Shapelets for shear Shapelets for shear surveyssurveys
Richard Massey (IoA, Cambridge)Richard Massey (IoA, Cambridge)
Alexandre Refregier (CEA Saclay), Richard Ellis Alexandre Refregier (CEA Saclay), Richard Ellis (CalTech), (CalTech),
Chris Conselice (CalTech), David Bacon Chris Conselice (CalTech), David Bacon (Edinburgh), & SNAP team:(Edinburgh), & SNAP team:
Jason Rhodes (GSFC), Justin Albert (CalTech), Jason Rhodes (GSFC), Justin Albert (CalTech), Mike Lampton (LBL), Alex Kim (LBL), Gary Mike Lampton (LBL), Alex Kim (LBL), Gary
Bernstein (U.Penn) & Tim McKay (Michigan)Bernstein (U.Penn) & Tim McKay (Michigan)
m =
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tional osc
illati
ons
(c.f
. Q
M L
r m
om
n)
n =radial oscillations (c.f. QM energy)
Shapelet (Gauss-Shapelet (Gauss-Hermite) basis fHermite) basis fnnss
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).
Refregier (2001)
n=radial oscillations (c.f. QM energy)
m=rotational oscillations (c.f. QM Lr momn)
Modelling HDF galaxy Modelling HDF galaxy shapesshapes
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Decomposition of a galaxy image into shape components:
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OrthogonalBasis functions
nma
Any image I(x) can be represented as a Taylor series (like a Fourier transform):
Refregier (2001)Refregier & Bacon (2001)
HDF galaxies in HDF galaxies in “shapelet space”“shapelet space”
Shapelet space(series is in practise truncated at finite
nmax)
< >|nma
Real space
Complete, 1-to-1
uniquely specified map
PSF deconvolutionPSF deconvolution
NB: logarithmic scale!
Bacon & Refregier (2001) Kim et al. (2002)Lampton et al. (2002) Rhodes et al. (2002)
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.
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
KSB uses Gaussian-weighted quadrupole moments of a galaxy
image to measure ellipticities, and octupole moments to convert into
shears.
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 ~2S/N.
Shapelets for shear Shapelets for shear measurementmeasurement
Kaiser, Squires & Broadhurst (1995) Refregier & Bacon (2001)
aw nmnmnm
In doing this, we’ve built up a catalogue of all HDF galaxy shapes in n parameters…
• 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!
Shapelet parameter space of Shapelet parameter space of HDF galaxiesHDF galaxies
-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.
First 10 principal components of morphology distribution in shapelet space.
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:
Galaxy morphology Galaxy morphology classification (PCA)classification (PCA)
Average HDF galaxy:
Galaxy morphology Galaxy morphology classification (estimators)classification (estimators)
aw
aw
aw
aw
nmnmC
nmnm
nmnmR
nmnm
nm
C
nm
nm
R
nm
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Gravitational lensing shear
Size
Chirality (asymmetry)
Concentration
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)
Invariant under flux change, rotations. Changes linearly with dilations.
SuperNova/Acceleration SuperNova/Acceleration ProbeProbe
Light Baffles
Door Assembly
Solar Array, ‘Sun side’
Secondary MirrorHexapodBonnet
Secondary Metering structure
Primary Mirror
Optical Bench
Instrument Metering Structure
Tertiary Mirror
Fold-Flat Mirror
Spacecraft
Shutter
Instrument Bay
Instrument Radiator
Solar Array, ‘dark side’
Hi Gain Antenna
Solid-state recorders
ACSCD & HCommPowerData
CCD detectorsNIR detectorsSpectrographFocal Plane guiders Cryo/Particle shield
Perlmutter et al. (2002) Lampton et al. (2002)
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.
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.
Simulated SNAP imagesSimulated SNAP images
A bit like IRAF.noao.artdata, but much better!
Fake Simulated ImageWho needs a real • telescope now?!
HDF galaxy morphology HDF galaxy morphology PDFPDF
The PDF is:
• kernel-smoothed (assume a smooth underlying PDF exists)
• Monte-Carlo sampled, to synthesise new ‘fake’ galaxies.
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.
Morphing in shapelet Morphing in shapelet spacespace
Use
d in
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s
R
eal H
DF
smoo
thin
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para
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pace
Use
d in
sim
s
R
eal H
DF
smoo
thin
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para
m s
pace
Oversmoothed galaxies become random junk… Massey et al. (2002)
Simulated imagesSimulated images
Animation shows0%-10% shear in 1% steps.
Objects have known positions, magnitudes and
added shear:
ftp://ftp.ast.cam.ac.uk/pub/rjm/simages/
Proof of the pudding IProof of the pudding IBlind test: run SExtractor on HDFs and simulated images.
Proof of the pudding IIProof of the pudding II
Concentration
Concentration
Asy
mm
etr
y
Conselice et al. (2000)
Even second-order statistics match, including morphology parameters.
Non-trivial that this should work: shapelet modelling is capturing something important, & simulations are realistic!
Simulating SNAP Simulating SNAP sensitivitysensitivity
Simulated images
Model HDF galaxies
Morphology parameter space
Catalogue of objects before observational noise
Add SNe on top of galaxies
Add instrumental distortions due to telescope optics
Add varying PSF, and stars from which to measure itTry to detect them
Gravitationally shear galaxies (by known amount)
Detection efficiency
Cosmological parameter constraints
SN
e
shear
SNAP Weak Lensing SNAP Weak Lensing sensitivitysensitivity
Mapping the Dark Mapping the Dark Matter Matter
Statistical errors based on realized performance
CDM (Jain, Seljak & White 1997)
WHT SNAP-deep
CDM: 1’ smoothed
Cosmological Cosmological constraints with SNAPconstraints 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