Revisions continued.

plasma-partition.tex

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@@ -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 $10^{-11}<b_0<10^{-3}$, we have $m_e/T\gg b_0$ in the temperature range we are interested in. Then the dimensionless variable $x_{-}$ can written as
- \begin{align}
- x_{-}&=\frac{m_e}{T}\sqrt{1+\left(\frac{2T^2b_0}{m^2_e}\right)}\approx x_{+}\left[1+\left(\frac{T^2b_0}{m^2_e}\right)\right].
- \end{align}
-In this case, the $\partial{\mathfrak M}_{-}/\partial b_0$ becomes
-\begin{align}
-\frac{\partial {\mathfrak M}_{-} }{\partial b_0}\approx&\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\xi^{-1}\cosh{\frac{\mu}{T}}\notag\\
-&\left[\left(\frac{1}{6}\right)K_0(x_{+})+\frac{b_0}{6x_{+}}K_1(x_{+})+{\cal O}\left(b_{0}^{2}\right)\right].
-\end{align}
-then the fluctuation of magnetized electron-positron plasma Eq.~(\ref{Fluctuation}) can be written as
- \begin{align}
-  \langle\Delta \mathcal{M}^2\rangle
-  &=\frac{T}{6}\left(\frac{e^{2}}{\pi^{2}}\right)\cosh{\frac{\mu}{T}}\cosh{\frac{\eta}{T}} K_0(x_+)\notag\\
-  &+b_0\left[\frac{T}{6}\left(\frac{e^{2}}{\pi^{2}}\right)\cosh{\frac{\mu}{T}}\xi^{-1}\frac{K_1(x_+)}{x_+}\right]+{\cal O}\left(b_{0}^{2}\right)
- \end{align} 
- In Fig.~\ref{Flu_fig} we plot the fluctuation $ \langle\Delta {\mathcal M}^2\rangle/m_e$ and  $\langle\Delta {\mathcal M}^2\rangle/\sqrt{\langle \mathcal M\rangle}$ as a function of temperature with $g=2$,  $\xi=1$, $b_0=10^{-11}$ and $b_0=10^{-3}$.
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{figure}[ht]
- \centering
- \includegraphics[width=0.5\textwidth]{plots/Fluctuation_Magnetization}
- \includegraphics[width=0.5\textwidth]{plots/Fluctuation_M002}
- \caption{The  dimensionless fluctuation $ \langle\Delta {M}^2\rangle/m_e$  and  $\langle\Delta {\mathcal M}^2\rangle/\sqrt{\langle \mathcal M\rangle}$ of the primordial $e^{+}e^{-}$-plasma as a function of temperature, with $g=2$,  $\xi=1$, $b_0=10^{-11}$ and $b_0=10^{-3}$.} 
- \label{Flu_fig} 
-\end{figure}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Spatial scale considerations}
-\label{sec:spatial}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%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~\cite{..}. While such locally acute magnetization would not survive until present day, their presence could be inferred from future observation of pre-recombination structure.
-
-\noindent 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 gravitational collapse. If the plasma were home to such localized magnetic domains, the nonzero local angular momentum within these domains would provide a natural mechanism for the formation of rotating galaxies today.
+In principle there are two field scales: The first is the correlation length~\cite{Kahniashvili:2012uj} associated with the `external' PMF $\lambda_{B}$, and the other related to the possibly spontaneously occurring magnetization in the plasma, $\lambda_\mathcal{M}$. Under the paramagnetic response described in \rsec{sec:magnetization}, the coherent length of the magnetization naturally mirrors that of the external PMF whatever mechanism is responsible for its generation. However, if magnetization is spontaneous as discussed in \rsec{sec:ferro}, then the two scales, $\lambda_{B}$ and $\lambda_\mathcal{M}$ may differ. Literature in general refers to $\lambda_{B}$ and in following discussion we address this quantity.
 
-As our model quantifies the relativistic paramagnetism of the $e^{+}e^{-}$ medium, the induced magnetization should then be subject to the variations and spectra of the external PMF present. If an observational signature of the $e^{+}e^{-}$ magnetization could be ascertained, then this would provide a way to characterize the length scale and coherence of the original PMF. If the polarization fugacity was nonzero as per \req{ferro} and \req{hiTferro}, then the spectra of the magnetization would match the variation in spatial spin polarization.}
+The observational restriction on the coherent length scale of IGMFs are not well constrained~\cite{Giovannini:2022rrl,Durrer:2013pga,AlvesBatista:2021sln} and are bounded by the range  $\lambda_{B}\sim10^{-2}-10^{3}$ Mpc for the external field strengths we 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.
 
-% Add new section describing the expected induced fluctuations in the primordial field as well as the coherent length scale of the induced (and external) primordial fields.
-% Fluctuation of magnetic fields: Standard deviations.
-% Magnetic pressure (B^2) versus thermodynamics pressure (T^3) critical point crossing which limits size of coherent magnetic fields.
-% Length scale (email Chris)
-% Look at length scale models in reference given by referee
+As our model quantifies the relativistic paramagnetism of the $e^{+}e^{-}$ medium, the induced magnetization should then be subject to the variations and spectra of the external PMF present. If an observational signature of the $e^{+}e^{-}$ magnetization could be ascertained, then this would provide a way to characterize the length scale and coherence of the original PMF. If the polarization fugacity was nonzero as per \req{ferro} and \req{hiTferro}, then the spectra of the magnetization would match the variation in spatial spin polarization $\eta(x)$. This would likely be associated with the long-range interaction length scale which generates a nonzero $\eta$. We have yet to consider the effect of the induced magnetization on the conservation of magnetic helicity~\cite{Boyarsky:2011uy}. We plan to return to the question of thermally induced spontaneous magnetization more fully in a future work.
+}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Summary and discussion}