An Effective Search Method for Gravitational Ringing of Black Holes - - PowerPoint PPT Presentation

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An Effective Search Method for Gravitational Ringing of Black Holes

Hiroyuki Nakano (Osaka City) Hirotaka Takahashi (Osaka, Niigata and YITP[Kyoto]) Hideyuki Tagoshi (Osaka) Misao Sasaki (YITP[Kyoto]) [Phys. Rev. D 68, 102003 (2003)] The 8th Gravitational Wave Data Analysis Workshop in Milwaukee, Wisconsin, USA (December 17th-20th, 2003)

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§1. Introduction

Formation of black hole：Quasinormal oscillation (Ringdown) Observation of gravitational waves (Direct observation of B.H.) Information of the mass and angular momentum of B.H. Search method: Using theoretical waveforms as templates Matched filtering method GOAL: Efficient method for tiling the template space Ringdown waveform: h(fc, Q, t0, φ0; t) = e−π fc (t−t0)

Q

cos(2 π fc (t − t0) − φ0) for t ≥ t0 , fc : Central frequency, t0 : Initial time Q : Quality factor, φ0 : Initial phase

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Relation between ringdown parameters (fc, Q) and (Mass, Spin) of Kerr B.H. The least damped mode of ringdown [Echeverria fitting formulas (’89)]: fc ≃ 32kHz [1 − 0.63(1 − a)3/10]

    M

M⊙

   

−1

, Q ≃ 2(1 − a)−9/20 . M: Mass of B.H. M⊙: Solar mass a: Non-dimensional spin parameter (0 ≤ a ≤ 1)

<Strategy>

* White noise case: Analytical treatment Fourier transformation (in the frequency domain) How do we tile the template points? (Templates are taken at the template points.) Introduce new parameters (Coordinate transformation) Efficient template spacing * Colored (detector’s) noise case: Check the efficiency.

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§2. Template space

Ringdown waveform (t ≥ t0): h(fc, Q, t0, φ0; t) = e−π fc (t−t0)

Q

cos(2 π fc (t − t0) − φ0) . h(fc, Q, t0, φ0; t) = hc(fc, Q, t0; t) cos φ0+hs(fc, Q, t0; t) sin φ0 , hc(fc, Q, t0; t) = e−π fc (t−t0)

Q

cos(2 π fc (t − t0)) , hs(fc, Q, t0; t) = e−π fc (t−t0)

Q

sin(2 π fc (t − t0)) . Fourier transformation: ˜ hc/s(f) =

∞

−∞ dt e2πifthc/s(t) .

Complex conjugation : ˜ h∗(f) = ˜ h(−f) . Inner product (in the white noise case): (a, b) =

fmax

−fmax d

f ˜ a(f)˜ b∗(f) . Normalized wave [Nc/s = (˜ hc/s, ˜ hc/s)]: ˆ hc/s(fc, Q, t0; f) = 1

Nc/s(fc, Q, t0)

˜ hc/s(fc, Q, t0; f) .

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Note! hc and hs are NOT orthogonal. (ˆ hc(fc, Q, t0), ˆ hs(fc, Q, t0)) = 1

• 2 (2 Q2 + 1) =: c(fc, Q, t0) .

Maximize for the initial phase φ0 [Mohanty (’98)]: Λ(x) ≡ max

φ0 (x, ˆ

h) =

• (x, ˆ

hc(fc, Q, t0))2 + (x, ˆ hs(fc, Q, t0))2 −2 c(fc, Q, t0)(x, ˆ hc(fc, Q, t0))(x, ˆ hs(fc, Q, t0))

• /
• 1 − c(fc, Q, t0)2
• .

Match C(d fc, dQ, dt0) (3-dim. space): Correlation between (fc, Q, t0) and (fc + d fc, Q + dQ, t0 + dt0) C(d fc, dQ, dt0) = Λ(ˆ hc(fc + d fc, Q + dQ, t0 + dt0)) ≤ 1 . Smaller the match C ——> Larger the distance between two signals

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Distance function (define the “metric”) [Owen (’96)]: ds2

(3) = 1 − C ,

ds2

(3) = g(3) ij dxidxj

= 2 Q4 (2 Q2 + 1) f 2

c

d f 2

c +

2 Q2 (4 Q2 + 5) (4 Q2 + 1)2(2 Q2 + 1) dQ2 − 2 Q3 fc (4 Q2 + 1)(2 Q2 + 1) d fc dQ + 4 π fc (1 + 4 Q2)fmax Q (2 Q2 + 1) dt2

0 +

π fc 2 Q2 + 1 dt0 dQ . Maximize the match for dt0 (orthogonal to t0 axis): g(2)

