Validity of the Effective Fisher matrix for parameter estimation analysis: Comparing to the analytic Fisher matrix
aa r X i v : . [ g r- q c ] O c t Validity of the Effective Fisher matrix for parameter estimation analysis: Comparingto the analytic Fisher matrix
Hee-Suk Cho and Chang-Hwan Lee
Department of Physics, Pusan National University, Busan 609-735, Korea (Dated: October 2, 2018)The effective Fisher matrix method recently introduced by Cho et al . [1] is a semi-analytic ap-proach to the Fisher matrix, in which a local overlap surface is fitted by using a quadratic fittingfunction. Mathematically, the effective Fisher matrix should be consistent with the analytic oneat the infinitesimal fitting scale. In this work, using the frequency-domain waveform (TaylorF2),we give brief comparison results between the effective and analytic Fisher matrices for several non-spinning binaries consisting of binary neutron stars with masses of (1.4, 1.4) M ⊙ , black hole-neutronstar of (1.4, 10) M ⊙ , and binary black holes of (5, 5) and (10, 10) M ⊙ for a fixed signal to noiseratio (SNR=20) and show a good consistency between two methods. We also give a comparisonresult for an aligned-spin black hole-neutron star binary with a black hole spin of χ = 1, where wedefine new mass parameters ( M c , η − , χ / ) to find good fitting functions to the overlap surface.The effective Fisher matrix can also be computed by using the time-domain waveforms which aregenerally more accurate than frequency-domain waveform. We show comparison results between thefrequency-domain and time-domain waveforms (TaylorT4) for both the non-spinning aligned-spinbinaries. PACS numbers: 04.30.–w, 04.80.Nn, 95.55.Ym, 02.70.–c, 07.05.Kf
I. INTRODUCTION
In gravitational wave data analysis, the parameter esti-mation methods are implemented to figure out the phys-ical parameters of the wave source. The relevant infor-mation is the distribution of the measured values and theerror bounds on their variances. There are several meth-ods for parameter estimation that are based on MonteCarlo simulations. They are able to search the whole pa-rameter space but in general computationally very expen-sive. One of those methods is the Markov chain MonteCarlo (MCMC) [2–6], which involves the Bayesian anal-ysis framework.The Fisher matrix method has been generally usedto estimate the error bounds [6–10]. The inverse ofthe Fisher matrix represents the covariance matrix, fromwhich the error bounds can be directly derived as well asthe correlations between the parameters. Although theFisher matrix is valid only for the high signal to noiseratio (SNR), this method is very useful because that cancompute measurement accuracies very quickly comparedto the MCMC.The Fisher matrix prediction has only used thefrequency-domain waveform, so called TaylorF2 (orSPA), because of the possibility of deriving analyticalexpression of the Fisher matrix. The TaylorF2 wave-form has only been applied for comparison betweenthe MCMC and Fisher matrix so far [11, 12]. Thefrequency-domain waveforms require much less compu-tational time for the MCMC runs. However, the time-domain waveforms are basically more accurate becausethat do not assume the stationary phase approximation.The MCMC methods have used various time-domainwaveforms for more accurate parameter estimation per-formance [1, 3, 6, 13, 14]. Motivated by that, first and foremost, Cho et al. [1] introduced an effective method,with which they calculated the Fisher matrices using thetime-domain waveform, TaylorT4. When using the time-domain waveforms for the Fisher matrix, it may be verycomplicated to obtain the derivatives of the waveformsbecause the Fourier transform of the time-domain wave-form is not the analytic function but numerical data. Theeffective method can avoid the difficulty by fitting thelocal overlap surface, where the Fisher matrix can be de-rived from the quadratic fitting function.The analytic Fisher matrix method is straightforwardbecause the result can be derived analytically from theanalytic waveform function except for overlap integra-tion. While, the effective method involves the fittingfunction which is manually calculated, and the overlapintegration can be done by the same manner as in the an-alytic method. These two methods should give the sameresults under certain physical conditions. Therefore, inorder to accept the effective Fisher matrix generally, thefaithfulness of that should be proved by