Gravitational Lensing of Gravitational Waves Ken Ng, Kaze Wong, Tom - - PowerPoint PPT Presentation

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]

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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

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Anomalous Event?

5 10 15 20 25 30 35 40 45 50

Mo in M⊙

0.00 0.05 0.10 0.15 0.20

P(Mo)

BH-BH

Dominik Askar

Dominik et al. [3] and Askar et al. [4]

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Gravitational Lensing!

Image: NASA/ESA

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Gravitational Lensing of Gravitational Waves

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Gravitational Lensing of Gravitational Waves

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Effect of Lensing on GW Signals

◮ Effect of lensing on the original waveform

h′(t) = √µ1h(t − ∆t1) + √µ2h(t − ∆t2)

◮ h(t) is the original signal ◮ µ1,2 are the magnification of the images ◮ ∆t1,2 are the delay in arrival time of the images.

⇒ Strong lensing changes the amplitude but not the frequency content

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How often do GWs get lensed?

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Gravitational-wave Detection

◮ Signal-to-noise ratio (SNR) indicates the loudness of GW event

ρ2 = fmax

fmin

|h(f)|2 Sn(f) d f ∼ ΘM5/6 dL

◮ M = (m1m2)3/5/(m1 + m2)1/5 is the chirpmass ◮ dL(zs) is the luminosity distance of the GW source ◮ 0 < Θ < 4 represents the detector response (sky location, orbital

• rientation)

◮ Lensing: ρ → ρ′ = √µρ

An observation requires ρ > ρth = 8

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Monte Carlo Simulation

Distribution Description (m1, m2) Numerical result from galaxy- synthesis simulations [3] zs (1 + zs)3 within z = 2.5, estimation from simulations and star formation rate Θ Uniform sky position, orbital

• rientation and polarization in aLIGO detector

τ(zs) ∼ 0.001(dC/dH)3, modelling from observation [5] µ 1/µ3 in high magnification µ > 2

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Source Distributions

0.00.51.01.52.02.53.03.54.0

Θ

0.0 0.1 0.2 0.3 0.4 0.5 0.6

P(Θ)

0.0 0.5 1.0 1.5 2.0

z

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Θ

2σ 2 σ 2 σ

0 5 10 15 20 25 30 35 40

Mo

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Θ

2σ 2σ 2σ

10 20 30 40 50

Mo

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

P(Mo)

0.0 0.5 1.0 1.5 2.0 2.5

z

10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101

P(z)

Lensed, P(Θ) Non-lensed, P(Θ) Intrinsic Lensed, avg Θ

0 5 10 15 20 25 30 35 40

Mo

0.0 0.5 1.0 1.5 2.0

z

2σ 2σ 2σ 2σ

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Rate of Lensed Signals

2 4 6 8 10 12 14

ρth w.r.t. aLIGO O1 noise

10−3 10−2 10−1 100 101 102 103 104 105 106

Absolute rates (yr−1)

Design O1 > 1 event per year

Original events Lensing events Tjonnie Li GWTW 2017 11

Rate of Lensed Signals

0.0 0.5 1.0 1.5 2.0

Source Redshift zs

10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 103

Differential rates (yr−1)

Designed, lensed Designed, total Middle, lensed Middle, total O1, lensed O1, total

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What do we expect to see?

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Effect of Lensing on GW Signals

◮ Effect of lensing on the original waveform

h′(t) = √µ1h(t − ∆t1) + √µ2h(t − ∆t2)

◮ The two quantities are given by

∆t = (1 + zL)DOS cDOLDLS 1 2(∇φ)2 − φ( r)

• ,

µ =

• (1 − ∂x∂xφ)(1 − ∂y∂yφ) − (∂x∂yφ)2−1

◮ where φ is the effective projected gravitational potential. Tjonnie Li GWTW 2017 14

Elliptical Galaxies

◮ Consider Blandford-Kochanek model for elliptical galaxies [6]

φ( r) = 2DLSDOLA DOSc2

• 1 + (1 − ǫ)

x s 2 + (1 + ǫ) y s 2 − 1

• ◮ 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

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Time & Magnification Maps

−0.03 −0.02 −0.01 0.00 0.01 0.02 x-position (Mpc) −0.03 −0.02 −0.01 0.00 0.01 0.02 y-position (Mpc) 0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 log(µ) −0.03 −0.02 −0.01 0.00 0.01 0.02 x-position (Mpc) −0.03 −0.02 −0.01 0.00 0.01 0.02 y-position (Mpc) 7.77 7.80 7.83 7.86 7.89 7.92 7.95 7.98 8.01 log(∆t + 1e8)

Calculate probability distribution of time differences and magnifications between images

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Time-delay Distribution

100 101 102 103 104 105 106 107 ∆t 0.0 0.2 0.4 0.6 0.8 1.0 P(< ∆t) Overlapping signal Multiple signals in a LIGO run

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Overlapping Signals

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

time (s)

−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8

h(t)

×10−20

• rig

lensed 101 102

freq (Hz)

10−25 10−24 10−23 10−22 10−21

—h(f)—

• rig

lensed

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Sub-threshold Signals

2σ 3σ 4σ 5.1σ > 5.1σ 2σ 3σ 4σ 5.1σ > 5.1σ

8 10 12 14 16 18 20 22 24

Detection statistic ˆ ρc

10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102

Number of events

GW150914 Search Result Search Background Background excluding GW150914

• 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]

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Diffraction Limit

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Diffraction Limit for Gravitational Waves

◮ Diffraction set by Fresnel number

F = a2 DOLλGW (1)

◮ a ∼ rE: Einstein radius

◮ Galaxy lens: a ∼ kpc → F ≫ 1 ◮ Stellar lens: a < 1 pc → F ∼ 1

Microlensing by stellar mass objects can diffract GW signals

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Diffracted Waveforms

101 102 Frequency (Hz) 10−25 10−24 10−23 10−22 10−21 10−20 10−19 |h(f)| Unlensed Lensed

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Effects of Diffraction on GW Data Analysis

30 60 90 120 150 180 Mc (M⊙) 100 200 300 400 500 600 700 ML (M⊙) 0.860 0.970 0.990

Match

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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

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Thank you!

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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].

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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.

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