Binary Neutron Star Mergers and Third Generation Detectors: Localization and Early Warning
Man Leong Chan Chris Messenger Ik Siong Heng Martin Hendry











                             
                             
                             



 
 
 
 
 
 

^1SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK





For third generation gravitational wave detectors, such as the Einstein 
Telescope, gravitational wave signals from binary neutron stars can last 
up to a few days before the neutron stars merge. To estimate the measurement 
uncertainties of key signal parameters, we develop a Fisher matrix approach 
which accounts for effects on such long duration signals of the time-dependent 
detector response and the earth’s rotation. 
We use this approach to characterize the sky localization 
uncertainty for gravitational waves from binary neutron stars at 40, 
200, 400, 800 and 1600Mpc, for the Einstein Telescope and Cosmic Explorer 
individually and operating as a network. We find that the Einstein Telescope alone 
can localize the majority of detectable binary neutron stars at a distance 
of ≤200Mpc to within 100deg^2 with 90% confidence. 
A network consisting of the Einstein Telescope and Cosmic Explorer can 
enhance the sky localization performance significantly – with the 90% credible region of 𝒪(1) deg^2 
for most sources at ≤200Mpc and ≤100deg^2 for most sources at ≤1600Mpc. 
We also investigate the prospects for third generation detectors identifying the presence of a signal 
prior to merger.
To do this, we require a signal to have a network signal-to-noise ratio of ≥12 and ≥5.5 for at least 
two interferometers, and to have a 90% credible 
region for the sky localization that is no larger than 100 deg^2. We find 
that the Einstein Telescope can send out such “early-warning” detection alerts 
1 - 20 hours before merger for 100% of detectable binary neutron stars 
at 40Mpc and for ∼58% of sources at 200Mpc. For sources at a distance of 400Mpc, a 
network of the Einstein telescope and Cosmic Explorer can produce 
detection alerts up to ∼3 hours prior to merger for 98% of detectable 
binary neutron stars.



Binary Neutron Star Mergers and Third Generation Detectors: Localization and Early Warning
    Man Leong Chan^1[m.chan.1@research.gla.ac.uk], Chris Messenger^1, Ik Siong Heng^1 & Martin Hendry^1
    Received: 24 October 2018 / Accepted: TBD
=======================================================================================================



GW[GW]gravitational wave
BNS[BNS]binary neutron star
BBH[BBH]binary black hole
NSBH[NSBH]neutron star black hole
ET[ET]Einstein Telescope
EM[EM]electromagnetic
H[H]LIGO Hanford
L[L]LIGO Livingston
V[V]VIRGO
SNR[SNR]signal to noise ratio
MCMC[MCMC]Monte Carlo Markov Chain
CE[CE]Cosmic Explorer





§ INTRODUCTION




The first detections of GW from BBH systems GW150914, GW151226,
GW170104 and GW170608 by the two LIGO detectors at Hanford and Livingston
<cit.> have
opened a new window on the universe and marked the beginning of GW
astronomy.  In 2017, VIRGO began observation and the first joint detection
GW170814 was made by LIGO and VIRGO together <cit.>.  Just a few days later, the three GW observatories detected the first BNS
merger event GW170817 <cit.>.  The detections of
multiple EM counterparts associated with GW170817 initiated the era of GW 
multi-messenger astronomy
<cit.>.  In the coming years, 
additional ground-based interferometric detectors such as KAGRA
and LIGO India are likely to join the global network
<cit.>.  Many more GW
detections can therefore be expected, from several different GW sources in the universe such as compact
binary mergers (i.e. BBH, BNS and NSBH), core-collapse supernovae, non-symmetric neutron stars and
the stochastic background <cit.>.



For systems such as BNS and NSBH, 
the presence of a neutron star component is expected to lead to the generation
of associated EM emission
accompanying the mergers of these systems – including gamma ray bursts, x-ray
emission, kilonovae in the optical and infrared
bands and radio afterglows
<cit.>.  Detecting an EM counterpart in coincidence with a GW trigger will increase the
detection confidence and improve the sky
localization of the GW detection.  In addition, a successful EM
follow-up observation can establish an association between the GW trigger 
and its progenitor and provide a better understanding of the progenitor and its local
environment<cit.>.  A coincident detection of an EM 
counterpart may also allow measurement of the redshift  of the source independently of the
GW signal; this can be used for cosmological tests
and for constraining the equation of state of dark
energy <cit.>.





Despite the low probability that
BBH mergers will be accompanied by EM
emission, identifying the host galaxy of a BBH will still hopefully
establish the association between the BBH and the galaxy.



One key factor to the success of EM follow-up observations triggered by
GW events, or that of identifying the host galaxy, is the
localization of the GW source. However, GW
interferometric detectors are not directional
instruments, and usually triangulation from a network of GW detectors is
the approach to localizing GW sources.  Therefore, their performances are
usually relatively poor when it comes to pinpointing sources of
GW.  For example, the 90% credible region for the sky positions
of the first detected BBH events are 𝒪(10^2) deg^2
<cit.>.


The situation improved with VIRGO joining the observing runs, and it will
continue to improve when KAGRA and LIGO India start operating in later years
<cit.>.  However, for sources at large
distances, EM follow-up observations will remain a difficult challenge. 
This is because a given type of GW source observed at a larger distance will also have a larger associated localization error on the sky.  Larger distances also mean that the telescopes performing EM 
follow-up observations will need to observe for longer to reach the sensitivity required to detect
the counterparts <cit.>. As a result, the telescopes
may be unable to scan the whole 90% credible region before the EM
counterpart fades below the detectable threshold.  For example, the brightness
of kilonovae may peak at 1 day after the merger and then start fading
<cit.> as was the case for the kilonova associated
with GW170817 <cit.>.







Third generation detectors such as the ET and
CE can bring significant improvements in GW source localization. 
Third generation detectors with enhanced sensitivity across the frequency band accessible to ground-based
detectors will be able to detect GW from BBH and
BNS sources located at distances far beyond the horizon of second generation
detectors.  In particular, the improved sensitivity from 1Hz to 10Hz of
these detectors will distinguish them from second generation detectors, as this
allows for an extended duration of in-band observation of the signal.  Depending on the distance, signals from BNS can be traced back
up to hours or days before merger. 



