Gravitational Lensing of Gravitational Waves Ken Ng, Kaze Wong, Tom Broadhurst Otto Hannuksela, Adrian Lai, Tjonnie G. F. Li Workshop on Gravitational Wave activities in Taiwan Institute of Physics, Academia Sinica, Taiwan 15 January 2017
Discovery of Gravitational Waves B. P. Abbott et al. “Observation of Gravitational Waves from a Binary Black Hole Merger”. Physical Review Letters 116.6, 061102 (Feb. 2016), p. 061102. arXiv: 1602.03837 [gr-qc] Tjonnie Li GWTW 2017 1
Characteristics B. P. Abbott et al. “Binary Black Hole Mergers in the First Advanced LIGO Observing Run”. Physical Review X 6.4, 041015 (Oct. 2016), p. 041015 Tjonnie Li GWTW 2017 2
Anomalous Event? BH-BH 0 . 20 Dominik Askar 0 . 15 P ( M o ) 0 . 10 0 . 05 0 . 00 5 10 15 20 25 30 35 40 45 50 M o in M ⊙ Dominik et al. [3] and Askar et al. [4] Tjonnie Li GWTW 2017 3
Gravitational Lensing! Image: NASA/ESA Tjonnie Li GWTW 2017 4
Gravitational Lensing of Gravitational Waves Tjonnie Li GWTW 2017 5
Gravitational Lensing of Gravitational Waves Tjonnie Li GWTW 2017 5
Effect of Lensing on GW Signals ◮ Effect of lensing on the original waveform h ′ ( t ) = √ µ 1 h ( t − ∆ t 1 ) + √ µ 2 h ( t − ∆ t 2 ) ◮ h ( t ) is the original signal ◮ µ 1 , 2 are the magnification of the images ◮ ∆ t 1 , 2 are the delay in arrival time of the images. ⇒ Strong lensing changes the amplitude but not the frequency content Tjonnie Li GWTW 2017 6
How often do GWs get lensed? Tjonnie Li GWTW 2017 7
Gravitational-wave Detection ◮ Signal-to-noise ratio (SNR) indicates the loudness of GW event � f max | h ( f ) | 2 f ∼ Θ M 5 / 6 ρ 2 = S n ( f ) d d L f min ◮ M = ( m 1 m 2 ) 3 / 5 / ( m 1 + m 2 ) 1 / 5 is the chirpmass ◮ d L ( z s ) is the luminosity distance of the GW source ◮ 0 < Θ < 4 represents the detector response (sky location, orbital orientation) ◮ Lensing: ρ → ρ ′ = √ µρ An observation requires ρ > ρ th = 8 Tjonnie Li GWTW 2017 8
Monte Carlo Simulation Distribution Description Numerical result from galaxy- ( m 1 , m 2 ) synthesis simulations [3] (1 + z s ) 3 within z = 2 . 5 , estimation z s from simulations and star formation rate Uniform sky position, orbital Θ orientation and polarization in aLIGO detector ∼ 0 . 001( d C /d H ) 3 , modelling from observation [5] τ ( z s ) 1 /µ 3 in high magnification µ > 2 µ Tjonnie Li GWTW 2017 9
Source Distributions 0 . 6 3 . 5 3 . 5 0 . 5 2 σ σ 3 . 0 2 3 . 0 0 . 4 2 . 5 2 . 5 P (Θ) 2 . 0 2 . 0 0 . 3 Θ Θ 2 σ 1 . 5 1 . 5 2 σ 0 . 2 σ 2 σ 1 . 0 1 . 0 2 0 . 1 0 . 5 0 . 5 0 . 0 0 . 0 0 . 0 0 . 00 . 51 . 01 . 52 . 02 . 53 . 03 . 54 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 0 5 10 15 20 25 30 35 40 Θ M o z 10 1 0 . 14 10 0 0 . 12 2 . 0 10 − 1 0 . 10 10 − 2 P ( M o ) 2 σ 1 . 5 0 . 08 P ( z ) 10 − 3 2 σ z 0 . 06 1 . 0 10 − 4 0 . 04 10 − 5 Lensed, P (Θ) 2 σ 0 . 5 Non-lensed, P (Θ) 0 . 02 10 − 6 Intrinsic 2 σ Lensed, avg Θ 10 − 7 0 . 00 0 . 0 0 10 20 30 40 50 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 0 5 10 15 20 25 30 35 40 M o M o z Tjonnie Li GWTW 2017 10
Rate of Lensed Signals 10 6 Original events Design O1 10 5 Lensing events 10 4 Absolute rates (yr − 1 ) 10 3 10 2 10 1 10 0 10 − 1 > 1 event per year 10 − 2 10 − 3 0 2 4 6 8 10 12 14 ρ th w.r.t. aLIGO O1 noise Tjonnie Li GWTW 2017 11
Rate of Lensed Signals 10 3 10 2 10 1 10 0 Differential rates (yr − 1 ) 10 − 1 10 − 2 10 − 3 10 − 4 Designed, lensed Designed, total 10 − 5 Middle, lensed Middle, total 10 − 6 O1, lensed O1, total 10 − 7 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 Source Redshift z s Tjonnie Li GWTW 2017 12
What do we expect to see? Tjonnie Li GWTW 2017 13
Effect of Lensing on GW Signals ◮ Effect of lensing on the original waveform h ′ ( t ) = √ µ 1 h ( t − ∆ t 1 ) + √ µ 2 h ( t − ∆ t 2 ) ◮ The two quantities are given by ∆ t = (1 + z L ) D OS � 1 � 2( ∇ φ ) 2 − φ ( � r ) , cD OL D LS (1 − ∂ x ∂ x φ )(1 − ∂ y ∂ y φ ) − ( ∂ x ∂ y φ ) 2 � − 1 � µ = ◮ where φ is the effective projected gravitational potential. Tjonnie Li GWTW 2017 14
