weak Gravitational Lensing Weak Lensing 1 Tom Kitching

(weak) Gravitational Lensing

Weak Lensing 1 Tom Kitching

Introduction • Lecture 1 : Introduction to w� eak lensing • Background to the subject • Basic equations and derivations • Lecture 2 : Observing weak lensing • How to observe weak lensing • Overview of simulations • Lecture 3 : Statistics of weak lensing • Cosmic shear • Fisher matrices

Part I : Introduction to Lensing • • Weak lensing Why is weak � lensing useful Brief History of Surveys Current and future cosmological applications

Introduction • What is weak lensing? • strong vs. weak lensing

What is Gravitational Lensing? • •

Kappa > 1 Strong Lensing Multiple Images z

Kappa<< 1 Weak Lensing Local Distortion © NASA z

What can we observe? z z

What can we observe? z •

z z

Why is weak lensing useful? • Low-z equivalent to high-z CMB

Weak Lensing Surveys • What we need • Large field, high resolution • Mostly in optical so far (some radio) • Weak Lensing by LSS First Detected in 2000 • • Bacon et al. Kaiser et al. van Waerbeke et al. Wittman et al.

Kaiser Kasier et al. 2000 Kaiser 2000

Weak Lensing Surveys • Current and on going surveys Euclid DES LSST Ki. DS* Pan-STARRS 1** CFHTLen. S** 05 10 ** complete or surveying * first light 15 20 25

Dark Energy • Current constraints, • Kitching et al, (2007) COMBO-17 2 sqdeg proof of concept • Sembolini et al. (2006) • Kilbinger et al. (2009) CFHTLS 170 sqdeg Preliminary results • Schrabback et al. (2010) COSMOS, 2 sqdeg

Dark Energy • Expect constraints of 1% from Euclid

Dark Matter • Clusters • Halo masses • 3 D DM mapping

Part II: Weak Lensing Foundations • • Lens Equation From General Relativity Derivation of the Distortion Matrix Complex Notation for shear

Geometry • The Born Approximation

The Lens Equation • Lens equation and define the terms Relates True Position to Observed Position

Deflection Angle from GR • Want to know what the deflection angle is

Deflection Angle from GR §

Deflection Angle from GR

Deflection Angle from GR • Equation of motion of deflected light §

GR-Newton Factor of 2 Confirmed in 1919 © R. Massey 2010

The Distortion Matrix • Go back to the lens equation • Every point is mapped to a new point via a shear mapping

The Distortion Matrix • Define a lensing potential • Projected Newtonian potential - Born approximation (linear transform) • Describes how unlensed point maps to lensed point • =0 implies no mass and no lensing

The Distortion Matrix • Define the Distortion Matrix as • Derivatives of the lensing potential • 11, 22, 21, 12

Shear and Convergence • Closer look at the Distortion matrix • Decompose into a trace and traceless part • Define

Shear and Covergence • What shape of distortion does this imply?

Disortion is Elliptical • For a general set of source positions X write the eigen equation • • AX= X (A- )X=0 Characteristic equation : |A- 1|=0 are the eigenvalues

Disortion is Elliptical • Solving the determinant

Disortion is Elliptical • The shear is symmetric under rotations of 180 degrees • Anew=RAold. RT • Distortion is elliptical • “Ellipticity” defined as note |e|< 1 or with r (see next lecture)

Distortion is elliptical 1 2

Complex Notation • Can write shear in complex form • Shear is a spin-weight 2 field • Symmetric under rotations of 180 deg. • Polarisation also an example of spin-2 • Kappa is is spin-wight 0 field (symmetric under any rotation) • Spin 0 = scalar • Spin 1 = vector

Raising and Lowering • Shear as spin 2, Kappa as spin 0 • is a scalar (spin-0) field • Define a complex derivative • A raising/lowering opertato • Kappa and shear are now defined as

Recap • Lensing useful for • Dark energy • Dark Matter • Lots of surveys covering 100’s or 1000’s of square degrees coming online now

Recap • Lensing equation • Local mapping • General Relativity relates this to the gravitational potential • Distortion matrix implies that distortion is elliptical : shear and convergence • Simple formalise that relates the shear and convergence (observable) to the underlying gravitational potential

Next Lecture How do we measure the lensing effect?