arXiv:astro-ph/0108483 v1 30 Aug 2001 - iucaa

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arXiv:astro-ph/0108483 v1 30 Aug 2001 - iucaa

Mon. Not. R. Astron. Soc. 000, 1–12 (2001) Printed 5 October 2002 (MN LATEX style file v2.2)Untangling the merger history of massive black holes withLISAarXiv:astro-ph/0108483 v1 30 Aug 2001Scott A. Hughes ⋆Institute for Theoretical Physics, University of California, Santa Barbara CA 93106, USA5 October 20021 INTRODUCTIONEvidence from quasars at redshifts z > 5 makes it clear thatmassive black holes have existed since the universe’s youth(Stern et al. 2000; Zheng et al. 2000; Fan et al. 2000). Inorder to be powering quasars at z ∼ 5, seed black holes withmasses of 10 5 M ⊙ or so had to exist at z ∼ 10 (Gnedin 2001).In hierarchical formation scenarios, these seed holes are producedby the infall and merger of baryonic and dark matterclumps, and are then associated with dark matter halos.As the halos interact and merge to form larger structuresand eventually galaxies, the black holes that they containcan merge as well [although there may be many more haloand structure mergers than there are black hole mergers; seeMilosavljević & Merritt (2001)]. This creates a populationof coalescing massive black holes at cosmological distances.These coalescing black holes will be strong sources ofgravitational radiation. It has been known for some timethat such coalescences will be detectable by the space-basedgravitational-wave antenna LISA (Thorne 1995; Haehnelt1998; Flanagan & Hughes 1998). This suggests that LISAcould be used to map the distribution of black holes as theuniverse evolves, tracing the growth and evolution of themassive black hole population. Such observations could shedlight on the merger of the black hole’s host structures, pro-⋆ E-mail: hughes@itp.ucsb.eduABSTRACTBinary black hole coalescences emit gravitational waves that will be measurable by thespace-based detector LISA to large redshifts. This suggests that LISA may be able toobserve black holes grow and evolve as the universe evolves, mapping the distributionof black hole masses as a function of redshift. An immediate difficulty with this idea isthat LISA measures certain redshifted combinations of masses with good accuracy: if asystem has some mass parameter m, then LISA measures (1+z)m. This mass-redshiftdegeneracy makes it difficult to follow the mass evolution. In many cases, LISA willalso measure the luminosity distance D of a coalescence accurately. Since cosmologicalparameters (particularly the mean density, the cosmological constant, and the Hubbleconstant) are now known with moderate precision, we can obtain z from D and breakthe degeneracy. This makes it possible to untangle the mass and redshift and to studythe mass and merger history of black holes. Mapping the black hole mass distributioncould open a window onto an early epoch of structure formation.Key words: gravitational waves – gravitation – black hole physics – cosmology:miscellaneousviding information about the development of structure inthe universe (Menou, Haiman, & Narayanan 2001).By tracking the gravitational-wave phase as the blackholes inspiral and merge, LISA accurately measures certainmass parameters of the binary system. Unfortunately, themasses that LISA will measure are redshifted. Modulo overallamplitude, the gravitational waves that we measure froma binary with masses [m 1, m 2] at redshift z are those of alocal binary with masses [(1 + z)m 1,(1 + z)m 2]. This is becauseany quantity m with the dimension of mass entersthe binary’s orbit evolution as a timescale Gm/c 3 , and thistimescale is redshifted. Thus, the redshift and mass of a binaryblack hole system are “tangled”, greatly complicatingthe mapping of the universe’s black hole mass distribution.The degeneracy between mass and redshift must be brokenor else we cannot tell the difference between a system withmass 5×10 4 M ⊙ at z = 1 and a system with mass 2×10 4 M ⊙at z = 4.One way to break this degeneracy would be to associatean electromagnetic event with the coalescence. If that eventhad clear emission or absorption lines, one could directlyread z and thus learn the system’s mass. In many cases,LISA will be able to measure the luminosity distance D of asource with good precision. A measurement that gives boththe luminosity distance and redshift could be used to provideadditional information about parameters such as the Hub-


2 Scott A. Hughesble constant and the cosmological constant (Cutler 1998;Marković 1993; Wang & Taylor 1997).It is likely that many more mergers will be measuredwith gravitational waves than are seen electromagnetically.Even if there is an electromagnetic signature to the coalescence,it might be missed because of effects such as beamingor absorption. Fortunately, cosmological parameters are nowknown well enough that we can, with good accuracy, invertthe luminosity distance as a function of redshift, D(z), tofind the redshift as a function of luminosity distance, z(D).Knowledge of the cosmological parameters allows us to breakthe mass-redshift degeneracy using gravitational-wave measurementsalone, making it possible to untangle the mergersand map the population of massive black holes.In this paper, we examine how well the mass-redshiftdegeneracy can be broken in practice. Our eventual goal isto understand how well LISA can measure the distributionof black hole masses as a function of redshift. Thus we willfocus on how well LISA can measure the mass of binaryblack hole systems at cosmological distances, and on howwell the luminosity distance and redshift can be determined.It is worth noting at this point that the masses andluminosity distance influence the gravitational waveform invery different ways. The masses impact the binary’s dynamics,particularly the orbital frequency and inspiral rate, andthus determine the waveform’s phase evolution. One measuresthe masses by tracking the phase and fitting to a template.If the template’s phase maintains coherence with thesignal for a large number of wave cycles, the phase — andhence the masses — can be measured quite accurately. Theluminosity distance, by contrast, simply sets the overall amplitudeof the wave.Several mass parameters can be measured with gravitationalwaves. The early inspiral portion of the waveform(when the holes are well-separated, distinct bodies) dependsmost strongly on the “chirp mass”, M = (m 1m 2) 3/5 /(m 1 +m 2) 1/5 = µ 3/5 M 2/5 (where µ is the binary’s reduced mass,and M its total mass). Measuring the inspiral measures M.The inspiral also depends (although rather more weakly) onthe binary’s reduced mass, µ = m 1m 2/(m 1 + m 2). The reducedmass is not measured as precisely as the chirp mass,but is often determined well enough to be useful. The lastwaves to be measured from the system’s coalescence will bethe “ringdown” waves, emitted as the merged system relaxesto a quiescent Kerr black hole. These waves depend only onthe final state’s mass and spin, and thus directly provide themass M f of the final merged remnant.As mentioned above, the major effect of the luminositydistance