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

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

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Part III : Measuring Lensing

• Measuring Moments• Model Fitting • PSF modelling



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Typical starUsed for finding Convolution kernel

Typical galaxyused for cosmicshear analysis

7/19

Cosmic Lensing

Real data:gi~0.03

gi~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

g1 g2

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 QuickTime™ and a


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 • e1, e2, position (x,y), brightness, size



e2

e1

|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



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• 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 • • •

• • • • •

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



KSB


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Quality Factor

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



7 non-lensing participantsQ~1000 in some regimes

GREAT08 : Stacking Procedure is Important



Winning Methods (Q=1000) Stacked the Data

Average Data

AverageEstimators

IndividualObject

Statistic

EnsembleStatistic


are needed to see this picture.STEP

2006 2008 2010



•

•

•





Massey et al. 2008 Fu et al. 2008







http://www.great10challenge.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 • GREAT08 • Currently LIVE(!) GREAT10

Next LectureCosmic Shear : the Statistics of Weak Lensing



Documents

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