Constraining Lorentz Violation of Gravitational Waves with Lensing - PowerPoint PPT Presentation

Introduction Method Parameter Estimation GW In CUHK Backup Slide Constraining Lorentz Violation of Gravitational Waves with Lensing Adrian K.W. Chung and Tjonnie G.F. Li 1 1 Department of Physics, The Chinese University of Hong Kong, Long-Term Workshop on Gravity and Cosmology, Yukawa Institute for Theoretical Physics, Kyoto University, 6th Febrauary, 2018 1/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide Introduction 1 Method 2 Parameter Estimation 3 GW In CUHK 4 2/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide Dispersion of Gravitational Waves With h = c = G = 1 Without Lorentz violation: ω = k (1) Isotropic dispersion [1]: E 2 = p 2 + m 2 g + Ap α ⇒ ω 2 = k 2 + m 2 g + Ak α (2) ⇒ v g ( f ) ≈ 1 − 1 2 m 2 f − 2 − 1 2 Af α − 2 3/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide Geometry of the Lensing 1 1 Figure 1. from R. Takahashi et al. " Arrival time differences between gravitational waves and electromagnetic signals due to gravitational lensing ". ApJ 835 (Jan. 2017), arXiv:1606.00458 4/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide Lensing (Diffraction) of Gravitational Waves Lensed waveform h L ( f ) = F ( f ; lensing parameters )˜ ˜ h ( f ) (3) Amplification function [2]: θ s ) ∝ (1 + z L ) f � F ( f ; � d 2 θ exp(2 πift d ( � θ, � θ s )) (4) i where t d is the arrival time delay between lensed and unlensed rays. Time delay: � � θ s ) = (1 + z L ) D L D S θ | 2 − ψ ( � t d ( � θ, � | � θ s − � θ s ) (5) c 2 D LS 5/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide Effect due to lensing 10 − 22 10 − 23 10 − 24 | h ( f ) | 10 − 25 10 − 26 10 − 27 unlensed 10 − 28 lensed by M L = 400 M ⊙ at y = 0 . 5 10 2 10 3 f (Hz) 6/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide The Central Question How would the lensing pattern look like if gravitational waves are with dispersions? 7/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide Arrival Time Delay With dispersion c t d → v g ( f ) t d (6) From now on β ( f ) = c/v g ( f ) Dispersion changes the phase differences along the rays. ⇒ lensing pattern is changed. 8/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide Amplification Functions For point mass lens � i w � π �� w 2 β F ( f ; y ) = exp 4 wβ 2 β (7) 1 − iw iw 2 β, 1; iw � � � 2 βy 2 � × Γ 2 β 1 F 1 where w = 8 πM L (1 + z L ) f , (7) can be reduced to known case [3] when there is no dispersion. 9/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide Image Pattern 3 . 0 10 − 22 Dispersionless, m = A = α = 0 . 0 Dispersive, m g = 0 . 0 , A = 100 . 0 , α = 0 . 5 2 . 5 10 − 23 10 − 24 2 . 0 | F ( f ) | | h ( f ) | 10 − 25 1 . 5 10 − 26 1 . 0 10 − 27 0 . 5 Dispersionless, m = A = α = 0 . 0 10 − 28 Dispersive, m g = 0 . 0 , A = 100 . 0 , α = 0 . 5 0 . 0 10 2 10 3 10 2 10 3 f (Hz) f (Hz) (a) | F ( f ) | (b) | h ( f ) | 10/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide Parameter Estimations λ g = � /m g c Cumulative Posterior Probability lensed by M L = 400 M ⊙ at y = 0 . 5 Cumulative Posterior Probability lensed by M L = 400 M ⊙ at y = 0 . 5 1 . 0 1 . 0 unlensed unlensed 0 . 8 0 . 8 0 . 6 0 . 6 0 . 4 0 . 4 0 . 2 0 . 2 0 . 0 0 . 0 15 . 0 15 . 5 16 . 0 16 . 5 17 . 0 17 . 5 18 . 0 15 . 0 15 . 5 16 . 0 16 . 5 17 . 0 17 . 5 18 . 0 log λ m g [m] log λ m g [m] (c) d L = 200 Mpc (d) d L = 100 Mpc 11/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide Advantages Relies solely on the lensed signals. SNR of signal is boost. Improved constraint on m g 12/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide Summary Lensing pattern of gravitational waves with dispersions ⇒ probe dispersion using lensing. Better constrains on m g Systematic run is on going. Will have more complete results soon. Incorporating the SIS. 13/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide Our Awesome Group! 14/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide References 1 Saeed Mirshekari et al. " Constraining Lorentz-violating, modified dispersion relations with gravitational waves ". Phys. Rev. D 85, 024041. (Jan. 2012) 2 Schneider, et al (1992). " Gravitational Lenses ".Springer’s Publications. ISBN: 0941-7834. DOI: 10.1007/978-3-662-03758-4 3 R. Takahashi et al. " Wave Effects in the Gravitational Lensing of Gravitational Waves from Chirping Binaries ". ApJ 595 (Oct. 2003), pp. 1039-1051. eprint: astro-ph/0305055. 15/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide Overlap Plots × 10 − 43 × 10 − 30 6 1 . 0 6 1 . 0 5 5 0 . 8 0 . 8 A ( c = h = 1) A ( c = h = 1) 4 4 0 . 6 0 . 6 Overlap Overlap 3 3 0.96 0 . 9 7 0 . 4 0 . 4 2 2 0 . 9 6 0 . 9 7 0.99 0 . 2 0 . 2 1 1 0.99 0 0 360 380 400 420 440 360 380 400 420 440 M L ( M ⊙ ) M L ( M ⊙ ) (e) α = 0 (f) α = 0 . 5 16/17

Introduction Method Parameter Estimation GW In CUHK Backup Slide Unlensed Dispersive GWs [1] Propagation time delay when A = 0 ∆ t e + m 2 � 1 − 1 � �� g ∆ t = (1 + z ) 2 D 0 (8) f ′ 2 f 2 e e This leads to a phase difference, δ Ψ( f ) = − πD 0 m 2 g (9) (1 + z ) f such that h disp ( f ) = h ( f ) e iδ Ψ( f ) (10) 17/17