is to set the amplitude of the two gravitationalwavepolarizations, h + and h ×. These amplitudes also dependupon the chirp mass, the binary’s sky position, andits orientation. The chirp mass is determined very preciselywhen the wave’s phase is measured. All that remains to getthe luminosity distance is to measure the sky position andorientation. These parameters are extracted by taking advantageof LISA’s orbital motion: as the spacecraft orbitthe sun, the gravitational waveform is modulated due to thechanging aspect of the source with respect to the instrument,and due to the detector’s changing antenna pattern. Thesemodulations depend upon, and thus encode, the sky positionand binary orientation. This makes it possible to measurethe luminosity distance to the source, in some cases with aprecision δD/D ∼ 1%.The luminosity distance D(z) is then inverted to givethe source’s redshift, z(D). If the relevant cosmological parameters(fraction of closure density in matter Ω M, cosmologicalconstant Λ, and Hubble constant H 0) were knownwith perfect accuracy, then the redshift would be determinedabout as well as the luminosity distance: δz/z ≃ δD/D. Infact, each of the cosmological parameters is presently knownto about 10% (Wang, Tegmark, & Zaldarriaga 2001), so itturns out that the redshift can at best be measured withabout 15% precision. As future cosmological measurementspin down the universe’s geometry more precisely, the inversionwill become more precise. By the time LISA flies (c.2010) it is likely that in many cases cosmological uncertaintieswill not be the major source of error in determining theredshift of a binary black hole coalescence.To estimate how well LISA will be able to untangle binaryblack hole mergers, we choose a range of system massesand randomly populate the sky with binaries having thesemasses, located at redshifts 1 z 9. For simplicity, we focuson equal mass binaries, m 1 = m 2. We also randomly distributethe binaries’ merger time within an assumed 3 yearLISA mission. We then use a maximum likelihood measurementformalism (Finn 1992; Finn & Chernoff 1993; Cutler& Flanagan 1994; Poisson & Will 1995) to estimate the precisionwith which each source’s redshifted masses and redshiftcan be measured. For each set of redshift and systemmasses, we produce the distribution of errors for the relevantparameters — the redshift z, and the redshifted chirp mass(1+z)M, reduced mass (1+z)µ, and final mass (1+z)M f .The mass-redshift degeneracy is thus broken for these particularmass combinations.The remainder of this paper is organized as follows: inSec. 2, we briefly describe how LISA measures gravitationalwaves [following the discussion of Cutler (1998)], and summarizethe formalism used to estimate parameter measurementaccuracies. Section 3 describes our models for the gravitationalwaveform, the universe’s geometry, and the detectornoise. Section 4 presents our main results, describing theaccuracy with which the redshift and mass parameters canbe measured by LISA. Using present uncertainties for ourknowledge of the cosmological parameters, we find that inmany cases LISA can measure the redshift of a system with15 − 30% accuracy. The mass parameters — particularlythe chirp mass — are often measured much more accuratelythan this, suggesting that the mass-redshift degeneracy willbe broken with 15−30% error. It should be strongly emphasizedat this point that the 15% lower limit on this error isdominated by present uncertainties in our knowledge of cosmologicalparameters. If instead luminosity distance errorsdominate, it could be reduced to 5% or less.At each redshift, there is a range of system masses forwhich the binary’s parameters are best determined. That isthe range of masses in which the signal lies in LISA’s mostsensitive frequency band. As we look out to larger redshifts,this “best mass” becomes smaller, since the cosmologicalredshift moves the relatively high frequency signal of thesmaller binary into LISA’s band. This is probably a veryuseful trend, since younger (high z) black holes will tend tobe smaller than older (low z) holes that have had more timeto grow. Section 5 summarizes our major results; we also


Untangling the merger history of massive black holes 3discuss work that could improve this analysis. Throughout,we set the speed of light c and Newton’s gravitational constantG to unity. A useful conversion factor in these units is1 M ⊙ = 4.92×10 −6 seconds. Any mass parameter m writtenwith a z subscript, m z, denotes (1 + z)m.2 GRAVITATIONAL-WAVE MEASUREMENTAND PARAMETER EXTRACTIONThe LISA gravitational-wave antenna (Danzmann et al.1998) consists of three spacecraft arranged in an equilateraltriangle orbiting the sun. The arms of the triangle areapproximately L = 5 ×10 6 km in length, and the triangle isinclined at an angle of 60 ◦ to the ecliptic. The entire triangularconfiguration spins as the antenna orbits the sun, rotatingonce during a single orbit (one year). A gravitationalwave interacting with the configuration causes the length ofthe three arms to oscillate.The gravitational-wave signal is reconstructed from thetime-varying armlength data, (δL 1, δL 2, δL 3). As describedby Cutler (1998), LISA can be regarded as two gravitationalwavedetectors. The armlength data can be combined to produceoutput equivalent to two detectors with 90 ◦ arms; theseequivalent detectors are rotated 45 ◦ with respect to one another.This “equivalent detector” viewpoint works well whenthe radiation’s wavelength is greater than the armlength ofthe detector, but is not so accurate when λ GW∼ < L — highfrequency structure in the LISA sensitivity (Larson, Hiscock& Hellings 2000) is ignored. We will use the equivalent detectorpicture throughout this analysis, accordingly introducingsome error 1 for f GW∼> 0.06 Hz. The sensitive band of thedetector is taken to run from 10 −4 Hz up to about 1 Hz.Following Cutler’s notation, we will label the equivalentdetectors “I” and “II”, so the data streams are s I(t)and s II(t). Each data stream consists of noise and (possibly)an astrophysical signal. We assume that the noises arestationary, Gaussian random processes with the same rmsvalue, 〈n I(t) 2 〉 = 〈n II(t) 2 〉, and that they are uncorrelated,〈n I(t)n II(t)〉 = 0. The two polarizations of the astrophysicalgravitational-wave, h +(t) and h ×(t), enter the datastreamsweighted by detector response functions, F + I,II and F × I,II :s I,II(t) =√32[ F+I,II(t)h +(t) + F × I,II(t)h ×(t) ] + n I,II(t) .