Phenomenological waveforms start with an analytical PN inspiral, which is stitched to a merger-ringdown signal with parameters tuned using NR methods. As such, the waveform can be expressed by the following equation: A(f)≡Cf1−7/6 f0−7/6(1 + Σ3 i=2αivi) iff < f1 wmf0−2/3(1 + Σ2i=1ivi) iff16f < f2 wrL(f, f2, σ) iff26f < f3, (2.21)
whereCis a numerical constant depending on sky-location, orientation, and masses; andf0 =f /f1. The
inspiral phase ends atf1, the merger phase is betweenf1andf2, and the ringdown phase is betweenf2and f3. According to post-Newtonian formalism,v = (πMtotalf)1/3can be compared toβin Equation (2.11),
except withG=c= 1, andf being the GW frequency rather than the orbital frequency. Lis a Lorentzian centered aroundf2with widthσ[50]. Theαiandiare tunable parameters, constructed as functions of the
mass ratio and an optional combined spin parameter [50], given by
χ= 1 + m1−m2 M χ 1 2 + 1 +m2−m1 M χ 2 2 , (2.22)
whereχi=Si/mi2is the dimensionless spin of black holei, projected onto the orbital angular momentum.
The full waveforms have been calibrated against NR for|χ| 6 0.85, and mass ratios between 1 and 4 and are recommended for mass ratios only up to 10 [50]; the inspiral portion of the waveform has also been checked to be consistent in the extreme mass ratio limit [50]. To match what we expect astrophysically, we would like to trust these up to mass ratios of 20; efforts in numerical and analytical relativity are currently underway to reach this goal.
This family of phenomenological waveforms was created in the frequency domain. Examples are shown in Figure 2.11, for an equal-mass system with a total mass of 25M, and in Figure 2.12 for an equal-mass
system with a total mass of 100M. Note that the distance of the system in Figure 2.11 is at 10 Mpc, but 100 Mpc in Figure 2.12. As the IMRPhenomB waveforms are created in the frequency domain, they do not have the same non-physical wiggles as the EOBNR waveforms in Figure 2.11 and Figure 2.12.
This thesis uses two sets of phenomenological waveforms — a nonspinning set and a set with aligned or anti-aligned spins (the waveform gets much more complicated when the precession effects are included, causing a vast increase in the parameter space needed to be searched over). Systems with aligned spins will always have χ > 0 and will produce longer waveforms in LIGO’s sensitive band than systems with the same mass andχ 60; see, for example, the equal-mass system in Figure 2.13. Systems with anti-aligned spins can have a range values of the combined spin parameter; Figure 2.14 shows the case of an equal-mass system with anti-aligned spins of equal magnitude — by Equation (2.22), this system has χ = 0. For a
system with a component mass ratio of 1:4 and a total mass of 50M, Figure 2.15 depicts theχ = 0 (non-spinning) andχ =.5(aligned spin) case. Figure 2.16 shows the anti-aligned spin cases for the same system (χ1,2 = ±0.5); if the more massive component has the positive dimensionless spin parameter, the
combined spin parameter is positive (likewise, if the more massive component has a negative dimensionless spin parameter, the combined spin parameter is negative). As is seen in Figure 2.15 and Figure 2.16, as the combined spin parameter increases, so does the length of the waveform in LIGO’s sensitive band.
Theχ= 0IMRPhenomB waveforms are also compared to their EOBNRv2 counterparts in Figure 2.7 and Figure 2.8. Although the two models used in the analysis described in this thesis are supposed to be similar, they differ in end time and phase evolution, which can make a big difference; therefore, it is important to use both — until we detect GWs, we do not know which one better matches reality. The IMRPhenomB waveforms, which are used in this thesis to assess our sensitivity, are not used in the official rate upper limit calculation as they are not trusted above a mass ratio of 10.
Figure 2.13: Time-domain waveforms for a 12.5M+ 12.5Msystem. Blue: neither black hole is spinning.
Red: dimensionless spins are aligned but unequal in magnitude (χ1 = 0.85,χ2 = 0.5), giving a combined
Figure 2.14: Time-domain waveforms for a 12.5M+ 12.5Msystem. Blue: neither black hole is spinning.
Red: dimensionless spins are anti-aligned, and equal in magnitude (χ1,2= 0.5). The red and blue curves lie
atop one another, as is expected — the combined spin parameterχ= 0for both systems.
Figure 2.15: Time-domain waveforms for a 10M+ 40 Msystem. Blue: neither black hole is spinning
Figure 2.16: Time-domain waveforms for a 10M+ 40Msystem. Both waveforms are from systems with
component black holes having anti-aligned spins. Blue: a 10Mblack hole withχ1=−0.5with a 40M
black hole withχ2 = 0.5, giving a combined spin parameter ofχ = 0.3. Red: a 10M black hole with
χ1= 0.5with a 40Mblack hole withχ2=−0.5, giving a combined spin parameter ofχ=−0.3.
Table 2.2: The number of full cycles in LIGO’s band for various non-spinning waveforms at the cor- ners of our search space. The starting frequency is of 40 Hz for the LIGO detectors. Cycles are listed for the inspiral-only portion of the waveform (TaylorT3 at 2 PPN), the full IMR waveform in the EOBNRv2 implementation, and the full IMR waveform in the IMRPhenomB implementation.
Component masses inspiral-only
(PPN) EOBNRv2 IMRPhenomB
12.5M+ 12.5M 36 46 48
24M+ 1M 219 204 231
99M+ 1M 0 12 38
50M+ 50M 0 12 2
Table 2.3: High frequency cutoff, duration, and number of cycles in the detector’s band of the different waveforms. The PPN inspiral column is taken from the 2nd PPN or- der of the inspiral (parametrized by the TaylorT4 family), which is taken to end at the in- nermost stable circular orbit. Because of design differences between the detectors, LIGO has a low frequency cutoff of 40 Hz while Virgo has a low frequency cutoff of 30 Hz.
Component masses high fre-
quency cutoff LIGO: dura-tion (number of cycles) in PPN inspiral Virgo: dura- tion (number of cycles) in PPN inspiral 12.5M+ 12.5M 175 Hz .6 s (36.1) 1.4 s (61.8) 24M+ 1M 157 Hz 3.8 s (219) 8.5 s (380) 99M+ 1M 38 Hz 0.6 (0) 1.7 s (46.8) 50M+ 50M 44 Hz .003 s (.5) .009 s (2.6)