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Expanding the LISA Horizon from the Ground

Expanding the LISA Horizon from the Ground 1 1 2, 3 1, 4 Kaze W. K. Wong, Ely D. Kovetz, Curt Cutler, and Emanuele Berti Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218 USA Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125, USA Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA Department of Physics and Astronomy, The University of Mississippi, University, MS 38677, USA The Laser Interferometer Space Antenna (LISA) gravitational-wave (GW) observatory will be limited in its ability to detect mergers of binary black holes (BBHs) in the stellar-mass range. A future ground-based detector network, meanwhile, will achieve by the LISA launch date a sensitivity that ensures complete detection of all mergers within a volume >O(10) Gpc . We propose a method to use the information from the ground to revisit the LISA data in search for sub-threshold events. By discarding spurious triggers that do not overlap with the ground-based catalogue, we show that the signal-to-noise threshold  employed in LISA can be significantly lowered, greatly boosting LISA the detection rate. The efficiency of this method depends predominantly on the rate of false-alarm increase when the threshold is lowered and on the uncertainty in the parameter estimation for the LISA events. As an example, we demonstrate that while all current LIGO BBH-merger detections would have evaded detection by LISA when employing a standard  = 8 threshold, this method LISA will allow us to easily (possibly) detect an event similar to GW150914 (GW170814) in LISA. Overall, we estimate that the total rate of stellar-mass BBH mergers detected by LISA can be boosted by a factor  4 (& 8) under conservative (optimistic) assumptions. This will enable new tests using multi-band GW observations, significantly aided by the greatly increased lever arm in frequency. Multi-band measurements of GWs [1] from coalescing measurements for multiple parameters in tandem. binary black holes (BBHs) can open the door to a wide The procedure is as follows: we first set an initial array of invaluable studies. Spanning a wider range of threshold, e.g.  = 8, and determine which (real) LISA frequencies will increase sensitivity to eccentric orbits, events in the Ground catalogue are detectable in LISA which can be used to distinguish between different binary with this threshold. The parameters of all LISA candidate formation channels, improve merger-rate estimation, allow events identified with this threshold are then compared for more precise tests of gravity and assist in instrument with those in the Ground list (taking into account the calibration. Better science will be enabled if many events LISA parameter-estimation uncertainty), and those that are detected in both a ground-based network (Ground) do not overlap with any real event are discarded. We and a space observatory such as LISA. lower the threshold and iterate this procedure until the Unfortunately, LISA will not be nearly as sensitive as probability that a random trigger is consistent with some Ground event becomes significant. the Ground detectors to stellar-mass BBH mergers. This issue affects in particular “multiband” inspiral events, for Figure 1 illustrates the concept of filtering spurious which the GW frequency drifts from the LISA to the triggers using only t , the time of coalescence, as the Ground band during the LISA observation window. This discarding parameter. Compared with the entire LISA condition determines a minimum frequency at which the observation time, O(1) years, the typical uncertainty on event can appear in LISA (typically & 10 Hz for stellar- t as determined by LISA is  7 orders of magnitude mass BBHs). Taking advanced LIGO (aLIGO) at design smaller, O(10) seconds. With O(1000) events expected to sensitivity as an example and adopting a similar signal-to- be detected from the Ground within the volume accessible noise threshold of  = 8 in both experiments, the fraction by LISA with  & 5, we should therefore be able to LISA of aLIGO events that will be detectable in LISA is less filter out roughly & 10 random triggers based on t alone. than 1%. This will allow a detection of events with   7, such LISA as GW150914 [3], over the LISA mission lifetime. We will If we can manage to lower the LISA signal-to-noise see that incorporating additional parameters may enable threshold, the horizon distance (which is the maximum a multi-band detection of events with   4, such as distance at which a source is detectable) will grow, and LISA GW170814 [4]. the increase in accessible volume will result in a rapid rise in the multi-band detection rate. Setting a lower In what follows we choose to focus on three waveform threshold, however, means that we increase the risk of ingredients: the source masses, sky location and merger classifying noise triggers as real events (false alarms). The time. We will test the efficiency of our proposed method false-alarm rate (FAR) is a steep function of  [2]. based on a Fisher matrix analysis