(1)(The factor √ 3/2 enters when the outputs of the real interferometerswith arms at 60 ◦ are converted to the datastreamof the effective interferometers, with 90 ◦ arms.) Theresponse functions F +,×I,II(t) depend on the source’s orientationand position on the sky relative to the detector. Becauseany astrophysical signal will be at fixed position in thebarycenter frame of the solar system, the detector’s motioninduces modulations in the signal’s phase and amplitude.The response functions are written as functions of time toillustrate this orbital-motion-induced modulation. See Cutler(1998) for further discussion and details.The gravitational waveform from a particular black hole1 We expect this error to be unimportant, since the luminositydistance and redshift are determined from the inspiral signal,which largely accumulates at lower frequencies.binary depends upon parameters θ which describe the systemand its evolution. (The symbol θ stands for a vectorwhose components θ a are the distance to the system, themasses of the black holes, their spins, the binary’s positionon the sky, etc.) Following Finn (1992), the probability thata signal with parameters θ is present in the data streamss I(t) and s II(t) is given byp(θ|s I, s II) = p (0) (θ)exp { − (H I(θ) − s I|H I(θ) − s I) /2− (H II(θ) − s II|H II(θ) − s II) /2 } . (2)In this equation, the inner product (a|b) is(a|b) = 4Re∫ ∞0df ã∗ (f)˜b(f)S h (f), (3)where S h (f) is the spectral density of noise in the detector(discussed further in Sec. 3), the superscript ∗ denotescomplex conjugation, and ã(f) is the Fourier transform ofa(t):ã(f) =∫ ∞dte 2πift a(t) . (4)−∞The functions H I,II(t; θ) are the gravitational waveformsmeasured in detectors I and II, including the motion-inducedmodulation:H I,II(t;θ) =√32[ F+I,II(t)h +(t; θ) + F × I,II(t)h ×(t; θ) ] . (5)For further discussion, see Cutler (1998). The functionp (0) (θ) is the prior probability distribution for the parametersθ. It encapsulates all information about the system’sparameters known before measurement [e.g., for each blackhole, the spin magnitude | ⃗ S| must be less than or equal tom 2 ; see Poisson & Will (1995)].The source parameters are estimated by finding the parametersˆθ that maximize p(θ|s I, s II); these are the “mostlikely” parameters. Operationally, this is done using matchedfiltering: a bank of model waveforms (“templates”) thatcover a range of parameters is assembled beforehand, andthe datastreams (s I, s II) are cross-correlated with filtersconstructed from the templates. The parameters of thebinary are estimated as those of the template with thelargest cross-correlation (i.e., the template that “matches”the data). See Cutler & Flanagan (1994), Balasubrahmanianet al. (1996), and Owen (1996) for details. The templatewhich maximizes the probability distribution (2) alsogives the largest value for the signal-to-noise ratio (SNR) ρ:ρ = √ ρ 2 I + ρ2 II ,ρ 2 I,II = (H I,II(θ)|H I,II(θ)) . (6)We next estimate the errors in the measured parametersby expanding the probability distribution functionp(θ|s I, s II) about θ = ˆθ. To do so, we expand the innerproduct: denoting ξ I,II(θ) = (H I,II(θ) − s I,II|H I,II(θ) −s I,II) (Poisson & Will 1995), we haveξ I,II(θ) = ξ I,II(ˆθ) + 1 2 ∂a∂ bξ I,II(ˆθ)δθ a δθ b . (7)Here, ∂ a means partial differentiation with respect to theparameter θ a . In the limit of large SNR (Finn 1992),∂ a∂ b ξ I,II/2 = (∂ aH I,II|∂ b H I,II), so the probability distributionbecomes


4 Scott A. Hughesp(θ|s I, s II) = p 0 (θ) exp[− 1 ]2 Γ abδθ a δθ b , (8)whereΓ ab ≡ (∂ aH I|∂ b H I) + (∂ aH II|∂ b H II) (9)is the Fisher information matrix. The variance-covariancematrix Σ ab is the inverse of this:Σ ab = 〈δθ a δθ b 〉 = (Γ −1 ) ab . (10)The angle brackets denote an average over the probabilitydistribution. Thus, the diagonal components, Σ aa =〈(δθ a ) 2 〉, are the expected squared errors in the parametersθ a . Off-diagonal components describe correlations betweenparameters. It is useful to introduce the correlation coefficientc ab =Σ ab√ . (11)ΣaaΣbb This coefficient lies between −1 and 1.Equation (10) will be the workhorse of this analysis. Itwill be used, along with models for the binary black holecoalescence waveform and the LISA noise spectrum, in orderto estimate how well LISA will be able to measure binaryparameters, particularly the luminosity distance andthe system’s masses.3 MODEL AND ASSUMPTIONSFollowing Flanagan & Hughes (1998), we will describe thecoalescence of the black hole binary in terms of three epochs:a slow inspiral, in which the black holes spiral towards oneanother driven by adiabatic gravitational-wave emission; afar more violent and dynamical merger, in which the individualblack holes plunge towards one another and mergeinto a single body; and a final ringdown, when the mergedremnant becomes well-described as a distorted Kerr blackhole.This characterization is rather crude. In particular, theinterface between “inspiral” and “merger” is not very clearcut when the members of the binary are of comparable mass.For our purposes, this characterization is useful because parameterizedwaveforms exist that describe the inspiral andringdown waves, and can thus can be used to study how wellwe will be able to determine the masses and redshifts of binaryblack hole coalescences. At present, the merger regimecannot be included, since its characteristics have not yetbeen modeled for astrophysically interesting coalescences.Numerical (Baker et al. 2001; Pfeiffer, Teukolsky, & Cook2000; Brandt et al. 2000; Grandclément, Gourgoulhon, &Bonazzola 2001) and analytic (Buonanno & Damour 2000;Damour, Jaranowski, & Schaefer 2000; Damour 2001) workin this field is very active, and hopefully will provide usefulinsight into the nature of the merger and the transitionfrom inspiral to merger by the time that LISA begins tomake observations.3.1 Inspiral waveformThe inspiral waveform used here is based on the post-Newtonian expansion of general relativity [see, e.g.,Blanchet, Iyer, Will, & Wiseman (1996) and referencestherein]. We will use waveforms computed to second-post-Newtonian (2PN) order [i.e., (v 2 /c 2 ) 2 beyond the leadingquadrupole result]. A very useful summary of the 2PN waveformis given in Poisson & Will (1995). It depends on sevenparameters: the luminosity distance to the source D(z); acoalescence time t c; a coalescence phase φ c; the redshiftedchirp mass M z; the redshifted reduced mass µ z; a spin-orbitparameter β; and a spin-spin parameter σ.The coalescence time and phase are essentially constantsof integration, specifying the time and orbital phase ofa binary at the end of inspiral. They are not physically interesting,but nonetheless must be fit for in a measurement, andthus influence the accuracy with which other parameters aremeasured. The chirp mass M = (m 1m 2) 3/5 /(m 1 + m 2) 1/5is the combination of masses that most strongly influencesthe gravitational-wave driven inspiral. As we shall see, thisparameter tends to be measured to very high precision. Thereduced mass µ = m 1m 2/(m 1+m 2) gives the next most importantcontribution to the inspiral rate. In principle, if onemeasures M and µ, one can solve for the individual massesof the holes in the binary [though it may turn out that µ isnot determined well enough for this to work well in practice(Cutler & Flanagan 1994)]. The spin-orbit parameter β describescouplings between the spins of the black holes andthe orbital angular momentum vector. It is given byβ = 1 2∑ [ 113(mi/M) 2 + 75µ/M ] ˆL · Si/m ⃗ 2 i , (12)12i=1where M = m 1 + m 2, and ˆL is the unit vector along theorbital angular momentum. The spin-spin parameter σ likewisedescribes couplings between the two spins, and is givenbyµ [σ =721(ˆL48M(m 2 1 m2 2 ) · ⃗S 1)(ˆL · ⃗S 2) − 247(⃗S 1 · ⃗S ] 2) . (13)Spin-spin and spin-orbit interactions lead to complicatedprecessional motions in the binary’s orbit, which inturn modulate the waveform (Apostolatos et al. 1994).These modulations may provide additional informationabout the binary’s spins and thereby reduce the effect ofcorrelations between various parameters. However, includingthe effect of these modulations is rather difficult. Weneglect the precession-induced modulation of the waveformin this analysis.In the barycenter frame of reference, the two polarizationsof the gravitational waveform described by these parametersis written (in the frequency domain)˜h +(f) =AD(z) [1 + (ˆL · ˆn) 2 ]f −7/6 exp[iΨ(f)] ,˜h ×(f) = − 2AD(z) (ˆL · ˆn)f −7/6 exp[iΨ(f)] . (14)Here, ˆn is the direction vector pointing from the center ofthe barycenter frame (i.e., the Sun) to the system beingmeasured. The amplitude isA =√596π π−2/3 M 5/6z , (15)and Ψ(f) is a rather complicated function of t c, φ c, M, µ,β, and σ; see Poisson & Will (1995), Eq. (3.6).This post-Newtonian description of A and Ψ(f) is more


Untangling the merger history of massive black holes 5properly called the restricted post-Newtonian approximation.The “full” post-Newtonian approximated waveform includescontributions from several (in principle, all) harmonicsof the binary’s orbital motion (Cutler & Flanagan 1994):h(t) = Re [ h 1e iΦ(t) + h 2e 2iΦ(t) + h 3e 3iΦ(t) + . . . ] . (16)Here, Φ(t) is the time domain orbital phase. Each amplitudeh i itself is described by a post-Newtonian expansion;the results rapidly get quite complicated [cf. Will & Wiseman(1996), Eqs. (6.10) and (6.11)]. Not surprisingly, thestrongest harmonic is h 2, associated with the quadrupolemoment of the source. In the restricted post-Newtonian approximation,we ignore the other harmonic contributions tothe waveform. Further, we ignore all but the leading “Newtonian”order contributions to that harmonic’s amplitude.High-order post-Newtonian information is used to describethe binary’s dynamics and hence to compute the phaseΦ. The Fourier transform of this restricted post-Newtoniantime domain signal then gives the frequency domain waveform(14).The location vector ˆn is given by the sky position coordinatesof the binary, (¯µ S, ¯φ S) (where ¯µ ≡ cos θ). Likewise,the binary orientation vector ˆL can be described using coordinates(¯µ L, ¯φ L). As LISA orbits the sun, the coordinatesas seen by LISA continuously change, modulating the waveform’samplitude and phase. Simultaneously, the detector is“rolling”, changing the profile of the arms as they “look” at asource, further modulating the waveform. For details of howthis modulation works and an elegant way to build it intothe waveform, see Cutler (1998). These modulations make itpossible to determine the sky position of the binary to withina solid angle δΩ S = 2πδ¯µ Sδ ¯φ S[1 − |c¯µ S ¯φ S|], and the orientationof the binary to within δΩ L = 2πδ¯µ Lδ ¯φ L[1 − |c¯µ L ¯φ L|].If the duration of a particular measurement is too short,these angles are determined very poorly — the waveformis not modulated enough. This can severely affect the measurementof other parameters, particularly the luminositydistance. Note that the amplitudes (h 1, h 2, h 3, . . .) definedschematically in Eq. (16) each depends upon these angles ina different manner. By ignoring all but h 2, the restrictedpost-Newtonian approximation throws away informationthat, in principle, could be used to improve gravitationalwavemeasurement accuracy. This is a natural point for animproved follow up to this analysis, as will be discussed furtherin Sec. 5.We terminate the inspiral when the binary’s membersare separated by a distance 6M; this very roughly correspondsto the point at which the post-Newtonian expansionceases to be accurate. The gravitational-wave frequency atthis point isf gw(r = 6M) = 2f orb(r = 6M)(1 + z)= 2Ω orb(r = 6M)2π(1 + z)=1(1 + z)π= 6−3/2πM z≃0.04 Hz√M(6M) 3( )10 5 M ⊙. (17)M zThe total number of inspiral parameters is eleven: thefour position and orientation coordinates, the distance tothe source, the constants of integration t c and φ c, and fourcombinations of the binary’s masses, spins, and orbital angularmomentum. In Sec. 4, we determine the accuracy withwhich these parameters can be measured for a wide varietyof interesting systems using the parameter measurementformalism discussed in Sec. 2, particularly Eq. (10). Notethat we confine our discussion to the masses, the luminositydistance, and the redshift, since our primary interest isto understand what LISA can say about mapping the blackhole mass distribution. In the course of this analysis we alsofit for and estimate the errors on all the other parametersdiscussed in this section.3.2 Ringdown waveformIn some cases, the waves from the final ringdown will bemeasurable as well. These waves are emitted as the systemsettles down to the stationary Kerr black hole solution. Theyare of relatively high frequency, so a coalescence with a veryinteresting inspiral may have a ringdown that is entirely lostin high frequency noise. Likewise, some systems have inspiralsthat are overwhelmed by low frequency noise, but emitringdown waves right in the band of maximum sensitivity.Weak distortions of Kerr black holes can be decomposedinto spheroidal modes, with spherical-harmonic-like indices land m (Leaver 1985). Each mode oscillates with a unique frequencyf lm and damping time τ lm , generating gravitationalwaves whose form is a damped sinusoid. The frequency anddamping time depend only on the mass and spin of the blackhole. Measuring ringdown waves thus measures the massand spin of the coalesced system. This provides additionalinformation about the black hole mass distribution. It canalso be combined with the chirp mass M and reduced massµ measured during the inspiral to infer the initial masses(m 1, m 2) of the binary’s members. (Because a fraction ofthe system’s mass is radiated away during the merger andringdown, some systematic error is necessarily introducedwhen mass determined from the ringdown is combined withmasses determined from the inspiral.)The ringing waves emitted at the endpoint of binarycoalescence will presumably be dominated by the l = m = 2mode. This is a bar-like mode that propagates about theequator in the same sense as the hole’s spin. It is likely todominate at late times because the coalescing system has ashape that nearly mimics an l = m = 2 distortion (so thatit should be preferentially excited), and also because thismode is more long-lived than any other. A more detailedunderstanding of the merger epoch is needed to better assessthe likely mixture of modes at the end of coalescence.A good fit to the frequency f 22 ≡ f ring and qualityfactor Q ≡ πf ringτ is (Echeverria 1989)f ring =12πM z[ 1 − 0.63(1 − a)3/10 ] , (18)Q = 2(1 − a) −9/20 , (19)where a = | ⃗ S|/M 2 is the dimensionless Kerr spin parameter.A merged remnant with mass 10 5 M ⊙ at z = 1 would emitringdown waves somewhere in the band from f = 0.06 Hz(a = 0) to f = 0.16 Hz (a = 1). The ringdown frequency isalways quite a bit higher than inspiral frequencies.


