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3
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0011490655
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(1992)
Science
, vol.256
, pp. 325
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Abramovici, A.1
Althouse, W.E.2
Drever, R.W.P.3
Gürsel, Y.4
Kawamura, S.5
Raab, F.J.6
Shoemaker, D.7
Sievers, L.8
Spero, R.E.9
Thorne, K.S.10
Vogt, R.E.11
Weiss, R.12
Whitcomb, S.E.13
Zucker, M.E.14
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5
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84927338844
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also in Gravitation 1990, Proceedings of the Banff Summer Institute, Banff, Alberta, edited by R. Mann and P. Wesson (World Scientific, Singapore, 1991).
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10
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0001706230
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(1993)
Phys. Rev. Lett.
, vol.70
, pp. 2984
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Cutler, C.1
Apostolatos, T.A.2
Bildsten, L.3
Finn, L.S.4
Flanagan, É. E.5
Kennefick, D.6
Marković, D.M.7
Ori, A.8
Poisson, E.9
Sussman, G.J.10
Thorne, K.S.11
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12
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84927272744
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edited by, H. Sato, T. Nakamura, World Scientific, Singapore
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(1986)
Gravitational Collapse and Relativity, Proceedings of the 14th Yamada Conference, Kyoto, Japan, 1986
, pp. 350-368
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Schutz, B.F.1
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19
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84927338843
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However eccentricities may not be negligible for binaries formed in dense star clusters in galactic nuclei;
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23
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84927338832
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A rapidly rotating neutron star will be somewhat oblate, and, therefore, compared to a point mass, its potential energy will be modified by a term proportional to its quadrupole moment times the second derivative of the gravitational potential at its center of mass, Ref. 18, However this correction to the orbital energy is of second post Newtonian order, and thus is of higher order than the other post Newtonian effects considered in this paper.
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27
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84927338829
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The noise spectrum to which Eq. ( ref{snf2}) is an approximate analytic fit is given by the sum of the following terms from, Ref. 14, : Eq. (4.1), with parameters η I0 = 60 W, L = 4 km, fc = 130 Hz, λ = 5.1 ×10-7 m, and A2 = 5 ×10-5; Eq. (4.3) with the modifications to Eq. (4.2) of f0 to f2/f0 in the numerator and f f0 / Q0 to f0sup 2 / Q0 in the denominator, and with parameters Q0 = 109, T = 300 K, f0 = 1 Hz; and Eq. (4.4), similarly modified, with parameters fint = 14 kHz and Qint = 106. Equations (4.2) and (4.4) are modified in order to describe structural damping, Ref. 25, which is now thought to be the likely dominant damping mechanism in the thermal modes, Ref. 23
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31
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84927338827
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Note that this definition differs by a factor of 2 from that found in, Ref. 26, Our definition is chosen to correspond to the quadratic form ``xi (Σ-1)ij xj'' which appears in finite dimensional Gaussian PDFs, cf. Eq. ( ref{noise_distribution}) above.
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33
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84927338825
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The last stable circular orbit is not exactly at r = 6 M because of the fact that r is the orbital separation in de Donder gauge and not the Schwarschild radius, and also because of the finite mass ratio. However, this orbit will be close to r = 6 M;
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35
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84927338824
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The calculation is simplified if one first defines the moments { overline {f^k}} equiv { ( f^k h ,|, h ) over ( h ,|, h )}, where the inner product ( | ) is given by Eq. ( ref{inner}). As pointed out by Finn and Chernoff, Ref. 14, all the elements of Γij can be expressed in terms of the orbital parameters and a few of the moments { overline {fk}}. This continues to hold true when post Newtonian corrections are added to the signal. A useful identity is ( fj h | i fk h) =0 for all real j and k, which follows from Eq. ( ref{inner}).
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43
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84927338823
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In fact, because of the spin induced precession of the orbital plane described in Sec. ref{spin_sec}, in general k will be a slowly varying function of time instead of a constant. We neglect this small effect.
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45
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84927338812
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Somewhat inconsistently, we neglect the δ function contribution to the derivatives partial tilde h(f)/ partial { cal M} and partial tilde h(f)/ partial μ that comes from varying the cutoff frequency. While a sharp cutoff gives an acceptable approximation to h(f) (and is easy to work with analytically), gives a terrible approximation for the derivatives of h(f). Our somewhat careless attitude toward the high frequency end of the waveform is justified by the fact that the detector noise Sn(f) rises steeply at high frequency, so very little signal to noise is accumulated there.
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46
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84927338811
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The pattern of how the predicted rms errors change, when one includes extra variables in the calculation of the Fisher matrix ( ref{sig}) which are strongly correlated with the original variables, can be simply understood by considering the approximation in which all but two of the variables are fixed. The predicted measurement accuracy for a variable x, when its correlations with other variables are neglected, is δ x = (h,x | h,x)-1/2. When we include the effects of correlations with another variable y, described by the correlation coefficient c_{xy} = {Σ^{xy} over Σ^{xxΣ^{yy}}} = { (h_{,x} | h_{,y}) over (h_{,x|h_{,x}),(h_{,y}|h_{,y})}}, then from Eqs. ( ref{sig}) and ( ref{bardx}) we find that (i) the rms error in x is now Δ x = δ x / 1 -cxy2, and thus is increased by a large factor if |cxy| is close to one and (ii) if 1 - cxy2 ll 1, the eigenvalues of the variance covariance matrix ( ref{sigma_def0}) are approximately (δ x-2 + δ y-2)-1 and (δ x2 + δ y2) / (1 - cxy2), where δ y = (h,y | h,y)-1/2. Thus, if δ x and δ y are comparable, one linear combination of x and y will be measurable with an accuracy comparable to δ x, i.e., that accuracy predicted when correlations are neglected; and the orthogonal linear combination will have an rms error that is larger than this by a factor sim 1 / 1 -cxy2.