to estimate the param- eter estimation uncertainty in the LISA band [5], and In this Letter we propose a method to discard spuri- report the potential improvement in the LISA event rate ous LISA triggers that show up as the signal-to-noise given different assumptions about the FAR and the BBH threshold is lowered, using information from the Ground. mass function. We show that a large number of random noise triggers can be filtered out by imposing consistency with Ground We assume the posteriors to be Gaussian, so a trigger arXiv:1808.08247v1 [astro-ph.HE] 24 Aug 2018 2 treated as the “true" values (neglecting any systematic bias). The consistent volume in parameter space of a par- GW170814 ticular source with parameters  will be well-approximated GW150914 by the ellipsoid GW151226 GW170104 GW170608 (p) ~ ~ V (; p) = (2) j ()j; (2) LISA (0:67) where j ()j denotes the determinant of the covari- LISA ance matrix given by LISA using the most recent noise power spectral density S (f ) [6], and  (0:67) corre- Trigger sponds to a bound at "1" level. The fraction of triggers which are consistent between the two detectors is then given by R R 0 0 ~ ~ ~ ~ d n (; ; T ) d n ( ) s ~ b V (;p) f (; T ) = ; (3) c R 0 0 ~ ~ FIG. 1. Illustration of our method to discard LISA triggers. n ( )d The waveforms are those of the gravitational events which were observed by aLIGO in its O1 and O2 runs (2015-2017). where n (; ; T ) is the number of astrophysical (real) GW150914 would have had the highest signal-to-noise in LISA, events which LISA is sensitive to (all of which are de- = 7, while GW170814 would have had  = 4:5 (as- LISA LISA tectable from the Ground) for a given vector  of source suming 4 years of integration time), both of which are below parameters, a signal-to-noise threshold  , and integra- the conventional =8 threshold. The red stripes indicate the LISA ~0 merger time of LISA triggers (their width set by the uncer- tion time T ; n ( ) is the number density of LISA triggers tainty). If a trigger does not agree with any of the events ~0 as a function of  in the search parameter space. detected from the Ground, it can be discarded as random The most important ingredient in our analysis is the noise (or as an astrophysical event whose merger will appear relationship between the threshold  and the number LISA in LIGO in the future and is thus irrelevant for our purposes). of expected background triggers, which we call the “FAR We show that if LISA had started observing in 2011, it would curve.” At this time, there is no reliable estimate for the have been possible to lower its signal-to-noise threshold and re- LISA FAR curve. We therefore use as a proxy the results cover GW150914, and potentially also GW170814. The other events would have been out of reach. of the LIGO Mock Data Challenge [2], which suggest that the number of background triggers increases by about two orders of magnitude when the signal-to-noise threshold is characterized by its k-dimensional vector of best-fit is decreased by one (we use their Experiment 3, which is parameter values ~ and covariance matrix . The problem the most relevant for our study). This agrees with the of consistency checking between the LISA and Ground recent findings of Ref. [7]. measurements corresponds to finding the overlap between We can then define the effective LISA threshold as two volumes in a multi-dimensional space given some eff 0 eff metric. We claim that two measurements taken by LISA (T ) =  + log (f ( ; T )); (4) LISA LISA LISA and the Ground agree with each other if they meet the following criterion: where  is the conventional signal-to-noise threshold, LISA 2 is the FAR and T is the integration time in LISA. Eq. (4) D(~ ; ~ ;  ;  )   (p); (1) LISA Ground LISA Ground is the crux of the method proposed in this work. where D is a function that gives the distance between two The FAR curve given in Ref. [2] has a slope  100 and points in the high-dimensional space under some metric, is not shown below  = 5:5. As a conservative estimate, 2 3(5:5) we impose an exponential cutoff e starting at and  (p) is the quantile function for probability p of the Chi-Squared distribution with k degrees of freedom.  = 5:5, essentially preventing any improvement beyond A typical source in LISA will be characterized by k = 9  = 5. We also consider a more optimistic case in which parameters (when taking into account the antenna pat- we extrapolate the FAR curve with a similar cutoff at tern), so the exact two-point distance problem would be  = 4. Given the volume permitted by a single source, solved in an 18-dimensional space, and hence it can be Eq. (2), the number density n of real sources in the computationally intensive. Instead of solving the problem parameter space and the FAR function, we are now ready exactly, we calculate the volume bounded by  (p) in the to obtain  by solving Eq. (4) self-consistently. eff parameter space centered at the best-fit value for each In order to compute the second integral in Eq. (3), we parameter that is given by the more precise measurement need to estimate  . We