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

sim

s

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eal H

DF

smoo

thin

g

para

m s

pace

Use

d in

sim

s

R

eal H

DF

smoo

thin

g

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