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\begin{document}


\title{Supplementary information for the letter: "Formation and Coalescence of Cosmological Supermassive Black Hole Binaries in Supermassive Star Collapse"}


\author{C. Reisswig}
\thanks{Einstein Fellow}
\email{reisswig@tapir.caltech.edu}
\affiliation{TAPIR, MC 350-17, California Institute of Technology, 1200 E California Blvd.,
Pasadena, CA 91125, USA}

\author{C. D.\ Ott}
\thanks{Alfred P. Sloan Research Fellow}
\affiliation{TAPIR, MC 350-17, California Institute of Technology, 1200 E California Blvd.,
Pasadena, CA 91125, USA}
\affiliation{Kavli Institute for the Physics and
 Mathematics of the Universe (Kavli IPMU), The University of Tokyo, Kashiwa, Japan}

\author{E. Abdikamalov}
\affiliation{TAPIR, MC 350-17, California Institute of Technology, 1200 E California Blvd.,
Pasadena, CA 91125, USA}

\author{R. Haas}
\affiliation{TAPIR, MC 350-17, California Institute of Technology, 1200 E California Blvd.,
Pasadena, CA 91125, USA}

\author{P. M\"osta}
\affiliation{TAPIR, MC 350-17, California Institute of Technology, 1200 E California Blvd.,
Pasadena, CA 91125, USA}

\author{E. Schnetter}
\affiliation{Perimeter Institute for Theoretical Physics, 31 Caroline
  St.\ N., Waterloo, ON N2L 2Y5, Canada}
\affiliation{Department of Physics, University of Guelph, 50 Stone
  Road East, Guelph, ON N1G 2W1, Canada}
\affiliation{Center for Computation \& Technology, 216 Johnston Hall,
  Louisiana State University, Baton Rouge, LA 70803, USA}

\pacs{04.25.D-, 04.30.Db, 97.60.Bw, 02.70.Bf}
  
\maketitle
  

\section{Numerical Convergence}

\begin{figure}
  \includegraphics[width=1.\linewidth]{H_conv}
  \caption{Convergence of the $L_2$-norm of the Hamiltonian constraint
    $\norm{H}_2$ of model M2G2 for the low $r0$, medium $r1$, and high
    resolution $r2$ (not rescaled).  As the resolution is increased,
    the error decreases with a convergence rate between first and
    second-order as expected.  The peaks occur when the black holes
    form, and when the black hole interior is not yet excluded from the 
    calculation of the $L_2$-norm.}
           \label{fig:H_conv}
\end{figure}


We show numerical convergence of our results by considering the
Hamiltonian constraint $H$.  $H = 0$ is one of the four elliptic
constraint equations which arise from the 3+1 ADM decomposition of the
Einstein equations (e.g.~\cite{alcubierre:11}).  Any solution to the
Einstein equation must satisfy these constraints.  At the continuum
level, if they are satisfied initially, they are automatically
satisfied at all times.  During a numerical evolution, however,
numerical error can lead to constraint violations.  As the resolution
is increased, the constraints must converge to zero at the expected
order of accuracy inherent to the numerical scheme.

As detailed in \cite{reisswig:13a}, the overall accuracy of our
simulations is limited by the finite-volume hydrodynamics scheme,
which is second-order accurate in regions where the fluid flow is
smooth, and reduces to first order near shocks and
discontinuities. Thus, we expect between first and second-order
convergence.  We note that the we evolve the spacetime geometry with a
fourth-order finite-difference scheme \cite{reisswig:13a}, but the
overall error is dominated by the fluid evolution.

In Fig.~\ref{fig:H_conv}, we show the $L_2$-norm of the Hamiltonian
constraint $\norm{H}_2$ of model $M2G2$ (which forms a pair of black
holes) for three resolutions labeled by $r0$, $r1$, and $r2$.  The
coarse resolution, $r0$, has $25\%$ reduced resolution compared to
baseline resolution $r1$, and $r2$ has $25\%$ increased resolution.
The $L_2$-norm is taken over the entire domain, excluding the interior
of the black hole horizons.
With increasing resolution, $\norm{H}_2$ decreases with a convergence
rate consistent with first and second order as expected. More
specifically, we find that $\norm{H}_2^{r2}\leqslant 1.25
\norm{H}_2^{r1} \leqslant 1.25\norm{H}_2^{r0}$.  We note that the
numerical evolution is initially non-convergent for the first
$t\leqslant250M$ due to the non-constraint preserving application of
an initial density perturbation.


\section{Conversion of Units}


For convenience, we provide the conversion factors from code units
$c=G=M=K=1$ to cgs units in Table~\ref{tab:units}.  As an example for
a $M=10^6M_\odot$ star, a code unit time of $T=1M$ corresponds to a
physical time $t\simeq4.93\,\rm{sec}$, the initial stellar equatorial
radius $r_e=80M$ corresponds to
$\simeq1.2\times10^8\,\rm{km}\simeq0.8\,\rm{AU}$, and the initial
central density $\rho_c=3.38\times10^{-6}M^{-2}$ is
$\simeq2.1\,\rm{g}\, \rm{cm}^{-3}$.

\begin{table}[t]
\caption{Conversion factors for 1 solar mass $M_\odot$ in code units $c=G=M=K=1$
  to cgs units.}
\label{tab:units}
\begin{ruledtabular}
\begin{tabular}{lll}
Quantity & code & cgs \\
\hline
\\ [-2ex]
Mass $M$       & $M$ & $1.9891\times10^{33}(M/M_\odot)$ g \\
Length $L$     & $M$ & $1.4771\times10^5(M/M_\odot)$ cm\\
Time $T$       & $M$  & $4.9271\times10^{-6}(M/M_\odot)$ sec \\
Density $\rho$ & $M^{-2}$ & $6.1716\times10^{17}(M_\odot/M)^2$ g cm${}^{-3}$ \\ 
\end{tabular}
\end{ruledtabular}
\end{table}




\bibliography{bibliography/cosmology_references,bibliography/bh_formation_references,bibliography/gw_references,bibliography/gr_references,bibliography/sn_theory_references,bibliography/grb_references,bibliography/stellarevolution_references,bibliography/methods_references,bibliography/gw_data_analysis_references,bibliography/NSNS_NSBH_references,bibliography/misc_references,bibliography/cs_hpc_references,bibliography/numrel_references,bibliography/publications-schnetter,bibliography/accretion_disk_references,bibliography/ns_references} 


\end{document}


