Fitting Sums of Gaussians

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

2. 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?

3. Simulating galaxies with elliptical isophotes

4. 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)

5. 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)

6. Modelling the galaxy with a single Gaussian: Gaussian PSF, small pixels Simulated galaxy True convolved image PSF = Gaussian Best-fit model Model = Gaussian PSF = true PSF Question: Is the measured galaxy ellipticity biased?

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

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

9. 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.

10. 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.

11. 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.

12. Modelling the galaxy with more than 1 Gaussian

13. Fitting the galaxy with multiple Gaussians:No PSF, small pixels Residuals Model = Gaussian Simulated galaxy = exponential 1 G 2 G • Tied parameters: x0,y0,e,phi • Free parameters: a,A 3 G

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

15. Fitting the galaxy with multiple Gaussians:Including the PSF

16. Χ2minimisation with multiple Gaussians • Modelling the galaxy with 3 Gaussians requires minimising 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.

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

18. Bias as a function of ellipticity m~5х10-4 C~7x10-6 • Includes PSF • Galaxy size = • 3 pixels/FWHM • No noise bias = e1m – e1t = m e1t + c

19. 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) e1o + c where e1o≈ e1i + γ1t • If we apply the same shear to all the galaxies in the sample then γ1t≈ < e1o >galaxies + < e1i >galaxies • We estimate the shear, γ1m, by averaging over e1m, so γ1m = <e1m >galaxies = (1 + m) <e1o >galaxies + c γ1m≈ (1 + m) γ1t + c

20. Simulating galaxies with non-elliptical isophotes

21. 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. PSF PSF convolved image Simulated galaxy

22. Fitting multiple Gaussians to Galaxies with non-elliptical isophotes

23. Fitting multiple Gaussians to Galaxies with non-elliptical isophotes

24. 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…