4.4 Kludge waveforms: Results and comparison
4.4.2 Time-domain comparison
-1 -0.5 0 0.5 1
0 500 1000 1500 2000
h+ D/µ
p=6.5M e=0.5 a=0.5M θd=90(deg)
-1 -0.5 0 0.5 1
h+ D/µ
p=6.5M e=0.5 a=0.5M θd=90(deg)
-1 -0.5 0 0.5 1
0 500 1000 1500 2000 2500 3000
h+ D/µ
p=6.5M e=0.5 a=0.5M θd=45(deg) TB waveform
-1 -0.5 0 0.5 1
h+ D/µ
p=6.5M e=0.5 a=0.5M θd=45(deg)
-1 -0.5 0 0.5 1
0 200 400 600 800
h+ D/µ
time (M)
p=3.5M e=0.4 a=0.99M θd=90(deg) NK waveform
Figure 4.2: Comparing TB and NK (Press-formula numerical kludge) waveforms (black and red curves, respectively) for equatorial orbits and for an observer at a latitudinal positionθ=45◦or 90◦. Orbital parameters are listed above each graph. The waveforms are scaled in units ofD/µ whereDis the radial distance of the observation point from the source andµis the test-body’s mass. Thex-axis measures retarded time (in units ofM) and we are showing the “+” polarization of the GW in each case. The overlaps between the NK and TB waveforms are 0.979,0.990,0.507 going from the top figure down.
-1 -0.5 0 0.5 1
0 300 600 900 1200
h+ D/µ
r=5M a=0.99M ι=60(deg) θd=90(deg)
-1 -0.5 0 0.5 1
0 200 400 600 800 1000 1200
h+ D/µ
time(M) r=5.05M a=0.99M ι=60(deg) θd=90(deg)
TB waveform NK waveforms
Figure 4.3: Comparing TB and NK (Press-formula numerical kludge) waveforms (black and red curves, respectively) for circular-inclined orbits and for an observer at a latitudinal positionθ = 90◦. Orbital parameters are listed above each graph. The waveforms are scaled in units ofD/µ whereDis the radial distance of the observation point from the source andµis the test-body’s mass. The x-axis measures retarded time (in units ofM) and we again show the plus polarization of the GW. The overlaps between the NK and TB waveforms are 0.888 for the top figure and 0.961 for the bottom figure.
the TB waveforms.
One also notices that for certain parts of the waveforms (e.g., Fig. 4.3), there is a disagreement in the amplitude while the phase is accurately reproduced. This amplitude discrepancy is periodic, i.e., the points where the amplitude is poorly reproduced occur at regular intervals. This phenomenon exists for all NK wave- forms (quadrupole, quadudrupole-octupole, and Press), but is less pronounced in the Press waveforms. The amplitude disagreement suggests that NK waveforms are missing some periodic components, which again can be attributed to the truncated expansions in multipole moments (for quadrupole and quadrupole-octupole waveforms), as well as the lack of back-scattering and other strong gravity features in all NK waveforms.
Nevertheless, in order to construct templates that have a high overlap with the true signals, it is much more important that the waveform phase is faithfully reproduced than the waveform amplitude. The waveform phase is determined by the orbit generating the gravitational radiation. The fact that the kludge waveforms are based on true geodesic orbits is presumably the reason that we find,post facto, such impressively high overlaps with TB waveforms, especially for those from circular-inclined orbits.
A comprehensive set of data for the overlaps between NK and TB waveforms is given in Tables 4.1, 4.2 and 4.3. These were computed using the overlap function described in Section 4.4.1 and assuming a central black hole mass ofM=106M. This was chosen since preliminary event rate estimates suggest the inspirals
-0.4 -0.2 0 0.2 0.4
0 2000 4000 6000 8000 10000
h+ D/µ
p=12M e=0.3 a=0.9M ι=140(deg) θd=60(deg)
-1 -0.5 0 0.5 1
0 1000 2000 3000 4000 5000 6000
h+ D/µ
time(M) p=6M e=0.7 a=0.9M ι=60(deg) θd=90(deg)
TB waveform NK waveforms
Figure 4.4: Comparing TB and NK (Press-formula numerical kludge) waveforms (black and red curves, respectively) for generic orbits. Orbital parameters are listed above each graph. The waveforms are scaled in units ofD/µwhereDis the radial distance of the observation point from the source andµis the test-body’s mass. The x-axis measures retarded time (in units ofM). The overlaps between the NK and TB waveforms are 0.994 and 0.970 for the top and bottom figures respectively.
