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\begin{document}

\mbox{\quad} \\
{\bf Notes on CPI, magnetogenesis, gravitational wave radiation, and baryogenesis} \\
{Andrew J. Long} \\
\today \\

%==========
{\bf Purpose of Note} \\
In these notes, I discuss the phenomenon known as chiral plasma instability (CPI).  
In the cosmological context, a primordial chiral asymmetry can lead to the generation of a primordial magnetic field (PMF), which may survive in the universe today as an as-yet undetected intergalactic magnetic field (IGMF).  
As the PMF gravitates in the early universe, it sources gravitational wave radiation, which can survive in the universe today as a stochastic gravitational wave background (SGWB).  
The PMF is expected to be helical, with the sign of its helicity linked to the sign of the progenitor chiral symmetry, and it induces chiral gravitational wave radiation.  
At the electroweak phase transition, the transfer of helicity from the helical hyper-magnetic field to a helical electromagnetic field induces anomalous baryon number generation, which contributes to the baryon asymmetry of the universe.  

%\vspace{-1cm} 

\begingroup
\setlength{\parskip}{-0.1cm}
\hypersetup{linkcolor=black}
\tableofcontents
\endgroup

\newpage
%==================================
% Setup
%==================================
\section{Setup}
\label{sec:Setup}

%-------------------------------------------
% The primordial plasma
%-------------------------------------------
\subsection{The primordial plasma}
\label{sub:primordial_plasma}

%====
Consider a time in our cosmic history when the universe was younger than $\sim 0.1 \ \mathrm{ns}$.  
At this time, the universe was filled with a nearly homogeneous thermal bath of Standard Model particles:  quarks, leptons, Higgs, and gauge bosons.  
These particles would participate in various reactions -- such as scatter, decay, and inverse decay -- and thereby attain thermal equilibrium at a common temperature.  
A Standard Model plasma at a temperature above $\approx 100 \GeV$ is in the phase of unbroken electroweak symmetry; the Higgs has a vanishing expectation value; the quarks, leptons, and gauge bosons are massless; the force carriers are $8$ gluons of $\SU{3}_c$, $3$ $W$-bosons of $\SU{2}_L$, and $1$ hyper-photon of $\U{1}_Y$; and there is no electromagnetism.  
It is possible that the primordial plasma sustained a long-range coherent magnetic field.  
In the phase of unbroken electroweak symmetry, this magnetic field is associated with the $\U{1}_Y$ force, and we talk about a \textit{hypermagnetic field} (rather than an electromagnetic field).  

%====
As the universe grew older, it expanded.  
The spatially homogeneous, isotropic, and flat universe corresponds to an FRW spacetime with metric $(\ud s)^2 = (\ud t)^2 - a(t)^2 \, |\ud \xvec|^2$ where $t$ is the temporal coordinate, $a(t)$ is the FRW scale factor, and $\xvec$ is the comoving spatial coordinate.  
We can also write $(\ud s)^2 = a(\eta)^2 \, [ (\ud \eta)^2 - |\ud \xvec|^2]$ where $\eta$ is conformal time.  
The two time coordinates are related by 
\ba{\label{eq:eta_and_t}
	\eta - \eta_1 = \int_{t_1}^t \! \ud t^\prime / a(t^\prime) 
	\qquad \text{or equivalently} \qquad 
	t - t_1 = \int_{\eta_1}^\eta \! \ud \eta^\prime \, a(\eta^\prime) 
	\per
}
The comoving rate of expansion at conformal time $\eta$ is given by 
\ba{\label{eq:H_def}
	H(\eta) = \dot{a}(\eta) / a(\eta)
	\com
}
and the physical expansion rate is $H/a$.  

%====
As space expanded, the primordial plasma contained within it cooled down.  
Let $T(\eta)$ be the comoving temperature of the primordial plasma at conformal time $\eta$.  
If the universe cooled in such a way that the comoving entropy density remained conserved, then the comoving temperature at conformal time $\eta$ is given by 
\ba{\label{eq:T}
	T(\eta) = \biggl( \frac{g_{\ast,S,0}}{g_{\ast,S}(\eta)} \biggr)^{1/3} \, T_0 
	\per
}
%In this formula, $a_0 = 1$, $T_0 \simeq 0.234 \meV$, $g_{\ast,S,0} \simeq 3.91$, $a(t)$ is the FRW scale factor at time $t$, and $g_{\ast,S}(t)$ is the effective number of relativistic species at time $t$.  
In this formula, $T_0 \simeq 0.234 \meV \, a_0$ is the comoving temperature today, $g_{\ast,S,0} \simeq 3.91$ is the effective number of relativistic species today, and $g_{\ast,S}(\eta)$ is the effective number of relativistic species at conformal time $\eta$.  
The physical temperature is $T / a$.  
%We refer to the time $t = t_\mathrm{ew}$ when $T$ was comparable to $T_\mathrm{ew}$ as the electroweak epoch.  

%====
The comoving energy density of the primordial plasma at conformal time $\eta$ is given by 
\ba{\label{eq:rho}
	\rho(\eta) = \frac{\pi^2}{30} \, g_\ast(\eta) \, T(\eta)^4 
	\com
}
where $g_\ast(\eta)$ is the effective number of relativistic species at conformal time $\eta$.  
It is an excellent approximation to take $g_\ast(\eta) = g_{\ast,S}(\eta)$.  
If the primordial plasma consists of only Standard Model particles, then $g_\ast(\eta) = g_{\ast,\text{\sc sm}} \equiv 106.75$ at early times when $T/a \gtrsim 100 \GeV$ and it decreases quickly when $T/a \lesssim 100 \GeV$.  

%====
The expansion of the universe is induced by the energy of the radiation and matter contained within it.  
During the epoch of radiation domination, the energy of the universe arises predominantly from the primordial plasma, and a homogeneous and isotropic cosmological expansion occurs.  
The comoving rate of expansion at conformal time $\eta$ is given by the first Friedmann equation, $3 a(\eta)^2 \Mpl^2 H(\eta)^2 = \rho(\eta)$, which is solved by 
\bes{\label{eq:H}
	& H(\eta) 
	= a(\eta)^{-1} \sqrt{\frac{\rho(\eta)}{3 \Mpl^2}} 
	= \underbrace{\frac{\pi \, g_{\ast,S,0}^{2/3}}{\sqrt{90}} \, \frac{T_0^2}{\Mpl} \, \biggl( \frac{g_{\ast}(\eta)^{1/2}}{g_{\ast,S}(\eta)^{2/3}} \biggr)}_{\equiv \ \Hcal} a(\eta)^{-1} \\ 
	& \qquad \text{where} \qquad 
	\Hcal \simeq \bigl( 8 \times 10^{-36} \eV a_0^2 \bigr) \biggl( \frac{g_\ast}{106.75} \biggr)^{-1/6} 
	\com
}
where we have also used \erefs{eq:T}{eq:rho} in the second and third equalities, and we have defined $\Hcal$.  

%====
The expansion factor $a(\eta)$ is determined by equating \erefs{eq:H_def}{eq:H} and solving the first-order differential equation.  
In this calculation, it is a good approximation to assume that $g_{\ast}(\eta)$ and $g_{\ast,S}(\eta)$ are slowly varying, which is the case for a plasma of Standard Model particles at a temperature above $\approx 100 \GeV$.  
A general solution is written as $a(\eta) = a_1 + \Hcal \, (\eta - \eta_1)$.  
Evaluating the integral in \eref{eq:eta_and_t} gives $t(\eta) = t_1 + a_1 \, (\eta - \eta_1) + \Hcal \, (\eta - \eta_1)^2 / 2$.  
We take $a_1 = \Hcal \eta_1$ and $t_1 = \Hcal \eta_1^2 / 2$.  
Therefore, in a radiation-dominated universe we have the relations 
\ba{\label{eq:Hat_to_eta}
	H(\eta) = 1/\eta
	\ , \qquad 
	H(\eta) / a(\eta) = 1 / \Hcal \eta^2
	\ , \qquad 
	a(\eta) = \Hcal \eta 
	\ , \qquad \text{and} \qquad 
	t(\eta) = \Hcal \eta^2 / 2
	\per
}
It follows that $H = 1/\eta = \sqrt{\Hcal/2t} = \Hcal/a$, $H/a = 1/\Hcal\eta^2 = 1/2t = \Hcal/a^2$, $a = \Hcal\eta = \sqrt{2\Hcal t}$, and $t = \Hcal \eta^2/2 = a^2 / 2\Hcal$.  
%Since $H(t) \propto a(t)^{-2}$ the FRW scale factor that solves this ODE is\footnote{} $a(t) = a_\mathrm{ew} (t / t_\mathrm{ew})^{1/2}$ where $t_\mathrm{ew}$ is the age of the universe at the electroweak epoch when $T \approx T_\mathrm{ew} = 100 \GeV$ and $a_\mathrm{ew}$ is the corresponding FRW scale factor.  

