# Networks of gravitational wave detectors and three figures of merit

Bernard F. Schutz1,2 1 Albert Einstein Institute, Potsdam, Germany  2 School
of Physics and Astronomy, University of Cardiff, Wales, UK
<EMAIL_ADDRESS>
###### Abstract

This paper develops a general framework for studying the effectiveness of
networks of interferometric gravitational wave detectors and then uses it to
show that enlarging the existing LIGO-VIRGO network with one or more planned
or proposed detectors in Japan (LCGT), Australia, and India brings major
benefits, including much larger detection rate increases than previously
thought. I focus on detecting bursts, i.e. short-duration signals, with
optimal coherent data-analysis methods. I show that the polarization-averaged
sensitivity of any network of identical detectors to any class of sources can
be characterized by two numbers – the visibility distance of the expected
source from a single detector and the minimum signal-to-noise ratio (SNR) for
a confident detection – and one angular function, the antenna pattern of the
network. I show that there is a universal probability distribution function
(pdf) for detected SNR values, which implies that the most likely SNR value of
the first detected event will be 1.26 times the search threshold. For binary
systems, I also derive the universal pdf for detected values of the orbital
inclination, taking into account the Malmquist bias; this implies that the
number of gamma-ray bursts associated with detected binary coalescences should
be 3.4 times larger than expected from just the beaming fraction of the gamma
burst. Using network antenna patterns, I propose three figures of merit that
characterize the relative performance of different networks. These measure (a)
the expected rate of detection by the network and any sub-networks of three or
more separated detectors, taking into account the duty cycle of the
interferometers, (b) the isotropy of the network antenna pattern, and (c) the
accuracy of the network at localizing the positions of events on the sky. I
compare various likely and possible networks, based on these figures of merit.
Adding any new site to the planned LIGO-VIRGO network can dramatically
increase, by factors of 2 to 4, the detected event rate by allowing coherent
data analysis to reduce the spurious instrumental coincident background.
Moving one of the LIGO detectors to Australia additionally improves direction-
finding by a factor of 4 or more. Adding LCGT to the original LIGO-VIRGO
network not only improves direction-finding but will further increase the
detection rate over the extra-site gain by factors of almost 2, partly by
improving the network duty cycle. Including LCGT, LIGO-Australia, and a
detector in India gives a network with position error ellipses a factor of 7
smaller in area and boosts the detected event rate a further 2.4 times above
the extra-site gain over the original LIGO-VIRGO network. Enlarged advanced
networks could look forward to detecting three to four hundred neutron star
binary coalescences per year.

###### pacs:

95.55.Ym,95.45.+i

††: Class. Quantum Grav.

##  1 Introduction: detector networks

###  1.1 Current and future networks of interferometers

The three large gravitational wave detectors of the LIGO project [1], located
at two sites, and the large instrument of the VIRGO project [2], all of which
are expected to reach their Advanced level of sensitivity around 2016,
represent the bare minimum required to realize the potential of gravitational
wave astronomy when detecting signals of short duration. Using gravitational
wave information alone, it is necessary to have at least three separated
detectors for locating such sources on the sky, measuring the intrinsic
amplitude and polarization of the incoming waves [3], and determining
distances to “standard-siren” coalescing compact-object binaries [4]. For
long-duration (continuous-wave) signals, a single detector can use the phase
modulation imprinted by the motion of the Earth to locate sources on the sky.
But if the signal is a “burst”, too short for modulation to be measurable,
then positions must be inferred by time-delay triangulation among at least
three separated detectors. Some of the most important expected signals will be
bursts, such as those from inspiraling and coalescing binaries of neutron
stars and/or black holes.

If one of these delicate interferometers temporarily falls out of observing
mode or experiences a period of unusually high noise, so that one of the three
sites has no functioning detector, or if an incoming gravitational wave
arrives from a location on the sky or with a polarization where one of the
detectors is significantly less sensitive, then an observation by the
remaining detectors will not be able to reconstruct the event completely
unless there is other associated information, for example from a gamma-ray
burst. Although two-detector observations can have enough significance to
identify an event and measure important physical parameters, such as the
stellar masses in a binary system, the aim of building detector networks is to
extract the greatest possible information from the weak and infrequent signals
that we expect to observe with Advanced detectors, and this requires all three
sites operated by LIGO and VIRGO.

Fortunately, this network will be enlarged on a short timescale. Funding has
begun for the LCGT detector in Japan. There are further proposals for
construction in Australia and India. Detectors in Asia or Australia help to
cover sky gaps and operational down-times of the basic three and bring an
added bonus of improved angular resolution, by increasing the length and
number of baselines among detectors of the network. There have been a number
of detailed studies of the observing benefits brought by one or another
detector [5, 6, 7, 8, 9, 10, 11]. These studies usually simulate network
detection by using Monte-Carlo techniques, which provide reliable comparisons
of specific configurations but little insight into what would happen with
other configurations. It would be helpful, therefore, to have general results
applicable to all networks as well as complementary and easily computed ways
of quantifying the extra science brought by one or more further detectors. To
this end I suggest here three relatively simple figures of merit (f.o.m.’s)
that measure the mean performance of different network configurations. They
compare networks’ overall event rates (including allowance for realistic duty
cycles of detectors), the isotropy of their joint antenna patterns, and the
precision with which the networks can measure sky positions of sources. I also
derive two general probability distributions for events detected by any
network: their observed signal-to-noise ratios, and the observed values of the
inclination angle of detected binary systems. The Nissanke et al [11] Monte-
Carlo study of coalescing-binary detection by various networks is particularly
close to the subject of this paper and will provide a useful reference
comparison for various analytic results derived below.

###  1.2 Network coherent analysis

The analysis in this paper assumes that a number of detectors observe
gravitational waves coherently, by combining their data in the most sensitive
way. The earliest detailed study for gravitational waves of what we now call
coherent detection was by Gürsel and Tinto [12]. The papers that placed
coherent network detection on a sound statistical basis were by Flanagan and
Hughes [13] and by Finn [14]. In this paper I shall concentrate on detecting
short-duration signals whose waveform is known in advance, using matched
filtering. Coherent detection can also be used to find signals whose waveform
is not known [15].

Coherent detection is not at present the default method of data analysis. All
the searches carried out so far by the LSC-VIRGO collaboration have involved
coincidence thresholding, which means selecting for further study only
stretches of data that appear to contain signals strong enough to pass a pre-
determined threshold in two or more detectors, where the signals occur within
a maximum time-separation equal to the light-travel time between the detectors
(the coincidence “window”). The experience of current searches has been that
most large events in the individual detector data streams are random
instrumental artifacts (sometimes called “glitches”), and the coincidence test
eliminates almost all of them because the glitches are not correlated in the
data streams of separated detectors. But thresholding is not the optimal
signal detection method against Gaussian noise, and in fact it can be very far
from optimum, as discussed in section 4.2 below. Thresholding is used because,
although most of the noise background in detectors is Gaussian, glitches make
the background far from Gaussian at amplitudes above a few standard
deviations. Interferometer-network searches that use thresholding extend
methods originally developed for networks of bar antennas [16].

However, networks containing three or more detectors – our subject in this
paper – have a degree of redundancy that allows them to veto glitches: once
the time-delays allow identification of the location of the source, the two
polarization waveforms are over-determined by the three or more detector
responses. This means that such networks have linear combinations of detector
outputs that contain no gravitational wave signal, often called null streams
[12, 17, 18, 19]. These can be used to test for and veto glitches, which do
not in general cancel out in the null streams.

Current searches for short-duration signals often follow the
thresholding/coincidence step by doing a coherent analysis of the coincident
events, in order to use the null-stream vetoes and to extract as much
information from them as possible [20, 21]. In fact, the very first analysis
of gravitational wave data from a network of interferometers – the so-called
“Hundred Hour Run” – applied a two-detector null-stream method (after
thresholding) to eliminate glitches and show that the strongest observed
coincident event had a high probability of occurring by chance [22], and
consequently that no gravitational wave event had been observed.

But the glitch vetoes provided by null streams in principle allow three-
detector networks to do fully coherent analysis, without prior thresholding. A
number of studies have therefore explored fully coherent detection or compared
it with coincidence thresholding [6, 10, 18, 19, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33]. It is now clear that coherent detection is already able to
discriminate real gravitational waves from glitches even in a general three-
detector network, and when there are four or more detectors this gets even
better.

Since we will see below that coherent methods are capable of detecting far
more events than coincidence methods, it seems reasonable to assume that fully
coherent detection will become the norm as the detector network grows. This
will not be entirely trivial: one of the principal challenges of introducing
coherent data analysis is that it is very demanding of computing, because one
has to do a signal search for each resolvable location on the sky. But the
payoffs will be worth the effort, especially with the computing power that can
be expected to be available by the time the current network is enlarged. The
purpose of this paper is, therefore, to characterize the performance of
different possible networks when they use coherent detection.

###  1.3 Assumptions and principal results

The detection sensitivity of a detector network is a function of the
sensitivity of the individual detectors and their placement on the earth. An
important part of the sensitivity is the network’s antenna pattern, which
defines up to a radial scaling the region of space around the earth within
which a source should be detected. The overall scale depends on the
sensitivity of the individual detectors and the detection threshold that is
set for discriminating real signals from noise impersonators. It is
conventional in the literature to combine threshold and sensitivity into a
radial measure called the horizon distance, the maximum distance a detector or
network can detect an event, allowing for an optimum alignment. In this paper
I separate threshold from sensitivity by measuring the sensitivity of a
detector or network to a given source in terms of a visibility distance, which
is the distance at which the given source would produce a mean response with a
signal-to-noise ratio 1, averaged over polarizations.

From the properties of the antenna pattern I define the three new f.o.m.’s,
for a rather general source population. The f.o.m.’s are meant to be simple to
compute and to use. They should give a broad-brush characterization of the
effectiveness of networks, but they won’t be precise enough to make fine
discriminations between similar networks. Although the f.o.m.’s can in
principle be computed for any network, I will keep the discussion in this
paper simple by making some assumptions.

  * •

Detectors. All the detectors are interferometers with identical sensitivity
and identical duty cycles. The detectors’ noise streams are not correlated
with one another, nor are the times when they drop out of observing mode. The
generalization to detectors with different sensitivity is not difficult.

  * •

Networks. The networks are made up of combinations of the Advanced upgrades of
the existing LIGO and VIRGO instruments plus planned and possible instruments
at the locations in Japan, Australia, and India that are given in table 1
below. Only networks containing three or more detectors in different locations
are considered, because, as noted above, fewer detectors do not return sky
position and polarization information from an observation unless there are
associated detections in, say, gamma or optical observatories.