comparing to theanalytic Fisher matrix. Several works [2, 11, 12] showedinconsistency between the Fisher matrix and MCMC re-sults. However, the authors emphasize that the mainpurpose of this work is not to prove the adequacy of theFisher matrix in parameter estimation analysis but toinvestigate the possibility of the effective method to cal-culate the Fisher matrix.In Sec. II, we review the TaylorF2 waveform and theoverlap formalism. We review the analytic and effectiveFisher matrix methods in Sec. III. We show our compari-son results between the analytic and effective methods aswell as the results between the TaylorF2 and TaylorT4waveforms. In Sec. V, we summarize our results andgive some discussions. Throughout this paper, all massparameters are in units of the solar mass ( M ⊙ ) unlessotherwise noticed, and we use a geometrized unit, where G = c = 1 . II. WAVE FUNCTION AND OVERLAP
The TaylorF2 waveform is given by˜ h ( f ) = Af − / e i Ψ( f ) , (1)where A ∝ M c / Θ(angle) / D, M c is a chirp mass, D is the luminosity distance of the binary, and Ψ( f ) is theorbital phase. Θ(angle) is a function of the orbital ori-entation with respect to the detector network in termsof the sky position (RA, DEC), orbital inclination ( ι ),and the wave polarization ( ψ ). If we assume the fixedSNR of the waveforms, all information of the waveformis coming from the wave phase. The phasing factor con-sists of the coalescence time ( t c ) and termination phase( φ ), and the remaining intrinsic parameters ( λ int ):Ψ( f ) = 2 πf t c − φ − π η F ( λ int , f ) , (2)where t c can be chosen arbitrarily, F ( λ int , f ) can be rep-resented by the post-Newtonian (pN) expansion, whichis provided in [8] up to 3.5 pN order for the non-spinningcase, where λ int = {M c , η } , η is a symmetric mass ratio.For the aligned-spin case, a dimensionless spin parame-ter χ is included, so λ int = {M c , η, χ } , and F ( λ int , f ) isprovided in [7, 15, 16].The termination phase ( φ ) is related to the coales-cence phase ( φ c ) by [17, 18]2 φ = 2 φ c − arctan (cid:18) F × F + ι ι (cid:19) , (3)where F × and F + are the antenna response functions de-pending on the angle parameters ( ι, ψ , RA, DEC). Forsimplicity, we consider a fixed binary position, then φ is a function of ι, ψ , and the coalescence phase φ c (thecoalescence phase can also be chosen arbitrarily). In sev-eral works (e.g., [7, 16]) φ has been assumed to be anarbitrary constant when calculating Fisher matrices, andthey considered only one angle parameter as componentsin the Fisher matrices. However, in order to take into ac-count more than two angle parameters among ( ι, ψ, φ c ),one should define the φ as a function of the angle param-eters ( ι, ψ, φ c ). For example, if the binary is optimallyplaced and orientated (i.e., ι =RA=DEC= 0), the phase φ is exactly degenerated into φ c and ψ by a function of φ = φ c − ψ . In this case, the Fisher matrix is singularand the inverse matrix can not be defined. The correla-tion between these two parameters becomes reduced asthe ι increases. If ι = π/ φ is equal to the arbitraryconstant φ c and other angle parameters can be removedfrom the wave phase equation. In this work, we only con-sider the fixed binary orientation, so φ is assumed to bethe same as in the previous works, then the wave phase in Eq. (2) is determined by a combination of the parame-ters ( λ int , t c , φ c ). However, when computing the analyticFisher matrix which considers both φ c and ψ with otherphysical parameters of interest, one should not set φ equal to φ c in general.The overlap between a signal ( h s ) and a template ( h t )is defined by h h s | h t i = 4Re Z ∞ ˜ h s ( f )˜ h t ( f ) ∗ S n ( f ) df, (4)where ˜ h ( f ) is the Fourier transform of h ( t ). Note that theinverse Fourier transform will compute the overlap for allpossible coalescence times of h t at once [18]. In addition,by taking the absolute value of the complex number wecan maximize the overlap over all possible coalescencephases [18],max t c ,φ c h h s | h t i ≡ (cid:12)(cid:12)(cid:12)(cid:12) Z ∞ ˜ h s ( f )˜ h t ( f ) ∗ S n ( f ) e πift df (cid:12)(cid:12)(cid:12)(cid:12) . (5)Finally, the normalized overlap is defined by P ( h s , h t ) = max t c ,φ c h h s | h t i p h h s | h s ih h t | h t i . (6)In Eq. (5), we assume the analytic initial LIGO sensitiv-ity curve [19, 20], which takes the form: S h ( f ) = S (cid:20)(cid:18) . ff (cid:19) (7)+ 0 . (cid:18) ff (cid:19) − . + 0 .