%====
We identify the electroweak epoch as the time when the physical plasma temperature was around $T/a \approx 100 \GeV$ and the effective number of relativistic species was $g_{\ast}(\eta) \approx g_{\ast,S}(\eta) \simeq 106.75$.  
Using \erefs{eq:T}{eq:Hat_to_eta} gives 
\bsa{}{
	(T/a)_\mathrm{ew} & \simeq 100 \GeV \\ 
	\eta_\mathrm{ew} & \simeq \bigl( 6 \times 10^4 \sec a_0^{-1} \bigr) \biggl( \frac{g_\ast}{106.75} \biggr)^{-1/6} \\ 
	t_\mathrm{ew} & \simeq \bigl( 2 \times 10^{-11} \sec \bigr) \biggl( \frac{g_\ast}{106.75} \biggr)^{-1/2} \\ 
	a_\mathrm{ew} & \simeq \bigl( 8 \times 10^{-16} \ a_0 \bigr) \biggl( \frac{g_\ast}{106.75} \biggr)^{-1/3} \\ 
	H_\mathrm{ew} & \simeq \bigl( 1 \times 10^{-20} \eV a_0 \bigr) \biggl( \frac{g_\ast}{106.75} \biggr)^{1/6} \\ 
	(H/a)_\mathrm{ew} & \simeq \bigl( 1 \times 10^{-5} \eV \bigr) \biggl( \frac{g_\ast}{106.75} \biggr)^{1/2} 
	\per
}

%-------------------------------------------
% Particle densities and chemical potentials
%-------------------------------------------
\subsection{Particle densities and chemical potentials}
\label{sub:chemical_potential}

%====
The primordial plasma contains particles drawn from each of the various Standard Model particle species.  
Some examples are right-chiral anti-muon, left-chiral down-type blue quark, red-antigreen gluon, and so on.  
Label the various particle species by an index $i$.  
We assume that each species reaches thermal equilibrium, and their phase space distribution functions take the form of either the Bose-Einstein or Fermi-Dirac distribution (depending on the particle's quantum statistics).  
The comoving number density of particles of species $i$ with comoving momentum between $\pvec$ and $\ud \pvec$ at conformal time $\eta$ is given by 
\ba{
	\ud n_i(\pvec,\eta) = g_i \frac{\ud^3 \pvec}{(2\pi)^3} f_i(\pvec,\eta) 
	\qquad \text{with} \qquad 
	f_i(\pvec,\eta) = \begin{cases}
	\bigl( e^{[ E_i(\eta) - \mu_i(\eta) ] / T(\eta)} - 1 \bigr)^{-1} & , \quad \text{if $i$ is a boson} \\ 
	\bigl( e^{[ E_i(\eta) - \mu_i(\eta) ] / T(\eta)} + 1 \bigr)^{-1} & , \quad \text{if $i$ is a fermion} 
	\end{cases}
}
where $E_i(\eta) = \sqrt{|\pvec|^2 + a(\eta)^2 m_i^2}$ and $m_i$ are the comoving energy and mass of particles of species $i$, and where $g_i$ is a multiplicity factor to account for redundancy.  
Here we assume a common comoving temperature $T(\eta)$ for every particle species, but we allow each species to have a different comoving chemical potential $\mu_i(\eta)$.  

%====
The comoving number density of particles of species $i$ is calculated by integrating $n_i(\eta) = \int \! \ud n_i(\pvec,\eta)$.  
For massless particles with small chemical potential $|\mu_i|/T \ll 1$ we can evaluate $n_i(\eta)$ as a power series: 
\ba{\label{eq:ni}
	n_i(\eta) = g_i \begin{cases}
	\frac{\zeta(3)}{\pi^2} T(\eta)^3 + \frac{1}{6} \mu_i(\eta) T(\eta)^2 + O(\mu_i^2 T) & , \quad \text{if $i$ is a boson} \\ 
	\frac{3}{4} \frac{\zeta(3)}{\pi^2} T(\eta)^3 + \frac{1}{12} \mu_i(\eta) T(\eta)^2 + O(\mu_i^2 T) & , \quad \text{if $i$ is a fermion} 
	\end{cases} 
	\per
}
Recall that $T(\eta)$ is given by \eref{eq:T} if the cosmological expansion is adiabatic, and prior to the electroweak epoch $T(\eta)$ is approximately constant.  
On the other hand, $\mu_i(\eta)$ may evolve as particles enter or leave thermal equilibrium.  

%====
Many Standard Model particles come in particle/antiparticle pairs.  
If $i$ denotes a particular species of particle, let $\bar{i}$ denote the $\textsf{CP}$-conjugate antiparticle.  
If the hypercharge interactions are in equilibrium, they impose $\mu_{\bar{i}} = - \mu_i$, so that conjugate particle/antiparticle pairs have opposite chemical potentials.  
We are often interested in the asymmetry in the number of particles and their conjugate antiparticles.  
We define the comoving asymmetry number density as $\Delta n_i(\eta) = n_i(\eta) - n_{\bar{i}}(\eta)$.  
Using \eref{eq:ni} the asymmetry number density at conformal time $\eta$ is given by 
\ba{\label{eq:ni}
	\Delta n_i(\eta) = g_i \begin{cases}
	\frac{1}{3} \mu_i(\eta) T(\eta)^2 + O(\mu_i^2 T) & , \quad \text{if $i$ is a boson} \\ 
	\frac{1}{6} \mu_i(\eta) T(\eta)^2 + O(\mu_i^2 T) & , \quad \text{if $i$ is a fermion} 
	\end{cases}
	\per
}
The hypercharge gauge interactions can create (or destroy) pairs of $i$ particle and $\bar{i}$ antiparticles, which does not affect $\Delta n_i$.  
However, other interactions may be able to change the relative abundances of $i$ particles and $\bar{i}$ antiparticles.  
When such reactions are out of thermal equilibrium, the comoving asymmetry number density is constant, which implies that $\mu_i(\eta)$ is approximately constant (since $T(\eta)$ is already approximately constant, see \eref{eq:T}).  

%====
Why is it reasonable to assume that all of the Standard Model particles share a common temperature $T$, but that they may have different chemical potentials $\mu_i$?  
At the electroweak epoch, elastic scattering reactions are rapid, and the associated momentum transfer brings all the Standard Model particle species to a common temperature.  
However, some particle-number-changing processes are in equilibrium, while others are out of equilibrium.  
Some of these reactions are out of equilibrium because they are strictly forbidden, such as reactions that would change the value of baryon-number minus lepton-number: $\textsf{B} - \textsf{L}$.  
Other reactions are out of equilibrium, because their would-be interaction time scale is long compared to the age of the universe; the universe is not yet old enough for these reactions to be occurring frequently.  
Out of equilibrium reactions correspond to exactly or approximately-conserved charges, which are parametrized by the comoving chemical potentials $\mu_i(\eta)$.  