adopt a modification of the LISA between the Ground and LISA. Since most of the sources Fisher matrix code from Ref. [5] to calculate the uncer- will be detected from the Ground with signal-to-noise tainties on source parameters. As explained above, we eff well above threshold, the Ground measurements can be calculate  (T ) using the three groups of parameters LISA 3 which contribute the most to the fraction of discarded black hole (assumed to be 5M ), and by default we set events f : the upper cutoff M = 40M [13–15]. To account for c cut (i) Time of coalescence t : we care only for events that uncertainty regarding these choices, we also calculate will merge in the Ground frequency band and assume that our results using two other mass functions: in one we noise triggers will be distributed uniformly in the LISA replace the Gaussian cutoff with a sharp step function observation window, which is determined by T . P (M ) / H(M M ), and in another with an expo- cut M =M 1 cut (ii) Component masses (M , M ): we assume that noise nential cutoff P (M ) / e . For all cases we limit 1 2 triggers will pick up a random template in the template the maximum component mass to 100M . Finally, given bank, and calculate the fraction f assuming noise triggers a value for M , we define the PDF of M as a uniform c 1 2 are distributed uniformly in the (M ; M ) plane. The distribution ranging from M to M [8, 12]: 1 2 gap 1 uncertainty on either component mass is normally  10% P (M j M )  A H(M M )H(M M ): (7) 2 1 M 2 gap 1 2 of the measured value, but due to the strong correlation between the two component masses [8], the allowed vol- For the sky locations, we assume sources are uniformly ume in the parameter space is typically much smaller than distributed on the celestial sphere. In principle one should 10%. This volume is related to the uncertainty in chirp generate a 6-dimensional sample in the mass–sky-location mass measurement, which is expected to be quite small parameters space, but this is quite computationally inten- in LISA (as BBHs spend many cycles in its frequency sive. In practice, we average over a reasonable amount band). Typically the probability of a noise trigger being of sources distributed across the sky and compress the consistent with one real event is  10 . calculation of hV Ti to two (mass) dimensions. (iii) Sky location ( ,  ): We assume that noise triggers S S th The next term we need is f (z; ;  ), which is related will be uniformly distributed across the sky. LISA will 2 to the horizon redshift of the source. The LISA signal-to- be able to localize sources to within O(10) deg [9]. Com- noise of a source with frequency-domain waveform h(f ) paring to the whole sky, the probability of a noise trigger 3 at some luminosity distance is given by [16] being consistent with one event is . 10 . Our figure-of-merit is the number of additional sources max ~ ~ h (f )h(f ) we can recover in LISA by replacing the conventional = 4 df; (8) 0 e S (f ) threshold  with  . This of course depends on min LISA LISA the astrophysical BBH merger rate. Multiband events where f and f are the initial and final frequencies. min max probed by LISA are in the local Universe, so we can th We get the horizon redshift, and hence f (z; ;  ), by assume the merger rate R to be constant in redshift. We th setting  =  . denote by  the mean rate of events of astrophysical When calculating the uncertainty and signal-to-noise origin above a certain signal-to-noise threshold, given by for a given source, we need to integrate the waveform = RhV Ti, where hV Ti is the time and population- over a certain frequency range. Since we are interested in averaged space-time volume accessible to the detector at sources which can in principle be detected in both LISA th the chosen threshold  , defined as [10] and the Ground, we set f = 1 Hz (the conventional max upper cutoff on the LISA noise curve). To determine f , dV 1 min th ~ ~ ~ hV Ti = T dzd s()f (z; ;  ); (5) we require that a source drifts from the LISA band to the dz 1 + z Ground band in less than a total time T . The chirp time of a source with chirp mass M (in the observer frame) is where V is the comoving volume, s() is the injected th ~ given by [17] distribution of source parameters, and 0  f (z; ;  )  1 is the fraction of injections detectable by the experiment. Z max 5 5c 5=3 11=3 In order to calculate hV Ti, we need to solve for the t = df (GM) f : (9) 8=3 horizon distance and redshifted volume as a function of f min source parameters [11], and then marginalize over an To determine f () we solve Eq. (9) setting t  T . ~ min input population s(). We consider sources characterized In Figure 2 we plot our main result:  = , the =8 eff by 9 parameters: the two component masses (M ; M ), 1 2 increase in detection rate compared to using the standard time of coalescence t , phase of coalescence  , luminosity c c = 8 threshold, under different assumptions. We see that distance D , sky locations of the source ( ;  ), and the L S S using the Ground information can boost the number of orbital angular momentum