From black holes to their progenitors: A full population study in measuring black hole binary parameters from ringdown signals
FFrom black holes to their progenitors: A fullpopulation study in measuring black hole binaryparameters from ringdown signals
Ioannis Kamaretsos
School of Physics and Astronomy, Cardiff University, Queens Buildings, CF24 3AA, Cardiff,United KingdomE-mail: [email protected]
Abstract.
A perturbed black hole emits gravitational radiation, usually termed the ringdownsignal, whose frequency and time-constant depends on the mass and spin of the black hole.I investigate the case of a binary black hole merger resulting from two initially non-spinningblack holes of various mass ratios, in quasi-circular orbits. The observed ringdown signal will bedetermined, among other things, by the black hole’s spin-axis orientation with respect to Earth,its sky position and polarization angle - parameters which can take any values in a particularobservation. I have carried out a statistical analysis of the effect of these variables, focusing ondetection and measurement of the multimode ringdown signals using the reformulated EuropeanLISA mission, Next Gravitational-Wave Observatory, NGO, the third generation ground-basedobservatory, Einstein Telescope and the advanced era detector, aLIGO. To the extent possibleI have discussed the effect of these results on plausible event rates, as well as astrophysicalimplications concerning the formation and growth of supermassive and intermediate mass blackholes.
1. Introduction
Astrophysical observations to date have provided sturdy evidence that black holes (BHs) mayexist and play an important role in many physical processes [1, 2, 3]. With direct evidencestill lacking, it is expected that observation of gravitational waves (GWs) from merged BHs willnot only provide indisputable evidence for the existence of BHs, but also the ability to extractaccurate information about the progenitor system and the BH.The ringdown radiation consists of a superposition of, in principle, an infinite number ofessentially damped sinusoids, termed quasi-normal modes. Their frequencies and time-constantsdepend only on the mass and spin of the BH – a consequence of the no-hair theorem [4]. Ina recent work [5] we have argued that the amplitude terms of the various quasi-normal modesencode important information about the origin of the perturbation that caused them, such asthe component masses of the progenitor binary. This allows performing parameter estimationon the system from the strong-field regime, as opposed to using the inspiral phase. However, inthat study, as well as in previous studies of parameter estimation from ringdown signals [4, 6],only a small region of the parameter space was explored. In a realistic scenario, a BH ringdownsignal can have any sky location and polarization. All the while, it could have originated froma BH with any spin-axis orientation with respect to Earth. These variables have a significant a r X i v : . [ g r- q c ] M a r ffect on the observed ringdown signal and a direct impact on the science we can achieve byobserving GWs from merged BHs.In the present study, I have investigated in detail how these angular parameters affect thedetection and measurement of the ringdown signals. To this end, I have varied the angularparameters over their full range, thereby considering a large population of BH mergers. I haveconsidered supermassive BHs (SMBHs of mass ≥ M (cid:12) ) visible in NGO and intermediate massBHs (IMBHs of mass ∼ M (cid:12) ) observable in ET and aLIGO. I have computed the probabilitydistribution functions of signal to noise ratios (SNR), as well as measurement errors of a chosenset of parameters, for a wide range of the BH mass and for mass ratios between 2 and 20.Finally, I translate these probabilities to proportions of observed events in NGO and ET thatwill yield parameter errors below certain thresholds and discuss how observations of ringdownsignals could help in dealing with open questions on the existence and history of SMBHs andIMBHs. frequency (Hz) po w e r s p ec t r a l d e n s it y ( H z - ) eLISA-NGOinitial LISACLISA1_P005_LPF10LISA1_P2_DRS Figure 1:
Noise power spectral densities for variousproposed configurations of LISA-like space detectors.The dashed red line corresponds to the original LISAmission, while the rest of them refer to Europeandesigns of LISA. In all cases, the galactic binarywhite dwarf confusion noise [7, 8] is included, whichhas a negligible visible effect on the newer LISAcurves though, due to their, almost two orders ofmagnitude worse sensitivity. Additionally, a lowfrequency cut-off - not shown - was induced at5 × − Hz . In this study, I am using the latestarrangement for the European mission of LISA,labeled eLISA-NGO in this graph. mass ratio r e l a ti v e a m p lit ud e s averaged valuevalue at 10 M
33 over 22 21 over 2244 over 22
Figure 2:
This plot shows the relative amplitudes ofmodes (3 , ,
1) and (4 ,
4) over (2 ,
2) as a functionof the mass ratio. The diamond points show thevalues that correspond to a time 10 M after the peakluminosity of the 22 mode in the equal mass case.(see also Fig.3 in [5]) Fits to these points were usedin this previous work. On the other hand, the circlepoints, which are used in the present study, werecomputed by taking into account all the points inthe waveform, in a time region starting at the peakluminosity of 22 and ending 30 M later. The solidlines shown, constitute fits to these circle points,given by expressions (6)-(8).