52 + 0 . (cid:18) ff (cid:19) (cid:19)(cid:21) , where f = 150 Hz, and S = 9 × − Hz − . III. EFFECTIVE FISHER MATRIX:SEMI-ANALYTIC APPROACH
The Fisher matrix is defined byΓ ij = (cid:28) ∂h∂λ i (cid:12)(cid:12)(cid:12)(cid:12) ∂h∂λ j (cid:29) , (8)where λ i = { λ int , φ c , t c } . In this equation, since the Tay-lorF2 is an analytic function, derivatives with respect tothe parameters can be analytically computed, and theFisher matrix can be calculated by integrating deriva-tives of two functions numerically. The analytic resultsof Fisher matrices in this work are computed by usingthe software package Mathematica .On the other hand, a Fourier transform of the time-domain waveform is only numerical data. In this case, itis very complicated to obtain the derivatives, so almostimpossible to compute the Fisher matrix. However, inEq. (8), the derivatives involves with respect to the physi-cal parameters, while the overlap integral with respect tothe frequency, therefore, both computations are formallycommutable. Then, the Fisher matrix can be directly de-rived by the log likelihood (ln L ), and the log likelihoodis a function of the overlap ( P ) [1, 21, 22]:Γ ij = − ∂ ln L ( λ ) ∂λ i ∂λ j = ρ ∂ (1 − P ) ∂λ i ∂λ j (cid:12)(cid:12)(cid:12)(cid:12) λ i ,λ j = λ , (9)where λ is the fiducial value of each parameter and ρ is the SNR. Using this expression, one can compute theFisher matrix by differentiating the overlap with respectto the corresponding parameters at the position of thefiducial values of parameters. We still have to calcu-late the derivatives of the overlap, which is also numer-ical data. However, for Gaussian noise and high SNR,since the likelihood function is a normal distribution as-suming a flat prior [23, 24], a quadratic fitting functionbest fits the log likelihood. So if we find an analytic fit-ting function ( P ∗ ) at a physically appropriate scale, thederivatives of the fitting function can be analytically ob-tained. Using this semi-analytic method, Cho et al. [1]introduced the effective Fisher matrix asΓ eff = − ∂ P ∗ ∂λ i ∂λ j (cid:12)(cid:12)(cid:12)(cid:12) λ i ,λ j = λ ≃ Γ ij /ρ . (10)Note that in the infinitesimal fitting scale, the effectiveFisher matrix exactly reflects the analytic result in Eq.(8). In this work, we choose very localized overlap sur-faces in the range of P > . ij ), and the measurementerror ( σ i ) of each parameter and correlation coefficient( c ij ) between two parameters are obtained by σ i = p Σ ii , c ij = Σ ij p Σ ii Σ jj . (11)Note that the covariance matrix is a inverse of theFisher matrix (not the effective Fisher matrix), so de-pends on both the SNR and effective Fisher matrix asΣ ij = ( ρ Γ eff ) − , consequently the parameter estimationerror also depends on SNR simply as σ i ( ρ ) = σ i (1) /ρ .In this work, we assume the SNR to be ρ = 20. Thecorrelation coefficient ( c ij ) is ρ -independent. IV. RESULT: COMPARISON TO THEANALYTIC FISHER MATRIX
In this work, we take into account four non-spinningbinary models, whose mass components are summarizedin Table. I. For the aligned-spin case, we consider theBHNS binary with a black hole spin of χ = 1 where χ is a dimensionless spin parameter, so the black hole ismaximally rotating.We use the TaylorF2 waveform that is implementedin the LIGO Algorithm Library (LAL) [25] to calculatethe overlap surfaces for the effective Fisher matrices. We BNS BHNS BBH1 BBH2 m , m M c η f LSO [Hz] 1570 386 440 220 T chirp [s] 53.54 11.97 6.42 2.02TABLE I: Fiducial values of the mass parameters forthe non-spinning binaries.
