Shapelet measurement of galaxy morphologies

1
Shapelet measurement of galaxy morphologies
gravitational lensing
Shapelet
measurement of
gravitational lensing
Richard Massey with Alexandre Refregier, Joel
Berge, David Bacon Richard Ellis
2
Outline
WHAT
WHAT - Introduction to the shapelets basis
functions WHY - Uses of shapelets and
advantages over KSB OUR - Specifics of our
shapelets shear measurement algorithm OTHER -
Differences between the various different
implementations
3
Shapelets basis functions
WHAT
Complete orthogonal basis Can represent any
isolated image as a (unique) weighted, linear sum
of shapelet basis functions. Weights/coefficients
can be calculated via inner product of image with
each basis function, or a least-squares
fit. Useful parameterisation Mathematically
convenient for analysis and manipulation of
images (e.g. convolution is a matrix
multiplication). Physical interpretation is often
intuitive. N radial osciallations M
rotational symmetry Mathematically
elegant Proper treatment of pixellisation
(different to convolution shear)
8 6 4 2 0 -2 -4 -6 -8
M
Truncation of high spatial frequencies
http//www.astro.caltech.edu/rjm/shapelets
0 2 4 6
8
N
4
Modelling arbitrary galaxy shapes
WHAT
5
Shapelets meta-parameters
WHAT
Scale size of Gaussian, ?
Truncation order, nmax
6
Optimisation of scale size truncation parameter
WHAT
POOR FIT
Scale size (pixels)
GOOD FIT
Maximum shapelet order
For a given number of coefficients (truncation
parameter), there is a clearly preferred scale
size for basis functions to get a good fit. For a
given scale size, a certain number of
coefficients are needed to ensure ?r21. Choose
scale size and centroid so that residual ?r21
using smallest possible number of coefficients.
Then de-noise by truncating higher order terms.
7
Interpreting shapelet basis functions
WHAT
8 6 4 2 0 -2 -4 -6 -8
M
Truncation of high spatial frequencies
http//www.astro.caltech.edu/rjm/shapelets
0 2 4 6
8
N
8
Interpreting shapelet basis functions
WHAT
8 6 4 2 0 -2 -4 -6 -8
A circular object has nonzero shapelet
coefficients only where M0
M
Radial profile (monopole)
0 2 4 6
8
N
9
Interpreting shapelet basis functions
WHAT
8 6 4 2 0 -2 -4 -6 -8
M
Radial profile (monopole)
0 2 4 6
8
N
10
Interpreting shapelet basis functions
WHAT
8 6 4 2 0 -2 -4 -6 -8
M
Centroid x2 (dipole)
Radial profile (monopole)
Centroid x1 (dipole)
0 2 4 6
8
N
11
Outline
WHY
WHAT - Introduction to the shapelets basis
functions WHY - Uses of shapelets and
advantages over KSB OUR - Specifics of our
shapelets shear measurement algorithm OTHER -
Differences between the various different
implementations
12
Shear susceptibility factor (Psh)
WHY

• Quadrople ellipticity is not a shear estimator,
  but requires a shear susceptibility factor
  (c.f. solved by im2shape via alternative
  definition of e).
• This creates three problems
• Division by noisy Psh is numerically unstable,
  and introduces Cauchy errors
• on the shear distribution.
• We need to know Psh before shear.
• Psh is a tensor!
• All three are helped by fitting to a large
  population ensemble as a function of galaxy size,
  but any fitting depends on the model and is
  unstable at few level.

13
Shear susceptibility factor (Psh)
WHY
Bright galaxies
Problems with KSB
Faint galaxies
Psh is what always causes problems! Varies
unexpectedly in WHT data as a function of galaxy
SN
14
PSF deconvolution (Psm)
WHY

• A shapelets expansion of a PSF and a galaxy
• Enables full analytic deconvolution from the PSF
  (rather than KSBs Psm correction)
• Avoids mismatch between galaxy size rg and PSF
  size r

• Full deconvolution is ambitous, but the resulting
  model
• Simultaneously creates several shear estimators
  per object
• Provides many other systematics checks from the
  shape information other than ellipticity
• Handles pixellisation properly