%-------------------------------------------
% Hypercharge-weighted chiral chemical potential
%-------------------------------------------
\subsection{Hypercharge-weighted chiral chemical potential}
\label{sub:muY5}

%====
Using the individual $\mu_i(\eta)$ for each particle species, one can define effective chemical potentials for different charge combinations.  
For instance, we could define $\mu_\textsf{B}(\eta)$ for baryon-number or $\mu_\mathrm{em}(\eta)$ for electromagnetic charge.  
The comoving hypercharge-weighted chiral chemical potential at conformal time $\eta$ is given by 
\ba{\label{eq:muY5}
	\mu_{Y,5}(\eta) = \sum_i \varepsilon_i g_i Y_i^2 \mu_i(\eta) 
	\per
}
In this expression, we sum over all the Standard Model particle species (indexed by $i$), and the various factors are as follows: $\varepsilon_i$ is equal to $\pm 1$ for a right/left-chiral particle (and $0$ otherwise), $g_i$ is a multiplicity factor (counting color, spin, etc), and $Y_i$ is the hypercharge of species $i$.  
In this way $\mu_{Y,5}$ is the sum of chemical potentials for right-handed particles minus the sum for left-handed particles, but with different weighting factors; roughly speaking, $\mu_{Y,5} \sim c_R \mu_R - c_L \mu_L$

%-------------------------------------------
% Chiral charge erasure
%-------------------------------------------
\subsection{Chiral charge erasure}
\label{sec:erasure}

%====
The hypercharge-weighted chiral chemical potential may be nonzero in the early universe, prior to the electroweak epoch.  
Since \eref{eq:muY5} involves a sum over many Standard Model particle species, one can ask: which term (or terms) in the sum contributes the most?  
The answer to this question changes with time.  
Each of the constituent comoving chemical potentials $\mu_i(\eta)$ evolves in response to the various particle physics reactions involving particles of species $i$.  
In particular, inverse-decay reactions involving the Higgs tend to erase these chemical potentials, driving them to zero.  
Heavier particles couple more strongly to the Higgs, and they have their chemical potentials erased earlier.  
The particle with the weakest coupling to the Higgs is the right-chiral electron (and its $\textsf{CP}$-conjugate antiparticle partner), since the electron is the lightest Standard Model fermion.  
Starting from some very high temperature $\gg 100 \TeV$, the plasma cools down, and different contributions to $\mu_{Y,5}(\eta)$ drop out.  
At a temperature of $\sim 100 \TeV$ only the right-chiral electron is left: $\mu_{Y,5}(\eta) = Y_{e_R}^2 \mu_{e_R}(\eta)$.  
As the temperature continues to decrease $\lesssim 100 \TeV$, even the asymmetry carried by the right-chiral electron is erased by Higgs-mediated scattering.  

%====
We can calculate the erasure of the comoving hypercharge-weighted chiral chemical potential as follows.  
We model the system with the following kinetic equation: 
\ba{\label{eq:dmuY5}
	\frac{d}{d\eta} \mu_{Y,5}(\eta) = - \Gamma_\mathrm{f}(\eta) \, \mu_{Y,5}(\eta)
}
where $\Gamma_\mathrm{f}(\eta)$ is the comoving charge-erasure rate at conformal time $\eta$.  
This rate has been estimated as\footnote{Here I'm using the results of \href{https://arxiv.org/abs/1902.07220v2}{B\"odeker \& Schr\"oder (2019)}, namely $\gamma = (4266/481) T^{-2} \Gamma$ from eq.~(2.22) and $\Gamma \simeq (1.5 \times 10^{-3}) h_e^2 T^3$ from tab.~1; note that their notation is different from this note.  Earlier estimates of this rate include eq.~(2.26) of \href{https://arxiv.org/abs/1606.08891}{Kamada \& Long (2016)} and eq.~(7) of \href{https://arxiv.org/abs/hep-ph/9302221}{Campbell, Davidson, Ellis, \& Olive (1993)}.  }
\ba{
	\Gamma_\mathrm{f}(\eta) 
	\sim \bigl( 1.3 \times 10^{-2} \bigr) y_e^2 T(\eta) 
	\sim \bigl( 1.1 \times 10^{-13} \bigr) T(\eta) 
	\simeq \bigl( 9.6 \times 10^{-19} \eV a_0 \bigr) \biggl( \frac{g_\ast(\eta)}{106.75} \biggr)^{-1} 
	\com
}
where we used \eref{eq:T} in the final equality.  
This rate corresponds to reactions involving the Higgs that create or destroy right-chiral electrons (or their $\textsf{CP}$-conjugate antimatter partner).  
The strength of these interactions is parametrized by the Yukawa coupling $y_e = \sqrt{2} m_e / v \simeq 3 \times 10^{-6}$.  
%We only attempt an order of magnitude estimate for $\Gamma_\mathrm{f}$, hence the factor of $10^{-2}$.  
Use use the subscript $\mathrm{f}$, since the reactions leading to charge erasure are also called spin-flip reactions, although in my opinion this is really a misnomer.  

%====
Prior to the electroweak epoch, $\Gamma_\mathrm{f}(\eta)$ is approximately constant, and \eref{eq:dmuY5} is solved by $\mu_{Y,5} \propto e^{- \Gamma_\mathrm{f} \eta}$.  
At early times, the charge-erasure reaction is out of equilibrium and $\mu_{Y,5}(\eta)$ is approximately conserved: $e^{- \Gamma_\mathrm{f} \eta} \approx 1$ for $\Gamma_\mathrm{f} \eta \ll 1$.  
However, once the age of the universe exceeds the interaction time scale, an exponential expression of $\mu_{Y,5}(\eta)$ will occur.  
We define the fiducial charge erasure conformal time $\eta_\mathrm{f}$ such that $\Gamma_\mathrm{f}(\eta_\mathrm{f}) \eta_\mathrm{f} = 1$.  
Solving for this time, and using also the relations in \eref{eq:Hat_to_eta}, gives\footnote{These estimates agree with earlier work by \href{https://arxiv.org/abs/hep-ph/9302221}{Campbell, Davidson, Ellis, \& Olive (1993)} who obtain $(T/a)_\mathrm{f} \simeq 1 \TeV$, whereas \href{https://arxiv.org/abs/astro-ph/9703005}{Joyce \& Shaposhnikov (1997)} find $(T/a)_\mathrm{f} \simeq 80 \TeV$, although I think the latter estimate is less reliable, since they drop important factors of $2$ and $\pi$ that build up to $\sim 10^2$.  }
\bsa{eq:f_estimates}{
	\eta_\mathrm{f} & \simeq \bigl( 8 \times 10^1 \sec a_0^{-1} \bigr) \biggl( \frac{g_\ast}{106.75} \biggr)^{1/3} \\ 
	t_\mathrm{f} & \simeq \bigl( 4 \times 10^{-17} \sec \bigr) \biggl( \frac{g_\ast}{106.75} \biggr)^{1/2} \\ 
	a_\mathrm{f} & \simeq \bigl( 1 \times 10^{-18} \ a_0 \bigr) \biggl( \frac{g_\ast}{106.75} \biggr)^{1/6} \\ 
	(T/a)_\mathrm{f} & \simeq \bigl( 8 \times 10^4 \GeV \bigr) \biggl( \frac{g_\ast}{106.75} \biggr)^{-1/2} 
	\per 
}
Most importantly, the physical temperature of the primordial plasma at the time when charge erasure occurs is estimated to be $(T/a)_\mathrm{f} \sim 100 \TeV$.  

%====
In summary: 
\begin{quotation}
\textit{If there is no mechanism present to sustain the hypercharge-weighted chiral chemical potential, it will be erased by Higgs-mediated reactions when the plasma temperature is around $\sim 100 \TeV$.  Consequently, it is not possible to generate a primordial magnetic field from the hypercharge-weighted chiral asymmetry via the chiral plasma instability after the plasma has cooled below a temperature of $\sim 100 \TeV$.}  
\end{quotation}
Nevertheless, the chiral plasma instability can be operative at higher temperatures, $\gtrsim 100 \TeV$, and provide a source of magnetogenesis, and we turn to that subject next in \sref{sec:mag_field}.  
Moreover, the conclusions drawn here regarding chiral charge erasure may be modified under different assumptions about the cosmological expansion history and particle content of the plasma; we discuss these points further in \sref{sec:modifications}.  

