% $Id: notes.tex,v 1.8 2021/01/24 05:18:52 brandenb Exp $
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\input macros
\def\EEK{{\Omega}_{\rm K}}
\def\EEM{{\Omega}_{\rm M}}
\def\EEGW{{\Omega}_{\rm GW}}
\def\red{\textcolor{red}}
\def\blue{\textcolor{blue}}
\title{Changing $\kf$}
\author{}
\date{\today,~ $ $Revision: 1.8 $ $}
\begin{document}
\maketitle

\section{}

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Hypothesis:
\begin{equation}
\EEGW=\left(q\EEM/\kf\right)^2
\end{equation}
Determine $q$; we find $q\propto\kf^{1/4}$; see \Fig{pres}.
The GW energy is high by comparison with earlier work; see \Fig{EEGW_vs_EEKM}.

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pcomp}
\end{center}\caption[]{
Black, blue, and red are for $\kf=6$, 20, and 60, respectively.
}\label{pcomp}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pres}
\end{center}\caption[]{
$\EEGW=5.2\times10^{-4}\,(\EEM/\kf)^{1/2}$
}\label{pres}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{EEGW_vs_EEKM}
\end{center}\caption[]{
Positions of the new runs in a diagram showing
$\Omega_{\rm GW}^{\rm sat}$ versus $\Omega_{\rm M}^{\max}$,
For orientation the old data points of the Ref.~\cite{Pol:2019yex}
are shown as gray symbols.
}\label{EEGW_vs_EEKM}\end{figure}

\begin{table}[htb]\caption{
All runs with $\eta=2\times10^{-3}$.
}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
$k_1$ & $f_0$ & $\EEM$ & $\EEGW$ \\%run
\hline
  1 & 6.0e-02 & 3.874e-02 & 4.189e-05\\%M64sig0_k6_wav1d
  3 & 3.2e-02 & 4.038e-02 & 2.313e-05\\%M64sig0_k6_wav3_3p2em2
 10 & 2.3e-02 & 3.831e-02 & 1.375e-05\\%M64sig0_k6_wav10_2p34em2a
\label{Ttimescale}\end{tabular}}\end{table}

%r e f
\begin{thebibliography}{99}

\bibitem{Pol:2019yex}
  A.~Roper~Pol, S.~Mandal, A.~Brandenburg, T.~Kahniashvili and A.~Kosowsky,
  ``Numerical Simulations of Gravitational Waves from Early-Universe Turbulence,''
%AB: should refer to published version:
  Phys. Rev. D {\bf 102}, 083512 (2020).
  doi:10.1103/PhysRevD.102.083512

\end{thebibliography}

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