3.2 Initial estimates
3.2.1 Detection range
In this section, we shall estimate the detection range of Advance LIGO detectors for “canonical” IMRI sources at a prescribed signal-to-nosie ratio (SNR) by assuming the binaries move along circular equatorial orbits. Under the same assumption, we shall also estimate on the number of wave cycles that sweep through the advanced LIGO band.
The detector output, i.e., the measured gravitational-wave strain, can be written as
s(t) = h(t)+n(t), (3.2)
whereh(t) is the (possibly present) gravitational-wave signal andn(t) the detector noise. The detector output of the gravitational-wave signalh(t) is a linear combination of the waveform from the source (h+andh×in TT gauge) multiplied by the detector beam-pattern functions (F+andF×). It can be written as (see, e.g., Eq. (103) in Ref. [39])
h(t)=F+(θN, φN, ψN)h+(t;ι, β)+F×(θN, φN, ψN)h×(t;ι, β), (3.3) where the anglesιandβdescribe the direction towards the detector in the source’s preferred local axes (see Figs. 9.2 and 9.9 in Ref. [39] for illustrations); the beam-pattern functions are given by
F+(θN, φN, ψN) = 1
2(1+cos2θN) cos 2φNcos 2ψN−cosθNsin 2φNsin 2ψN, (3.4a) F×(θN, φN, ψN) = 1
2(1+cos2θN) cos 2φNsin 2ψN+cosθNsin 2φNcos 2ψN. (3.4b) Here the anglesθNandφNdescribe the direction of the source with respect to the detector; andψNcharacter- izes the orientation of the polarization axes (see Fig. 3.7 in Appendix B).
We shall assume that the detector noisen(t) follows a Gaussian distribution, and is statistically charac- terized by the autocorrelation functionCn(τ)=hn(t)n(t+τ)i, where angular brackets denote a time average.
The one-sided noise spectral density is the Fourier transform of the autocorrelation function Sn(f) = 2
Z ∞
−∞
Cn(τ)e2πi fτdτ , f >0. (3.5)
In this chapter, we shall adopt the numerical Advanced LIGO strain noise spectrum from [40], and simplify it by means of an analytic fit.5 Our fit to the one-sided spectral density functionSn(f) is given by [34]
Sn(f) = 1.6×10−49
300 (f/15)−17+7 (f/50)−6+24 (f/90)−3.45−3.5 (f/300)−2 (3.6) +
561/5−(33/10)(f/50)2+(22/30)(f/100)4 1+(7/3)(f/1000)2−1 Hz−1. Figure 3.1 shows the fitted noise strain p
Sn(f) as well as the simulation data: the two match each other well.
10 50 100 500 1000 5000
FrequencyHHzL -23.5
-23 -22.5 -22
Log10ISnM
Figure 3.1: Equivalent strain noise versus frequency for the nominal Advanced LIGO interferom- eter, using fused silica test masses; the signal recyling parameters are tuned to optimize the NS-NS inspiral range. Blue dots are drawn according to numeric data from [40]. The red curve represents the analytic fit defined in Eq. (3.6).
For ground-based interferometers, the amplitude of the expected IMRI gravitational wave signals will be close to, or more likely, below the instrumental noise level in the detector output data. To distinguish the signal contribution from the noise background, a pattern recognition technique of matched filtering is widely used in LIGO data analysis. To perform matched filtering, the detector outputs(t) is first convolved against a Wiener optimal filter4(t) whose Fourier transform is proportional to 1/Sn(f). This procedure is meant to suppress the those frequency components of the output at which the detector noise is dominant. The filtered data are then matched to a bank of theoretical template waveforms{hT(t;α)}, each characterized by a different parameter setα. The SNR for a particular waveform template is defined by (cf. Eq. (1.2) in Ref. [43])
S N
hT =
R hT(t)4(t−τ)h(τ)dτdt rmsR
hT(t)4(t−τ)n(τ)dτdt , (3.7)
5The LIGO-II noise curve fit in Owen and Sathyaprakash [41], which is often used by theorists, predates the work of Buonanno and Chen [42], which has changed the noise curves significantly. Reference [41] also assumes sapphire mirrors, which are no longer used;
and it ignores coating thermal noise—so its thermal noises are incorrect. This has motivated us to construct the new fit, Eq. (3.6).
where “rms” means root-mean-square value of the denominator and the average is taken over an ensemble of realizations of the noisen(t). When the template waveform matches the incoming gravitational-wave signal exactly, the SNR becomes
S N
2
= 4 Z ∞
0
|h(˜ f)|2
Sn(f)d f . (3.8)
In the following discussion of detection ranges at a prescribed SNR, we shall refer to Eq. (3.8) as our defini- tion. It can be shown that the SNR is approximately proportional to the square-root of the number of wave cycles contained in the data, and is inversely proportional to the distance between the source and the detector.
For sources at a fixed distancerfrom the detector, the squared signal-to-noise ratio [Eq. (3.8)] averaged over sky positions and source orientations is given by6
S N
2
avg = 4hF2+(θs, φs, ψs)i Z ∞
0
h|h˜+(f)|2+|h˜×(f)|2i
Sn(f) d f, (3.9a)
where ˜h+,×(f) are the Fourier transformation of the time-domain waveform h+,×(t). The average over the beam-pattern function (i.e., average over sky positions), in the case of LIGO’s L-shaped interferometers, is (Eq. (110) in [39])
hF+2i = hF2×i = 1
5. (3.9b)
The average over the source orientation can be written as (Eq. (44) in [39])
h|h˜+|2+|h˜×|2i = π 12
m r
2M3 m
1
(πMf)7/3 , (3.9c)
whereM =M+µis the total mass of the binary andm=Mµ/(M+µ) is the reduced mass. Note that Eq. (3.9c) contains only the leading quadrupole-radiation contribution to the evolution of the waveform strength. In computing the integration in Eq. (3.9a), we shall take the starting frequency to be flower = Flower = 10Hz, the lower cut-offfrequency of advanced LIGO interferometers; and take the ending frequency to be fupper= Fisco= Ωisco/π, whereΩiscois the orbital angular frequency at the innermost-stable-circular-orbit (isco) of the central black hole. The formula to computeΩisco(and henceFisco) can be found from, e.g, Eq. (3.20)-(3.21) in [44]. In Fig. 3.2, we plotFiscoas a function of the black-hole spin for three different black-hole masses. It can be seen thatFiscoincreases sharply as the black-hole spin approaches its maximal value 1, which has to do with the fact that the ISCO radius shrinks for rapidly spinning holes, and can get very close to the horizon radiusrH(the two becomerisco=rH=Mwhena/M=1).7
Using the above formulas we have computed the SNR curves shown in Fig. 3.3—angle-averaged SNR as a function of BH massMfor an inspiralingµ=1.4MNS at a distancer=100Mpc from Earth as observed
6This is obtained by combining Eq. (26) and (29) in [39]. Note that there is a factor-of-2 error in Eq. (29) in [39].
7The proper distance between the isco and horizon approaches infinity asa→M, but radii become the same. See Fig. 2 in Ref. [45].
-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.