  * •

Sources. The gravitational waves all come from an identical population that
are randomly and uniformly distributed in (Euclidean) space and in
polarization. The waves are short bursts, in that the detectors do not move
significantly during the observations, and they are emitted at random times.
The waveforms are identical except that they have different overall
amplitudes, inversely proportional to the distance to the source; they all
have the same polarization evolution (as a function of time) except for a
random rotation in the plane of the sky at the start of the burst. Note that,
according to this definition, binary systems with different inclinations to
the line of sight (different amounts of elliptical polarization) are members
of different populations, but binaries with the same inclination but different
orientations (rotations in the plane of the sky: the angle $\psi$ in figure 1)
are members of the same population. We do not consider stochastic signals or
long continuous-wave signals from GW pulsars.

  * •

Analysis. The data are analyzed coherently with a matched filter family
capable of matching the incoming signal perfectly. The data analysis finds the
ideal match by maximizing the log likelihood. Detector noise is purely
Gaussian, at least at the times when events arrive.

Given these assumptions, I summarize here the principal results of this paper:

  1. 1.

The sensitivity of a network to a population of identical but randomly
oriented and randomly located sources depends on the signal waveform, the
sensitivity of the detectors (all assumed the same), and the geometry of the
network. The signal and sensitivity contribute only to a scaling factor that
multiplies the antenna pattern, which depends only on the network geometry
(14). Therefore the relative performance of any two networks of similar
detectors observing any given source population is independent of the nature
of the source. This allows us to compare the advantages and disadvantages of
networks without needing to specify much about the signal.

  2. 2.

The population of detected events has a universal signal-to-noise
distribution, with a probability density function (p.d.f.) proportional to
$\rho^{-4}$ above the detection threshold, where $\rho$ is the amplitude
signal-to-noise ratio (SNR). The p.d.f. (2.4) depends only on the detection
threshold $\rho_{\rm min}$ set on $\rho$, not on the geometry or sensitivity
of the network.

  3. 3.

From this p.d.f it is possible to deduce that the median amplitude SNR of any
detected population will be $2^{1/3}\simeq 1.26$ times the detection threshold
$\rho_{\rm min}$. As we wait for the first detection, this is the most likely
SNR of the first event, provided that coherent data analysis is used for the
search. Similarly, the mean amplitude SNR of the detected population will be
1.5 times the threshold.

  4. 4.

Binaries with different inclinations have different maximum detection ranges,
which biases the expected observed distribution of inclinations. I compute the
universal probability distribution for detected inclinations, independent of
the network configuration (28). It peaks around $\pm 30^{o}$.

  5. 5.

From this distribution of inclinations one can also deduce another bias,
namely that – provided that mergers involving neutron stars give rise to
narrow-beamed gamma-ray bursts – the number of gamma-ray bursts that will be
detected in association with gravitational wave signals will be 3.4 times
larger than one would expect if there was no correlation between burst
direction and the maximum-power direction of a binary.

  6. 6.

The first figure of merit (f.o.m.) is called Triple Detection Rate [3DR]
(section 3.1). It measures the rate at which a network can detect events in
detectors at three or more separated locations. The rate at which events of a
given source population are detected depends of course on the detection volume
accessible to the network, but it also depends on the duty cycle of detectors,
which is the fraction of time they spend in observation mode. The first
introduction of figures of merit into the discussion of networks seems to have
been by Searle, et al [5], who defined a measure of detection rate that
depends effectively only on the detection volume. (See also Searle, et al
[7].) However, especially at the beginning of the operation of Advanced
Detectors, the duty cycle of the detectors will not be 100%. For the full
reconstruction of information about the source, we require at least three
separated detectors to observe the event, so a three-detector subnet of a
larger network can still return detections. Therefore, Triple Detection Rate
as defined here is designed to compute how many events can be detected by sub-
networks of three or more separated detectors, even when some other detectors
in the network may be off the air.

  7. 7.

The second f.o.m. is called Sky Coverage [SC] (section 3.2). It measures the
isotropy of the network’s antenna pattern. It is defined as the fraction of
the $4\pi$ sphere that is covered by the network’s antenna pattern at a range
that is $1/\sqrt{2}$ of the maximum. For a given number of detectors of a
standard sensitivity, there is a trade-off between isotropy and overall
detection volume: if the antenna patterns of individual detectors reinforce
each other, then the volume they include will be larger than if they fill in
each other’s directional “holes”. But isotropy might be a desirable thing in
itself. For example, if the source population is anisotropic (perhaps biased
toward the Galactic plane) then an isotropic network might do better than one
with a larger range. Or if the sources are expected to be associated with
objects that can be detected also by a non-gravitational signal, but only if
they are relatively nearby compared to the maximum range of the network (e.g.
supernovae seen with neutrinos), then an isotropic network could do better.

  8. 8.

The third f.o.m. is called Directional Precision [DP] (section 3.3). It
measures how well the network localizes events on the sky, its directional
accuracy. Generally speaking, longer baselines improve direction-finding.
Directional Precision uses the measure of solid-angle error introduced by Wen
and Chen [8]. It is proportional to an average over the antenna pattern, not
of the size of the error box, but of its inverse. This prevents the measure
being distorted by small regions where direction-finding is poor; instead it
is weighted more by the regions of the sky where direction-finding is
particularly good.

  9. 9.

Enlarging the basic LIGO-VIRGO network with detectors in Japan and/or
Australia also provides a less obvious but perhaps even more important
benefit: it makes coherent data analysis more robust and allows the detection
of events that would not pass the coincidence threshold tests used in the
current LIGO-VIRGO data analysis (section 4.2), where fully coherent analysis
is difficult because of the geometry of the network. This could lead to an
improvement of as much as a factor of 4 in the recovery of signals within a
given detection volume, depending on the effectiveness with which coherent
methods can be introduced into the data analysis of the basic LIGO-VIRGO
network.

By comparing these measures for various possible networks, some simple lessons
emerge. First, if one takes as a baseline the performance of the original
network of Advanced detectors – LIGO Hanford with two full-size
interferometers, LIGO Livingston, and VIRGO – using coherent detection, then
there is a big win in event rate from putting another large detector anywhere
in Asia or Australia. This comes partly from adding more detection volume and
partly from providing greater coverage when individual detectors randomly drop
out of observing mode. A Japanese detector (LCGT) makes the antenna pattern
more isotropic; an extra Australian detector (AIGO) makes its reach go deeper.
If instead of building an extra detector in Australia, one of the LIGO Hanford
instruments is placed in Australia (LIGO Australia), the improvement in
detection rate is not quite as dramatic. The big change then is a significant
improvement in direction-finding. If we take the network that includes LIGO
Australia and LCGT, again there is a very big improvement in the event rate,
and of course it becomes more isotropic as well. This is pretty much a “dream
configuration” in terms of present opportunities. If a project gains traction
in India and a large INDIGO detector is built, then this produces even further
gains that the f.o.m.’s quantify. On top of these improvements due to detector
numbers and geometry, the robustness of coherent detection in an enlarged
network will lead to further striking gains in event rate over the current
coincidence style of analysis.

It is important to remark that these f.o.m.’s should be regarded as rules of
thumb, not as exact measures of the performance of any network. But treated
with a small amount of caution, the measures show how big the science gains
can be from adding further Advanced detectors to the existing three sites.

##  2 Network antenna patterns and the amplitude distribution of detected
events

###  2.1 Antenna pattern and detection volume of a single interferometer

The f.o.m.’s are based on the antenna patterns of the detectors, which
describe their relative sensitivity in different directions. Each detector is
linearly polarized and has a quadrupolar antenna pattern. In the notation of
Sathyaprakash and Schutz [34], we consider a detector in the $x-y$ plane with
arms along the axes, and a gravitational wave coming from a direction given by
the usual spherical coordinates $\theta$ and $\phi$ relative to the detector’s
axes, whose two polarization components $h_{+}$ and $h_{\times}$ are referred
to axes in the plane of the sky that are rotated by an angle $\psi$ relative
to the detector axes (see figure 1, which is taken from figure 3 of
Sathyaprakash and Schutz [34]). Then the strain $\delta L/L$ of the
interferometer is

$\frac{\delta
L(t)}{L}=F_{+}(\theta,\,\phi,\,\psi)h_{+}(t)+F_{\times}(\theta,\,\phi,\,\psi)h_{\times}(t),$
(1)

where the $F_{+}$ and $F_{\times}$ are the antenna pattern functions for the
two polarizations. Using the geometry in figure 1, one can show that

$\displaystyle F_{+}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1+\cos^{2}\theta\right)\cos 2\phi\cos
2\psi-\cos\theta\sin 2\phi\sin 2\psi,$ $\displaystyle F_{\times}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left(1+\cos^{2}\theta\right)\cos
2\phi\sin 2\psi+\cos\theta\sin 2\phi\cos 2\psi.$ (2)

Figure 1: The relative orientation of the sky and detector frames. From [34].

These are the antenna pattern response functions of the interferometer to the
two polarizations of the wave as defined in the sky plane [35]. Note that the
maximum value of both $F_{+}$ and $F_{\times}$ is 1.

Sometimes the angle $\eta$ between the arms of a detector is not exactly
$\pi/2$, for reasons of local geography or by design. For that reason it is
helpful to orient the detector in the $x$-$y$ coordinate plane by aligning the
bisector of the angle between the arms with the bisector of the angle between
the axes [36]. One also has to multiply the functions $F_{+}$ and $F_{\times}$
in (1) by $\sin\eta$. When we discuss networks we will define the orientation
of the detector to be the geographical direction of the arm bisector.

The expected power signal-to-noise ratio (SNR) of the signal in the detector’s
data stream is, if it can be discovered by ideal matched filtering,

$\rho^{2}=4\int_{0}^{\infty}\frac{|\tilde{\delta L}(f)/L|^{2}}{S_{h}(f)}{\rm
d}f,$ (3)

where $S_{h}(f)$ is the one-sided spectral noise density normalized to the
gravitational wave amplitude, and the time-series strain $\delta L(t)/L$ in
(1) has been Fourier-transformed into $\tilde{\delta L}(f)/L$, which then
depends on the Fourier transforms $\tilde{h}_{+}(f)$ and
$\tilde{h}_{\times}(f)$ of the incoming waves. I will assume from now on that
we are detecting a short burst of gravitational waves, so that the detector
does not change its orientation during the observation. A discussion of
network detection of long-duration signals, such as those from gravitational
wave pulsars, may be found in Cutler and Schutz [37, 38].