2. Full population Analysis
I have closely followed the procedure of Ref. [5] for estimating the signal-to-noise ratios andmeasurement errors. In this Section I will discuss the signal model and the parameter spacecovered in this study.In the generic case where we have a network of detectors, we write the response to a ringdownsignal as: h a ( t ) = (cid:88) (cid:96),m,n ≥ B a(cid:96)mn e − t/τ (cid:96)mn cos ( ω (cid:96)mn t + γ a(cid:96)mn ) , (1)here the superscript a is an index denoting the detector in question and ω (cid:96)mn , τ (cid:96)mn arethe frequencies and time-constants of each mode, which are functions of the mass and spinmagnitude of the BH. For further reference, see [9, 10]. In this study I neglect modes withovertone index n ≥
1, thereby considering only the least damped modes. From now on, the n index ( n =0) is omitted. The terms B (cid:96)m and γ (cid:96)m are the following combinations of the antennapattern functions F a + , F a × , amplitude factors α (cid:96)m and the angular functions Y (cid:96)m + ( ι ), Y (cid:96)m × ( ι ) : B a(cid:96)m = M α (cid:96)m D L (cid:113)(cid:0) F a + Y (cid:96)m + (cid:1) + (cid:0) F a × Y (cid:96)m × (cid:1) , (2) γ a(cid:96)m = φ (cid:96)m + m φ + tan − (cid:34) F a × Y (cid:96)m × F a + Y (cid:96)m + (cid:35) . (3)Here, φ (cid:96)m are arbitrary constant phases of each mode. The effective amplitudes B (cid:96)m varyinversely with the luminosity distance and proportionally to the intrinsic amplitudes α (cid:96)m of themodes, which are determined by the numerical simulations. The angular functions Y (cid:96)m + , × ( ι ) arethe following combinations of the spin-weighted spherical harmonics [6]: Y (cid:96)m + ( ι ) ≡ − Y (cid:96)m ( ι,
0) + ( − (cid:96) − Y (cid:96) − m ( ι, ,Y (cid:96)m × ( ι ) ≡ − Y (cid:96)m ( ι, − ( − (cid:96) − Y (cid:96) − m ( ι, . (4)The antenna pattern functions are functions of the sky location coordinates, θ and φ andthe polarization angle ψ , that is, F a + ( θ, ϕ, ψ ), F a × ( θ, ϕ, ψ ). The spheroidal harmonics are angularfunctions of the inclination angle, ι and the azimuth angle φ . The first refers to the angle formedby the BH’s spin angular momentum and the line-of-sight, while the latter is the azimuth angledefined in a non-rotating frame fixed to the BH. The ringdown waveform used is of the form described by Eqs. (1)-(3). It is a signal comprised offour modes, with mode indices ( (cid:96), m, n ) = (2 , , , (3 , , , (2 , , , (4 , , α (cid:96)m of the various modes in Eq.(2), as well as theirfrequencies and time-constants (see also Table I of [5]). These simulations involve the case of initially non-spinning BHs in quasi-circular orbits and for different mass ratios of the binary.For the mass ratios, q = { , , , } the simulation results were first presented in [14, 15, 16],while an additional simulation of a q = 11 binary was carried out in [5].The amplitude terms α (cid:96)m in Eq. 2, are given by the expressions: α ( q ) = 0 . e − q/ . , (5) α ( q ) = 0 . α ( q ) ( q − . , (6) α ( q ) = 0 . α ( q ) ( q − . , (7) α ( q ) = 0 . α ( q ) q . . (8)These constitute fits, that were produced by fitting the merger-ringdown part of the numericalsimulations data, taking into account all the different mass ratios for which these were performed.All points in a time region beginning at the peak luminosity of 22 and ending 30 M later wereconsidered, for each mass ratio. As opposed to the method that was applied in our previouswork [5], where the relative amplitude values at 10 M were used, this approach is expected tobe more robust and to average out any numerical noise that might be present in this part of total SNR q = 2q = 10q = 20 fractional error in D L q = 2q = 10q = 20 fractional error in M q = 2q = 10q = 20 fractional error in j q = 2q = 10q = 20 fractional error in q q = 2q = 10q = 20 fractional error in ι q = 2q = 10q = 20 Figure 3:
