
\newpage

\section{Scientific outcomes from GW observation}

Scientific outcomes obtained from GW observation highly depend on the sensitivity 
and observation band of a detector.
In this Section, we survey possible GW sources which would be targets of LCGT,
and show advantages and disadvantages of each candidate interferometer configurations.

The most important criteria in the comparison is to increase the possibility to 
achieve the minimum success of LCGT: 
detect at least a few GW events within one year operation.
In addition, extracted information from observed waveforms 
(for example, binary mass parameters) and those accuracies should be considered.
The primary target of LCGT is coalescences of neutron-star binaries.
Thus, we at first estimate the observable range, expected event rates, 
required observation period for the first detection,
and measurement accuracy of binary mass parameters for these sources.
After that, we survey scientific outcomes from the other GW sources:
black-hole binaries, black-hole ringdown, supernovae with stellar-core collapses,
pulsars.
We also compare the candidate configurations from a viewpoint of international observation network.


%We would like to summarize possibility of observation: range for event search, expected event rates, 
%required observation period for the first detection. 
%We also need to compare some possibility in data analysis 
%as measurement accuracy of binary mass parameters. 

%In the view of the observation, we must choose the configuration which ensures the minimal success of first stage of LCGT project:
%it will be able to detect at least a few GW events
%within one year operation. All proposed detector configurations have enough neutron star detection range as exceed 200 Mpc. 
%However, 10\% difference of the range will make about 30\%
%in event rate, we suspect that it is better to choose DRSE as possible.
%On the other hand, possibility of various GW sources and analysis on GW sources require wider frequency band data. 
%BRSE becomes absolutely necessary.
%Therefore in these reasons, {\bf we recommend varBRSE (variable configuration in prior BRSE) or maxBRSE
% for observation and data analysis.}

%In following subsections, we display estimations for possible GW sources/events.


\subsection{Neutron star binary coalescences}

Coalescences of neutron star (NS) binary are most promising target sources
with known waveforms and reasonable event rates. 
Here, we estimate the detection probability of these sources.

\vspace*{2mm}
\noindent
{\bf Expected event rate and detector ranges}


The expected detection rate for these events in one year observation is given as 
\begin{equation}
\overline{N_{\rm ev}}\ =\ \frac{4}{3}\pi \ d^3 /(1+z)^3 \ \rho_{\rm galaxy}\ {\cal R} ,
\label{eq:nev}
\end{equation}
where $ d $ is an observable distance of the detector, 
$ \rho_{\rm galaxy} $ is a number density of galaxies,
$ {\cal R} $ is a binary merger rate for milky-way equivalent galaxy, and
$z$ is a redshift at range distance. 
$(1+z)^3$ is a correction for commoving volume.
A galactic binary merger rate ${\cal R}$ are expected as 82 events per mega-years 
for milky-way equivalent galaxy \cite{kalogera2004erratum,kalogera2004}:
$$
{\cal R}\ =\ 83.0\ ^{+209.1}_{-66.1} \ {\rm (C.I.\,95\%)} \ {\rm [events\ Myr^{-1}].}
$$
The detection range of the LCGT is determined by the GW event's signal-to-noise (${\rm S/N}$),
which is calculated from detector noise spectrum%
\footnote{In the case of the optimal direction (azimuthal incident, best polarization), 
${\rm S/N}$ will be given in following relations (In detail and notations, see \cite{tagoshi2001}):
\begin{equation}
{\rm S/N} = \sqrt{2} \ A \ \left[ 4 \int \frac{f^{-\frac{7}{3}} }{S_{n}(f)} df\ \right]^{\frac{1}{2}}
\label{eq:SNR}
\end{equation}
where, 
$$
A = T_{\odot} \ \frac{c}{d} \ \left( \frac{5 \mu}{96 M_{\odot}} \right)^{\frac{1}{2}} 
\left( \frac{m}{\pi^{2} M_{\odot}}\right)^{\frac{1}{3}} T_{\odot}^{-\frac{1}{6}},\ \ 
T_{\odot} = \left( \frac{G}{c^{3}}\right) M_{\odot}, \ \ \ m = m_1 + m_2,
\ \ \ \mu = \frac{m_1 m_2}{m},
\label{eq:nrm}
$$
and $S_{n}(f)$ is average noise spectrum of the detector.}.
We define the detection range (the observable distance) $d$ as where GW event's signal-to-noise ratio 
will be larger that 8 (${\rm S/N} > 8$). 
The maximum range $d_{\rm  max}$ is derived from Eq.\,(\ref{eq:SNR}).
According to the antenna pattern of the IFO, a detection range $d_{\rm avg}$ 
for whole sky average is given as
$$
d_{\rm avg} \ =\ d_{\rm max} \times 0.44.
$$
The factor 0.44 convert $d_{\rm max}$ to the radius $d_{\rm avg}$ of a sphere which 
has same volume of IFO antenna pattern%
\footnote{The factor also include the effect of GW polarization and inclination of binary orbit plane.}$^{,}$%  
\footnote{The factor 0.44 is determined by the numerical simulation study \cite{tatsumiH19}, 
which assume the threshold of S/N for the detection.
The factor is 0.4 with RMS average \cite{kandaH16}, and $1/\sqrt{5}$ for non-polarized case.}.
Figure \ref{fig:range} displays the whole sky average detection range in cases of four proposed IFO configurations.
With detuned configuration (DRSE), detection range is better than others for binary mass below $40\,M_{\odot}$. 
%To estimate the detection rate in LCGT, 
We employ the number density of the galaxy as
$$
\rho_{\rm galaxy}\ =\ 1-1.5 \times 10^{-2} \ {\rm [Mpc^{-3}],}
$$
which determined from the B-band luminosity \cite{Phinney1991,Kalogera2001}.
Figure\,\ref{fig:range} shows $\overline{N_{\rm ev}}$ as a function of sky average detection range.