The long in-band duration of a signal will introduce several effects.
Firstly, it allows the detector to accumulate SNR
over a significantly longer period of time.  As mentioned, one of the
difficulties in EM follow-up observations of BNS merger triggers at
large distance will be the time available to scan the 90% credible region
associated with the BNS trigger before the EM counterpart becomes too
faint.  If the SNR of a BNS trigger can be accumulated to a
statistically significant level prior to the merger, prompt detection alerts could be made possible – thus increasing the probability of detecting the EM signature of the trigger <cit.>.  The long duration also enables the detector to observe the source from different positions and directions as the earth rotates. This effect
is important in localizing the source of GW as it result in the
time-dependency of the antenna pattern.  The GW will also be Doppler shifted as the detector moves relative to the source as it rotates with the Earth's spin. The long duration of a signal thus requires the consideration of the earth's rotation when estimating the localization error.



In the literature, most studies of localization errors of GW from
compact binary coalescences are based on the
assumption that the duration of the signal is short enough
that the rotation of the earth is negligible <cit.>.  This is justifiable for the reason that the in-band durations of the signals in those
studies are only seconds to minutes in length.  
Mills, et al, 2017 <cit.> consider 
localization of short transient signals from BNS mergers 
with a network of both second and three generation detectors.
More recently, there has
been work considering the long in-band duration and the rotation of the earth
<cit.>. 
In this latter work the authors have modeled the GW signal using the stationary phase approximation – essentially the leading-order term in an expansion in powers of the small quantity that is the ratio of the radiation-reaction time scale and the orbital period.

In this work, we take into account the earth's rotation and, using a Fisher matrix approach, estimate the localization of GW from BNS sources observed by the ET and CE individually and as a
network.  We perform a series
of tests to estimate the localization capabilities of these detectors for
BNS at 40, 200, 400, 800 and 1600Mpc and for
a population of BNS sources that are distributed uniformly in the comoving volume. We then investigate the feasibility of “early-warning” detection by setting requirements on localization error and accumulated SNR before merger for an alert to be released.


Throughout this work, we focus our analysis on BNS where the in-band duration of their signals can be days' long, and thus the
effect of earth's rotation is important. 



This paper is structured as follows: in Section<ref>, we briefly
introduce the configuration of the detectors we
considered for this work, and the
technologies employed to achieve their sensitivities.  The methodology is
presented in Section<ref> with the results and simulations
presented in Section <ref> 
together with a discussion.  We then provide our conclusions
in Section <ref>.




§ THIRD GENERATION DETECTORS




There are currently two proposed  third
generation detectors: the ET and CE. For
the ET, we employed the geometrical configuration known as ET-D as
discussed in <cit.> where the detector consists
of 3 individual interferometers. Each interferometer has an
opening angle equal to 60^∘ and is rotated by 120^∘ relative to the others, thus forming an equilateral triangle. The lengths of the arms of the interferometers are 10km. 



The current design for CE is an interferometric detector with a
90^∘ opening angle and the arms as long as
40km <cit.>.



 §.§ Location and Antenna Pattern




The exact geographic locations at which the detectors will be built are
still unknown.  In this work, the location adopted for the ET is
(Longitude, Latitude) = (10.4^∘, 43.7^∘), and for CE
is (-119.41^∘, 46.45^∘).  Due to the differences between the ET and CE configurations, 
the antenna patterns of these detectors will be computed differently.  The antenna pattern of each
interferometer of the ET can be expressed as <cit.>



    F_+^1(θ, ϕ, ψ) =    -√(3)/4[(1+cos^2θ)sin2ϕcos2ψ
       +2cosθcos2ϕsin2ψ], 
    
    F_×^1(θ, ϕ, ψ) =   √(3)/4[(1+cos^2θ)sin2ϕsin2ψ
       - 2cosθcos2ϕcos2ψ],


    F_+,×^2(θ, ϕ, ψ) =    F_+,×^1(θ, ϕ + 2π/3, ψ), 
    
    F_+,×^3(θ, ϕ, ψ) =    F_+,×^1(θ, ϕ - 2π/3, ψ),

where the above equations assume that the detector is at the center of a spherical coordinate system in which θ and ϕ are the azimuthal angle and polar angle of the source respectively, 
and ψ is the GW polarization angle. The superscripts indicate the 3 interferometers in the configuration. 

A single interferometer with arms separated by 60^∘
will have a response
smaller by a factor of √(3)/2 compared to an 90^∘
interferometer.  However, the sum of three 60^∘ separated
interferometers will enhance the response by a factor of
√(3), giving an overall factor of 3/2 larger
than a single 90^∘ interferometric detector
configuration<cit.>.  The CE's
antenna pattern can be written as

    F_+(θ, ϕ, ψ) =   1/2(1 + cos^2θ)cos2ϕcos2ψ
       - cosθsin2ϕsin2ψ, 
    
    F_×(θ, ϕ, ψ) =    -1/2(1 + cos^2θ)cos2ϕsin2ψ
       -cosθsin2ϕcos2ψ.





















































 §.§ Duration of Binary Neutron Star Signals




In a GW detector, different noise sources limit the sensitivity of the
detector in different frequency bands <cit.>. For
example, from 1Hz to 10Hz, seismic noise and gravity
gradient noise are the main contributors. From 10Hz to 200Hz, the
performance is limited by
quantum noise and thermal noise from the suspension and mirror coatings.  For frequencies
beyond 200Hz, shot noise is the primary source of
noise <cit.>.



There are many proposed technologies designed to improve the
sensitivity of the ET <cit.>.  
The ET is expected to have improved sensitivity across the frequency band accessible to ground-based detectors compared to 
second generation detectors.  
For CE, the exact technologies that will be employed are still undergoing
research and study. The current sensitivity curve of CE is obtained using
the existing technologies and well defined extrapolations from
them. As CE will make use of the available technologies at the time that
it is built, the sensitivity curve should not be considered the design target
of CE <cit.>. The estimated amplitude spectrum density of the
ET <cit.> and CE are presented in Figure
<ref>. The design sensitivity of aLIGO and Advanced VIRGO are also
shown for comparison. Compared to aLIGO and Advanced Virgo, the ET's
sensitivity from 10Hz is improved by at least a factor of 10, the
improvement can even be up to a factor of 10^6 in the frequency band below
10Hz. CE's sensitivity is also improved by up to a factor of 10^3 from
1Hz to 10Hz compared to second generation detectors, and is improved
by ∼30 times above 10Hz.