IJ = g(3) IJ − g(3) It0g(3) Jt0

g(3)

t0t0

, ds2

(2) = g(2) IJ dxIdxJ

= 2 Q4 (2 Q2 + 1) f 2

c

d f 2

c +

2 Q2 (4 Q2 + 5) (4 Q2 + 1)2(2 Q2 + 1) dQ2 − 2 Q3 fc (4 Q2 + 1)(2 Q2 + 1) d fc dQ .

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Diagonalization of the metric: Using Q ≥ 2, and Coordinate transformation. X = F + ln 2 + 1 8 1 Q2 − 1 64 1 Q4 + 1 384 1 Q6 − 1 2048 1 Q8 , Y = 1 2 1 Q + 1 24 1 Q3 − 3 160 1 Q5 + 1 128 1 Q7 − 17 4608 1 Q9 . (F = ln fc, Accuracy up to O(1/Q8) inclusive.) F = X − ln 2 − 1 2Y 2 + 7 12Y 4 − 67 45Y 6 + 1769 360 Y 8 , Q = 1 2 1 Y + 1 6Y − 37 90Y 3 + 166 135Y 5 − 5917 1350Y 7 .

ds2 = Ω(Y )

• dX2 + dY 2
• Ω(Y ) = 1

4 1 Y 2 − 1 3 + 37 60Y 2 − 85 54Y 4 + 13069 2700 Y 6

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§3. Tiling method

Signal to noise ratio (SNR) loss: ds2

max = 0.02.

Search region: fc,min ≤ fc ≤ fc,max , Qmin ≤ Q ≤ Qmax . How do we place the template points (TPs)?

• 1. Put TPs along the first Y =constant line.

Y0 = Y (Qmin), vmax(Y ) = X(Fmax, Q) (X0, Y0) = (vmax(Y0), Y0) First TP: (p1, q1) = (X0 − r1/ √ 2, Y0 − r1/ √ 2) Intersecting point of two circles lies on Y = Y0. (This Fig. shows a part of whole region.)

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r2 = ds2

max/Ω(q)

Most effective tiling: d = r/ √ 2

• 2. Cover the second line.

Y1 = Y0 − √ 2r1 (X1, Y1) = (vmax(Y1), Y1) (p2, q2) = (X1 − r2/ √ 2, Y1 − r2/ √ 2)

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• 3. Repeat until tiling the whole template space.

Ex.) 102Hz ≤ fc ≤ 104Hz , 2 ≤ Q ≤ 20(22.9)，ds2

max = 0.02

118M⊙ ≥ M ≥ 1.18M⊙, (a = 0) 277M⊙ ≥ M ≥ 2.77M⊙, (a = 0.994)

Number of templates: 1148

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Original coordinates：The contours show ds2

max = 0.02.

How effective? Ratio of Sum of the circle areas Scir to Parameter space Spar: η = Scir Spar = 1.57 . cf.) η = 2.12 (N. Arnaud et al. (’03))

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§4. Varidity of tiling method

TAMA Data Taking 7 fitting noise power spectrum (Fig. 3.4.1 in TAMA REPORT 2002) Valid between 60Hz and 40000Hz. Sn(|f|) =

   85

f

   

63

+ 1 2

   220

f

   

10

+ 1 9

   710

f

   

3

+ 3 20 + 1 20

   f

2000

  

2

+ 1 5

   f

5500

  

6

. * Ignore the overall amplitude of Sn. Using 2500 signals Maximize for t0 and φ0. 100Hz ≤ fc ≤ 104Hz，2 ≤ Q ≤ 20

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Horizontal axis: Match Vertical axis: Number of signals Mean: 99.3% * We can detect the most of the signals without losing of the SNR.

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* Preparation of Searching Gravitational Ringing

Orthonormal template waveforms ˜ h1 = ˜ hc C , ˜ h2 = C˜ hs − (c/C)˜ hc √ C2S2 − c2 , where (˜ hc(fc, Q, t0), ˜ hc(fc, Q, t0)) = C2 , (˜ hs(fc, Q, t0), ˜ hs(fc, Q, t0)) = S2 , (˜ hc(fc, Q, t0), ˜ hs(fc, Q, t0)) = c . Definition of inner product in the colored noise case: (a, b) = 2

fmax

d f ˜ a(f)˜ b∗(f) + ˜ a∗(f)˜ b(f) Sn(|f|) , Signal to noise ratio: x is the real data. ρ =

• (x, h1)2 + (x, h2)2 .

ρ2 = Mohanty’s Λ(x)

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§5. Conclusion and Discussion

* Parameters of ringdown waves: (t0, φ0, fc, Q)

• 1. Maximize for (t0, φ0).
• 2. Parameter space of (fc, Q) in the white noise case.

* Detector’s noise case Valid in the fitting TAMA noise spectrum. Check by using the real data. —–> Tsunesada (NAOJ) et al. <Future Works> * Search for the real data. Remove fake events. * Coincidence analysis by using many detectors.