Are stellar-mass black-hole binaries too quiet for LISA?
AAre stellar-mass black-hole binaries too quiet for LISA?
Christopher J. Moore (cid:63) , Davide Gerosa, Antoine Klein
School of Physics and Astronomy and Institute for Gravitational Wave Astronomy, University of Birmingham,Birmingham, B15 2TT, UK
22 July 2019
ABSTRACT
The progenitors of the high-mass black-hole mergers observed by LIGO and Virgoare potential LISA sources and promising candidates for multiband GW observations.In this letter, we consider the minimum signal-to-noise ratio these sources must haveto be detected by LISA bearing in mind the long duration and complexity of thesignals. Our revised threshold of ρ thr ∼
15 is higher than previous estimates, whichsignificantly reduces the expected number of events. We also point out the importanceof the detector performance at high-frequencies and the duration of the LISA mission,which both influence the event rate substantially.
Key words: gravitational waves – black holes – data analysis
Until recently, stellar-mass black holes (BHs) were onlythought to be as heavy as ∼ M (cid:12) , as inferred from X-raybinary measurements (e.g. Wiktorowicz et al. 2014). The firstdetections of gravitational-waves (GWs) by the LIGO/Virgodetectors point to a population of BHs with masses up to ∼ M (cid:12) (Abbott et al. 2018a). This difference has importantconsequences for the formation and evolution of BH binaries–for instance, proving that low-metallicity environments playa vital role (Belczynski et al. 2010a; Abbott et al. 2016c). Itis also crucial for the future of GW astronomy.BH binaries with components of ∼ M (cid:12) might emitGWs strongly enough at mHz frequencies to be within reachof LISA (Sesana 2016). This opens up the exciting possibilityof performing multiband GW astronomy: a single sourcebeing observed by both LISA and LIGO. Following thisrealisation, stellar-mass BH binaries started to be exploredas an important part of the LISA science case, in terms ofastrophysics (Belczynski et al. 2010b; Breivik et al. 2016;Nishizawa et al. 2016, 2017; Kyutoku & Seto 2016; Sesana2016, 2017; Samsing et al. 2018; D’Orazio & Samsing 2018;Samsing & D’Orazio 2018, 2019; Kremer et al. 2018, 2019;Randall & Xianyu 2019; Fang et al. 2019; Gerosa et al. 2019),fundamental physics (Barausse et al. 2016; Chamberlain &Yunes 2017; Tso et al. 2018), cosmology (Kyutoku & Seto2017; Del Pozzo et al. 2018), and data analysis (Vitale 2016;Wong et al. 2018; Mangiagli et al. 2019; Cutler et al. 2019;Tanay et al. 2019)In this letter, we add a point of caution. Stellar-mass BHbinaries can emit in the LISA band for the entire durationof the mission, generating millions of GW cycles with acomplex, chirping signal morphology. These will need to be (cid:63) [email protected] extracted from the LISA datastream. If this were to be doneusing templates, we estimate the size of the template bankrequired and, consequently, the threshold signal-to-noise ratio(SNR) where events are loud enough to be detected. We findan expected SNR threshold ρ thr ∼
15 for systems mergingwithin 10 yr. Previous work has assumed a threshold ρ thr ∼ ρ − . Therefore,increasing the SNR threshold by nearly a factor of ∼ a posteriori (Wong et al. 2018).Detections from ground-based interferometers will allow usto revisit past LISA data hunting for signals with knownparameters. In this case, we find ρ thr ∼ GW detection is routinely performed using template banks.These searches involve matching sets of precomputed wave-form templates against the observed data. The threshold SNRabove which a signal can be confidently detected dependson the number of templates in the bank. This relationship isderived for an idealised search in this section using methods c (cid:13) a r X i v : . [ a s t r o - ph . H E ] J u l Moore et al. similar to those in Buonanno et al. (2003) and Chua et al.(2017).Let us assume that the GW signals may be written as h α ( t ) = ρ ˆ h α ( t ) exp( iφ s ) , (1)where the ˆ h α are the normalised template waveformswith | ˆ h α | = 1. A hypothetical template bank { ˆ h α | α =1 , , . . . , N bank } may be constructed spanning all the sourceparameters except for the SNR, ρ , and the phase shift, φ s ,which may be searched over for each ˆ h α at negligible addi-tional cost. In practice one would not usually use a templatebank to search over the time-of-arrival parameter, as thiscan be handled more efficiently using fast Fourier transformtechniques (Brady et al. 1998); however, for our hypotheticalsearch it is convenient to imagine treating this the same asthe other parameters.The detection statistics are the phase-maximised innerproduct between the data, s , and the templates, σ α = max φ s (cid:104) s | ˆ h α (cid:105) . (2)When the data contains only stationary, Gaussian noise( s = n ) the statistics σ α follow a Rayleigh distribution withprobability density f ( σ α ) = σ α exp (cid:18) − σ α (cid:19) , (3)with σ α ≥