Different relative improvements in sensitivity as a function of frequency 
can lead to different impacts on the sky localization capability. Better
sensitivity in the medium to high frequency band can effectively increase the
SNR for a GW event and thus reduce the 
localization error.  On the other hand, improvement in the low frequency band
might not increase the SNR as much, but it will substantially extend the
in-band duration of the signal from the order of seconds/minutes to
the order of hours/days. We can express the time remaining prior to merger for a compact binary system as
a function of the instantaneous GW frequency<cit.>
using









    τ_c = 5/256c^5/G^5/3(π f_s)^-8/3/ℳ^5/3,
 
where τ_c is the time to merger, c the speed of light, G the gravitational constant, ℳ
the chirp mass and f_s the starting
frequency considered for the GW. Figure
<ref> shows the time to merger for a
1.4M_⊙ - 1.4M_⊙ merger as a function of f_s.  Also plotted are 10M_⊙ - 10M_⊙ and 30M_⊙
- 30M_⊙ BBH mergers for comparison.  The in-band duration of a GW from a given compact binary system in a detector can be obtained by replacing the
starting frequency f_s with the low frequency cut-off 
of the detector in Eq.<ref>. 
As indicated in Figure <ref>, for BNS systems, if
the detector's low frequency cut-off is reduced
to 2Hz, the in-band duration of the signal will be close to 1 day,
and will be more than 5 days if the low frequency cut-off is 1Hz.


This is substantially longer than the in-band duration for aLIGO and Advanced
VIRGO, where the low frequency cut-off is 10Hz. 
The in-band duration of BBH signals are expected to be shorter. 
For the ET with low cut-off frequency at 1Hz,  
10M_⊙ - 10M_⊙ BBH signals will last for ∼ 5 hours. 






Such a long duration allows the detector to observe the signal along the detector's 
trajectory on earth as the earth rotates, and therefore
makes the detector's response explicitly time-dependent. To illustrate this time-dependence, in Figure <ref> two source sky locations are selected and the change over 5 days of the ET and CE detector response to sources at those locations is shown.































§ METHODOLOGY




Using the Fisher matrix, we aim to provide a lower
bound on source sky position error for GW sources and examine the feasibility of early warning.

It is often reported that the Fisher matrix approach produces estimates that are more optimistic than those methods that completely explore the likelihood such as MCMC.  However, this is often due to the misuse of the Fisher matrix in situations where the SNR is too low.  In the moderate to high SNR regime where the Cramer–Rao lower bound is valid <cit.>,
estimates from the Fisher matrix are a good indicator of the
expected uncertainty on parameters.  Furthermore, complete
MCMC simulations are usually too computationally expensive
to carry out, while the Fisher matrix is low to moderate in computational cost. 



For an incoming GW, the strain observed by the
I^th detector can be expressed as h_I(θ, t) in
the time domain.  It is a linear
combination of the wave's two polarizations h_+(θ,
t), h_×(θ, t) and the detector response
F_I^+(θ, ϕ, ψ, t), F_I^×(θ, ϕ, ψ, t) as 
<ref>


    h_I(θ, t) = F^+_I(θ, ϕ, ψ,
    t)h_+(θ, t) + F^×_I(θ, ϕ, ψ,
    t)h_×(θ, t),


where the vector θ represents the unknown
signal parameters: sky position, distance,
time of arrival at the center of the earth, binary masses, initial phase
of the wave when it arrives at the center of the earth, inclination angle and
polarization angle. The time at the
detector is denoted by t which is equal to the arrival
time t_0 of the incoming wave at the center of the earth, plus the time τ required for the wave to travel from the center of the earth to the detector, given by


    τ = n·r/c,


where n is the GW propagation direction and r is the location vector of the detector
relative to the center of the Earth. 
The Fourier transform of h_I(θ, t) is then
defined as


    h̃_I(θ, f) = ∫_-∞^∞
    h_I(θ, t) e^-2iπ ft dt,




The mathematical definition of the Fisher matrix is given by


    Γ_ij = ∑_I=1^N 2  ∫_0^∞∂h̃_I^*(θ, f)/∂θ_i∂h̃_I(θ, f)/∂θ_j 
    + ∂h̃_I^*(θ, f)/∂θ_j∂h̃_I(θ, f)/∂θ_i/S_I(f)df.


where  ∂h̃(θ,
f)/∂θ_i is the partial derivative of
h̃(θ, f) with respect to the i^th unknown
parameter θ_i. The power spectrum density
of the I^th detector is denoted by S_I(f).
We also sum over the number of detectors, or
in the case of the ET, over the number of individual interferometers.  The
optimal SNR, ρ, of the incoming GW can be expressed as


    ρ^2 = 4  ∫_0^∞|h̃_I(θ,
    f)|^2/S_I(f) df.





In this work, we construct the Fisher matrix for the following
unknown parameters: right ascension, α; declination, δ; arrival time, t_0, at the center of the earth; the log of the distance, log_10d; polarization angle, ψ; the log of the total binary masses, 
log_10M; the cosine of the inclination angle, cosι; 
the symmetric mass ratio, η = M_1 × M_2 / M^2, of the masses of the two bodies in the binary; the initial phase, ϕ_0, of the wave when it arrives at the center of the earth. 



When computing the GW localization error 
for a source at a particular sky
location,  we divide the entire wave into pieces,
each of which is 100 seconds long – with the final piece
≤100 seconds depending on the specific in-band duration of the
signal. The total number of pieces N_p is then equal to
τ_c/100, rounded towards positive infinity.
For each piece of the wave, we employ the formalism described above to compute
the Fisher matrix Γ_ij^k and the optimal SNR ρ^k. The
superscript k indicates the k^th piece of the wave. 




