![Page 1: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/1.jpg)
Weak Lensing 2
Tom Kitching
![Page 2: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/2.jpg)
Recap• Lensing useful for • Dark energy• Dark Matter
• Lots of surveys covering 100’s or 1000’s of square degrees coming online now
![Page 3: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/3.jpg)
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
QuickTime™ and a decompressor
are needed to see this picture.
![Page 4: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/4.jpg)
Part III : Measuring Lensing
• Measuring Moments• Model Fitting • PSF modelling
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressorare needed to see this picture.QuickTime™ and a decompressor
are needed to see this picture.QuickTime™ and a
decompressorare needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
![Page 5: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/5.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
![Page 6: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/6.jpg)
Typical starUsed for finding Convolution kernel
Typical galaxyused for cosmicshear analysis
![Page 7: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/7.jpg)
7/19
Cosmic Lensing
Real data:gi~0.03
gi~0.2
![Page 8: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/8.jpg)
8/19
Atmosphere and Telescope
Convolution with kernel
Real data: Kernel size ~ Galaxy size
![Page 9: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/9.jpg)
9/19
Pixelisation
Sum light in each square
Real data: Pixel size ~ Kernel size /2
![Page 10: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/10.jpg)
10/19
Noise
Mostly Poisson. Some Gaussian and bad pixels.Uncertainty on total light ~ 5 per cent
![Page 11: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/11.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
Need to measure shear to 10-3
![Page 12: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/12.jpg)
Intrinsic Ellipticity• Have introduced here the notion that the
sources themselves are already elliptical
g1 g2
![Page 13: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/13.jpg)
Quadrupole Moments• Most common implementation called KSB • Unwieghted quadrupole moments
• Sum over all pixels and find the 2nd moments
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
![Page 14: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/14.jpg)
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
decompressorare needed to see this picture.
![Page 15: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/15.jpg)
Moments
• Want the lensed ellipticity
• Rotation of the unlensed quadrupoles (exercise to show this)
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
![Page 16: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/16.jpg)
• Schneider & Seitz (1995)
• Allows the observed ellipticity to be related to the unlensed ellipticity and shear
• Reduced shear g=/(1-)
QuickTime™ and a decompressor
are needed to see this picture.
![Page 17: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/17.jpg)
• Also Bonnet & Mellier (1995)
• Different normalisation of the moments
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
![Page 18: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/18.jpg)
Moments• The weak lensing limit• g<<1
QuickTime™ and a decompressor
are needed to see this picture.
![Page 19: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/19.jpg)
The Weak Lensing Assumption• When we average over (enough) galaxies in
the universe the intrinsic ellipticity is randomly orientated such that
QuickTime™ and a decompressor
are needed to see this picture.
![Page 20: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/20.jpg)
Moments
• Taking into account the PSF• Additional Quadrupole
• For practical implementation (KSB, 95)
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
![Page 21: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/21.jpg)
![Page 22: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/22.jpg)
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
![Page 23: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/23.jpg)
Model Fitting• Minimum set of parameters we need are • e1, e2, position (x,y), brightness, size
QuickTime™ and a decompressor
are needed to see this picture.
e2
e1
|e|=1
QuickTime™ and a decompressor
are needed to see this picture.
![Page 24: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/24.jpg)
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
QuickTime™ and a decompressor
are needed to see this picture.
![Page 25: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/25.jpg)
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
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
![Page 26: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/26.jpg)
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
QuickTime™ and a decompressor
are needed to see this picture.
![Page 27: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/27.jpg)
Model Fitting
• Accounting for this effect (noise bias)
• Can add extra weight if needed
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
![Page 28: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/28.jpg)
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)
![Page 29: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/29.jpg)
Model Fitting• Shapelets
• Complex model based on a QM formalism
• Similar to raising lowering operators (see L1)
• Noisy on real data • Not regularised
![Page 30: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/30.jpg)
PSF Modelling
• For model fitting methods need to model the PSF as well
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
![Page 31: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/31.jpg)
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
![Page 32: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/32.jpg)
Part IV : Lensing Simulations
• Shear Testing Programme
• GRavitational lEnsing Accuracy Testing
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressorare needed to see this picture.QuickTime™ and a decompressor
are needed to see this picture.QuickTime™ and a
decompressorare needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
![Page 33: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/33.jpg)
• 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
![Page 34: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/34.jpg)
STEP : Shear Testing Programme • • •
• • • • •
QuickTime™ and a
decompressorare needed to see this picture.
![Page 35: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/35.jpg)
Heymans et al., 2006; Massey et al., 2007 & Kitching et al., 2008
QuickTime™ and a decompressor
are needed to see this picture.
KSB
![Page 36: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/36.jpg)
QuickTime™ and a decompressor
are needed to see this picture. QuickTime™ and a decompressor
are needed to see this picture.
![Page 37: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/37.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
•
•
• • • • •
![Page 38: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/38.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
![Page 39: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/39.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
Quality Factor
Kitching et al., 2008 (form filling functions); Amara & Refregier (2007)
![Page 40: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/40.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
7 non-lensing participantsQ~1000 in some regimes
![Page 41: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/41.jpg)
GREAT08 : Stacking Procedure is Important
QuickTime™ and a decompressor
are needed to see this picture.
Winning Methods (Q=1000) Stacked the Data
Average Data
AverageEstimators
IndividualObject
Statistic
EnsembleStatistic
![Page 42: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/42.jpg)
QuickTime™ and a decompressor
are needed to see this picture.STEP
2006 2008 2010
QuickTime™ and a decompressor
are needed to see this picture.
![Page 43: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/43.jpg)
•
•
•
QuickTime™ and a decompressor
are needed to see this picture.
![Page 44: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/44.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
Massey et al. 2008 Fu et al. 2008
QuickTime™ and a decompressor
are needed to see this picture.
![Page 45: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/45.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
![Page 46: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/46.jpg)
http://www.great10challenge.info
![Page 47: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/47.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
![Page 48: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/48.jpg)
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
![Page 49: 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](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f2a5503460f94c44a12/html5/thumbnails/49.jpg)
Next LectureCosmic Shear : the Statistics of Weak Lensing
QuickTime™ and a decompressor
are needed to see this picture.