Evaluating Errors in Hybrid Waveforms

Evaluating Errors in Hybrid Waveforms

Ilana MacDonald1,2, Harald Pfeiffer2, Samaya Nissanke2,3,4

Motivation★ Gravitational wave (GW) detectors like Advanced LIGO need

waveform templates for optimal sensitivity.★ Usual approach: Hybrid waveforms, composed of a post-

Newtonian (PN) inspiral and a numerical relativity (NR) late inspiral, merger and ringdown.

★ Both NINJA and NRAR collaborations require knowledge of hybrid-errors to ensure sufficient accuracy.

★ OUR GOAL: Comprehensive error analysis of hybrid waveforms. • Choice of matching region of the PN and NR waveforms.• Systematic errors in NR waveform.• Choice of PN approximants.

★ Current NR waveforms are likely too short.

1 Constructing hybrid waveforms2★ Align PN waveform by minimizing phase-difference over matching interval Mω ± 5% Mω,

where ω is the gravitational wave frequency.★ TaylorT1, T3 & T4 PN waveform (3.5PN phase, 3.0PN amplitude) [1].★ NR waveform (equal mass, non-spinning) computed with Caltech/Cornell/CITA SpEC

code [2], [3].PN waveform

NR waveform

matching region

Calculating waveform differences3

1 Department of Astronomy and Astrophysics, 50 St. George street, University of Toronto, Toronto, ON, M5S 3H82 Canadian Institute for Theoretical Astrophysics, 60 St. George street, University of Toronto, Toronto, ON, M5S 3H83 Theoretical Astrophysics, California Institute of Technology, Pasadena, California, 911254 Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California, 91109

[1] Blanchet, L., "Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries", Living Rev. Relativity 9, (2006)[2] Boyle, M. et al. “High-accuracy comparison of numerical relativity simulations with post-Newtonian expansions”, Physical Review D, vol. 76, Issue 12[3] http://www.black-holes.org/SpEC.html[4] Lindblom, L., Owen, B., & Brown, D., “Model waveform accuracy standards for gravitational wave data analysis”, Physical Review D, vol. 78, Issue 12

Results II: Effect of Matching Regionand PN approximant

4 Results I: Systematic Errors in the NR Waveform

★ Systematic errors in the NR waveform will affect the hybrid.

★ Illustration: waveform extracted at finite radius. This leads to systematic phase-errors at low frequencies (right figure).

5

★ Consider a waveform template h which differs from the exact waveform by an amount δh<<h. The error δh does not affect data-analysis if [4]:

★ Define error criterion ||δh|| ∕ ||h||:

★ Properties of ||δh|| ∕ ||h||: • linear in error δh• independent of source distance• unifies bounds for event detection • and param. estimation

★ Approximate δh by difference between low-quality hybrid and higher-quality “reference hybrid”, to estimate error of low-quality hybrid.

★ ||δh|| ∕ ||h|| depends on total mass M, therefore output is plot of ||δh|| ∕ ||h|| vs. M.

Conclusions★ Quantifying errors of hybrid waveforms is difficult:

• Depending on total mass, different portions of the hybrid are in the GW-detector frequency band.

• There are many potentially relevant sources of error.★NR with 15 orbits likely not long enough for hybrids:

• Need a longer NR waveform if matching with Taylor T3.• Taylor T4 hybrids work well for simple case, but not

necessarily in general.★ Future work:

• Extend to non-zero spins and unequal mass ratios.

6

★ Compare hybrid matched at high frequency Mω with reference hybrid matched at a lower frequency.

★ Hybridization at lower Mω decreases error in hybrid (as expected).

★ Error also depends on difference between trial Mω and reference Mω.

★ As reference Mω decreases, we asymptote towards error between true gravitational waveform and hybrid.

★ Assume true waveform is hybrid of NR and Taylor T4 PN waveforms.

★ Hybrids constructed with Taylor T3 waveforms are insufficient for detection.

★ Hybrids constructed with Taylor T4 waveforms are sufficient for parameter estimation at a matching interval of [0.064 ± 0.0032].