The final Fisher matrix Γ_ij^f is then 


    Γ_ij^f = ∑_k = 1 ^N_pΓ_ij^k,


where we sum over the Fisher matrix contributions from each piece of the waveform, and the superscript f indicates the resultant Fisher matrix.
The matrix inverse of the Fisher matrix then gives the covariance matrix of the unknown parameters as 


    cov_ij = Γ_ij^-1,


from which the localization error is extracted using 

    ΔΩ = 2 π√(λ_αλ_δ)cosδ,

where ΔΩ is the localization uncertainty, λ_α and λ_δ 
are the eigenvalues of the matrix cov_ij corresponding to the α and δ of the source respectively.
The following expression can be used to convert ΔΩ to any desired confidence level,

    ΔΩ_p = -2log(1-p)ΔΩ,

where p is a value between 0 and 1 indicating the confidence level.
Similarly, the accumulated SNR is given by


    (ρ^f)^2 = ∑_k=1^N_p (ρ^k)^2.





§ RESULTS OF SIMULATION AND DISCUSSION






 §.§ Localization




To test the localization capabilities of third generation detectors, we have
simulated GW signals from 1.4M_⊙-1.4M_⊙ BNS sources at
distances of 40, 200, 400, 800, and 1600Mpc. The masses are defined in the local frame, i.e. M_Local,
which is related to the observed masses M_Obs by


    M_Obs = M_Local(1 + z),


We use z to denote redshift.
All the masses defined earlier refer to the observed masses M_Obs.
The inclination angle ι, polarization angle
ψ and the sky position (α, δ) are randomized. For each
specific distance, we have simulated 500 BNS signals. To determine
whether a source is detectable, we have employed different SNR cuts for
each network configuration. For networks with more
than one interferometer such as the ET or the
ET and CE, we have applied an SNR requirement similar to that in
<cit.>. A detection is achieved if the network SNR is
larger than or equal to 12 and the SNR
in at least two interferometers are no less than 5.5. For a network
with only one interferometer, namely, CE, we
require that the accumulated SNR is no less than
12 to claim a detection [We require a single detector to achieve 
SNR ≥ 12 for a detection for consistency. The results shown here for a single detector may therefore be more pessimistic than the reality.].
The results of the simulations are presented as cumulative distributions in
Figures <ref>.




For  BNS at 40Mpc using only the ET, 50% of the
detectable sources can be localized with 90% confidence to within
2.0deg^2, and 90% of the detectable sources to within
7.5deg^2. For the best localized 10% of sources, the
90% credible region is within
0.2deg^2 and these correspond to the
best located and orientated sources. For BNS at 200Mpc, 50% and
90% of the detectable sources can be localized with 90% confidence to within
42deg^2 and 183deg^2 respectively.
Assuming EM follow-up observations are achievable for sources that are
localized to within 100deg^2, this indicates 100%  (74%) of the
detectable sources at 40Mpc (200Mpc), suggesting many opportunities for joint
EM observations provided by the ET for BNS within 200Mpc. 



For sources located at 400Mpc, the upper limit of
the size of 90% credible region increases to
187deg^2 (812deg^2) for the best localized 50% (90%)
of the detectable sources. 
This still leaves 36% of the detectable sources localized
to within 100deg^2 with 90% confidence.  For sources located at
larger distances, i.e. 800Mpc and 1600Mpc, 
the upper limit of the size of 90% credible regions for the best localized 50% (90%) of the detectable
sources increases substantially to 764deg^2 (3485deg^2) and 3994deg^2 (1.7×10^4deg^2)
respectively. Moreover, only 11% and 5% of the
detectable sources can be localized to within 100deg^2. This is
because the amplitude of the signals from sources at greater distances will be
weaker. Also, the observed M_Obs as defined in Eq.<ref> will be
larger, meaning that the in-band duration will be shorter.
This suggests that localization of a BNS at such distances by the ET alone will still be poor and EM follow-up observations will remain a challenge if the ET is the only operating detector. 



Since the sensitivity of CE at low
frequencies is limited, the in-band durations of the signals are shorter than that in the
ET. As shall be seen later, the time-dependent modulation of detector response is
the main factor contributing to improved localization.
Consequently, the localization of a BNS by CE alone is worse.  For example,
50% (90%) of the detectable BNS at 40Mpc can be localized to only within
252deg^2 (2212deg^2), a factor of ∼126 and ∼ 295 larger than using only the ET. 
Only 30% of the detectable sources can be localized to within 100deg^2 with 90%
confidence.



For sources at distances ≥ 400Mpc, the upper limits of localization error
for the best localized 50% and 90% are larger than the whole sky.  This
means that for some sources, despite accumulating
enough SNR to claim a detection, no localization information is available. 



Combining the ET and CE together as a network greatly improves 
the localization since it vastly increases the geographical baseline of the
network.  This greatly improves triangulation between the detectors in the network and will take advantage of the high frequency,
high SNR components of the waveform, i.e., the final seconds. This will complement the localization information gained from the long duration and changing antenna patterns. All sources within 200Mpc are
localized to within 30deg^2 with 90% confidence.  Importantly, at 40Mpc and 200Mpc, the 90% credible region upper limit for the best localized 90% of the detectable sources are only 𝒪(10^-2)deg^2 and 𝒪(1)deg^2 respectively. 
For the detectable sources at 1600Mpc, there are
still 92% localized to within 100deg^2 with 90% confidence.
This shows a great promise for BNS
multimessenger astronomy even at relatively large cosmological distances.




















To present a more general picture, we have also simulated the localization of a population of BNS that are distributed uniformly in the comoving
volume. The results are presented in Figure <ref> as
cumulative distributions. Using the ET alone, 

the farthest detectable source is at z = 1.7. Of the
detectable sources, 50% can be localized to within ∼1.7 ×
10^4deg^2 with 90% confidence. 
The cumulative distribution for the ET reaches 68%
when the value at the x-axis is the size of the entire sky – i.e.
this indicates that for up to 32% of the detectable
sources, essentially no localization information is available. 
For CE, the situation is worse. The farthest detectable source is located at z = 4.9, 
but only ∼2% of the detectable sources will have any localization information available. 
Again, a network with the ET and CE can bring a
huge improvement to the localization performance. 
For example, compared to using the ET only, the
upper limit of the 90% credible region for the best localized 90% of the detectable sources
has been reduced by a factor >100 to 123deg^2. 
The fraction of detectable sources that can be
localized to within 100deg^2 with 90% confidence has increased by
more than 10 times to 43%. Interestingly, the farthest detectable source with a network of the ET 
and CE is located at z = 2.2. This is because for a network of more than one interferometer, we require an SNR ≥ 5.5 in at least two of the interferometers, besides also requiring a network SNR ≥ 12.  For sources located at z > 2.2, only the CE is able to accumulate enough SNR – leading to a failure to meet the detection criterion.