%==================================
% Magnetic field generation
%==================================
\section{Magnetic field generation}
\label{sec:mag_field}

%====

%-------------------------------------------
% Chiral magnetohydrodynamics
%-------------------------------------------
\subsection{Chiral magnetohydrodynamics}
\label{sub:chiral_MHD}

%====
We're interested in the dynamics of a long-range coherent magnetic field and its associated electric field.  
Since the plasma is in a phase of unbroken electroweak symmetry, this field is associated with the $\U{1}_Y$ gauge group.  
We denote the comoving hyperelectric and hypermagnetic fields by $\Evec_Y(\eta,\xvec)$ and $\Bvec_Y(\eta,\xvec)$.  
The dynamics of these fields are governed by the equations of chiral magnetohydrodynamics.  
In an FRW spacetime, these equations are written as 
\ba{
	0 = \dot{\Bvec}_Y + \dvec \times \Evec_Y 
	\qquad \text{and} \qquad 
	0 = \dvec \times \Bvec_Y - \Jvec_Y 
	\com
}
where we have neglected the displacement current term $\propto \dot{\Evec}_Y$.  
The comoving hypercharge current density field is given by 
\ba{
	\Jvec_Y(\eta,\xvec) = \sigma_Y(\eta) \bigl[ \Evec_Y(\eta,\xvec) + \vvec(\eta,\xvec) \times \Bvec_Y(\eta,\xvec) \bigr] + \frac{2}{\pi} \alpha_Y \, \mu_{Y,5}(\eta) \, \Bvec_Y(\eta,\xvec) 
	\com
}
where $\sigma_Y(\eta)$ is the comoving hypermagnetic conductivity, $\vvec(\eta,\xvec)$ is the fluid velocity, $\alpha_Y \approx 0.01$ is the hypermagnetic fine structure constant, and $\mu_{Y,5}(\eta)$ is the comoving hypercharge-weighted chiral chemical potential.  
The conductivity and chemical potential are assumed to be spatially homogeneous.  
If we neglect the fluid velocity, $\vvec(\eta,\xvec) \approx 0$, then the system of equations reduces to 
\bsa{eq:chiral_MHD}{
	\Evec_Y & = \frac{1}{\sigma_Y} \dvec \times \Bvec_Y - \frac{2}{\pi} \alpha_Y \frac{\mu_{Y,5}}{\sigma_Y} \Bvec_Y \\ 
	0 & = \dot{\Bvec}_Y - \frac{1}{\sigma_Y} \nabla^2 \Bvec_Y - \frac{2}{\pi} \alpha_Y \frac{\mu_{Y,5}}{\sigma_Y} \dvec \times \Bvec_Y 
	\com
}
where we've used $\dvec \cdot \Bvec_Y = 0$.    
The term containing $\mu_{Y,5}$ is associated with the chiral magnetic effect, and it leads to the chiral plasma instability.  

%-------------------------------------------
% Fourier representation
%-------------------------------------------
\subsection{Fourier representation}
\label{sub:Fourier}

%====
We move to Fourier space by writing 
\ba{
	\Bvec_Y(\eta,\xvec) = \int \! \! \frac{\ud^3 \kvec}{(2\pi)^3} \, \Bvec_{Y,\kvec}(\eta) \, e^{i \kvec \cdot \xvec} 
	\com
}
where $\kvec$ is the comoving wavevector $\kvec$ and $k = |\kvec|$ is the corresponding comoving wavenumber.  
The equation of motion \pref{eq:chiral_MHD} becomes 
\ba{
	0 & = \dot{\Bvec}_{Y,\kvec} + \frac{k^2}{\sigma_Y} \Bvec_{Y,\kvec} - \frac{2}{\pi} \alpha_Y \frac{\mu_{Y,5}}{\sigma_Y} i \kvec \times \Bvec_{Y,\kvec} 
	\per
}
Next we decompose the mode function onto a basis of left- and right-circularly polarized modes: 
\ba{
	\Bvec_{Y,\kvec}(\eta) = B_{Y,k}^{(R)} \, \epsilon_{\hat{\kvec}}^{(R)} + B_{Y,k}^{(L)} \, \epsilon_{\hat{\kvec}}^{(L)} 
	\com
}
where the polarization unit vectors obey 
\bsa{}{
	\kvec \cdot \epsilon_{\hat{\kvec}}^{(R)} & = 0 \\ 
	\kvec \cdot \epsilon_{\hat{\kvec}}^{(L)} & = 0 \\ 
	\epsilon_{\hat{\kvec}}^{(R)\dagger} \cdot \epsilon_{\hat{\kvec}}^{(R)} & = 1 \\ 
	\epsilon_{\hat{\kvec}}^{(R)\dagger} \cdot \epsilon_{\hat{\kvec}}^{(L)} & = 0 \\ 
	\epsilon_{\hat{\kvec}}^{(L)\dagger} \cdot \epsilon_{\hat{\kvec}}^{(R)} & = 0 \\ 
	\epsilon_{\hat{\kvec}}^{(L)\dagger} \cdot \epsilon_{\hat{\kvec}}^{(L)} & = 1 \\ 
	i \kvec \times \epsilon_{\hat{\kvec}}^{(R)} & = + k \epsilon_{\hat{\kvec}}^{(R)} \\ 
	i \kvec \times \epsilon_{\hat{\kvec}}^{(L)} & = - k \epsilon_{\hat{\kvec}}^{(L)} 
	\per
}
Then the right- and left-mode functions satisfy 
\bsa{eq:B_eom}{
	0 & = \dot{B}_{Y,k}^{(R)} + \frac{k^2}{\sigma_Y} B_{Y,k}^{(R)} - \frac{2}{\pi} \alpha_Y \frac{\mu_{Y,5}}{\sigma_Y} k B_{Y,k}^{(R)} \\ 
	0 & = \dot{B}_{Y,k}^{(L)} + \frac{k^2}{\sigma_Y} B_{Y,k}^{(L)} + \frac{2}{\pi} \alpha_Y \frac{\mu_{Y,5}}{\sigma_Y} k B_{Y,k}^{(L)} 
	\per
}
We can study these equations to identify the chiral plasma instability.  

%====
Although not necessary, we can also express the equations of motion in terms of the comoving spatially-averaged magnetic energy density and the comoving spatially-averaged helicity density, which are defined by 
\bsa{}{
%---
	\rho_B(\eta) 
	& = \frac{1}{V} \int_V \! \ud^3 \xvec \, \frac{1}{2} \Bvec_Y(\eta,\xvec) \cdot \Bvec_Y(\eta,\xvec) 
	= \int_0^\infty \! \ud k \, \rho_{B,k}(\eta) \\ 
%---
	h_B(\eta) 
	& = \frac{1}{V} \int_V \! \ud^3 \xvec \, \Avec_Y(\eta,\xvec) \cdot \Bvec_Y(\eta,\xvec) 
	= \int_0^\infty \! \ud k \, h_{B,k}(\eta) 
	\per
}
They admit the Fourier representations, 
\bsa{}{
	\rho_{B,k}(\eta) & = \frac{k^2}{2\pi^2} \times \frac{1}{2} \Bigl( |B_{Y,k}^{(R)}(\eta)|^2 + |B_{Y,k}^{(L)}(\eta)|^2 \Bigr) \\ 
	h_{B,k}(\eta) & = \frac{k^2}{2\pi^2} \times k^{-1} \, \Bigl( |B_{Y,k}^{(R)}(\eta)|^2 - |B_{Y,k}^{(L)}(\eta)|^2 \Bigr) 
	\per
}
Notice that these quantities obey the inequality $|h_{B,k}(\eta)| \leq 2 \rho_{B,k}(\eta) / k$, and the inequality is saturated for a maximally helical magnetic field configuration with either $B_{Y,k}^{(R)} = 0$ or $B_{Y,k}^{(L)} = 0$. 
It follows that 
\bsa{eq:lin_sys}{
	\dot{\rho}_{B,k} & = - \frac{2 k^2}{\sigma_Y} \rho_{B,k} + \frac{2}{\pi} \alpha_Y \frac{\mu_{Y,5}}{\sigma_Y} k^2 h_{B,k} \\ 
	\dot{h}_{B,k} & = - \frac{2 k^2}{\sigma_Y} h_{B,k} + \frac{8}{\pi} \alpha_Y \frac{\mu_{Y,5}}{\sigma_Y} \rho_{B,k} 
	\per
}
In the absence of a chemical potential, $\mu_{Y,5} = 0$, the solutions are exponentially decaying in time; whereas, a nonzero chemical potential opens a tachyonic instability toward the growth of a helical magnetic field.  