direction ( ;  ). In practice, L L detections in LISA by a factor  4, under the conservative we sample over the two component masses and four sky choice for the FAR. locations, with t and  arbitrarily set to zero. c c Since our figure-of-merit compares total rates, and we For the injected mass distribution, we follow Ref. [12] assume a constant merger rate density per comoving and define the probability density function (PDF) of M volume, the uncertainties in the merger rate cancel out. (M =M ) 1 cut The dominant uncertainty in our result stems from the P (M )  A M H(M M )e ; (6) 1 M 1 1 gap FAR. With a more optimistic choice of FAR the boost where A is a normalization constant, H is the Heav- factor can increase up to  8: the LISA sampling rate [18] iside function, M is the minimum mass of a stellar sets a lower limit on the threshold. gap 4 the LISA and Ground templates are qualitatively similar, Conservative this replacement should increase the discarding power at Optimistic the higher-mass end compared to the uniform case, and 8 4 Mass sharp cutoff therefore improve the boost in rate. Mass single exp cutoff Another approximation we made was to extrapolate our Fisher matrix calculation into the low signal-to-noise regime, where it generally serves only as a lower bound of the uncertainties [22]. A more realistic estimate of the uncertainties can be achieved with other parameter estimation approaches, such as the Markov Chain Monte Carlo method [23]. We hope that our work will motivate participants in the ongoing LISA Data Challenges [24] to verify and improve our FAR estimates. To conclude, while the idea to use LISA detections to 1m 3m 0.5yr 1yr 4yr alert ground-based experiments about pending mergers Integration time has been explored before [1], we have investigated for the first time the potential of exploiting the opposite route. FIG. 2. The boost in the LISA detection rate enabled by We have introduced in this Letter a method to recover our method, compared to setting the standard signal-to-noise sub-threshold stellar-mass BBH merger events from the threshold of  = 8, and assuming that all sources are observed for the integration time T given in Eq. (9). The blue solid LISA data stream using information from the subsequent line shows the rate increase using a FAR function with a ground-based measurements of these events. Our analysis cutoff at  = 5 and a mass function with a Gaussian cutoff. forecasts a remarkable increase – by a factor of 4 to 8, The dashed-blue line corresponds to a more optimistic FAR depending on the assumptions – in the number of LISA function, where the cutoff is at  = 4. For comparison, we detections. While our estimate was restricted to multi- show in red and green the result when using a mass function band sources whose merger is detected from the Ground with a sharp cutoff at 50M and a single-exponential cutoff during the LISA lifetime, the same algorithm can be at 40M , respectively. continuously applied for events that merge after LISA has finished its mission, yielding more detections. The increase in number of multi-band GW detections can bring forth a plethora of rewards. For example, im- The next source of uncertainty is due to the choice of provements in parameter estimation and modeling con- mass function. The increase in detection rate is biased straints will enable novel tests of extreme gravity theo- toward the lower end of the mass function, and so it is ries [25–29]. Most notably, discrimination between differ- more significant for mass functions that favor lower mass ent BBH-formation channels using eccentricity [30–35], events. This uncertainty amounts to  5%. A uniform-in- spins [36–40], and other waveform features [41–43] will log mass function should yield similar results [7]. greatly benefit from the larger lever arm in frequency Various assumptions we have made here can be im- garnered from these measurements. proved upon. For example, in checking for consistency between LISA and the Ground we considered only the volume allowed by the LISA covariance matrix, instead of ACKNOWLEDGMENTS solving the exact two-point problem. This is a reasonable assumption, based on the expected sensitivity of Ground It is our pleasure to thank Ilias Cholis, Marc observatories by the time LISA flies. Kamionkowski, Johan Samsing and Fabian Schmidt for We also took the distribution of noise triggers to be useful discussions. K.W.K.W. and E.B. are supported by uniform in the parameters of interest. This assumption is NSF Grants No. PHY-1841464 and AST-1841358. E.D.K. valid for time of coalescence and sky location, but it may was supported by NASA grant NNX17AK38G. C.C.’s not be accurate for the two component masses. Search work was carried out at the Jet Propulsion Laboratory, template banks for ground-based detectors typically have California Institute of Technology, under contract to the more templates at the low-mass end [19–21]. More re- National Aeronautics and Space Administration. 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Expanding the LISA Horizon from the Ground

