Method to estimate ISCO and ring-down frequencies in binary systems and consequences for gravitational wave data analysis
Chad Hanna, Miguel Megevand, Evan Ochsner, Carlos Palenzuela
MMethod to estimate ISCO and ring-downfrequencies in binary systems and consequences forgravitational wave data analysis
Chad Hanna , Miguel Megevand , Evan Ochsner , and CarlosPalenzuela , Department of Physics and Astronomy, Louisiana State University Baton Rouge,LA 70802, USA Department of Physics, University of Maryland, College Park, MD 20742 Max-Planck-Institut fur Gravitationsphysik, Albert-Einstein-Institut, Golm,GermanyE-mail: [email protected], [email protected], [email protected],[email protected]
Abstract.
Recent advances in the description of compact binary systems have producedgravitational waveforms that include inspiral, merger and ring-down phases.Comparing results from numerical simulations with those of post-Newtonian (PN),and related, expansions has provided motivation for employing PN waveforms innear merger epochs when searching for gravitational waves and has encouraged thedevelopment of analytic fits to full numerical waveforms. The models and simulationsdo not yet cover the full binary coalescence parameter space. For these yet un-simulatedregions, data analysts can still conduct separate inspiral, merger and ring-downsearches. Improved knowledge about the end of the inspiral phase, the beginning of themerger, and the ring-down frequencies could increase the efficiency of both coherentinspiral-merger-ring-down (IMR) searches and searches over each phase separately.Insight can be gained for all three cases through a recently presented theoreticalcalculation, which, corroborated by the numerical results, provides an implicit formulafor the final spin of the merged black holes, accurate to within 10% over a largeparameter space. Knowledge of the final spin allows one to predict the end of theinspiral phase and the quasinormal mode ring-down frequencies, and in turn providesinformation about the bandwidth and duration of the merger. In this work we willdiscuss a few of the implications of this calculation for data analysis. a r X i v : . [ g r- q c ] J a n
1. Introduction
Gravitational radiation emitted during the inspiral, merger and ring-down of twocompact objects with a total mass below ∼ M (cid:12) is a likely detection source forinterferometric gravitational wave detectors [1]. Maximal signal-to-noise ratio (SNR)is achieved in searches for compact binary coalescence with matched filtering banks oftemplate waveforms [2, 3, 4, 5, 6] allowing faint signals to be detected reliably. TheSNR depends on having a template waveform that matches the true signal correctly[2]. Because templates are weighted by the noise power spectral density of the detectorwith non-trivial frequency dependence it is important when in the binary evolution thetemplate matches well. Numerical simulations are improving our knowledge of the fullwaveforms emitted during the inspiral, merger and ring-down phases of binary blackhole coalescence [7, 8, 9], yet they still do not cover an adequate parameter space forconstructing filter banks [10, 9]. When the full parameter space is simulated it willbe useful to have fully parameterized models of these waveforms and even now manyhave considered hybrid waveforms that patch together PN and numerical simulations[11, 12, 7, 8, 9] as data analysis templates. Others have introduced a phenomenologicalfourth order correction to make effective one-body (EOB) waveforms match numericalrelativity results near the merger [11, 7] known as pseudo • The calculation of the final spin given in [13] contains an implicit reference to theinnermost stable circular orbit (ISCO) of a test particle orbiting the Kerr blackhole that results from the merger. We will also discuss the impact of using thenew ISCO frequency as a cut off for current gravitational wave searches that aretargeting coalescing compact objects including the impact on SNR and the abilityto probe neutron star properties. • Knowing the final spin gives an estimate for the quasi-normal mode ring-downfrequencies. These frequencies could be used in a separate matched filter search[26]. • We can also use the final spin to calculate the light ring (smallest, unstable circularorbit for a photon orbiting a Kerr black hole) and quasinormal mode ring-downfrequencies of the final black hole; we will see that the l=2, m=2, n=0 QNM is quiteclose to the light ring frequency. We will define a merger epoch as the evolutionfrom ISCO to light ring and dicuss how to obtain its time frequency content for yetunsimulated systems.We define the merger epoch as everything that occurs between the ISCO andi ring-down frequencies. We show that the duration of the radiation seems to be welldescribed by the infall duration of a test particle orbitting the merged black holejust below the ISCO. Knowing the duration and bandwith gives a time-frequencyvolume, which is useful in data analysis for unmodeled signals.