0. If a signal is present ( s = h α + n ), the statisticfor the corresponding template follows a Rice distributionwith offset ρ . This has probability density f ( σ α , ρ ) = σ α exp (cid:18) − σ α + ρ (cid:19) I ( ρσ α ) , (4)where I denotes the zeroth-order modified Bessel functionof the first kind.A detection is claimed if at least one of the σ α exceeda predetermined threshold, σ thr . This threshold is set byrequiring a certain (small) false-alarm probability: P F ( σ thr ) = (cid:90) ∞ σ thr d σ α f ( σ α ) ⇒ σ thr ( P F ) = √− P F . (5)A typical choice for P F across the bank might be 10 − .Approximating the statistics { σ α | α = 1 , , . . . , N bank } asindependent random variables, the false-alarm probabilityin a single template is approximately reduced by a factor N bank . Hence we set P F = 10 − N bank . (6)The detection probability (i.e. the probability that, inthe presence of a signal, the statistic for the correspondingtemplate exceeds the threshold; σ α > σ thr ) is given by P D ( ρ ) = (cid:90) ∞ σ thr d σ α f ( σ α , ρ ) ≈ Θ( ρ − ρ thr ) . (7)This detection probability rises from zero to unity across anarrow range ∆ ρ ≈ For example, Abbott et al. (2016a) use a false-alarm ratethreshold of FAR = 0 .
01 yr − for an observation period of T =51 . P F = 1 − e − T FAR ≈ . × − . N bank ρ t h r L I G O S O B H ( t m e r g e r > y r) S O B H ( t m e r g e r < y r) E M R I Figure 1.
The threshold SNR as a function of template bank size.Solid vertical lines indicate the bank sizes for stellar origin blackhole (SOBH) binaries with component masses in the range (5 -50) M (cid:12) ; these are split into those which merge in <
10 yrs (fastchirping) and >
10 yrs (slow chirping). The thresholds are ρ thr ∼ ∼
14 respectively (see Table. 1). For comparison, two otherclasses of GW are also indicated. Binary BHs in LIGO/Virgo canbe detected with single-detector SNRs as low as ∼ ∼ × templates (Dal Canton & Harry 2017; thisnumber does not include the time-of-arrival parameter; includingthis enlarges the effective size of the bank by a factor of ∼ ,the number of cycles in a typical template). At the other extreme,EMRIs in LISA have ρ thr (cid:38)
16 and would require very largetemplate banks (Chua et al. 2017). As discussed in the text,we do not propose to actually use such huge template banks inpractical searches; these are estimates of the numbers required bya hypothetical, optimal search and provide lower bounds on thethreshold for a practical, possibly suboptimal search. step function, Θ. Here we are assuming that all sources with ρ > ρ thr are recovered whilst all other sources are missed.The threshold SNR depends on the size of the tem-plate bank through the trials factor N bank in Eq. (6); thisdependence is plotted in Fig. 1.The above discussion considered an idealised templatebank search and gave no consideration to computationalcosts. Some of the template banks indicated in Fig. 1 arefar too large to be practically implemented. In those casesit is necessary to use an alternative procedure. For example,when searching for extreme mass-ratio inspirals (EMRIs) inLISA data, a semi-coherent search strategy has been pro-posed (Gair et al. 2004) that involves splitting the datainto segments, searching each segment separately with smalltemplate banks, and combining the results into an overall de-tection statistic. Multiband binaries might require a similarapproach. This computationally viable alternative is sub-optimal compared to a full template bank search and thiscan further raise the detection threshold (in the EMRI case,from ρ thr ∼
16 to (cid:38)
