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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

20 40 −0.01 −0.005 0.005 0.01

t/M rM|ψ4(t)|

0.5 1 10

−1

Mω rM|ψ4(ω)|

ω22

Hair Loss: (q, S1, S2, ...) → (Mf , jf ) Kerr,Teukolsky Perturbation Theory: Ringdown should be well approximated by a discrete set

• f QNMs, {Anlmei ˜

ωnlmt} Teukolsky, Others

˜ ωnlm ≡ ωnlm + i/τnlm Teukolsky’s Equations: (Mf , jf ) ← → ˜ ωnlm

Ringdown Analysis: (q, S1, S2, ...) → {Anlm} 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) rψ4 =

l,m

ψNR

lm Ylm(θ, φ)

ψNR

lm =

• Ω rψ4 ¯

YlmdΩ Perturbation Theory: Spheroidal Multipoles (Not Orthogonal in l) Overtones labeled by n rψ4 =

nlm

ψPT

nlm Slm(˜

ωnlmjf , θ, φ) ψPT

nlm = Anlm [e−i ˜ ωnlmt]

Ylm Decomposition ⇒ NR Ringdown Is a Sum of QNMs

ψNR

lm = nl′m

Anlm [e−i ˜

ωnl′mt] [

• Ω ¯

YlmSnl′mdΩ] 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) → (Mf = 0.9516, jf = 0.6862)

10 30 50 −1 −0.5 0.5 1 x 10

−4

t/M rM|ψ4(t)|

0.5 1 1.5 10

−4

10

−3

Mω rM|ψ4(ω)|

ω22 ω32

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) → (Mf = 0.9978, jf = 0.1895)

10 30 50 −2 2 x 10

−5

t/M rM|ψ4(t)|

0.2 0.4 0.6 0.8 1 10

−4

Mω rM|ψ4(ω)|

ω22 ω32

Figure: The l = 3 m = 2 spherical harmonic multipole for a 15:1, nonspinning black hole binary (GaTech MAYA). Snlm = Snlm(jf ˜ ωnlm, θ, φ), where for jf = 0, Snlm = Ylm

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: ωnlm, τnlm, Mf , jf , 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(tn), but in the basis of QNMs ( eiωnlmt )

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

1

Given NR Data, Choose Fitting Region:

Ringdown: tstart ≤ t ≤ tstop Model: ψNR

lm (t) = nl′m

[Anlm

• Ω ¯

YlmSnl′mdΩ] e−i ˜

ωnl′mt

2

Apply Fit: [Multimode Fit](ψNR

lm ) −

→ AFit

nlm,l′ ≡ Anlm [

• Ω ¯

YlmSnl′mdΩ]

3

Estimate Spheroidal Amplitudes: Anlm = AFit

nlm,l′ /

• Ω ¯

YlmSnl′mdΩ

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(ψNR

lm −ψFIT lm )

RMSE(ψNR

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 1.5 2 2.5 3 3.5 4 4.5 10

−2

10

−1

10 10

1

q fractional rmse (l,m)=(3,2) MultiMode SingleMode

Figure: Multimode analysis of ψNR

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?

1 1.5 2 2.5 3 3.5 4 4.5 10

−3

10

−2

10

−1

10

q fractional rmse (l,m)=(2,2) MultiMode SingleMode

Figure: Multimode analysis of ψNR

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?

1 1.5 2 2.5 3 3.5 4 4.5 10

−2

10

−1

10

q fractional rmse (l,m)=(4,4) MultiMode SingleMode

Figure: Multimode analysis of ψNR

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)

0.6 0.8 1 0.1 0.15 0.2 0.25

2 3 4 2 3 4 2,1 2,1

ω 1/τ

0.5 1 1.5 10

−5

10

−4

ω |ψ(ω)|

Figure: Multimode analysis of ψNR

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

0.5 0.6 0.7 0.8 0.9 0.1 0.15 0.2 0.25

2 3 2 3 2,1 2,1

ω 1/τ

0.5 1 10

−5

10

−4

10

−3

10

−2

ω |ψ(ω)|

Figure: Multimode analysis of ψNR

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

1.1 1.2 1.3 0.1 0.15 0.2 0.25

4 5 4 5 2,2 2,2 2,2 3,2

ω 1/τ

0.6 0.8 1 1.2 1.4 1.6 10

−5

10

−4

10

−3

ω |ψ(ω)|

Figure: Multimode analysis of ψNR

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

1 1.5 2 2.5 3 3.5 4 4.5 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

q Afit (2,2,n) (l,m,n) = (2,2,0) (l,m,n) = (2,2,1)