We now apply the assumption that the wave has a randomly oriented
polarization. Consider a source which emits wave components $H_{+}(f)$ and
$H_{\times}(f)$, referred to its own frame, defined perhaps by some preferred
axis or plane in the source. Suppose that at the start of the observation this
source frame is different from the detector frame as projected onto the sky by
a rotation angle $\alpha$. During the observation the polarization will rotate
in some way determined by $H_{+}(f)$ and $H_{\times}(f)$. This is of no
interest to us here. The important point is that the ensemble of sources at
the same position in space contains systems with all possible initial angles
$\alpha$. When we average the power SNR in (3) over the ensemble, we will
simply be changing in a uniformly random way the projection of the source’s
intrinsic $+$ and $\times$ components onto the detector’s. The result is that
the mean power SNR over the ensemble (denoted by $\left<\;\right>$) depends
only on the sum of the squares of the sensitivity functions of the detector to
both polarizations:

$\left<\rho^{2}\right>=2\left[F_{+}(\theta,\phi,\psi)^{2}+F_{\times}(\theta,\phi,\psi)^{2}\right]\int_{0}^{\infty}\frac{|H(f)|^{2}}{S_{h}(f)}{\rm
d}f,$ (4)

where $|H(f)|^{2}=|H_{+}|^{2}+|H_{\times}|^{2}$. We call the function

$\displaystyle P(\theta,\phi)$ $\displaystyle=$ $\displaystyle
F_{+}(\theta,\phi,\psi)^{2}+F_{\times}(\theta,\phi,\psi)^{2}$ (5)
$\displaystyle=$
$\displaystyle\frac{1}{4}(1+\cos^{2}\theta)^{2}\cos^{2}2\phi+\cos^{2}\theta\sin^{2}2\phi$

the antenna power pattern of a single interferometer. Note that, from (2.1),
the antenna power pattern is independent of the angle $\psi$ that is the
reference angle for the wave’s polarization, as one would expect after our
ensemble polarization average. It is plotted in the detector coordinate frame
in figure 2. This is often referred to as the “peanut diagram”.

Figure 2: The antenna power pattern (left panel) and its square-root
(amplitude pattern: right panel) of a single interferometer oriented with axes
in the $x$-$y$ plane, averaged over polarizations of the incoming wave. The
amplitude pattern represents the shape of the detection volume of the
instrument, or its maximum detection reach in different directions.

If, for a single detector, there is a detection threshold $\rho_{\rm min}$ on
the amplitude SNR, then a signal from a direction $(\theta,\,\phi)$ can be
expected to be detected if

$2P(\theta,\phi)\int_{0}^{\infty}\frac{|H(f)|^{2}}{S_{h}(f)}{\rm
d}f\geq\rho_{\rm min}^{2}.$ (6)

For the purposes of our discussion, we suppose that the gravitational wave
source has a standard intrinsic amplitude, so that its received amplitude
$H(f)$ is inversely proportional to the distance $r$ to the source. We also
suppose that these sources are randomly distributed in space. Let us normalize
the amplitude by defining (arbitrarily) a standard reference distance $r_{s}$
at which our source would have amplitude $H_{s}(f)$, so that a source at a
distance $r$ has amplitude

$H(f)=\frac{r_{s}}{r}H_{s}(f).$ (7)

This is much the way astronomers distinguish between absolute and apparent
magnitudes, by defining the absolute magnitude to equal the apparent magnitude
of the source if it were at a fixed fiducial distance (10 pc).

Explicitly separating $r$ out in $H(f)$ will be helpful for the volume
integrals below. For example, we can now rewrite (4) as

$\left<\rho^{2}\right>=\frac{2}{r^{2}}P(\theta,\phi)\int_{0}^{\infty}\frac{|r_{s}H_{s}(f)|^{2}}{S_{h}(f)}{\rm
d}f.$ (8)

Note that the product $rH=r_{s}H_{s}$ is independent of the distance to the
source. We use this to define the visibility distance of the source $D_{V}$:

${D_{V}}^{2}=2\int_{0}^{\infty}\frac{|r_{s}H_{s}(f)|^{2}}{S_{h}(f)}{\rm d}f.$
(9)

This is the distance at which the source would have SNR = 1 in a single
detector at its most sensitive location in the sky, namely directly overhead
at $\theta=0$ or $\pi$, after averaging over the sky polarization angle
$\psi$. All the details of filtering and the detector noise curve are hidden
in the single parameter $D_{V}$. This leads to a simple way of writing (8)

$\left<\rho^{2}\right>=P(\theta,\phi)\frac{{D_{V}}^{2}}{r^{2}}.$ (10)

Similarly, I will define the mean horizon distance $R_{0}$ for a single
detector observing this source to be the distance at which the source is on
average just at the detection threshold $\rho_{\rm min}$ when it is overhead,
so that $R_{0}=D_{V}/\rho_{\rm min}$. Then the reach $R$ of the single
detector in any direction ${\theta,\phi}$ is

$R(\theta,\phi)=R_{0}[P(\theta,\phi)]^{1/2}=\frac{D_{V}}{\rho_{\rm
min}}[P(\theta,\phi)]^{1/2}.$ (11)

We call the square-root of the antenna power pattern the antenna amplitude
pattern. The volume bounded by the reach $R(\theta,\phi)$ is called the
detection volume. Its size is determined by the antenna amplitude pattern,
scaled by the mean horizon distance $R_{0}$. The mean horizon distance is
smaller than what is conventionally called the horizon distance, which is the
distance at which an optimally polarized source exactly overhead can just be
detected at threshold.

Note that we are making an approximation here when we define a detection
volume by polarization averaging. Sources at the edge of the volume have only
a 50% chance of being detected, while those that are well inside are detected
with higher probability. Moreover, a number of sources outside this volume
will be detected if they have a favorable polarization. Our approximation is
to replace the real detection probability distribution in space with a fixed
volume that has a hard edge: everything inside is detected, everything outside
is missed. We use this approximation only to study the gross properties of
detection, such as numbers detected, typical position accuracies, and so on,
and only to compare different networks. The comparisons are likely to be
better than the accuracy of the approximation for any single network, since
the errors will systematically affect all networks the same way. The test of
how accurate this approximation is for any specific network is whether it
matches up with Monte-Carlo studies of the real detection problem for that
network.

###  2.2 Antenna pattern of a network of detectors

We now need to generalize these concepts to networks of more than one
detector. I will assume here that the detectors’ noise streams are
uncorrelated. This is a good assumption for all networks except those that
include two detectors at the Hanford LIGO site. Even there, experience has
shown that the correlations can be reduced to a very low level with careful
experimental design. A full treatment of the theory of detection in networks
that have detectors with correlated noise may be found in Finn [14].

When computing the joint antenna pattern of the entire network, the antenna
patterns of the individual detectors must of course be transformed to a common
celestial coordinate system. We take this to be the Earth-based spherical
coordinates, and from now on we denote them by $(\theta,\phi)$. In addition,
there must be a common definition of the incoming wave polarization. I use
here the formulation given in [39], whose expressions were developed for the
problem of long-term observations, where the detector changes orientation with
time. For the present paper we merely need to set $t=0$ in their formulation,
and we shall use conventional spherical sky coordinates rather than
declination and right-ascension.

As shown by Finn [14], the network power SNR is just the sum of the power SNRs
of the individual detectors

$\rho^{2}_{N}=\sum_{k=1}^{N_{D}}\rho^{2}_{k},$ (12)

where $N_{D}$ is the number of detectors and where we define the individual
power SNRs as

$\rho^{2}_{k}=2\int_{0}^{\infty}\frac{|H_{k}(f)|^{2}}{S_{h}(f)}{\rm d}f,$ (13)

where $H_{k}(f)$ is the waveform projected onto the $k$-th detector. Averaging
as before over the random polarization angle, we have

$\left<\rho^{2}_{N}\right>=2\sum_{k}(F_{+,k}^{2}+F_{\times,k}^{2})\int_{0}^{\infty}\frac{|H(f)|^{2}}{S_{h}(f)}{\rm
d}f,$ (14)

where $F_{+,k}$ and $F_{\times,k}$ are the antenna patterns of the individual
detectors. Note that the integral in this equation does not depend on $k$ and
is therefore taken outside the sum. The sum is then the function

$P_{N}(\theta,\phi)=\sum_{k}(F_{+,k}^{2}+F_{\times,k}^{2}),$ (15)

which is called the network antenna power pattern.111If the detectors were not
identical, then one could modify the network antenna pattern simply by
including a single weighting factor consisting of the ratio of $\rho^{2}$ for
each detector to a standard detector $\rho^{2}$ for the particular signal
waveform being considered. The network antenna pattern would then be waveform-
dependent. This is our analytic approximation to the detection sensitivity
found in [7] from Monte-Carlo studies of randomly oriented binary systems. In
terms of $P_{N}$ the network SNR takes the simple and useful form

$\left<\rho^{2}_{N}\right>=P_{N}(\theta,\phi)\frac{D_{V,L}^{2}}{r^{2}},$ (16)

where $D_{V,L}$ is the visibility distance of a single detector, labelled here
as the Livingston detector. Remember, all detectors are assumed identical so
all have the same visibility distance. This assumption is easily dropped if
necessary, but it makes the discussion in the present paper simpler.

It is worth remarking that the polarization-averaged network antenna pattern
does not depend on the local orientation of each detector, since it is the sum
of the individual detector power patterns (15), and for each detector the sum
of the squares of the antenna pattern components is invariant under rotations
of the detector in its plane. It might seem counterintuitive that two co-
located detectors with orthogonal orientation make the same average
contribution to the signal power received by the network as they would if they
were perfectly aligned. When aligned they work well together but miss many
events that one of them would catch when not aligned. When searching for a
stochastic gravitational wave signal, of course, alignment is crucial.
Moreover, even for bursts, the ability to determine polarization and sky
position of a signal will be affected by the relative alignment of the
detectors. I will return to this point later.