20; Chua et al. 2017). On the other hand,it might be possible to compress the template bank; i.e. toaccurately cover the signal space of interest with a reducedbasis (Cannon et al. 2010; Field et al. 2011). Such a compres-sion scheme would reduce the effective size of the templatebank potentially lowering the detection threshold from thatestimated here. Future work should address the practicalimplementation of a search and assess the sensitivity via theblind injection and recovery of signals into mock LISA data. re stellar-mass BH binaries too quiet for LISA? Let us now estimate the size of the template bank N bank required to detect stellar-mass BH binaries with LISA. Weconsider the following parameters λ µ ∈ { ln m , ln m , cos θ N , φ N , cos θ L , φ L , e , φ e , χ eff , t merger } , where m i is the mass of object i , θ N and φ N are anglesdescribing the source’s sky location, θ L and φ L are anglesdescribing the direction of the sources orbital angular mo-mentum, e is the eccentricity at t = 0, φ e is the argument ofperiastron at t = 0, χ eff is the effective spin parameter, and t merger is the time to merger from t = 0. The LISA missionstarts collecting data at t = 0.We adopt a conservative approach and do not includespin components other than χ eff when estimating the sizeof the bank. If these parameters are significant for a frac-tion of the source population– in particular systems withsmall t merger (Mangiagli et al. 2019)– they will provide anadditional contribution to the overall size of the bank.The Fisher matrixΓ µν = (cid:42) ∂ ˆ h∂λ µ (cid:12)(cid:12)(cid:12) ∂ ˆ h∂λ ν (cid:43) (8)provides a natural metric on parameter space to guidewhere templates should be placed (Owen 1996; Owen &Sathyaprakash 1999; Sathyaprakash & Dhurandhar 1991;Dhurandhar & Sathyaprakash 1994). The diagonal compo-nents of Γ are the squared reciprocals of the natural lengthscale for the template separation along each parameter direc-tion. In order to ensure that there is at least one templatealong each dimension; we employ a modified Fisher matrix˜Γ µν = max (cid:18) Γ µν , δ µν [∆ λ µ ] (cid:19) , (9)where ∆ λ µ is the prior range on the parameter λ µ . Thismodification is only important for parameters which havevery little effect on the waveform (e.g. χ eff for systems farfrom merger).The total number of templates in the bank is foundby integrating over the parameter space (Gair et al. 2004;Cornish & Porter 2005) N bank ≈ (cid:90) d λ (cid:112) det ˜Γ . (10)The square root of det ˜Γ gives the template number densityrequired such that the mismatch between adjacent templatesis ∼ independent templates inthe bank, as required by Eq. (6).We evaluate this integral using Monte Carlo integration.We use templates described by Klein et al. (2018), setting thespins to S i = m i χ eff ˆ L . We compute the determinant of theFisher matrices using the noise curve given by Robson et al.(2019), being careful to remove near-singular matrices. Wefocus on binaries observed by LIGO/Virgo and set m , m ∈ [5 M (cid:12) , M (cid:12) ] (Abbott et al. 2018a). We consider both fast-(0 < t merger <
10 yr) and slow-chirping (10 yr < t merger <
100 yr) sources and set a range of eccentricities 0 < e < . . (cid:46) ρ thr (cid:46) . m , m [ M (cid:12) ] N (fast)bank N (slow)bank ρ (fast)thr ρ (slow)thr −
50 10 . . . . −
50 10 . . . . −
50 10 . . . . . – 9 . Table 1.
Total effective number of templates in the bank andcorresponding threshold SNR. We consider different lower-masslimits, as well as a representative archival search for a GW150914-like event. Superscripts (fast) and (slow) correspond to fast- (0 10 yr) and slow-chirping (10 yr < t merger < 100 yr)binaries, respectively. The results for the row highlighted in grayare shown in Fig. 1. Slow chirping sources, on the other hand, are easier to detect;we find 13 . (cid:46) ρ thr (cid:46) . 9. The lower (upper) edge of theseranges correspond to heavier (lighter) systems, with fewer(more) cycles in band. These estimates are significantly higherthan the threshold ρ thr ∼ ρ thr on N bank in Fig. 1is rather flat. Although tweaking the parameter ranges tobe covered by a search changes the number of templatesrequired, it has only a modest impact on the threshold SNR. Revisiting past LISA data in light of ground-based observa-tions is a promising avenue to detected more events (Wonget al. 2018). In such a scenario, the targeted template bankcan be restricted given prior knowledge on the source. Forconcreteness, we consider an archival search correspondingto a GW150914-like event detected by a third generationground-based detector 4 years after the start of the LISAmission. The integral in Eq. (10) is computed restricting itsparameter range to the measurements errors of GW150914(Abbott et al. 2016b) reduced by a