Fitting Sums of Gaussians Can model fitting using

Fitting Sums of Gaussians Can model fitting using sums of Gaussians provide an unbiased estimate of galaxy shear? Lisa Voigt & Sarah Bridle UCL

Talk Overview q Can modelling with sums of Gaussians provide an unbiased estimate of the ellipticity of galaxies with elliptical isophotes? q Investigate how the following factors affect the bias: • PSF convolution • Pixellisation • Number of Gaussians used to model the galaxy • Noise q 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 q 15 pixels per FWHM along minor axis q Best-fit Gaussian to exponential found by minimising the Χ 2 between the images with respect to the 6 model parameters: x 0, y 0, 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 q 15 pixels per FWHM along minor axis q Best-fit Gaussian to exponential found by minimising the Χ 2 between the images with respect to the 6 model parameters: x 0, y 0, 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 Best-fit model Question: Is the measured galaxy ellipticity biased?

Modelling the galaxy with a single Gaussian: Gaussian PSF, small pixels Simulated galaxy PSF = Gaussian True convolved image Model = Gaussian PSF = true PSF Best-fit model Answer: Yes! - bias on galaxy ellipticity measured > 1%

Modelling the galaxy with a single Gaussian: Gaussian PSF, small pixels Simulated galaxy PSF = Gaussian True convolved image Model = Gaussian PSF = true PSF Best-fit model Answer: Yes! - bias on galaxy ellipticity measured > 1% Need to model the galaxy with more than 1 Gaussian!

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 Model = Gaussian Simulated galaxy = exponential Residuals 1 G 2 G q Tied parameters: x 0, y 0, e, phi q Free parameters: a, A 3 G

Fitting the galaxy with multiple Gaussians: No PSF, small pixels Residuals=Σ (Iitrue - Iimodel)2/(Σ Iitrue)2 x 100%

Fitting the galaxy with multiple Gaussians: Including the PSF

Χ 2 minimisation with multiple Gaussians Modelling the galaxy with 3 Gaussians requires minimising over 10 parameters … takes a long time. q Solution: image is linear in A, so solve for A’s analytically! q Removes thin line of degeneracy between A 1 and A 2. q

Adding in noise Galaxy size = 5 pixels/FWHM along minor axis

Bias as a function of ellipticity m~5 х10 -4 C~7 x 10 -6 q Includes PSF q Galaxy size = 3 pixels/FWHM q No noise bias = e 1 m – e 1 t = m e 1 t + c

Relationship between the bias on the ellipticity and the bias on shear • Express the relationship between the measured and observed e 1 using the equation: e 1 m = (1 + m) e 1 o + c where e 1 o ≈ e 1 i + γ 1 t • If we apply the same shear to all the galaxies in the sample then γ 1 t ≈ < e 1 o >galaxies + < e 1 i >galaxies • We estimate the shear, γ 1 m, by averaging over e 1 m, so γ 1 m = <e 1 m >galaxies = (1 + m) <e 1 o >galaxies + c γ 1 m ≈ (1 + m) γ 1 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 convolved image

Fitting multiple Gaussians to Galaxies with non-elliptical isophotes

Fitting multiple Gaussians to Galaxies with non-elliptical isophotes

Summary q 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… q 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…