\begin{figure}[tbp]
\begin{center}
%
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\includegraphics[width=0.95\textwidth]{range_Nev.eps}
\end{center}
\end{minipage}
%
\hspace{0.02\textwidth}
%
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\includegraphics[width=0.95\textwidth]{ranges.eps}
\end{center}
\end{minipage}
%
\end{center}
\vspace*{-5mm}
\caption{{\it Left}: Expected event rate $\overline{N_{\rm ev}}$ as a function of sky average detection range.
Solid thick balck line is $\overline{N_{\rm ev}}$ according to the Eq.\,(\ref{eq:nev}), 
and gray shaded zone is a confidence band of 95\%.
Solid thin lines show the range for binary coalescence. Colors are corresponding to the IFO configurations.
{\it Right}: Sky-averaged detection ranges as a function of mass of the source.
Solid lines show the range for binary coalescence. Dashed lines show the range for ringdown GW from black-hole quasi-normal mode. 
Colors are corresponding to the IFO configurations.}
%\label{fig:range_Nev}
\label{fig:range}
\end{figure}



\begin{figure}[tbp]
\begin{center}
%
%\begin{minipage}[t]{0.48\textwidth}
%\begin{center}
\includegraphics[width=10cm]{range_P1510ev.eps}
%\end{center}
%\end{minipage}
%
%\hspace{0.02\textwidth}
%
%\begin{minipage}[t]{0.48\textwidth}
%\begin{center}
%\includegraphics[width=0.95\textwidth]{obsT_P1ev.eps}
%\end{center}
%\end{minipage}
%
\end{center}
\vspace*{-5mm}
\caption{Probability of NS-NS event detection as a function of sky average detection range.
Solid thick balck line is a probability of NS-NS event detection, and gray shaded zone is a confidence interval of 95\%.
Solid thin lines show the range for binary coalescence. Colors are corresponding to the IFO configurations.}
\label{fig:range_P1510ev}
%\label{fig:obsT_P1ev}
\end{figure}

\vspace*{2mm}
\noindent
{\bf Detection probabilities and extracted information}

We estimate the probability that LCGT achieve its minimum success of detecting at least one GW signal
in one-year observation, assuming Poisson statistics with mean expected events.
%The results are presented in Figures\,\ref{fig:range_P1510ev}. 
Figure\,\ref{fig:range_P1510ev} shows the results:
the probability of detection for at least one (and 5, 10) event(s) 
within one year observation as a function of a detection range. 
All candidate IFO configurations reach a minimal success requirement; 
at least one event per year.
Left graph of Fig.\,\ref{fig:contour_range_obsT} shows the detection probability as a function of observation period.
Table \ref{table:NSNS} summarize ranges, expected event rates, required observation period
for event detection, etc. 