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1 1 2, 3 1, 4 Kaze W. K. Wong, Ely D. Kovetz, Curt Cutler, and Emanuele Berti Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218 USA Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125, USA Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA Department of Physics and Astronomy, The University of Mississippi, University, MS 38677, USA The Laser Interferometer Space Antenna (LISA) gravitational-wave (GW) observatory will be limited in its ability to detect mergers of binary black holes (BBHs) in the stellar-mass range. A future ground-based detector network, meanwhile, will achieve by the LISA launch date a sensitivity that ensures complete detection of all mergers within a volume >O(10) Gpc . We propose a method to use the information from the ground to revisit the LISA data in search for sub-threshold events. By discarding spurious triggers that do not overlap with the ground-based catalogue, we show that the signal-to-noise threshold  employed in LISA can be significantly lowered, greatly boosting LISA the detection rate. The efficiency of this method depends predominantly on the rate of false-alarm increase when the threshold is lowered and on the uncertainty in the parameter estimation for the LISA events. As an example, we demonstrate that while all current LIGO BBH-merger detections would have evaded detection by LISA when employing a standard  = 8 threshold, this method LISA will allow us to easily (possibly) detect an event similar to GW150914 (GW170814) in LISA. Overall, we estimate that the total rate of stellar-mass BBH mergers detected by LISA can be boosted by a factor  4 (& 8) under conservative (optimistic) assumptions. This will enable new tests using multi-band GW observations, significantly aided by the greatly increased lever arm in frequency. Multi-band measurements of GWs [1] from coalescing measurements for multiple parameters in tandem. binary black holes (BBHs) can open the door to a wide The procedure is as follows: we first set an initial array of invaluable studies. Spanning a wider range of threshold, e.g.  = 8, and determine which (real) LISA frequencies will increase sensitivity to eccentric orbits, events in the Ground catalogue are detectable in LISA which can be used to distinguish between different binary with this threshold. The parameters of all LISA candidate formation channels, improve merger-rate estimation, allow events identified with this threshold are then compared for more precise tests of gravity and assist in instrument with those in the Ground list (taking into account the calibration. Better science will be enabled if many events LISA parameter-estimation uncertainty), and those that are detected in both a ground-based network (Ground) do not overlap with any real event are discarded. We and a space observatory such as LISA. lower the threshold and iterate this procedure until the Unfortunately, LISA will not be nearly as sensitive as probability that a random trigger is consistent with some Ground event becomes significant. the Ground detectors to stellar-mass BBH mergers. This issue affects in particular “multiband” inspiral events, for Figure 1 illustrates the concept of filtering spurious which the GW frequency drifts from the LISA to the triggers using only t , the time of coalescence, as the Ground band during the LISA observation window. This discarding parameter. Compared with the entire LISA condition determines a minimum frequency at which the observation time, O(1) years, the typical uncertainty on event can appear in LISA (typically & 10 Hz for stellar- t as determined by LISA is  7 orders of magnitude mass BBHs). Taking advanced LIGO (aLIGO) at design smaller, O(10) seconds. With O(1000) events expected to sensitivity as an example and adopting a similar signal-to- be detected from the Ground within the volume accessible noise threshold of  = 8 in both experiments, the fraction by LISA with  & 5, we should therefore be able to LISA of aLIGO events that will be detectable in LISA is less filter out roughly & 10 random triggers based on t alone. than 1%. This will allow a detection of events with   7, such LISA as GW150914 [3], over the LISA mission lifetime. We will If we can manage to lower the LISA signal-to-noise see that incorporating additional parameters may enable threshold, the horizon distance (which is the maximum a multi-band detection of events with   4, such as distance at which a source is detectable) will grow, and LISA GW170814 [4]. the increase in accessible volume will result in a rapid rise in the multi-band detection rate. Setting a lower In what follows we choose to focus on three waveform threshold, however, means that we increase the risk of ingredients: the source masses, sky location and merger classifying noise triggers as real events (false alarms). The time. We will test the efficiency of our proposed method false-alarm rate (FAR) is a steep function of  [2]. based on a Fisher matrix analysis to estimate the param- eter estimation uncertainty in the LISA band [5], and In this Letter we