Frequency distributions for the total SNR and the measurement errors in NGO, for a 5 × M (cid:12) BH situated at a luminosity distance of 6.73 Gpc ( z (cid:39) . The top left plot shows the probability tohave a detector-BH configuration which will yield the SNR shown, while the rest of the plots concerncumulative frequency distributions for the measurement errors. In each graph, the comparison is shownamong different mass ratios, q , specifically taking the values of 2, 10 and 20. For q=2 - black solidlines - the parameter estimation for the BH mass and spin is outstanding in all configurations. Thiseffectiveness degrades considerably with increasing q. For q of around 10 it is still acceptable, while atq (cid:39)
20, all parameters except the mass are very likely to have huge errors, of the order of 200%. the waveform. Nevertheless, as can be seen from Fig. 2, these two methods do not give verydifferent results. Note also that the above fitting functions, as well as the mode frequenciesand time-constants, may be less accurate in the higher mass ratio values of around 20, whereextrapolation has been performed.
In Ref. [5] we had ignored the effect of the various angles { θ, ϕ, ψ, ι } on the quasi-normalmode spectrum and their impact on the detection and measurement of ringdown signals. Toassess this effect, the aforementioned analysis was repeated by varying the angular parameters { θ, ϕ, ψ, ι } . Specifically, six uniformly spaced values were chosen for these angles. This resultsin 6 = 1296 distinct relative orientations between the detector and the BH and its spin axis.Additionally, a couple of simulations with eight uniformly spaced values were performed, thatis 8 = 4096 configurations, to allow the comparison. Hence, it was decided that six values ineach parameter was acceptable in capturing the behaviour of the observable quantities.The values in the polarization angle, ψ and the azimuth sky location angle, ϕ were linearlysampled in the ranges [0 , π ] and [0 , π ] respectively. Whereas, the values in the inclination angle ι and sky position, θ - which range from 0 to π - are deduced from the uniformly spaced valuesof cos( ι ) and cos( θ ) in the range [ − , +1]. Note that this excludes configurations of optimallyoriented binaries, that is of ι = 0 and ι = π . The parameter set of the ringdown signal in the case of a non-spinning black hole binary,consists of the following nine parameters: { M, j, q, D L , θ, ϕ, ψ, ι, φ } . Namely, the mass M ,the dimensionless Kerr parameter or spin magnitude, j of the BH, the mass ratio q of the total SNR q = 2q = 10q = 20 fractional error in D L q = 2q = 10q = 20 fractional error in M q = 2q = 10q = 20 fractional error in j q = 2q = 10q = 20 fractional error in q q = 2q = 10q = 20 fractional error in ι q = 2q = 10q = 20 Figure 4:
Same as in Fig. 3, but concerning a 25 × M (cid:12) BH. This figure concerns a BH 5 times moremassive than that of Fig. 3, but some of the results are actually slightly worse, as the multimodal signal’spower spectrum is shifted away from the lowest part of NGO’s sensitivity curve. progenitor binary, the sky location vector ( D L , θ, ϕ ) of the BH with respect to Earth, thepolarization angle ψ , the inclination angle ι and the BH azimuth angle φ . Note however, that inthis case, the final spin of the BH is directly mapped to the mass ratio of the progenitor binary,therefore q and j are not treated as independent.The above parameters are the standard ones, as they pertain to all of the modes. Additionalparameters can be introduced, that are characteristic to each mode, such as an initial phasefactor φ (cid:96)m , see Eq. (3). Therefore, the total number of parameters can increase with the numberof modes. We are considering a four mode signal, therefore we have a total of 13 parameters.One thing to note is that the (cid:96) = m modes have a nearly consistent rotational phasing, whilethe (cid:96) (cid:54) = m modes seem to have somewhat distinct associated dynamics, with differentiatedamplitude and phasing during the merger process [11].By virtue of the large number of parameters involved, it was unmanageable to treat the effectof all of them in this analysis. Thus, some of the above mentioned parameters had to be fixed.Specifically, the initial phase angles, φ (cid:96)m in all the four modes considered, were plainly chosenas zero. In addition, the luminosity distance is chosen to be 6.73 Gpc for NGO, and 1 Gpc forET and aLIGO. Lastly, the BH azimuth angle, φ was given the value π/