2. Formalism
The inspiral phase of two compact objects is generally considered to end when the binaryhas evolved to an innermost stable circular orbit (ISCO), should it exist. The ISCO canbe defined by waveform models and numerical relativity [17, 18, 7]. Unfortunately thereis a difference in the numbers obtained from different methods. Some LIGO searchesfor non-spinning systems have taken the conservative approach to use the ISCO definedfor a test particle orbiting a Schwarzschild black hole.Intuitively the space-time of an inspiralling comparable mass binary system willnot be well described by a Schwarzschild space-time, and thus the Schwarzschild ISCOwill not capture the dynamics correctly. If the merger product is a black hole, it will bea Kerr-like black hole with a final spin that is a function of the masses and spins of thecomponents. Recently [13] has proposed a simple set of assumptions that predict thefinal spin of the black hole merger product to well within 10% of numerical simulationsbased on first principle arguments. Implicitly in the derivation of [13] is the ISCO radiusof a test particle orbiting a Kerr black hole having a final spin equal to the spin of themerger product. The fact that the ISCO radius used gives the correct answer for thefinal space-time suggests that it plays an important role in the pre-merger dynamics.We hypothesize that the test particle ISCO of the merged Kerr black hole may describea way to define the end of the inspiral phase for binaries that produce black holes.The ISCO solution for a test particle orbiting a Kerr black hole is [20], Z ≡ (cid:18) − a f M (cid:19) / (cid:104) (1 + a f M ) / + (1 − a f M ) / (cid:105) Z ≡ (cid:18) a f M + Z (cid:19) / r ISCO = M { Z ∓ [(3 − Z )(3 + Z + 2 Z )] / } , (1)where M is the total mass of the black hole and a f is the angular momentum. Thefinal spin is required in order to compute this ISCO. In [13], assuming the amount ofmass and angular momentum radiated beyond the ISCO is small, the following implicitformula for the final angular momentum of a black hole a f with component spins alignedwith the orbit is calculated, a f M = L orb M (cid:16) q, rM = r ISCO
M , a f M (cid:17) + q χ + χ (1 + q ) , (2)where χ i = a i /m i , q = m /m ∈ [0 ,
1] and M = m + m is the total mass. Theimplicitly found a f agrees well with numerical simulations [13] and the analysis canbe modified to include arbitrary spin angles. L orb is the orbital angular momentumcontribution calculated from the orbital angular momentum of a particle at the ISCOof a Kerr black hole with spin parameter a f , which has the following expression [20], L orb M (cid:16) q, rM , a f M (cid:17) = q (1 + q ) ± ( r ∓ a f M / r / + a f ) M / r / ( r / − M r / ± a f M / ) / , (3)where the upper/lower signs correspond to prograde/retrograde orbits. In order toagree with numerical simulations this function has to be evaluated at r = r ISCO givenby equation (1). Alternative radii in the above prescription – like the photon-ringradius or Schwarzschild radius – give a prediction that is off the numerically obtainedvalue. This fact, combined with the same analysis extended to elliptic binaries, againdepending sensitively on the ISCO location [23], suggests the ISCO radius determinedby this approach is relevant[24].In order to get the gravitational wave frequency at ISCO we use the coordinateangular velocity of a circular orbit [20],Ω = ± M / r / ± a f M / , (4)with a f given by the implicit equation (2). The gravitational wave frequency at a givenradius is then f = Ω /π , and so we define f ISCO [BKL] as the frequency obtained by solvingthe system of equations (1-4) at the Kerr ISCO radius.The solution space for f ISCO [BKL] can be written as a function of the final unknownspin a f . For convenience, we prefer to extend it as a surface parametrized by ( a f , q, χ ),as it is shown in figure 1. This way the mass ratio dependence of the lines correspondingto different individual spins can be seen explicitly. For the equal mass case without spinthe approximate expression for the Kerr test particle ISCO frequency is, f ISCO [BKL] ( M, q ) ≈ (cid:0) . q − . q + 2 . q + 1 (cid:1) × π (6 M ) / , (5)which can be compared to the Schwarzschild test particle ISCO f ISCO [SCH] = π − (6 M ) − / .Once the final spin is known it is possible to compute the last stable photon orbit(“light ring”) and the quasi-normal mode ring-down frequencies [16]. The light ring fora Kerr black hole is, [20] r light = 2 M { / − ( ∓ a f /M )] } (6)Ω light = ± M / r / ± a f M / , (7)An approximate fit to the quasinormal mode ring-down frequencies as a function of thefinal spin is given in [25, 26, 27]. For the l=2, m=2, n=0 mode we have,Ω QNM ≈ . M (cid:0) − . − a f ) . (cid:1) , (8)Others ring-down modes could could be computed as necessary.Figure 2 compares the BKL ISCO for non-spinning binaries with the ISCOscomputed from the minimum energy conditions at 3PN order [19] as well as 2PN and
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 0.000.250.500.750.90 f I S C O [ BK L ] x f − I S C O [ S CH ] a f q χ=χ=χ=χ=χ= Figure 1.
