Numerical Relativity Ringdown Waveforms: From Spherical to Spheroidal Mode Decomposition Lionel London, James Healy, Deirdre Shoemaker The Georgia Institute of Technology Center for Relativistic Astrophysics NRDA 2013 September 20, 2013 Motivating Questions Numerical Relativity(NR) waveforms are decomposed into spin -2 spherical multipoles. On the other hand, the quasi-normal mode(QNM) ringdown of Kerr black holes is naturally described by spin -2 spheroidal multipoles. Can we estimate spheroidal information from NR ringdown? If we can, so what? London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 1 / 30
QNM Analysis Seeks to Decode Hair Loss 0.01 ω 22 0.005 −1 10 rM| ψ 4 ( ω )| rM| ψ 4 (t)| 0 −0.005 −0.01 0 20 40 0 0.5 1 M ω t/M Hair Loss: ( q , � S 1 , � S 2 , ... ) �→ ( M f , j f ) Kerr,Teukolsky Perturbation Theory: Ringdown should be well approximated by a discrete set of QNMs, { A nlm e i ˜ ω nlm t } Teukolsky, Others ω nlm ≡ ω nlm + i /τ nlm ˜ Teukolsky’s Equations: ( M f , j f ) ← → ˜ ω nlm Ringdown Analysis: ( q , � S 1 , � S 2 , ... ) �→ { A nlm } Kelly (2013), Kamerestos (2012), Buonanno (2007), Berti, and others ... London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 2 / 30
Numerical Relativity Meets Perturbation Theory Numerical Relativity: Spherical Multipoles (Orthogonal in l and m ) ψ NR r ψ 4 = � lm Y lm ( θ, φ ) l , m Ω r ψ 4 ¯ ψ NR lm = � Y lm d Ω Perturbation Theory: Spheroidal Multipoles (Not Orthogonal in l ) Overtones labeled by n ψ PT r ψ 4 = � nlm S lm ( ˜ ω nlm j f , θ, φ ) nlm nlm = A nlm [ e − i ˜ ψ PT ω nlm t ] Y lm Decomposition ⇒ NR Ringdown Is a Sum of QNMs Ω ¯ ψ NR A nlm [ e − i ˜ ω nl ′ m t ] [ lm = � � Y lm S nl ′ m d Ω] nl ′ m Kelly et al (2012), Bounanno et al (2006), Berti et al (2005), Teukolsky (1972) London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 3 / 30
NR Ringdown is a Sum of QNMs Example: ( q = 1 , Nonspinning ) → ( M f = 0 . 9516 , j f = 0 . 6862) −4 1 x 10 ω 22 −3 10 ω 32 0.5 rM| ψ 4 ( ω )| rM| ψ 4 (t)| 0 −0.5 −4 10 −1 10 30 50 0 0.5 1 1.5 M ω t/M Figure: The l = 3 m = 2 spherical harmonic multipole for an equal mass, nonspinning black hole binary (GaTech MAYA). London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 4 / 30
NR Ringdown is a Sum of QNMs Example: ( q = 15 , Nonspinning ) → ( M f = 0 . 9978 , j f = 0 . 1895) −5 x 10 ω 22 ω 32 2 rM| ψ 4 ( ω )| rM| ψ 4 (t)| 0 −4 10 −2 10 30 50 0.2 0.4 0.6 0.8 1 M ω t/M Figure: The l = 3 m = 2 spherical harmonic multipole for a 15:1, nonspinning black hole binary (GaTech MAYA). S nlm = S nlm ( j f ˜ ω nlm , θ, φ ), where for j f = 0, S nlm = Y lm See also Kelly et al (2012) London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 5 / 30
NR Ringdown is a Sum of QNMs What does Perturbation Theory suggest that we should find in NR Ringdown? ℓ -mixing � Overtones ? Price, Teukolsky, ... Mirror Modes ? Price, Teukolsky, Leaver ... Second Order QNMs ? Ioka, Okuzumi, Campanelli, Lousto, ... Sum and Difference tones Figure: Artist’s depition of a closed box. London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 6 / 30
What needs to be done? Multimode Fitting Fit Parameters: ω nlm , τ nlm , M f , j f , Number of modes A N × 2 × 2 dimensional optimization problem Prony Methods, NLLS Methods are computationally expensive, and do not always allow for trivial association between fit frequencies with QNM frequencies. Berti, Others ... Our approach : Linear least squares fitting, not in the basis of polynomials( t n ), but in the basis of QNMs ( e i ω nlm t ) Frequency Domain Greedy Efficient Error Analysis London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 7 / 30
Multimode Fitting − → Spheroidal Decomposition Algorithm Given NR Data, Choose Fitting Region: 1 Ringdown: t start ≤ t ≤ t stop Ω ¯ Y lm S nl ′ m d Ω] e − i ˜ Model: ψ NR � ω nl ′ m t lm ( t ) = � [ A nlm nl ′ m Ω ¯ Apply Fit: [Multimode Fit]( ψ NR → A Fit � lm ) �− nlm , l ′ ≡ A nlm [ Y lm S nl ′ m d Ω] 2 Ω ¯ Estimate Spheroidal Amplitudes: A nlm = A Fit � nlm , l ′ / Y lm S nl ′ m d Ω 3 London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 8 / 30
