TESTING THE VALIDITY OF THE SINGLE-SPIN APPROXIMATION IN - - PowerPoint PPT Presentation

TESTING THE VALIDITY OF THE SINGLE-SPIN APPROXIMATION IN INSPIRAL-MERGER-RINGDOWN WAVEFORMS

Michael Pürrer1, Mark Hannam1, P . Ajith2,3, Sascha Husa4

1Cardiff, 2LIGO Lab, Caltech, 3TAPIR, Caltech, 4UIB

• M. Pürrer, et al., PRD 88, 064007 (2013)

NRDA 2013 Mallorca

Monday, 23 September 13

OVERVIEW

• Motivation
• Single spin models
• Configurations
• Matches
• Biases and uncertainties
• Conclusions

Monday, 23 September 13

MOTIVATION

• GW signals from black-hole binaries with non-precessing spins are described by four

parameters – each black hole’s mass and spin.

• Dominant spin effects can be modeled by a single spin parameter, leading to the

development of several three-parameter waveform models.

• Previous studies indicate that these models should be adequate for GW detection.
• Their advantage is a great reduction of cost for searches over double spin templates.
• We show that the single spin approximation is also sufficient for parameter estimation

with low-mass binaries, but leads to significant bias in the spin at high masses.

• Our results suggest that it may be possible to accurately measure both black-hole spins

in intermediate-mass binaries.

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Monday, 23 September 13

MOTIVATION

• An effective total spin parameter χIMR = (m1 χ1 + m2 χ2) / M has been

used in the construction of the phenomenological inspiral-merger-ringdown (IMR) models “IMRPhenomC” [Santamaria et al., Phys. Rev. D82, 064016 (2010)].

• It models waveforms for black-hole binaries with non-precessing spins.
• It parametrizes the waveforms by their mass M, mass ratio q = m2/m1,

and the effective total spin parameter, χIMR.

• It incorporates a PN description of the inspiral, while the merger and

ringdown regimes are tuned using the results of numerical relativity (NR) simulations.

NRDA 2013 Mallorca

Monday, 23 September 13

MOTIVATION

• A recent study [P

. Ajith, PRD 84, 084037 (2011)] has addressed how well

a related reduced spin parameter, χPN, works for inspiral searches.

• This PN model has been shown to be faithful (match > 0.97) when

either the spins or the masses are equal and effectual (FF > 0.97).

• It uses a parameter motivated by the leading order PN spin-orbit

coupling:

χPN = χIMR − 76η 113(χ1 + χ2)/2 η = m1m2 (m1m2)2

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Monday, 23 September 13

CONFIGURATIONS

• Production of NR waveforms is costly.
• Consider waveforms at a single mass-

ratio (q=4) and effective spin only.

• The configurations we choose lie on

lines of constant χIMR =0.45 and χPN ≈0.4.

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Monday, 23 September 13

MATCHED FILTERING

• Overlap (noise-weighted inner product) between two waveforms, h1(f) and h2(f):
• Match between two normalized waveforms is then defined as their overlap, maximized
• ver time and phase shifts of the waveform:
• Given a signal waveform h(λ) with physical parameters λ and a template x(Λ) with

physical parameters Λ we define the fitting factor:

• We consider l=2 modes only and assume optimal orientation for SNRs.
• We use the “zero-detuned high-power” noise curve with fmin = 15Hz, and fmax = 8kHz.

ˆ h(f) ⌘ ˜ h(f)/ p hh|hi match(h1, h2) ⌘ max

∆t,∆φhˆ

h1|ˆ h2i

hh1|h2i = 4Re Z fmax

fmin

˜ h1(f)˜ h∗

2(f)

Sn(f) d f

FF = max

∆t,∆φ,Λhˆ

x(Λ)|ˆ h(λ)i

ρ ⌘ p hh, hi

where SNR:

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Monday, 23 September 13

SINGLE SPIN MODELS

• Start with the mapping where

is an arbitrary effective spin parameter.

• To build a single spin model an inverse of this mapping is needed which

requires a relation between χ1 and χ2. We choose to use only the symmetric part of the input spins χs= (χ1 +χ2)/2.

• We define the frequency domain single spin model strain as
• With this definition the model represents equal spin configurations exactly. For

the choice χeff=χPN this model is identical to the one defined in [Ajith, 2011].

(χ1, χ2, η; f) 7! (χeff, η; f) χeff = χeff(χ1, χ2, η)

˜ hm(χeff(χ1, χ2, η), η; f) := ˜ hF2(χs, χs, η; f)

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Monday, 23 September 13

MATCHES

• The noise-weighted inner product (match) between PN single-spin models and

TaylorF2 signals with χIMR or χPN = const degrades when moving away from the equal spin line. A model built with χPN is more faithful than χIMR .

• We construct IMR waveforms as TaylorF2 frequency domain PN-NR hybrids; the

same approximant used in IMRPhenomC.

