1. STEP: THE SHEAR TESTING PROGRAMME
Konrad Kuijken
Leiden

2. Gravitational shear
Gradient of deflection  coherent distortions of galaxy shapes
e1=e cos(2 PA)
e2=e sin(2 PA)

3. Accurate shear measurements
Weak lensing measures
Power spectrum
Cluster abundance
Light-mass correlation
Why precision?
Trace growth of structure (cosmology probe)
Dark matter/galaxy connection
Halo shapes

4. Growth of structure
Subtle tracer of expansion history

5. Lensing vs distance Lens bending angle depends on distance
Integral measurement of mass distribution along sightline
 =  (Dls / Ds)
Tomography needed

6. Requirements High fidelity shear measurement
Well-controlled distance measurements (understand errors in redshifts)
Accurate predictions of power spectrum
Matter power spectrum
Consequent shear power spectrum

7. Averaging ellipticities
Shear = ell. of intrinsically round galaxy
Response of ellipticity to shear depends on ellipticity
Extreme example: e1=1  e1=1 under any 1
‘Ring response’ e2 e1

8. Measuring galaxy ellipticity
Intrinsic shape is altered by
Lensing shear
PSF smearing
Pixellation (=boxcar smoothing+subsampling)
Noise
Some of the information loss irretrievable
Sub-pixel information
Photon noise

9. <x2>, <xy>, <y2>
Ideally…
Measure 2nd moments of light distribution
<x2>, <xy>, <y2>
Subtract PSF 2nd moments
Form ellipticity
(Ixx-Iyy, 2Ixy)
Alas…

10. Techniques Ellipticity from: PSF / pixellation correction from:
Weighted 2nd moments
Model fitting
PSF / pixellation correction from:
Model fitting of PSF effect
Pre-smoothing images with kernel

11. Kaiser, Squires, Broadhurst 95
Weighted 2nd moments
Gaussian wt. function
PSF correction 2 stages:
PSF anisotropy correction
Assumes anisotropic part of PSF is compact
Polarizability depends on 4th weighted moments
PSF circularization correction (LK97)
Very succesful, but imperfect
Very good for Gaussian PSF
 dx dy x2 W(r) I(x,y)

12. KSB PSF model Compact anisotropic core  circular PSF Gaussian: OK:
(separable in x,y)
Moffat func: not OK
Radial PSF profile implicitly determines ellipticity profile =   =

13. The weight function
Hoekstra et al. 1998
KSB formalism works for any Gaussian wt. function.
Pick one that is optimal for S/N
Radius of PSF weight function matters if ellipticity of PSF depends on radius
Empirically, best results for radius that matches galaxy

14. Direct modelling Model sources as full PSF  elliptical model
Read off ellipticity
Different galaxy models:
Multiple elliptical Gaussians (KK99, Bridle Im2shape)
Shapelets series (Refregier et al, Massey et al, KK06)
Sheared shapelets (Bernstein & Jarvis)
Disk+bulge models (Mandelbaum et al)

15. Stacking Average galaxy is intrinsically round Stack observed sources
Write as [sheared round source]  PSF
Characterized by single ellipticity and radial brightness profile
Subtlety: centroid errors
Extra smearing term. Not necessarily isotropic!

16. Shapelets
Direct modelling of PSF and sources as Gaussians x polynomials (QHO!)
Ellipticity measurement and PSF effects analytic
Model a galaxy as
PSF  [1+1 S1+2 S2]  C
All operations linear, matrix multiplications of shapelet coefficients.

17. Shapelets PRO: CON Shapelet coeffs replace pixels (compression)
Error propagation simple
Simple combinations of coeffs. mimic weighted moments
Can be extended to flexions
CON
Galaxies are not Gaussian!

18. Sech-shapelets Gaussian  poly Sech  poly
(radial orders 0,2,4,6,8,10)

19. Many methods! Shear PSF corr. Moments Model Subtract KSB B&J Shapelets
Im2shape

20. STEP Confront all methods and software with uniform datasets
Large enough to draw significant conclusions on accuracy
Blind simulations
Involve most of the community
Meet regularly to discuss progress

22. STEP 1 Heymans et al 2005 CFHT-like simulations (single colour) Task:
Skymaker, Van Waerbeke
Galaxies exp. disk + de Vauc bulge
Random orientations, axis ratio distribution
Stars added into the images
100,000 galaxies per PSF/shear set (30 sets)
Task:
Model PSF
Deduce average shear in the images
End-to-end pipeline tests

23. (CFHT + coma, astigm., defocus, m=3, m=4)
STEP1: Results
6 different PSF types
(CFHT + coma, astigm., defocus, m=3, m=4)
5 sets of images, shears
1=0, 0.005, 0.01, 0.05, 0.1
Quantify results as
out = (1+m) in + c

24. STEP1: Results
~7% calibration bias
Good PSF anisotropy correction

25. STEP1: Results Some unexplained trends with magnitude Noise effect?
Driven by size-mag relation in simulation?
Polarizability error (Kaiser flow)?

26. STEP2 Massey et al 2006 More complex galaxies More complex PSF
Shapelet reconstructions
‘Evolving’ galaxy pop
More complex PSF
suppress shape noise artificially:
Include each galaxy with 2 PA, 90 deg apart (i.e. at (e1,e2) and (-e1,-e2) )

28. STEP2: Results
All improved! 1-2% calibration errors

29. STEP3 Space-like PSFs Results under analysis.
Investigate subsampling (pixel size)
Very non-Gaussian PSFs (ACS & SNAP - like)
Results under analysis.

30. STEP4 Back to Basics Take out effects of
Source detection
Source overlap
Star/galaxy separation
FWHM 1.4” (gals), 0.7” (PSF), 0.2” pixels
Simulations of grids of galaxies, 32 different shears, enough galaxies to get statistical error on measured m, c parameters down to 0.001
Blind analysis

31. STEPWEB

32. Outstanding issues Colour effects Ellipticity-dependent selection bias
PSF SED different from galaxies
Ellipticity-dependent selection bias
Pixel noise correlations
Non-linear shear effects?
2nd order light bending

33. 2nd order light bending Failure of single lens plane approx.
Eg singular isothermal sphere
1st order:
2nd order: accel. bigger by [1+O()]
At most 10-3 effect, usually much smaller
~  b/2 b
 c

34. Multiple lens planes 2 deflections at different z: ~1arcsec ~5kpc
~1Gpc

35. Further steps (& STEPs)
Key issues:
Modelling PSF accurately, including interpolation
Source selection independent of intrinsic ellipticity
Propagating errors & covariances
PSF ‘Gaussianization’ + KSB: how far will it get you?
Photo-z accuracy experiment (PACE) ?