The surface of solutions of the frequencies at the ISCO as a function ofthe mass ratio q and the final spin a f for components with spins that are aligned withthe orbital angular momentum. Also shown, are curves corresponding to the solutionof the equal spin case χ = χ = χ . p4PN EOB ISCOs [15, 16]. BKL and EOB agree exactly with the Schwarzschild ISCO atthe extreme mass ratio limit, but the PN calculations are known to be inconsistent [19]in that regime. Predicting the final spin gives the l = m = 2 ring-down mode frequencyby (8) and the light ring frequency. Both are shown in figure 2. The ISCO and light ringset a natural “merger” epoch which can be searched for by burst search techniques [28] inbetween the inspiral and ring-down matched filter searches. The ring-down frequenciesshown in [16] agree well with these predictions. Although figure 2 is plotted for thenon-spinning case the formalism described above can be easily generalized.It has been shown [29, 30, 32] to be useful to understand un-modeled time-frequencyevolutions in gravitational wave analysis in terms of time-frequency volumes, ∆ f ∆ t .This concept can provide insight into the merger epochs of various systems. We willconsider three timescales for the merger 1) the Newtonian free fall time scale t ff , 2) thequadrupole radiation time scale t quad and 3) the time scale of a test particle followingits geodesic just inside the ISCO of the Kerr space-time (corresponding to the mergerproduct). For comparison we also plot the time-frequency volumes found from the Mass ratio, q M Ω BKL QNMBKL light ring3PN ISCOBKL ISCOEOB p4PN λ = 150 ISCOEOB p4PN λ = 60 ISCOEOB 2PN ISCOSchwarz. ISCO INSPIRALMERGERRING DOWN
Figure 2.
ISCO and QNM frequency estimates for non-spinning binaries as a functionof mass ratio using different methods. Knowing the final spin of the black hole gives theexpected ring-down frequency which agrees with the light ring frequency. The ISCOfrequencies are very different depending on the method. The PN minimum energycondition gives an inconsistent result in the extreme mass ratio limit whereas the theother methods (EOB, BKL) agree with the Schwarzschild ISCO at small values of q.The ISCO and QNM frequencies define a natural merger epoch which can be analyzed,albeit sub-optimally, even without knowing the numerical waveform. figures in [16]. The Newtonian free fall time-frequency volume is,∆ f ISCO ∆ t ff = 2 − / √ M Ω ISCO . (9)The quadrupole time-frequency volume is calculated by assuming that an ISCO doesn’texist and that the system continues to be driven by radiation. To calculate this we usethe quadrupole approximation to the inspiral waveform given by [31], which gives thefrequency evolution as, M Ω = (cid:20) q ) q Mt − t o (cid:21) / (10)This leads to the time frequency volume,∆ f BKL ISCO ∆ t quad = 5 (1 + q ) π q (cid:2) ( M Ω ISCO ) − / − ( M Ω light ) − / (cid:3) × [( M Ω light ) − ( M Ω ISCO )] , (11)where the ISCO frequencies and light ring frequencies are obtained from the BKLapproximation as in figure 2.The merger evolution will not be purely dominated by radiation by virtue thatthere are no circular orbits below the ISCO. The numbers obtained by this estimateshould greatly over predict the time scale for extreme mass ratios where particles wouldtake an asymptotically infinite time to fall in. The free fall time scale will under predictthe time since the system may still complete some (unstable) orbits before merging.The estimate that agrees best with numerical relativity simulations is the infall timeof a test particle falling from a circular orbit just below the ISCO to the light ringof the the merged Kerr black hole. Figure 3 shows the time-frequency volume forthe merger epochs as a function of mass ratio for non-spinning binaries. The