The frequencies at the last sta-ble orbit ( f LSO ) are calculated by Eq. (12), the “chirp time” T chirp is determined by Eq. (14). consider only the restricted waveforms with the phaseterm up to 3.5 pN for both the non-spinning and aligned-spin waveforms, where only the 1.5 pN spin-orbit phasecorrection is contained in the aligned-spin case. Themaximum frequency cutoff for the TaylorF2 waveform istaken when the binary hits the “last stable orbit (LSO)”.For simplicity, we assume the expression for a test par-ticle orbiting a Schwarzschild black hole with a mass of M = m + m for both the non-spinning and aligned-spincases: f LSO = 16 / πM . (12)In many previous works, the minimum frequency hasbeen taken to be 40 Hz according to the noise sensitivitycurve, but we choose 30 Hz not to lose any additional in-formation in the lower frequency region. O’Shaughnessy et al. [6] showed a non trivial difference between the min-imum frequencies of 30 and 40 Hz (e.g., see figure. 8theirin) using the time-domain waveform, TaylorT4. Wealso give some similar results for the TaylorF2 waveformincluding a dependence on the upper frequencies in Ap-pendix A.The multivariate covariance matrix is very sensitive tothe components of Fisher matrix, so high precision num-bers in the Fisher matrix should be presented to producethe exact inverse Fisher matrix. In this work, it is moreconvenient to represent the covariance matrices insteadof Fisher matrices. Exact components of the Fisher ma-trix can be determined by an inverse of the covariancematrix. A. Non-spinning binaries
As stated in the previous section, we find that the localoverlap surfaces are almost quadratic at the region of
P > .
99, so the Fisher matrix does not depend on thefitting scale. As an example, for the BHNS binary thefitting function is P ∗ = 1 − [1738( δ M c ) − δ M c )( δη ) + 6676( δη ) ] . (13)Figure. 1 shows the overlap contours and correspond-ing fitting ellipses using this fitting function. At the - - - ∆ M C @ M Ÿ D ∆ Η FIG. 1:
Comparison between overlap contours andcorresponding quadratic fittings for BHNS binary,showing the quadratic overlap surface. δλ is definedby a difference from the fiducial value in the Table I. Thedotted lines correspond to the overlap contours and the solidlines correspond to the fitting ellipses using eq.(13). scale of P > .
999 the overlap surface is almost exactlyquadratic. In this work, unless otherwise noted, we as-sume the fitting scale of
P > .
999 to calculate the fittingfunctions ( P ∗ ).In Table. II, we summarize the comparison results be-tween the effective and analytic Fisher matrices for thenon-spinning binary models. For all binary models, wefind a very good agreement between two methods. Es-pecially, for the BNS and BHNS models the correlationsand measurement errors for both methods are almost ex-actly the same.On the other hand, although we assumed the sameSNR ( ρ = 20) for all binary systems, the fractional errorsof parameter measurement are distributed quite broadlyby ranges of 0 . − .
7% and 0 . − .