3. Note that this angledoes not have an effect on the SNR, but of course needs to be considered in the waveform.
The BH mass and the binary mass ratio constitute key parameters and the results depend quitestrongly on them. The reasoning behind the choice of mass values is the following. First of all,the low and high mass end is limited by the sensitivity band of the detectors. The existence ofBHs in the mass range 10 − M (cid:12) , is highly predicted by the mass - velocity dispersion inthe galactic bulge of low-mass, low-luminosity galaxies, as well as in a number of galaxies whichcontain active galactic nuclei [17, 18]. The evidence for SMBHs ranging from 10 M (cid:12) to 10 M (cid:12) ,is quite abundant. They are thought to dwell at the centers of most galaxies. For various recentSMBHs mass estimation results and methods see for instance [19, 20, 21, 22, 23]. The SNR inNGO is quite low for BH masses of less than about 5 × M (cid:12) . Therefore, I take the lower end total SNR q = 2q = 10q = 20 fractional error in D L q = 2q = 10q = 20 fractional error in M q = 2q = 10q = 20 fractional error in j q = 2q = 10q = 20 fractional error in q q = 2q = 10q = 20 fractional error in ι q = 2q = 10q = 20 Figure 5:
Same as in Figs. 3 and 4, but for a 10 M (cid:12) BH. The results slightly deteriorate with respect tothe lower mass BH of Fig. 4. of the mass range to be 5 × M (cid:12) and consider two other values of 25 × M (cid:12) and a 10 M (cid:12) BH, to cover the interesting range of masses potentially observable in NGO.In the case of ET and aLIGO I have considered three IMBHs of mass 200 M (cid:12) , 600 M (cid:12) and1000 M (cid:12) . It is believed that BHs in this range are situated in the centers of many globularclusters. However, their existence is being questioned, the evidence is thought to be strong butcircumstantial [24, 18, 25, 26, 27].Concerning the mass ratios, only unequal mass binaries are presented, except for one casein aLIGO. We do not consider the equal mass case as such systems are not as likely to occurin nature as slightly asymmetric ones. For NGO we examine the mass ratios of 2, 10 and 20,whilst lower mass ratios of 2, 5 and 10 are considered for ET. We, therefore, have in total, 9different sets of simulations for each detector. Let me emphasize here that there is a possibilitythat massive BH binaries in the early universe ( z ≤
10) will have a mass ratio significantlylarger than one. For instance, in [28] it is suggested that low-redshift massive BH mergers occurpredominantly with a mass ratio of 10 or higher.
3. Signal detectability and Parameter estimation
I will discuss the total signal to noise ratio, as well as the fractional errors in estimating thefollowing 5 parameters: { M, j, q, D L , ι } . These errors are actually the quantities, σ λ = √ C λλ ,which are computed from the covariance matrix, C (cid:96)m [29, 30, 31, 32].The results from all the distinct arrangements of the system (see Section 2.2) are presentedvia cumulative frequency distribution plots. That is to say, the different system configurationsare classified according to the value they render in the error of the observable quantity inquestion. The proportion of a number of occurrences in a small width of values should, to agood approximation, equal the probability that a randomly placed observer will measure thatquantity to take this range of values. total SNR q = 2q = 5q = 10 fractional error in D L q = 2q = 5q = 10 fractional error in M q = 2q = 5q = 10 fractional error in j q = 2q = 5q = 10 fractional error in q q = 2q = 5q = 10 fractional error in ι q = 2q = 5q = 10 Figure 6:
Similar graphs as those in Figs. 3 - 5, although for the ET detector, concerning a IMBH of200 M (cid:12) . The luminosity distance of the BH is now picked closer, at 1 Gpc ( z (cid:39) . The sensitivity curve that we use is what is thought to be possible for NGO and it is contrastedwith other LISA-like sensitivity curves in Fig. 1. It refers to a 4-link interferometer, comprised ofone mother and two daughter spacecraft, having armlengths of 10 m and trailing a few degreesbehind the Earth, in heliocentric orbit.The main noise contributions are the acceleration noise, the shot noise, as well as some othermeasurement noise. These are respectively: S acc,m ( f ) = 1 . × − (1 + 10 − f ) f − m Hz − ,S SN,m = 5 . × − m Hz − ,S OMN,m = 6 . × − m Hz − . The formula for the amplitude sensitivity curve is, (cid:113) S h ( f ) = √ √ T ( f ) √ S acc + S SN + S OMN
L Hz − / , (9)while the transfer function is T ( f ) = (cid:118)(cid:117)(cid:117)(cid:116) (cid:32) f . (cid:0) c L (cid:1) (cid:33) . (10)with L = 10 m and c = 299 , ,
458 metres per second.
