1.2 Detecting compact binary objects with first-generation interferometric gravitational-wave detectorsinterferometric gravitational-wave detectors
1.2.5 The Modified Apostolatos Ansatz and DTF for spinning binaries
In Chapter 9, Buonanno, Vallisneri and I study the adiabatic dynamics of precessing binaries, mostly “heavy” BBH cases (with 5M¯ < m1, m2 < 20M¯), but also including one NS-BH case [(10 + 1.4)M¯], including up to 2 PN orbital motion and radiation reaction, without restricting to the ACST special cases. We do not study the PN failure in this case, but instead use only the 2 PN adiabatic model, as given by Kidder in a comprehensive paper, Ref. [54], in which he derived the equations of motion (direct and adiabatic, in the sense of Sec. 1.2.1) for precessing binaries and expressions for the waveforms, and studied the dynamics and waveforms qualitatively. As we show in Sec. 9.3, the dynamics of these precessing binaries is qualitatively the same as found by ACST in lower-PN-order computations of the ACST special cases. In particular:
• The accumulated orbital phase Ψ(t), obtained by integrating the instantaneous angular fre- quency,
Ψ(t) = Z t
ω(t0)dt0, (1.33)
12 One should be cautioned against saying that ansatz II is bad, since the low fitting factor might be due to the imperfectness of the two-step hierarchy. Nevertheless, since the two-step hierarchy could be the only obvious way of lowering computational cost, it is plausible to say that ansatz II, in its original form, isnot yet practical for use in data analysis.
although affected by PN contributions that involve spins (which ACST did not take into account), deviates largely monotonically from the non-spinning phasing, and can be fit well with a polynomial in orbital frequency.
• The total angular momentum J still remain roughly constant in orientation for most of the configurations, while the precession phase of LaroundJis still roughly described by a power law in orbital frequency, except for the very rare cases of transitional precessions.
These were the starting points of the ansatz II of Apostolatos. However, we have developed a modificationto this ansatz that allows the construction of DTFs that improve both the computational efficiency and the fitting factor:
We start by looking at the response of a gravitational-wave detector, which (at leading PN order) can be put into the following form
h(t)∝I¨ij(t)
| {z } Qij
[T+(Θ, ϕ)F+(θ, φ, ψ) +T×(Θ, ϕ)F×(θ, φ, ψ)]ij
| {z }
Pij
, (1.34)
in which Iij(t) is the instantaneous quadrupole moment of the binary, T+,× are the polarization tensors of waves propagating in the (Θ, ϕ) direction and F+× are the antenna patterns of the detector, which depend on the detector orientation [see Fig. 1.8]. Moreover, in the leading PN order, we have [see Sec. 9.4]
Qij(t)≡I¨ij(t)∝ω2(t) [e+(t) cos 2Ψ(t) +e×(t) sin 2Ψ(t)]ij , (1.35) where Ψ(t) is the accumulated orbital phase [which does not oscillate], and
e+(t)≡e1(t)⊗e1(t)−e2(t)⊗e2(t), e×(t)≡e1(t)⊗e2(t) +e2(t)⊗e1(t), (1.36) with e1,2(t) a time-dependent orthonormal basis of the precessing orbital plane that follows the precession in a non-rotational way (see Sec. 9.4.1, Appendix 9.9, in particular Eqs. (9.71) and (9.72), for the specific meaning of this). Sincee1,2(t) follow the orbital precession, it is plausible to modify the Apostolatos ansatz into
[eK]ij(t)∼αK ij+βK ijcos(Bft−p+δK, ij), (p= 1,2/3), K= +,×, (1.37)
whereftis twice the orbital frequency at timet. By inserting Eqs. (1.35)–(1.37) into Eq. (1.34) and Fourier transforming in the Stationary-Phase Approximation, we obtain
˜h(f) ∝ £
C1+iC2+ (C3+iC4) cos(Bf−p) + (C5+iC6) sin(Bf−p)¤ h
f−7/6exp(iΨNM)i
≡ X6 j=1
CjAj(f) exp(iΨNM)≡ A(f) exp(iΨNM), (1.38)
where theCj’s are real constants. Here ΨNMstands for a non-modulatedphasing, with the form of Eq. (1.29). Again, we propose to use only two free parametersin ΨNM: ψ0 andψ3/2. It should be noted that, although the modulations are all added formally in the amplitudeA(f), the fact that Aj can be complex means that the modulations act both on amplitude and phase. We call this template family themodulated BCVtemplate family.