Measurement of the source characteristics is an important task in the data analysis. 
Narrow band configuration (DRSE) is worse as roughly twice of  broadband configurations about the accuracy of binary parameters
as listed in Table\,\ref{table:NSNS}.
Especially, arrival time accuracy will become important in the determination of the GW source direction.

Estimations in Table\,\ref{table:NSNS} suggest that switching between broadband and detuned in variable configuration,
VRSE (B/D), has no significant merit and demerit as far as noise spectrum. We must consider with accuracy of the calibration, 
stableness, commissioning schedule, etc.

\begin{table}[tdp]
\caption{Summary of comparison of interferometer configurations for  NS-NS binary observations}
\vspace*{-5mm}
\begin{center}
\begin{tabular}{r||c|cc|c}
& \multicolumn{4}{c}{IFO configurations} \\
& BRSE & VRSE (B) & VRSE (D) & DRSE\\ 
\hline
\parbox{17zw}{\strut{}NS Binary ($1.4-1.4 M_{\odot}$) :\strut} & & & &\\
Detection range $d_{\rm avg}$ [Mpc] & 114 &  112  &   123    & 132 \\
(maximum. $d_{\rm max}$ [Mpc]) & (259) & (255) & (281) & (299)\\
& & & &\\
\parbox{15zw}{\strut{}Expected observed event rate $\overline{N_{\rm ev}}$ (C.I.\,95\%) [events/yr] \strut} 
& $ 5.4^{+13.7}_{-4.3}$ & $5.2^{+13.0}_{-4.1}$ & $6.9^{+17.3}_{-5.5}$ & $8.2^{+20.6}_{-6.5} $\\
& & & &\\
Probability of event detection & & & & \\
 within 1 year   & 99.6\% & 99.4\% & 99.9\% & 99.9\% \\
 within 6 months & 93.4\% & 92.5\% & 96.8\% & 98.3\% \\
 within 3 months & 74.3\% & 72.5\% & 82.1\% & 87.0\% \\
& & & &\\
\parbox{15zw}{\strut{}Observation time for 90\% probability of event detection [months] \strut}
 & 5.1 & 5.3 & 4.0 & 3.4\\
& & & &\\
\parbox{15zw}{\strut{}Accuracy of source parameters for the event at 200\,Mpc away\strut}& & & &\\ 
S/N & 10.4 & 10.2 & 11.3 & 12.0\\
arrival time $\delta t_{c}$ [msec]& 0.254 & 0.220 & 0.255 &1.08\\
chirp mass $\frac{\delta{\cal M}}{\cal M}$ & $2.34 \times 10^{-5}$ & $2.14 \times 10^{-5}$ & $2.77 \times 10^{-5}$ & $3.72 \times 10^{-5}$\\
$\frac{\delta \eta}{\eta}$ & 0.0045 & 0.0041 & 0.0048 & 0.0098\\
\hline
\end{tabular}
\end{center}
\label{table:NSNS}
\end{table}



\begin{figure}[htbp]
\begin{center}
%
\begin{minipage}[t]{0.58\textwidth}
\begin{center}
\includegraphics[width=0.95\textwidth]{obsT_P1ev.eps}
\end{center}
\end{minipage}
%
\hspace{0.02\textwidth}
%
\begin{minipage}[t]{0.35\textwidth}
\begin{center}
\includegraphics[width=0.95\textwidth]{contour_range_obsT.eps}
\end{center}
\end{minipage}
%
\caption{{\it Left}: Probability of NS-NS event detection as a function of observation periods.
Solid thin lines show the range for binary coalescence. Colors are corresponding to the IFO configurations.
{\it Right}: Expected rate $\overline{N_{\rm ev}}$ as a function of sky average detection range.
Solid thin lines show the range for binary coalescence. Colors are corresponding to the IFO configurations.}
\label{fig:contour_range_obsT}
%
\end{center}
\end{figure}


\subsection{Black-hole binary coalescences}

Black-hole (BH) binary coalescences are also interest GW sources. 
LCGT is able to detect GW's from stellar mass BH binaries (BH-BH or BH-NS). 
Table\,\ref{table:BHrelated} summarize the range for these binaries, etc.