6 Scott A. HughesAs mentioned above, the time domain gravitationalwaveform for the ringdown waves are damped sinusoids.They can be writtenh +(t) = A + exp(−πf ringt/Q)cos(2πf ringt + ϕ) ,h ×(t) = A × exp(−πf ringt/Q) sin(2πf ringt + ϕ) . (20)It is not easy to estimate the polarization amplitudes A +,×;they will depend upon the detailed evolution of the mergerepoch, as well as variables such as the orientation of the finalmerged remnant. A reasonable hypothesis is that their ratiofollows the ratio of the inspiral polarization amplitudes:A + = A ring[ 1 + (ˆL · ˆn)2 ] ,A × = −2A ring(ˆL · ˆn) . (21)We set the overall amplitude A ring by requiring that theringdown radiate some fraction ǫ of the system’s total mass[see Fryer, Holz, & Hughes (2001), Sec. 2.5]. The result isA ring = 1D(z)√5ǫMz4πf ringQ . (22)It seems likely that ǫ will vary rather strongly dependingupon the constituents of the binary, particularly the blackholes’ spins and spin orientations. For concreteness, we willuse ǫ = 1% in all calculations. This is in accord with thefraction of system mass radiated in recent numerical simulations[see, for example, Baker et al. (2001) and referencestherein]. The phase ϕ essentially tells us the configuration ofthe merged remnant when the l = m = 2 mode dominatesits dynamics. We will take it to be randomly distributed overmergers.Whereas the inspiral waveform may be measured byLISA over the course of several months or years, the ringdownfor any source considered here will last no more thana few hours (and in many cases, only a few minutes). Thiscan be seen by using Eqs. (18) and (19) with τ = Q/πf ring.The motion-induced modulation of the ringdown waveformmay therefore be ignored. The ringdown provides no informationabout the binary’s location on the sky — the onlycharacteristics of the binary we are likely to learn from theringdown are the mass and spin of the remnant hole.Because the ringdown waves are fairly narrow in frequencyspace, we may make some useful approximationswhen computing the expected SNR from a ringdown measurement.Suppose the ringdown begins at time T ring. DefineH I,II(t) = √ 3[F + I,II (Tring)h+(t − Tring) + F × I,II (Tring)h×(t −T ring)]/2 (since we ignore detector motion, the responsefunctions are only evaluated at t = T ring). Then, we haveρ 2 I,II = 4= 2≃≃≃∫ ∞0∫ ∞df | ˜H I,II(f)| 2S h (f)df | ˜H I,II(f)| 2S−∞ h (|f|)2S h (f ring)2S h (f ring)2S h (f ring)∫ ∞df | ˜H I,II(f)| 2−∞∫ ∞dt H I,II(t) 2−∞∫ ∞T ringdt H I,II(t) 2 (23)On the third line, we have used the fact that most of thesignal power is at f = f ring to pull the noise out of theintegral, on the fourth line we have used Parseval’s theoremto go from a frequency domain to a time domain integral,and on the fifth line we have used the fact that ringdownwaves by definition are zero before T ring. This final integralis simple to evaluate using Eq. (20).We could now apply the full parameter estimation formalismdiscussed in Sec. 2 to see how well the mass M andspin a of the remnant black hole can be determined. Such ananalysis has in fact already been performed (Finn 1992). Becausethe detector motion has no important impact, Finn’sresults carry over directly to this analysis. In particular, thefinal mass is determined to an accuracyδM f,zM f,z= F(a)ρ≃ 2 ρ = 2√ρ2I+ ρ 2 II. (24)The function F(a) slowly varies from roughly 2.5 for aSchwarzschild black hole to 0.5 for an extreme Kerr hole.Since the goal of this analysis is a simple estimate, andalso since the initial conditions for ringing waves from coalescencesare not well known, we consider approximatingF(a) ≃ 2 to be as accurate as is warranted.3.3 Cosmological modelThe cosmological parameters enter this analysis through theneed to convert between redshift and luminosity distance.We will assume a flat cosmology, with matter and cosmologicalconstant contributions to the total density given byΩ Λ = 0.65 and Ω M = 1 − Ω Λ = 0.35. These choices are inaccord with recent observational evidence (e.g., Netterfieldet al. 2001). The luminosity distance to a source at redshiftz is given by [Hogg (1999) and references therein]D(z) =where(1 + z)cH 0∫ z0dz ′E(z ′ ) , (25)E(z) = √ Ω M(1 + z) 3 + Ω λ . (26)The Hubble constant H 0 = 100 h 0 km/(Mpc s); we assumeh 0 = 0.65 (Wang, Tegmark, & Zaldarriaga 2001).This expression is very easy to invert numerically; wedo so with a simple bisection, obtaining z(D). For a particulargravitational-wave measurement, we get D with someerror δD. This error, and also errors in the cosmological parametersΩ Λ and h 0, mean that z is measured to a precisionδz:δz = ∂z∂D[δD 2 +where( )∂z ∂D −1∂D = ,∂z[∂D c (1 + z)=+∂z H 0 E(z)∂D∂Ω λ=(1 + z)c2H 0( ) ∂D 2 ( )δΩ 2 ∂D 2] 1/2Λ + δh 2 0 , (27)∂Ω Λ ∂h 0∫ z0∫ z0]dz ′,E(z ′ )dz ′ [ (1 + z ′ ) 3 − 1 ]E(z ′ ) 3 ,∂D∂h 0= − D(z)h 0. (28)


Untangling the merger history of massive black holes 7Note that δΩ M is not included since we have assumed thatthe universe is precisely flat, and therefore δΩ M = −δΩ Λ.Following the discussion in Wang, Tegmark, & Zaldarriaga(2001), we will put δΩ Λ = 0.1 and δh 0 = 0.1. WhenδD/D ≪ 10%, δz is dominated by δΩ Λ and δh 0; in thesecases, it typically turns out that δz/z ≃ 15%. We willalso look at redshift measurement accuracy assuming thatδΩ Λ = 0 = δh 0; this demonstrates how well the redshiftcould be measured in principle if the universe’s geometrywere known perfectly.This assumed cosmology is adequate for the purposesof this paper. As our knowledge of the universe’s geometryimproves, this prescription can be readily generalized (forexample, if the “dark energy” turns out to have an equationof state other than that of a cosmological constant).3.4 Detector noiseFinally, we need a model for the noise in the LISA detectorto proceed. As discussed in Sec. 2, we assume that the noisein each of the effective detectors is a stationary, Gaussianrandom process. We also assume that their noises are uncorrelated.As discussed by Cutler (1998), one can combinethe data streams in such a way that this second assumptionis true by construction.The detector noise is characterized by the spectral densityS h (f). This quantity is the Fourier transform of theautocorrelation of the time domain noise:∫ ∞S h (f) = 2 dτ C n(τ)e 2πifτ ,−∞C n(τ) = 〈n(t)n(t + τ)〉 , (29)where angle brackets denote ensemble averaging. We assumethat the spectral density is the same for each of the effectivedetectors.LISA’s datastream will include instrumental noise intrinsicto the detector, and confusion noise arising primarilyfrom the large number of gravitational wave producingwhite dwarf binaries in the galaxy. Especially at low frequencies,one cannot resolve these binaries individually —for a mission lasting T ∼ several 10 7 seconds, there could beas many as 10 3 binaries contributing power in a single frequencybin δf ∼ 1/T (Cutler 1998). The unresolved whitedwarf binaries constitute a stochastic background that, fromthe perspective of measuring black hole binary waves, is asource of noise.A good fit (Phinney, private communication) to the projectedinstrumental noise for LISA (Folkner 1998) is givenbyS insth (f) =where{s 1[1 + f/(0.01 Hz)] + s 2/f 2+ s 3/f 5/2 + s 4(10 −4 Hz/f) 20 } 2, (30)s 1 = 4 × 10 −21 Hz −1/2 ,s 2 = 3 × 10 −26 Hz 3/2 ,s 3 = 5 × 10 −28 Hz 2 ,s 4 = 1 × 10 −16 Hz −1/2 . (31)Notice that this noise shoots up very rapidly at frequenciesless than 10 −4 Hz. In fact, LISA could have good sensitivityat lower