Fermi-Dirac and Bose-Einstein statistics instead of the Boltzmann approximation here employed. +Our results can be both improved and lead to extension of present day paradigms. Addressing first high temperature domain, an improvement on the here presented results would be to introduce a full inventory of particles including neutrinos before these disappear or decouple at lower temperatures and to do this using Fermi-Dirac and Bose-Einstein statistics instead of the Boltzmann approximation here employed. Beyond the $e^+e^-$-plasma of great interest is further exploration of the electromagnetic properties of the ultrarelativistic quark-$uds$-plasma consisting of pairs of up ($u$), down ($d$) and strange ($s$) quarks, where magnetization is dominated by the lightest and most charged up-quark. A connection from quark plasma to the $e^+e^-$ plasma requires understanding of the impact of the hadronization process on magnetization. We note that contribution of $\mu^+\mu^{-}$-plasma to magnetization is reduced by a factor $\simeq 200$ compared to $e^+e^-$ plasma due to $1/m$-magnetic moment behavior. Near to $T=80$\,keV just prior to BBN we have $4.47\times10^{8}$ $e^{+}e^{-}$ pairs per baryon and a magnetic field in range $10^{-4}$--$10^4$ Tesla. 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 presence of a primordial magnetic field impacts BBN. The $e^{+}e^{-}$ pair plasma has been considered before, for contemporary study see Ref.\,\cite{Grayson:2023flr}. Below $T=20$\,keV the universe contents inventory is dominated by electrons, protons, and $\alpha$-particles. In order to conserve in time 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 homogeneouse polarized $e^{+}e^{-}$ pair plasma universe into polarization domains along with substitution of Gilbertian magnetic dipole moment source of magnetic field by Amp{\`e}rian kinetic current curl response. + The effort to connect the bulk magnetization due to Gilbertian by Amp{\`e}rian magnetization sources generated through currents and inhomogeneous flows will require creation of kinetic matter flow model allowing for spin polarization. A mechanism could be found in the kinetic coupling between the two chemical potentials of the $e^{+}e^{-}$ plasma. The local charge neutrality than creates response of the baryon density to magnetization. This model could provide mechanism necessary to generate collapsing (and rotating) cosmic structure and galaxies. The consideration of such primordial magnetic flux related large scale structure is supported by the recognition of the very early large-scale structure, galactic, and supermassive black hole formation. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Matter inhomogeneities in the cosmic plasma} +\label{sec:inhomogeneous} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\noindent In general, an additional physical constraint is required to fully determine $\mu$ and $\eta$ simultaneously as both potentials have mutual dependency (see \rsec{sec:ferro}). We note that spin polarizations are not required to be in balanced within a single species to preserve angular momentum. For magnetized domains or finite volumes, boundary/surface conditions would also need to be considered. + +The CMB~\cite{Planck:2018vyg} indicates presence in the early universe of domains with slightly higher and lower baryon density. This is usually associated with the following development of galactic super-clusters, cosmic filaments, and great voids observed today. + +However, the CMB, studied at present day angular distribution is insensitive to the more localized inhomogeneities required for gravity to begin at very galaxy and supermassive black hole formation. The mechanism for the required inhomogeneity is very likely a novel one: Recent measurements by the James Webb Space Telescope (JWST)~\cite{Yan:2022sxd,adams2023discovery,arrabal2023spectroscopic} indicate that the galaxy formation began surprisingly early at a large redshift value of $z\gtrsim10$ within the first 500 million years of the universe. This is requiring gravitational collapse to begin in a hotter environment than expected. Additionally the observation of supermassive (with millions of solar masses) black holes already present~\cite{CEERSTeam:2023qgy} in this same high redshift period indicates the need for exceptionally local high density regions in the early universe. + + +whose generation is not yet explained and likely need to exist long before the recombination epoch. + +Such acute inhomogeneity distributed like a dust~\cite{Grayson:2023flr} in the plasma would make the proton density sharply and spatially dependant $n_{p}\rightarrow n_{p}(x)$ which would directly affect the potentials $\mu(x)$ and $\eta(x)$ and thus the density of electrons and positrons locally. This suggests that $e^{+}e^{-}$ may play a role in the initial seeding of gravitationally collapsing systems. If the primordial plasma were home to such small localized magnetic domains, the associated nonzero local angular momentum within these domains would provide a natural mechanism for the formation of rotating galaxies today. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \acknowledgments \label{sec:ack}