The resulting expression for the antenna pattern of an arbitrarily located and
oriented interferometer in our notation is as follows. The source position is
given by the spherical coordinates $(\theta,\phi)$ on the sky, and the frame
for the wave polarization angle $\psi$ is defined to be aligned with this
spherical-coordinate grid. The detector is at latitude $\beta$ and longitude
$\lambda$. It is an interferometer oriented such that the bisector of its arms
points in the direction $\chi$, measured counter-clockwise from East. Its arms
have an opening angle of $\eta$. The celestial coordinates $(\theta,\phi)$ are
aligned with latitude and longitude, so that the equators of both systems
coincide and the celestial point $(\theta=\pi/2,\,\phi=0)$ is in the zenith
direction above the geographic location $(\beta=0,\,\lambda=0)$. The antenna
pattern functions are

$\displaystyle F_{+}$ $\displaystyle=$
$\displaystyle\sin\eta[a\cos(2\psi)+b\sin(2\psi)],$ (17) $\displaystyle
F_{\times}$ $\displaystyle=$
$\displaystyle\sin\eta[b\cos(2\psi)-a\sin(2\psi)],$ (18)

where the functions $a$ and $b$ are given by

$\displaystyle
a=\frac{1}{16}\sin(2\chi)[3-\cos(2\beta)][3-\cos(2\theta)]\cos[2(\phi+\lambda)]+$
$\displaystyle\frac{1}{4}\cos(2\chi)\sin(\beta)[3-\cos(2\theta)]\sin[2(\phi+\lambda)]+$
$\displaystyle\frac{1}{4}\sin(2\chi)\sin(2\beta)\sin(2\theta)\cos(\phi+\lambda)+$
$\displaystyle\frac{1}{2}\cos(2\chi)\cos(\beta)\sin(2\theta)\sin(\phi+\lambda)+\frac{3}{4}\sin(2\chi)\cos^{2}(\beta)\sin^{2}(\theta),$
(19) $\displaystyle
b=\cos(2\chi)\sin(\beta)\cos(\theta)\cos[2(\phi+\lambda)]-\frac{1}{4}\sin(2\chi)[3-\cos(2\beta)]\cos(\theta)\sin[2(\phi+\lambda)]+$
$\displaystyle\cos(2\chi)\cos(\beta)\sin(\theta)\cos(\phi+\lambda)-\frac{1}{2}\sin(2\chi)\sin(2\beta)\sin(\theta)\sin(\phi+\lambda).$
(20)

###  2.3 Detection volume of a network of detectors

The detection volume $V_{N}$ of the network is defined as the region enclosed
by its reach in any direction, which as before is

$R_{N}(\theta,\phi)=R_{0}[P_{N}(\theta,\phi)]^{1/2}=\frac{D_{V,L}}{\rho_{N,{\rm
min}}}P_{N}^{1/2},$ (21)

where $R_{0}$ is defined as before to be the mean horizon distance (maximum
reach) of a single detector for this source at the chosen network detection
threshold SNR $\rho_{N,{\rm min}}$, and where (as before) $D_{V,L}$ is the
single-detector visibility distance (maximum range at SNR = 1). I will assume
that when we compare networks, all of them have the same detection threshold.

We can compute the detection volume explicitly:

$\displaystyle V_{N}$ $\displaystyle=$ $\displaystyle\int{\rm
d}\Omega\int_{0}^{R_{N}(\theta,\phi)}r^{2}dr=\frac{1}{3}\int{\rm d}\Omega
R_{N}^{3}(\theta,\phi)$ (22) $\displaystyle=$
$\displaystyle\frac{1}{3}R_{0}^{3}\int{\rm
d}\Omega[P_{N}(\theta,\phi)]^{3/2}.$

Figure 3: The antenna power patterns of the LIGO and VIRGO detector network
with two detectors at Hanford (HHLV: left panel) and of the network after
including the Japanese detector LCGT (HHJLV: right panel). All detectors are
assumed to be identical. As in Figure 2, the sensitivity is averaged over
polarizations of the incoming wave. Top row: The coordinate system is oriented
with $z$ aligned with geographic North and the $x$-axis at geographic
longitude 0o. In all such plots from now on, the viewer is located at
longitude 40oW and 20oN, above the mid-Atlantic. Note that all antenna
patterns are reflection symmetric through the center of the earth, so that the
hidden side is a mirror image of the side shown in the diagram. Bottom row:
The same data plotted as contour plots. Contours are labeled with values
relative to the maximum. For HHLV on the left, the maximum is 3.03 (square of
mean horizon distance from table 2). For HHJLV on the right, the maximum is
3.31.

Table 1 gives the important parameters of the detector locations that will be
considered in this paper, including the one-letter abbreviation by which the
detectors will be denoted in naming the various networks.

As an illustration, in figure 3 the network antenna power patterns are plotted
for two networks: the planned Advanced network of two LIGO detectors at
Hanford, one at Livingston, and VIRGO; and the same network plus the LCGT
detector in Japan. Notice that the hole in the southwest direction has been
filled in by the Japanese detector.

###  2.4 Universal distribution of detected amplitudes

Because the angular sensitivity of the detectors is totally decoupled from the
dependence of SNR on the distance of the source, which resides in $H(f)$ in
(14), we can work out the expected distribution of SNR for detected events
analytically for any detector network and source population. To do this we
make explicit in (22) the fact that $R_{N}$ is inversely proportional to the
detection threshold $\rho_{N,{\rm min}}$, by using (21):

$V=\frac{D_{V,L}^{3}}{3\rho_{N,{\rm min}}^{3}}\int{\rm
d}\Omega[P_{N}(\theta,\phi)]^{3/2}.$ (23)

The number of detections with SNR larger than any given $\rho_{N}$ is
proportional to the detection volume with $\rho_{N,{\rm min}}$ set equal to
this $\rho_{N}$. This scales as $\rho_{N}^{-3}$. This is a cumulative
distribution: the number of detections with SNR larger than $\rho_{N}$ scales
as $\rho_{N}^{-3}$. It is straightforward from this to show that the universal
probability density function for the distribution of detected SNR values is

$\displaystyle p(\rho_{N}){\rm d}\rho_{N}$ $\displaystyle={}3\rho^{3}_{N,{\rm
min}}\rho_{N}^{-4}{\rm d}\rho_{N},$ $\displaystyle\qquad\rho>\rho_{N,{\rm
min}}$ $\displaystyle={}0,$ $\displaystyle\qquad\rho_{N}<\rho_{N,{\rm min}}.$

From this simple universal distribution one can deduce any of the moments one
wishes. For example, the mean expected amplitude SNR is $1.5\rho_{N,{\rm
min}}$. The mean expected power SNR2 is $3\rho_{N,{\rm min}}^{2}$.

The median of this distribution is of particular interest and can also be
deduced from a simple argument: it is the value of the threshold for which the
detection volume is one-half of the full volume. Since the volume scales as
the inverse cube of the threshold, the median amplitude SNR value will be
$2^{1/3}\rho_{N,{\rm min}}$. The median power SNR2 is $2^{2/3}\rho_{N,{\rm
min}}^{2}$. The importance of the median is that it is the most likely SNR
value of the first signal that will be detected. It has often been remarked
that the rapid increase of volume with distance means that the first source is
likely to be near the detection limit. Here we quantify that statement: the
most likely amplitude SNR of the first detection is $2^{1/3}\simeq 1.26$ times
the threshold of the search. The median source is weaker than either the
amplitude mean or the power mean. That is because the universal distribution
has a peak at the lowest values (at threshold) and has a long tail of strong
but rare events.

Of course, this argument has been made in the context of our antenna pattern
detection criterion, which is an approximation. However, I believe one can
expect that the distribution should be close to the distribution of real
observations, provided the detection criterion depends on coherent addition of
signals against mainly Gaussian noise.

###  2.5 Detection volumes for binary systems

As remarked in the definition of sources in section 1, binary systems with
different inclinations belong to different source populations as far as our
detection volumes are concerned, because the strength of their emitted
radiation depends on inclination, and their own radiation patterns are
anisotropic; in fact, if we were to average the power pattern shown in the
peanut diagram (figure 2) over circles around its long axis we would get a
plot of the radiation power pattern of a binary system. But binaries with
different inclinations are all members of the same physical family, just seen
from different and random directions. Therefore it is interesting here to
consider binary detection as a function of inclination angle $\iota$.

The maximum power is radiated along the rotation axis of the binary, defined
as $\iota=0$, and the minimum power in its orbital plane, $\iota=\pi/2$. For a
general inclination angle it is easy to show from, e.g., Sathyaprakash and
Schutz [34] that the radiated power depends on inclination in the following
way:

$P_{{\rm rad}}(\iota)=F_{\rm rad}(\iota)P_{{\rm rad}}(\iota=0),$ (25)

with

$F_{\rm rad}(\iota)=\frac{1}{8}(1+6\cos^{2}\iota+\cos^{4}\iota).$ (26)

We call this function the binary radiation pattern. As remarked above, this is
the $\phi$-average of the interferometer’s antenna pattern (5). The detection
volume will depend on $\iota$ as

$V_{N}(\iota)=\left[F_{\rm rad}(\iota)\right]^{3/2}V_{N}(\iota=0),$ (27)

This predicts the relative numbers of sources that will be detected, i.e. it
quantifies the bias toward small inclination angles created by the stronger
radiation pattern in those directions. We can derive the probability
distribution function of detected values of $\iota$ by normalizing $F_{\rm
rad}^{3/2}$ over the intrinsic distribution of angles, which has the
probability distribution function $\sin\iota$. The normalizing integral is

$\int_{0}^{\pi}\left[F_{\rm rad}(\iota)\right]^{3/2}\sin\iota\,{\rm
d}\iota=0.58092.$

The probability distribution of detected values of $\iota$ is therefore

$p_{\rm det}(\iota)=0.076076(1+6\cos^{2}\iota+\cos^{4}\iota)^{3/2}\sin\iota.$
(28)

This is plotted in figure 4. Note that this, also, is a universal
distribution, in that it applies to any network doing coherent analysis. As
with the distribution of detected values of SNR, this result is exact only
within the approximation we are making that the polarization-averaged antenna
pattern defines a detection volume with a sharp boundary. This pdf is
completely consistent with the Monte-Carlo result of Nissanke et al [11]
(their figure 3), when allowance is made for the difference between using
$\iota$ as the independent variable (here) and $\cos\iota$ ([11]).

Figure 4: The probability distributions of inclination angle $\iota$ (in
radians) for randomly oriented binaries (the single-peaked curve, which is
just $\sin\iota$) and for detected binaries (the double-peaked curve, from
(28)). The selection bias (essentially the Malmquist bias) toward low
inclinations due to the anisotropic radiation pattern of a binary is clear.

The mean value of $V_{N}(\iota)$ in (27) is $0.29046V_{N}(\iota=0)$. This
means that the expected number of binaries detected, allowing for random
inclination and polarization angles, is about 29% of the number that would be
expected if all the systems were face-on.

Figure 4 also has implications for coincidences between gravitational wave
detections and gamma-ray bursts. If we accept the popular model in which a
coalescence of two neutron stars or a neutron star and a black hole is
accompanied by a gamma-ray burst that is emitted in a narrow cone around the
binary’s rotation axis, then events where the cone points toward us are also
stronger gravitational wave emitters, and so we will see relatively more of
them [40]. The slope of the distribution of detected binaries in figure 4 at
$\iota=0$ is about 1.72, compared with 0.5 for the true distribution, a ratio
of 3.44. Therefore, a coincidence between a gravitational wave event and a
gamma burst with a narrow cone (so that only the linear behavior of the curves
in the figure is relevant) is about 3.4 times more likely than one would
expect by just naively computing the solid angle of the jet. For example, if
jets have a solid angle of $4\pi/100$, then only one out of every one hundred
coalescences would point its jet toward us. But we could expect that one in
every 29 detected coalescences would be accompanied by a gamma-ray burst.

