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 1

⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ Summary of optics in curved spacetime Time delay / Fermat potential η Source plane Source D d D s η ) = 1 ξ η | 2 − ψ ( D ds T ( ξ , | − ξ ) 2 D ds D d D s Lens ξ Lens plane D s ξ ) = 1 2 ∫ Γ n i ̂ n j + 2 h 0 i ̂ n i ) ψ ( d λ ( h 00 + h ij ̂ β D d ξ θ Observer Geometric limit: Wave optics: 1 η ) ∝ | ∫ d 2 η ) } | 2 ξ exp { i ω T ( H i ( η ) = H ( ξ , det( ∂ a ∂ b T ( ξ i , η )) ∇ T ( ξ i , η ) = 0 2

When do wave effects matter? - wave effects matter for point sources of coherent radiation - point source means smaller than Fresnel scale, θ F = 1/ ω D Fresnel Angular Pulse Brightness Source Frequency Distance scale scale width temp. FRB ~ GHz ~ Gpc 10 35 K ∼ ms ∼ 10 − 3 μ as ∼ 10 − 12 μ as Pulsar ~ GHz ~ 10 kpc ∼ μ as ∼ ms 10 25 − 10 30 K ∼ 10 − 6 μ as Star in ~ 100 THz ~ kpc - ∼ μ as MW 10 3 K ∼ 10 − 2 μ as - wave optics will be important for lensing of FRBs and pulsars - also, important for lensing of gravitational waves 3

⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ EXAMPLE: THE POINT LENS ds 2 = − (1 − 2 GM ) dt 2 + (1 + 2 GM ) dr 2 r r η ) = 1 ξ η | 2 − 4 GM log | ⟹ T ( ξ , 2 D | ξ | − D d D s y | 2 2 π i ∫ d 2 s | x − 2 x exp [ is { x | }] ⟹ H ( y ) = − log | 2 s = θ 2 4 GM E where and y = β / θ E , x = θ / θ E , θ E = θ 2 D F 4

⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ EXAMPLE: POINT LENS y | 2 | 2 π i ∫ d 2 s x − 2 x exp [ is { x | }] H ( y ) = − log | 2 x 2 + y 2 − 2 xy cos ϕ 2 π i ∫ xdxd ϕ exp [ is { s 2 − log x }] = 2 ∞ ∫ 1 2 2 x 2 − log x ]} − ise isy 2 /2 dx x J 0 ( sxy )exp { is [ = 0 π s 1 F 1 (1 2 is , 1; 1 2 2 isy 2 ) = 1 − e − π s y 2 + 2 y Geometric: 4 + y 2 ) , 2 y ( y ± x 1,2 = μ = y 2 + 4 y 5

Moving source, μ rel τ = t − t 0 t E = θ E y ( τ ) = ( τ 2 + y 2 0 ) 1/2 t E μ rel - geometric light curve depends only on (and , ). The full wave t E y 0 t 0 optics result has an additional dependence on , through which it s depends on the frequency of light 6

Dynamic spectrum of point lens 7

Why wave effects matter: frequency dependence breaks degeneracies 8

Why wave effects matter: boosted cross sections H wave ∼ 2 H geom. ∼ 2 y 2 y 4 9

⃗ Why wave effects matter: boosted cross sections H wave ∼ 2 H geom. ∼ 2 y 2 y 4 σ A = μ ( { | H ( y ) − 1 | > A } ) ̂ y Geometric cross section: σ geom. Let y * = H − 1 , then = π ( y *) 2 θ 2 geom. ( A ) E A Wave-optics cross section: σ wave = π ( y *) 4 θ 2 E A ≈ 12 σ geom. Example: For A = 10 − 2 , then y * ≈ 3.5 ⟹ σ wave A A 10

FOR RADIO SOURCES, WHAT OBJECTS WILL ACT AS POINT LENSES IN THE FULL WAVE-OPTICS REGIME? - Full wave-optics regime when , i.e. s ≲ 1 θ E ≲ θ F - For , we have when (roughly ω ∼ 1 GHz s ≲ 1 M ≲ 100 M ⊕ Jupiter mass) - Condition on amplitude of signal ( A > 10 − 1 ) gives lower bound on mass of M ≳ 0.1 M ⊕ - Point objects in with mass include free-floating [0.1,100] M ⊕ planets (FFPs) 11

FFP MICROLENSING EVENT RATE From Mróz et al. (2017), arxiv 1707.07634 - observed microlensing of stars consistent with prediction of FFPs ∼ 1 M ⊕ as numerous as stars - Using σ wave , optical depth of FFPs: τ FFP ∼ 10 − 8 toward the galactic centre - For pulsars as background source: Γ = 10 − 6 per pulsar per day 12

⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ BINARY MICROLENSING - other potential lenses of interest are two point masses, e.g. planets bound to stars, LISA white dwarfs + gravitational waves y | 2 2 π i ∫ d 2 s | x − 1 q 2 x exp [ is { x 2 | }] H ( y ) = − 1 + q log | x − x 1 | − 1 + q log | x − 2 s = θ 2 E , = 4 GM ω q = M 2 / M 1 θ 2 F - In geometric lensing, magnification depends only on mass ratio . In q wave optics, depends on total mass through . s 13

Caustic structure and geometric limit well-studied; used to detect exoplanets From Gaudi (2017), arxiv 1002.0332 14

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- order wave effects beyond the geometric limit are captured in the Eikonal or “semi-classical” limit. H ( η ) ≈ ∑ ( η ) + 2 ∑ H geom. | H geom. ( η ) H geom. ( η ) | 1/2 cos [ ω ( T ( ξ j , η ) − T ( ξ i , η )) − π ( n j − n i ) ] i i j i i < j 15

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Beyond the Eikonal limit - the Eikonal limit is only valid when all the images are well-separated (relative to ). θ F - the Eikonal limit gives unphysical results near caustics (infinite magnifications). - to get the full wave-optics result, we need to compute highly oscillatory integrals that rarely have analytic answers. How? Use Picard-Lefschetz theory. - PL theory gives a recipe for numerically evaluating integrals of the ∫ ℝ n form e iS d n x ( see Job Feldbrugge’s earlier talk ) 17

∫ ℝ Picard-Lefschetz theory. Example: dxe ix 2 18

Lefschetz thimbles are found by gradient descent along h = Re( iS ) e iS = ∫ ℝ ∫ ℝ e i ( x 3 Example: 3 − x ) Animation courtesy of Fang Xi Lin 19

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 20