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 Pürrer 1 , Mark Hannam 1 , P . Ajith 2,3 , Sascha Husa 4 1 Cardiff, 2 LIGO Lab, Caltech, 3 TAPIR, Caltech, 4 UIB 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. NRDA 2013 Mallorca Monday, 23 September 13

MOTIVATION • An effective total spin parameter χ IMR = (m 1 χ 1 + m 2 χ 2 ) / 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 = m 2 /m 1 , and the effective total spin parameter, χ IMR . • 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, χ PN, 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: χ PN = χ IMR − 76 η m 1 m 2 η = 113( χ 1 + χ 2 ) / 2 ( m 1 m 2 ) 2 NRDA 2013 Mallorca 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 χ IMR =0.45 and χ PN ≈ 0.4. NRDA 2013 Mallorca Monday, 23 September 13

MATCHED FILTERING • Overlap (noise-weighted inner product) between two waveforms, h 1 (f) and h 2 (f): Z f max h 1 ( f )˜ ˜ 2 ( f ) h ∗ SNR: p h h 1 | h 2 i = 4Re d f ρ ⌘ h h, h i S n ( f ) f min • Match between two normalized waveforms is then defined as their overlap, maximized over time and phase shifts of the waveform: ∆ t, ∆ φ h ˆ h 1 | ˆ h ( f ) ⌘ ˜ ˆ p where match( h 1 , h 2 ) ⌘ max h 2 i h ( f ) / h h | h i • Given a signal waveform h( λ ) with physical parameters λ and a template x( Λ ) with physical parameters Λ we define the fitting factor : x ( Λ ) | ˆ FF = ∆ t, ∆ φ , Λ h ˆ max h ( λ ) i • We consider l=2 modes only and assume optimal orientation for SNRs. • We use the “zero-detuned high-power” noise curve with f min = 15Hz, and f max = 8kHz. NRDA 2013 Mallorca Monday, 23 September 13

SINGLE SPIN MODELS • Start with the mapping where ( χ 1 , χ 2 , η ; f ) 7! ( χ e ff , η ; f ) is an arbitrary effective spin parameter. χ e ff = χ e ff ( χ 1 , χ 2 , η ) • To build a single spin model an inverse of this mapping is needed which requires a relation between χ 1 and χ 2 . We choose to use only the symmetric part of the input spins χ s = ( χ 1 + χ 2 )/2. • We define the frequency domain single spin model strain as ˜ h m ( χ e ff ( χ 1 , χ 2 , η ) , η ; f ) := ˜ h F2 ( χ s , χ s , η ; f ) • With this definition the model represents equal spin configurations exactly . For the choice χ e ff = χ PN this model is identical to the one defined in [Ajith, 2011] . NRDA 2013 Mallorca Monday, 23 September 13

MATCHES • The noise-weighted inner product (match) between PN single-spin models and TaylorF2 signals with χ IMR or χ PN = const degrades when moving away from the equal spin line. A model built with χ PN is more faithful than χ IMR . • We construct IMR waveforms as TaylorF2 frequency domain PN-NR hybrids; the same approximant used in IMRPhenomC. Matches between IMR waveforms with Matches between single spin PN-models the equal-spin (0.45, 0.45) configuration. and TaylorF2 signals at 7M ⊙ . 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 match( h m ( θ ) , h m ( θ 0 )) ≥ 1 − 3 . 12 / ρ 2 • We find the parameter bias by locating the model waveform for which the match with the given signal is maximized (fitting factor). NRDA 2013 Mallorca Monday, 23 September 13

BIASES AND UNCERTAINTIES: PN R EGIME Want to assess systematic parameter biases that are due to the effective single-spin approximation and contrast them with statistical errors ( uncertainties ). Other markers: parameters of Red star: actual value of model parameters best fit single spin value ( η =0.16, χ PN =0.401575) Signal waveforms are not exactly We show here the represented by deviation from the PN 7M ⊙ model! recovered parameters for the equal-spin configuration. Ellipse shows the statistical uncertainty in measuring the Signals: TaylorF2 parameters (90% Model: TaylorF2 single-spin χ PN confidence region) at a given SNR Monday, 23 September 13

BIASES AND UNCERTAINTIES: PN R EGIME • 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 PN 7M ⊙ 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. • χ IMR leads to larger spread. Monday, 23 September 13

BIASES AND UNCERTAINTIES: IMR R EGIME IMR regime • For IMRPhenomC we find a significant systematic bias for all IMR Signals: TaylorF2-NR hybrids waveforms. This is a problem of the model which can be removed in the Model: IMRPhenomC future. • At intermediate masses (around 50 IMR 50M ⊙ M ⊙ ) the spread in recovered spin values is far larger than the statistical uncertainty in χ IMR , even at an SNR of 10, which is close to the detection threshold. • Single spin approximation is valid (spread in χ IMR <uncertainty) for χ IMR up to SNR 10 for masses 20, 50M ⊙ and up to SNR 20 for M>100M ⊙ . Monday, 23 September 13

FINAL SPIN VS EFFECTIVE SPIN 1.0 a f ê M f • The waveform from the ringdown of the final BH will be characterized by the final spin , and not by either χ PN or χ IMR . 0.8 0.9 ‡ Ê • Results of various final-spin formulas 0.8 ‡ 0.6 agree to within a few percent with our Ê c 2 NR results for the final spins. ‡ Ê 0.7 Ê ‡ 0.4 Ê ‡ • Final spins for our family of χ IMR =0.45 0.6 NR simulations range from 0.68 up to 0.2 0.84. c PN = 0.401575 0.5 Ê c IMR = 0.45 ‡ • The effective single spin approximation 0.0 must get worse as we approach the - 1.0 - 0.5 0.0 0.5 1.0 merger. c 1 NRDA 2013 Mallorca Monday, 23 September 13

SENSITIVITY OF BIASES TO ERRORS 0 0 - 5 - 10 - 10 - 20 - 15 Bias @ % D Bias @ % D - 20 - 30 m m - 25 q q - 40 c IMR c IMR - 30 - 50 - 35 10 15 20 30 50 70 100 150 200 10 15 20 30 50 70 100 150 200 M ü M ü Variation in biases caused by changing the Variation in biases caused by changing the resolution of the numerical waveform. hybridization frequency ω m . N=80 (solid) vs N=64 (dashed) gridpoints M ω m ∼ 0.07 (solid) vs M ω m ∼ 0.08 (dashed) M ω m ∼ 0.07 NRDA 2013 Mallorca Monday, 23 September 13