5% for M c and η ,respectively. We see that the accuracy is related with the“chirp time” of the binary system, which is the amountof time that the binary system will take from a minimumfrequency ( f min ) to coalescence. At the leading pN order,the chirp time is given by [18] T chirp = 5256 M ( πf min M ) − / η = 5256 ( πf min ) − / M c / . (14)The total number of cycles of a binary is proportional to the chirp time for a given f min [8]: N cycle = 132 ( πf min M ) − / πη = 132 f min ( πf min ) − / M c / = 85 f min T chirp . (15)Roughly speaking, as the the time of inspiral is longer,the more number of cycles the signal has, and the moreinformation of a wave signal can be accumulated, con-sequently the measurement accuracy can be improved.Note that, the chirp mass is a function of not only a to-tal mass but also a symmetric mass ratio, so the higher f max does not always mean the longer T chips . For ex-ample, the total mass of the BHNS is similar to that ofBBH1 but the chirp time is about twice longer, givingsmaller errors by a factor of a half for both M c and η .To investigate the correlation of parameter estima-tion performance with detector characteristics, Arun etal. [8] considered the number of “useful cycles” instead ofthe N cycle . They showed various results using advancedLIGO, initial LIGO (where f min = 40 Hz), and VIRGOdetectors for NSNS, BHNS, and BBH2 models varyingpost-Newtonian orders of the TaylorF2 waveform. In ad-dition to these three models, Cokelaer [11] also providederror for a BBH2 system assuming the initial LIGO sen-sitivity curve (where f min = 40 Hz). B. Aligned-spin BHNS binary
We select the BHNS binary model in Table. I for thecase of aligned-spin binary. In general, the NS spin canbe neglected, and we need one additional parameter χ in the function of waveforms, so the number of intrinsicparameters of interest is three, then the fitting functionshould be a 3-dimensional ellipsoid. We assume a max-imally rotating black hole ( χ = 1), then a shape of theoverlap surface is a very long and thin ellipsoid. We find,unfortunately, the surface is not symmetric even at thefine scale of P = 0 . P > . φ c and t c . That means we need longer computing timefor smaller fitting scales.On the other hand, fortunately, by choosing a newparameter coordinate, the overlap surface can be sym-metric at the scale of P > . η → η − and χ → χ / . Fig. 2 shows projections of the overlap el-lipsoid, where P = 0 . M c , η, χ )and ( M c , η − , χ / ) coordinates. Like the case of non-spinning binary in Fig. 1, the fitting function is in good ææ M C @ M Ÿ D Η ææ M C @ M Ÿ D Η - ææ Χ Η ææ Χ (cid:144) Η - ææ Χ M C @ M Ÿ D ææ Χ (cid:144) M C @ M Ÿ D FIG. 2:
Projections of the overlap ellipsoid ( P = 0 . )and their fitting functions for the aligned-spin BHNSbinary. Dotted lines indicate the overlap contours and solidlines correspond to their quadratic fitting ellipses. Large dotsindicate the fiducial parameter values. Note that the fittingfunctions are in good agreement with the overlap contourswhen using the new parameters ( M c , η − , χ / ). agreement with the overlap surface by using the new co-ordinate. In Table. III, we give the results of the effec-tive method at different scales, and using different coor-dinates of ( M c , η, χ ) and ( M c , η − , χ / ), also give thecomparison to the analytic results. We can get a moreexact fitting function by decreasing the fitting scale. But,by using new parameters instead of decreasing the fittingscale, the consistency between the effective and analyticmethods can be improved. Using the new parameters,the results for both methods are in perfect agreement atthe scale of P > . C. Comparison to the time-domain waveform
O’Shoughnessy et al. [6] investigated the effectiveFisher matrices using the time-domain waveform, Tay-lorT4 implemented in the LAL [25], for the non-spinningand aligned-spin BHNS binaries. They used the initialLIGO sensitivity curve and assumed f min = 30 Hz as thesame conditions as in this work. We summarize someof their results with our results for comparison of Fishermatrices between the TaylorF2 and TaylorT4 waveformsin Table. IV. We find small differences between TaylorF2and TaylorT4 waveforms for both the non-spinning andaligned-spin binaries.The TaylorT4 waveform is terminated when the binaryreaches the “minimum energy circular orbit (MECO)”[26, 27], so f max = f MECO . With this waveforms, theMECO frequencies are different depending on the BHspin. For example, f MECO ∼
560 and ∼
890 Hz for thenon-spinning and aligned-spin (where only the 1.5 pNspin-orbit phase correction is contained) BHNS binaries,respectively. For exact comparison, in Table. IV, we setthe f max of the TaylorF2 waveform to be the same fre-quency as the f MECO of TaylorT4 waveform instead of f LSO . Note that, since we have already shown good con-sistency between the effective and analytic Fisher matri-ces for the BHNS binaries, we used the analytic methodfor the TaylorF2 waveform in this Table.