of∼10MBHs into∼106MSMBHs will dominate the LISA detection rate [21]. These tables indicate that if the orbital periastron isrp &5M, the overlap between TB waveforms and both the quadrupole-octupole and Press waveforms stays above∼0.95. We also find that both these expressions have better performance than the pure quadrupole waveforms (4.17), but there is little difference between the quadrupole-octupole and Press waveforms. The NK and TB waveforms begin to deviate significantly for strong-field, ultra-relativistic orbits withrp . 4M, with the overlap dropping to∼ 50% for orbits that come very close to the horizon.
Disappointingly, the Press waveforms do not seem to do much better than the quadrupole-octupole waveforms in this strong field regime, despite the inclusion of additional multipole components. The Press waveforms do perform consistently better for circular orbits and weak-field eccentric orbits, but the difference between the two approaches is usually small. We therefore conclude that the quadrupole-octupole waveform model is sufficient and there is not much gain from using the computationally more intensive Press formula.6
To summarize, we find that NK waveforms are accurate—and very quick to generate—substitutes for TB waveforms for all orbits around a Schwarzschild black hole withe.1/3 right up until the final plunge. This result follows from applying the conditionrp&5Mat the Schwarzschild separatrixpS =(6+2e)M. Compu-
6For detection purpose, one important practical reason for not using the Press formula has to do with the separation of intrinsic and extrinsic parameters. In [21], by employing a pure quadrupole gravitational waveform, the source orientation angles and the azimuthal phase of the inspiraling body can be treated as extrinsic parameters, which do not affect the intrinsic radiation of the source, but only how it projects onto the detector. By constrast, the Press formula [Eq. (4.16), (4.31)], which is valid for an extended source, inherently contains those parameters vianaand the retarded time expressiont0p=t− |x−x0p|, making them “more expensive” to search over.
p/M1e2ι3(deg)a/M4Θ5(deg)overlap(+)6overlapwithWD(+)duration(M)8
QuadQuad-OctPressQuadQuad-OctPress1.70.100.99900.840.7720.74120001.70.300.99900.760.5570.5007001.90.500.99900.5440.5700.5477001.90.500.99900.5230.4840.44520002.110.700.99450.5620.5660.5627002.20.700.99900.5260.4960.4587002.50.100.99900.9060.8530.82720002.50.500.99900.6710.6650.6517003.50.400.99900.5880.5240.50720003.50.400.99450.6240.5980.59350005.10.500.5900.8560.9620.9670.8560.9610.9687005.50.500.5900.8640.9640.9730.8620.9620.97320006.00.400.5900.8710.9700.9800.8640.9670.97920006.00.500.5900.8580.9660.9740.8550.9630.97320006.50.500.5900.8700.9700.9790.8640.9680.97820006.50.500.5450.9370.9870.9900.9320.9860.990800010.00.31800.99900.8640.9610.9660.8060.9430.954800010.00.31800.99450.9220.9710.9690.8830.9570.956800010.40.51800.9900.9980.9980.9990.9970.9990.998200010.50.51800.99900.8780.9750.9820.8560.9680.978200015.00.400.5900.8240.9630.9680.600.8780.881800015.00.400.99900.8240.9610.9630.6030.8740.8608000
Table4.1:NumericaldataforoverlapsbetweenTBandkludgewaveforms:Equatorial-eccentricKerrorbits.DataisnotshownwithWDconfusionnoiseforthetenorbitsintheverystrongfieldregime.Inthisregime,noneofthekludgewaveformsreproducetheTBwaveformsverywell,andthisiscompoundedwhenthedominantharmonicsaresuppressedbywhitedwarfconfusionnoise.Theoverlapsareuniformlypooranduninformative,sowedonotincludethem.