Figure: Multimode analysis of ψNR

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: 2nd Order QNMs

1 1.5 2 2.5 3 3.5 4 4.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10

−3

q Afit (l,m)=(5,4) (l1,m1,n1)(l2,m2,n2) = (2,2,0)(2,2,0) (l1,m1,n1)(l2,m2,n2) = (2,2,0)(3,2,0)

Figure: Multimode analysis of ψNR

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

10

−1

10 10

1

10

−27

10

−26

10

−25

10

−24

10

−23

10

−22

10

−21

ω Sn(ω)1/2 and 2*|h(ω)|ω1/2 M = 500 (MSol), D = 20 (Mpc), (θ,φ) = (0.00,0.0) (2,2,0) (2,2,1) (3,2,0) (4,2,1) (3,2,1) (2,1,0)(2,1,0) advLIGO

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

10

−1

10 10

1

10

−27

10

−26

10

−25

10

−24

10

−23

10

−22

10

−21

ω Sn(ω)1/2 and 2*|h(ω)|ω1/2 M = 500 (MSol), D = 20 (Mpc), (θ,φ) = (0.79,0.0) (2,2,0) (2,2,1) (4,4,0) (2,2,0)(2,2,0) (4,4,1) (3,2,1) advLIGO

London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 20 / 30

SNR Scenarios: q = 2, Source Directly Over Detector

10

−1

10 10

1

10

−27

10

−26

10

−25

10

−24

10

−23

10

−22

10

−21

ω Sn(ω)1/2 and 2*|h(ω)|ω1/2 M = 500 (MSol), D = 20 (Mpc), (θ,φ) = (0.00,0.0) (2,2,0) (2,2,1) (3,2,0) (4,2,1) (2,1,0)(2,1,0) (4,2,0) advLIGO

London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 21 / 30

SNR Scenarios: q = 2, θ = π/4, φ = 0

10

−1

10 10

1

10

−27

10

−26

10

−25

10

−24

10

−23

10

−22

10

−21

ω Sn(ω)1/2 and 2*|h(ω)|ω1/2 M = 500 (MSol), D = 20 (Mpc), (θ,φ) = (0.79,0.0) (2,2,0) (3,3,0) (2,2,1) (3,3,1) (2,2,0)(2,2,0) (4,4,1) advLIGO

London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 22 / 30

From Spherical to Spheroidal

Can we robustly estimate spheroidal content from NR ringdown? Yes, Multimode Fitting, but it is non-trivial. What does Multimode fitting reveal about NR ringdown? QNMs beyond the fundamental allow better ringdown modelling. Overtones, 2nd Order QNMs, but no significant sight of ”Mirror Modes”. What is suggested about detection scenarios? No time soon, as expected. However, the l = m = 2 overtone is the second most dominant near optimal

• rientation.

Possible detection of many subdominant modes by future detectors and subsequent applications in signal analysis: system parameters, GR tests, etc.

London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 23 / 30

Discussion

London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 24 / 30

Spheroidal Harmonic Analysis

ℓ = m

1 1.5 2 2.5 3 3.5 4 4.5 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

q Afit

(l,m,n) = (2,2,0) (l,m,n) = (3,3,0) (l,m,n) = (4,4,0)

London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 25 / 30

Spheroidal Harmonic Analysis

ℓ = m − 1

1 1.5 2 2.5 3 3.5 4 4.5 1 2 3 4 5 6 7 8 x 10

−3

q Afit

(l,m,n) = (2,1,0) (l,m,n) = (3,2,0) (l,m,n) = (4,3,0)

London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 26 / 30

Spherical Harmonic Analysis: More Multipoles

0.15 0.17 0.2 0.23

α22

Kamaretsos et al MAYA

0.08 0.16 0.24 0.32

α33

1 2 3 4 0.02 0.04 0.06 0.08

α44/α22

q

1 2 3 4 0.08 0.16 0.24 0.32

α21/α22

q

London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 27 / 30

Final Spin Estimation (Faster than Horizon Calculation)

London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 28 / 30

Inner Products

London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 29 / 30

Perturbation Theory Model (eg: arxiv1212.5553v1)

Ratio of Inner-Products For Different q Agrees With Perturbation Theory Calculation.

London, Healy et. al. (Georgia Tech, CRA) NR Ringdown Waveforms: Spherical to Spheroidal NRDA 2013 September 20, 2013 30 / 30