V. SUMMARY AND DISCUSSION
In this work, we have investigated the effective andanalytic Fisher matrices using TaylorF2 waveform forvarious non-spinning binary models and an aligned-spinBHNS binary. The effective Fisher matrix is computedby differentiating the fitting function to the local over-lap surface, so mathematically that should be the sameas the analytic Fisher matrix at the infinitesimal fittingscale. We have shown that the two methods are in verygood agreement at the scale of
P > .
999 for all binarymodels concerned in this work, introducing a new co-ordinate ( M c , η − , χ / ) for the aligned-spin case. Wehave also shown some results for the time-domain wave-form, TaylorT4, and given comparison to the TaylorF2results for the non-spinning and aligned-spin BHNS bina-ries. We have found small differences between TaylorF2and TaylorT4 waveforms.For the time-domain waveforms, the effective methodcan avoid the inconvenience in computing the derivativesof the numerical waveform data. The effective methodsimply computes the derivatives by fitting the local over-lap surface. The best advantage of the effective method isthat we can easily compute the Fisher matrix using var-ious time-domain waveforms implemented in LAL. Formore accurate performances of the parameter estimationanalysis, the MCMC methods have used the time-domainwaveforms, that require very long computational timedepending on the binary model. By using the effectiveFisher matrices, however, we can investigate the parame-ter estimate performances for various binary models priorto the real MCMC runs. Waveform models that containthe merger-ring down phase, can also be usable for theeffective Fisher matrix.The measurement errors derived from the analyticFisher matrix is independent of the SNR, that just falloff as the inverse of SNR. This behavior is similar forthe effective Fisher matrix if we use the TaylorF2 wave-form. When using the time-domain waveforms, however,one can investigate the dependence on SNR by choos-ing the fitting scales physically adjusted. Cho et al. [1]found that the fitting functions are pretty dependent onthe SNR (e.g., see Fig. 2 theirin). Their subsequentwork [6] showed good consistency of the effective Fishermatrices with MCMC parameter estimation results forthe non-spinning and aligned-spin BHNS binaries usingTaylorT4 waveforms with the SNR of 20.For the precessing binary cases, time-domain wave-forms have been developed and already implemented inthe LAL [25]. Cho et al . [1] showed preliminary results ofthe effective Fisher matrices for precessing BHNS bina-ries. For comparison to their results, the MCMC runs areon going. Recently, a frequency-domain waveform for theprecessing cases was developed by [28] and implementedin the LAL, that may be available for an extension of thiswork. VI. ACKNOWLEDGMENTS
This study was financially supported by the 2013 Post-Doc. Development Program of Pusan National Univer-sity. H. S. C. and C. H. L. are supported in part by theNational Research Foundation Grant funded by the Ko-rean Government (No. NRF-2011-220-C00029) and theBAERI Nuclear R & D program (No. M20808740002) ofKorea.
Appendix A: Dependence on the cufoff frequencies:30 Hz versus 40 Hz and f LSO versus f MECO
We show overlap contours with various cutoff frequen-cies using TaylorF2 waveform in Fig. 3 and summarizethe corresponding results in Table. V. To see the depen-dence on the minimum frequency we choose three dif-ferent f min with the same f max as { f mim , f max } [Hz] = { , f LSO } , { , f LSO } , { , f LSO } , where f LSO = 386 Hz. The result shows a difference between 40 (dashed line)and 30 Hz (black solid line) by about 30 −
50 % for thefractional errors, that are (∆ M c / M c , ∆ η/η )=(0 . , . f min = 30 Hz and (0 . , .