propose a method to discard spuri- report the potential improvement in the LISA event rate ous LISA triggers that show up as the signal-to-noise given different assumptions about the FAR and the BBH threshold is lowered, using information from the Ground. mass function. We show that a large number of random noise triggers can be filtered out by imposing consistency with Ground We assume the posteriors to be Gaussian, so a trigger arXiv:1808.08247v1 [astro-ph.HE] 24 Aug 2018 2 treated as the “true" values (neglecting any systematic bias). The consistent volume in parameter space of a par- GW170814 ticular source with parameters  will be well-approximated GW150914 by the ellipsoid GW151226 GW170104 GW170608 (p) ~ ~ V (; p) = (2) j ()j; (2) LISA (0:67) where j ()j denotes the determinant of the covari- LISA ance matrix given by LISA using the most recent noise power spectral density S (f ) [6], and  (0:67) corre- Trigger sponds to a bound at "1" level. The fraction of triggers which are consistent between the two detectors is then given by R R 0 0 ~ ~ ~ ~ d n (; ; T ) d n ( ) s ~ b V (;p) f (; T ) = ; (3) c R 0 0 ~ ~ FIG. 1. Illustration of our method to discard LISA triggers. n ( )d The waveforms are those of the gravitational events which were observed by aLIGO in its O1 and O2 runs (2015-2017). where n (; ; T ) is the number of astrophysical (real) GW150914 would have had the highest signal-to-noise in LISA, events which LISA is sensitive to (all of which are de- = 7, while GW170814 would have had  = 4:5 (as- LISA LISA tectable from the Ground) for a given vector  of source suming 4 years of integration time), both of which are below parameters, a signal-to-noise threshold  , and integra- the conventional =8 threshold. The red stripes indicate the LISA ~0 merger time of LISA triggers (their width set by the uncer- tion time T ; n ( ) is the number density of LISA triggers tainty). If a trigger does not agree with any of the events ~0 as a function of  in the search parameter space. detected from the Ground, it can be discarded as random The most important ingredient in our analysis is the noise (or as an astrophysical event whose merger will appear relationship between the threshold  and the number LISA in LIGO in the future and is thus irrelevant for our purposes). of expected background triggers, which we call the “FAR We show that if LISA had started observing in 2011, it would curve.” At this time, there is no reliable estimate for the have been possible to lower its signal-to-noise threshold and re- LISA FAR curve. We therefore use as a proxy the results cover GW150914, and potentially also GW170814. The other events would have been out of reach. of the LIGO Mock Data Challenge [2], which suggest that the number of background triggers increases by about two orders of magnitude when the signal-to-noise threshold is characterized by its k-dimensional vector of best-fit is decreased by one (we use their Experiment 3, which is parameter values ~ and covariance matrix . The problem the most relevant for our study). This agrees with the of consistency checking between the LISA and Ground recent findings of Ref. [7]. measurements corresponds to finding the overlap between We can then define the effective LISA threshold as two volumes in a multi-dimensional space given some eff 0 eff metric. We claim that two measurements taken by LISA (T ) =  + log (f ( ; T )); (4) LISA LISA LISA and the Ground agree with each other if they meet the following criterion: where  is the conventional signal-to-noise threshold, LISA 2 is the FAR and T is the integration time in LISA. Eq. (4) D(~ ; ~ ;  ;  )   (p); (1) LISA Ground LISA Ground is the crux of the method proposed in this work. where D is a function that gives the distance between two The FAR curve given in Ref. [2] has a slope  100 and points in the high-dimensional space under some metric, is not shown below  = 5:5. As a conservative estimate, 2 3(5:5) we impose an exponential cutoff e starting at and  (p) is the quantile function for probability p of the Chi-Squared distribution with k degrees of freedom.  = 5:5, essentially preventing any improvement beyond A typical source in LISA will be characterized by k = 9  = 5. We also consider a more optimistic case in which parameters (when taking into account the antenna pat- we extrapolate the FAR curve with a similar cutoff at tern), so the exact two-point distance problem would be  = 4. Given the volume permitted by a single source, solved in an 18-dimensional space, and hence it can be Eq. (2), the number density n of real sources in the computationally intensive. Instead of solving the problem parameter space and the FAR function, we are now ready exactly, we calculate the volume bounded by  (p) in the to obtain  by solving Eq. (4) self-consistently. eff parameter space centered at the best-fit value for each In order to compute the second integral in Eq. (3), we parameter that is given by the more precise measurement need to estimate  . We adopt a modification of the LISA between the Ground and LISA. Since most of the sources Fisher matrix