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50
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0001348922
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of% assuming a uniform a priori PDF in (M1,M2), integrating% { cal M}, and taking the limit Δ { cal M} to 0 (true to a% % {μ^{ 3/2}, Thetaμ), Theta(2^{ 4/5} { cal M}_0 μ) over% (μ μ_0)^2 / Δ μ^2}.% Upper and lower 95 % confidence limits for μ calculated% Eqs. ( ref{conf_limits}) (replacing { tilde μ} pm 2 Δ μ)% method used to generate Fig. ref{m1m2}.
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(1991)
Phys. Rev. D
, vol.43
, pp. 2470
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Krolak, A.1
Lobo, J.A.2
Meers, B.J.3
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52
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84927338809
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In this section we interpret the parameter tc to be the time at which the coalescence would be observed by a hypothetical detector at the origin of spatial coordinates x.
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53
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84927338808
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Other detector network parameters, such as the distances between the detectors, affect strongly the angular resolution Δ n of measurements of sky location, but affect only weakly the distance measurement accuracies. This is due to the decoupling discussed in Sec. ref{draza_approx} and Appendix ref{decouple}.
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54
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84927338807
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We have here ignored the fact that a factor of { cal M}5/6 appears in the signal amplitudes; this is unimportant because { cal M} will be measured to much higher relative accuracy ( sim 10-3) than the amplitudes ( sim 10-1). In other words we in fact calculate Δ D1 / D1, where D1 equiv D { cal M}-5/6; for all practical purposes this is the same as Δ D / D.
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55
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84927338806
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In fact we have compared the values of Δ D given by the approximation used in, Ref. 9, to those given by Eq. ( ref{final_ans}), and found that they never differ by more than 10 % for any values of σD, varepsilonD, v, and ψ.
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56
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84927338805
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Another way to think about this is to associate a signal to noise ratio with each polarization component [with respect to fiducial axes determined by the detector network, cf. Eq. ( ref{deltapsi}) above] of the incident waves; the distance measurement accuracy will essentially be determined by the smaller of the two signal to noise ratios.
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57
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84927338804
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It is simplest to calculate this prefactor using the variables α and β defined by Eq. ( ref{transform}) instead of D and v. In fact diverges at v=1, because of the fact that partial h / partial ψ propto partial h / partial φc in this limit. This divergence would seem to contradict our claim that the prefactor does not depend strongly on v and D. In the special case where varepsilonD=0, a more careful calculation of the integral over ψ and φc (without first expanding to quadratic order in ψ - ψ0 and φc - φc0) shows that the effective prefactor remains finite at v =1; we expect similar behavior for varepsilonD ne 0. In the case varepsilonD =0 we obtain p(v,D) , propto , p^{(0)}(v,D) F(ω { hat ω}) , F(ζ { hat ζ}) , e^{ (Δ ω^2 + Δ ζ^2)/2}, where ω = (α + β) / 2 σD r0, ζ = -(α - β) / 2 σD r0, and { hat ω}, Δ ω, etc., are similarly defined in terms of { hat α}, { hat β} [cf. Eqs. ( ref{hatalpha}) and ( ref{hatbeta}) above] and Δ α equiv α - { hat α}, Δ β equiv β - { hat β}. The prefactor function F is nonumber (Omitted equation) Thus the prefactor is regular and slowly varying despite the apparent divergence propto 1 / ζ propto 1 / (1 - v) that would be obtained by doing a Gaussian integration over the angles ψ and φc.
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59
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84927338803
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M. H. A. Davis, in Gravitational Wave Data Analysis [35].
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64
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84927338792
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F. Echeverria, Ph.D. thesis, California Institute of Technology, 1993, addendum to Chap. 2.
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65
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84927338791
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In fact this method can determine all the parameters theta = ( theta1, ldots, thetak), except for the overall amplitude of the signal which drops out of Eq. ( ref{snr_def_1}). To obtain the overall amplitude one must use Eq. ( ref{bestfit}); see, Ref. 58, for more details.
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66
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84927338790
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Throughout this paper we use the term Fisher matrix to refer to the matrix ( ref{gamma_def}); strictly speaking, this term as defined in, e.g., Ref. 55, refers to a quantity which coincides with the matrix ( ref{gamma_def}) only when the noise is Gaussian. The Cramer Rao inequality is usually stated in terms of this more general Fisher matrix.
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67
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84927338789
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For an example of such a Monte Carlo simulation, see K. Kokkotas, A. Krolak, and G. Tsegas (unpublished).
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