6) for f min = 40 Hz. Onthe other hand, we do not see any difference between30 and 12 Hz. This is because the sensitivity curve in-creases very rapidly below 30 Hz due to the seismic noise.Therefore, one should choose f min = 30 Hz not to loseany additional information in the frequency range lowerthan 40 Hz, no additional waveform is necessary lowerthan 30 Hz for the initial LIGO. We consider one ad-ditional case as { f mim , f max } [Hz] = { , f MECO } , where f MECO = 560 Hz, and find that the overlap also dependson the maximum frequency as shown in Table. V. - - - - ∆ M C @ M Ÿ D ∆ Η FIG. 3:
Overlap contours ( P = 0 . ) with var-ious cutoff frequencies for the non-spinningBHNS binary. We assume that { f mim , f max } [Hz] = { , f LSO } (black solid) , { , f LSO } (dotted) , { , f LSO } (dashed),where f LSO = 386 Hz, to see the dependence on the minimumfrequencies, and { , f MECO } (blue), where f MECO = 560 Hz,on the maximum frequencies. There is a significant changebetween 40 and 30 Hz, while no change between 30 and 12Hz (see, Fig. 8 in [6] for the TaylorT4 waveform), this isbecause the sensitivity curve increases very rapidly below 30Hz due to the seismic noise. The overlap also depends onthe maximum frequencies. Exact numbers are summarizedin Table. V.[1] H. -S. Cho, E. Ochsner, R. O’Shaughnessy, C. Kim, andC. -H. Lee, Phys. Rev. D , 024004 (2013).[2] N. J. Cornish and E. K. Porter, Class. Quantum Grav.2006, , S761.[3] M. van der Sluys, I. Mandel, V. Raymond, V. Kalogera,C. R¨over, and N. Christensen, Class. Quantum Grav. 2009, , 204010.[4] N. Cornish, L. Sampson, N. Yunes, and F. Pretorius,Phys. Rev. D 84, 062003 (2011).[5] J. Veitch, I. Mandel, B. Aylott, B. Farr, V. Raymond,C. Rodriguez, M. van der Sluys, V. Kalogera, and A.Vecchio, Phys. Rev. D 85, 104045 (2012). Binary BNS BHNS BHBH1 BHBH2Method Effective Analytic Effective Analytic Effective Analytic Effective AnalyticParameter M c η M c η M c η M c η M c η M c η M c η M c η Σ ij × M c η - 0.04501 - 0.04519 - 1.667 - 1.666 - 31.75 - 32.65 - 174.2 - 188.6 c ij M c η - 1.00 - 1.00 - 1.00 - 1.00 - 1.00 - 1.00 - 1.00 - 1.00 σ i × σ i /λ i [%] 0.0133 0.849 0.0133 0.850 0.0798 1.20 0.0797 1.20 0.148 2.25 0.149 2.29 0.668 5.28 0.691 5.49TABLE II: Comparison between the effective and analytic Fisher matrices for the non-spinning binaries.