code from Ref. [5] to calculate the uncer- will be detected from the Ground with signal-to-noise tainties on source parameters. As explained above, we eff well above threshold, the Ground measurements can be calculate  (T ) using the three groups of parameters LISA 3 which contribute the most to the fraction of discarded black hole (assumed to be 5M ), and by default we set events f : the upper cutoff M = 40M [13–15]. To account for c cut (i) Time of coalescence t : we care only for events that uncertainty regarding these choices, we also calculate will merge in the Ground frequency band and assume that our results using two other mass functions: in one we noise triggers will be distributed uniformly in the LISA replace the Gaussian cutoff with a sharp step function observation window, which is determined by T . P (M ) / H(M M ), and in another with an expo- cut M =M 1 cut (ii) Component masses (M , M ): we assume that noise nential cutoff P (M ) / e . For all cases we limit 1 2 triggers will pick up a random template in the template the maximum component mass to 100M . Finally, given bank, and calculate the fraction f assuming noise triggers a value for M , we define the PDF of M as a uniform c 1 2 are distributed uniformly in the (M ; M ) plane. The distribution ranging from M to M [8, 12]: 1 2 gap 1 uncertainty on either component mass is normally  10% P (M j M )  A H(M M )H(M M ): (7) 2 1 M 2 gap 1 2 of the measured value, but due to the strong correlation between the two component masses [8], the allowed vol- For the sky locations, we assume sources are uniformly ume in the parameter space is typically much smaller than distributed on the celestial sphere. In principle one should 10%. This volume is related to the uncertainty in chirp generate a 6-dimensional sample in the mass–sky-location mass measurement, which is expected to be quite small parameters space, but this is quite computationally inten- in LISA (as BBHs spend many cycles in its frequency sive. In practice, we average over a reasonable amount band). Typically the probability of a noise trigger being of sources distributed across the sky and compress the consistent with one real event is  10 . calculation of hV Ti to two (mass) dimensions. (iii) Sky location ( ,  ): We assume that noise triggers S S th The next term we need is f (z; ;  ), which is related will be uniformly distributed across the sky. LISA will 2 to the horizon redshift of the source. The LISA signal-to- be able to localize sources to within O(10) deg [9]. Com- noise of a source with frequency-domain waveform h(f ) paring to the whole sky, the probability of a noise trigger 3 at some luminosity distance is given by [16] being consistent with one event is . 10 . Our figure-of-merit is the number of additional sources max ~ ~ h (f )h(f ) we can recover in LISA by replacing the conventional = 4 df; (8) 0 e S (f ) threshold  with  . This of course depends on min LISA LISA the astrophysical BBH merger rate. Multiband events where f and f are the initial and final frequencies. min max probed by LISA are in the local Universe, so we can th We get the horizon redshift, and hence f (z; ;  ), by assume the merger rate R to be constant in redshift. We th setting  =  . denote by  the mean rate of events of astrophysical When calculating the uncertainty and signal-to-noise origin above a certain signal-to-noise threshold, given by for a given source, we need to integrate the waveform = RhV Ti, where hV Ti is the time and population- over a certain frequency range. Since we are interested in averaged space-time volume accessible to the detector at sources which can in principle be detected in both LISA th the chosen threshold  , defined as [10] and the Ground, we set f = 1 Hz (the conventional max upper cutoff on the LISA noise curve). To determine f , dV 1 min th ~ ~ ~ hV Ti = T dzd s()f (z; ;  ); (5) we require that a source drifts from the LISA band to the dz 1 + z Ground band in less than a total time T . The chirp time of a source with chirp mass M (in the observer frame) is where V is the comoving volume, s() is the injected th ~ given by [17] distribution of source parameters, and 0  f (z; ;  )  1 is the fraction of injections detectable by the experiment. Z max 5 5c 5=3 11=3 In order to calculate hV Ti, we need to solve for the t = df (GM) f : (9) 8=3 horizon distance and redshifted volume as a function of f min source parameters [11], and then marginalize over an To determine f () we solve Eq. (9) setting t  T . ~ min input population s(). We consider sources characterized In Figure 2 we plot our main result:  = , the =8 eff by 9 parameters: the two component masses (M ; M ), 1 2 increase in detection rate compared to using the standard time of coalescence t , phase of coalescence  , luminosity c c = 8 threshold, under different assumptions. We see that distance D , sky locations of the source ( ;  ), and the L S S using the Ground information can boost the number of orbital angular momentum direction ( ;  ). In practice, L L detections in LISA by a factor  4, under the conservative we sample