figureshows that the numbers taken from the simulations presented in [16] agree well withthe predictions from the test particle in the Kerr space-time. By these estimates themergers from yet un-simulated evolutions (low mass ratio) are likely to have the largesttime frequency content and searches may benefit from the numerical simulations of these[30]. This information should be useful in conducting IMR searches and also in guidingthe construction of analytic full waveform models. It is of course possible to repeatthis analysis for arbitrary spin configurations using the above prescription and others[13, 21, 22, 33]. We will leave further discussion of these matters to future work andwill spend the rest of this work considering the implications of using the ISCO foundfrom the above set of equations as a termination frequency for PN waveforms in initialLIGO data analysis. Mass ratio, q T i m e − f r equen cy v o l u m e Quadrupole radiationKerr particle evolutionNetwonian free fallNR simulations
Figure 3.
An estimate of the time-frequency volume for the merger epoch of non-spinning binary black holes. Numerical relativity results agree well with the time-frequency volume of a test particle falling into the merged Kerr black hole from acircular orbit just inside the ISCO.
3. Possible impact on data analysis for the inspiral phase of compactbinary coalescence searches
As mentioned previously, there is ambiguity on how to define when the inspiral phaseends. However, what is really necessary is the characteristic frequency at which a giventemplate waveform ceases to resemble the numerical simulations. Some searches for lowmass binary systems of non-spinning component masses use the ISCO of a test particleorbiting a Schwarzschild black hole [14]. Other searches use the ISCOs calculated fromexplicit PN energy considerations, and some abandon the use of an ISCO altogetherand use the Schwarzschild light ring as a termination frequency [15].Recall that for low mass ratio ( q ∼
0) systems of non-spinning objects, theSchwarzschild test particle limit is a good approximation for the expected ISCOfrequency since the merger product will be a Schwarzschild-like black hole. However, forsystems with comparable masses and/or spin the true ISCO frequency may be differentsince the non-trivial contribution from the orbital angular momentum will have a strongimpact on the final black hole’s spin, and the space-time in the near merger epoch. Manyhave addressed how well various PN approximants stay faithful to numerical relativitysolutions[11, 41, 42, 43]. Most approximants stay faithful through the SchwarzschildISCO frequency [43]. Some approximants fare extremely well beyond this point [43].The fact that some approximants do remain faithful far beyond the SchwarzschildISCO leads us to examine using the ISCO frequency described in the previous section,equation (5), as the termination frequency for inspiral data analysis. As shown, thisfrequency is consistent with some of the PN and EOB models for predicting the ISCO inthe equal mass regime (fig. 2) and is consistent with exact solutions in the test particlelimit. It also has the advantage of being waveform model, or fit, independent (basedon first principles) and is easy to calculate. We will conservatively model the impactof possible phase errors incurred by using an approximant that doesn’t quite matchnumerical relativity results in our assessment.We use the stationary phase approximation waveforms, which are often employedin searches, and can be defined as,˜ h ( f ) ∝ f − / e i Ψ( f ; M,µ ) , (12)where M is the total mass and µ is the reduced mass [14]. In order to ascertain thephase fidelity we turn to comparisons given in [11] which show for equal mass binarysimulations the phase difference between numerical