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

1 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)

2 § 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 ( f c , Q, t 0 , φ 0 ; t ) = e − π fc ( t − t 0) cos(2 π f c ( t − t 0 ) − φ 0 ) for t ≥ t 0 , Q f c : Central frequency, t 0 : Initial time Q : Quality factor, φ 0 : Initial phase

3 Relation between ringdown parameters ( f c , Q ) and (Mass, Spin) of Kerr B.H. The least damped mode of ringdown [Echeverria fitting formulas (’89)]: − 1  M   Q ≃ 2(1 − a ) − 9 / 20 . f c ≃ 32kHz [1 − 0 . 63(1 − a ) 3 / 10 ] ,     M ⊙  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.

4 § 2. Template space Ringdown waveform ( t ≥ t 0 ): h ( f c , Q, t 0 , φ 0 ; t ) = e − π fc ( t − t 0) cos(2 π f c ( t − t 0 ) − φ 0 ) . Q h ( f c , Q, t 0 , φ 0 ; t ) = h c ( f c , Q, t 0 ; t ) cos φ 0 + h s ( f c , Q, t 0 ; t ) sin φ 0 , h c ( f c , Q, t 0 ; t ) = e − π fc ( t − t 0) cos(2 π f c ( t − t 0 )) , Q h s ( f c , Q, t 0 ; t ) = e − π fc ( t − t 0) sin(2 π f c ( t − t 0 )) . Q Fourier transformation: � ∞ ˜ −∞ dt e 2 πift h c/s ( t ) . h c/s ( f ) = h ∗ ( f ) = ˜ ˜ Complex conjugation : h ( − f ) . Inner product (in the white noise case): � f max a ( f )˜ b ∗ ( f ) . ( a, b ) = − f max d f ˜ Normalized wave [ N c/s = (˜ h c/s , ˜ h c/s )]: 1 ˆ ˜ h c/s ( f c , Q, t 0 ; f ) = h c/s ( f c , Q, t 0 ; f ) . � N c/s ( f c , Q, t 0 )

5 Note! h c and h s are NOT orthogonal. 1 (ˆ h c ( f c , Q, t 0 ) , ˆ h s ( f c , Q, t 0 )) = 2 (2 Q 2 + 1) =: c ( f c , Q, t 0 ) . � Maximize for the initial phase φ 0 [Mohanty (’98)]: φ 0 ( x, ˆ Λ( x ) ≡ max h ) h c ( f c , Q, t 0 )) 2 + ( x, ˆ ( x, ˆ h s ( f c , Q, t 0 )) 2 � = − 2 c ( f c , Q, t 0 )( x, ˆ h c ( f c , Q, t 0 ))( x, ˆ � h s ( f c , Q, t 0 )) 1 − c ( f c , Q, t 0 ) 2 � � / . Match C ( d f c , dQ, dt 0 ) (3-dim. space): Correlation between ( f c , Q, t 0 ) and ( f c + d f c , Q + dQ, t 0 + dt 0 ) f c , dQ, dt 0 ) = Λ(ˆ C ( d h c ( f c + d f c , Q + dQ, t 0 + dt 0 )) ≤ 1 . Smaller the match C —— > Larger the distance between two signals

6 Distance function (define the “metric”) [Owen (’96)]: ds 2 (3) = 1 − C , (3) = g (3) ds 2 ij dx i dx j 2 Q 2 (4 Q 2 + 5) 2 Q 4 f 2 (4 Q 2 + 1) 2 (2 Q 2 + 1) dQ 2 = d c + (2 Q 2 + 1) f 2 c 2 Q 3 − f c (4 Q 2 + 1)(2 Q 2 + 1) d f c dQ + 4 π f c (1 + 4 Q 2 ) f max π f c dt 2 0 + 2 Q 2 + 1 dt 0 dQ . Q (2 Q 2 + 1) Maximize the match for dt 0 (orthogonal to t 0 axis): IJ − g (3) It 0 g (3) g (2) IJ = g (3) Jt 0 , g (3) t 0 t 0 (2) = g (2) ds 2 IJ dx I dx J 2 Q 2 (4 Q 2 + 5) 2 Q 4 f 2 (4 Q 2 + 1) 2 (2 Q 2 + 1) dQ 2 = d c + (2 Q 2 + 1) f 2 c 2 Q 3 − f c (4 Q 2 + 1)(2 Q 2 + 1) d f c dQ .

