Fitting Sums of Gaussians

Can model fitting using sums of Gaussians provide an unbiased estimate of galaxy

shear?

Lisa Voigt & Sarah BridleUCL

Talk Overview

Can modelling with sums of Gaussians provide an unbiased estimate of the ellipticity of galaxies with elliptical isophotes?

Investigate how the following factors affect the bias:• PSF convolution• Pixellisation• Number of Gaussians used to model the galaxy• Noise

Does the method provide an unbiased estimate of the ellipticity of galaxies with non-elliptical isophotes?

Simulating galaxies with elliptical isophotes

Modelling the galaxy with a single Gaussian: No PSF, small pixels

Simulated galaxy Model

Exponential e=0.2

Gaussian

15 pixels per FWHM along minor axis Best-fit Gaussian to exponential found by minimising the Χ2 between the images with respect to the 6 model parameters: x0,y0,e,phi,a,A

Question: Is the measured galaxy ellipticity biased?(Bias = measured – true ellipticity)

Modelling the galaxy with a single Gaussian: No PSF, small pixels

Simulated galaxy Model

Exponential e=0.2

Gaussian

15 pixels per FWHM along minor axis Best-fit Gaussian to exponential found by minimising the Χ2 between the images with respect to the 6 model parameters: x0,y0,e,phi,a,A

Answer: No - bias on ellipticity measured is < 0.1 %(Bias = measured – true ellipticity)

Modelling the galaxy with a single Gaussian: Gaussian PSF, small pixels

Simulated galaxy PSF = Gaussian True convolved image

Model = Gaussian PSF = true PSF

Question: Is the measured galaxy ellipticity biased?

*

*Best-fit model

Modelling the galaxy with a single Gaussian: Gaussian PSF, small pixels

Simulated galaxy PSF = Gaussian True convolved image

Model = Gaussian PSF = true PSF

Answer: Yes! - bias on galaxy ellipticity measured > 1%

*

*Best-fit model

Modelling the galaxy with a single Gaussian: Gaussian PSF, small pixels

Simulated galaxy PSF = Gaussian True convolved image

Model = Gaussian PSF = true PSF

Answer: Yes! - bias on galaxy ellipticity measured > 1%Need to model the galaxy with more than 1 Gaussian!

*

*Best-fit model

Modelling the galaxy with a single Gaussian: Including the effects of pixellisation

• Pixellisation is a convolution followed by a sampling.• Convolution with the PSF dominates over convolution with the pixels if the PSF is larger than the pixel size.

Pixel integration

• Modelling pixellated images requires pixel integration … takes a long time!

• Use analytic approximation to pixel integration for a Gaussian image.

• How good is this approximation?• Test by fitting a Gaussian to a Gaussian.• In simulated Gaussian use numerical pixel integration,

with each pixel split into 50 x 50 sub-pixels.• Compare bias measured using analytic pixel integration

in the model image with that measured using numerical pixel integration.

Pixel integration

• Analytic approximation to pixel integration equivalent to ~10 x 10 sub-pixels. • To obtain a bias ~ 0.1% need approximately 3 x 3 sub-pixels.

Modelling the galaxy with more than 1 Gaussian

Fitting the galaxy with multiple Gaussians:No PSF, small pixels

Tied parameters: x0,y0,e,phi Free parameters: a,A

Model = Gaussian

Simulated galaxy = exponential

Residuals

1 G

2 G

3 G

Fitting the galaxy with multiple Gaussians:No PSF, small pixels

Residuals=Σ (Iitrue - Ii

model)2/(Σ Iitrue)2 x 100%

Fitting the galaxy with multiple Gaussians:Including the PSF

Χ2 minimisation with multiple Gaussians

Modelling the galaxy with 3 Gaussians requiresminimising over 10 parameters … takes a long time. Solution: image is linear in A, so solve for A’s analytically! Removes thin line of degeneracy between A1 and A2.

Adding in noise

Galaxy size = 5 pixels/FWHM along minor axis

Bias as a function of ellipticity

m~5х10-4

C~7x10-6

bias = e1m – e1

t = m e1t + c

Includes PSF Galaxy size = 3 pixels/FWHM No noise

Relationship between the bias on the ellipticity and the bias on shear

• Express the relationship between the measured and observed e1

using the equation:

e1m = (1 + m) e1

o + c

where

e1o ≈ e1

i + γ1t

• If we apply the same shear to all the galaxies in the sample then

γ1t ≈ < e1

o >galaxies + < e1i >galaxies

• We estimate the shear, γ1m, by averaging over e1

m, so

γ1m = <e1

m >galaxies

= (1 + m) <e1o >galaxies + c

γ1m ≈ (1 + m) γ1

t + c

Simulating galaxies with non-elliptical isophotes

Fitting multiple Gaussians to galaxies with non-elliptical isophotes

• Simulate galaxies with a sum of 2 Gaussians, each with the same flux, but different axis ratios.

• First Gaussian represents the bulge: e fixed at 0.• Second Gaussian represents the disk: e varied up to 0.4• Perform ‘ring test’ to obtain bias on shear when galaxy is

modelled with a sum of Gaussians.• Plot m and c as a function of disk ellipticity.

Simulated galaxy PSF PSF convolved image

Fitting multiple Gaussians to Galaxies with non-elliptical isophotes

Fitting multiple Gaussians to Galaxies with non-elliptical isophotes

Summary

Using Gaussians to model galaxies with elliptical isophotes:

– Using 3 Gaussians to model the galaxy reduces the bias on the ellipticity measured to < 0.1%.

– The ellipticity measured from noisy images is also biased by less than 0.1 %.

– This method should do well on STEP 4 simulations! But…

A first look at modelling galaxies with non-elliptical isophotes suggests that using sums of Gaussians to measure shear may not be good enough for future surveys…