GRAVITATIONAL LENSING LECTURE 2 Docente: Massimo Meneghetti AA - - PowerPoint PPT Presentation

GRAVITATIONAL LENSING

LECTURE 2

Docente: Massimo Meneghetti AA 2015-2016

CONTENTS

➤ Lens equation ➤ Lensing potential

DEFLECTION OF LIGHT IN GENERAL RELATIVITY

How to define the effective diffraction index? From last lecture

DEFLECTION OF LIGHT IN GENERAL RELATIVITY

SCHWARZSCHILD METRIC

ds2 = ✓ 1 − 2GM Rc2 ◆ c2dt2 − ✓ 1 − 2GM Rc2 ◆−1 dR2 − R2(sin2 θdφ2 + dθ2) R = r 1 + 2GM rc2 r

x = r sin θ cos φ

y = r sin θ sin φ

z = r cos φ

dl2 = [dr2 + r2(sin2 θdφ2 + dθ2)]

In the weak field limit:

✓ 1 − 2GM Rc2 ◆ = 1 − 2GM c2r 1 q 1 + 2GM

c2r

≈ 1 − 2GM c2r ✓ 1 − GM c2r ◆ ≈ 1 − 2GM c2r

✓ 1 − 2GM Rc2 ◆−1 ≈ 1 + 2GM c2R = 1 + 2GM c2r 1 q 1 + 2GM

c2r

≈ 1 + 2GM c2r ✓ 1 − GM c2r ◆ ≈ 1 + 2GM c2r

SCHWARZSCHILD METRIC IN THE WEAK FIELD LIMIT

ds2 = ✓ 1 − 2GM Rc2 ◆ c2dt2 − ✓ 1 − 2GM Rc2 ◆−1 dR2 − R2(sin2 θdφ2 + dθ2)

ds2 = ✓ 1 − 2GM rc2 ◆ c2dt2 − ✓ 1 + 2GM rc2 ◆ dl2

Φ = −GM r

DEFLECTION OF LIGHT BY A BLACK HOLE

suggested reading: http://arxiv.org/pdf/0911.2187v2.pdf

Generic static spherically symmetric metric: u=impact parameter rm=minimum distance between the photon and the BH

DEFLECTION OF LIGHT BY A BLACK HOLE

3 √ 3GM c2

for the Schwarzschild metric

DEFLECTION ANGLE OF A POINT MASS

ˆ ~ ↵ = 2 c2 Z +∞

−∞

~ r⊥Φdz

Φ = −GM r

AGAIN ON THE EDDINGTON EXPEDITION

➤ The goal of Eddington

expeditions was to measure a shift in the position of the Hyades stars due to solar deflection

➤ What is the exact shift we

should expect to measure?

➤ What is the relation

between the intrinsic and the apparent positions of the stars?

LENS EQUATION

S L

COMOVING DISTANCE

suggested reading: http://arxiv.org/pdf/astro-ph/9905116v4.pdf comoving distance (along the line of sight) = distance between two points which remains constant if the two points are moving with the Hubble flow proper distance = distance between the two points measured by rulers at the time they are being observed

Dc = Dpr(1 + z)

ANGULAR DIAMETER DISTANCE

O A(z) B(z)

DMδθ

DMδθ = comoving transversal distance

= angular diameter distance = ratio of the physical (proper) transverse size to its angular size

LENS EQUATION

S L

β = θ − DLS DS ˆ α

LENS EQUATION

S L

β = θ − DLS DS ˆ α

LENS EQUATION

S L

β = θ − DLS DS ˆ α

LENS EQUATION

S L

β = θ − DLS DS ˆ α

LENS EQUATION

S L

β = θ − DLS DS ˆ α

DEFLECTION BY EXTENDED LENSES

➤ Remaining in the weak field

limit, one can use the superposition principle

➤ The deflection angle by a

system of point masses is the vectorial sum of the deflection angles of the single lenses

➤ This can be easily generalized

to the case of a continuum distribution of mass

➤ Assumption: thin screen

approximation

ˆ ~ ↵ = 2 c2 Z +∞

−∞

~ r⊥Φdz

DEFLECTION ANGLE OF AN AXIALLY SYMMETRIC LENS

DEFLECTION ANGLE OF AN AXIALLY SYMMETRIC LENS

2π ξ ξ0 < ξ

LENS EQUATION

~ ✓ = ~ ⇠ DL

~ = ~ ✓ − DLS DS ˆ ~ ↵(~ ✓)

~ = ~ ⌘ DS

~ ↵(~ ✓) = DLS DS ˆ ~ ↵(~ ✓)

~ = ~ ✓ − ~ ↵

OTHER NOTATIONS

~ ✓ = ~ ⇠ DL

~ = ~ ⌘ DS ~ ↵(~ ✓) = DLS DS ˆ ~ ↵(~ ✓) ~ = ~ ✓ − ~ ↵ θ0 = ξ0 DL = η0 DS

~ y = ~ x − ~ ↵(~ x) ~ ↵(~ x) = ~ ↵(✓) ✓0 = DL ⇠0 DLS DS ˆ ~ ↵(~ ✓)

LENSING POTENTIAL

ˆ ~ ↵ = 2 c2 Z +∞

−∞

~ r⊥Φdz

This formula tells us that the deflection is caused by the projection of the Newtonian gravitational potential on the lens plane. We introduce the effective lensing potential

LENSING POTENTIAL

ˆ ~ ↵ = 2 c2 Z +∞

−∞

~ r⊥Φdz

This formula tells us that the deflection is caused by the projection of the Newtonian gravitational potential on the lens plane. We introduce the effective lensing potential

1

the lensing potential is the projection of the 3D potential

LENSING POTENTIAL

ˆ ~ ↵ = 2 c2 Z +∞

−∞

~ r⊥Φdz

This formula tells us that the deflection is caused by the projection of the Newtonian gravitational potential on the lens plane. We introduce the effective lensing potential

1

the lensing potential is the projection of the 3D potential

2

the lensing potential scales with distances

OTHER PROPERTIES OF THE LENSING POTENTIAL

The deflection angle is the gradient of the lensing potential The laplacian of the lensing potential is twice the convergence

OTHER PROPERTIES OF THE LENSING POTENTIAL

The deflection angle is the gradient of the lensing potential The laplacian of the lensing potential is twice the convergence