%-------------------------------------------
% Chiral plasma instability
%-------------------------------------------
\subsection{Chiral plasma instability}
\label{sub:CPI}

%====
The magnetic field equations of motion \pref{eq:B_eom} can be written as 
\bsa{}{
	\dot{B}_{Y,k}^{(R)} & = \frac{k^2}{\sigma_Y} \biggl( - 1 + \frac{k_\cpi}{k} \times \mathrm{sgn}(\mu_{Y,5}) \biggr) B_{Y,k}^{(R)} \\ 
	\dot{B}_{Y,k}^{(L)} & = \frac{k^2}{\sigma_Y} \biggl( - 1 - \frac{k_\cpi}{k} \times \mathrm{sgn}(\mu_{Y,5}) \biggr) B_{Y,k}^{(L)} 
	\per
}
The comoving CPI wavenumber at conformal time $\eta$ is given by $k_\cpi(\eta) = 2 \alpha_Y |\mu_{Y,5}(\eta)| / \pi$.  
The corresponding comoving length scale is given by $\lambda_\cpi = 2\pi / k_\cpi = \pi^2 / \alpha_Y |\mu_{Y,5}|$.  
For small-scale modes with $k_\cpi(\eta) < k$, the two magnetic field mode functions decay exponentially quickly $\propto e^{- k^2 \eta / \sigma_Y}$.  
However, for the large-scale modes with $0 < k < k_\cpi(\eta)$, the chiral chemical potential opens a tachyonic instability that allows one of the two polarization modes to grow exponentially.  
If $\mathrm{sgn}(\mu_{Y,5}) > 0$ then the right-handed polarization mode experiences the instability, and if $\mathrm{sgn}(\mu_{Y,5}) < 0$ it is the left-handed polarization mode.  
The amplitudes of the unstable modes grow exponentially as $\propto e^{k k_\cpi \eta / \sigma_Y}$
The fastest growing mode has $k = k_\text{\sc cpi} / 2$, and this mode grows on a time scale $\eta_\cpi = 2 \sigma_Y / k_\cpi^2 = \pi^2 \sigma_Y / 2 \alpha_Y^2 |\mu_{Y,5}|^2$.  
This means that even if the chemical potential is generated very early, the growth of the magnetic field will be ineffective until the age of the universe becomes larger than $\eta_\cpi$.  
In a radiation-dominated universe, this corresponds to the time when the physical plasma temperature is $(T/a)_\cpi = 2 g_{\ast,S,0} T_0 \alpha_Y^2 \mu_{Y,5}^2 / g_{\ast,S} \Hcal \pi^2 \sigma_Y$.  

%====
Using the formulas above, here I provide some numerical estimates.  
The largest comoving wavenumber of the modes that experience the CPI is 
\begin{subequations}\label{eq:cpi_estimates}
\ba{\label{eq:kcpi}
	k_\text{\sc cpi} 
	\simeq 
	\bigl( 3 \times 10^{-5} \cm^{-1} a_0 \bigr) 
	\biggl( \frac{\alpha_Y}{0.01} \biggr) 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{-1/3} 
	\biggl( \frac{|\mu_{Y,5}|/T}{10^{-3}} \biggr)
	\per
}
The corresponding comoving length scale is 
\ba{\label{eq:lambda_cpi}
	\lambda_\text{\sc cpi} 
	\simeq 
	\bigl( 3 \times 10^5 \cm \, a_0^{-1} \bigr) 
	\biggl( \frac{\alpha_Y}{0.01} \biggr)^{-1} 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{1/3}
	\biggl( \frac{|\mu_{Y,5}|/T}{10^{-3}} \biggr)^{-1} 
	\per
}
For the modes with $k = k_\cpi / 2$, the instability develops at a time when 
\ba{
%---
	\eta_\cpi 
	& \simeq \bigl( 4 \times 10^1 \sec \, a_0^{-1} \bigr) 
	\biggl( \frac{\alpha_Y}{0.01} \biggr)^{-2} 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{1/3}
	\biggl( \frac{\sigma_Y}{100 T} \biggr)^{}
	\biggl( \frac{|\mu_{Y,5}|/T}{10^{-3}} \biggr)^{-2} \\ 
%---
	t_\cpi 
	& \simeq \bigl( 1 \times 10^{-17} \sec \bigr) 
	\biggl( \frac{\alpha_Y}{0.01} \biggr)^{-4} 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{1/2}
	\biggl( \frac{\sigma_Y}{100 T} \biggr)^{2}
	\biggl( \frac{|\mu_{Y,5}|/T}{10^{-3}} \biggr)^{-4} \\ 
%---
	a_\cpi 
	& \simeq \bigl( 5 \times 10^{-19} a_0 \bigr) 
	\biggl( \frac{\alpha_Y}{0.01} \biggr)^{-2} 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{1/6}
	\biggl( \frac{\sigma_Y}{100 T} \biggr)^{}
	\biggl( \frac{|\mu_{Y,5}|/T}{10^{-3}} \biggr)^{-2} 
	\com
}
and the corresponding plasma temperature at this time is 
\ba{\label{eq:T_cpi}
	(T/a)_\cpi 
	& \simeq 
	\bigl( 1 \times 10^5 \GeV \bigr) 
	\biggl( \frac{\alpha_Y}{0.01} \biggr)^{2} 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{-1/2} 
	\biggl( \frac{\sigma_Y}{100 T} \biggr)^{-1} 
	\biggl( \frac{|\mu_{Y,5}|/T}{10^{-3}} \biggr)^{2} 
	\per
}
\end{subequations}
The fiducial value of $|\mu_{Y,5}|/T = 10^{-3}$ was chosen to give $(T/a)_\cpi = 100 \TeV$.  

%====
In order for these calculations to be self-consistent, it's necessary that the processes that would lead to erasure of the hypercharge-weighted chiral chemical potential are out of equilibrium at the time when the chiral plasma instability develops.  
This condition is expressed as the inequality $\eta_\cpi < \eta_\mathrm{f}$ or equivalently $(T/a)_\mathrm{f} < (T/a)_\cpi$.  
Recall that $(T/a)_\mathrm{f}$ was given by \eref{eq:f_estimates}.  
This self-consistency condition can be expressed as a lower bound on the chemical potential:\footnote{See eq.~(1.2) of \href{https://arxiv.org/abs/1902.07220v2}{B\"odeker \& Schr\"oder (2019)}.}  
\ba{\label{eq:erasure_avoidance}
	\text{CPI occurs before chiral charge erasure} 
	\qquad \Rightarrow \qquad 
	|\mu_{Y,5}| / T 
	& > 
	\bigl( 7 \times 10^{-4} \bigr) 
	\biggl( \frac{\alpha_Y}{0.01} \biggr)^{-1} 
	\biggl( \frac{\sigma_Y}{100 T} \biggr)^{1/2} 
	\per
}
If the hypercharge-weighted chiral chemical potential is smaller than this value, the chiral charge will be erased before the CPI develops, and no magnetic field will be generated.  

%-------------------------------------------
% Magnetogenesis
%-------------------------------------------
\subsection{Magnetogenesis}
\label{sub:maggen}

%====
As the chiral plasma instability develops, the growing helical hypermagnetic field is accompanied by a depletion of the hypercharge-weighted chiral chemical potential.  
The two quantities are linked via a chiral anomaly.  
The evolution of the comoving asymmetry number density $\Delta n_{Y,5}(\eta)$ is expressed as\footnote{\draftnote{Check the $O(1)$ coefficient.  It can be derived from eq.~(2.9) of 1606.08891.}}
\ba{
	\Delta \dot{n}_{Y,5}(\eta) = - \frac{\alpha_Y}{\pi} \, \dot{h}_B(\eta) + \cdots 
	\com
}
where $\Delta n_{Y,5}(\eta) = \mu_{Y,5} T^2 / 6$.  
We have dropped additional terms associated with the charge erasure processes, since we assume that these terms are negligible.  