over the two component masses and four sky choice for the FAR. locations, with t and  arbitrarily set to zero. c c Since our figure-of-merit compares total rates, and we For the injected mass distribution, we follow Ref. [12] assume a constant merger rate density per comoving and define the probability density function (PDF) of M volume, the uncertainties in the merger rate cancel out. (M =M ) 1 cut The dominant uncertainty in our result stems from the P (M )  A M H(M M )e ; (6) 1 M 1 1 gap FAR. With a more optimistic choice of FAR the boost where A is a normalization constant, H is the Heav- factor can increase up to  8: the LISA sampling rate [18] iside function, M is the minimum mass of a stellar sets a lower limit on the threshold. gap 4 the LISA and Ground templates are qualitatively similar, Conservative this replacement should increase the discarding power at Optimistic the higher-mass end compared to the uniform case, and 8 4 Mass sharp cutoff therefore improve the boost in rate. Mass single exp cutoff Another approximation we made was to extrapolate our Fisher matrix calculation into the low signal-to-noise regime, where it generally serves only as a lower bound of the uncertainties [22]. A more realistic estimate of the uncertainties can be achieved with other parameter estimation approaches, such as the Markov Chain Monte Carlo method [23]. We hope that our work will motivate participants in the ongoing LISA Data Challenges [24] to verify and improve our FAR estimates. To conclude, while the idea to use LISA detections to 1m 3m 0.5yr 1yr 4yr alert ground-based experiments about pending mergers Integration time has been explored before [1], we have investigated for the first time the potential of exploiting the opposite route. FIG. 2. The boost in the LISA detection rate enabled by We have introduced in this Letter a method to recover our method, compared to setting the standard signal-to-noise sub-threshold stellar-mass BBH merger events from the threshold of  = 8, and assuming that all sources are observed for the integration time T given in Eq. (9). The blue solid LISA data stream using information from the subsequent line shows the rate increase using a FAR function with a ground-based measurements of these events. Our analysis cutoff at  = 5 and a mass function with a Gaussian cutoff. forecasts a remarkable increase – by a factor of 4 to 8, The dashed-blue line corresponds to a more optimistic FAR depending on the assumptions – in the number of LISA function, where the cutoff is at  = 4. For comparison, we detections. While our estimate was restricted to multi- show in red and green the result when using a mass function band sources whose merger is detected from the Ground with a sharp cutoff at 50M and a single-exponential cutoff during the LISA lifetime, the same algorithm can be at 40M , respectively. continuously applied for events that merge after LISA has finished its mission, yielding more detections. The increase in number of multi-band GW detections can bring forth a plethora of rewards. For example, im- The next source of uncertainty is due to the choice of provements in parameter estimation and modeling con- mass function. The increase in detection rate is biased straints will enable novel tests of extreme gravity theo- toward the lower end of the mass function, and so it is ries [25–29]. Most notably, discrimination between differ- more significant for mass functions that favor lower mass ent BBH-formation channels using eccentricity [30–35], events. This uncertainty amounts to  5%. A uniform-in- spins [36–40], and other waveform features [41–43] will log mass function should yield similar results [7]. greatly benefit from the larger lever arm in frequency Various assumptions we have made here can be im- garnered from these measurements. proved upon. For example, in checking for consistency between LISA and the Ground we considered only the volume allowed by the LISA covariance matrix, instead of ACKNOWLEDGMENTS solving the exact two-point problem. This is a reasonable assumption, based on the expected sensitivity of Ground It is our pleasure to thank Ilias Cholis, Marc observatories by the time LISA flies. Kamionkowski, Johan Samsing and Fabian Schmidt for We also took the distribution of noise triggers to be useful discussions. K.W.K.W. and E.B. are supported by uniform in the parameters of interest. This assumption is NSF Grants No. PHY-1841464 and AST-1841358. E.D.K. valid for time of coalescence and sky location, but it may was supported by NASA grant NNX17AK38G. C.C.’s not be accurate for the two component masses. Search work was carried out at the Jet Propulsion Laboratory, template banks for ground-based detectors typically have California Institute of Technology, under contract to the more templates at the low-mass end [19–21]. More re- National Aeronautics and Space Administration. 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Journal

General Relativity and Quantum Cosmology – arXiv (Cornell University)

Published: Aug 24, 2018

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