A summary of the results is given in Table <ref>.  
Given the success of the EM follow-up observations of GW170817, where the
localization error at 90% confidence is 28deg^2
<cit.>, also presented in the 
table is a column showing the percentage of detectable
sources that can be localized to within 30deg^2 with 90% confidence.




























 
 
 



























 §.§ Early Warning




In the era of third generation detectors, due to the extended
in-band duration of detectable signals, it is possible that signals will accumulate SNR such that the trigger may be considered significant before the merger occurs. In
this section we investigate the feasibility of issuing early warnings prior to
binary coalescence. We assume that if the SNR for a GW event
can be accumulated before merger, up to a
level that satisfies the detection requirement as defined in
Section <ref>, the event will be deemed significant. As the purpose of releasing an early warning is to increase the chance of successful EM follow-up observation, releasing an alert too
early may result in a localization error too large to carry out any meaningful
follow-ups. We therefore require two criteria to be met before an alert can be released. Firstly, the signal has to satisfy the SNR requirement for detection and secondly, the 90% credible region has to be no larger than 100deg^2 at the moment the alert is sent. We will refer to these two requirements as early warning criteria in the remaining of this paper.  As early warning is mostly made possible due to the improvement in the sensitivity in the low frequency band, we focus our analysis on the ET, and the ET and CE as a network. The BNS systems are distributed at specific distances and uniformly in the comoving volume as discussed before.



We present the results for the ET in Figure <ref> and the
ET and CE operating together as a network in Figure <ref>. These histograms show the distribution of the fraction of
detectable events as a function of the time before merger at which the events meet the early warning criteria.



Using the ET, all the signals at 40Mpc meet the early warning criteria
between 1 and 20 hours before merger, with the mode of the distribution at ∼5 hours. 
At 200Mpc, 58% of the detectable signals have accumulated enough SNR for early warning between 1
to 6 hours prior to merger. This represents a
significant advantage that can be provided by the ET in EM
follow-up observations for sources within 200Mpc.


As the distance increases, the fraction of detectable sources that meet the early warning criteria continues to drop.
Of the detectable sources at 400Mpc,  only ∼ 27% can meet the early warning criteria and the fraction further drops to ∼ 9% and ∼ 3% for sources at 800Mpc and 1600Mpc respectively.
Moreover, at 1600Mpc, the times prior to merger when the signals meet the early warning criteria drop to ≤50 minutes.











































As would be expected, an additional third generation detector
will improve the performance significantly and provide much improved early warning capability. 
In Figure <ref>, it can be seen that for distances ≥200Mpc and ≤ 1600Mpc,
the distributions of early warning times have become
noticeably skewed to larger times compared to using only the
ET. This suggests that a network of the ET and CE detectors
will provide better early warning capability for sources at relatively large distance. 
For example, the fractions of the detectable sources at 400Mpc, 800Mpc and 1600Mpc
that can meet the early warning criteria are 98%, 51% and 5% respectively.
At 40Mpc, since using only the ET all the
sources will have already met the early warning criteria at a time when
the frequency of the signal is still relatively low, an additional detector of
CE does not alter the distribution significantly.
At 1600Mpc, the result may seem to suggest that a network of the ET and CE does not perform much better 
than using the ET alone.
However, this is because a network of the ET and CE 
will be able to detect sources that are undetectable to the ET alone.
These sources will not contribute much to the number of events that meet the early warning criteria but will contribute to the number of detectable events.


Finally, to provide a more general picture, we present in Figure <ref> the results for a population of BNS distributed uniformly in comoving volume.

In line with the results shown in the previous sections, a network of the ET and CE will increase the number of events that meet the early warning criteria. With the ET alone, ∼ 2% of detectable sources can have their alerts released prior to merger. This ratio is ∼ 4% after CE joining the observation. However, the reason for the small increase in the fraction is because a network of the ET and CE will be able to detect sources that are undetectable to the ET alone, and sources located at greater distances.

In Table <ref>, 
we present a summary of the results in terms of early warning.














































































As discussed, modulations of the Doppler effect and time-dependent detector responses are the two main consequences that will be seen 
in long in-band duration signals. 
Zhao and Wen, 2017 <cit.> 
has tested thoroughly the difference in localizations with or without including the time-dependencies of these two effects, for networks of third generation detectors. However, it is still not clear which of these two factors has a more important role in terms of localizing BNS mergers.
We here investigate the relative importance of these two factors.



























To test this, we repeat the simulations for the ET shown in Section <ref>.
While we still enable a time-dependent detector response, we fix the time delay between the center of the earth and the ET at the beginning of the signals.
This is because turning on and off the Doppler shift should allow us to see more easily its importance. The results are shown in Figure <ref>.

It can be seen that at all distances, the cumulative distributions are almost identical, with only marginal discrepancy.
This suggests that the Doppler effect is not important and the modulation of the detector response is the main cause of improved sky localization.




 §.§ Calibration Errors




Previous studies have dealt with calibration errors in the context of second
generation detectors <cit.>. We present here a brief discussion
of the impact of calibration errors on localization for third generation detectors. 
It is recognized that bias in the output of a detector can be introduced by errors 
in its calibration – i.e., differences between the actual
response function and the measured response function of the detector.


These differences can then affect the noise and cause
amplitude errors and timing errors in the gravitational wave strain used
for analysis. Inference on the location of
the source of a GW from the output strain can therefore be
biased.