factor of 10. We also as-sume a perfect measurement of t merger and do not integrateover it. Prior information from the ground allows to decreasethe size of the template bank by a factor of ∼ , reducingthe threshold to ρ thr (cid:39) . ∼ ρ thr forarchival searches. Our results are largely consistent with thisimprovement factor. We now assess the impact of our revised SNR threshold on asimple, but realistic astrophysical population of stellar-massBH binaries. Our procedure closely mirrors that of Gerosaet al. (2019), to which we refer for further details.The number of multiband detections is estimated by N multib = (cid:90) d z d ζ d θ d t merger R ( z ) p ( ζ ) p ( θ ) d V c ( z )d z 11 + z × Θ[ ρ ( ζ, θ, t merger ) − ρ thr ] F p det ( ζ, z )Θ( T wait − t merger ) . (11)Here ζ collectively denotes BH masses, spins, and binaryeccentricity, p ( ζ ) is their probability distribution function, z is the redshift, V c is the comoving volume, R ( z ) is theintrinsic merger rate density, θ collectively denotes the angles Moore et al. θ N , φ N , θ L and φ L , and p ( θ ) is the corresponding probabilitydistribution function.For simplicity, we consider non-spinning BHs on quasi-circular orbits, i.e. ζ = { m , m } . We assume m , m ∈ [5 M (cid:12) , M (cid:12) ] distributed according to p ( m ) ∝ m − . and p ( m | m ) = const. For this mass spectrum, Abbott et al.(2018a) measured R = 57 +40 − Gpc − yr − . We stress thatuncertainties in both R and p ( ζ ) affect our predictions.Gerosa et al. (2019) used a sky-averaged LISA noisecurve to compute SNRs. Here we perform a more genericcalculation where we compute ρ as a function of θ . Thisallows us to capture individual events that are expected tobe above threshold only for favorable orientations or positionsin the sky. We use the low-frequency LISA response by Cutler(1998) and waveforms by Santamar´ıa et al. (2010). Binariesare distributed uniformly in cos θ N , φ N , cos θ L and φ L . Theinitial frequency is set by t merger . LISA SNRs are computedusing the mission specification of Robson et al. (2019) and amission duration T obs of 4 or 10 yrs. Events are then selectedusing a cut in SNR at ρ thr .The term p det ( ζ, z ) in Eq. (11) encodes selection effectsof the ground based detector, and is estimated using thesingle-detector approximation by Finn & Chernoff (1993).As stressed by Gerosa et al. (2019), the multiband detec-tion rate is largely independent of the specifications of theground-based network. For concreteness we assume a LIGOinstrument at design sensitivity (Abbott et al. 2018b), but wehave also verified that identical results are obtained if a thirdgeneration detector is used instead. For multiband scenarios,one might be interested only in binaries merging within agiven timeframe T wait , and thus limit the detection rate tosources with t merger < T wait . For simplicity, we assume aground-based network with a duty cycle F = 1.Figure 2 shows the number of multiband detectionsmerging within T wait = 10 yr as a function of the SNRthreshold. Multiband sources will be restricted to the thelocal Universe, where the probability distribution functionof the SNRs assumes the universal form p ( ρ ) ∝ ρ − (Schutz2011; Chen & Holz 2014). The number of detections abovethreshold, therefore, scales as N ( ρ thr ) ∝ (cid:90) ρ>ρ thr ρ ∝ ρ . (12)This severe scaling means that even a modest increase of thethreshold SNR can push the number of sources below unity.Unfortunately, this turns out to be the case in most of thesemodels. Using ρ thr = 15, we predict LISA will not provideforewarnings to ground-based detectors for this populationof stellar-mass BHs.As shown in Sec. 3, archival searches require smallertemplate banks, lowering the SNR threshold to ∼ 9. In thiscase, we find 0 . (cid:46) N multib (cid:46) 2. Revisiting past LISA data,as first put forward by Wong et al. (2018), might well be ouronly chance to observe stellar-mass BH binaries with LISA.Some events from the population of binaries in the earlyinspiral might also be above threshold (Fig. 3). If the missionis long enough, we find that a few sources with merger times t merger (cid:46) 100 yr will be observable by LISA. For these slowlychirping signals, LISA will be able to provide forewarningsof a small number of events a very long time into the future. ρ thr − N m u l t i b T wait = 10 yr T obs = 4 yr T obs = 10 yr Figure 2. Number of stellar-mass BH binaries jointly detectableby the