##  3 Figures of merit

###  3.1 Triple Detection Rate: Relative effectiveness of a network

The first of the figures of merit measures the relative effectiveness of a
network at detecting the short bursts of gravitational waves that we assume in
our signal model, using enough detectors to extract the full information
available in the gravitational wave signal. Since all detectors are assumed
identical and the source waveform is the same in each case, only the network
detection volume and the duty cycle need to be used to provide a realistic
measure of the relative rates at which events will be detected by different
networks.

The relative detection volumes of various networks calibrate the volume of
space accessible to the network (often given in current LSC-Virgo papers in
units of MWEG: Milky Way Equivalent Galaxies). But adding extra detectors to a
network does more than increase its detection volume. It also ensures that
there is less time when there are fewer than three detectors in operational
mode. Current interferometers need exquisitely tuned control systems to keep
the interferometry locked on a fringe. During the recent S5 science run [41],
the two big LIGO detectors achieved a duty cycle of about 80%. When the
detectors start up at the advanced level of sensitivity, around 2016, the duty
cycle may well be similar. In principle there is no reason that the duty cycle
could not ultimately be pushed well above 90%, but this will require time and
effort. (The smaller GEO600 detector achieved a 95% duty cycle during S5 and
VIRGO operated at close to that efficiency during its several-month
participation at the end of S5.) If one requires an observation to be
performed by all instruments in a three-detector network with a duty cycle of
80% then they will be observing simultaneously only $0.8^{3}\simeq 51\%$ of
the time. If one adds a fourth detector, the amount of time at least three
detectors will be in observing mode dramatically increases to
$(0.8)^{4}+4(0.2)(0.8)^{3}\simeq 82\%$. Adding a fifth raises this to
$(0.8)^{5}+5(0.2)(0.8)^{4}+10(0.2)^{2}(0.8)^{3}\simeq 94\%$, a further
significant increase. We can expect that these numbers will be realistic
during the first few years of the operation of Advanced detectors, until the
experimental teams can focus their efforts on improving duty cycle instead of
raw sensitivity.

The Triple Detection Rate figure of merit for a given network sums the
detection volumes of all sub-networks containing detectors in three or more
locations, each weighted by the probability that the given sub-network will be
the only one observing at a given time. We do not include the amount of time
that only two detectors are in operation because these cannot fully
reconstruct the event in the absence of other information. Specifically, then,
consider a network of 4 separated detectors, called A, B, C, and D, all of
which are in observing mode for a fraction $f$ of the data-taking time, and
whose down-times are not correlated with one another. We define the Triple
Detection Rate of this network to be the effective available volume, with the
scaling factor $(D_{V,L}/\rho_{N,{\rm min}})^{3}$ removed:

$\displaystyle[3DR]_{\rm ABCD}$ $\displaystyle=$
$\displaystyle\left(\frac{D_{V,L}}{\rho_{N,{\rm
min}}}\right)^{-3}\left[f^{4}V_{\rm ABCD}+(1-f)f^{3}(V_{\rm ABC}+V_{\rm
BCD}+V_{\rm ACD}+V_{\rm ABD})\right],$ (29) $\displaystyle=$
$\displaystyle\frac{f^{4}}{3}\int{\rm
d}\Omega[P_{ABCD}(\theta,\phi)]^{3/2}+\frac{(1-f)f^{3}}{3}\int{\rm
d}\Omega\left\\{[P_{ABC}(\theta,\phi)]^{3/2}+\right.$
$\displaystyle\left.[P_{BCD}(\theta,\phi)]^{3/2}+[P_{ACD}(\theta,\phi)]^{3/2}+[P_{ABD}(\theta,\phi)]^{3/2}\right\\}.$

Triple Detection Rate is thus a measure of the effective three-site detection
volume averaged over a long observing run. The number of events detected by
three or more separated detectors in a network during a given observing period
will be proportional to the network’s value of Triple Detection Rate. The
generalization of (29) to networks with other numbers of detectors is obvious.

The definition of Triple Detection Rate specifies detectors at different sites
because a network of 3 detectors involving two at Hanford cannot resolve sky
positions, and hence cannot infer polarizations, distances, and other
parameters. Therefore, in computing $[3DR]_{\rm HHLV}$, the original four-
detector Advanced network, I do not use (29). Instead of all four three-
detector subnetworks, I include only two, both having the antenna power
pattern HLV, but involving different Hanford detectors. With this assumption
and an 80% duty cycle, we get $[3DR]_{\rm HHLV}=4.86$. This serves as a
reference value for other networks, since it is the basic coverage available
from the presently funded Advanced detectors with a realistic duty cycle for
the initial operation.

By contrast, if one of the Hanford detectors is placed in Australia, we get
the network AHLV, which has $[3DR]_{\rm AHLV}=6.06$ with a duty cycle of 80%.
The rate of events whose locations can be measured goes up by 25% simply by
separating the two Hanford detectors, because doing this creates two more
useful three-detector sub-networks. On the other hand, with a 95% duty cycle,
the difference is not so pronounced: $[3DR]_{\rm HHLV}=7.81$ while $[3DR]_{\rm
AHLV}=8.28$. In this case, most detections occur with all four detectors
working, for which in both configurations there is always a subset of three at
separate locations. We return to compare other interesting specific networks
in section 4 below.

To convert [3DR] back to an effective detection volume in space, multiply by
$(D_{V,L}/\rho_{N,{\rm min}})^{3}$, where $D_{V,L}$ is the visibility distance
of the source for the Livingston detector (the distance at which an optimally
located source has unit SNR), and $\rho_{N,{\rm min}}$ is the network
detection threshold SNR. To convert this effective volume into an expected
detection rate one multiplies by the volume rate of events of this population.

###  3.2 Isotropy

If the antenna patterns of detectors in a network are well-aligned, they
increase the detection volume nonlinearly, since the detection volume of a
small solid angle in any direction depends on the $3/2$ power of the total
antenna power pattern. Where the antenna patterns do not overlap
significantly, they make the network more isotropic. Increasing the detection
volume is obviously an important gain, but there may also be merit in a
network that is more isotropic. Isotropic antenna patterns are better for
coincidence observations with other all-sky survey instruments, particularly
those that are significantly flux-limited with a range shorter than that of
the gravitational wave detectors, as for example neutrino detectors searching
for gravitational collapse events [42, 43]. In such a coincidence observation
the events will be relatively nearby, so the isotropy of the antenna pattern
is more important than its total volume. This illustrates the key point that
the importance attached to different values of the f.o.m.’s depends on one’s
priorities in building a new detector, a point also made in [7].

We define the f.o.m. Sky Coverage to be the fraction of the sky over which the
network’s antenna power pattern is greater than half of its maximum value. By
cutting the sky at this value we are accepting all directions where the reach
of the network is at least $1/\sqrt{2}\simeq 71\%$ of its mean horizon
distance $R_{N}$. The concept of sky coverage was discussed for single
detectors in Sathyaprakash and Schutz [34], but the sky cut was done there at
50% of the mean horizon distance. The place where the cut is made is clearly
arbitrary, but since detection is based on computing SNR2, I use the 50% power
level in defining this f.o.m..

Networks differ greatly in their isotropy. For a single interferometer, [SC]
is just 34%. Aligning antenna patterns keeps them anisotropic, so networks
including the LIGO detectors and an Australian detector tend to have low
values of [SC], while adding in VIRGO or LCGT increases isotropy. Again, this
is illustrated for specific interesting networks in section 4. The AHJLV
network, with detectors in Australia and Japan, reaches 85%, and adding a
detector in India pushes the sky coverage over 90%.

Detector | Label | Longitude | Latitude | Orientation
---|---|---|---|---
LIGO Livingston, LA | L | 90o 46’ 27.3” W | 30o 33’ 46.4” N | 208.0o(WSW)
LIGO Hanford, WA | H | 119o 24’ 27.6” W | 46o 27’ 18.5” N | 279.0o(NW)
VIRGO, Italy | V | 10o 30’ 16” E | 43o 37’ 53” N | 333.5o(NNW)
LCGT, Japan | J | 137o 10’ 48” E | 36o 15’ 00” N | 20.0o(WNW)
AIGO, Australia | A | 115o 42’ 51” E | 31o 21’ 29” S | 45.0o(NE)
INDIGO, India | I | 74o 02’ 59” E | 19o 05’ 47” N | 270.0o(W)
Table 1: Name, abbreviation, geographic location, and orientation of the
various detector positions considered in this paper. The abbreviations will be
used to label functions and diagrams. When there are two instruments at
Hanford we will use HH. The orientation is the geographic compass angle,
measured clockwise from North, of the line bisecting the arms of the detector.
(This decouples the orientation from opening angle for detectors that may not
have perpendicular arms.) For the averages performed in this paper, however,
the orientation will not matter. The data for the LIGO and VIRGO detectors are
for the actual detectors. The data for LCGT are for the planned orientation.
The data for AIGO are from the Australian group (private communication) and
place the detector at Gin-gin. The data for INDIGO are essentially arbitrary;
they correspond to the location of GMRT and an arbitrary orientation. Opening
angles $\eta$ are not listed because all detectors are assumed to have
$\eta=\pi/2$.

###  3.3 Accuracy

The biggest benefit of adding one or more detectors in Asia or Australia is
that they add longer baselines to the existing three detectors, and it is the
baseline that determines the accuracy with which the source can be located on
the sky. Source resolution is achieved by time-delay triangulation, so that
for fixed errors in measuring the time-of-arrival of a signal at different
detectors, longer baselines provide better relative accuracy and smaller sky-
position errors. Position accuracy in turn affects the determination of other
parameters: if the position is wrong then the inferred intrinsic amplitude of
the signal and its polarization will be wrong. This issue has been studied for
specific networks, particularly those containing a detector in Australia,
which offers the longest baselines [8, 44, 45]. These studies sometimes
provide detailed sky maps of error ellipses under various assumptions, and
they show that for any network the angular resolution varies considerably over
the sky. The purpose here is instead to develop a single measure that captures
the general difference in resolution when one compares two different networks.
The f.o.m. called Directional Precision attempts to provide a simple sky-
averaged measure of the relative accuracy with which a given network can
determine positions.