%====
As the chemical potential is depleted, eventually the CPI is shut off.  
Typically the system equilibrates when the chemical potential is depleted by an $O(1)$ factor, and the helicity density grows by a commensurate factor.  
This observation lets us estimate the comoving helicity density at the time of the chiral plasma instability as 
\ba{
	h_{B,\cpi} \equiv |h_B(\eta_\cpi)| \sim \frac{\pi}{6\alpha_Y} \, |\mu_{Y,5}(\eta_\cpi)| \, T(\eta_\cpi)^2 
	\per 
}
Using the estimate of $T_\cpi$ from \eref{eq:cpi_estimates}, we obtain a numerical estimate for the comoving hypermagnetic helicity density at the time of the chiral plasma instability  
\ba{
	h_{B,\cpi}
	& \simeq \bigl( 4 \times 10^{-40} \Gauss^2 \Mpc \, a_0^3 \bigr) 
	\biggl( \frac{\alpha_Y}{0.01} \biggr)^{-1} 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{-1} 
	\biggl( \frac{\mu_{Y,5}/T}{10^{-3}} \biggr) 
	\per
}
The corresponding comoving coherence length $\xi_{B,\cpi}$ and comoving field strength $B_\cpi$ at the time of the chiral plasma instability are given by $\xi_{B,\cpi} \approx \lambda_\cpi$ and $B_\cpi \approx \sqrt{h_{B,\cpi} / \xi_{B,\cpi}}$, where the second relation assumes that the field is maximally helical.  
Numerical estimates are\footnote{\draftnote{Note to self:  Is this a good initial condition for the inverse cascade scaling?  Note that it doesn't satisfy $v_{A,\cpi} \sim \xi_{B,\cpi} / \eta_\cpi \sim \sqrt{2 \rho_{B,\cpi}/\rho_\cpi}$.  }}   
\ba{
\label{eq:xiB_cpi}
	\xi_{B,\cpi} 
	& \simeq 
	\bigl( 3 \times 10^5 \cm \, a_0^{-1} \bigr) 
	\biggl( \frac{\alpha_Y}{0.01} \biggr)^{-1} 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{1/3}
	\biggl( \frac{|\mu_{Y,5}|/T}{10^{-3}} \biggr)^{-1} 
	\\ 
\label{eq:B_cpi}
	B_\cpi
	& \simeq \bigl( 7 \times 10^{-11} \Gauss \, a_0^2 \bigr) 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{-2/3} 
	\biggl( \frac{\mu_{Y,5}/T}{10^{-3}} \biggr) 
	\per
}
The comoving magnetic energy density is $\rho_{B,\cpi} = B_\cpi^2 / 2$, and compared to the comoving critical density $\rho_{c,\cpi} = 3 a_\cpi^2 \Mpl^2 H_\cpi^2$ their ratio is $\Omega_{B,\cpi} = \rho_{B,\cpi} / \rho_{c,\cpi}$, and it evaluates to 
\ba{\label{eq:OmegaB_cpi}
	\Omega_{B,\cpi} 
	& \simeq \bigl( 8 \times 10^{-10} \bigr) 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{-1} 
	\biggl( \frac{\mu_{Y,5}/T}{10^{-3}} \biggr)^2 
	\per
}
The energy carried by the newly-generated magnetic field is tiny compared to the total energy density, which is mostly carried by the thermal plasma.  

%====
During the electroweak epoch, the Higgs field develops its nonzero vacuum expectation value (vev).  
In the Standard Model, this process is a continuous crossover, and in theories with new physics it may be a first-order phase transition.  
As the Higgs vev becomes nonzero, the hypermagnetic field is converted into an electromagnetic field.  
If this process occurs sufficiently slowly (e.g., if it's a crossover), it's reasonable to expect that the magnetic energy density is conserved, and the strength of the electromagnetic field after the conversion is comparable to the strength of the hypermagnetic field before the conversion.  
That is what I have assumed for the estimates above.  
On the other hand, if the conversion occurs quickly (e.g., if it's a first-order phase transition), it's possible that some magnetic energy will be lost (or generated), and more careful study is required to derive a reliable estimate for the electromagnetic field strength today.  

%-------------------------------------------
% Inverse cascade evolution
%-------------------------------------------
\subsection{Inverse cascade evolution}
\label{sub:inverse}

%====
After its generation the helical PMF evolves along with the fluid velocity subject to the non-linear equations of motion provided by the theory of MHD.  
Previous studies of helical MHD turbulence reveal that a maximally-helical magnetic field evolves into a scaling regime where power is transported from small length scales to larger scales.  
During this \textit{inverse cascade} scaling, the comoving magnetic field strength and comoving coherence length evolve as $B(\eta) \propto \eta^{-1/3}$ and $\xi_B(\eta) \propto \eta^{2/3}$ such that the comoving helicity density is approximately conserved $h_B(\eta) = \xi_B B^2 \propto \eta^0$.  
For order of magnitude estimates, it is reasonable to assume that the inverse cascade scaling begins immediately at the time of magnetogenesis $\eta = \eta_\cpi$, that it is maintained until recombination $\eta = \eta_\mathrm{rec}$, and that afterward the comoving field strength and coherence length are constant until today $\eta = \eta_0$.
This implies 
\ba{
	B_0 = B_\cpi \biggl( \frac{\eta_\mathrm{rec}}{\eta_\cpi} \biggr)^{-1/3}
	\qquad \text{and} \qquad 
	\xi_{B,0} = \xi_{B,\cpi} \biggl( \frac{\eta_\mathrm{rec}}{\eta_\cpi} \biggr)^{2/3}
	\com
}
and we define $h_{B,0} = \xi_{B,0} B_0^2$.  
We identify the fiducial plasma temperature at recombination as $(T/a)_\mathrm{rec} = 0.3 \eV$.  
%\ba{
%	\eta_\mathrm{rec} - \eta_\cpi 
%	& = \int_{t_\cpi}^{t_\rec} \! \frac{\ud t^\prime}{a(t^\prime)} \\ 
%	& = \int_{a_\cpi}^{a_\rec} \! \frac{\ud a^\prime}{\partial_t a(a^\prime) \, (a^\prime)} \\ 
%	& \qquad \qquad H(\eta) = (\partial_\eta a)/a = (\partial_t a) \\ 
%	& = \int_{a_\cpi}^{a_\rec} \! \frac{\ud a^\prime}{H(a^\prime) \, (a^\prime)} \\ 
%	& \qquad \qquad 3 a^2 \Mpl^2 H(a)^2 = \frac{\pi^2}{30} g_\ast(a) T(a)^4 \\ 
%	& = \sqrt{\frac{90}{\pi^2}} \Mpl \, \int_{a_\cpi}^{a_\rec} \! \frac{\ud a^\prime}{g_\ast(a^\prime)^{1/2} T(a^\prime)^2} \\ 
%	& \qquad \qquad g_{\ast,S}(a) T(a)^3 = g_{\ast,S,\rec} T_\rec^3 \\ 
%	& = \sqrt{\frac{90}{\pi^2}} \frac{\Mpl}{g_{\ast,S,\rec}^{2/3} T_\mathrm{rec}^2} \, \underbrace{\int_{a_\cpi}^{a_\rec} \! \ud a^\prime \frac{g_{\ast,S}(a^\prime)^{2/3}}{g_\ast(a^\prime)^{1/2}}}_{\ \approx \ a_\rec \, g_{\ast,S,\rec}^{2/3} \, g_{\ast,\rec}^{-1/2}} \\ 
%	& = \sqrt{\frac{90}{\pi^2}} \frac{\Mpl \, a_\rec}{g_{\ast,\rec}^{1/2} T_\mathrm{rec}^2} \\ 
%	& = \sqrt{\frac{90}{\pi^2}} \frac{\Mpl}{g_{\ast,\rec}^{1/2} \, a_\rec \, (T/a)_\mathrm{rec}^2} 
%}
To calculate the conformal time at recombination, we assume a radiation-dominated universe as in \erefs{eq:H}{eq:Hat_to_eta} but with only $g_{\ast,\rec} \simeq 3.36$ relativistic degrees of freedom.  
These estimates give 
\ba{\label{eq:eta_rec}
	\eta_\mathrm{rec} 
	\approx a_\rec / \Hcal_\rec
	\approx \frac{T_\rec / \Hcal_\rec}{(T/a)_\mathrm{rec}} 
	\simeq \bigl( 4 \times 10^{16} \sec \, a_0^{-1} \bigr) 
	\biggl( \frac{(T/a)_\mathrm{rec}}{0.3 \eV} \biggr)^{-1} 
}
where the comoving plasma temperature at recombination is the same as the comoving plasma temperature today $T_\rec = T_0 = (T/a)_0 \, a_0 \simeq (0.234 \meV) \, a_0$.  
Using these formulas, we estimate the physical field strength and coherence length today to be\footnote{Durrer and Neronov say that the inverse cascade evolution stops at recombination when the comoving field strength and coherence length are related by $B / 10^{-8} \Gauss \approx \xi_B / 1 \Mpc$; see Eq.~(91) on page 30 of the journal version.  However, \eref{eq:B_estimates} corresponds to $(B_0 / 10^{-8} \Gauss) \approx 100 \times (\xi_{B,0} / 1 \Mpc)$, which differs from Durrer \& Neronov's formula by a factor of $100$.  I think their calculation is too rough, and mine is more trustworthy.  My calculation could also be improved by accounting for the transition into a matter-dominated era, which occurs slightly before recombination; although, I can't see how this would account for a factor of $100$.}   
\bsa{eq:B_estimates}{
%---
	(B/a^2)_0 
	& \simeq \bigl( 7 \times 10^{-16} \Gauss \bigr) 
	\biggl( \frac{\alpha_Y}{0.01} \biggr)^{-2/3} 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{-5/9} 
	\biggl( \frac{\sigma_Y}{100 T} \biggr)^{1/3} 
	\biggl( \frac{\mu_{Y,5}/T}{10^{-3}} \biggr)^{1/3} 
	\biggl( \frac{(T/a)_\mathrm{rec}}{0.3 \eV} \biggr)^{1/3} \\ 
%---
	(a \xi_B)_0 
	& \simeq \bigl( 8 \times 10^{-4} \pc \bigr) 
	\biggl( \frac{\alpha_Y}{0.01} \biggr)^{1/3} 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{1/9} 
	\biggl( \frac{\sigma_Y}{100 T} \biggr)^{-2/3} 
	\biggl( \frac{\mu_{Y,5}/T}{10^{-3}} \biggr)^{1/3} 
	\biggl( \frac{(T/a)_\mathrm{rec}}{0.3 \eV} \biggr)^{-2/3} \\ 
%---
	(h_B / a^3)_0 
	& \simeq \bigl( 4 \times 10^{-40} \Gauss^2 \Mpc \bigr) 
	\biggl( \frac{\alpha_Y}{0.01} \biggr)^{-1} 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{-1} 
	\biggl( \frac{\mu_{Y,5}/T}{10^{-3}} \biggr)^{} 
	\per
}
Comparing with \eref{eq:B_cpi} shows that the comoving field strength drops by about five decades due to the inverse cascade evolution.  
Note that the field strength and coherence length have only a weak dependence on the hypercharge-weighted chiral chemical potential: increasing its value to $\mu_{Y/5}/T = O(1)$ would only boost $B$ and $\xi_B$ by one decade.  

