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

7/19 Cosmic Lensing Real data: g i ~0.03 g i ~0.2

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

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

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

Need to measure shear to 10 -3

Intrinsic Ellipticity Have introduced here the notion that the sources themselves are already elliptical g1g1 g2g2

Quadrupole Moments Most common implementation called KSB Unwieghted quadrupole moments Sum over all pixels and find the 2nd 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 e2e2 e1e1 |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 PS1, CFHTLenS)

Model Fitting Shapelets Complex model based on a QM formalism Similar to raising lowering operators (see L1) 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 lEnsing 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

GREAT08 : Stacking Procedure is Important Winning Methods (Q=1000) Stacked the Data Average Data Average Estimators Individual Object Statistic Ensemble Statistic

STEP

                      

Massey et al. 2008Fu et al. 2008

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 GREAT08 Currently LIVE(!) GREAT10

Next Lecture Cosmic Shear : the Statistics of Weak Lensing