Weassume ρ = 20, then Σ = ( ρ Γ eff ) − , σ ( ρ ) = σ (1) /ρ and c ij is ρ -independent. The frequency range for the overlap integralis [30 , f LSO ] Hz, where each number of f LSO is presented in Table. I. For all binary models, two methods show a very goodagreement. Due to the differences of “chirp time”, the errors are distributed quite broadly between the binary models. Forcomparison, we give fractional errors (∆ σ i /λ i ). Method Effective Effective (
P > . M c η χ M c η χ M c η χ M c η − χ / M c η − χ / Σ ij × M c M c ×
10 4292 221.7 2677 ×
10 4291 η - 712.8 -2565 - 470.7 -1777 - 457.0 -1733 η − - 3395 × × - 3393 × × χ - - 9461 - - 6936 - - 6801 χ / - - 8331 ×
10 - - 8332 × c ij M c M c η - 1.00 -0.988 - 1.00 -0.983 - 1.00 -0.983 η − - 1.00 0.983 - 1.00 0.983 χ - - 1.00 - - 1.00 - - 1.00 χ / - - 1.00 - - 1.00 σ i × TABLE III:
Comparison between the effective and analytic Fisher matrices for the aligned-spin BHNS binary .We assume ρ = 20. The maximum frequency is taken to be f LSO (386 Hz). For a smaller fitting scale, we need higher samplingrates for the overlap integration, consequently longer computing time. Instead, by choosing a new coordinate ( M c , η − , χ / ),consistency between the effective and analytic methods can be improved at the scale of P > . M c η M c η M c η χ M c η χ Σ ij × M c η - 0.994 - 1.966 - 91.81 -439.3 - 40.47 -256.4 χ - - - - - - 2208 - - 1756 c ij M c η - 1.00 - 1.00 - 1.00 -0.976 - 1.00 -0.962 χ - - - - - - 1.00 - - 1.00 σ i × Comparison between the TaylorF2 and TaylorT4 waveforms for the BHNS binaries : We assume ρ = 20.We only consider the 1.5 pN spin-orbit phase correction for the aligned-spin case. The results for TaylorF2 are analyticallycomputed and those for TaylorT4 are taken from Tables. VII and VIII of [6]. For consistency, we assume the same maximumfrequencies for both waveforms, as 560 Hz and 890 Hz for the non-spinning and aligned-spin binaries, respectively.[6] R. O’Shaughnessy, B. Farr, E. Ochsner, H. -S. Cho, C.Kim, and C. -H. Lee, (arXiv:1308.4704) (2013).[7] E. Poisson and C. M. Will, Phys. Rev. D , 848 (1995).[8] K. G. Arun, B. R. Iyer, B. S. Sathyaprakash, and P. A.Sundararajan, 2005, Phys. Rev. D , 084008.[9] R. N. Lang and S. A. Hughes, 2006, Phys. Rev. D ,122001.[10] C. Van den Broeck and A. S. Sengupta, Class. QuantumGrav. 2007, , 1089. [11] T. Cokelaer, Class. Quantum Grav. , 184008 (2008).[12] C. L. Rodriguez, B. Farr, W. M. Farr, and I. Mandel,(arXiv:1308.1397) (2013).[13] M. van der Sluys, V. Raymond, I. Mandel, C. R¨over,V. Kalogera, R. Meyer, and A. Vecchio, Class. QuantumGrav. 2008, , 184011.[14] J. Aasi et al. (LIGO-Virgo Scientific Collaboration).Phys. Rev. D , 062001 (2013).[15] K. G. Arun, A. Buonanno, G. Faye, and E. Ochsner, f max
386 Hz ( f LSO ) 560 Hz ( f MECO ) f min
40 Hz 30 Hz 30 HzParameter M c η M c η M c ηc ij M c η - 1.00 - 1.00 - 1.00 σ i × σ i /λ i [%] 0.123 1.58 0.0798 1.20 0.0685 0.926TABLE V: Results using various cutoff frequencies forthe non-spinning BHNS binary, showing the depen-dence on the f min and f max . We assume ρ = 20. The resultfor { f mim , f max } = { , f LSO } Hz is taken from Table. III of[1], where ρ = 10.2009, Phys. Rev. D , 104023.[16] C. Cutler and E. ´E. Flanagan, Phys. Rev. D , 2658(1994).[17] B. S. Sathyaprakash and S. V. Dhurandhar, Phys. Rev. D , 3819 (1991).[18] B. Allen, W. G. Anderson, P. R. Brady, D. A. Brown andJ. D. E. Creighton, Phys. Rev. D , 122006 (2012).[19] T. Damour, B. R. Iyer, and B. S. Sathyaprakash, Phys.Rev. D , 044023 (2001).[20] P. Ajith and S. Bose, Phys. Rev. D , 084032 (2009).[21] P. Jaranowski and A. Kr´olak, 1994, Phys. Rev. D ,1723.[22] M. Vallisneri, 2008, Phys. Rev. D , 042001.[23] L. S. Finn, Phys. Rev. D , 5236 (1992).[24] C. Cutler and M. Vallisneri, Phys. Rev. D , 104025.[27] L. Blanchet, 2002, Phys. Rev. D65