Fit convolved, pixellised basis functions to image
15
Astrometry
WHY
Input Shapelet model Subset of good shapelet
models
16
Photometry
WHY
Input Shapelet model Subset of good shapelet
models
17
Photometry
WHY
Input Shapelet model Subset of good shapelet
models
18
Ellipticity measurement
WHY
Without deconvolution
With deconvolution
Input Shapelet model Subset of good shapelet
models
19
Ellipticity measurement
WHY
Without deconvolution
With deconvolution
Input Shapelet model Subset of good shapelet
models
20
Size measurement
WHY
Input Shapelet model Subset of good shapelet
models
21
Diagnostics that are always available from a
complete model
WHY
22
Outline
OUR
WHAT - Introduction to the shapelets basis
functions WHY - Uses of shapelets and
advantages over KSB OUR - Specifics of our our
shapelets shear measurement algorithm OTHER -
Differences between the various different
implementations
23
Effect of gravitational lensing shear
OUR
8 6 4 2 0 -2 -4 -6 -8
M
Truncation of high spatial frequencies
http//www.astro.caltech.edu/rjm/shapelets
0 2 4
6 8
N
24
Effect of gravitational lensing dilation
OUR
S(O)2 operations simply expressed in shapelet
space as mixing of power between a minimal number
of adjacent coefficients.
8 6 4 2 0 -2 -4 -6 -8
M
e.g. c20 c20 constant ? c40
0 2 4
6 8
N
25
Effect of gravitational lensing shear
OUR
S(O)2 operations simply expressed in shapelet
space as mixing of power between a minimal number
of adjacent coefficients.
8 6 4 2 0 -2 -4 -6 -8
M
e.g. c22 c22 constant ? c42
0 2 4
6 8
N
26
Effect of gravitational lensing shear
OUR
8 6 4 2 0 -2 -4 -6 -8
Circularly symmetric object (or ensemble of
objects)
(KSB)
M
0 2 4
6 8
N
27
Effect of gravitational lensing shear
OUR
8 6 4 2 0 -2 -4 -6 -8
Elliptical object
M
There are several independent shear estimators
for any given galaxy. These can be used
separately, or combined to remove biases - like
dependence on the choice of scale size.
0 2 4
6 8
N
28
Shear susceptibilty factor
OUR
8 6 4 2 0 -2 -4 -6 -8
Elliptical object
(KSB)
Shear estimator ?1
M
Shear estimator ?2
0 2 4
6 8
N
29
KSB expressed in terms of shapelets
OUR
8 6 4 2 0 -2 -4 -6 -8
Off-diagonal components of Psh
M
Diagonal components of Psh
Off-diagonal components of Psh
0 2 4 6
8
N
30
Shapelets n2 shear estimator
OUR
Results shown are derived from a set of
simulations like STEP2 PSF A (shapelet galaxies),
but with uncorrelated background
noise. Correlated noise is easy to handle in
principle, but frustratingly slow in practice.
ngal31/arcmin2 ??0.25 per cmpnt
31
Shapelets n4 shear estimator
OUR
Completely independent shear estimator from the
n4 moments. So far somewhat disappointing
results from the ground (from space we got about
half as much signal as in the n2 estimator). But
this is rather preliminary.
ngal9/arcmin2 ??0.39 per cmpnt
32
Shapelets multiple multipole shear estimator
OUR

• Options for combining independent
• shear estimators include
• Minimum variance estimator
• Weighted by signal expected from radial
• profile
• ?-independent shear estimator
• Estimator with shear-invariant Psh (Pshflux)

ngal31/arcmin2 ??0.19 per cmpnt
33
Shapelets in space
OUR
Given half a chance, astronomers think only to
first order.
34
Mass maps
OUR
E modes

B modes

KSB
Shapelets
35
Outline
OTHER
WHAT - Introduction to the shapelets basis
functions WHY - Uses of shapelets and
advantages over KSB OUR - Specifics of our
shapelets shear measurement algorithm OTHER -
Differences between the various different
implementations
36
Getting a simple form for PshPros and cons of
different methods
OTHER
Shear circular model until match (or shear
model until circular)
Measure observed moments

• Can fit even irregular shapes with
• non-concentric profiles
• Can use ?2 to optimise ? and nmax
• Automatically obtain several independent
• shear estimators per object
• Tractable goal of a good decomposition
• gives more possible checks for systematics

• Shear susceptibility known explictly

• Need to distort models by strong shear
• in order to match observations - no
• longer weak lensing
• Shear estimators formed from a difference
• of two large non-linear quantities
• How to choose ? and nmax?
• PSF deconvolution less stable away from
• optimum values of ? and nmax

• Shear susceptibility measured after shear
• (need to fit ensemble, or form a more
• complicated shear estimator see later)
• Shear estimators typically a (non-linear)
• ratio of moments, giving Cauchy errors
• Do shear estimators that avoid these probs
• converge sufficiently quickly in practice?

37
2nd order gravitational lensing
8 6 4 2 0 -2 -4 -6 -8
M
0 2 4
6 8
N
38
2nd order gravitational lensing
Terms of order ? create ellipticity.
8 6 4 2 0 -2 -4 -6 -8
Terms of order create
Multiply imaged
M
0 2 4
6 8
N
Also see poster by Dave Goldberg
39
2nd order gravitational lensing
Terms of order ? create ellipticity.
8 6 4 2 0 -2 -4 -6 -8
Terms of order create
Multiply imaged
Terms of order ? 2 create
M
arclets
0 2 4
6 8
N
Also see poster by Dave Goldberg
40
Gamma2 terms
8 6 4 2 0 -2 -4 -6 -8
M
0 2 4
6 8
N
41
Reference
Terms of order are not very interesting.
However, if there is a significant gradient of
the shear signal, the next terms to be excited
are of order . e.g. substructure/gal-gal
lensing
8 6 4 2 0 -2 -4 -6 -8
A shapelet decomposition contains all of this
information!
M
Centroid shift and
0 2 4
6 8
N
42
Reference
8 6 4 2 0 -2 -4 -6 -8
M
0 2 4
6 8
N