Our results for NGO are plotted in Figs. 3-5. For the 5 × M (cid:12) , 25 × M (cid:12) and 10 M (cid:12) BHsthe SNR curves (top left subplots of Figs. 3-5) peak at around 300, 600 and 1700 respectively. see https://lisa-light.aei.mpg.de/bin/view/DetectorConfigurations total SNR q = 2q = 5q = 10 fractional error in D L q = 2q = 5q = 10 fractional error in M q = 2q = 5q = 10 fractional error in j q = 2q = 5q = 10 fractional error in q q = 2q = 5q = 10 fractional error in ι q = 2q = 5q = 10 Figure 7:
As in Fig. 6, but for a 600 M (cid:12) black hole. Relative frequency distributions were chosen for their plotting, as they portray clearly where themaximum occurs, as well as how they fall off. The higher mass ratio SNR curves have a similaroutline. The fact that all of the curves resemble log-normal distributions, with steep risings andlong tails, is mostly attributed to the fact that only a small fraction of the configurations, thoseclose to the optimal orientation of the binary, will yield the highest SNRs.Regarding the rest of the plots in Figs. 3-5, the general trend is that the mass and spin havecomparable, low fractional errors, whilst the other group of parameters, namely the luminositydistance, mass ratio and inclination angle, yield an order of magnitude higher error values.This is not surprising, considering that the mass and spin have a direct effect on the modes’frequencies, ω (cid:96)mn and time-constants, τ (cid:96)mn , quantities that determine to first order the shapeof the ringdown waveform.Quoting a few numbers for a progenitor of mass ratio 2, the probability to get a binary mergerevent that will yield a mass error less than 1% is correspondingly 58%, 55% and 10%, as we gofrom the lower mass to the higher mass value. The lowest mass value actually gives the bestresults, whereas the signal’s main power content takes place near the lowest sensitivity area ofthe NGO detector. The spin magnitude exhibits a similar trend with the probabilities to fallbelow a 10% fractional error being 96%, 90% and 67% respectively.For the second group of parameters, I again quote how likely it is to do better than 10%.For the luminosity distance, the values are 38%, 41% and 20%, corresponding to the BH masses5 × M (cid:12) , 25 × M (cid:12) and 10 M (cid:12) . The mass ratio is harder to determine accurately, withthe likelihoods being 32%, 15% and only 2% respectively. Lastly, there is a 40%, 37% and 13%likelihood of achieving an accuracy better than 10% in the inclination angle. ET’s very low sensitivity curve accounts for impressive results in the mass range ∼ M (cid:12) to ∼ M (cid:12) and for mass ratios between 1 and 5. I consider the sensitivity curve designatedET-B [33], whose noise power spectral density is given by S h ( f ) = 10 − h n ( f ) Hz − , with: h n ( f ) = 2 . × − x − . + 0 . x − . + 1 . x − . + 0 . x . , (11)here x = f /