\begin{table}[tdp]
\caption{BH observations}
\begin{center}
\begin{tabular}{r||c|cc|c}
& \multicolumn{4}{c}{IFO configurations} \\
& BRSE & VRSE (B) & VRSE (D) & DRSE\\ 
\hline
\parbox{19zw}{\strut{}Binary coalescences NS-BH (1.4-10$M_{\odot}$) \strut}  & & & &\\
Detection range $d_{\rm avg}$ [Mpc]& 240 &  235  &   261    & 278 \\
Expected event rate [events/yr]&  \multicolumn{4}{c}{$0.006 - 6$} \\
\hline
\parbox{19zw}{\strut{}Binary coalescences BH-BH (10-10$M_{\odot}$) \strut}  & & & &\\
Detection range $d_{\rm avg}$ [Mpc] & 570 &  557  &   615    & 677 \\
Expected event rate [events/yr]&  \multicolumn{4}{c}{$0.07 - 7$} \\
\hline
\hline
\parbox{19zw}{\strut{}Quasi-normal mode :\strut} & & & &\\
Detection range $d_{\rm avg}$ [Gpc] for $200\,M_{\odot}$& 2.1 & 2.0 & 2.3 & 3.0\\
Survey mass region at 1\,Gpc [$M_{\odot}$]& 110 - 910 & 115 - 760 & 100 - 490 & 100 - 450\\
\hline
\end{tabular}
\end{center}
\label{table:BHrelated}
\end{table}


\subsection{BH quasi-normal mode}

The ringdown GW from quasi-normal mode of black-hole will make it possible to search and to study BH physics. 
LCGT's detection range and possible survey mass region is listed in Table\,\ref{table:BHrelated}%
\footnote{ In this estimation, we assume the amplitude of ringdown GW corresponding to 3\% of mass. 
See reference \cite{bib:ana_flanagan} in detail.}.
All four IFO configuration has long detection range around $200\,M_{\odot}$ of BH mass. 
Broadband configurations, BRSE and VRSE (B), has slightly wider survey region of mass.


\subsection{Supernovae with stellar-core collapse}

In supernovae explosion, various scenario of GW emission are proposed. 
Stella-core collapse is a reasonable process to emit the burst gravitational waves with a bounce of the rotating core. 
As shown in Fig.\,\ref{fig:supernovaGWcore}, recent studies estimate its frequency is around 500 - 1000\,Hz. 
For the study of GW from supernovae, we need good sensitivity above a few 100\,Hz.

Rate of the supernovae is estimated as Fig.\,\ref{fig:supernovaPop}. 
If LCGT has a detection range of 
$3\,-\,5$\,Mpc, an expected rate of the observation will jumps to 0.5\,event/year.

\begin{figure}[tbp]
\begin{center}
%
\begin{minipage}[t]{0.55\textwidth}
\begin{center}
\includegraphics[width=0.95\textwidth]{SNGWcore.eps}
\end{center}
\end{minipage}
%
\hspace{0.02\textwidth}
%
\begin{minipage}[t]{0.41\textwidth}
\begin{center}
\includegraphics[width=0.95\textwidth]{SNpop.eps}
\end{center}
\end{minipage}
\caption{{\it Left}: GW spectrum of stellar-core collapse.
Figure\,17 in the reference \cite{bib:supernovaGWcore} by Dimmelmeier, et.al.
{\it Right}: Population of supernovae.
Figure\,1 in the reference \cite{bib:supernovaPop} by S. Ando et al.}
\label{fig:supernovaPop}
\label{fig:supernovaGWcore}
%
\end{center}
\end{figure}


\subsection{Pulsar and LMXB}

Figures\,\ref{fig:pulsar_maxBRSE} and \ref{fig:pulsar_maxDRSE} shows the strain sensitivity for continuous waves, 
with integration of long period. Table \ref{table:Pulsar} summarize the sensitivity and possibility of GW detection from the pulsars.  
LCGT sensitivity will reach at the theoretical maximum arrowed amplitude due to spin down rate for several ten pulsars in known catalogue. 
Comparing to DRSE, other configurations have $30\sim40\%$ better chance for the candidate pulsars. 
For LMXB, sensitivity around several 100\,Hz is necessary.