Amplitude errors will affect the localization by introducing a bias in the measurement of distance, inclination and polarization angles.  With second generation detectors, these parameters cannot be measured precisely. For example, the uncertainty on distance can be ∼40% for an event with SNR ∼ 8 <cit.>. Therefore, systematic errors caused by amplitude uncertainties are not expected to be dominant. However, the fractional uncertainty on distance of a GW from BNS is inversely proportional to SNR. It is conceivable that when the ET and CE are operational, higher SNR and the extended in-band duration will increase the accuracy with which these parameters can be determined. The amplitude error-induced bias may therefore be comparable to the uncertainty on the measurement of the parameters. Moreover, we assumed in this work that the actual value of the detector response will agree with the theoretical calculation. 
As shown in Figure <ref>, the time evolution of the detector response is crucial for localization of BNS mergers with 
third generation detectors. Any uncertainty in the amplitude of the waves 
or the detector response will certainly affect that.
As a result, the inference without
accounting for these errors may systematically shift the probable locations of
the source away from its true location. Amplitude errors are therefore
expected to have a larger effect in parameter estimation for third generation detectors and need to be quantified.



Localization can also be affected by timing errors of a signal through timing triangulation.

The accuracy with which the arrival time of a signal is determined is inversely proportional 
to the SNR of a wave cycle at the frequencies at which the detectors are most sensitive. 
For advanced detectors, such as aLIGO and Advanced VIRGO, this happens at ∼ 100Hz giving a timing accuracy 
𝒪(10^-3) seconds.
Timing errors (i.e., the errors intrinsic to timing when the data sample is taken) therefore would have to be comparable to a millisecond in order to be significant. However, as third generation detectors will have improved sensitivity, the SNR for a fraction of detectable sources 
will therefore be high enough that timing error may be significant.
It is therefore necessary to quantify timing errors for third generation detectors.




§ CONCLUSION




The ET and CE are two currently proposed
third generation detectors. Due to the huge improvement in the sensitivity
in the frequency band below 10Hz, the in-band durations of the gravitational waves detected from BNS mergers will be hours or even days long. Therefore the Earth's rotation will become important, leading to several effects that become relevant for such long in-band duration signals. 
The long in-band duration allows us to observe the signal from different positions along the detector trajectory as the earth rotates.
This in turn leads to a time-dependent detector response
during the signal and also causes the wave to be Doppler
modulated. 


Using the Fisher matrix and taking the earth's rotation into consideration, 
we have estimated the localization capabilities of the ET and CE
individually and as a network for BNS sources at distances equal to
40, 200, 400, 800 and 1600Mpc and for a population of BNS sources that is distributed uniformly in comoving volume. We have found that for BNS at 40 and 200Mpc, the ET alone will be able to localize most of the signals to within 100deg^2 with 90% confidence. If we assume
EM follow up observation is achievable for BNS whose associated
90% credible region is ≤ 100deg^2, this means the ET alone will be able to provide support for multi-messenger astronomy for BNS mergers within 200Mpc. However,
for distances ≥ 400Mpc, localization from
the ET alone will still be poor. This is consistent with the localization
performance for a population of BNS distributed uniformly in comoving
volume. Of the detectable sources, only ∼ 32% can be localized with 90% to within a region less than the size of the whole sky.



Combining the ET and CE can dramatically boost the
performance in localization. Almost all the sources within 1600Mpc can be
localized to within 100deg^2 with 90% confidence. In particular, the upper limit of the 90% credible region for the best localized 90% of the detectable sources at 40 and 200 Mpc has reduced by ∼ 100 times compared to using only the ET. Similar or greater improvements are seen for sources at greater distances.
For a population of BNS uniformly distributed in the comoving volume, the improvement is equally impressive.  The upper limit of the 90% credible region for the best localized 90% of the detectable sources as derived from the Fisher matrix shrinks from an area larger than the entire sky to ∼500deg^2.



Regarding the ability to send event alerts prior to merger,
the trend is similar.  Using the ET alone, alerts for most BNS within
200Mpc can be sent a few hours prior to merger, while for BNS at ≥
400Mpc, a large fraction of sources do not meet our early warning
criteria. Those which do meet the criteria do so at a time relatively close to
merger (𝒪 (10) - 𝒪 (10^2) minutes). A network with both the ET and CE substantially increases the number of signals at distances ≥400Mpc that meet the early warning criteria. This
highlights the desirability and potential of such a network for BNS at relatively large distances.  By turning on and off the Doppler effect in the simulation, we also established that the modulation of detector responses 
during the in-band duration is the main cause for improved localization.




§ ACKNOWLEDGEMENTS

We are grateful to Prof. Yanbei Chen for his help with the methodology of the paper,
and to Prof. Steve Fairhurst for his constructive comments on the paper. 
We are thankful to Dr. Xilong Fan and Teng Zhang for constructive discussions of this work. 
We are also grateful for computational resources provided by Cardiff University, 
and we are funded by an STFC grant supporting UK Involvement in the Operation of Advanced LIGO. This research is supported by The Scottish Universities Physics
Alliance and Science and Technology Facilities Council. I.S.H., C.M. and M.H. are supported by the Science and Technology Research Council (grant No. ST/L000946/1)




10

abbott2016observation
B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese,
  K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, and
  et al.
Observation of Gravitational Waves from a Binary Black Hole Merger.
Physical Review Letters, 116(6):061102, February 2016.

abbott2016gw151226
B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese,
  K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, and
  et al.
GW151226: Observation of Gravitational Waves from a 22-Solar-Mass
  Binary Black Hole Coalescence.
Physical Review Letters, 116(24):241103, June 2016.

scientific2017gw170104
B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley,
  C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya, and
  et al.
GW170104: Observation of a 50-Solar-Mass Binary Black Hole
  Coalescence at Redshift 0.2.
Physical Review Letters, 118(22):221101, June 2017.

2017ApJ...851L..35A
B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley,
  C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya, and
  et al.
GW170608: Observation of a 19 Solar-mass Binary Black Hole
  Coalescence.
The Astrophysical Journal Letters, 851:L35, December 2017.

abbott2017gw170814
B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley,
  C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya, and
  et al.
GW170814: A Three-Detector Observation of Gravitational Waves from a
  Binary Black Hole Coalescence.
Physical Review Letters, 119(14):141101, October 2017.