100 Hz . As for advanced LIGO, the noise spectral density is S h ( f ) = 10 − (cid:34) − f − . + 0 . x − . + 123 .
35 1 − . x + 0 . x . x (cid:35) Hz − , (12)where x = f /
215 Hz . The results obtained for ET and advanced LIGO are plotted in Figs. 6-9. We fix the luminositydistance of the BH to be 1 Gpc. For the lowest mass considered (a 200 M (cid:12) BH) although theSNR could be pretty high (in the range 30-200), errors in the estimation of parameters are poor(see Fig. 6). { D L , q, ι } have 50% probability to be measured to an accuracy of ∼ { M, j } are 90% and 60% likely to be below 10%. For the higher mass ratios the results,as expected, are worse.The results for 600 M (cid:12) and 1000 M (cid:12) BHs are shown in Figs. 7 and 8. The parameterestimation accuracies for these systems observed with ET is almost as good as that for a SMBHwith NGO. Referring to the heaviest BH and mass ratio in the range 2-5, it is 100%-99% and96%-53% likely to acquire errors below 10% for the BH mass and spin. For the { D L , q, ι } , theefficiencies are correspondingly 50%-15%, 23%-0% and 60%-17%. The results are similar for the600 M (cid:12) case (see Fig. 7).Fig. 9 shows two examples for aLIGO: a BH resulting from an equal mass binary and one forwhich the mass ratio is 5. Most configurations give ringdown SNR values in the range 10 to 30,not large enough for a good estimation of parameters. This translates to a 30% likelihood fora fair measurement of 10% accuracy in the mass and mass ratio, while the luminosity distance,spin and inclination angles are all measured to accuracies far worse than 10%.
4. Astrophysical implications
The presence of SMBHs in the centers of massive galaxies seems to be a well established fact.Detecting their gravitational wave signals can give additional clues on their spatial and massdistribution, as well as help discriminate among the different scenarios of their formation andgrowth. This could be done, for instance, by measuring the BH mass function as a function ofred-shift. Additionally, determining the mass ratios [5] of these early universe merger events willbe an important piece of information in selecting out the current models on SMBH formation.Note that, in NGO, BHs of mass higher than 10 − M (cid:12) are visible almost entirely due to theringdown signal that they emit rather than the inspiral signal.Several studies have been realized in the field of predicting the coalescence rates of SMBHs[34, 35, 36, 37, 38, 39, 40, 41]. Let us admit an event rate of ∼ yr − at z (cid:39)
1, which ascertainsa scenario of most efficient BH coalescence [42]. Then, assuming that the BH masses are botharound 5 × M (cid:12) , Figs. 3 and 4, there is a good chance that in 6 of the events, the BH masswill be measured more accurately than 1% and that in 9 of the events the BH spin magnitudewill have an accuracy better than 10%. As for the parameters { D L , q, ι } , in approximately 3-4of the events they will feature errors lower than 10%, while in 7 of the events, the errors will liebelow 30% for D L and below 20% for q and ι . This fit was provided by C. Capano, Syracuse University and is tuned for detecting binary neutron stars. Unless the inspiral phase is inside NGO’s band, it will not be possible to determine the sky position of thesignal. The only hope in this case would be the existence of an electromagnetic counterpart to the merger event. total SNR q = 2q = 5q = 10 fractional error in D L q = 2q = 5q = 10 fractional error in M q = 2q = 5q = 10 fractional error in j q = 2q = 5q = 10 fractional error in q q = 2q = 5q = 10 fractional error in ι q = 2q = 5q = 10 Figure 8:
As in Figs. 6 and 7, but for a 1000 M (cid:12) BH, a sweet spot in ET. The results are very encouragingfor this size of BH and distance, with a very high probability to get errors lower than 10% in most of theparameters at mass ratios 2-5.
