Measuring shear using

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Measuring shear using
Kaiser, Squires Broadhurst (1995) Luppino
Kaiser (1997)
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Definition of shape
If we have an object with axis ratios a and
b Ellipticity e 1 - b / a Polarisation e
(a2-b2) / (a2b2) Shear/stretch/distortion g
(a-b) / (ab)
These are equivalent, but often called the wrong
name
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Quantifying shapes
Many techniques quantify galaxy shapes in terms
of the quadrupole moments of the image f And
combine them into the spin-2 polarisation
and
The weight function W(q) is necessary because of
noise. We use a Gaussian with a dispersion rg
matched to the size of the galaxy.
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How to obtain the real shape?
Galaxies are typically convolved with an
anisotropic point spread function. We also
introduced a weight function. How do undo their
effects? Brute force deconvolution of the
galaxy deals with PSF anisotropy and size at
the same time Approximate the problem (e.g.,
KSB95) Separates PSF anisotropy and size
correction Each has its own advantages/disadvant
ages
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Shear polarisability
The first order shift in polarisation due to a
shear is
where the shear polarisability is given by a
combination of higher order moments of the true
image
This tensor is close to diagonal and the diagonal
terms are similar.
Choice 1 Can we use the observed image?
Choice 2 Use diagonal terms or full tensor?
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How to deal with the PSF?

KSB assumption PSF is convolution of a an
istotropic function and a compact anisotropic
function.
Choice 3 Do we believe this?
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Correction for PSF anisotropy
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Correction for PSF anisotropy
The shift in polarisation due to PSF anisotropy is
The smear polarisability is a combination of
higher order moments
and
Choice 4 Use diagonal terms or full tensor?
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Correction for PSF anisotropy
PSF unweighted moments are not useful in
practice But the correction should work for
stars are well, which should have zero
polarisation after correction. So an alternative
choice is to assume
Choice 5 Use this assumption?
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Correction for PSF anisotropy
pa depends on the width of the adopted weight
function with the width matched to the size of
the object, we see different parts of the PSF
Choice 6 Which weight function to use?
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Correction for PSF anisotropy
KSB
Unweighted PSF
H98
Hoekstra et al. (1998) size matters
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Correction for PSF anisotropy
Hoekstra et al. (1998)
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Correction for PSF size
Luppino Kaiser showed how in the KSB formalism
one can rescale the anisotropy corrected
polarisations to obtain the pre-seeing shear.
with
Choice 7 Which weight function to use?
Choice 8 Use diagonal terms or full tensor?
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Correction for PSF size
LK97
H98
Hoekstra et al. (1998)
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Correction for PSF size
The pre-seeing shear polarisability is a noisy
quantity for an individual object. It depends on
the galaxy size, profile and shape. To reduce
noise, one can average the values of galaxies
with similar properties or use the raw values.
Choice 9 How does one implement this?
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How to deal with noise?
The images contain noise. Hence the polarisation
of each galaxy has an associated measurement
error. Hoekstra et al. (2000) showed how this
can be estimated from the data. The noise
estimate depends on higher order moments of the
image. In addition the noise adds a small bias
in the polarisation and polarisabilities because
only the quadropole moments are linear in the
noise.
Choice 10 Should we correct for noise bias?
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How to deal with noise?
The pre-seeing shear polarisability approaches
zero for objects that are comparable in size to
the PSF. Noise in the polarisation is enhanced by
a factor 1/Pg. Also residual systematics are
scaled by this factor. We need to weight
galaxies accordingly. The obvious choice is to
use the inverse variance of the shear
Choice 11 Should we use this weighting?
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How to deal with noise?
By coincidence, the scatter in the polarisation
is almost constant with apparent magnitude, but
at the faint end one effectively measures noise.
This becomes even more apparent when looking at
the shear after correction for the size of the
PSF.
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How to deal with noise?
We should also use effective source densities
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How to deal with noise?
A proper weighting should also account for the
fact that more distant galaxies are lensed more
efficiently. Hence, we need to modify the source
redshift distribution to account for the
weighting scheme.
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Conclusions
There are many choices on can make in the
implementation of KSB. Although not always well
defined, most choices are fairly obvious, or
result in only minor differerences. The
underlying assumption regarding the PSF appears
silly, but remember that we are interested in
ensemble averages. The ensemble averaged galaxy
is close to a Gaussian and higher order effects
tend to be averaged out due to the random
orientation of galaxies. KSB is wrong for any
given galaxy, but appears to do quite well in the
ensemble average.