frequencies than this [cf. Fig. 5 of Larson, Hiscock,& Hellings (2000)]. The very low-frequency sensitivity willbe limited by LISA’s ability to maintain “drag-free” motionon long timescales (that is, motion driven solely by gravity,without interference from external forces due to, for example,the solar wind). In cutting off LISA’s sensitivity at10 −4 Hz, we assume that drag-free behavior will be difficultto maintain on timescales longer than about 3 hours.The galactic confusion noise is well described byS galh (f) = 5 × 10−44 Hz −1 ( 1 Hzf) 7/3× [1 − exp(−δfdN/df)] . (32)The factor in square brackets gives the fraction of frequencybins near f that have a contribution from galactic whitedwarf binaries. This fraction decreases with increasing frequency— the population of white dwarf binaries thins outin frequency space at high f. This is because the inspiralrate df/dt grows with f, so a population that is initiallyclustered near some frequency spreads out as it evolves. Inthis equation, we put δf = 3/T, where T is the duration ofthe LISA mission, and the factor 3 roughly accounts for thesmearing of a white dwarf binary’s signal due to detectormotion. The function(dN1Hzdf = 2 × 10−3 Hz −1 f) 11/3(33)is the number of white dwarf binaries per unit gravitationalwavefrequency. The fit (32) was provided by Phinney (Phinney,private communication); the prefactor 5×10 −44 and thevalue of dN/df correspond closely to the results presentedin Webbink & Han (1998).At high enough frequencies, there will be on averagefewer than one white dwarf binary per bin. The confusionnoise goes from a smooth continuum to a series of nonconfusedlines. For galactic binaries, these lines should bestrong enough that they can be fit and subtracted from thedata stream, reducing the overall noise. A simple estimateof the resulting noise (Phinney, private communication) is[S inst+galh(f) = minShinst (f)/exp(−δfdN/df),S insth (f) + 5 × 10 −44 Hz −1 (1Hz/f) 7/3 ]. (34)The factor exp(−δfdN/df) is the fraction of empty bins. Forthe total LISA noise, we take (34) plus a contribution fromextragalactic binaries (Folkner 1998):( ) 7/3ex. galSh(f) = 1.1 × 10 −46 Hz −1 1 Hz. (35)fFinally, it should be noted that Phinney has recentlyre-examined the issue of confusion limited backgrounds invery general terms (Phinney 2001a), and will soon producea paper pointing out some overlooked sources of confusionnoise (Phinney 2001b). These noises are not included in thisanalysis, but could be very easily. Their main effect willbe to augment the low-frequency noise, thus reducing thesignal-to-noise ratio. The accuracy with which parameters


8 Scott A. Hughescan be measured will be reduced as well, but very likelythat reduction will nearly scale with the reduction in SNR.4 RESULTSTo determine how well LISA will be able to measure massesand redshift, we have performed a large number of MonteCarlo simulations of binary black hole coalescence measurement.We choose binary masses and a redshift, and thenrandomly distribute 100 such binaries over sky position andorientation. We also randomly distribute the binary members’spins, and the magnitude of the final merged hole’sspin. (Of course, the final spin should be found by conservingthe initial spin and orbital angular momentum of thesystem, less that which is lost to radiation. The details ofthis are complicated and depend to some degree on poorlyunderstood physics. For the purpose of estimating how wellthe final system mass is determined, randomly choosing thefinal spin should be fine.) Finally, we assume that the LISAmission lasts three years, and we uniformly distribute thecoalescence time of the binary during that time. As a consequence,the total observation time may vary quite a bitamong inspirals in a particular run, which can have a largeimpact on parameter determination. The routine ran2()(Press et al. 1992) was used to generate random numbers.The binaries chosen were placed at z = 1, 3,5, 7,9, andhad masses m 1 = m 2 = 10 3,4,5,6,7 M ⊙. For each redshiftand mass, we develop the distribution of parameter determinationerrors. Two examples of such distributions are illustratedin Figures 1 and 4. Figure 1 shows the distributionof errors in z, M z, µ z, and M f,z expected for measurementsof a binary with m 1 = m 2 = 10 5 M ⊙ at z = 1.This is a case where most parameters are measured ratherwell. Note in particular the extreme precision with which theredshifted chirp mass is determined: the distribution peaksat δM z/M z ∼ 1.5 × 10 −4 . The chirp mass is measuredso precisely because it most strongly determines the inspiralrate, and hence has the greatest impact on the gravitationalwave phase evolution. The reduced mass also impactsthe inspiral, but not as strongly, and hence is measuredwith less precision: the peak in the distribution is atδµ z/µ z ∼ 2.5 × 10 −2 . Finally, note that the redshift errorspeak at δz/z ∼ 15%. This is the value expected when theredshift error is dominated by the present uncertainty inour knowledge of cosmological parameters. The luminositydistance is actually determined far better than this, indicatingthat improved knowledge of cosmological parameterswill greatly reduce errors in the redshift. This is illustratedin Fig. 2. The top panel shows the distribution in luminositydistance errors, and the bottom panel illustrates the redshifterror that would be achieved if we knew the cosmologicalparameters exactly. Comparing the two panels shows thatδz/z ≃ δD/D when the cosmological parameters are knownaccurately. Both of these distributions peak near 1% relativeerror, and are largely confined to less than 10% error.The rule of thumb “δz/z ≃ δD/D when δ(cosmology) = 0”holds more or less independent of redshift. As cosmologicalparameters become better determined, our ability to determinethe redshift (and thus the masses) of coalescing binaryblack holes will be greatly improved.Table 1 is a somewhat massaged representation of theFigure 1. Distribution of errors in z, M z, µ z, and M f,z for LISAmeasurement of a binary with m 1 = m 2 = 10 5 M ⊙ at z = 1. Thetypical inspiral SNR is about 1000; the ringdown SNR is around65. This mass is nearly optimal at this redshift for determiningthe binary’s parameters. The error in z is dominated by errors inthe cosmological parameters; the luminosity distance is actuallydetermined quite precisely for this source (cf. Fig. 2).covariance matrix Σ ab for a typical inspiral in this distribution— diagonal components are actually the mean error〈(δθ a ) 2 〉 1/2 = √ Σ aa , off-diagonal components are the correlationcoefficient c ab defined in Eq. (11). Note the strongcorrelations between the luminosity distance and the positionand orientation angles. These parameters are ratherstrongly entangled since they set the amplitude of the measuredwaveform: the angles through the detector responsefunctions F + and F × and the ratio of the wave’s polarizations;the luminosity distance through the overall amplitudeA [cf. Eq. (15)]. In order to make a good measurementof the luminosity distance, we must determine the binary’ssky position and orientation very accurately. The sky positionand orientation are encoded in the modulations inducedby LISA’s orbital motion; they are well determined whenthe waveform is subject to a large amount of this motioninducedmodulation. Hence, luminosity distance is only welldetermined when the waveform is significantly modulatedby the detector motion. This accounts for the rather largetails apparent in Figs. 1 and 2: though most of the distributionfor δD/D (for example) is confined to small error, someof the coalescences in the sample are measured with muchlarger δD/D, up to 60%. These large error measurementsoccur when the inspiral is shorter than usual — because werandomly distribute the merger time during LISA’s mission,some events occur very near the mission’s start. The positionand orientation angles are poorly determined for theseshort inspirals, so the luminosity distance is also determinedpoorly.The impact of these correlations on the accuracy withwhich D is measured can be assessed by imagining that thebinary’s sky position is measured with zero error. This is a


