Weak Lensing 2 Tom Kitching Recap Lensing useful

Weak Lensing 2 Tom Kitching

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 conformal 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

Part III : Measuring Lensing • Measuring Moments • Model Fitting • PSF modelling

Typical galaxy used for cosmic shear analysis Typical star Used for finding Convolution kernel

Cosmic Lensing gi~0. 2 Real data: gi~0. 03 7/19

Atmosphere and Telescope Convolution with kernel Real data: Kernel size ~ Galaxy size 8/19

Pixelisation Sum light in each square Real data: Pixel size ~ Kernel size /2 9/19

Noise Mostly Poisson. Some Gaussian and bad pixels. Uncertainty on total light ~ 5 per cent 10/19

Need to measure shear to 10 -3

Intrinsic Ellipticity • Have introduced here the notion that the sources themselves are already elliptical g 1 g 2

Quadrupole Moments • Most common implementation called KSB • Unwieghted quadrupole moments • Sum over all pixels and find the 2 nd moments

Moments • In the same way as the derivation of the shear have a traceless part of the matrix • Define a source ellipticity such that

Moments • Want the lensed ellipticity • Rotation of the unlensed quadrupoles (exercise to show this)

• Schneider & Seitz (1995) • Allows the observed ellipticity to be related to the unlensed ellipticity and shear • Reduced shear g= /(1 - )

• Also Bonnet & Mellier (1995) • Different normalisation of the moments

Moments • The weak lensing limit • g<<1

The Weak Lensing Assumption • When we average over (enough) galaxies in the universe the intrinsic ellipticity is randomly orientated such that

Moments • Taking into account the PSF • Additional Quadrupole • For practical implementation (KSB, 95)

Model Fitting • Idea of model fitting • Instead of measuring a quantity from the data we can fit a model to the data • The model can contain elements that • Model the galaxy (intrinsic shape) • Model the PSF • The model can be convolved with the PSF • Bayesian • Prior elliticity distribution

Model Fitting • Minimum set of parameters we need are • e 1, e 2, position (x, y), brightness, size e 2 e 1 |e|=1

Model Fitting • Bayesian Model Fitting • Prior in this case is the probability distribution of the intrinsic ellipticity distribution • Can iteratively extract this from the data by summation of the posteriors

Model Fitting • How to estimate ellipticity and shear using model fitting • We know (from quadrupoles) that in the weak lensing limit • For probability (model fitting) this is the expectation value

Model Fitting • Need prior to correctly weight ellipticity • However the ellipticity prior can bias individual shear values if they are low signal-to-noise • But a Bayesian method can exactly account for this • Other terms <<1 • Define shear sensitivity

Model Fitting • Accounting for this effect (noise bias) • Can add extra weight if needed

Model Fitting • Lensfit • Miller et al. (07) • Kitching et al. (08) • Bayesian Model fitting • Uses emperical models (bulge+disk) • Analytically marginalises over brightness and galaxy position • Best performing shape measurement method to date (used on PS 1, CFHTLen. S)

Model Fitting • Shapelets • Complex model based on a QM formalism • Similar to raising lowering operators (see L 1) • Noisy on real data • Not regularised

PSF Modelling • For model fitting methods need to model the PSF as well

PSF Modelling • Two main ways of PSF modelling • 1) Direct : Model the PSF in each exposure using a fitted model to either pixel intensity, ellipticity, size of stars • 2) Indirect : Use multiple exposures to extract the model from the data -- a PCA-like approach • Also deconvolution : remove the PSF from the data by deconvolving the data

Part IV : Lensing Simulations • Shear Testing Programme • GRavitational l. Ensing Accuracy Testing

• Lots of shape measurement codes and approaches • • • KSB Lensfit Shapelets DIEMOS Seclets Sersiclets HOLICS Sextractor … • We don’t know the true shear (no “spectra”) • So need simulations

STEP : Shear Testing Programme • • •

Heymans et al. , 2006; Massey et al. , 2007 & Kitching et al. , 2008 KSB

Quality Factor Kitching et al. , 2008 (form filling functions); Amara & Refregier (2007)

7 non-lensing participants Q~1000 in some regimes

GREAT 08 : Stacking Procedure is Important Average Data Individual Object Statistic Ensemble Statistic Average Estimators Winning Methods (Q=1000) Stacked the Data

STEP 2006 2008 2010

Massey et al. 2008 Fu et al. 2008

http: //www. great 10 challenge. info

Recap • Observed galaxies have instrinsic ellipticity and shear • Reviewed shape measurement methods • Moments - KSB • Model fitting - lensfit • Still an unsolved problem for largest most ambitous surveys • Simulations • STEP 1, 2 • GREAT 08 • Currently LIVE(!) GREAT 10

Next Lecture Cosmic Shear : the Statistics of Weak Lensing