results and 3.5 PN waveforms asa function of frequency. The figures in [11] indicate a good fidelity in phase throughthe Schwarzschild ISCO frequency. However substantial phase errors may accumulatebetween the Schwarzschild and BKL ISCO frequencies. In order to accurately predictthe impact of integrating PN waveforms to the BKL ISCO we decided to modelthe possibility that secular phase errors could be ± π or more radians between theSchwarzschild and BKL ISCOs. We propose calculating the SNR ratio of an ensembleof waveforms where phase errors are allowed to accumulate linearly between theSchwarzschild ISCO f ( M, q ) and the BKL ISCO f ( M, q ) given by,SNR f SNR f = (cid:42)(cid:18)(cid:90) f f low e πiR n ( f − f ) /f Θ( f − f ) d ff / S n ( f ) (cid:19) / (cid:18)(cid:90) f f low d ff / S n ( f ) (cid:19) − / (cid:43) , (13)where R n is a normally distributed variable with mean 0 and variance 1. f low is 40Hz [6]. ‡ We note that the above calculation is a conservative estimate of the SNR gain for tworeasons: 1) A template with different parameters may match the numerical waveformsbetter, producing better phase agreement. Since search results are maximized over SNRthis is an important possibility. 2) The phase errors do not accumulate linearly betweenthese frequencies, most of the phase error occurs near the BKL ISCO. Using the { M, q } dependent ISCO frequencies as f ( M, q ) and the Schwarzschild ISCO frequencies as f ( M ) we plot the expected SNR ratio for initial LIGO [35] in figure 4, which shows thatfor some total mass and mass ratio combinations there is an appreciable gain in SNR byintegrating to the new ISCO frequencies despite modeled phase errors. This calculationis not intended to suggest that SPA templates cut at the BKL ISCO produce the optimalgain in SNR, since other approximants or frequency cut offs could do better. Instead,it aims to simply illustrate the effect of integrating to the BKL ISCO as a function of M and q for present searches. The mass range is limited to 80 M (cid:12) so that at least twocycles exist in the waveform between 40Hz and the Schwarzschild ISCO frequency.
4. Probing the tidal disruption of neutron stars
If the BKL ISCO truly marks the transition from an inspiral, radiation-dominatedevolution to a dynamical one then it does have some consquence for the ability toprobe the tidal disruption of neutron stars through gravitational waves. Estimation ofgravitational-wave frequencies for tidal disruption of NS-BH binaries may be useful fordetermining various properties of the neutron star, such as the radius [36, 37]. Knowingthe radius of the neutron star would in turn provide information about its equationof state [38, 39]. However, as pointed out in [36], it would be difficult to extractinformation about the disruption unless it occurs before the binary reaches the ISCO.If the disruption occurs afterwards, its signature in the produced gravitational wavesmight be too weak for extracting accurate information. Thus, the “useful” cases arethose for which the tidal-disruption frequency is less than the frequency at the plunge( f td < f plunge ≡ f ISCO ).A more accurate calculation of f plunge would imply a better estimate of the rangeof NS-BH binaries for which the disruption is more plausible to be measured. In thissection we show how that range changes when the formalism of section 2 is applied, ascompared to that in [36]. ‡ We take S n ( f ) to be approximated analytically as [34], S n ( f ) ≈ × − (cid:34)(cid:18) . f (cid:19) − + 0 . (cid:18) f (cid:19) − . + 0 . (cid:18) f (cid:19) + 0 . (cid:35) . (14) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
01 1 . . . . . . . . . . . . . . . . . Total Mass (M sun ) M a ss R a t i o , q
20 30 40 50 60 70 8000.10.20.30.40.50.60.70.80.91
Figure 4.
