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\title{Technical note for Galactic Binary Searches of the Second Round of the Mock LISA Data Challenge}

\author{Jeff Crowder}
\affiliation{Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109}
\author{Neil J. Cornish}
\affiliation{Department of Physics, Montana State University, Bozeman, MT 59717}
\author{Tyson B. Littenberg}
\affiliation{Department of Physics, Montana State University, Bozeman, MT 59717}


\begin{abstract}

This work briefly discusses the searches performed on the blind data set prepared by the {working group} of the LISA International Science Team for Round $2$ of the the Mock LISA Data Challenge (MLDC). These results address searches for the monochromatic binary systems in Challenges 2.1 and Challenge 2.2. The algorithm used to perform these searches is the Blocked-Annealed Metropolis Hasting algorithm (BAM). While the search was cut short due to restrictions of time and access to necessary computing resources, we were able to identify $19324$ candidate sources in Challenge 2.1 and $18461$ candidate sources in Challenge 2.2.

\end{abstract}

\maketitle

\section{Updates to the BAM}

As the BAM algorithm has been discussed in previous work~\cite{BAM,BAM_MLDC} we will not reiterate many of the details here, but will instead give a flavor of some of the updates that have been implemented in the algorithm.

The first of these updates is the ability to automatically determine the noise levels of the data stream. This is done by introducing three search parameters, $N_A$, $N_E$, $N_AE$, which represent a multiplicative factor for the noise level in the pseudo- A \& E channels, and the noise correlation between them. The likelihood function, with the noise factors explicitly written is given by:
\begin{equation}\label{likelihood_equation}
p(s \vert \vec{\lambda}) = C \exp\left( -\frac{\chi^2}{2} - N_b \ln \left(\frac{{\rm det}(S)}{S_n^2}\right)\right),
\end{equation}
where $N_b$ is the number of frequency bins spanned by the window, 
\begin{equation}
\chi^2 = \frac{4}{Tobs} \sum_i \left(S^{-1}\right)^{\alpha\beta}(s_alpha-h_alpha)_i(s_beta-h_beta)_i^*
\end{equation}
The noise correlation matrix, $S$, is given by:
\begin{equation}
S_{\alpha\beta} = S_n \left(  \begin{array}{cc}  N_A & N_{AE} \\ N_{AE} & N_E \\  \end{array} \right)
\end{equation}
where $S_n$ is a reference noise level, chosen here to be the standard LISA instrument noise plus the theoretical contribution of the galactic confusion limited background~\cite{TRC}.

The second update is the implementation of a new model selection criteria. We now use the Bayesian Information Criterion (BIC) to determine whether to accept an increase in the model number of the search. The BIC is calculated according to the following:
\begin{equation}
{\rm BIC} = (\ln{p(s \vert \vec{\lambda})}+N_b \ln{(N_A N_E - N_{AE}^2)} + (7 N_s + 3) \ln (2 N_b),
\end{equation}
where $N_s$ is the number of galactic binary system in the current model. An increase in the model is accepted if the value of the BIC decreases by more than $2$. 

The third update was the selection for the best fit values of the chain. We now use the best Maximum A Posteriori (MAP) estimate of the chain. This is determined by product:
\begin{equation}
{\rm MAP} = p(\vec{\lambda}) \ln{p(s \vert \vec{\lambda})},
\end{equation}

Our use of the BAM for these searches followed a hierarchical approach, where we first searched the entire band using large windows ($\sim 10 \mu$Hz across). The searches of each window started at $0$ and continued looking for up to $2$ candidate sources. This allowed us to identify extremely bright sources in the data that might bleed power into frequency bins far removed from the true source's frequency, and would affect the fit to other candidate sources in neighboring search windows. The next round of searches used smaller windows ($\sim 2 \mu$Hz across) and looked for up to $6$ candidate sources. If a window contained a candidate source from the previous round, its parameters were given as the starting point for one of the candidate sources in the next round. As the number of candidate sources grew the window sizes were decreased (particularly the sizes of the wings in each window, which increases the size of the acceptance window relative to the total window size), until the windows were $\sim 1.5 \mu$Hz across, and we were looking for up to $15$ candidate sources per window.

Our results returned $19324$ candidate sources for Challenge 2.1 and $18461$ for Challenge 2.2. The main reason for this difference was the time allocated to each search. As Challenge 2.1 did not involve the presence of any other sources such as Supermassive Black Hole Binaries (SMBHBs) we were able to begin immediately searching for galactic sources across the entire LISA band. Challenge 2.2, on the other hand, contained bright SMBHBs that needed to be regressed from the data stream before galactic binaries could be pulled out at lower frequencies. This lessened the time available to ``dig deep'' into the source rich regions between $1$ and $4$ mHz. Also, due to the presence of ``higher priority'' users on the supercomputing cluster where most of our runs were performed, we were limited in our searches. So while we are pleased at the performance to this point of the BAM algorithm, this should not be seen as the best that it can do. In the future we will be taking these results and expanding them to see where the actual limits of the BAM algorithm lay.

\begin{thebibliography}{99}

\bibitem{BAM} Crowder J. \& Cornish N. J., Phys. Rev. D {\bf 75} 043008 (2007).
\bibitem{BAM_MLDC} Crowder J. \& Cornish N. J., gr-qc/07042917 (2007).
\bibitem{TRC} S. Timpano, L. J. Rubbo \& N. J. Cornish, PRD {\bf 73} 122001 (2006). 
\bibitem{tyson} N. J. Cornish \& T. B. Littenberg, gr-qc/07041808 (2007).

\end{thebibliography}

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