LISA space mission and ground-based interferometers asa function of the threshold SNR. Blue (red) curves assume aLISA mission duration T obs = 4 yr (10 yr). We only considerbinaries merging within T wait = 10 yr. For each set of parame-ters, the shaded areas captures the current uncertainties in thelocal BH merger rate; in particular, we set R = 97 Gpc − yr − (32 Gpc − yr − ) for the upper (lower) line in each set. Verticalsolid lines mark the SNR thresholds estimated in this letter forboth forewarnings ( ρ thr ∼ 15, c.f. Fig. 3) and targeted archivalsearches ( ρ thr ∼ T wait [yr] − − N m u l t i b ρ thr = 15 T obs = 4 yr T obs = 10 yr Figure 3. Number of stellar-mass BH binaries merging within T wait observable by LISA with ρ ≥ 15. Blue (red) curves assumea LISA mission duration T obs = 4 yr (10 yr). For each set ofparameters, the shaded areas captures the current uncertainties inthe local BH merger rate; in particular, we set R = 97 Gpc − yr − (32 Gpc − yr − ) for the upper (lower) line in each set. The verticalline marks T wait = 10yr, as used in Fig. 2. In this letter we have considered the LISA detectability of BHbinaries with component masses in the range (5 − M (cid:12) .Wefind that, due to the complexity of the signal space, anidealised template bank search has a threshold SNR of ρ thr ≈ − 15. This is significantly higher than previousassumed. Because the expected number of events scales as ρ − , our revised estimate implies that LISA might not pro-vide forewarnings for any ground-based detectors within a10 yr timescale. From a data analysis perspective, stellar-mass BH binaries in LISA are in some respects more similar re stellar-mass BH binaries too quiet for LISA? to EMRIs than to LIGO/Virgo binary BHs. Our estimateapplies to an optimal template-bank search; in practice, asub-optimal approach may be required, further raising ρ thr .We stress that our calculation is only a preliminary esti-mate and will need to be corroborated with future injectioncampaigns.Because the expected numbers of events is so low, it iscrucial to maximise our sensitivity to these events using allthe tools at our disposal. Figure 3 shows the importance ofa long mission duration. With T = 4 yrs, LISA might notbe able to provide forewarnings even 100 yrs into the future.Conversely, a 10 yr mission might deliver a few sources with t merger (cid:46) 100 yr. These stellar-mass events exists at the highfrequency end of the LISA sensitivity window. Therefore itis crucial to preserve the detector performance in this region.It is possible the high-frequency noise level might turn outto be up to ∼ . . ≈ . ∼ ∼ M (cid:12) BHs in wide orbits (Elbert et al.2018; Lamberts et al. 2018). Because these systems are slowlychirping and closer to being monochromatic, the signal spaceis considerably simpler allowing for a lower threshold SNR.We defer an analysis of this population to future work.The rate estimates of Sec. 4 depends on the largestBH mass considered, here taken to be 50 M (cid:12) . This value ismotivated by current LIGO/Virgo observations as well astheoretical predictions of supernova instabilities (Barkat et al.1967). If, however, a population of BHs with masses ∼ M (cid:12) were to be present, such systems would be prominent multi-band sources. Their larger mass would imply both largerLISA SNRs and shorter merger times. Repeating the analy-sis of Sec. 4 with a cutoff of 100 M (cid:12) yields 0 . (cid:46) N multib (cid:46) ρ thr = 15 and T wait = 10 yr.Stellar-mass BHs binaries in the mHz regime are in-trinsically quiet and their observation with LISA will bechallenging. A combination of detector-sensitivity improve-ments, data-analysis advancements, and possibly a pinch ofluck, might all turn out to be necessary. ACKNOWLEDGEMENTS We thank A. Vecchio, A. J. K. Chua, P. McNamara, E. Berti,K. Wong and B. S. Sathyaprakash for discussions. Computa-tional work was performed on the University of Birmingham’sBlueBEAR cluster and at the Maryland Advanced ResearchComputing Center (MARCC). REFERENCES Abbott B. P., et al., 2016a, PRX, 6, 041015 (arXiv:1606.04856)Abbott B. P., et al., 2016b, PRL, 116, 241102 (arXiv:1602.03840)Abbott B. P., et al., 2016c, ApJ, 818, L22 (arXiv:1602.03846)Abbott B. P., et al., 2016d, ApJ, 833, L1 (arXiv:1602.03842)Abbott B. P., et al., 2018a, (arXiv:1811.12907)Abbott B. P., et al., 2018b, LRR, 21, 3 (arXiv:1304.0670)Ade P. A. 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