asemi-latusrectumbeccentricitycinclinationangledspineobservationpoint,φ=0alwaysfoverlapbetween“+”polarizationofTBwaveformwithquadrupole(“Quad”),quadrupole-octupole(“Quad-Oct”)andPress(“Press”)kludgewaveformsgoverlapbetween“×”polarizationshwaveformduration
p/Meι(deg)a/MΘ(deg)overlap(+)overlap(×)overlapwithWD(×)duration(M)QuadQuad-OctPressQuadQuad-OctPressQuadQuad-OctPress5.00300.5450.9440.9840.9900.9460.9840.9900.9450.9840.99030005.00300.5900.8990.9740.9840.8910.9710.9840.890.9690.98330005.00300.99450.9290.9690.9750.930.9690.9750.9270.9670.97430005.00300.99900.9040.9580.9640.90.9570.9660.8950.9540.96530005.050600.5450.9240.9670.9730.9220.9690.9770.9220.9680.97630005.050600.5900.910.9550.9610.9110.9630.9730.9070.9610.97230005.00600.99450.8570.9120.9170.860.9170.9250.8570.9130.92130005.00600.99900.8540.8820.8880.870.9120.9250.8620.9070.922300010.00300.5450.930.9890.9950.9360.990.9940.7510.9570.975800010.00300.5900.890.9810.9900.9010.9810.9900.630.930.969700010.00300.99450.920.980.9860.930.980.9860.6740.920.946800010.00300.99900.8840.9740.9820.9150.9750.9830.3710.8920.963700010.00450.7450.9320.9800.9810.9360.9830.9870.750.9350.953800010.00450.7900.9220.9720.9720.920.980.9870.680.9290.969800010.00600.5450.9140.9810.9870.9120.9820.9900.6910.9370.969800010.00600.5900.9450.980.9810.9320.9810.9860.7320.9250.948700010.00600.99450.8730.9530.9580.8750.9570.9650.6110.8750.906800010.00600.99900.9230.9410.9370.9150.9590.9650.6750.860.893700020.00300.5450.9340.9870.9820.9380.9890.9890.9570.9870.9025000020.00300.5900.8890.9700.9220.8930.9740.9620.920.9840.9003000020.00300.99450.930.980.9510.940.9880.9920.9510.9870.9775000020.00300.99900.8950.9680.9070.9140.9790.9880.9330.9790.9513000020.00600.5450.920.9750.9660.9160.9810.9950.9520.9580.9905000020.00600.5900.950.9730.9580.9340.9760.9720.950.9670.9573000020.00600.99450.8950.9630.9620.8920.9670.9730.9340.9510.8965000020.00600.99900.9540.960.9380.9370.9720.9780.9630.9460.94330000
Table4.2:NumericaldataforoverlapsbetweenTBandkludgewaveforms:Inclined-circularKerrorbits.
p/Meι(deg)a/MΘ(deg)overlap(+)overlap(×)overlapwithWD(×)duration(M)QuadQuad-OctPressQuadQuad-OctPressQuadQuad-OctPress6.00.120.13640.9900.9120.9820.9930.910.9840.9960.8940.980.995150006.00.120.13640.9600.9350.9880.9960.9390.990.9970.9290.9890.997150006.00.120.13640.9300.9720.9950.9980.9720.9950.9980.9680.9950.998150006.00.520.1030.9900.890.9670.9730.890.9720.9800.8750.9680.979150006.00.520.1030.9600.9160.9750.9800.9120.9780.9820.9130.9760.981150006.00.520.1030.9300.9580.9850.9850.9590.9850.9860.9550.9840.985150006.00.160.14610.9900.9510.9890.9930.9410.9880.9950.9330.9870.995100006.00.160.14610.9600.9580.9890.9940.950.9880.9950.9460.9870.996100006.00.160.14610.9300.9430.9860.9960.9420.9870.9960.9410.9860.996100006.00.560.11080.9900.9340.9750.9800.9190.9740.9820.9120.9710.981150006.00.560.11080.9600.9390.9740.9820.9270.9720.9840.9240.970.984150006.00.560.11080.9300.9170.9710.9780.9160.9710.9790.9150.9710.979150006.00.760.07550.9900.9260.9660.9700.9110.9670.9740.9060.9660.975200006.00.760.07550.9600.9280.9670.9710.9190.9660.9740.9160.9660.974200006.00.760.07550.9300.8970.9630.9660.8970.9630.9670.8970.9630.9682000012.00.1119.95860.9900.9160.9870.9940.9110.9880.9980.6270.9220.9851500012.00.1119.95860.9600.9260.9880.9910.920.9890.9980.6340.9260.9851500012.00.1119.95860.9300.90.9830.9940.8990.9860.9940.5770.910.9561500012.00.5119.96860.9900.9380.9880.9920.9340.9880.9950.8890.9790.9931500012.00.5119.96860.9600.9420.9880.9930.9360.9870.9940.8890.9770.9911500012.00.5119.96860.9300.9240.9840.9930.9240.9860.9940.8620.9710.9881500012.00.7119.97860.9900.9350.9860.9920.9330.9870.9930.9040.9810.9912500012.00.7119.97860.9600.9400.9870.9920.9340.9850.9920.9040.9780.9902500012.00.7119.97860.9