%==================================
% Gravitational wave radiation
%==================================
\section{Gravitational wave radiation}
\label{sec:GWs}

%====
The formation of a primordial magnetic field due to the onset of the chiral plasma instability also provides a source of energy for gravitational wave radiation.  
Recall that $\lambda_\cpi = \pi^2 / \alpha_Y |\mu_{Y,5}|$ from \eref{eq:lambda_cpi} was the comoving wavelength of the fastest-growing mode when the CPI develops.  
As a rough order-of-magnitude estimate, we assume that this length scale also sets the wavelength of the gravitational wave radiation, $\lambda_\mathrm{gw}(\eta_\cpi) \approx \xi_B(\eta_\cpi) = \lambda_\cpi$.  
The corresponding frequency of this radiation today is $f_\mathrm{gw}(\eta_0) = c / a_0 \lambda_\cpi = \alpha_Y |\mu_{Y,5}| / a_0 \pi^2$.  
Using \eref{eq:lambda_cpi} gives 
\ba{\label{eq:nu_cpi}
	f_{\mathrm{gw},0}
	& \simeq 
	\bigl( 1 \times 10^5 \Hz \bigr) 
	\biggl( \frac{\alpha_Y}{0.01} \biggr)^{} 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{-1/3} 
	\biggl( \frac{\mu_{Y,5}/T}{10^{-3}} \biggr)^{} 
	\per
}
So this is a $\sim 10 \kHz$ signal.  
For reference, the LIGO interferometer's sensitivity is maximal around $100 \Hz$.  
LIGO's sensitivity is worsening at higher frequencies due to quantum shot noise, but there is still some sensitivity at $10 \kHz$.  
Moreover, there is growing effort to develop technologies that might be able to probe higher-frequency gravitational waves.  
Space-based interferometers like LISA are targeting lower frequencies $\sim \mathrm{mHz}$, and they will not be useful.  

%====
The amplitude of the gravitational wave signal is parametrized by $\Omega_{\mathrm{gw},0} = \rho_{\mathrm{gw},0} / \rho_{c,0}$, which is the ratio of the comoving gravitational wave energy density today $\rho_{\mathrm{gw},0}$ to the comoving critical density today $\rho_{c,0} = 3 a_0^2 \Mpl^2 H_0^2$.  
I can't be bothered to calculate $\Omega_{\mathrm{gw},0}$ right now for these notes.  \draftnote{note to self:  include a calculation of the efficiency}
But we obtain a ridiculously generous over-estimate by imposing $\Omega_{\mathrm{gw},0} < \Omega_{B,0}$.  
In other words, the energy carried by the gravitational wave radiation cannot exceed the energy carried by the magnetic field that was sourcing it.  
This is a generous over-estimate, since it neglects important factors of $G_N \sim 1 / \Mpl^2$.  
Since the comoving gravitational wave energy density is conserved after production, we can estimate $\Omega_{\mathrm{gw},0} = \Omega_{\mathrm{gw},\cpi} \ll \Omega_{B,\cpi}$, and using \eref{eq:OmegaB_cpi} gives 
\ba{\label{eq:Omegagw_cpi}
	\Omega_{\mathrm{gw},0} 
	& \ll \bigl( 8 \times 10^{-10} \bigr) 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{-1} 
	\biggl( \frac{\mu_{Y,5}/T}{10^{-3}} \biggr)^2 
	\per
}
The corresponding strain $h_c$ is calculated using $\Omega_{\mathrm{gw}} = (2 \pi^2 f^2 h_c^2(f) / 3 (H_0/a_0)^2$.  
Using the frequency in \eref{eq:nu_cpi} gives 
\ba{
	h_c 
	& \ll \bigl( 2 \times 10^{-28} \bigr) 
	\biggl( \frac{\alpha_Y}{0.01} \biggr)^{-1} 
	\biggl( \frac{g_\ast}{106.75} \biggr)^{-1/6} 
	\biggl( \frac{(H/a)_0}{70 \km / \mathrm{sec} / \mathrm{Mpc}} \biggr)^{} 
	\per
}
To reiterate, these are generous overestimates for the gravitational wave strength, since they assume a $100\%$ efficient conversion of magnetic field energy into gravitational wave energy.  
I should come back and do a more accurate estimate that properly accounts for the $G_N$ suppression factors.  


%==================================
% Baryogenesis and baryon over-production
%==================================
\section{Baryogenesis and baryon over-production}
\label{sec:BG}

%====
The Standard Model predicts that baryon number $\textsf{B}$ and lepton number $\textsf{L}$ are not exactly conserved quantities, but rather both are violated by non-perturbative effects.  
Most famously, the electroweak sphaleron process is operative in a Standard Model plasma at temperatures above $\approx 100 \GeV$ where the electroweak symmetry is restored.  
However, more relevant to the present discussion is a second effect associated with helical hypermagnetic fields.  
Changes in hypermagnetic helicity are a source (or sink) of baryon and lepton number.  
The relation is expressed by the current conservation equation $\partial_\mu j_\textsf{B}^\mu = \partial_\mu j_\textsf{L}^\mu \propto \Evec_Y \cdot \Bvec_Y$.  
After integrating over volume, the right-hand side is proportional to the time rate of change of the hypermagnetic helicity $\dot{h}_B \sim \Evec_Y \cdot \Bvec_Y$.  
Consequently, changes in hypermagnetic helicity provide a source of baryon number.  