The SNR ratio defined by equation (13) for the non-spinning case χ = 0.The solid lines include phase errors of the form (13) whereas the dashed lines assumeperfect phase coherence for comparison. The frequencies obtained in [36] correspond to the extreme mass ratio case, q = 0,and non-spinning neutron star, χ NS = 0. In figure 5 we reproduce those results(compare to figure 2 of [36]), together with the values obtained using the formalismpresented in this work. We also assume in this case that the neutron star is non-spinning, but with q = m NS /m BH (cid:54) = 0. The curves of R vs f td are calculated usingthe formula given in equation. (3) of [36], for the same parameters as in that work: m NS = 1 . M (cid:12) ; m BH = 2 . , , , , M (cid:12) . The circles over each curve specify wherethe plunge occurs for different values of χ BH . Filled circles correspond to the extrememass ratio approximation. The meaningful part of each curve will be that to the left ofthe white circles ( f td < f plunge ).The plunge frequencies calculated in [36] for the case with m BH = 2 . M (cid:12) (blackcircles in the right-hand-side curve of figure 5) correspond to retrograde orbits. However,for that mass ratio ( q = 0 . f BKL for thesix cases considered in in that work, models A-F. The BKL frequencies are shown intable 1, together with the corresponding parameters q and M ≡ M BH + M NS ( χ BH and1 f (Hz)400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 R ( K m ) m=40m=80m=20m=10 m=2.5 Figure 5.
Radius of the neutron star vs. the disruption frequency. The circlesindicate the points at which the plunge would occur; the filled circles correspond to f plunge as calculated in [36], included here for comparing. The value of χ BH is indicatednext to the corresponding circle where possible, otherwise a dashed line was used toconnect the value to the circle. Each curve is labeled with the value of the BH mass: m BH = 2 . − (cid:12) . χ NS are set to zero as in [40]). Although the models studied in [40] expand only a smallrange of parameters; we see that, at least in that range, the frequencies obtained withthe BKL approach are consistent with the NR results (See figure 7 of [40]) - that is theypredict in every case a higher value than the measured f cut caused by the tidal effects. Table 1.
BKL frequencies for the same parameters as models A-F of [40]. Thesevalues are consistent with the NR results - that is the BKL ISCO frequency is higherthan the observable f cut from the simulations [40] Model q M ( M (cid:12) ) f BKL (kHz) f cut from [40]A 0.327 5.277 1.38 1.16B 0.327 5.244 1.39 1.41C 0.328 5.311 1.37 0.92D 0.392 4.623 1.65 1.14E 0.392 4.594 1.66 1.40F 0.281 5.929 1.18 1.092
5. Conclusion
The estimate for the spin of the black hole that results from BH-BH, BH-NS, andsome NS-NS mergers found in [13] leads to an estimate for the ISCO, light ring, andquasinormal mode ring-down frequencies of a general compact binary system that iswaveform model independent. Both frequencies have impact on searches for the inspiral,merger and ring-down epochs of the gravitational waves emitted by these systems. Wehave shown that the most interesting merger-epoch time-frequency volumes are forextreme mass ratios. More work may have to be done to model merger epochs forthese systems. The formalism presented here to describe inspiral, merger and ring-down epochs can be extended to arbitrary spin configurations and may help guide theconstruction of better analytic models.As for inspiral epoch searches alone, we have shown that for symmetric mass,non-spinning systems with greater than 45 M (cid:12) total mass there can be a 30% gain inSNR over using the Schwarzschild ISCO radius as is done currently [14] even includingconservative phase errors. It is worth noticing that more general fittings can becomputed in order to cover the full range of parameters ( q, χ , χ ) within the BKLapproximation. Additionally we discussed cases where neutron star tidal disruptionmay be observed more easily through gravitational waves. We would like to stress thatthis work is open ended. Here we have only considered spins that are aligned and anti-aligned with the orbital angular momentum. It is certainly possible to extend this workto make predictions about more general cases, all of which should be compared withnumerical relativity results as they are produced.
6. Acknowledgements
The authors would like to acknowledge L. Lehner for suggesting this project andfor invaluable discussions. The authors also thank L. Lehner and F. Pretorius forproviding the test particle time of flight used in section 2. Gabriela Gonz´alez, PatrickBrady, Alessandra Buonanno and Jolien Creighton provided motivating discussions andinsightful comments. C. Hanna would like to thank the LIGO Scientific CollaborationCompact Binary Coalescence (LIGO CBC) working group. This work was supported inpart by NSF grants PHY-0605496, PHY-0653369 and PHY-0653375 to Louisiana StateUniversity and PHY-0603762 to the University of Maryland. C. Hanna would like tothank the Kavli Institute for Theoretical Physics, for their hospitality, where some ofthis work was completed. The Kavli Institute is supported by NSF grant PHY05-51164.
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