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\newcommand{\neff}{N_{\rm eff}}
\newcommand{\dneff}{\Delta N_{\rm eff}}
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\newcommand{\nv}{N_{\nu}}
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\usepackage{nicefrac}


\begin{document}

\title{Supplementary Material to ``Big Bang Nucleosynthesis Limits and Relic Gravitational Waves Detection Prospects"}

\date{\today}
\preprint{NORDITA-2022-0XX}



\maketitle

\section{Analytical Approach/Gravitational Waves}

We follow the standard description of the gravitational waves (GWs) generation by the early universe sources, see 
Chapter 22 of \cite{MaggioreV2}, and \cite{Gogoberidze:2007an}: 
The solution for the transverse-traceless tensor metric perturbations in the weak field limit sourced by the transverse-traceless source $S_{ij}({\bf x}, t)=\lambda_{ij,lm} T_{lm}$ (with $\lambda_{ij,lm} = $ is tensor projection operator) is given through the following {\it retarded} integral, 
\begin{eqnarray}
h_{ij}^{TT} ({\bf x}, t) &= & 4G \int d^3 {\bf x}^\prime 
\frac{S_{ij}({\bf x}^\prime, t-|{\bf x} - {\bf x}^\prime|)}{|{\bf x} - {\bf x}^\prime|} 
\label{GWs-eq}
\end{eqnarray}




\section{Numerical Set-up/Gravitational Waves}



\section{Lepton Asymmetry}
The lepton asymmetry $\eta_L$ can be described by the asymmetry between neutrinos and antineutrinos, since it is generally assumed that any asymmetry in \textit{charged} leptons is very small (of the order of the baryon asymmetry $\eta_B \sim 10^{-10}$) due to the charge neutrality of the universe. The asymmetry between massless neutrinos and antineutrinos of flavor $\alpha = (e,\mu,\tau)$ can be written in terms of the chemical potential $\mu_\alpha$, or the dimensionless variable $\xi_\alpha \equiv \mu_\alpha/kT$ \cite{Simha:2008mt}
\begin{equation}
    \eta_L \equiv \Sigma_\alpha \frac{n_{\nu_\alpha}-n_{\bar{\nu}_\alpha}}{n_\gamma}=\frac{\pi^3}{12\zeta(3)}\Sigma_\alpha\bigg[\bigg(\frac{\xi_\alpha}{\pi}\bigg)+\bigg(\frac{\xi_\alpha}{\pi}\bigg)^3\bigg].
\label{eq:etaL}
\end{equation}


The neutrinos begin decoupling from the photon-$e^\pm$ plasma when the photon temperature drops below $\sim$ 2 MeV. Electron neutrinos continue interacting with neutrons and protons through charged-current weak interactions
%($n+\nu_e \rightleftarrow p + e^-$, $p + \bar{\nu}_e \rightleftarrow n + e^+$, $n \rightleftarrow p + e^- + \bar{\nu}_e$)
and the existence of a nonzero electron neutrino chemical potential shifts the neutron-to-proton equilibrium ratio and affects the abundance of $^4$He.  This makes the $^4$He primordial mass fraction $\Yp$ sensitive to $\xi_e$. 
Following \cite{Simha:2008mt}, we express $\xi_e$ in terms of parameters constrained by the CMB, BBN, and primordial abundances;
\begin{equation}
    \xi_e = \frac{4}{574}\bigg(\etab - \etaHe + 100\bigg[\sqrt{1 + \frac{7}{43}\dneff} \,- 1\bigg]\bigg).
\label{eq:xie}
\end{equation}
where $\etab = 10^{10}\eta_{B}$, $\dneff$ are the additional relativistic degrees of freedom, and $\etaHe$ is related to $\Yp$ through the analytical approximation \cite{Kneller:2004jz}
\begin{equation}
    \Yp = (0.2482\pm0.0006) + 0.0016(\etaHe - 6).
\end{equation}

%Combining all of this, we can compute values of $\eta_L$ around the 95\% contour on the joint probability distribution for $\neff$ and $\etab$ (Specifically, the \texttt{CMB + BBN + $Y_p$ + D} contour on Figure 18 of \cite{Fields:2019}). Along this contour, we obtain a value of $\xi_e =0.0173$.
%EC: reworded, please check this is correct
We compute an upper limit on $\xi_e$ from equation \ref{eq:xie} using joint constraints from CMB, BBN, $Y_p$, and deuterium abundance measurements on $\neff$ and $\etab$ at the 95\% confidence level and using the mean value of $\Yp$ as found in \cite{Fields:2019}. We obtain a value of $\xi_e =0.0173$.

In general, neutrino oscillations equilibrate the neutrino flavor asymmetries prior to BBN. However, depending on the neutrino mixing angles and size of the initial neutrino flavor asymmetries, flavor equilibrium may not be reached.
For simplicity, here we assume the case of flavor equilibrium where $\xi_e=\xi_\mu=\xi_\tau$. With this assumption and our value of $\xi_e$, we use equation \ref{eq:etaL} to find a lepton asymmetry of
\begin{equation}
    \eta_L \lesssim 0.0355. %\quad \text{(At 95\% certainty)}
\end{equation}
%EC: I think we can leave out the 95% part since we already mentioned how xi_e is computed...?
%EC: note on improvement





\bibliography{Chirality-BBN}
\end{document}
The data for $\EEGW$ follow a power law scaling, $\propto\EEM^n$,
where $n=2.7$ for the points with the smallest viscosity.
This is steeper than the quadratic scaling found in the work of
%%TK cite Roper Pol +
where the driving was applied for a much shorter time
interval, $1\leq t\leq1.1$.
Furthermore, for fixed values of $\nu$, we find smaller local values of
$n$, at least for the larger magnetic energies shown in \Fig{ppe}.
We also checked that these scalings are not significantly
affected if the driving was turned off abruptly after $t=2$.
This is shown as the dotted line in \Fig{ppe} for $\nu=5\times10^{-5}$.

Comparing the lines for $\nu=5\times10^{-5}$ and $\nu=10^{-4}$,
we see that the decline of $\EEM$ is stronger than that of $\EEGW$.
This suggests that $\EEM$ suffers more strongly from
the increase of viscosity and magnetic diffusivity, and that
$\EEGW$ is less sensitive to the change of $\nu$.
However, one has to remember that GWs are solely the results of the
magnetic and hydrodynamic stresses.
One sees that the runs with smaller values of $\nu$ all have a faster
rise of $\EEM(t)$ early on, which also translates into a rapid increase
of $\EEGW(t)$.
It is unclear, however, whether this aspect of the model with applied
magnetic driving is realistic and whether this would also be borne out
by a more physical implementation of a magnetogenesis model.