Let me clarify that ringdown signals in NGO, will be detectable out to about z (cid:39)
5, but thedistance of z = 1 was chosen to make the quote on the errors. If we take the distance at z = 3,an optimistic rate of merger events will be of the order of 100 [42]. If we assume that they allinvolve BHs of masses (cid:39) × M (cid:12) , then in around 90 of them the error in the mass will bebelow 10% and in 60 of them the spin magnitude error will fall below the 10% threshold. Forthe rest of the parameters, about 50 of the observed events should yield measurements betterthan 50% accuracy. Although this looks poor at the outset, it should suffice for a statistical testof different models of BH formation and growth. The existence of intermediate-mass black holes remains uncertain, as is their mass distribution.Colliding globular clusters in interacting galaxies could be a mechanism to obtain a compactbinary IMBH system [43, 3]. Another possibility could be the formation of a binary IMBHinside a young dense stellar cluster, especially when the fraction of binary stars is adequatelyhigh [26].Estimates of IMBH-IMBH coalescence rates can be found in [44, 26, 3, 45]. A relativelyoptimistic rate is R opt = 0.007 GC − Gyr − , where it has been assumed that 10% of star clustersare sufficiently massive and have a sufficient stellar binary fraction to form an IMBH-IMBHbinary once in their lifetime, taken at 13.8 Gyr. The maximum possible rate would come fromassuming that all of the star clusters satisfied the above conditions. The corresponding ratevalue is then R max = 0.07 GC − Gyr − .If the number of relatively luminous galaxies within a distance of 1 Gpc is approximately5 . × , [46] and the number of young dense stellar clusters per such galaxy is of the order of100, then the optimistic estimate gives 0.037 events per year, while the maximum rate would be0.37 events per year. An event within this distance will in some likelihood, involve IMBHs withmasses between ∼ × M (cid:12) and ∼ M (cid:12) which means a relatively fair chance for errors tobe low for several of the parameters using Einstein Telescope (see Figs. 7, 8). An additional simulation at z = 3, of a 5 × M (cid:12) BH, with a progenitor mass ratio of 2 was performed. total SNR q = 1q = 5 fractional error in D L q = 1q = 5 fractional error in M q = 1q = 5 fractional error in j q = 1q = 5 fractional error in q q = 1q = 5 fractional error in ι q = 1q = 5 Figure 9:
Frequency distributions involving the most optimistic scenario for advanced LIGO: a 1000 M (cid:12) BH which is the merger of an equal mass binary, that is q = 1. The BH is again situated at 1 Gpc. Amass ratio of 5 is shown as well for contrasting.
5. Conclusions
The present study constitutes a sensible and realistic approach to the subject of parameterestimation from a multimodal ringdown signal, inasmuch as it is supposed to be emitted froma merged binary in a generic configuration. Parameters such as the inclination angle ι , alongwith the sky location θ and ϕ and the signal polarization ψ have an effect on the observedquasi-normal mode spectrum. Their impact on the detectability and parameter estimation hasbeen assessed, by performing a large sample of Fisher-matrix analysis simulations, allowing fora simple statistical analysis of the results.I am quoting frequency distributions for the errors at the representative distances of z (cid:39) z (cid:39) . × M (cid:12) BH at z (cid:39)
3. The results are quite satisfactory in determining the mass and spin,especially for the low mass ratios from 1 to ∼
5, where in typically 90% of the cases theirerrors fall below 10%. The effectiveness in measuring the luminosity distance, mass ratio andinclination angle is almost an order of magnitude worse.The effects of these results on, as much as possible, realistic event rates in NGO and ET arediscussed. The likelihood to have a waveform parameter measured to an accuracy of a certainthreshold translates to the same proportion of observed events featuring error values below thatthreshold. As an example, if supermassive BHs within a luminosity distance of z = 1 coalesceat a rate of (cid:39)
10, then NGO could act as a SMBH dynamics probe, as almost all of these eventswill yield very low errors 1% −
10% in the BH mass and spin, while in half of the events theerrors in the luminosity distance, mass ratio and inclination angle will be of the order of 20%.
6. Acknowledgements
I am grateful to B.S. Sathyaprakash for his supervision and exceptional advice throughout thisstudy. I would like to thank Mark Hannam for discussions and him and Sascha Husa for makingavailable numerical relativity simulations used in this study. I am thankful to Mr AntoinePetiteau for clarifications on the sensitivity response of the European mission of LISA. eferences [1] Nandra K, Aird J, Alexander D, Ballantyne D, Barcons X et al.
Preprint )[2] Somerville R S 2008 * Brief entry * (
Preprint )[3] Amaro-Seoane P and Santamaria L 2010
Astrophys.J.
Preprint )[4] Berti E, Cardoso V and Will C M 2006
Phys. Rev. D Preprint gr-qc/0512160 )[5] Kamaretsos I, Hannam M, Husa S and Sathyaprakash B S 2012
Phys. Rev. D (2) 024018 URL http://link.aps.org/doi/10.1103/PhysRevD.85.024018 [6] Berti E, Cardoso J, Cardoso V and Cavagli´a M 2007 Phys. Rev.
D76
Preprint )[7] Nelemans G, Yungelson L and Portegies Zwart S 2004
Mon.Not.Roy.Astron.Soc.
Phys.Rev.