PhysRevLett.119.161101
B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley,
  C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya, and
  et al.
GW170817: Observation of Gravitational Waves from a Binary Neutron
  Star Inspiral.
Physical Review Letters, 119(16):161101, October 2017.

abbott2017gravitational
BP Abbott, R Abbott, TD Abbott, F Acernese, K Ackley, C Adams, T Adams,
  P Addesso, RX Adhikari, VB Adya, et al.
Gravitational waves and gamma-rays from a binary neutron star merger:
  Gw170817 and grb 170817a.
The Astrophysical Journal Letters, 848(2):L13, 2017.

2017ApJ...848L..20M
R. Margutti, E. Berger, W. Fong, C. Guidorzi, K. D. Alexander, B. D.
  Metzger, P. K. Blanchard, P. S. Cowperthwaite, R. Chornock,
  T. Eftekhari, M. Nicholl, V. A. Villar, P. K. G. Williams,
  J. Annis, D. A. Brown, H. Chen, Z. Doctor, J. A. Frieman, D. E.
  Holz, M. Sako, and M. Soares-Santos.
The Electromagnetic Counterpart of the Binary Neutron Star Merger
  LIGO/Virgo GW170817. V. Rising X-Ray Emission from an Off-axis Jet.
The Astrophysical Journal Letters, 848:L20, October 2017.

goldstein2017ordinary
A Goldstein, P Veres, E Burns, MS Briggs, R Hamburg, D Kocevski,
  CA Wilson-Hodge, RD Preece, S Poolakkil, OJ Roberts, et al.
An ordinary short gamma-ray burst with extraordinary implications:
  Fermi-gbm detection of grb 170817a.
The Astrophysical Journal Letters, 848(2):L14, 2017.

abbott2017multi
B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley,
  C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya, and
  et al.
Multi-messenger Observations of a Binary Neutron Star Merger.
Astrophysical Journal Letters, 848:L12, October 2017.

tanvir2017emergence
NR Tanvir, AJ Levan, C Gonzalez-Fernandez, O Korobkin, I Mandel, S Rosswog,
  J Hjorth, P D’Avanzo, AS Fruchter, CL Fryer, and et al.
The Emergence of a Lanthanide-rich Kilonova Following the Merger of
  Two Neutron Stars.
The Astrophysical Journal Letters, 848:L27, October 2017.

aasi2016prospects
B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese,
  K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, and
  et al.
Prospects for Observing and Localizing Gravitational-Wave Transients
  with Advanced LIGO and Advanced Virgo.
Living Reviews in Relativity, 19:1, February 2016.

sathyaprakash1200219scientific
BS Sathyaprakash, S Fairhurst, BF Schutz, J Veitch, S Klimenko, Dave Reitze,
  and Stan Whitcomb.
Scientific benefits of moving one of ligo hanford detectors to india.
LIGO Document, 1200219:2012.

aso2013interferometer
Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa,
  T. Sekiguchi, D. Tatsumi, and H. Yamamoto.
Interferometer design of the KAGRA gravitational wave detector.
Physical Review D, 88(4):043007, August 2013.

abbott2017all
B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese,
  K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, and
  et al.
All-sky search for short gravitational-wave bursts in the first
  Advanced LIGO run.
Physical Review D, 95(4):042003, February 2017.

damour2005gravitational
T. Damour and A. Vilenkin.
Gravitational radiation from cosmic (super)strings: Bursts,
  stochastic background, and observational windows.
Physical Review D, 71(6):063510, March 2005.

cowperthwaite2015comprehensive
P. S. Cowperthwaite and E. Berger.
A Comprehensive Study of Detectability and Contamination in Deep
  Rapid Optical Searches for Gravitational Wave Counterparts.
The Astrophysical Journal, 814:25, November 2015.

metzger2012most
B. D. Metzger and E. Berger.
What is the Most Promising Electromagnetic Counterpart of a Neutron
  Star Binary Merger?
The Astrophysical Journal, 746:48, February 2012.

tanvir2013kilonova
N. R. Tanvir, A. J. Levan, A. S. Fruchter, J. Hjorth, R. A. Hounsell,
  K. Wiersema, and R. L. Tunnicliffe.
A `kilonova' associated with the short-duration γ-ray burst
  GRB 130603B.
Nature, 500:547–549, August 2013.

berger2013r
E. Berger, W. Fong, and R. Chornock.
An r-process Kilonova Associated with the Short-hard GRB 130603B.
The Astrophysical Journal Letters, 774:L23, September 2013.

chassande2011multimessenger
E. Chassande-Mottin, M. Hendry, P. J. Sutton, and S. Márka.
Multimessenger astronomy with the Einstein Telescope.
General Relativity and Gravitation, 43:437–464, February 2011.

2009astro2010s20B
J. S. Bloom, D. E. Holz, S. A. Hughes, and K. Menou.
Coordinated Science in the Gravitational and Electromagnetic Skies.
In astro2010: The Astronomy and Astrophysics Decadal Survey,
  volume 2010 of Astronomy, 2009.

kanner2008looc
J. Kanner, T. L. Huard, S. Márka, D. C. Murphy, J. Piscionere,
  M. Reed, and P. Shawhan.
LOOC UP: locating and observing optical counterparts to
  gravitational wave bursts.
Classical and Quantum Gravity, 25(18):184034, September 2008.

schutz1986determining
Bernard F Schutz.
Determining the hubble constant from gravitational wave observations.
Nature, 323(6086):310–311, 1986.

sathyaprakash2010cosmography
B. S. Sathyaprakash, B. F. Schutz, and C. Van Den Broeck.
Cosmography with the Einstein Telescope.
Classical and Quantum Gravity, 27(21):215006, November 2010.