Untangling the merger history of massive black holes 9Figure 2. The top panel shows the distribution of error inluminosity distance D for LISA measurement of binaries withm 1 = m 2 = 10 5 M ⊙ at z = 1; the bottom panel shows how wellthe redshift could be measured if cosmological parameters wereknown perfectly. In both cases, the peak of the distribution is ata relative error near 1%. This indicates that in many cases themass-redshift degeneracy will be broken with very good precisionwhen cosmological parameters are better determined. Note,though, the very large tail in the distribution, extending out torelative error of about 60%. The distance to the source is poorlydetermined when a source’s sky position is poorly determined.This typically happens if the observation time is short — themerger occurs near the beginning of LISA’s mission.reasonable description of what might be achieved by coordinatedelectromagnetic and gravitational-wave measurementsof binary coalescence — for instance, if the merger is accompaniedby a gamma or x-ray flare. The electromagneticmeasurement will likely determine the merger’s sky positionmuch more accurately than can be done with gravitationalwaves. Figure 3 compares how well D is measured when thesky position is known precisely to the case of sky positiondetermined from the gravitational waves. Knowledge of thebinary’s sky position has an enormous impact, improvingthe accuracy with which D is measured by about an orderof magnitude.Note that Table 1 also shows strong correlations amongthe mass and spin parameters, M c, µ, β, and σ. This isa well-known feature of measurement with post-Newtoniantemplates, arising because the impact of the various parametersupon the phase evolution are not strongly different fromone another. For further discussion, see Cutler & Flanagan(1994) and Poisson & Will (1996).Figure 4 shows the distribution of measurement errorswhen m 1 = m 2 = 10 4 M ⊙ and z = 7. In all cases,the distributions peak at larger error values than whenm 1 = m 2 = 10 5 M ⊙ and z = 1. This isn’t too surprisingsince these sources are much fainter and thus harder to measure.The redshift determination in particular is quite a bitworse, so that the mass-redshift degeneracy will be brokenrather less accurately for these holes. Table 2 shows the pa-Figure 3. Comparison of measurement errors in luminosity distancefor LISA measurement of a binary with m 1 = m 2 = 10 5 M ⊙at z = 1. The top panel shows the error distribution when thesky position is determined using gravitational waves; the bottompanel shows the distribution assuming the sky position is measuredwith no error. Because of strong correlations between theluminosity distance and the sky position angles, improving theaccuracy with which the position is measured has a big impacton the distance determination. In this case, D is measured withroughly an order of magnitude less error.rameter errors and correlations for a typical coalescence inthis set. Because of the signal’s weakness, the measurementerrors (diagonal components of the matrix) are rather largerthan in the case z = 1, m 1 = m 2 = 10 5 M ⊙. However, thecorrelations (off-diagonal components) are not much different.This is typical: correlations between inspiral parametersdo not depend strongly on signal-to-noise ratio, though theinspiral time can have a big effect.Rather than show distribution histograms for all theremaining cases that we examined, we summarize their contentsin Tables 3 – 7. These tables give the “most likely”error values — the errors found at the peaks of the measurementdistribution. The reader should bear in mind thatthe distributions from which these values were taken alsohave long, high-error tails, as in the Figures.Several interesting features can be seen across the tables.Only Table 3 includes data for the merger of holes withm 1 = m 2 = 10 7 M ⊙. At redshifts higher than z = 1, the inspiralsignal from such binaries is radiated at frequenciesentirely below 10 −4 Hz, out of LISA’s band. Indeed, even atz = 1 this binary barely radiates in band, lasting less than2 hours before merger. The signal is so short that little inspiralphase accumulates, so D, M z, and µ z are determinedvery poorly. The ringdown, by contrast, is quite strong, soM f,z is measured with very good accuracy.The tables show that, in general, measurement accuracydegrades with increasing redshift. This isn’t surprising sincesuch sources are fainter. However, at each redshift, there aresome sources and parameters that can be measured with atleast moderate precision. At low redshift, there is a fairly


10 Scott A. Hughesbroad range of masses in which at least two of the masses(M z, µ z, M f,z ) can be determined with good precision, andin some cases all three are well measured. Out to z = 5,there are cases for which δD/D < δz/z, indicating that inthose cases the redshift distribution is skewed high becauseof the present error in h 0 and Ω Λ. Setting the cosmologicalparameter error to zero, we find δz/z ≃ δD/D, as expected.Improved knowledge of cosmological parameters will havea big impact on measurement in those cases. At redshiftsz > 5, δz becomes dominated by error in D — improvedknowledge of h 0 and Ω Λ will not have much effect.Finally, note that the tables confirm the trend discussedabove that D (and hence z) is not well determined for veryshort measurement times: if the inspiral does not last “longenough”, the motion-induced modulation of the waveformis not sufficient to determine the source’s sky position andorientation very accurately. As a consequence, D is poorlydetermined. A rough necessary condition to measure D wellseems to be that the LISA constellation must move througha radian or so of its orbit. This is not, however, a sufficientcondition — D can be determined poorly from long inspiralsif the signal is too weak.In all cases except for the largest masses, the best determinedparameter is the redshifted chirp mass. It is oftenmeasured with precision δM z/M z∼< 0.1% or better, evenfrom sources with z ∼ 9. LISA’s ability to break the massredshiftdegeneracy for the chirp mass will therefore be limitedby redshift error: we will measure the actual chirp massof distant coalescences to within 15 −30% assuming presentcosmological parameters, and perhaps as well as 5 − 30%when LISA actually flies. The reduced mass is often measuredwith a precision of about 10% or better, and so providesuseful additional information. The final mass of thesystem is only determined from more massive systems, sincethe ringdown waves are relatively high frequency. In thosecases, it can be measured with accuracy δM f,z /M f,z∼ < 5%.5 SUMMARY AND