1.2 Detecting compact binary objects with first-generation interferometric gravitational-wave detectorsinterferometric gravitational-wave detectors
1.2.1 Post-Newtonain waveforms of late-stage inspirals with vanishing spinsspins
Let us first recognize the importance of late-stage inspirals, using the example of non-spinning compact binaries. In the leading Post-Newtonian approximation, the chirp signal has a frequency- domain magnitude of
|h(f˜ )| ∝ M5/6
d f−7/6Θ(fISCO−f), M=M η3/5, (1.21) whereMis thechirp mass(withM =m1+m2the total mass andη =m1m2/M2 the mass ratio, i.e., ratio of reduced mass to total mass), anddis the distance from the source to the detector. A cutoff is made at the ISCO frequency, which can be estimated roughly from the Schwarzchild ISCO frequency, with the mass of the Schwarzchild black hole equal to the total mass (an estimate that is
correct only forη →0):
fISCO= 4400 Hz µM¯
M
¶
. (1.22)
Note that the higher the total massM, the lower the ISCO frequency, as the waveform only depends on the ratio t/M in the time domain, or f M in the frequency domain. The signal-to-noise ratio (SNR) given by optimal matched filtering is
SNR = s
2 Z +∞
−∞
df|˜h(f)|2
Sh(f) ∝ M5/6 d
sZ 4400 HzM/M
¯
0
dff−7/3
Sh(f) =M5/6η1/2 d
sZ 4400 HzM/M
¯
0
dff−7/3
Sh(f). (1.23) Here Sh(f) is the (sigle-sided) noise spectral density in h. [See notation in, e.g., Eq. (1.13).] From Eq. (1.23), we can see that, for binaries at a fixed distance (oriented in the same way with respect to the detector) [38, 5]:
• The higher the mass ratio η, the higher the SNR.
• In the low-mass regime, wherefISCOlies above the detection band (higher than∼240 Hz for LIGO-I, which corresponds to total mass lower than ∼20M¯), higher SNRs are obtained by increasing the total massM, with SNR∝M5/6.
• For binaries with high enough masses such thatfISCOlies within the detection band, increasing further the total mass will eventually result in lower SNRs.
These features of the SNR can be represented more quantitatively in terms of the volume of the universe that is visible with a fixed SNR (thus a fixed false-alarm probability) as a function of the binary’s masses:
V ∝d3∝ M5/2
"Z 4400 HzM/M
¯
0
dff−7/3 Sn(f)
#3/2
=M5/2η3/2
"Z 4400 HzM/M
¯
0
dff−7/3 Sn(f)
#3/2
. (1.24)
This relative visible volume is plotted (for equal-mass binaries) in Fig. 1.7. As we can see from the graph, the visible volume is dramatically larger for heavier binaries, peaking at a total mass of ≈ 35M¯ (with a visible range of ≈ 200 times the value for neutron-star binaries), for which the Schwarzchild ISCO frequency is 126 Hz, which lies right in the middle of the LIGO-I detection band. It should be noted that different Post-Newtonian expansions will give different predictions for the cutoff frequency, and hence differentV-M curves (see Figs. 8.16 and 8.17), which give different turning points in total masses. Nevertheless, it is clear that binary black holes whose inspirals end within the detection band are favored by the detector, which means total masses ∼ 20 – 60M¯ for LIGO-I detectors. Current astrophysical theories suggest that in binaries only black holes with masses smaller than ∼ 15M¯ can form directly from the collapse of stellar objects, but a recent study by Miller and Hamilton has suggested that higher-mass black holes can form from four-body
2 5 10 20 50 100
M
0.1 1 10 100
Visible volume
Figure 1.7: Relative visible volumes of LIGO-I interferometers for non-spinning, comparable-mass binaries with various total masses as observed by LIGO-I interferometers. These visible volumes are based on the leading-order waveform and the Schwarzchild-ISCO cutoff in the frequency-domain amplitude of the signal, with noise spectral density that of the LIGO-I design. The relative visible volume is set to unity for neutron-star binaries (with neutron-star masses set to 1.4M¯ each). The visible volume peaks atM ≈35M¯, with a maximum of≈200.
interactions in globular clusters [40].
Having appreciated the importance of these “heavy” binary black holes, in Chapter 8, Buo- nanno, Vallisneri and I study effects associated with the failure of Post-Newtonian calculations for nonspinning “heavy” binary black holes. We study black holes with masses in the range of 5–20M¯.
There exist three main approaches to Post-Newtonian expansions:
• The Direct Approach, where the equation of motion is obtained in the harmonic gauge, by expanding in powers of the orbital velocity divided by the speed of light,v/c[1 PN = (v/c)2]:
a = aN+a1 PN+a2 PN+a3 PN+. . .
+a2.5 PN+a3.5 PN+. . . . (1.25)
Integer PN orders (N, 1PN, 2PN, etc.) give dynamics with conserved orbital energy and angu- lar momentum (“conservative dynamics”) [57, 60], while the half-odd-number orders (starting from 2.5 PN) give the radiation reaction [58, 59], which drives the secular evolution of the orbit. Unfortunately, the PN-expanded acceleration has only been derived up to 3.5 PN order (with an undetermined regularization parameter in 3 PN), so the waveforms obtained from this approach can only be accurate up to 1 PN (since radiation reaction starts at 2.5 PN). A
Lagrangian can be derived from the conservative acceleration terms, so this approach is also called theLagrangianapproach in Chapter 8.
