Measuring shear using…Measuring shear using…

Kaiser, Squires & Broadhurst (1995)Luppino & Kaiser (1997)

Definition of shapeDefinition of shape

If we have an object with axis ratios a and b:

Ellipticity: e = 1 - b / a

Polarisation: e = (a2-b2) / (a2+b2)

Shear/stretch/distortion: = (a-b) / (a+b)

These are equivalent, but often called the wrong name

Quantifying shapesQuantifying shapes

Many techniques quantify galaxy shapes in terms of the quadrupole moments of the image f

And combine them into the spin-2 polarisation

)()(2

fWdI jiij

2211

22111 II

IIe

and2211

122

2

II

Ie

The weight function W() is necessary because of noise.We use a Gaussian with a dispersion rg matched to the size of the galaxy.

How to obtain the “real” shape?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 galaxydeals 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/disadvantages

Shear polarisabilityShear polarisability

The first order shift in polarisation due to a shear is

shPe

where the shear polarisability is given by a combination of higher order moments of the “true” image

eeXP shshsh This tensor is close to diagonal and the diagonal terms are

similar.

Choice #2: Use diagonal terms or full tensor?

Choice #1: Can we use the observed image?

How to deal with the PSF?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?

Correction for PSF anisotropyCorrection for PSF anisotropy

In KSB the correction for PSF anisotropy and the size of the PSF are separated. The former is a shift in polarisation and the latter a rescaling.

The anisotropic PSF changes the quadrupole moments

qlm are the unweighted quadrupole moments of the PSF and Wlmij depend on higher order moments of the galaxy light distribution.

lmlmijijij qZII '

Correction for PSF anisotropyCorrection for PSF anisotropy

The shift in polarisation due to PSF anisotropy is

pPe smThe smear polarisability is a combination of higher order moments:

eeXP smsmsm

),( 122211 qqqp

and

Choice #4: Use diagonal terms or full tensor?

Correction for PSF anisotropyCorrection 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

smPep /

Choice #5: Use this assumption?

Correction for PSF anisotropyCorrection for PSF anisotropy

Choice #6: Which weight function to use?

p 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

Correction for PSF anisotropyCorrection for PSF anisotropy

Hoekstra et al. (1998): size matters…

KSB

Unweighted PSF

H98

Correction for PSF anisotropyCorrection for PSF anisotropy

Hoekstra et al. (1998)

Correction for PSF sizeCorrection for PSF size

Luppino & Kaiser showed how in the KSB formalism one can rescale the anisotropy corrected polarisations to obtain the `pre-seeing’ shear.

eP 1][ sm

sm

shsh P

P

PPP

*

*with

Choice #7: Which weight function to use?

Choice #8: Use diagonal terms or full tensor?

Hoekstra et al. (1998)

LK97

H98

Correction for PSF sizeCorrection for PSF size

Correction for PSF sizeCorrection 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?

How to deal with noise?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?

How to deal with noise?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/PAlso 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:

2220

2

2 )(

)(1

eP

Pwn

Choice #11: Should we use this weighting?

How to deal with noise?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?How to deal with noise?

We should also use “effective” source densities

How to deal with noise?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.

ConclusionsConclusions

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.