★ Note: TaylorT4 is known to agree unreasonably well for this special case (equal mass, no spin). The behavior of TaylorT3 is more representative for generic parameters.

★ Sufficient for GW detection, insufficient for parameter estimation at SNR > 18.

★ Even when NR waveform seems sufficient for parameter estimation, hybrid isn’t necessarily.

Two hybrids aligned at peak amplitude. Note large phase-

difference at early time.

Fourier transform of hybrid waveform and AdLIGO noise curve

(Zero_Det_high_P).

Assessing errors of numerical waveforms for LIGO data-analysis 15

10 100

M/Msun

0.001

0.01

0.1

|!h

| /

|h|

M"m

= 0.042

M"m

= 0.046

M"m

= 0.049

M"m

= 0.052

M"m

= 0.058

M"m

= 0.064

SNR = 10

SNR = 30

ref. M"m

= 0.038

0.04 0.05 0.06

M"ref

0

0.1

0.2

0.3

0.4

0.5

[ |!

h| /

|h| ] m

ax

M"trial

= 0.046

M"trial

= 0.049

M"trial

= 0.052

M"trial

= 0.058

M"trial

= 0.064

T4 vs T3, "m

= 0.046

T4 vs T3, "m

= 0.049

T4 vs T3, "m

= 0.052

T4 vs T3, "m

= 0.058

T4 vs T3, "m

= 0.064

Figure 7. Investigation of the errors of a TaylorT3 hybrid, as a functionof matching frequency ωm. Left: ||δh||/||h|| as a function of mass fordifferent trial hybrid waveforms with gravitational wave frequencies Mω =[0.042, 0.046, 0.049, 0.052, 0.058, 0.064] compared to a reference hybrid waveformwith matching frequency Mω = 0.038. The dotted lines represent the upperlimit of ||δh||/||h|| for detection at SNR = 10 and parameter estimation at SNR= 30. Right: Maximum values of ||δh||/||h|| as a function of the matchingfrequency of the reference hybrid for different trial hybrids at matching frequenciesof Mω = [0.046, 0.049, 0.052, 0.058, 0.064]. The dashed lines are the maximumvalues of ||δh||/||h|| for each trial hybrid compared to a reference hybrid createdwith a Taylor T4 PN waveform matched at Mω = 0.042. Ilana: problem withωtrial = 0.050. Redo or remove?

it would be necessary to use a longer numerical waveform than is currently available.We look at how early one must hybridize these waveforms when using different

post-Newtonian approximants in section 4.5. The results can be seen in figure 8,where, in the left panel, ||δh||/||h||max for the T1 hybrids does not converge to anyvalue, and in the right panel, ||δh||/||h||max for the T4 hybrids clearly does, and at avalue that is well within our limits for detection and parameter estimation. We canthus conclude that for hybrid waveforms which use a Taylor T1 PN approximant, itwould be necessary to have a much longer numerical waveform for it to be sufficentlyaccurate, and that for hybrid waveforms which use a Taylor T4 PN approximant, itis sufficient to match the post-Newtonian and numerical waveforms at a gravitationalwave frequency of Mω = 0.064.

4.5. What is the effect of changing PN approximant or PN order?

SamayaIt is widely accepted that different PN approximants, though modelling identical

physical gravitational wave emission, can have significant effect on measurementaccuracies. This section presents the effects of on ||δh||/||h|| computations of changingeither the PN approximant or the PN order.

• As mentioned earlier, we have so far considered only the T3 approximant. Here,we vary the Taylor approximant of the reference hybrid waveform. Figure ***shows ***. We also compare the different Taylor approximants with each other.


0.03 0.04 0.05 0.06 0.08 0.1 0.14 0.2

0

0.2

0.4

0.6

0.8

1

M!