7 Diagonalization of the metric: Using Q ≥ 2, and Coordinate transformation. X = F + ln 2 + 1 Q 2 − 1 1 Q 4 + 1 1 1 1 1 Q 6 − Q 8 , 8 64 384 2048 Y = 1 Q + 1 1 Q 3 − 3 1 Q 5 + 1 1 Q 7 − 17 1 1 Q 9 . 2 24 160 128 4608 ( F = ln f c , Accuracy up to O (1 /Q 8 ) inclusive.) F = X − ln 2 − 1 2 Y 2 + 7 12 Y 4 − 67 45 Y 6 + 1769 360 Y 8 , Q = 1 Y + 1 1 6 Y − 37 90 Y 3 + 166 135 Y 5 − 5917 1350 Y 7 . 2 ds 2 = Ω( Y ) dX 2 + dY 2 � � 60 Y 2 − 85 54 Y 4 + 13069 Ω( Y ) = 1 Y 2 − 1 1 3 + 37 2700 Y 6 4

8 § 3. Tiling method Signal to noise ratio (SNR) loss: ds 2 max = 0 . 02. Search region: f c, min ≤ f c ≤ f c, max , Q min ≤ Q ≤ Q max . How do we place the template points (TPs)? 1. Put TPs along the first Y =constant line. Y 0 = Y ( Q min ), v max ( Y ) = X ( F max , Q ) ( X 0 , Y 0 ) = ( v max ( Y 0 ) , Y 0 ) First TP: √ √ ( p 1 , q 1 ) = ( X 0 − r 1 / 2 , Y 0 − r 1 / 2) Intersecting point of two circles lies on Y = Y 0 . (This Fig. shows a part of whole region.)

9 r 2 = ds 2 max / Ω( q ) √ Most effective tiling: d = r/ 2 2. Cover the second line. √ Y 1 = Y 0 − 2 r 1 ( X 1 , Y 1 ) = ( v max ( Y 1 ) , Y 1 ) √ √ ( p 2 , q 2 ) = ( X 1 − r 2 / 2 , Y 1 − r 2 / 2)

10 3. Repeat until tiling the whole template space. Ex.) 10 2 Hz ≤ f c ≤ 10 4 Hz , 2 ≤ Q ≤ 20(22 . 9) ， ds 2 max = 0 . 02 118 M ⊙ ≥ M ≥ 1 . 18 M ⊙ , ( a = 0) 277 M ⊙ ≥ M ≥ 2 . 77 M ⊙ , ( a = 0 . 994) Number of templates: 1148

11 Original coordinates ： The contours show ds 2 max = 0 . 02. How effective? Ratio of Sum of the circle areas S cir to Parameter space S par : η = S cir = 1 . 57 . S par cf.) η = 2 . 12 (N. Arnaud et al. (’03))

12 § 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. 63 10 3  85   + 1   220  + 1  710   + 3 S n ( | f | ) =             f 2 f 9 f 20    2 6 + 1  f + 1  f     .       20 2000 5 5500 * Ignore the overall amplitude of S n . Using 2500 signals Maximize for t 0 and φ 0 . 100Hz ≤ f c ≤ 10 4 Hz ， 2 ≤ Q ≤ 20

13 Horizontal axis: Match Vertical axis: Number of signals Mean: 99 . 3% * We can detect the most of the signals without losing of the SNR.

14 * Preparation of Searching Gravitational Ringing Orthonormal template waveforms ˜ h 2 = C ˜ h s − ( c/C )˜ h c h c ˜ ˜ √ h 1 = C , , C 2 S 2 − c 2 where h c ( f c , Q, t 0 )) = C 2 , (˜ h c ( f c , Q, t 0 ) , ˜ h s ( f c , Q, t 0 )) = S 2 , (˜ h s ( f c , Q, t 0 ) , ˜ (˜ h c ( f c , Q, t 0 ) , ˜ h s ( f c , Q, t 0 )) = c . Definition of inner product in the colored noise case: a ( f )˜ a ∗ ( f )˜ b ∗ ( f ) + ˜ f ˜ b ( f ) � f max ( a, b ) = 2 d , 0 S n ( | f | ) Signal to noise ratio: x is the real data. ( x, h 1 ) 2 + ( x, h 2 ) 2 . � ρ = ρ 2 = Mohanty’s Λ( x )

15 § 5. Conclusion and Discussion * Parameters of ringdown waves: ( t 0 , φ 0 , f c , Q ) 1. Maximize for ( t 0 , φ 0 ). 2. Parameter space of ( f c , 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.