3.2 Initial estimates
3.2.2 Choice of IMRI waveform templates
-1 -0.5 0 0.5 1 aMBH
0 100 200 300 400 500 600
FiscoHHzL
MBH200 MBH100 MBH50
Figure 3.2: The ISCO wave frequency as a function of black-hole spin.
by Advanced LIGO. (Note that core-collapsed globular clusters, which are the most likely location for IMRIs, have a space density of∼0.7Mpc−3.) The four curves shown in Fig. 3.3 correspond to different black hole spin parameters. The figure shows that for rapidly spinning black holes withq&0.9, the accumulated SNR can reach∼40 to∼50 at 100Mpc; even for slowly rotating holesq.0.3, the accumulated SNR can be∼20 to∼30.
200 400 600 800 1000 MBH!M!
10 20 30 40 50
"#S!N$avg2 a!0.99
a!0.9 a!0.5 a!0
Figure 3.3: Square-root ofh(S/N)2ifor IMRI waves from an inspirling 1.4MNS at a distance of 100Mpc as a function of the central black hole’s mass. In computing the integral Eq. (3.9a), we take the starting frequency to be 10Hz (the lower cut-offfrequency of the Advanced LIGO interferometers), and the ending frequency to be the wave frequency (Fisco) corresponding to the innermost-stable-circular-orbit (isco) of the central black hole. The formula to computeFiscocan be inferred from, e.g, Eq. (3.20)-(3.21) in [44]. The ISCO wave frequency as a function of spin is shown in Fig. 3.2 for three different masses. The estimated square-root ofh(S/N)2iis inversely proportional to the IMRI distancer, and scales approximately proportional to the massµof the small object.
and that the full inspiral can be approximated as a flow through a sequence of geodesic orbits.
To date, the leading-order adiabatic EMRI waveforms are computed via the so-called ”Radiation Reaction without Radiation Reaction Forces” program [46, 47] for special classes of orbits. It involves computing the time-averaged rates of the change of the constants of motion (E,Lz) from the leading-orderO(v2) graviational- wave fluxes. When augmented with Mino’s [48] adiabatic self-force rule to evaluate the (time-averaged) rate of change of the Carter constant, this program can be used to evolve generic orbits [49]. However, it is also important to determine the effect of conservative finite mass ratio correctionsO(v2) (sometimes called
“non-dissipative self-force effects”), which contribute to the secular change in orbital (and waveform) phase but do not change the constants of motion. The self-force issue is more pronounced for the IMRIs , whose mass ratio lies in the range∼ 10−3 to∼ 10−2, than for LISA’s EMRIs withµ/M ∼10−6. This is because, if we ignore theO(v2) conservative self-force, we can only accurately track the phase of the waveform up to O(M/µ) cycles [50].
By constrast, the standard post-Newtonian (PN) formulation, the tool for studying comparable-mass in- spirals, has built-in self-force ingredients. It is based on the expansion in the parameterM/r(1) and the symmetric mass ratioη =m/M (0< η ≤1/4). For circular equatorial orbits and nonspinning bodies, the orbital energy and the energy flux have been determined by PN-expansion techniques up to 3.5PN order [51], and the spin effects have been calculated up to 2.5PN order [16, 52, 53, 54]. To assess the importance of theO(v2) conservative self-force and the efficiency of adiabatic Teukolsky waveforms as search templates for IMRI waves, Brown [34] has computed the mismatch between restricted PN stationary phase teimplates that contain all knownηterms (contributed by the leading-order radiation reaction as well as conservative
self-force), and the same templates containing only theO(η−1) terms (i.e. the terms of leading order in the waves’ phase evolution; contributed only by the leading-order radiation reaction). He finds that the mismatch falls to less than 10% in all except the most rapid spinning cases. Therefore it is reasonable to believe that the adiabatic Teukolsky waveforms will lose no more than∼10% of the SNR due to the absence ofO(v2) conservative self-force terms (corresponding to no more than∼30% loss of event rate).
Although the PN formulation is capable of including higher-order contributions from the self-force, it can become inaccurate at a rather early stage after the IMRI wave enters the Advanced LIGO band at flower = 10Hz. In [55], Brady et al. analyzed the failure point of the PN expansion (defined as the stage when the PN series carried to 3PN makes a 2% error in the energy loss rate) during the binary inspiral phase. They estimated that the PN failure point is at the orbital speed3≡(πMF)1/3 '0.3, corresponding to separation distancer'10M. This failure causes problem for IMRIs in the Advanced LIGO band.
Table 3.1 lists the basic profile for three canonical IMRIs, assuming the system evolves along circular equatorial orbits. This includes: (i) the separation of the neutron star and the central black hole when the wave frequency reaches 10Hz; (ii) the wave frequencyF10M when their separation is 10M; (iii) the ISCO frequency of the central black hole; (iv) the ratio (∆N>10M%) of the wave cycles spent between 10Hz and 10M, to the total wave cycles spent in the advanced LIGO band (i.e., from flower =10Hz to fupper = Fisco) [The calculation of (iv) is based on Eq. (3.14)]. Table 3.1 shows that for central black holes with masses M >∼200M, very few wave cycles (if any) are spent atr>10Min the Advanced LIGO band. By contrast, for black holes with masses M ∼ 50-100M, a sizable portion of the wave cycles is spent atr > 10M.
However, this will not bring a proportional increase in the accumulated SNR in the corresponding frequency band (i.e., from flowertoF10M). Because the noise spectrum is steep at low frequencies (∼10Hz).
a/M M(M) r10Hz(M) F10M(Hz) Fisco(Hz) ∆N>10M%
0 50 25.6 40.9 87.9 92.6%
100 16.1 20.4 44.0 75.5%
200 10.1 10.2 22.0 4.8%
0.9 50 25.4 39.7 291.4 90.0%
100 16.0 19.9 145.7 68.6%
200 10.0 9.9 72.8 −
Table 3.1: Profile of IMRIs in advanced LIGO band. The column “∆N>10M%” lists the ratio of the wave cycles spent between 10Hz and 10M, to the total wave cycles spent in the advanced LIGO band (i.e., from 10Hz toFisco) [The calculation of is based on Eq. (3.14)]. For descriptions of other columns, see text above.
The above argument suggests that both the (adiabatic) Teukolsky waveforms and the standard PN wave- forms have their limitations. By comparison, the Teukolsky waveforms should by far the most accurate tem- plates available for IMRIs. In the following two subsections, we shall mainly use results from the Teukolsky formalism (in conjunction with 2PN formulas) to estimate the number of IMRI wave cycles in the Advanced LIGO band.