The problem of determining how accurately a network can measure positions has
a long history. Triangulation should produce angular position errors
proportional to the time-of-arrival measurement error divided by the baseline
between two detectors, measured in light-travel time [3]. But since three
detectors need to be involved in order to narrow down the position to a single
location on the sky (or at most two locations), the geometry of the detector
array is key. The first quantitative conjecture on the solid-angle uncertainty
for a network of three gravitational wave detectors appeared in Gürsel and
Tinto [12], who refer to a private communication by K S Thorne. The geometric
characteristic they use is the area $A_{\perp}$ of the triangle of the
detectors projected perpendicular to the direction to the source. The solid
angle error $\delta\Omega$ for a source in a particular direction is,
according to Gürsel and Tinto,

$\delta\Omega=2\frac{(c\delta t_{12})(c\delta t_{13})}{A_{\perp}},$ (30)

where $\delta t_{12}$ and $\delta t_{13}$ are the rms timing errors on two of
the arms of the triangle. This improves when the SNR improves because the
timing errors decrease. No proof of this expression seems to have appeared in
the literature until the recent work of Wen, Fan, and Chen [46, 8], who give a
much more general exact result that reduces to this when the network consists
of three identical detectors. I will base Directional Precision on a
simplification of the Wen-Fan-Chen expressions, which in their full form allow
the exact computation of position errors for networks of any number of non-
identical detectors.

Wen and Chen [8] show that the solid angle uncertainty is given by

$\displaystyle(\delta\Omega)^{-2}=\frac{\sum_{j,k,\ell,m}\xi_{j}\xi_{k}\xi_{\ell}\xi_{m}|(\mathbf{r}_{kj}\times\mathbf{r}_{m\ell})\cdot{\mathbf{n}}|^{2}}{\left[4\sqrt{2}\pi
c^{2}\sum_{j}\xi_{j}\right]^{2}},$ (31)

where the sum is over detectors in the network, $\mathbf{n}$ is the direction
to the source, and $\mathbf{r}_{kj}$ is the vector from detector $k$ to
detector $j$. (It follows that in the sum, $k$ and $j$ are distinct, as are
$m$ and $\ell$.) The symbol $\xi_{j}$ provides the timing accuracy, and for
our case, where we assume we can do perfect matched filtering, it is:

$\xi_{j}=\left<\omega^{2}\right>_{j}\rho^{2}_{j}=(\delta t_{arr,j})^{-2},$
(32)

where $\rho^{2}_{j}$ is the squared SNR in detector $j$, where
$\left<\omega^{2}\right>_{j}$ is the mean squared frequency in the signal,
averaged over the signal waveform in the detector weighted inversely by the
detector noise, and where $\delta t_{arr,j}$ is the r.m.s. time-of-arrival
measurement error in detector $j$ when there are no covariances with other
measurement errors [3]. Notice that (31) depends on the projected areas of all
the various triangles formed by the inter-detector vectors. If there are only
three detectors, there is only one triangle, and this expression essentially
reduces to (30).

The measure (31) is the inverse of one element of the error covariance matrix,
and is therefore an estimate of the inverse of the area of the 1-$\sigma$
error ellipse. It is also related to the Fisher information matrix element for
solid angle. My definition of Directional Precision in (34) below inherits
this: it is to be regarded as an indicator of the 1-$\sigma$ errors in area.
This is an important point to bear in mind when comparing with other authors,
who often quote 90th percentile or 2-$\sigma$ errors.

If we assume that all detectors are identical, then all the
$\left<\omega^{2}\right>_{j}$’s are the same and all the $\xi_{j}$’s are
proportional to the squares of their respective detector antenna pattern,
multiplied by factors that are common to all detectors. Our first
simplification will be to ignore the polarization-dependence of the antenna
patterns for the sources and take

$\xi_{j}=\left<\omega^{2}\right>P_{j}D_{L}^{2}/r^{2}.$

This is not strictly equivalent to taking a polarization average of the solid
angle uncertainty, but when using the expression to compare different networks
on average this should be a small correction. The sum $\sum_{j}\xi_{j}$ is
then proportional to the network power pattern $P_{N}$.

The next simplification is that I will replace each individual detector power
pattern $P_{j}$ by the average of the network power pattern, $P_{N}/N_{D}$.
Again this is in the spirit of finding a simple measure associated with the
network as a whole. It is equivalent to saying that the network power SNR is
equally shared by all detectors.

For the final step we have to decide what it is that we integrate to get a
measure of accuracy. Is it appropriate to find a measure of $|\delta\Omega|$,
$|\delta\Omega|^{2}$, $|\delta\Omega|^{-1}$, $|\delta\Omega|^{-2}$, …? Any of
these might be useful for comparing different networks. I shall opt for
something proportional to (with the previously mentioned simplifications) an
average value of $|\delta\Omega|^{-1}$, mainly for reasons of ease of
computation. This measure is more sensitive to locations where
$|\delta\Omega|$ is small, that is, where the network gives particularly good
directional information. An average of $|\delta\Omega|$ itself would be
dominated by the regions where directions are poor. Given the relationship
between (31) and the Fischer information, the measure used here can also be
thought of as an indicator (no more than that) of the directional information
contained in the measurement: the larger the value of Directional Precision
the more directional information we get.

It follows from these assumptions that

$\left<\left|(\delta\Omega)^{-2}\right|^{1/2}\right>\simeq\frac{R_{\oplus}^{2}\left<\omega^{2}\right>}{4\pi
c^{2}}\rho_{N,{\rm min}}^{2}[DP],$ (33)

where I define, for any network of $N_{D}$ detectors, the Directional
Precision of the network to be

$[DP]=N_{D}^{-2}(V_{N})^{-1}\int{\rm d}\Omega
P_{N}^{3/2}\left[\sum_{k>j,m>\ell}|(\tilde{\mathbf{r}}_{kj}\times\tilde{\mathbf{r}}_{m\ell})\cdot{\mathbf{n}}|^{2}\right]^{1/2}.$
(34)

Here $V_{N}$ the network’s total detection volume, normalized in such a way
that a single interferometer has maximum range 1 ((22) with $R_{0}=1$),
$R_{\oplus}$ is the Earth’s radius, and
$\tilde{\mathbf{r}}_{kj}=\mathbf{r}_{kj}/R_{\oplus}$ is the vector connecting
the locations of detectors $j$ and $k$ on the unit sphere (i.e. in latitude
and longitude).

Larger values of [DP] indicate better direction accuracy. The scale factor in
(33) evaluates straightforwardly to give

$\left<\left|(\delta\Omega)^{-2}\right|^{1/2}\right>\simeq 14\rho^{2}_{N,{\rm
min}}\left(\frac{\left<\omega^{2}\right>}{(2\pi\times 100\;\rm
Hz)^{2}}\right)[DP]\quad\mbox{sr}^{-1}.$ (35)

Note that, in the sum over detectors in (34), the sum is restricted to pairs
where $k$ exceeds $j$ and $m$ exceeds $\ell$. This is justified because, as
noted above, these indices cannot be equal and because including values where
$k<j$ would simply count the same detector pair twice. The coefficient in
front of the sum has been increased by a factor of $\sqrt{2}$ to compensate.
Terms for which $k$ equals $m$ and $j$ equals $\ell$ also vanish because they
involve the cross product of a vector with itself. The sum shown therefore has
$(N_{D}+1)N_{D}(N_{D}-1)(N_{D}-2)/4$ nonvanishing terms. This number of terms,
inside the square root, is roughly compensated by the factor of $N_{D}^{-2}$
outside the integral, which arose from our simplification in which we replaced
each individual detector power pattern $P_{j}$ by the average of the network
power pattern $P_{N}/N_{D}$. The fact that $N_{D}$ roughly cancels out means
that [DP] depends more on the size of the detector triangles than on the
number of detectors in the network: extending the baselines in a network has
more effect on angular accuracy than does adding more detectors with similar
baselines to the existing ones.

It should be noted that [DP] measures the average position accuracy of
detected signals, not the accuracy on a given signal with a fiducial
amplitude. If network A is more sensitive than network B, so that A has a
bigger detection volume, then its position accuracy will be averaged over a
population that includes more distant and weaker sources than those of B. If
we only asked how network A would perform on the detection volume of network
B, its mean direction accuracy would be better than one might guess just by
comparing $[DP]_{A}$ with $[DP]_{B}$. So when using [DP] to compare the
performance of different networks, it is somewhat easier to interpret when it
is used to compare networks with the same number of detectors but different
geometries.

When combined with the values of [DP] we compute in table 2, this is not
inconsistent with the plots of error ellipses in the literature [8, 9, 45,
10]. The dependence on threshold $\rho_{N,{\rm min}}$ is interesting: the
higher the threshold, the stronger the ensemble of detected SNRs, so the
larger the value of [DP], and the better the direction-finding.

Network |  Mean Horizon Distance |  Detection Volume |  Volume Filling Factor |  Triple Detection Rate (at 80%) |  Triple Detection Rate (at 95%) |  Sky Coverage |  Directional Precision
---|---|---|---|---|---|---|---
L |  1.00 |  1.23 |  29% |  - |  - |  33.6% |  -
HLV |  1.43 |  5.76 |  47% |  2.95 |  4.94 |  71.8% |  0.68
HHLV |  1.74 |  8.98 |  41% |  4.86 |  7.81 |  47.3% |  0.66
AHLV |  1.69 |  8.93 |  44% |  6.06 |  8.28 |  53.5% |  3.01
HHJLV |  1.82 |  12.1 |  48% |  8.37 |  11.25 |  73.5% |  2.57
HHILV |  1.81 |  12.3 |  50% |  8.49 |  11.42 |  71.8% |  2.18
AHJLV |  1.76 |  12.1 |  53% |  8.71 |  11.25 |  85.0% |  4.24
HHIJLV |  1.85 |  15.8 |  60% |  11.43 |  14.72 |  91.4% |  3.24
AHIJLV |  1.85 |  15.8 |  60% |  11.50 |  14.69 |  94.5% |  4.88
Table 2: Comparison of various networks. A: AIGO or LIGO Australia; H: LIGO
Hanford single detector; HH: LIGO Hanford two detectors; I: INDIGO; J: LCGT;
L: LIGO Livingston; V: VIRGO. Mean Horizon Distance is the maximum detection
distance, scaled to the mean horizon distance (maximum range) of a single
detector observing at the same threshold. Detection Volume is the volume
inside the antenna pattern, on the same scale. Volume Filling Factor is the
ratio between the Detection Volume in column 3 and the volume of a sphere with
radius equal to the Maximum Range in column 2. The remaining columns are the
figures of merit. Triple Detection Rate measures the overall detection rate
and is given for two different values of the duty cycle: 80% to represent a
likely figure at the start of operations, and 95% to represent a reasonable
long-term operation goal. The values of Triple Detection Rate are smaller than
the Detection Volume by factors representing the loss of 3-site observing time
to duty cycle downtime. Sky Coverage measures how isotropic the network
antenna pattern is. Directional Precision reflects angular accuracy: the
typical solid angle uncertainty is inversely proportional to Directional
Precision, so that larger values denote more accurate networks. The first row
of the table is for a single detector, to facilitate comparisons.

