WAVE OPTICS IN GRAVITATIONAL LENSING Dylan L. Jow, Simon Foreman, - - PowerPoint PPT Presentation

WAVE OPTICS IN GRAVITATIONAL LENSING

Dylan L. Jow, Simon Foreman, Ue-Li Pen, Wei Zhu

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Source Observer Lens Lens plane Source plane

β θ η ξ Dd Ds Dds

Wave optics:

H( ⃗ η ) ∝ |∫ d2 ⃗ ξ exp{iωT( ⃗ ξ , ⃗ η )}|2

Geometric limit:

Hi( ⃗ η ) = 1 det(∂a∂bT( ⃗ ξ i, ⃗ η )) ∇T( ⃗ ξ i, ⃗ η ) = 0

Summary of optics in curved spacetime

T( ⃗ ξ , ⃗ η ) = 1 2 DdDs Dds | ⃗ ξ Dd − ⃗ η Ds |2 − ψ( ⃗ ξ )

ψ( ⃗ ξ ) = 1 2 ∫Γ

⃗ ξ

dλ(h00 + hij ̂ ni ̂ nj + 2h0i ̂ ni)

Time delay / Fermat potential

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When do wave effects matter?

• wave effects matter for point sources of coherent radiation
• point source means smaller than Fresnel scale,

θF = 1/ωD

Source Frequency Distance Fresnel scale Angular scale Pulse width Brightness temp. FRB ~ GHz ~ Gpc Pulsar ~ GHz ~ 10 kpc Star in MW ~ 100 THz ~ kpc

• ∼ μas

∼ 10−3μas

∼ 10−12μas

∼ 10−6μas

• wave optics will be important for lensing of FRBs and pulsars
• also, important for lensing of gravitational waves

∼ ms ∼ ms

1035 K

1025 − 1030 K

103 K

∼ μas

∼ 10−2μas

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EXAMPLE: THE POINT LENS

ds2 = − (1 − 2GM r )dt2 + (1 + 2GM r )dr2

⟹ T(

⃗ ξ , ⃗ η ) = 1 2 D| ⃗ ξ Dd − ⃗ η Ds |2 − 4GM log| ⃗ ξ |

⟹ H( ⃗ y ) = s 2πi ∫ d2 ⃗ x exp[is{ | ⃗ x − ⃗ y |2 2 − log| ⃗ x |}]

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where and

s = θ2

E

θ2

F

y = β/θE, x = θ/θE, θE = 4GM D

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EXAMPLE: POINT LENS

H( ⃗ y ) = s 2πi ∫ d2 ⃗ x exp[is{ | ⃗ x − ⃗ y |2 2 − log| ⃗ x |}]

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= s 2πi ∫ xdxdϕ exp[is{ x2 + y2 − 2xy cos ϕ 2 − log x}]

2

= − iseisy2/2 ∫

∞

dx x J0(sxy)exp{is[ 1 2 x2 − log x]}

2

= πs 1 − e−πs

1F1(1

2is, 1; 1 2 isy2)

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Geometric: ,

⃗ x 1,2 = ⃗ y 2y(y ± 4 + y2) μ = y2 + 2 y y2 + 4

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Moving source,

μrel

τ = t − t0 tE tE = θE μrel

y(τ) = (τ2 + y2

0)1/2

• geometric light curve depends only on (and , ). The full wave
• ptics result has an additional dependence on , through which it

depends on the frequency of light

tE y0 t0 s

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Dynamic spectrum of point lens

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Why wave effects matter: frequency dependence breaks degeneracies

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Why wave effects matter: boosted cross sections

Hwave ∼ 2 y2

• Hgeom. ∼ 2

y4

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Why wave effects matter: boosted cross sections

Hwave ∼ 2 y2

• Hgeom. ∼ 2

y4 Geometric cross section: ̂

σA = μ({ ⃗ y |H(y) − 1| > A})

Let , then

y* = H−1

geom.(A)

σgeom.

A

= π(y*)2θ2

E

Wave-optics cross section: σwave

A

= π(y*)4θ2

E

Example: For , then

A = 10−2 y* ≈ 3.5 ⟹ σwave

A

≈ 12σgeom.

A

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FOR RADIO SOURCES, WHAT OBJECTS WILL ACT AS POINT LENSES IN THE FULL WAVE-OPTICS REGIME?

• Full wave-optics regime when

, i.e.

• For

, we have when (roughly Jupiter mass)

• Condition on amplitude of signal (

) gives lower bound on mass of

• Point objects in with mass

include free-floating planets (FFPs)

s ≲ 1 θE ≲ θF ω ∼ 1 GHz s ≲ 1 M ≲ 100 M⊕ A > 10−1 M ≳ 0.1 M⊕ [0.1,100] M⊕

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FFP MICROLENSING EVENT RATE

From Mróz et al. (2017), arxiv 1707.07634

• observed microlensing of

stars consistent with prediction of FFPs as numerous as stars

∼ 1 M⊕

• Using

, optical depth of FFPs: toward the galactic centre

σwave τFFP ∼ 10−8

• For pulsars as background source:

per pulsar per day

Γ = 10−6

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BINARY MICROLENSING

• other potential lenses of interest are two point masses, e.g. planets

bound to stars, LISA white dwarfs + gravitational waves

H( ⃗ y ) = s 2πi ∫ d2 ⃗ x exp[is{ | ⃗ x − ⃗ y |2 2 − 1 1 + q log| ⃗ x − ⃗ x 1| − q 1 + q log| ⃗ x − ⃗ x 2|}]

2

,

s = θ2

E

θ2

F

= 4GMω q = M2/M1

• In geometric lensing, magnification depends only on mass ratio . In

wave optics, depends on total mass through .

q s

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Caustic structure and geometric limit well-studied; used to detect exoplanets

From Gaudi (2017), arxiv 1002.0332

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Wave optics: Eikonal limit

• the full wave-optics integral for a binary lens does not have an

analytic result like the single lens case, and the numerical evaluation of oscillatory integrals is complicated. However, first-

• rder wave effects beyond the geometric limit are captured in the

Eikonal or “semi-classical” limit.

H(η) ≈ ∑

i

Hgeom.

i

(η) + 2∑

i<j

|Hgeom.

i

(η)Hgeom.

j

(η)|1/2cos[ω(T(ξj, η) − T(ξi, η)) − π(nj − ni)]

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• the Eikonal limit gives unphysical results near caustics (infinite

magnifications).

Beyond the Eikonal limit

• the Eikonal limit is only valid when all the images are well-separated

(relative to ).

θF

• to get the full wave-optics result, we need to compute highly
• scillatory integrals that rarely have analytic answers. How? Use

Picard-Lefschetz theory.

• PL theory gives a recipe for numerically evaluating integrals of the

form (see Job Feldbrugge’s earlier talk)

∫ℝn eiSdnx

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Picard-Lefschetz theory. Example:

∫ℝ dxeix2

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Lefschetz thimbles are found by gradient descent along

h = Re(iS)

Example:

∫ℝ eiS = ∫ℝ ei( x3

3 −x)

Animation courtesy of Fang Xi Lin

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SUMMARY

• wave-optical effects beyond the geometric limit will matter for the

gravitational lensing of coherent sources like pulsars and FRBs

• wave-optical effects can be useful (not just a computational

headache), providing more information about the lens, and boosting cross sections

• the simple analytic result for the point lens can already be applied to

looking for FFPs and other compact objects in the lensing of pulsars

• understanding the wave optics of more complicated lenses will

require more sophisticated numerical techniques, such as Picard- Lefschetz theory

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