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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,

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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

1Department 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

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Introduction

Method

Parameter Estimation

GW In CUHK

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Dispersion of Gravitational Waves With h = c = G = 1 Without Lorentz violation: ω = k (1) Isotropic dispersion [1]: E2 = p2 + m2

g + Apα

⇒ ω2 = k2 + m2

g + Akα

⇒ vg(f) ≈ 1 − 1 2m2f−2 − 1 2Afα−2 (2)

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Geometry of the Lensing1

1Figure 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

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Lensing (Diffraction) of Gravitational Waves Lensed waveform ˜ hL(f) = F(f; lensing parameters)˜ h(f) (3) Amplification function [2]: F(f; θs) ∝ (1 + zL)f i

d2θ exp(2πiftd(

θ, θs)) (4) where td is the arrival time delay between lensed and unlensed rays. Time delay: td( θ, θs) = (1 + zL) c

DLDS

2DLS | θs − θ|2 − ψ( θs)

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Effect due to lensing

102 103

f (Hz)

10−28 10−27 10−26 10−25 10−24 10−23 10−22

|h(f)|

unlensed lensed by ML = 400M⊙ at y = 0.5

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The Central Question

How would the lensing pattern look like if gravitational waves are with dispersions?

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Arrival Time Delay With dispersion td → c vg(f)td (6) From now on β(f) = c/vg(f) Dispersion changes the phase differences along the rays. ⇒ lensing pattern is changed.

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Amplification Functions For point mass lens F(f; y) = exp π 4 wβ w 2 β i w

2 β

× Γ

1 − iw

2 β

2 β, 1; iw 2 βy2 (7) where w = 8πML(1 + zL)f, (7) can be reduced to known case [3] when there is no dispersion.

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Image Pattern

102 103

f (Hz)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

|F(f)|

Dispersionless, m = A = α = 0.0 Dispersive, mg = 0.0, A = 100.0, α = 0.5

(a) |F(f)|

102 103

f (Hz)

10−28 10−27 10−26 10−25 10−24 10−23 10−22

|h(f)|

Dispersionless, m = A = α = 0.0 Dispersive, mg = 0.0, A = 100.0, α = 0.5

(b) |h(f)|

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Parameter Estimations λg = /mgc

15.0 15.5 16.0 16.5 17.0 17.5 18.0

log λmg [m]

0.0 0.2 0.4 0.6 0.8 1.0

Cumulative Posterior Probability

lensed by ML = 400M⊙ at y = 0.5 unlensed

(c) dL = 200 Mpc

15.0 15.5 16.0 16.5 17.0 17.5 18.0

log λmg [m]

0.0 0.2 0.4 0.6 0.8 1.0

Cumulative Posterior Probability

lensed by ML = 400M⊙ at y = 0.5 unlensed

(d) dL = 100 Mpc

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Advantages Relies solely on the lensed signals. SNR of signal is boost. Improved constraint on mg

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Summary Lensing pattern of gravitational waves with dispersions ⇒ probe dispersion using lensing. Better constrains on mg Systematic run is on going. Will have more complete results soon. Incorporating the SIS.

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Our Awesome Group!

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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.

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Overlap Plots

360 380 400 420 440

ML(M⊙)

1 2 3 4 5 6

A (c = h = 1)

×10−43 0.96 . 9 7 0.99 0.2 0.4 0.6 0.8 1.0

Overlap

(e) α = 0

360 380 400 420 440

ML(M⊙)

1 2 3 4 5 6

A (c = h = 1)

×10−30 . 9 6 . 9 7 0.99 0.2 0.4 0.6 0.8 1.0

Overlap

(f) α = 0.5

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Unlensed Dispersive GWs [1] Propagation time delay when A = 0 ∆t = (1 + z)

∆te + m2

2 D0 1 f2

− 1 f′2

This leads to a phase difference, δΨ(f) = − πD0m2

(1 + z)f (9) such that hdisp(f) = h(f)eiδΨ(f) (10)

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