M!m

=0.045

"infty

- "R=225M

"infty

- "R=385M

10 100M/Msun

0.01

0.03

0.1

|#h| /

|h|

Hybrid

NR

SNR = 30

Infty vs R = 385M

Infty vs R = 225M

Figure 4. Effect of systematic errors in the numerical waveform, illustrated witha finite extraction radius. Left: Phase-difference between GW’s extracted atradius R = 225M , and the waveform extrapolated to infinite extraction radius.Right: Error criterion ||δh||/||h|| computed between these two waveforms, whenboth are matched to a PN waveform at Mωm = 0.042. We show that ||δh||/||h||is sufficient for detection, but insufficient for parameter estimation. The dashedline shows ||δh||/||h|| for the numerical waveforms only, while the solid line shows||δh||/||h|| for the hybrid waveforms.

10 100M/Msun

0.01

1e-5

1e-4

0.005

4.5e-4

0.001

mis

mat

ch

1/2*[ |#h| / |h| ]2

mismatch

SNR = 30

Infty vs R = 225M

Infty vs R = 385M

SNR = 10

0.03 0.05 0.10.1 0.2

1.5

1.6

1.7

1.8

10 100

1.5

1.6

1.7

1.8

M!

M/Msun

Ratio of phase-errorsR=225M vs. R=385M

Ratio of |#h|/|h|R=225M vs. R=385M

385/225

385/225

Figure 5. Left: We show here that for phase errors, mismatch and ||δh||/||h||are equivalent. That is, h‖ can be ignored in equation 14. Right: The top panelshows the ratio of phase errors from the left panel of figure 4, and the bottompanel shows the ratio of the ||δh||/||h|| values from the right panel of figure 4,and shows that these ratios are very close to 385/225. That is, ||δh||/||h|| isproportional to the phase errors.

Error in waveform extracted at finite radii R=225M and R=385M

when compared to waveform extrapolated to infinity.


0.04 0.05 0.06

M!ref

0

0.1

0.2

0.3

0.4

0.5

[ |"

h| /

|h| ] m

ax

M!trial

= 0.046

M!trial

= 0.049

M!trial

= 0.052

M!trial

= 0.058

M!trial

= 0.064

T4 vs T1, !m

= 0.046

T4 vs T1, !m

= 0.049

T4 vs T1, !m

= 0.052

T4 vs T1, !m

= 0.058

T4 vs T1, !m

= 0.064

0.04 0.05 0.06

M!ref

0

0.01

0.02

[ |"

h| /

|h| ] m

ax

M!trial

= 0.046

M!trial

= 0.049

M!trial

= 0.052

M!trial

= 0.058

M!trial

= 0.064

Figure 8. Matching region comparison for Taylor T1 (left panel) and T4approximants (right panel). Ilana: problem with ωtrial = 0.050. Redo or remove?

0.03 0.04 0.05 0.06 0.07

M!match

0

0.1

0.2

0.3

0.4

0.5

[ |"

h| /

|h| ] m

ax

T1T3

Figure 9. Maximum ||δh||/||h|| for Taylor T3 hybrids matched at differentfrequencies compared to a T4 waveform matched at Mωm = 0.042. Ilana: redoωtrial = 0.038 and 0.042.

• PN order.

4.6. Effect of noise-curve?

SAMAYALess sensitive detectors should place less stringent requirements on the waveform

accuracy. Can we illustrate this?

Maximum error between hybrids matched at different frequencies as a function of the

matching frequency of the reference hybrid. Top: PN waveform is Taylor T3. Bottom: PN waveform is Taylor T4.


10 100M/Msun

0.0001

0.001

0.01

|!h

| / |h

|

Med. vs High

Low vs Med.SNR = 30

Figure 6. To assess the effect of the numerical resolution of the NR waveform,we calculate ||δh||/||h|| between Lev5 & Lev6, and between Lev4 & Lev5. Thesehybrids were matched at Mω = 0.042 and the numerical waveforms have anextraction radius of R = 385M.

4.4. How early to match?

One of the more interesting questions to be answered is how early the post-Newtonianand numerical waveforms must be matched. There is an advantage to grafting thesetwo waveforms closer to merger as fewer inpiral cycles would be required of thenumerical waveform which would be less computationally expensive. However, post-Newtonian waveforms become increasingly inaccurate closer to merger, and so onemust find the optimal trade-off between accuracy and computational expense.