The amplitude-modulation form of the modulated BCV template (1.38) is very advantageous for search purposes. At each set of (ψ0, ψ3/2,B), the linear coefficients C1,...,6 parametrize a six- dimensional linear template subspace, in which the optimization of template-signal correlations over the entire linear subspace can be obtained by taking the correlations between the signal and a set of basis vectors [independent templates], and then combining, by taking the square root of the sum of the squares of each individual correlation. As a consequence, the one-by-one search only need be done in a three-dimensional parameter space, (ψ0, ψ3/2,B). In the terminology of gravitational-wave data analysis, C1,...,6, which do not need to be searched over one-by-one, are calledextrinsic parameters, while the parameters (ψ0, ψ3/2,B), which need to be searched over one- by-one, areintrinsic parameters. By employing a lot of extrinsic parameters, the modulated BCV template family is computationally very efficient. A well-known example of this extrinsic-parameter technique in gravitational-wave data analysis is the optimization for the initial orbital phase of inspiral templates (see e.g., Ref. [67]), where the linear template space is two-dimensional. A four- dimensional version has also been proposed for the combined search over the initial phase and the detector orientation, in both spinning binaries [78] and for Pulsars [79]. The Jenet-Prince Fast Chirp Transformation (FCP) is another (rather different) way of converting intrinsic parameters (ψn) into extrinsic parameters.
In Sec. 9.6, we test the performance of our modulated BCV template family in terms of fitting factors. In Table 1.3 (excerpted from Tables 9.8 and 9.9), the averaged fitting factors, assuming a uniform spatial distribution of the orientations of the initial spins and angular momenta, weighted by the cube of signal strength (measured by the optimal SNR at a fixed distance), is given (in the line labeled BCV2) for maximally spinning binary black holes with masses (15 + 15)M¯, (10 + 10)M¯, (20 + 10)M¯, (20 + 5)M¯ and (7 + 5)M¯, and a neutron-star–black-hole binary with masses (10 + 1.4)M¯ with maximally spinning black hole and nonspinning neutron star. For comparison,
(7 + 5)M¯ (10 + 10)M¯ (15 + 15)M¯ (20 + 5)M¯ (20 + 10)M¯ (10 + 1.4)M¯
SPAc 0.937 0.956 0.955 0.910 0.946 0.813
BCV1 0.962 0.970 0.973 0.921 0.963 0.832
BCV2 0.983 0.990 0.989 0.979 0.988 0.945
Table 1.3: Average fitting factors of nonspinning templates (SPAc, the standard Stationary-Phase- Approximated nonspinning template with free higher cutoff frequency, and BCV1, the frequency- domain DTF proposed in Chapter 8) and modulated BCV templates (BCV2) for spinning binary black holes and neutron-star–black-hole binaries. Black holes are assumed to be maximally spinning, while neutron-star spins are neglected. A uniform distribution of initial spin and orbital orientations is assumed.
Table 1.3 also shows fitting factors of the Stationary-Phase-Approximated nonspinning templates [Eq. (1.34), withα= 0, free cutoff frequency, andψnvalues provided by PN calculations; denoted by SPAc] and of the frequency-domain DTF (for nonspinning binaries) proposed in Chapter 8 [defined by Eq. (1.34) and the text that follows, denoted by BCV1]. In the binary black hole case, the average fitting factor obtained by nonspinning templates were already higher than 0.9 (usually higher than 0.95). This is consistent with the findings of Apostolatos using his first ansatz [52].
The use of modulated BCV templates is shown to increase the average fitting factor, up to∼0.98.
