1. Shapelets for shear surveys
Taking shear and faking
images :
Shapelets
for shear surveys
Richard Massey (IoA, Cambridge)
Alexandre Refregier (CEA Saclay), Richard Ellis (CalTech),
Chris Conselice (CalTech), David Bacon (Edinburgh), & SNAP team:
Jason Rhodes (GSFC), Justin Albert (CalTech), Mike Lampton (LBL), Alex Kim (LBL), Gary Bernstein (U.Penn) & Tim McKay (Michigan)

2. Shapelet (Gauss-Hermite) basis fns
Orthonormal basis set of 2D ‘Gauss-Hermite’ functions; AKA the eigenfunctions of the Quantum Harmonic Oscillator.
Fourier transform invariant
(easy image manipulation
e.g. convolution).
Powerful bra-ket notation
already exists.
Gaussian-weighted
multipole moments
(many astronomical uses).
m =rotational oscillations (c.f. QM Lr momn)
n=radial oscillations (c.f. QM energy)
m=rotational oscillations (c.f. QM Lr momn)
n =radial oscillations (c.f. QM energy)
Refregier (2001)

3. Modelling HDF galaxy shapes
Any image I(x) can be represented as a Taylor series (like a Fourier transform):
> |
= a00
+ a01 +…
Orthogonal
Basis functions
Decomposition of a galaxy image into shape components:
<
> |
Refregier (2001)
Refregier & Bacon (2001)

4. HDF galaxies in “shapelet space”
Real space
Shapelet space
(series is in practise truncated at finite nmax)
Complete, 1-to-1
uniquely specified map
<
> |

5. PSF deconvolution
Shapelets are not necesarily convenient for the physics of galaxy morphology, but the mathematics of image manipulation.
PSF convolution is trivial in shapelet space: a bra-ket multiplication. Can implement deconvolution from the WFPC2 PSF via a matrix inversion.
Dilations, translations and shears can also be represented as QM ladder operations (â, â†) in shapelet space.
SNAP PSF
Shapelet
NB: logarithmic scale!
Circular core in the m=0 coefficients.
6-fold symmetry due to refraction around the 3 secondary support struts appears as power in the m=±6, ±12 coefficients.
Bacon & Refregier (2001) Kim et al. (2002)
Lampton et al. (2002) Rhodes et al. (2002)

6. Shapelets for shear measurement
Shapelets is a logical extension of traditional methods. We automatically deconvolve the PSF and can use extra information to form a more robust shear estimator with ~2S/N.
KSB uses Gaussian-weighted quadrupole moments of a galaxy image to measure ellipticities, and octupole moments to convert into shears.
Kaiser, Squires & Broadhurst (1995) Refregier & Bacon (2001)

7. Shapelet parameter space of HDF galaxies
Do combinations of shapelet parameters correlate with ‘eye-balled’ morphological Hubble types?
Quantitative galaxy morphology classification?
Parameter space of galaxy morphology c.f. Hubble tuning fork!
In doing this, we’ve built up a catalogue of all HDF galaxy shapes in n parameters…
-functions representing every HDF galaxy are placed into an n-dimensional parameter space, with each axis corresponding to a (polar) shapelet coefficient or size/magnitude.

8. Galaxy morphology classification (PCA)
Average HDF galaxy:
First 10 principal components of morphology distribution in shapelet space:
As before but with galaxies rotated and flipped, so that they are aligned with the x-axis and have the same chirality, before being stacked:
First 10 principal components of morphology distribution in shapelet space.

9. Galaxy morphology classification (estimators)
Gravitational lensing shear
Size
Chirality (asymmetry)
Concentration
Invariant under flux change, rotations. Changes linearly with dilations.
Invariant under dilations, rotations. Changes sign under parity flip.
Invariant under rotations, shears. Slope of “shapelet power spectrum”
Conselice et al. (2000) Bershady et al. (1998)

10. SuperNova/Acceleration Probe
Door Assembly
Secondary Mirror
Hexapod
Bonnet
300sq.degree optical +NIR survey is planned to R=28: yielding 150 million resolved galaxies!
Need new high precision, robust
shear measurement algorithm
(current limiting factor in
weak shear surveys).
Algorithms need calibrating:
old methods also required
this, using simulated images
and entire mock DR pipeline.
Light Baffles
Secondary Metering structure
Solar Array, ‘Sun side’
Primary Mirror
Optical Bench
Instrument Metering Structure
Tertiary Mirror
Solar Array, ‘dark side’
Fold-Flat Mirror
Instrument Radiator
Spacecraft
Instrument Bay
ACS
CD & H
Comm
Power
Data
CCD detectors NIR detectors Spectrograph Focal Plane guiders
Cryo/Particle shield
Solid-state recorders
Perlmutter et al. (2002) Lampton et al. (2002)
Shutter
Hi Gain Antenna

11. Simulated SNAP images
HDF is deep (R=28.6), but too small to do this. Most importantly, the properties of objects in it are not known. We need deep images, containing realistic galaxies – but with known sizes, magnitudes & shears.
Calibrate future shear (astrometry)
measurement algorithms.
Optimise telescope design (SNAP,
GAIA) and survey strategy.
Estimate errors upon cosmological
parameter constraints which will
be possible with the real data.
Fake Simulated Image
Who needs a real • telescope now?!
A bit like IRAF.noao.artdata, but much better!

12. HDF galaxy morphology PDF
Parameter space of galaxy morphology c.f. Hubble tuning fork!
-functions representing every HDF galaxy are placed into an n-dimensional parameter space, with each axis corresponding to a (polar) shapelet coefficient or size/magnitude.
The PDF is:
kernel-smoothed (assume a smooth underlying PDF exists)
Monte-Carlo sampled, to synthesise new ‘fake’ galaxies.

13. Morphing in shapelet space
smoothing
param space
smoothing
param space
Used in sims Real HDF
Used in sims Real HDF
Oversmoothed galaxies become random junk…
Massey et al. (2002)

14. Objects have known positions, magnitudes and added shear:
Simulated images
Objects have known positions, magnitudes and added shear:
Animation shows
0%-10% shear in 1% steps.
ftp://ftp.ast.cam.ac.uk/pub/rjm/simages/

15. Proof of the pudding I
Blind test: run SExtractor on HDFs and simulated images.

16. Proof of the pudding II
Even second-order statistics match, including morphology parameters.
Non-trivial that this should work: shapelet modelling is capturing something important, & simulations are realistic!
Asymmetry
Concentration
Concentration
Conselice et al. (2000)

17. Simulating SNAP sensitivity
Model HDF galaxies
Morphology parameter space
Simulated images
SNe
Catalogue of objects before observational noise
shear
Add SNe on top of galaxies
Add varying PSF, and stars from which to measure it
Try to detect them
Detection efficiency
Add instrumental distortions due to telescope optics
Cosmological parameter constraints
Gravitationally shear galaxies (by known amount)

18. SNAP Weak Lensing sensitivity

19. Mapping the Dark Matter
CDM (Jain, Seljak & White 1997)
CDM: 1’ smoothed
WHT
SNAP-deep
Statistical errors based on realized performance

20. Cosmological constraints with SNAP
Combining constraints based on DM power spectrum with those from CMB (and SNe) breaks degeneracies between M and w and directly tests for growth of structure