The left panel of figure 7 shows ||δh||/||h|| as a function of total mass forhybrid waveforms matched at different gravitational wave frequencies, Mω =[0.042, 0.046, 0.049, 0.052, 0.058, 0.064], compared to a reference hybrid matched atMω = 0.038, using a Taylor T3 post-Newtonian approximant. One might concludefrom this that it is sufficient to match the hybrids at Mω = 0.049 for event detectionand just barely sufficient for parameter estimation at Mω = 0.042.

However, in the right panel of figure 7 is shown the maximum value of ||δh||/||h||for different trial hybrid waveforms as a function of the matching frequency of thereference waveform. As would be expected, as Mωref approaches Mωtrial, ||δh||/||h||becomes smaller. Nevertheless, as the difference between the matching frequencies ofthe hybrids becomes larger, ||δh||/||h||max asymptotes towards some maximum valuewhich would be the true error in the trial hybrid waveform if our numerical waveformcould be arbitrarily long.

Since Taylor T4 is a much more accurate post-Newtonian approximant thanTaylor T3, we can take a hybrid waveform matched at Mω = 0.042, for example, witha T4 PN waveform and compare this to the different trial T3 hybrids. The maximumvalues of ||δh||/||h|| for this comparison are show as the dashed lines in the right panelof figure 7, and ||δh||/||h||max for the T3 hybrids appears to be converging on thesevalues. We can conclude that for hybrid waveforms using the Taylor T3 approximant,


0.03 0.04 0.05 0.06 0.08 0.1 0.14 0.2

0

0.2

0.4

0.6

0.8

1

M!

M!m

=0.045

"infty

- "R=225M

"infty

- "R=385M

10 100M/Msun

0.01

0.03

0.1

|#h| /

|h|

Hybrid

NR

SNR = 30

Infty vs R = 385M

Infty vs R = 225M

Figure 4. Effect of systematic errors in the numerical waveform, illustrated witha finite extraction radius. Left: Phase-difference between GW’s extracted atradius R = 225M , and the waveform extrapolated to infinite extraction radius.Right: Error criterion ||δh||/||h|| computed between these two waveforms, whenboth are matched to a PN waveform at Mωm = 0.042. We show that ||δh||/||h||is sufficient for detection, but insufficient for parameter estimation. The dashedline shows ||δh||/||h|| for the numerical waveforms only, while the solid line shows||δh||/||h|| for the hybrid waveforms.

10 100M/Msun

0.01

1e-5

1e-4

0.005

4.5e-4

0.001

mis

mat

ch

1/2*[ |#h| / |h| ]2

mismatch

SNR = 30

Infty vs R = 225M

Infty vs R = 385M

SNR = 10

0.03 0.05 0.10.1 0.2

1.5

1.6

1.7

1.8

10 100

1.5

1.6

1.7

1.8

M!

M/Msun

Ratio of phase-errorsR=225M vs. R=385M

Ratio of |#h|/|h|R=225M vs. R=385M

385/225

385/225

Figure 5. Left: We show here that for phase errors, mismatch and ||δh||/||h||are equivalent. That is, h‖ can be ignored in equation 14. Right: The top panelshows the ratio of phase errors from the left panel of figure 4, and the bottompanel shows the ratio of the ||δh||/||h|| values from the right panel of figure 4,and shows that these ratios are very close to 385/225. That is, ||δh||/||h|| isproportional to the phase errors.

Phase error Δϕ

★ Changing the numerical resolution has no impact.

〈δh, δh〉 <

{0.01(SNR)2, event detection1, param. estimation

‖δh‖‖h‖ =

√〈δh, δh〉√〈h, h〉

<

{0.1, event detection1/(SNR), param. estimation

http://www.black-holes.org/SpEC.html

http://www.black-holes.org/SpEC.html

Documents

Evaluating Errors in Hybrid Waveforms