The average fitting factor obtained by nonspinning templates for the NS-BH binary is much lower, around 0.8, which means a nearly 50% loss in event rate, if in reality the angular momenta are distributed uniformly. The modulated BCV templates (BCV2) increase the average fitting factor up to ∼ 0.95. We have explained fitting factors for NS-BH binaries further, by looking at the average fitting factors for different initial misalignment between the black-hole spin and orbital angular momentum (averaged over the relative orientations between the binary as a whole and the detector), see Fig. 1.10. As expected, the nonspinning templates can catch the waveform well only for nearly aligned and anti-aligned configurations. The modulated BCV templates have a similar bias toward aligned and anti-aligned binaries, but much less, by improving significantly the fitting factor for misaligned binaries.
Unfortunately, the improvements in fitting factors do not come for free. Because (when one takes account of all theCj’s) there are many more modulated BCV templates than nonspinning templates, the noise would cause many more false-alarm events, if the same detection threshold on SNR were imposed. This must be compensated by raising the threshold, which, unfortunately, will decrease the visible range and therefore counteract the increase in overlap achieved by including the more diverse templates. A rigorous study of the false alarm rate of a template family usually requires a Monte Carlo simulation, which is very computationally intensive. In Chapter 9, we only give a very rough overestimate of the false-alarm probability. This overestimate yields a requirement of 8.5%
in overlap increase in order to justify the use of a six-dimensional linear template space. This is only met in the NS-BH case. However, we should not simply rule out the modulated BCV template
(ψ0ψ3/2)2 (ψ0ψ3/2B)6
-1.0 -0.6 -0.2 0.2 0.6 1.0
0.60 0.70 0.80 0.90 1.00
FF
SPAs
κeff/κeffmax
Figure 1.10: The dependence of the neutron-star–black-hole average fitting factor on the spin-orbit misalignment,κ≡LˆN·Sˆ (which is conserved throughout the evolution). Here SPAs stands for the standard, Stationary-Phase Approximated templates for nonspinning binaries, (ψ0ψ3/2)2 stands for the frequency-domain DTF proposed in Chapter 8 for nonspinning binaries (BCV1 templates), and (ψ0ψ3/2B)6stands for the modulated BCV templates.
family, since:
• A Monte Carlo simulation has to be done to determine whether this pessimistic estimate is accurate enough.
• Even if that is done, a realistic astrophysical distribution of the orientations of the spins and orbital angular momenta has to be known to answer the ultimate question of whether the modulated BCV templates can increase the event rate. Such knowledge is not likely in the near future.
• Even though the event rate might not be higher, having a less biased template bank might be beneficial.
An interesting conceptual problem will also arise in the study of the modulated BCV template family, if a Monte Carlo study confirms the high false-alarm probability: did the high false alarm originate from a “non-physical” reason, due to the inclusion of signals in the template bank that cannot be generated by a precessing binary, or from a “physical” reason, due to the diversity of precessing waveforms. This problem can be studied by including only the true, physical signals in the template bank, which has been regarded as impractical. However, it is not impractical for NS-BH binaries. Let us begin by asking how many parameters are absolutely required to besearched over, one by one,(i.e., how many intrinsic parameters are there) in the signal (1.34), for a generic binary:
• It is plausible that the parameters (Θ, ϕ) and (θ, φ, ψ) can be converted into extrinsic param- eters, since the waveform depends on them only through the linear coefficientsPij.
• In order to computeQij, it might seem that all 10 binary parameters (Table 1.2) are needed.
However, the overall orientation of the binary is not absolutely needed, in the following sense:
we can choose an arbitrary frame, say, one in which L is along the z axis andS1 is in the x-z plane atf = 30 Hz, and computeQij — this is already enough, since the arbitrariness in the orientation of this frame will be automatically accounted for by the (Θ, ϕ) and (θ, φ, ψ) parameters. In the end, 7 is the number of relevant parameters, which we call “basic” and
“local parameters” in Table 1.2.
Although 7 is still too large for the dimensionality of the intrinsic parameter space, 3 of them are absent when the spin of one of the two bodies is unimportant, e.g., in NS-BH binaries. A four-dimensional intrinsic-parameter space is needed in this case, which is not extremely big. An exploration of this parameter space (now physical) will not only help clarify conceptually the ori- gin of the high false alarm probability, but for the first time provide a practical way of searching over the physical templates of precessing binaries, which so far has been regarded as non-practical.
Investigation of this approach is currently being made by Pan, Buonanno, Vallisneri and me.