%====
As the magnetic field evolves, its hypermagnetic helicity is very nearly conserved.  
However, it's not exactly conserved, since there is always a slow decay due to Ohmic losses, $\Evec_Y \sim \dvec \times \Bvec_Y / \sigma_Y$.  
This slow decrease in hypermagnetic helicity has a negligible impact on the magnetic field, but it is nevertheless very important for the generation of baryon number.  
In addition to the continuous slow decay of hypermagnetic helicity, there is a second (more important) place where hypermagnetic helicity changes.  
As the hypermagnetic field is converted into an electromagnetic field during the electroweak epoch, the full hypermagnetic helicity is depleted, and there is a corresponding growth in baryon number.  

%====
This baryon-number generation at the electroweak epoch is potentially dangerous.  
Observations of the cosmic microwave background radiation tells us that the baryon asymmetry of the universe is $n_\textsf{B} / s \sim 10^{-10}$.  
If a larger baryon asymmetry were generated by hypermagnetic helicity decay, there would be a problem.  
\href{https://arxiv.org/abs/1610.03074}{Kamada \& Long (2016)} argued that\footnote{Along the trajectory in parameter space where $(B/a^2)_0 / 10^{-8} \Gauss = (a \xi_B)_0 / \mathrm{Mpc}$.  These formulas don't show the dependence on $\alpha_Y$, $\sigma_Y$, and $g_\ast$.  \draftnote{Note to self:  I should revisit these estimate to see if they can be expressed more precisely.  In particular, these limits assume $\frac{(B/a^2)_0}{10^{-8} \Gauss} \approx \frac{(a \xi_B)_0}{1 \Mpc}$, but this relation doesn't hold for \eref{eq:B_estimates}.} }
\ba{
	(B/a^2)_0 \lesssim 10^{-17} \Gauss 
	\qquad \text{and} \qquad 
	(a \xi_B)_0 \lesssim 10^{-3} \pc 
}
is required to avoid baryon overproduction if the primordial magnetic field is generated before the electroweak epoch and if it evolves subject to the inverse cascade scaling.  
Comparing with the magnetogenesis estimates in \eref{eq:B_estimates}, we find that avoidance of baryon-number overproduction requires the hypercharge-weighted chiral chemical potential to satisfy $\mu_{Y,5}/T \lesssim 10^{-2}$.  
Taken in combination with the erasure-avoidance condition \pref{eq:erasure_avoidance}, we find a via range of values for the hypercharge-weighted chiral chemical potential that allows for magnetogenesis without running into either problems with chiral charge erasure or baryon overproduction: 
\ba{
	10^{-3} \lesssim |\mu_{Y,5}|/T \lesssim 10^{-2} 
	\per
}

%==================================
% Dark radiation constraints
%==================================
\section{Dark radiation constraints}
\label{sec:dark_rad}

%====
Observations of the cosmic microwave background radiation constrain the presence of dark radiation in the universe.  
Dark radiation is a hypothetical additional energy component that redshifts like radiation, but doesn't couple (strongly) to ordinary radiation and baryonic matter.  
Whereas radiation and baryons are tightly coupled prior to recombination and experience baryon acoustic oscillations, the dark radiation does not participate.  
The CMB constraints are conventionally expressed as an upper limit on the effective number of relativistic species $N_\mathrm{eff}$ defined by $\rho_\mathrm{rad} = N_\mathrm{eff} (7/8) (4/11)^{4/3} \rho_\gamma$; the standard cosmological model has $N_\mathrm{eff} \simeq 3.046$ arising from the cosmic neutrino background, but the presence of dark radiation would allow larger $N_\mathrm{eff}$.  
Constraints from current-generation telescopes (Planck 2018 + BAO) are at the level of $\Delta N_\mathrm{eff} \lesssim 0.3$.  



This limit can be recast as a limit on chiral asymmetry.  
See Sec.~6.1 of \href{https://arxiv.org/abs/1405.7654}{Long, Lunardini, \& Sabancilar (2014)}.  

%==================================
% Summary of viable parameter space
%==================================
\section{Summary of viable parameter space}
\label{sec:summary}

%====
\begin{center}\includegraphics[width=0.80\textwidth]{fig_summary.pdf}\end{center}

%==================================
% Modifications
%==================================
\section{Modifications}
\label{sec:modifications}

%====
Can the calculation be modified so as to yield a more favorable outcome for CPI at the EW epoch?  
Here are some initial thoughts, which aren't very developed.  

%-------------------------------------------
% Additional relativistic particles
%-------------------------------------------
\subsection{Additional relativistic particles}
\label{sub:extra_rel}

%====
In this note, the effective number of relativistic species has been kept as a free parameter apart from the (weak) assumption that $g_\ast(\eta) = g_{\ast,S}(\eta)$.  
The Standard Model primordial plasma at temperatures $T \gtrsim 100 \GeV$ contributes $g_{\ast,\text{\sc sm}} = 106.75$, and so we have used the fiducial value $g_\ast = 106.75$ for numerical estimates.  
However, the presence of additional relativistic particles (e.g., Standard Model superpartners) would imply $g_\ast > g_{\ast,\text{\sc sm}}$.  
The quantity $g_\ast(\eta)$ is constrained to be $O(1)$ at a time when the plasma temperature is $T \sim \mathrm{MeV}$, since the plasma physics at this time impacts observables such as the abundances of light elements (BBN) and the CMB via the cosmic neutrino background.  
However, at earlier times $g_\ast(\eta)$ is unconstrained.  
It may be interesting to consider a modified cosmology in which $g_\ast \gg g_{\ast,\text{\sc sm}}$ in the early universe at plasma temperatures $T \gg \mathrm{MeV}$.  
The formulas in this note already reveal how the various predictions change when $g_\ast$ is allowed to vary away from $106.75$.  

%-------------------------------------------
% Modified cosmic history
%-------------------------------------------
\subsection{Modified cosmic history}
\label{sub:mod_cosmic}

%====
The calculations presented here have assumed that the universe is radiation-dominated around the electroweak epoch.  
However, the cosmological expansion history prior to nucleosynthesis ($T \sim \mathrm{MeV}$) is entirely unknown.  
One can assume that the universe experienced an anomalous early period of matter domination ($w=0$), kination domination ($w=1$), or some other phase.  
By making the universe expand more quickly, this typically changes the comparison of particle interaction rate and Hubble expansion rate, $\Gamma \sim H$, so as to make the two balance earlier or later than in a radiation-dominated cosmology.  
This might help us to get CPI occurring before the hypercharge-weighted chiral asymmetry is erased.  

%-------------------------------------------
% Additional particle asymmetries
%-------------------------------------------
\subsection{Additional particle asymmetries}
\label{sub:extra_asym}

%====
The hypercharge-weighted chiral asymmetry is ultimately erased because it's stored in the right-chiral electron (and their $\textsf{CP}$-conjugate antiparticle partners), and these particles are created and destroyed by their Yukawa interactions with the Higgs.  
If there existed a new fermionic particle (beyond the Standard Model) that carries hypercharge but does not have a Yukawa coupling with the Higgs, then this particle could store the chiral asymmetry, even after the asymmetry from right-chiral electrons is erased.  
This could allow the CPI to occur at the electroweak epoch.  
To make this work, the new particle would have to carry a hypercharge, which means that it's electromagnetically charged today.  
And it would have to have a mass of no more than $\sim 100 \GeV$ to ensure that it's in equilibrium and relativistic at the electroweak epoch.  
This means that there will be constraints from LHC searches for new charged particles.  
There are constraints, but it might still be viable.  


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%Bibliography
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\bibliography{asdf}

\end{document}