Does MultiMode Fitting Yield Better Residuals? Fractional Root Mean Square Error Fractional Root Mean Square Error ≡ | RMSE ( ψ NR lm − ψ FIT lm ) | RMSE ( ψ NR lm ) London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 9 / 30
Does Multimode Fitting Yield Better Residuals? 1 10 MultiMode fractional rmse (l,m)=(3,2) SingleMode 0 10 −1 10 −2 10 1 1.5 2 2.5 3 3.5 4 4.5 q Figure: Multimode analysis of ψ NR 32 , initially nonspinning runs. London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 10 / 30
Does Multimode Fitting Yield Better Residuals? 0 10 MultiMode fractional rmse (l,m)=(2,2) SingleMode −1 10 −2 10 −3 10 1 1.5 2 2.5 3 3.5 4 4.5 q Figure: Multimode analysis of ψ NR 22 , initially nonspinning runs. London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 11 / 30
Does Multimode Fitting Yield Better Residuals? 0 10 MultiMode fractional rmse (l,m)=(4,4) SingleMode −1 10 −2 10 1 1.5 2 2.5 3 3.5 4 4.5 q Figure: Multimode analysis of ψ NR 44 , initially nonspinning runs. London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 12 / 30
Multimode Fitting Example: ( l , m ) = (3 , 2) ( l , m , n ): (2 , 2 , 0), (3,2,0), (2,2,1) 4 3 2 0.25 −4 0.2 | ψ ( ω )| 10 1/ τ 2,1 2,1 0.15 0.1 −5 4 10 3 2 0.6 0.8 1 0 0.5 1 1.5 ω ω Figure: Multimode analysis of ψ NR 32 , q = 1 . 5, initially nonspinning run. London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 13 / 30
Multimode Fitting Example: l = m = 2 3 2 −2 0.25 10 −3 0.2 10 | ψ ( ω )| 1/ τ 2,1 2,1 −4 0.15 10 −5 0.1 10 3 2 0.5 0.6 0.7 0.8 0.9 0 0.5 1 ω ω Figure: Multimode analysis of ψ NR 22 , q = 1 . 5, initially nonspinning run. London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 14 / 30
Multimode Fitting Example: l = m = 4 5 4 −3 10 0.25 −4 0.2 10 | ψ ( ω )| 1/ τ 2,2 2,2 3,2 2,2 0.15 −5 10 0.1 5 4 1.1 1.2 1.3 0.6 0.8 1 1.2 1.4 1.6 ω ω Figure: Multimode analysis of ψ NR 44 , q = 1 . 5, initially nonspinning run. London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 15 / 30
Multimode Fitting Analysis: l = m = 2 0.08 (l,m,n) = (2,2,0) 0.07 (l,m,n) = (2,2,1) 0.06 A fit (2,2,n) 0.05 0.04 0.03 0.02 0.01 0 1 1.5 2 2.5 3 3.5 4 4.5 q Figure: Multimode analysis of ψ NR 22 , initially nonspinning run. London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 16 / 30
Multimode Fitting Analysis: 2 nd Order QNMs −3 1.8 x 10 (l 1 ,m 1 ,n 1 )(l 2 ,m 2 ,n 2 ) = (2,2,0)(2,2,0) 1.6 (l 1 ,m 1 ,n 1 )(l 2 ,m 2 ,n 2 ) = (2,2,0)(3,2,0) 1.4 A fit (l,m)=(5,4) 1.2 1 0.8 0.6 0.4 0.2 0 1 1.5 2 2.5 3 3.5 4 4.5 q Figure: Multimode analysis of ψ NR 54 , initially nonspinning run. London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 17 / 30
Many Modes ... Overtones � Mirror Modes ✗ (not significantly) Second Order QNMs � Figure: An artists depiction Pandora opening box, and evil flowing out. But do they matter?? In other words, under what circumstances might they be relevant for detection? Parameter Estimation? London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 18 / 30
SNR Scenarios: q = 1, Source Directly Over Detector M = 500 (M Sol ), D = 20 (Mpc), ( θ , φ ) = (0.00,0.0) (2,2,0) (2,2,1) (3,2,0) −21 10 (4,2,1) (3,2,1) (2,1,0)(2,1,0) −22 advLIGO 10 S n ( ω ) 1/2 and 2*|h( ω )| ω 1/2 −23 10 −24 10 −25 10 −26 10 −27 10 −1 0 1 10 10 10 ω London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 19 / 30
SNR Scenarios: q = 1, θ = π/ 4, φ = 0 M = 500 (M Sol ), D = 20 (Mpc), ( θ , φ ) = (0.79,0.0) (2,2,0) (2,2,1) (4,4,0) −21 10 (2,2,0)(2,2,0) (4,4,1) (3,2,1) −22 advLIGO 10 S n ( ω ) 1/2 and 2*|h( ω )| ω 1/2 −23 10 −24 10 −25 10 −26 10 −27 10 −1 0 1 10 10 10 ω London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 20 / 30
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