\begin{table}[htdp]
\caption{Pulsars and LMXB searches}
\begin{center}
\begin{tabular}{r||c|cc|c}
& \multicolumn{4}{c}{IFO configurations} \\
& BRSE & VRSE (B) & VRSE (D) & DRSE\\ 
\hline
\parbox{19zw}{\strut{}The number of known pulsars which GW amplitude might reach the sensitivity\strut}  & & & &\\
1 year    & 35 & 38 & 35 & 25\\
10 years  & 59 & 63 & 57 & 46\\
\hline
\parbox{19zw}{\strut{} Strain $h_c$ sensitivity for typical sources\strut}  & & & &\\
\parbox{17zw}{\strut{} Crab (B0531+21, at 60Hz) \strut}  & & & &\\
1 year  & $8.5 \times 10^{-28}$ & $8.5 \times 10^{-28}$ & $8.3 \times 10^{-28}$ & $5.9 \times 10^{-28}$\\
10 years& $2.7 \times 10^{-28}$ & $1.9 \times 10^{-28}$ & $2.6 \times 10^{-28}$ & $1.9 \times 10^{-28}$\\
\parbox{17zw}{\strut{} Vela (PSR J0835-4510, at 22Hz) \strut}  & & & &\\
1 year  & $6.9 \times 10^{-27}$ & $6.0 \times 10^{-27}$ & $9.3 \times 10^{-27}$ & $1.0 \times 10^{-26}$\\
10 years& $2.2 \times 10^{-27}$ & $2.7 \times 10^{-27}$ & $2.9 \times 10^{-27}$ & $3.3 \times 10^{-27}$\\
\parbox{17zw}{\strut{} LMXBs (at 600Hz) \strut}  & & & &\\
1 weeks  & $1.1 \times 10^{-26}$ & $9.5 \times 10^{-27}$ & $1.1 \times 10^{-26}$ & $1.4 \times 10^{-25}$\\
\hline
\end{tabular}
\end{center}
\label{table:Pulsar}
\end{table}


\begin{figure}[htbp]
\begin{center}
%
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\includegraphics[width=1.0\textwidth]{pulsar_maxBRSE.eps}
\caption{Sensitivity (BRSE) and known pulsars. 
}
\label{fig:pulsar_maxBRSE}
\end{center}
\end{minipage}
%
\hspace{0.02\textwidth}
%
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\includegraphics[width=1.0\textwidth]{pulsar_maxDRSE.eps}
\caption{Sensitivity (DRSE) and known pulsars.
}
\label{fig:pulsar_maxDRSE}
\end{center}
\end{minipage}
%
\end{center}
\end{figure}



It is known that a pulsar period may shift suddenly,
which is called as a 'glitch'. During 1981-2006, more than 50
pulsar glitches were observed. There are some proposal to explain
glitch mechanism. If f-mode oscillation is excited at the glitches,
a gravitational wave radiation is expected as

\begin{equation}
h\ =\ 13.2 \times 10^{-23} \left(\frac{E}{10^{-12} M_{\odot} c^2}\right)^{1/2} 
\left(\frac{287\, {\rm pc}}{R}\right) \left(\frac{2\,{\rm kHz}}{f}\right) \left(\frac{0.2\,{\rm s}}{\tau}\right).
\end{equation}

For Vela pulsar, $h=10^{-22}$, which should be detectable for LCGT
except DRSE with high confidence. 16 glitches were observed in 18 years,
which means event rate for 0.89\,/yr.

$h=10^{-22}$ for exponential-decay signal at 2\,kHz corresponds to $h_{rss}=1.3 \times 10^{-21}$.
BRSE and VRSE(B/D) will be able to search these signal. 



\subsection{International Network of GW observation}

In the most cases of GW searches, we need to employ the world wide network of GW detectors.
A simultaneous detection will make possible to determine the source direction, to solve the
GW polarization, to exclude spurious events.
Hence, we take note affects of IF configuration in the world network analysis.

The accuracy of arrival time in NS-NS binary event is important to decide the source direction.
In rough estimation \cite{bib:kandaCQG2003}, the angular resolution is given as

\begin{equation}
\delta \theta \simeq 4.4 \ {\rm[degree]} \times \sqrt{\delta t_{c}\ {\rm[msec]}}
\end{equation}

Thus the arrival time accuracy of sub-msec order in one of the detector might
 dominate the accuracy of the network.




%\pagebreak

%\begin{thebibliography}{9}

%\end{thebibliography}




%\end{document}  