##  4 Lessons

###  4.1 Discussion of specific networks

At the present time the only network of Advanced detectors that is fully
approved and funded consists of two LIGO detectors at Hanford and one at
Livingston, plus VIRGO in Italy: HHLV. Working together, these four detectors
have a detection volume of 8.98, more than 7 times that of a single detector
at the same network threshold. But when the duty cycle is 80% the effective
volume [3DR] falls to 4.86. The network covers 47% of the sky at half-power.
Its value of 0.66 for [DP] is the starting point for comparisons of network
accuracy.

In addition to these detectors, funding has started for the LCGT detector in
Japan, so it is reasonable to expect that the Advanced network will include
detectors at Hanford and Livingston in the USA, in Italy, and in Japan. If the
current proposal to move one of the Hanford detectors to Australia becomes
reality, then we should have the network AHJLV. If not, then we are likely to
have HHJLV. In addition, if a proposal to build a detector in India succeeds,
then in the long run we could have AHIJLV or HHIJLV.

To understand the capabilities of these networks it is useful to compare them
with the basic HHLV and with LIGO’s own variant AHLV. These comparisons will
show clearly the considerable benefits brought by the ongoing investment in
Japan and the proposed investment in India.

First we ask, how does AHLV compare to HHLV? The range and volume of AHLV are
very similar to those of HHLV. Its effective detection rate, [3DR], however,
is 25% larger: 6.06 (compared to 4.86) at a duty cycle of 80%, simply because
there are more three-site sub-networks in this array. AHLV is slightly more
isotropic than HHLV, with [SC] equal to 53.5%. This reflects the fact that the
position of the Australian detector at Gingin is very close to being antipodal
to the LIGO detectors. So far these network characteristics are not very
different from HHLV. But the real improvement is in direction finding. The
value of [DP] for AHLV is 3.01, compared with 0.66 for HHLV. This suggests
that the typical error ellipses will be reduced in area by more than 4 if the
detector is moved to Australia. These numbers are consistent with the results
of the much more extensive comparison of these two networks in an unpublished
internal technical report of the LIGO Scientific Collaboration [9] and in a
recent study of coherent detection involving LIGO Australia [10], and they
give a very strong scientific reason for placing the LIGO instrument in
Australia, independently of other detector developments.

Next we examine the improvements brought by the LCGT detector in Japan, with
the simplifying assumption that it will have identical sensitivity to the
other Advanced detectors. If there is no detector in Australia then we will
have the network HHJLV. Its overall detection volume, at 12.1 (figure 5), is
significantly greater that that of HHLV (8.98) and AHLV (8.93), reflecting the
fact that there is one further detector. The improvement in the detection rate
as measured by Triple Detection Rate is even greater: with a Japanese detector
and duty cycles of 80% the rate of detection would be more than 70% higher
than for the basic HHLV, and more than a third higher than AHLV. The network
is also significantly more isotropic as well, with [SC] at 73.5% (figure 6).
Adding the baseline to Japan also greatly improves the direction-finding,
although not by as much as the longer Australian baselines would: for HHJLV
the value of [DP] is 2.57, much better than the 0.66 turned in by HHLV but a
bit below the 3.01 value of AHLV. Nevertheless, the improvement over the basic
HHLV still represents a 4-fold reduction in the typical area of the error
ellipses.

Figure 5: Three network amplitude patterns, which show the true spatial shape
of the detection volumes. As in figure 3, two views are shown, one in
perspective and the other as a contour plot. The networks are: (top row) the
basic network of two instruments at Hanford, one at Livingston, and one at
Pisa; (middle row): the basic network with LCGT in Japan added; (bottom row)
the same after moving one of the Hanford detectors to Australia. Notice that
all these networks have roughly the same maximum range (HHLV: 1.74; HHJLV:
1.82; AHJLV: 1.76), and these are the values to which the contour levels are
scaled. They have different volumes (HHLV: 8.98; HHJLV: 12.1; AHJLV: 12.1)
because of their different isotropy, shown in figure 6. (The numbers are taken
from table 2.)

The Japanese detector may instead operate with a LIGO detector in Australia.
To see the difference with the characteristics we found in the previous
paragraph, we compare AHJLV with HHJLV. In detection volume and event rate the
two networks are essentially indistinguishable (figure 5). Sky coverage goes
up a noticeable amount with the Australian option, from 73.5% to 85% (figure
6). And, as might be expected, the extra baselines to Australia and between
Japan and Australia improve the direction finding. The value of [DP] for AHJLV
is 4.24, compared with 2.57 for HHJLV. So also here the improvement in angular
position information provides a strong reason for putting the LIGO detector in
Australia. Conversely, if one takes the Australian detector as a given and
asks what improvement is brought by the detector in Japan, the comparison is
between AHJLV and AHLV. Here not only is direction-finding significantly
better (4.24 compared to 3.01), but there is a dramatic increase in isotropy
(from 53.5% to 85%) and a factor of 1.4 increase in event rate (from 6.06 to
8.71 at 80% duty cycle).

On the basis of these numbers the network with the Australian option and the
LCGT instrument in Japan looks close to the ideal use of the resources being
invested by the various countries involved. It will have nearly twice as many
detections per year as the basic HHLV would if it could operate in coherent
detection mode (see below), at 80% duty cycle. It will cover nearly twice the
sky area. And its typical direction error ellipses can be a factor of 6
smaller in area. These benefits are brought simply by building one further
detector in Japan and moving a detector from the US to Australia.

Figure 6: Three network isotropy patterns, which show the parts of the unit
sphere where the amplitude sensitivity of the detector is better than
$\sqrt{2}$ of its best sensitivity. The networks are the same as in figure 5.

A nascent project in India might also succeed in building a detector. I have
included it in networks by placing it rather arbitrarily at the site of the
Giant Metrewave Radio Telescope (GMRT) radio telescope. It is interesting to
ask what the properties of networks containing this detector would be. I
include the Japanese detector and consider the two LIGO options: HHIJLV and
AHIJLV. Adding the Indian detector to the existing HHJLV network increases the
event rate by roughly 1/3, regardless of duty cycle. Considering that this is
achieved by adding one detector to a network of 5, which is an investment of
20% on top of the existing expenditure, getting a return of 33% in terms of
science still makes a strong case for this development. The detector in India
also improves isotropy, from 74% to 91%. And the extra baselines improve
position error ellipses, as measured by [DP], by 30%. If the Australian
detector is also built, then we compare AHJLV with AHIJLV. Again the Indian
detector brings an improvement of around 1/3 in event rate and it achieves
nearly complete isotropy, with a value of [SC] of 95%. It brings a 15%
improvement in position error ellipses, as measured by [DP], simply by adding
more baselines to the network.

Several of these networks have recently been studied also by Fairhurst [45],
who concentrated on the localization ability, using a different approach than
that adopted here, and one that is closer to the present methods of data
analysis based on thresholding. His results on comparisons of the localization
abilities of different networks are broadly in agreement with the relative
values of [DP] in table 2, and the typical ellipse areas that the present
treatment gives using (35) are within factors of two of the typical values
obtained by Fairhurst. This gives us confidence that our figures of merit can
be used not only to compare networks but also, to within factors of two, to
characterize the performance of individual networks.

Another useful comparison is between our analytic results and the Monte-Carlo
simulations for coalescing binaries performed by Nissanke, et al [11]. They
take HLV as their baseline network, i.e. assuming only one detector at
Hanford, and they do not allow for duty cycle down-time. They find that AHLV
will detect 1.48 times more events than HLV. In table 2 the appropriate
comparison is between the full detection volumes of HLV and AHLV, whose ratio
is 1.55. We take this to be excellent agreement. Moreover, they measure the
isotropy of various networks by plotting detected event distributions on the
sky (their figure 2). Their conclusions are qualitatively in agreement with
ours in figure 6, and they remark that networks that include LCGT are
noticeably more isotropic, a conclusion also in agreement with our values of
Sky Coverage in table 2.

Note, however, that the true “default” network is HHLV, and in table 2 it is
clear that moving one of the H detectors to Australia hardly changes the total
detection volume. When network duty cycle is taken into account, there is a
net event rate gain (for three-site detections) of up to a factor of $1.24$.
On top of that there is an event rate gain from being able to do coherent data
analysis better, so that the LIGO Australia option not only has better angular
resolution but also a significantly higher detected event rate. This is the
subject of the next section.

###  4.2 Coherent versus coincidence data analysis: implications for event
detection rates

The assumption of this paper is that data analysis is done by fully coherent
combination of the different detectors’ data streams. This is not yet the
practice in the LSC-VIRGO data analysis, mainly because coherent analysis
normally assumes a Gaussian background of instrumental noise, and is therefore
vulnerable to what are often called “glitches”, bursts of noise from
instrumental effects that can masquerade as real signals. Because in present
detectors there is a significant glitch background, data analysis usually
includes a coincidence step, in which events of a sufficient size in single
data streams that occur in coincidence (within a time-window equal to the
light travel times among the various detectors) with events in other detectors
are selected and studied further. This coincidence test eliminates most of the
glitch background.

But a purely coincident analysis also eliminates most of the potentially
detectable signals, i.e. signals that could reliably be detected if the
background noise were ideally Gaussian. The penalty is easy to compute. In a
recent review of the astrophysical evidence for the rates of compact object
binary coalescences, the LSC and VIRGO collaborations predicted a detected
event rate for the HHLV network of Advanced detectors [47]. Their method was
to take the number of events that occur inside the detection volume of a
single detector above the detection threshold $\rho_{\rm min}=8$. They took
the most likely value of the rate of neutron-star coalescences to be
$1\,\rm{Mpc}^{-3}\,{Myr}^{-1}$, or equivalently 100 events per Milky Way
Equivalent Galaxy per million years. With this volume event rate, the most
likely detection rate for these systems came out to be 40 per year. The reason
for counting only events that occur in one detector’s detection volume despite
the fact that the network contains four detectors is to approximate in a rough
(and conservative) way the coincidence criterion.

For the same network, but with coherent data analysis using a network
threshold of the same value ($\rho_{\rm N,\,min}=8$.), the data in table 2
show that the rate would be higher by the ratio of [3DR] which is 4.86
(allowing for an 80% duty cycle), to the volume for a single detector, 1.23.
This ratio is 3.95, which implies that the HHLV network, with perfectly
Gaussian noise, could detect about 160 events per year if it did coherent
analysis. The difference in detection effectiveness between coherent and
coincidence analysis for coalescing binary signals in this basic network is a
factor of 4 in detection rate.  This difference is illustrated graphically by
comparing the volumes of space covered by fully coherent analysis and pure
coincidence analysis, in figure 7.