Matches between single spin PN-models and TaylorF2 signals at 7M⊙. Matches between IMR waveforms with the equal-spin (0.45, 0.45) configuration.

Monday, 23 September 13

METHOD FOR COMPUTING CONFIDENCE REGIONS

• The model parameters for the waveform that best matches the signal correspond to the

parameters that are most likely to be recovered in a GW measurement.

• Range of parameters that would be recovered in 90% of observations at a given SNR, i.e., the

90% confidence region for that SNR, illustrates the statistical uncertainty in the measurement.

• Can obtain a good approximation to the correct confidence region by computing matches

between the model waveform with the physical parameters of the signal, and model waveforms with a range of neighboring parameters [Baird, Fairhurst, Hannam, Murphy, PRD 87, 024035 (2013)].

• All neighboring waveforms that have a match greater than some threshold are within the 90%

confidence region. The threshold for a given SNR ρ assuming a 3-dim parameter space is

• We find the parameter bias by locating the model waveform for which the match with the given

signal is maximized (fitting factor).

match(hm(θ), hm(θ0)) ≥ 1 − 3.12/ρ2

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BIASES AND UNCERTAINTIES: PN REGIME

Want to assess systematic parameter biases that are due to the effective single-spin approximation and contrast them with statistical errors (uncertainties).

Ellipse shows the statistical uncertainty in measuring the parameters (90% confidence region) at a given SNR

PN 7M⊙

Red star: actual value of model parameters (η=0.16, χPN =0.401575) Other markers: parameters of best fit single spin value

Signal waveforms are not exactly represented by model!

Signals: TaylorF2 Model: TaylorF2 single-spin χPN

We show here the deviation from the recovered parameters for the equal-spin configuration.

Monday, 23 September 13

BIASES AND UNCERTAINTIES: PN REGIME

• Our aim is to assess systematic parameter biases that are due to the

effective single-spin approximation and contrast them with statistical errors (uncertainties).

PN regime

• Systematic biases from the

single-effective-spin models are much smaller than the statistical errors, even at high SNR.

• The reduced-spin model [Ajith,

2011] is likely to be sufficient for parameter estimation of low- mass signals from aLIGO and AdV.

• χIMR leads to larger spread.

PN 7M⊙

Monday, 23 September 13

BIASES AND UNCERTAINTIES: IMR REGIME

IMR regime

• For IMRPhenomC we find a

significant systematic bias for all IMR

• waveforms. This is a problem of the

model which can be removed in the future.

• At intermediate masses (around 50

M⊙) the spread in recovered spin values is far larger than the statistical uncertainty in χIMR, even at an SNR of 10, which is close to the detection threshold.

• Single spin approximation is valid

(spread in χIMR<uncertainty) for χIMR up to SNR 10 for masses 20, 50M⊙ and up to SNR 20 for M>100M⊙.

IMR 50M⊙

Signals: TaylorF2-NR hybrids Model: IMRPhenomC

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FINAL SPIN VS EFFECTIVE SPIN

• The waveform from the ringdown of

the final BH will be characterized by the final spin, and not by either χPN or χIMR.

• Results of various final-spin formulas

agree to within a few percent with our NR results for the final spins.

• Final spins for our family of χIMR=0.45

NR simulations range from 0.68 up to 0.84.

• The effective single spin approximation

must get worse as we approach the merger.

Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ Ê

cPN=0.401575

‡

cIMR=0.45

• 1.0
• 0.5

0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 c1 c2 afêMf 0.5 0.6 0.7 0.8 0.9

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SENSITIVITY OF BIASES TO ERRORS

m q cIMR 10 100 50 20 200 30 15 150 70

• 35
• 30
• 25
• 20
• 15
• 10
• 5

Mü Bias @%D m q cIMR 10 100 50 20 200 30 15 150 70

• 50
• 40
• 30
• 20
• 10

Mü Bias @%D

Variation in biases caused by changing the resolution of the numerical waveform. N=80 (solid) vs N=64 (dashed) gridpoints Mωm∼0.07 Variation in biases caused by changing the hybridization frequency ωm. Mωm∼0.07 (solid) vs Mωm∼0.08 (dashed)

NRDA 2013 Mallorca

Monday, 23 September 13

CONCLUSIONS

• The single-effective-spin approximation holds well at low masses (small parameter bias).

Therefore, it will be difficult to measure the component spins in low-mass binaries.

• Since we observe large parameter biases at intermediate masses, this implies that in these

cases we may be able to measure the individual spins:

• We expect a strong degeneracy between the two spins and the mass ratio at all stages in

the binary’s evolution.

• Inspiral and ringdown are characterized by a different total spin: χPN, and the final spin.
• It is likely that to describe the full waveform we require knowledge of both black-hole spins.

If this is the case, then it follows that accurate measurements of both spins may be possible.

• This will be investigated in future work.

NRDA 2013 Mallorca

Monday, 23 September 13