From 18e4881b608e7b62f2f59970cd14d6ab2b947017 Mon Sep 17 00:00:00 2001 From: Andrew James Steinmetz Date: Tue, 7 Nov 2023 17:08:45 -0700 Subject: [PATCH] Revisions continued. --- plasma-partition.tex | 104 ++++----------------------- plots/Fluctuation_MagneticMoment.jpg | Bin 0 -> 164558 bytes plots/Fluctuation_Spin.jpg | Bin 0 -> 162063 bytes refs-plasma-partition.bib | 37 ++++++++++ 4 files changed, 49 insertions(+), 92 deletions(-) create mode 100644 plots/Fluctuation_MagneticMoment.jpg create mode 100644 plots/Fluctuation_Spin.jpg diff --git a/plasma-partition.tex b/plasma-partition.tex index 4b0c064..18338f0 100644 --- a/plasma-partition.tex +++ b/plasma-partition.tex @@ -1,6 +1,5 @@ %\documentclass[aps,prd,floatfix]{revtex4-2} \documentclass[aps,prd,floatfix,reprint]{revtex4-2} -% ANDREW: Change this above back to move to two-column style. % Leave uncommented if the LaTeX file is uploaded to arXiv.org \pdfoutput=1 \pdfminorversion=7 @@ -95,11 +94,11 @@ \section{Introduction} 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. -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}) PMFs created in the dense cosmic $e^{+}e^{-}$-pair plasma by magnetic dipole moment paramagnetism competing with Landau's diamagnetism. +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\'erian = 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 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\'erian mechanisms of magnetization are distinct 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 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\'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}. +{\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}. 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. @@ -112,7 +111,7 @@ \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\'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{\`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. @@ -415,7 +414,7 @@ \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}. In the context of MHD, primordial magnetogenesis from fluid flows in the electron-positron epoch was considered in~\cite{Gopal:2004ut,Perrone:2021srr}. +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}. We introduce dimensionless units for magnetization ${\mathfrak M}$ by defining the critical field strength \begin{align} @@ -609,97 +608,18 @@ \subsection{Self-magnetization} {\xblue %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Magnetization coherent length scale and fluctuations} +\subsection{Macroscopic magnetization length scale and statistical fluctuations} \label{sec:lengthscale} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\noindent It is of interest to consider the coherent length scale $\lambda_\mathfrak{M}$ of the Gilbertian induced magnetization in the $e^{+}e^{-}$ medium and the size of thermal fluctuations $\langle(\Delta\mathfrak{M})^{2}\rangle$ present. This is a first look at characterizing these quantities and further effort is required in the future. As mentioned prior in \rsec{sec:introduction}, we expect that the two different mechanisms for magnetogenesis produce different spectra of magnetic fields across differing length scales. +\noindent It is of interest to consider in the $e^{+}e^{-}$ medium the spatial length scale $\lambda_\mathcal{M}$ over which the dipole induced magnetization is constant, referred in literature on Amp{\`e}rian magnetization as coherence length. Similarly, we are interested to understand the thermal fluctuations $\langle(\Delta\mathcal{M})^{2}\rangle$ present. As mentioned prior in \rsec{sec:introduction}, we expect that +The two different mechanisms for magnetogenesis (through Amp{\`e}rian matter currents, or through Gilbertian magnetic moment alignment produce different spectra of magnetic fields across differing length scales. Moreover prior Amp{\`e}rian work was considering ionic plasma and not $e^{+}e^{-}$ -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{General limitations} -\label{sec:limitations} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\noindent In principle there are