Figure 7: The antenna patterns of the LIGO-VIRGO detectors for (a) coherent
and (b) coincidence analysis methods. The coherent pattern is the HHLV
amplitude pattern. The coincidence pattern is the region in which, for random
polarizations, an event crosses threshold in at least two of the detectors
(but not allowing events that appear only in two Hanford detectors). The
thresholds are assumed to be the same, e.g. if the individual detector
thresholds for the coincidence analysis is 8, then the coherent data analysis
threshold is also set at 8, as discussed in the text.

Naturally this comparison depends on the threshold assumed for the two kinds
of data analysis. The comparison shown in the figure is for equal thresholds:
if, as in [47], the coincidence observation is done with a threshold SNR of 8
in each detector, then we assume that the network coherent threshold is set at
8 as well. This is not unreasonable, since the coherent analysis essentially
fights only against Gaussian noise, where events at $8\sigma$ occur only once
in $10^{5}$ years at an effective sampling rate of 300 Hz. This works well if
coherent methods can eliminate glitches. This may not be fully possible for
HHLV (see below) but it should be possible for the enlarged networks,
including AHLV. Therefore the comparison shown in this figure is relevant for
extrapolations of event rates to the larger networks.

Now, for the existing detectors, instrumentalists are working hard to reduce
the glitch rate, and the LSC-VIRGO analysis teams are bringing in coherent
analysis [28, 19, 29, 30, 33]. Networks containing three or more detectors can
also use their null streams to test for and veto glitches, as described in
section 1.2.

In practice, the analysis teams will begin by mixing coincidence and coherence
methods, by setting a low threshold on coincidences to obtain a population of
possible events, and then using coherent methods (including null streams) to
eliminate the glitch coincidences. Such methods are computationally much less
demanding than purely coherent methods, and they can presumably bridge some of
the gap of the factor of four between pure coincidence and full coherent
methods.

However, the basic HHLV network may not be completely amenable to coherent
analysis, because of the near-perfect alignment of the LIGO Hanford and LIGO
Livingston detectors. While this allows good discrimination against glitches
in one of the LIGO detectors, it reduces the information recoverable from real
events: polarization and sky location can be determined only if VIRGO is
excited comparably strongly to the LIGO detectors, and without a sky location
one cannot define a null stream. This in turn leads to more opportunities for
false alarms, and lowers the significance of real events. It remains to be
seen how much of the full factor of 4 computed above can be recovered by
introducing some degree of coherent analysis into the HHLV network, but
clearly it is a very important step to take. If the nominal detection rate of
40 events per year can be raised even to 80, this will be the most cost-
effective way to improve the baseline network.

It is worth noting that the LSC study of the LIGO Australia option [9] made a
strong recommendation to move to coherent data analysis. The move of one LIGO
detector to Australia breaks the degeneracy of the LIGO instruments,
especially if the new detector is anti-aligned with the existing LIGO
detectors. It should therefore allow fully robust coherent analysis, coming
close to the maximum possible event detection rate of 200 NS-NS events per
year, assuming the most likely rate quoted in [47], and assuming an 80% duty
cycle. The improvement of a factor of up to 5 in the detection rate is
probably the strongest reason for placing a LIGO detector in Australia.

The LCGT detector will add a third null stream to the HHLV or AHLV networks,
and make coherent analysis even more robust. If the most likely coalescence
rates prove to be accurate, and if the network detection threshold is set to
8, the HHJLV network can expect to detect 270 NS-NS coalescences per year, and
the AHJLV network 280. Adding a detector in India raises these numbers to
around 370 events per year. Improving the duty cycle to 95%, which seems
feasible after a few years of operation, increases the five-detector NS-NS
rates to around 360 per year and the six-detector rate nearly to 500 per year.
These rate improvements would qualitatively change the kind of science
obtainable from Advanced detectors.

For coalescences of neutron stars with black holes, the LSC and VIRGO paper
[47] quotes a “best” rate of 10 per year for HHLV with coincidence analysis.
The expected rates for larger networks can therefore be obtained from the NS-
NS rates just quoted by dividing by 4. Similarly, the rates for binary black
hole mergers are expected to be half of the NS-NS rates; black holes have a
much lower number density in the universe, but they can be detected much
further away because of their higher mass. The NS-BH and BH-BH rates are, of
course, much less secure than the NS-NS rates, because there are no observed
binary systems of those types; the rates used in [47] depend exclusively on
population simulations. However, the recent identification of two possible
X-ray binary precursors of BH-BH binaries provides a much-needed observational
normalization of the population. Bulik et al [48] conclude from these systems,
in which a black hole is in a close binary with a Wolf-Rayet star, that the
BH-BH detection rate might in fact be much higher and could even significantly
exceed the NS-NS rate.

One further item is worth noting. Searches for binary signals are optimal if
they incorporate as much prior information as possible, and Bayesian analysis
techniques that do this are becoming standard in the current LSC-VIRGO data
analysis methods. The present study provides three such priors: the network
antenna pattern (a prior on the sky location of the source) and the two
p.d.f.’s: the expected distribution of SNR values (2.4), which is a prior on
the signal amplitude; and (for binaries) the expected distribution of detected
inclination angles ((28) and figure 4), which is a prior that affects the
relative amplitudes and phases of the signal in different detectors. The use
of the antenna pattern as a prior needs to be done with care, because as noted
above there will be a number of sources detected that are outside the “hard”
edge of the detection volume. A polarization-dependent prior is of course even
better than the polarization-averaged antenna pattern computed here.

##  5 Conclusions

In this paper I have developed a framework in which it is possible to compare
networks of gravitational wave interferometers consisting of different numbers
of detectors in different geographical configurations. I have shown that, for
any network, the expected SNR distribution of detected events, once the data
analysis can be done by optimal coherent methods, is a universal $\rho^{-4}$
power law that falls to zero for $\rho$ smaller than the detection threshold.
It follows from this distribution that the most likely SNR of the first
detected signal will be about 1.26 times the threshold of the search. I have
derived the (similarly universal) probability distribution of the inclination
angle of detected binary systems, and I have shown that, if coalescing
binaries are associated with narrowly beamed gamma-ray bursts, then because
the radiated gravitational wave power is correlated with the direction of the
gamma-ray cone, we can expect 3.4 times more detected coincidences than if
they were not correlated. I have suggested three figures of merit that can be
computed for any network and which measure average properties of the network:
its expected event detection rate, its isotropy, and the accuracy of its sky
position measurements. These figures of merit are inevitably crude averages,
and they should not be a substitute for detailed comparisons of networks as
part of the planning for specific new detectors. But they give a clear
indication of the merit of enlarging the network from the originally planned
LIGO and VIRGO detectors to include detectors in Asia and Australia.

It is worth stepping back from the many different options that exist for
enlarging the worldwide interferometer network to consider the net
improvements that are possible if current plans are realized. Consider the
network AHJLV, consisting of LIGO with one detector in Hanford and one in
Livingston, VIRGO in Italy, LCGT in Japan, and LIGO Australia. The numbers in
table 2 show how much more science that network can do than the originally
planned HHLV. Its event rate, with detectors operating on 80% duty cycles,
would be nearly twice as high for all categories of burst sources. It would
cover nearly twice as much of the sky, making it a better bet for coincidence
observations with neutrino detectors. And our measure of the areas of angular
position measurement error ellipses improves by a factor of 6.4, from 0.66 to
4.24, indicating that the typical error ellipse goes down in area by a factor
of more than 6. This will make a huge improvement in follow-up studies with
optical and other telescopes. This network offers much more science than had
been promised in the initial proposals for the existing four large detectors,
at the cost of building only one more detector and moving another to a better
location. The impact of the single extra detector in Japan is so large because
robust gravitational wave astronomy requires a minimum of three detectors in
different locations, so the marginal impact of increases to four and five is
large.

If the project in India gains support and, on a longer timescale, leads to a
sixth Advanced detector, it would create the network AHIJLV, an even bigger
improvement on HHLV. Its event rate would be 2.4 times higher, on a duty cycle
of 80%. It would cover 95% of the sky at half power, and its sky localization
error ellipses would be fbetter than 7 times smaller in area than those of the
presently planned LIGO-VIRGO network.

It is important to realize that both of these enlarged networks have maximum
detection distances that are within 5% of the maximum range of HHLV. Their
large event rate gains come partly from increased isotropy and partly from
having more three-site sub-networks that can detect and localize events even
when one or more detectors has fallen out of observing mode. They survey the
same volume of space more completely than HHLV can. But the big improvements
in sky localization are perhaps the strongest arguments for pursuing these
enlarged networks. The values of Directional Precision we compute here suggest
(using the conversion to steradians given above) that the typical error box in
either network would be smaller than a degree on a side. This not only makes
searching with electromagnetic telescopes for counterparts easier but it
reduces the probability of chance coincidences in a large field of view.

The conclusions in this paper depend strongly on the assumption of coherent
data analysis. If coincidence data analysis is used, where events are selected
for further study only if they cross a particular threshold in each
participating detector, there is no guarantee that the properties described
here will still hold for the different networks. Coherent analysis produces
networks whose antenna patterns are the sum of the power patterns of the
network members. Coincidence analysis produces antenna patterns that are
basically determined by the intersections of the power patterns of network
members. Performing a first cut at the noise by coincidence analysis, even if
it is followed by a coherent follow-up, will not reproduce the assumptions
used here. The reason for coincidence analysis is, of course, to eliminate
rare but strong non-Gaussian noise events, but these can also be identified by
using network null streams, whose number increases with the number of
detectors in the network.

Moving from coincidence to coherent analysis can increase detection rates by
factors of four or more. It is to be expected that network data analysis will
move to fully coherent analysis as the number of detectors increases and as
experimenters manage over time to reduce the frequency and amplitude of non-
Gaussian noise glitches. With such analysis techniques, the full potential of
the enlarged networks, as illustrated by the figures of merit calculated here,
can eventually be realized.

It is a pleasure to acknowledge discussions of the network problem with many
colleagues, including B Krishnan, S Dhurandhar, J Hough, K Kuroda, A
Lazzarini, and J Marx. Special thanks to S Klimenko, S Nissanke, M-A Papa, P
Sutton, B Sathyaprakash, and L Wen for detailed discussions and comments. This
work was stimulated by a kind invitation from K Kuroda to the 58th Fujihara
Seminar in 2009. DFG grant SFB/TR-7 is gratefully acknowledged.

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