Path Integrals for Radio Astronomy Job Feldbrugge (PI and CMU) - - PowerPoint PPT Presentation

Path Integrals for Radio Astronomy

Job Feldbrugge (PI and CMU) Ue-Li Pen and Neil Turok

ArXiv: 1909.04532

Interference

• Interference is a universal phenomenon in physics. In

radio astronomy, the scintillation of pulsars and potentially Fast Radio Bursts

• However, multi-dimensional oscillatory integrals are

expensive to evaluate numerically

• We present a new integration scheme, based on Cauchy’s

integration theorem and Picard-Lefschetz theory

• The algorithm runs in polynomial time, and becomes more

efficient as the integrand becomes more oscillatory

• From the path integral to

the Fresnel integral

Z x(1)=xobs

x(0)=xs

Dx eiS[x] = Z dx⊥e

i ω

2c



(x⊥−µ)2 d

− R dz

ω2 p(x⊥,z) ω2

• 1

d = 1 dsl + 1 dlo !2

p = ne(x)e2

✏0me

Ψ(µ, ν) = ⇣ν π ⌘D/2 Z dDx eiν[(x−µ)2+φ(x)]

• In dimensionless units:
• A multi-dimensional oscillatory

integral with the imaginary exponent

Φ(x) = iν ⇥ (x − µ)2 + φ(x) ⇤

Fresnel Integral

• Multi-image regions separated

by caustics, where intensity spikes

• In geometric optics, the

integral is approximated with the real saddle points of the exponent

• In wave optics, we need to

evaluate the integral. Nontrivial behaviour near caustics

Φ(x)

Geometric Optics

Catastrophe theory

[Arnol’d 1973, 1975, Berry and Upstill 1980, many others]

Picard-Lefschetz Theory

• Picard-Lefschetz theory: any

meromorphic oscillatory integral I = Z

RD dDx eif(x;µ)

if(x; µ) = h(x; µ) + iH(x; µ)

I = X

i

ni Z

Ji

dDx eif(x;µ)

ni = hRD, Kii

• can be expressed as a sum of

convex integrals

• with the intersection numbers

J J

σ σ

Kσ Kσ

σ

• The Fresnel integral

Z

R

dx eix2

• can be written as a Gaussian

integral by a deformation in the complex plane

1 + i √ 2 Z

R

du e−u2

• Consider the lens

φ(x) = α 1 + x2

• Alternatively we can flow the

integration domain Φ(x, µ) = h(x, µ) + iH(x, µ) ∂γλ(x0) ∂λ = rh(γλ(x0)) γ0(x0) = x0

lim

λ→∞ γλ(RN) = J =

X

i

niJi

The Cusp

The Swallowtail

Localized lenses

Ψ(µ) = Z

R2 eiν[(x−µ)2−φ(x)]dx

φ(x) = α 1 + x2

1 + 2x2 2

Ψ(µ) = Z

R2 eiν[(x−µ)2−φ(x)]dx

φ(x) = α 1 + x2

1 + 2x2 2

• Caustics:
• Folds
• Cusps
• Hyperbolic
• Two dimensional

lens consisting of a blob

• Complicated lens

φ(x) = α 1 + x4

1 + x2 2

• Caustics:
• Folds
• Cusps
• Hyperbolic

Ψ(µ) = Z

R2 eiν[(x−µ)2−φ(x)]dx

• Complicated lens
• Caustics:
• Folds
• Cusps
• Elliptic

Ψ(µ) = Z

R2 eiν[(x−µ)2−φ(x)]dx

φ(x) = α(x3

1 − 3x1x2 2)

1 + x2

1 + x2 2

Summary

• Picard-Lefschetz theory: a new method to evaluate

multi-dimensional oscillatory integrals

• A useful tool in wave optics, especially near caustics
• Gravitational microlensing by Dylan Jow.
• Can we detect caustics in scintillation

measurements, how much amplifications?

[Main et al. 2018]

• Given an interference pattern, can we reconstruct

the lens?

Catastrophe theory

[Arnol’d 1973, 1975, Berry and Upstill 1980, many others]

Catastrophe theory