two field scales, the one associated with support of `external' field which we call $\lambda_B$ and the other related to the possibly spontaneously occurring magnetization in the plasma, $\lambda_\mathfrak{M}$. Should the spontaneous and external fields be the same these scales and associated spectra would be the same. Literature in general refers to $\lambda_B$ and in following discussion we address this quantity. - -The observational restriction on the length scale of both Gilbertian and Amperian type IGMF is not well constrained~\cite{Giovannini:2022rrl,Durrer:2013pga,AlvesBatista:2021sln}, a range of coherence lengths $\lambda_{B}\sim10^{-2}-10^{3}$ Mpc for the field strengths were considered. Inhomogeneous PMFs of $\lambda_{B}\lesssim400$ pc are subject to dissipating effects~\cite{Jedamzik:1999bm}. Inhomogeneous field dissipation would affect not only the external PMFs, but also the induced magnetization as well restricting the spectrum of fluctuations. However, length scales above $\lambda_{B}\gtrsim10^{3}$ are not disallowed, if generated during an sufficiently early epoch such as inflation, but would require that the coherence of the PMF is beyond the size of the present day visible universe. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Magnetic thermal field fluctuations} -\label{sec:fluc} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\noindent We return to consider \req{defmagetization} which originates in the inherent extensive magnetic moment $\tilde\mu$. The statistical average of the magnetization density $\mathcal{M}$ of the quantum system is defined as -\begin{align} -\langle V \mathcal M\rangle\equiv \langle \tilde\mu\rangle = T\frac{\partial \ln\mathcal Z}{\partial B}\;. -\end{align} -In statistical mechanics, the mean-square fluctuation of any extensive quantity including magnetic moment $\tilde\mu$ can be written as a second derivative in the conjugate variable, here $\beta B$ -\begin{align} -\langle\Delta \tilde\mu^2\rangle=\langle \tilde\mu^2\rangle-\langle \tilde\mu\rangle^2=T^2\frac{\partial^2 \ln\mathcal Z }{\partial B^2} -\end{align} -The fluctuation in magnetization (density) thus is -\begin{align} -%{\langle V\Delta \mathcal M^2\rangle}= -\langle\Delta \tilde\mu^2\rangle& -%=T^2\frac{\partial^2 \ln\mathcal Z }{\partial B^2} -=T\frac{\partial\langle V\mathcal{M}\rangle}{\partial B}. -\end{align} - -Using \req{g2magplus} and \req{g2magminus} we turn to consider dimensionless magnetization ${\mathfrak M}={\mathfrak M}_{+}+{\mathfrak M}_{-}$ for the case $g=2$. The fluctuation of magnetized electron-positron plasma becomes -\begin{align}\label{Fluctuation} - \langle\Delta \tilde\mu^2\rangle& - %\langle\Delta \mathcal{M}^2\rangle& - =T\frac{\partial b_0}{\partial B}\frac{\partial {\langle V\mathcal{M}\rangle} }{\partial b_0} - =V\frac{m_e^2}{T}\left(\frac{\partial {\mathfrak M}_{+} }{\partial b_0}+\frac{\partial {\mathfrak M} _{-}}{\partial b_0}\right) - \end{align} - In the last step we assumed that in the homogeneous Universe fluctuations in volume are constrained. The dimensionless magnetization $\partial{\mathfrak M}_{\pm}/\partial b_0$ are given by - \begin{align} - \frac{\partial {\mathfrak M}_{+} }{\partial b_0}=\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\xi\cosh{\frac{\mu}{T}}\,\left[\frac{1}{6}K_{0}(x_{+})\right],\quad x_{+}=\frac{m_{e}}{T} - \end{align} - and - \begin{align} - &\frac{\partial {\mathfrak M}_{-} }{\partial b_0}=\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\xi^{-1}\cosh{\frac{\mu}{T}}\bigg[\left(\frac{1}{6}+\frac{b^2_0}{12x^2_{-}}\right)K_0(x_{-})\notag\\ - &+\left(\frac{b_0}{6x_{-}}+\frac{b^2_0}{6x^3_{-}}\right)K_1(x_{-})\bigg],\quad x_{-}\!\!=\!\sqrt{\frac{m_{e}^{2}}{T^{2}}+2b_{0}}. - \end{align} - - Given the magnetic field $10^{-11}k!_DHsqotWh9=mYe{HCeiF=?yow+xMRt${N%pU<2=rfjHp?9z1`Lx_cyiS_AYGUmGM zyS<=upnbTwc(^$G@bK{V@85@ikc8mifddCAh>sAF&`{FR(oj-UA7fx=I(FhTJvB8m z|EbewIC*$@=$HgW1-L}mxp}xgUIb(R{{06J93&?oAm=(xeVps3e|BGj4)4Q$gU5h{ zaRP*S7z67t#_nqn9WX&0j88wHFFzQVSlBqYc>DI_9{@g(dkBPyfrW*MjfI1QjSYO- 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