2017arXiv171005835A
B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley,
  C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya, and
  et al.
A gravitational-wave standard siren measurement of the Hubble
  constant.
Nature, 551:85–88, November 2017.

fairhurst2011source
Stephen Fairhurst.
Source localization with an advanced gravitational wave detector
  network.
Classical and Quantum Gravity, 28(10):105021, 2011.

nissanke2011localizing
S. Nissanke, J. Sievers, N. Dalal, and D. Holz.
Localizing Compact Binary Inspirals on the Sky Using Ground-based
  Gravitational Wave Interferometers.
The Astrophysical Journal, 739:99, October 2011.

nissanke2013identifying
S. Nissanke, M. Kasliwal, and A. Georgieva.
Identifying Elusive Electromagnetic Counterparts to Gravitational
  Wave Mergers: An End-to-end Simulation.
The Astrophysical Journal, 767:124, April 2013.

veitch2012estimating
J. Veitch, I. Mandel, B. Aylott, B. Farr, V. Raymond, C. Rodriguez,
  M. van der Sluys, V. Kalogera, and A. Vecchio.
Estimating parameters of coalescing compact binaries with proposed
  advanced detector networks.
Physical Review D, 85(10):104045, May 2012.

rodriguez2014basic
C. L. Rodriguez, B. Farr, V. Raymond, W. M. Farr, T. B. Littenberg,
  D. Fazi, and V. Kalogera.
Basic Parameter Estimation of Binary Neutron Star Systems by the
  Advanced LIGO/Virgo Network.
The Astrophysical Journal, 784:119, April 2014.

cannon2011toward
K. Cannon, R. Cariou, A. Chapman, M. Crispin-Ortuzar, N. Fotopoulos,
  M. Frei, C. Hanna, E. Kara, D. Keppel, L. Liao, S. Privitera,
  A. Searle, L. Singer, and A. Weinstein.
Toward Early-warning Detection of Gravitational Waves from Compact
  Binary Coalescence.
The Astrophysical Journal, 748:136, April 2012.

chen2017facilitating
H.-Y. Chen and D. E. Holz.
Facilitating follow-up of LIGO-Virgo events using rapid sky
  localization.
The Astrophysical Journal, August 2015.

2017arXiv170800806M
J. Mills, V. Tiwari, and S. Fairhurst.
Localization of binary mergers with gravitational-wave detectors of
  second and third generation.
ArXiv e-prints, August 2017.

zhao2017localization
Wen Zhao and Linqing Wen.
Localization accuracy of compact binary coalescences detected by the
  third-generation gravitational-wave detectors and implication for cosmology.
arXiv preprint arXiv:1710.05325, 2017.

sathyaprakash2012scientific
B. Sathyaprakash, M. Abernathy, F. Acernese, P. Ajith, B. Allen,
  P. Amaro-Seoane, N. Andersson, S. Aoudia, K. Arun, P. Astone, and
  et al.
Scientific objectives of Einstein Telescope.
Classical and Quantum Gravity, 29(12):124013, June 2012.

LIGOW
LIGO Scientific Collabration.
Ligo instrument science white paper.
LIGO-T1700231–v2, 2017.

regimbau2012mock
T. Regimbau, T. Dent, W. Del Pozzo, S. Giampanis, T. G. F. Li,
  C. Robinson, C. Van Den Broeck, D. Meacher, C. Rodriguez, B. S.
  Sathyaprakash, and K. Wójcik.
A Mock data challenge for the Einstein Gravitational-Wave
  Telescope.
Physical Review D, 86(12):122001, December 2012.

punturo2010einstein
M Punturo, M Abernathy, F Acernese, B Allen, Nils Andersson, K Arun, F Barone,
  B Barr, M Barsuglia, M Beker, et al.
The Einstein Telescope: a third-generation gravitational wave
  observatory.
Classical and Quantum Gravity, 27(19):194002, October 2010.

2011LRR....14....5P
M. Pitkin, S. Reid, S. Rowan, and J. Hough.
Gravitational Wave Detection by Interferometry (Ground and Space).
Living Reviews in Relativity, 14:5, July 2011.

hild2011sensitivity
S Hild, M Abernathy, F Acernese, P Amaro-Seoane, N Andersson, K Arun, F Barone,
  B Barr, M Barsuglia, M Beker, et al.
Sensitivity studies for third-generation gravitational wave
  observatories.
Classical and Quantum Gravity, 28(9):094013, May 2011.

hild2009xylophone
S. Hild, S. Chelkowski, A. Freise, J. Franc, N. Morgado,
  R. Flaminio, and R. DeSalvo.
A xylophone configuration for a third-generation gravitational wave
  detector.
Classical and Quantum Gravity, 27(1):015003, January 2010.

hild2008pushing
S. Hild, S. Chelkowski, and A. Freise.
Pushing towards the ET sensitivity using 'conventional' technology.
ArXiv e-prints, October 2008.

bassan2014advanced
Massimo Bassan.
Advanced Interferometers and the Search for Gravitational Waves:
  Lectures from the First VESF School on Advanced Detectors for Gravitational
  Waves, volume 404.
Springer, 2014.

vallisneri2008use
M. Vallisneri.
Use and abuse of the Fisher information matrix in the assessment of
  gravitational-wave parameter-estimation prospects.
Physical Review D, 77(4):042001, February 2008.

zanolin2010application
M. Zanolin, S. Vitale, and N. Makris.
Application of asymptotic expansions for maximum likelihood
  estimators errors to gravitational waves from binary mergers: The single
  interferometer case.
Physical Review D, 81(12):124048, June 2010.

cho2014application
H.-S. Cho and C.-H. Lee.
Application of the effective Fisher matrix to the frequency domain
  inspiral waveforms.
Classical and Quantum Gravity, 31(23):235009, December 2014.

fairhurst2009triangulation
S. Fairhurst.
Triangulation of gravitational wave sources with a network of
  detectors.
New Journal of Physics, 11(12):123006, December 2009.

2012PhRvD..85f4034V
S. Vitale, W. Del Pozzo, T. G. F. Li, C. Van Den Broeck, I. Mandel,
  B. Aylott, and J. Veitch.
Effect of calibration errors on Bayesian parameter estimation for
  gravitational wave signals from inspiral binary systems in the advanced
  detectors era.
Physical Review D, 85(6):064034, March 2012.

2009PhRvD..80d2005L
L. Lindblom.
Optimal calibration accuracy for gravitational-wave detectors.
, 80(4):042005, August 2009.

2010PhRvD..82h4020L
L. Lindblom, J. G. Baker, and B. J. Owen.
Improved time-domain accuracy standards for model gravitational
  waveforms.
, 82(8):084020, October 2010.