Elliptical Galaxies ◮ Consider Blandford-Kochanek model for elliptical galaxies [6] �� � r ) = 2 D LS D OL A � x � y � 2 � 2 φ ( � 1 + (1 − ǫ ) + (1 + ǫ ) − 1 D OS c 2 s s ◮ A is constant related to the depth of potential well ◮ s is core size ◮ ǫ is the ellipticity. Study the effects of lensing by elliptical galaxies Tjonnie Li GWTW 2017 15
Time & Magnification Maps 3 . 0 8 . 01 2 . 7 7 . 98 0 . 02 0 . 02 2 . 4 7 . 95 2 . 1 0 . 01 0 . 01 y-position (Mpc) y-position (Mpc) 7 . 92 log (∆ t + 1 e 8) 1 . 8 log ( µ ) 7 . 89 0 . 00 1 . 5 0 . 00 7 . 86 1 . 2 − 0 . 01 − 0 . 01 7 . 83 0 . 9 0 . 6 7 . 80 − 0 . 02 − 0 . 02 0 . 3 7 . 77 − 0 . 03 − 0 . 03 0 . 0 − 0 . 03 − 0 . 02 − 0 . 01 0 . 00 0 . 01 0 . 02 − 0 . 03 − 0 . 02 − 0 . 01 0 . 00 0 . 01 0 . 02 x-position (Mpc) x-position (Mpc) Calculate probability distribution of time differences and magnifications between images Tjonnie Li GWTW 2017 16
Time-delay Distribution 1 . 0 Overlapping signal Multiple signals in a LIGO run 0 . 8 0 . 6 P ( < ∆ t ) 0 . 4 0 . 2 0 . 0 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 ∆ t Tjonnie Li GWTW 2017 17
Overlapping Signals × 10 − 20 10 − 21 0 . 8 orig orig lensed lensed 0 . 6 0 . 4 10 − 22 0 . 2 —h(f)— 0 . 0 h(t) 10 − 23 − 0 . 2 − 0 . 4 10 − 24 − 0 . 6 − 0 . 8 − 1 . 0 10 − 25 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 10 1 10 2 time (s) freq (Hz) Tjonnie Li GWTW 2017 18
Sub-threshold Signals 2 σ 3 σ 4 σ 5 . 1 σ > 5 . 1 σ 2 σ 3 σ 4 σ 5 . 1 σ > 5 . 1 σ 10 2 Search Result 10 1 Search Background Background excluding GW150914 10 0 Number of events 10 − 1 10 − 2 GW150914 10 − 3 10 − 4 10 − 5 10 − 6 10 − 7 10 − 8 8 10 12 14 16 18 20 22 24 Detection statistic ˆ ρ c B. P. Abbott et al. “Observation of Gravitational Waves from a Binary Black Hole Merger”. Physical Review Letters 116.6, 061102 (Feb. 2016), p. 061102. arXiv: 1602.03837 [gr-qc] Tjonnie Li GWTW 2017 19
Diffraction Limit Tjonnie Li GWTW 2017 20
Diffraction Limit for Gravitational Waves ◮ Diffraction set by Fresnel number a 2 F = (1) D OL λ GW ◮ a ∼ r E : Einstein radius ◮ Galaxy lens: a ∼ kpc → F ≫ 1 ◮ Stellar lens: a < 1 pc → F ∼ 1 Microlensing by stellar mass objects can diffract GW signals Tjonnie Li GWTW 2017 21
Diffracted Waveforms Unlensed 10 − 19 Lensed 10 − 20 10 − 21 | h ( f ) | 10 − 22 10 − 23 10 − 24 10 − 25 10 1 10 2 Frequency (Hz) Tjonnie Li GWTW 2017 22
Effects of Diffraction on GW Data Analysis Match 0.860 700 0.970 600 500 M L ( M ⊙ ) 400 0.990 300 200 100 30 60 90 120 150 180 M c ( M ⊙ ) Tjonnie Li GWTW 2017 23
Concluding Remarks ◮ The era of gravitational-wave astronomy has begun ◮ Lensing is a realistic expectation in the Advanced LIGO era ◮ Opens up a wealth of possibilities Tjonnie Li GWTW 2017 24
Thank you! Tjonnie Li GWTW 2017 25
References I [1] B. P. Abbott et al. “Observation of Gravitational Waves from a Binary Black Hole Merger”. Physical Review Letters 116.6, 061102 (Feb. 2016), p. 061102. arXiv: 1602.03837 [gr-qc] . [2] B. P. Abbott et al. “Binary Black Hole Mergers in the First Advanced LIGO Observing Run”. Physical Review X 6.4, 041015 (Oct. 2016), p. 041015. [3] M. Dominik et al. “Double Compact Objects. II. Cosmological Merger Rates”. ApJ 779, 72 (Dec. 2013), p. 72. arXiv: 1308.1546 [astro-ph.HE] . [4] A. Askar et al. “MOCCA-SURVEY Database - I. Coalescing binary black holes originating from globular clusters”. MNRAS 464 (Jan. 2017), pp. L36–L40. arXiv: 1608.02520 [astro-ph.HE] . Tjonnie Li GWTW 2017 26
References II [5] Y. Wang et al. “Caustics, critical curves and cross-sections for gravitational lensing by disc galaxies”. MNRAS 292 (Dec. 1997), p. 863. eprint: astro-ph/9702078 . [6] R. D. Blandford et al. “Gravitational imaging by isolated elliptical potential wells. I - Cross sections. II - Probability distributions”. ApJ 321 (Oct. 1987), pp. 658–675. Tjonnie Li GWTW 2017 27
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