300.9260.9830.9920.9240.9830.9900.8840.9730.9862500012.00.3139.95970.9900.90.9850.9920.8910.9840.9960.7510.9550.9911500012.00.3139.95970.9600.9380.9910.9940.9310.990.9970.8270.9720.9931500012.00.3139.95970.9300.9480.9920.9940.9490.9930.9980.8590.9790.9941500012.00.1159.97280.9900.8360.9730.9920.8380.9740.9930.4940.8870.9681500012.00.1159.97280.9600.8810.9840.9950.8890.9870.9980.5490.9260.9901500012.00.1159.97280.9300.9470.9940.9970.9470.9950.9970.7260.960.9681500012.00.5159.97930.9900.8760.9760.9910.8770.9770.9950.8080.9610.9931500012.00.5159.97930.9600.9120.9850.9940.9190.9880.9960.8620.9770.9941500012.00.5159.97930.9300.9610.9940.9960.9620.9950.9980.9320.9910.99615000
Table4.3:NumericaldataforoverlapsbetweenTBandkludgewaveforms:GenericKerrorbits.
tations of inspirals into Schwarzschild black holes [11] indicate that, in many cases, the residual eccentricity at plunge will be small, so that kludge waveforms will be suitable for the majority of Schwarzschild inspi- rals. For retrograde orbits around Kerr black holes (90◦ ≤ι≤180◦), the periapse moves out to even larger radii, so that even weaker restrictions can be imposed on the eccentricity. In contrast, for prograde orbits (0◦ ≤ι≤90◦), an increased black hole spin allows stable orbits to exist much deeper in the strong field. As a→M, the separatrix of equatorial orbits asymptotically goes topproK (e)=(1+e)M[16]. Kludge waveforms are not very good in this regime, with overlaps∼50%; fortunately, this corresponds to a comparatively small region of parameter space.
If the overlap between a given signal and the best-fit template in a search bank is less than 1, this leads to a decrease in the maximum distance to which that signal can be detected, and a corresponding reduction in event rate. For the purposes of detection, overlaps as low as 50% might be considered good enough, if the astrophysical event rate is sufficiently large [21]; but for very rare events it is conventional to seek overlaps of 97% or higher (rate losses of no more than 10%). For the purposes of parameter estimation, overlaps
>∼95% will be required in general. It is clear from the results in this paper that overlaps>∼97% or 95%
are only partially achievable by the existing family of kludge waveforms. Nonetheless, these waveforms might be useful for LISA data analysis as search or detection templates over some (perhaps a large part) of the astrophysically relevant portion of the{a/M,p,e, ι}parameter space. The waveforms may also provide sufficiently accurate estimation of the source parameters (in certain regions of parameter space) that they could be used as the first stage in a hierarchical search. The purpose of such a search would be to identify
“interesting” regions of parameter space for follow up with more accurate waveforms.
One should bear in mind, however, that the regions where kludge waveforms are good enough must be identified more carefully than in this paper by comparison to accurateinspiralwaveforms. As we have discussed earlier, the flat-space emission formulas used in the construction of the NK waveforms ignore all effects of scattering from the background curvature. These “tail” terms make a significant contribution to the waveform structure, and build up over the course of an inspiral. Although we have found good overlaps with geodesic waveforms here (see [1] for overlap between circular-inclined inspiral waveforms), comparisons to generic inspiral orbits are required to properly assess the importance of the tail terms. Accurate, self-force waveforms for such orbits will not be available for a few years and only then will it be possible to firmly demarcate the regime of usefulness of the present, or further improved, NK waveforms.