• The Adiabatic Approach, where energy and angular momentum fluxes in generic orbits have been calculated up to 3.5 orders beyond the quadrupole (2.5 PN) order [i.e., 6 orders after the leading Newtonian order, which corresponds to (v/c)12!] but with undetermined regularization parameters in 3 PN [61, 62, 63]. The energy and angular momentum fluxes can then be used to construct a sequence of adiabatic orbits accurate up to 3.5 PN, by using the balance equation,
E˙(v) =−F(v), (1.26)
whereE(v) is the orbital energy corresponding to the Keplerian orbital velocityv≡(πM f)1/3 (withf twicethe orbital frequency andG=c= 1), andF(v) is the corresponding energy flux.
• The Hamiltonian approach, where a Hamiltonian for the conservative dynamics has been de- rived up to 3 PN order, starting from the 3+1 decomposition formalism [64]. This Hamil- tonian has also been shown to be equivalent to the one derived from the Lagrangian in the Direct Approach derived in harmonic gauge [65] (the undetermined 3 PN regularization pa- rameter in the Direct-Approach conservative dynamics can be determined in the Hamiltonian Approach by means of dimensional regularization). Radiation reaction can be added to the Hamiltonian equations of motion as a generalized force which gives the correct energy and angular-momentum losses. The Hamiltonian approach can probe certain non-adiabatic ef- fects, but cannot give the complete picture, since the radiation reaction is added assuming certain adiabaticity.
[See Secs. 8.4.2, 8.3.1 and 8.4.1 for more details on the three approaches respectively.] For binary black holes with total mass 10–40M¯, the detection band off = 40–240 Hz corresponds to Keplerian orbital speedsv/c= 0.18–0.53. At such high speeds, the Taylor-expanded fluxF(v) (for example) at adjacent PN orders can differ a lot, as we see in Fig. 8.2, and does not seem to have converged at the currently available PN orders (up to 3.5 PN).8As for the Taylor-expanded Hamiltonian models (up to 3 PN), they usually cannot give the ISCO structure of dynamics (except at 1 PN, where the given ISCO frequency is obviously too small, see Table 8.2). With these signs of PN failure identified at the late-stage inspiral,resummationtechniques have correspondingly been developed to improve the convergence of the PN expansions, in the absence of further inputs from higher PN orders. In Chapter 8, we discuss the following two prescriptions:
• Pad´e approximants [67, 68], where a Pad´e expansion is used to enhance the convergence of the Taylor-expanded forms given directly by the Post-Newtonian expansion, and the Pad´e
8 In fact, for such high speeds, the PN expansion might not converge at all, since it could be an asymptotic expansion in nature [66].
expansion takes account of the expectation that the flux function F(v) should have a pole at thelight ring. The Pad´e-expanded flux out to the orders that have been computed has been shown to converge much better than the Taylor-expanded versions, see Fig. 8.5.
• Effective One-Body (EOB) Approach [69, 70, 71, 72], where the two-body Hamiltonian dy- namics is matched to that of a single test particle in a deformedSchwarzchild metric — with the deformation of the metric Post-Newtonian expanded in v/c. This approach recovers, in the resummed Hamiltonian, the late-stage dynamical features such as the ISCO and the light ring. Dynamics beyond the ISCO can be probed to a better extent by this approach than by the Taylor-expanded Hamiltonian approach.
[See Secs. 8.3.2 and 8.4.4 for further details on these resummation techniques.]
Data analysis oriented comparisons between different PN waveforms are based on the overlap (defined in Sec. 8.2.1) between the target signal and the template waveform. This overlap is equal to the fraction of optimal SNR achievable by an imperfect template, and therefore never exceeds unity.
[The loss in visible volume with a fixed SNR, and hence event rate, is then (overlap)3, see arguments around Eq. (1.24).] A systematic comparison of PN waveforms in the Adiabatic approaches (Taylor and Pad´e) and the Effective One-body (EOB) approach (as the fiducial “exact” waveform) has been made by Damour, Iyer and Sathyaprakash (DIS) [67, 68, 55, 56], with a detailed numerical study carried up to 2.5 PN order. They formulated two types of tests:
• Comparison of physical predictions of the PN models (approach, prescription and order), in terms of the overlap between waveforms of the same binaryas predicted by different PN models. This can be regarded as an internal convergence test of PN expansions, since if the PN expansion converges, all models should give similar results. In particular, if the two waveforms are generated from the same prescription (i.e., Taylor, or Pad´e, or EOB), but at different orders, this test is similar to a Cauchy convergence test of that approach.
• Setting waveforms from one PN model as the fiducial target signal (for which DIS use only EOB waveforms), and test whether using waveforms generated by another model (the template model) can successfully mimic the target, regardless of whether the optimal (M, η) used (in the template model) is the same as the target one. At the end, if the overlap is high, then the template family is regarded aseffectual. If the resulting optimal (M, η) is close to the original one, then the template family is regarded asfaithful.
In Chapter 8, we first (among other things) confirm the results of DIS, but with more models added, and without taking the EOB as the fiducial exact signal. We first test the Cauchy convergence, which was shown to be rather poor for Adiabatic Taylor (unless we skip the 2.5 PN order, see Table 8.3), and Taylor-expanded Hamiltonian models (Table 8.5) and better but not perfect for Adiabatic Pad´e
(Table 8.4) and EOB (see Table 8.8) models. Then, we go on to take the overlaps between 11 different typical models to test the effectualness and faithfulness of them against each other (see Table 8.11). Neither the effectualness nor the faithfulness is satisfactory, especially when Taylor- expanded Hamiltonians and 2.5 PN Taylor fluxes are involved. Nevertheless, overlaps obtained here are much higher than those obtained in the Cauchy tests. As a consequence, it is reasonable to conjecturethat the function space spanned by different PN waveforms are approximately the same, although the same waveform in the function space might correspond to different(M, η)’s in different PN models.