D81
Preprint )[9] Berti E, Cardoso V and Starinets A O 2009
Class. Quant. Grav. Preprint )[10] Kokkotas K D and Schmidt B G 1999
Living Rev.Rel. Preprint gr-qc/9909058 )[11] Baker J G, Boggs W D, Centrella J, Kelly B J, McWilliams S T and van Meter J R 2008
Phys.Rev.
D78
Preprint )[12] Br¨ugmann B et al.
Phys. Rev.
D77
Preprint gr-qc/0610128 )[13] Husa S, Gonz´alez J A, Hannam M, Br¨ugmann B and Sperhake U 2008
Class. Quant. Grav. Preprint )[14] Berti E, Cardoso V, Gonzalez J, Sperhake U, Hannam M, Husa S and Brugmann B 2007
Phys. Rev.
D76
Preprint gr-qc/0703053v2 )[15] Hannam M, Husa S, Sperhake U, Bruegmann B and Gonzalez J A 2008
Phys.Rev.
D77
Preprint )[16] Hannam M, Husa S, Ohme F, Muller D and Bruegmann B 2010
Phys.Rev.
D82
Preprint )[17] Valluri M, Ferrarese L, Merritt D and Joseph C L 2005
Astrophys. J.
Preprint astro-ph/0502493 )[18] Safonova M and Shastri P 2010
Astrophys. Space Sci.
Preprint )[19] Beifiori A, Courteau S, Corsini E M and Zhu Y 2011 (
Preprint )[20] Li Y R, Ho L C and Wang J M 2011
Astrophys. J.
33 (
Preprint )[21] Rafiee A and Hall P B 2011
Astrophys. J. Suppl.
42 (
Preprint )[22] Burkert A and Tremaine S 2010
Astrophys. J.
Preprint )[23] Harris G L H and Harris W E 2010 (
Preprint )[24] Miller M C 2009
Class. Quant. Grav. Preprint )[25] Noyola E and Baumgardt H 2011 (
Preprint )[26] Fregeau J M, Larson S L, Miller M C, O’Shaughnessy R W and Rasio F A 2006
Astrophys. J.
L135–L138(
Preprint astro-ph/0605732 )[27] Bash F N, Gebhardt K, Goss W M and Bout P A V 2008
The Astronomical Journal
182 URL http://stacks.iop.org/1538-3881/135/i=1/a=182 [28] Volonteri M, Madau P, Quataert E and Rees M J 2005
Astrophys. J.
Preprint astro-ph/0410342 )[29] Wainstein L A and Zubakov V D 1962
Extraction of Signals from Noise (Englewood Cliffs: Prentice-Hall)[30] Finn L S 1992
Phys. Rev. D Phys. Rev. D , 1860 (1996) ( Preprint gr-qc/9508011 )[32] Sathyaprakash B S and Schutz B F 2009
Living Rev. Rel. Preprint arXiv:0903.0338 )[33] Hild S, Chelkowski S, Freise A, Franc J, Morgado N et al.
Class.Quant.Grav. Preprint )[34] Filloux C, Pacheco J A d F, Durier F and de Araujo J C N 2011 (
Preprint )[35] Berti E 2006
Classical Quantum Gravity S785 (
Preprint astro-ph/0602470 )[36] Haehnelt M G 2003
Classical and Quantum Gravity S31 URL http://stacks.iop.org/0264-9381/20/i=10/a=304 [37] Rhook K J and Wyithe J S B 2005
Mon. Not. Roy. Astron. Soc.
Preprint astro-ph/0503210 )[38] Erickcek A L, Kamionkowski M and Benson A J 2006
Mon. Not. Roy. Astron. Soc.
Mon. Not. Roy. Astron. Soc.
Astrophys. J.
29 (
Preprint )[41] de Freitas Pacheco J A, Filloux C and Regimbau T 2006
Phys. Rev.
D74
Astrophys.J.
Preprint astro-ph/0101196 )[43] Amaro-Seoane P and Freitag M 2006
Astrophys. J.
L53–L56 (
Preprint astro-ph/0610478 )[44] Abadie J et al. (LIGO Scientific Collaboration) 2010 (
Preprint arXiv:1003.2480 )[45] Mandel I, Gair J R and Miller M C 2009 (
Preprint )[46] White D J, Daw E and Dhillon V 2011
Class.Quant.Grav. Preprint1103.0695