The overlap between waveforms

4.4 Kludge waveforms: Results and comparison

4.4.1 The overlap between waveforms

slow-motion approximation) essentially truncate the sum over multipoles. The Press formula (4.16) includes contributions at all multipoles and so might be expected to handle these contributions quite well. However, it turns out not to perform much better than the quadrupole-octupole waveform in this regime. While it includes contributions at all multipoles, the lack of background curvature in the waveform modelmightcause the amplitudes of the higher modes to be much lower than for true EMRI waveforms.

Fortunately, these two deficiencies become important for the same class of orbits—those that allow the body to approach very close to the black hole. As a rule of thumb we shall find that NK waveforms are reliable (in terms of the overlap discussed in the next section) as long as the closest orbital approach (periastron) is rp&5M.

whereSh(f) is the detector’s one-sided noise power spectral density (PSD) (in this case, the noise PSD for the LISA detector) and an asterisk denotes complex conjugation. Consideringh(t, λ^α) as a template with parametersλ^αwe can approximate the SNR by [63]

S N

hh(λ^α)i

= h(s|h)i_n

ph(n|h)²i_n = (s|h)

√

(h|h) . (4.36)

The notationhfi_nmeans to ensemble average the function f over all possible noise realizationsn. The PSD in the denominator of Eq. (4.35) serves to suppress those frequency components of the signal at which the detector noise is large.

The main tool which will be used in this paper is not the SNR, but rather the overlap functionO. The overlap is defined as an inner product between two normalized vectors/signals:

O=( ˆs|h)ˆ . (4.37)

The normalization is chosen so that ( ˆs|s)ˆ = (ˆh|h)ˆ = 1. The overlap can be regarded as the inner product between two unit vectors; it varies within [−1,1]. The overlap is equal to 1 if the two waveforms are identical, and it equals zero if the two waveforms are orthogonal (for example, cosine and sine signals). The overlap is an appropriate measure of “goodness-of-fit” since we are interested in knowing how well the NK approximate the behavior of TB EMRI waveforms. In this context, it is important to include the noise properties of the detector since it is no problem if a template has poor performance at frequencies where detector noise is large.

For a fair comparison, we should choose the signal (TB waveform) and template to have the same pa- rameters.⁵ In practice, the start times of our signals and templates were slightly mismatched, so we allow for maximisation over starting time (“time of arrival”). This maximization is commonly done in GW data analysis, and can be accomplished very efficiently in the Fourier transform, since the time offset corresponds to a phase shift in the frequency domain. Accordingly, we modify the inner product (x|h) slightly:

(x|h)=max_t Re Z ∞

0

4 ˜x(f)˜h^∗(f) Sh(f) e^{i2πf t}d f

!

(4.38)

wheretcorresponds to the time difference. We find that the maximum usually occurs with a time offset close to zero (typically a few frequency bins).

We conclude our discussion of the overlap by describing our approximation to the LISA noiseSh(f). We have used an analytic approximation to the numerically generated sensitivity curve given in Ref. [66]. The agreement between these two curves is excellent as clearly illustrated in Fig. 4.1. Our approximateSh(f) function can be easily calculated according to the following simple prescription. Defineu = 2πfτ, where

5Note that here we study not the fitting factors but the faithfulness of the NK waveforms as compared to TB-based ones, i.e., how well an approximate waveform with given parameters reproduces the “true” waveform with thesamephysical parameters. A faithful bank of waveforms could be used for parameter estimation, while for detection all that is required is an “effectual” template bank, i.e., one in which every true waveform is well represented by one template, even if that template has very different parameters [65].

0.0001 0.001 0.01 0.1

frequency (Hz)

1e-21 1e-20 1e-19 1e-18 1e-17 1e-16 1e-15

(S

h

(f))

1/2

LISA Average sensitivity, SNR=1 Analytic fit

0.0001 0.001 0.01 0.1

frequency (Hz)

1e-21 1e-20 1e-19 1e-18 1e-17 1e-16 1e-15

With white dwarf confusion noise

Figure 4.1: Expected sensitivity curve p

Sh(f) for LISA; black curve: numerical curve as gener- ated in [66], red curve: analytic approximation used in this paper; see text for details.

f is frequency andτ = (5×10⁶km)/c = 50/3 sec is the light travel time down one of LISA’s arms. For u<utrans=0.25 we set

Sh(f)="8.08×10⁻⁴⁸

(2πf)⁴ Hz⁴+5.52×10⁻⁴¹

#

Hz⁻¹, (4.39)

while foru≥utrans,

Sh(f)= 1 R

2.88×10⁻⁴⁸

(2πf)⁴ Hz⁴+5.52×10⁻⁴¹

!

Hz⁻¹, (4.40a)

where

R= 1 u²

"

(1+cos²(u)) 1 3− 2

u²

!

+sin²(u)+4 sin(u) cos(u) u³

#

. (4.40b)

Note that LISA’s characteristics are incorporated in the light travel timeτand in the numerical constants in the above expressions, so a similar mission but with different noise characteristics would still be described by the functional form defined by Eqs. (4.39) – (4.40).

In Fig. 4.1, we have also included noise from a confused population of galactic white dwarf binaries following the prescription outlined in [32].

-1 -0.5 0 0.5 1

0 500 1000 1500 2000

h+ D/µ

p=6.5M e=0.5 a=0.5M θ_d=90(deg)

-1 -0.5 0 0.5 1

h+ D/µ

p=6.5M e=0.5 a=0.5M θ_d=90(deg)

-1 -0.5 0 0.5 1

0 500 1000 1500 2000 2500 3000

h+ D/µ

p=6.5M e=0.5 a=0.5M θ_d=45(deg) TB waveform

-1 -0.5 0 0.5 1

h+ D/µ

p=6.5M e=0.5 a=0.5M θ_d=45(deg)

-1 -0.5 0 0.5 1

0 200 400 600 800

h+ D/µ

time (M)

p=3.5M e=0.4 a=0.99M θ_d=90(deg) NK waveform

Figure 4.2: Comparing TB and NK (Press-formula numerical kludge) waveforms (black and red curves, respectively) for equatorial orbits and for an observer at a latitudinal positionθ=45^◦or 90^◦. Orbital parameters are listed above each graph. The waveforms are scaled in units ofD/µ whereDis the radial distance of the observation point from the source andµis the test-body’s mass. Thex-axis measures retarded time (in units ofM) and we are showing the “+” polarization of the GW in each case. The overlaps between the NK and TB waveforms are 0.979,0.990,0.507 going from the top figure down.