CONCLUSIONFigure 4. Distribution of errors in z, M z, µ z, and M f,z for LISAmeasurement of a binary with m 1 = m 2 = 10 4 M ⊙ at z = 7. Thetypical inspiral SNR is about 45; the ringdown SNR is around0.7. Because of the weak ringdown waves, the final mass is ratherpoorly determined. In this case, measurement error and cosmologicalparameter error contribute to the redshift error about equally.Setting the cosmological parameter errors to zero in this case doesnot have a large effect.By combining gravitational-wave measurements with informationabout cosmological parameters, LISA will be able tomeasure the redshift of coalescing binaries with moderateprecision (relative error of 15 − 30% using present uncertaintiesin cosmological parameters, perhaps 5 −30% by thetime that LISA flies). The redshifted chirp mass is typicallymeasured far more accurately than this, and in many cases,either the binary’s reduced mass or the final mass of theremnant black hole produced when the binary merges canbe measured with a precision of 5 −20%. This precision willallow LISA to untangle mass and redshift, making it possibleto track the merger history of massive black holes in theuniverse.These measurements work best for signals whose frequencyrange lies in LISA’s band of maximum sensitivity.From Tables 3 – 7, we see that the best measurement sensitivitiesare for systems that have M z ∼ 10 5 M ⊙ or so.This isn’t too surprising: recall that inspiral ends when thebodies are separated by a distance r ∼ 6M, and that thegravitational-wave frequency at that point is( )10 5 M ⊙f GW ≃ 0.04 Hz(36)M z[cf. Eq. (17)]. This frequency is right about where LISA’ssensitivity begins to degrade due to high frequency noise.When M z ∼ 10 5 M ⊙ most of the inspiral signal accumulatesin a band where LISA has very good sensitivity. A goodrule of thumb seems to be that LISA will measure at leasttwo of the redshifted mass combinations M z, µ z, and M f,zwith good precision when the total redshifted system massis within a factor of ten to twenty of 10 5 M ⊙. Even outsidethat range, at least one mass (either the chirp mass or thefinal mass) can be measured well.To measure the redshift precisely, we first must determinethe luminosity distance. This correlates most stronglywith the amount of time over which the inspiral signal ismeasured. Very short inspirals do not experience enoughdetector-motion-induced modulation of the gravitationalwaveform to pin down a source’s location on the sky very accurately,and as a consequence the luminosity distance canbe poorly determined. This is the reason the largest systemsin our sample have such poor precision in D and z:they enter LISA’s band already very close to merging, andquickly evolve to merger. The distance is determined wellwhen LISA moves through at least a radian or so of its orbit,corresponding to at least 2 months of observation.This analysis is essentially just a first cut, proof-ofprincipledemonstration of how cosmological informationand gravitational-wave measurements can be combined tostudy the merger history of massive black holes. We havemade several simplifying assumptions that, if lifted, maymodify some of our conclusions. One example is our use


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Untangling the merger history of massive black holes 13Table 1. Measurement accuracy and correlations for coalescences at z = 1 with m 1 = m 2 = 10 5 M ⊙ . Diagonal elements inthis matrix are the mean error expected in the parameter; off-diagonal elements are the correlation coefficient c ab . Elementscontaining · can be found by symmetry. The errors in φ L , φ S , and φ c are in radians; those in t c are in units of 10 4 seconds.All other entries are dimensionless.lnD µ L µ S φ L φ S t c φ c ln M z lnµ z β σln D 0.0095 0.8969 0.8258 0.6692 0.8185 -0.0532 0.3224 0.1608 -0.2196 0.2182 -0.2076µ L · 0.0114 0.8215 0.2972 0.8881 -0.1140 0.2273 -0.0143 -0.0625 0.0502 -0.1048µ S · · 0.0006 0.4474 0.7591 -0.1786 0.2526 0.0823 -0.1402 0.1353 -0.1483φ L · · · 0.0362 0.3280 0.0777 0.3351 0.3884 -0.3883 0.4063 -0.2879φ S · · · · 0.0051 -0.0802 0.2074 0.0067 -0.0703 0.0612 -0.0100t c · · · · · 0.0002 0.8837 0.7522 -0.8287 0.7980 -0.9189φ c · · · · · · 0.7179 0.8691 -0.9473 0.9247 -0.9918ln M z · · · · · · · 0.0001 -0.9798 0.9904 -0.8828ln µ z · · · · · · · · 0.0334 -0.9976 0.9556β · · · · · · · · · 0.4171 -0.9331σ · · · · · · · · · · 0.3251Table 2. Measurement accuracy and correlations for coalescences at z = 7 with m 1 = m 2 = 10 4 M ⊙ .lnD µ L µ S φ L φ S t c φ c ln M c ln µ z β σln D 0.2036 -0.2030 0.8774 -0.6732 -0.6861 -0.3873 -0.2672 -0.4736 0.3933 -0.4331 0.2200µ L · 0.6189 -0.5836 -0.4715 -0.4980 -0.7733 0.0455 0.0608 -0.0245 0.0258 -0.0177µ S · · 0.3647 -0.4104 -0.3818 -0.0090 -0.2387 -0.4250 0.3413 -0.3758 0.1907φ L · · · 0.7262 0.9927 0.9066 0.2173 0.3933 -0.3429 0.3792 -0.1863φ S · · · · 0.4652 0.9261 0.2182 0.3898 -0.3412 0.3771 -0.1861t c · · · · · 0.0083 0.1932 0.2900 -0.2768 0.2992 -0.1778φ c · · · · · · 4.0390 0.9008 -0.9562 0.9354 -0.9960ln M z · · · · · · · 0.0005 -0.9848 0.9931 -0.9012ln µ z · · · · · · · · 0.1479 -0.9977 0.9589β · · · · · · · · · 2.1417 -0.9373σ · · · · · · · · · · 1.9260Table 3. Summary of measurement accuracies for binary black hole coalescences at z = 1. The redshift error δz 1 assumesδh 0 = 0.1, δΩ Λ = 0.1; the error δz 2 assumes the cosmological parameters are known perfectly.m 1 (= m 2 ) ρ insp T insp ρ ring δD/D δz 1 /z δz 2 /z δM z/M z δµ z/µ z δM f,z /M f,z10 3 M ⊙ 20 575 days 10 −3 0.08 0.15 0.05 5 × 10 −5 0.05 250010 4 M ⊙ 150 550 days 0.25 0.05 0.15 0.04 5 × 10 −5 0.03 1010 5 M ⊙ 1000 430 days 60 0.02 0.15 0.02 1 × 10 −4 0.04 0.0310 6 M ⊙ 200 15 days 3500 0.2 0.2 0.2 5 × 10 −3 0.5 5 × 10 −410 7 M ⊙ 40 100 minutes 612 70 70 70 150 300 4 × 10 −3Table 4. Summary of measurement accuracies for binary black hole coalescences at z = 3.m 1 (= m 2 ) ρ insp T insp ρ ring δD/D δz 1 /z δz 2 /z δM z/M z δµ z/µ z δM f,z /M f,z10 3 M ⊙ 10 575 days 10 −3 0.2 0.2 0.2 1 × 10 −4 0.08 100010 4 M ⊙ 75 530 days 0.35 0.1 0.15 0.1 5 × 10 −4 0.1 510 5 M ⊙ 400 200 days 85 0.1 0.15 0.1 1 × 10 −3 0.3 0.0210 6 M ⊙ 70 5 days 1300 0.8 0.8 0.8 1.5 × 10 −2 0.6 1 × 10 −3Table 5. Summary of measurement accuracies for binary black hole coalescences at z = 5.m 1 (= m 2 ) ρ insp T insp ρ ring δD/D δz 1 /z δz 2 /z δM z/M z δµ z/µ z δM f,z /M f,z10 3 M ⊙ 9 540 days 2 × 10 −3 0.25 0.3 0.25 2 × 10 −4 0.1 100010 4 M ⊙ 55 560 days 0.6 0.2 0.2 0.2 2 × 10 −4 0.2 410 5 M ⊙ 250 100 days 100 0.15 0.2 0.15 3 × 10 −3 0.5 0.0210 6 M ⊙ 30 2.5 days 350 2 2 2 0.05 0.6 0.01


14 Scott A. HughesTable 6. Summary of measurement accuracies for binary black hole coalescences at z = 7.m 1 (= m 2 ) ρ insp T insp ρ ring δD/D δz 1 /z δz 2 /z δM z/M z δµ z/µ z δM f,z /M f,z10 3 M ⊙ 9 540 days 3 × 10 −3 0.3 0.3 0.3 2.5 × 10 −4 0.15 50010 4 M ⊙ 46 560 days 0.7 0.25 0.25 0.25 7 × 10 −4 0.25 1.510 5 M ⊙ 150 65 days 120 0.5 0.5 0.5 5 × 10 −3 0.6 0.01510 6 M ⊙ 24 1.5 days 100 6 6 6 0.1 0.7 0.03Table 7. Summary of measurement accuracies for binary black hole coalescences at z = 9.m 1 (= m 2 ) ρ insp T insp ρ ring δD/D δz 1 /z δz 2 /z δM z/M z δµ z/µ z δM f,z /M f,z10 3 M ⊙ 8 580 days 4 × 10 −3 0.5 0.5 0.5 3 × 10 −4 0.16 50010 4 M ⊙ 43 530 days 0.9 0.3 0.3 0.3 7 × 10 −4 0.25 1.510 5 M ⊙ 100 45 days 130 1 1 1 7 × 10 −3 0.6 0.01510 6 M ⊙ 17 1 day 40 20 20 20 0.6 1.5 0.05

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