Binary Neutron Star Mergers and Third Generation Detectors: Localization and Early Warning

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 earths 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 $\leq200$Mpc to within $100\text{deg}^2$ 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 $\mathcal{O}(1) \text{deg}^2$ for most sources at $\leq200$Mpc and $\leq100\text{deg}^2$ for most sources at $\leq1600$Mpc. 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 $\geq12$ and $\geq5.5$ for at least two interferometers, and to have a 90% credible region for the sky localization that is no larger than $100 \text{deg}^2$. 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 $\sim58\%$ 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 $\sim 3$ hours prior to merger for 98% of detectable binary neutron stars.

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 earths 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 100deg 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.

I. INTRODUCTION
The first detections of gravitational waves (GWs) from binary black hole (BBH) systems GW150914, GW151226, GW170104 and GW170608 by the two LIGO detectors at Hanford and Livingston [1][2][3][4] 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 [5]. Just a few days later, the three GW observatories detected the first binary neutron star (BNS) merger event GW170817 [6]. The detections of multiple electromagnetic (EM) counterparts associated with GW170817 initiated the era of GW multimessenger astronomy [7][8][9][10][11]. In the coming years, additional ground-based interferometric detectors such as KAGRA and LIGO India are likely to join the global network [12][13][14]. 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 neutron star black hole (NSBH)), core-collapse supernovae, non-symmetric neutron stars and the stochastic background [15,16].
For systems such as BNSs and NSBHs, 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, * m.chan.1@research.gla.ac.uk x-ray emission, kilonovae in the optical and infrared bands and radio afterglows [17][18][19][20]. 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 [21][22][23]. 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 [24][25][26]. 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 GWs. For example, the 90% credible region for the sky positions of the first detected BBH events are O(10 2 ) deg 2 [1,2]. 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 [12,13,[27][28][29][30][31]. However, for sources at large distances, EM follow-up obser-vations 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 [29]. 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 [18] as was the case for the kilonova associated with GW170817 [11].
Third generation detectors such as the Einstein Telescope (ET) and Cosmic Explorer (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 GWs 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 signal to noise ratio (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 [32]. 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 GWs 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 GWs 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 [12,27,33]. 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 [34] 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 [35]. 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 GWs 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 BNSs 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 BNSs where the inband 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 SectionII, 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 SectionIII with the results and simulations presented in Section IV together with a discussion. We then provide our conclusions in Section V.

II. 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 [36] 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 [37].

A. 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 [38] F 1 + (θ, φ, 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 [38]. The CE's antenna pattern can be written as

B. Duration of Binary Neutron Star Signals
In a GW detector, different noise sources limit the sensitivity of the detector in different frequency bands [39]. 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 [40].
There are many proposed technologies designed to improve the sensitivity of the ET [41][42][43]. 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 [37]. The estimated amplitude spectrum density of the ET [43] and CE are presented in Figure 1. 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 and Advanced Virgo (Orange). Both aLIGO and Advanced Virgo are at their respective design sensitivities.
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 [44] using 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 2 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 inband 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.4. As indicated in Figure 2, 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 inband 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 timedependence, in Figure 3 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.

III. 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 Monte Carlo Markov Chain (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 CramerRao lower bound is valid [45][46][47], 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 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 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 ash The mathematical definition of the Fisher matrix is given by (8) where ∂h(θ, f )/∂θ i is the partial derivative ofh(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 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 10 d; polarization angle, ψ; the log of the total binary masses, log 10 M ; the cosine of the inclination angle, cos ι; 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 longwith 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 k ij and the optimal SNR ρ k . The superscript k indicates the k th piece of the wave. The final Fisher matrix Γ f ij is then 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 from which the localization error is extracted using where ∆Ω 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 ∆Ω to any desired confidence level, where p is a value between 0 and 1 indicating the confidence level. Similarly, the accumulated SNR is given by IV.

A. 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 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 [27]. A detection is achieved if the network SNR is larger than or equal to 12 and the SNRs 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 1 . The results of the simulations are presented as cumulative distributions in Figures 4.
For BNSs 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 BNSs 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 BNSs 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 4 deg 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.15 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 BNSs 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. 1 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.
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 5 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 4 deg 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 I. Given the success of the EM follow-up observations of GW170817, where the localization error at 90% confidence is 28deg 2 [6], 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.

B. Early Warning
In the era of third generation detectors, due to the extended in-band duration of detectable signals, it is possi- ble 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 IV A, 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 6 and the ET and CE operating together as a network in Figure 7. 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  A brief statistical summary of our results for sky localization. In the first row, we use d to denote distance and n the number of injections. The third and the fourth columns indicate the upper limit of the size of 90% credible regions for the best localized 50% and 90% of the detectable sources. The fifth column shows the percentage of the detectable sources that can be localized to within 100deg 2 with 90% confidence, and the last column the percentage within 30deg 2 with 90% confidence.
detector will improve the performance significantly and provide much improved early warning capability. In Figure 7, 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 8 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  II, 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 [35] 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 The x-axes indicate the time to merger when the signal meets the early warning criteria. The y-axes indicate the fraction of detectable events that achieve these early warning criteria. Note that at distances ≥ 400Mpc, since a large fraction of the times until merger will fall within 1 hour, for greater clarity the scale of the axes varies from panel to panel. Only those signals which achieve the early warning criteria at least 100 seconds prior to merger will be counted.
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 IV A. While we still enable a timedependent 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 9. 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.  Figure 6 but using the ET and CE as a network. For the same source distance, the scale of the panels is the same as is shown in Figure 6, to allow a convenient comparison.

C. Calibration Errors
Previous studies have dealt with calibration errors in the context of second generation detectors [48][49][50][51]. 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 [49]. 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 9, 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 SNRs 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.

V. 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 as-tronomy 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 BNSs within 200Mpc can be sent a few hours prior to merger, while for BNSs 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.