(last updated 2022-03-07)
This note first shows how to transform galaxy shapes under shear distortion in Section 1. Then, it shows the condition that the shear responsivity derived from the intrinsic ellipticity is biased in Section 2 based on Section 5.6 of Shirasaki et. al (2019). The notes mainly try to understand what assumptions are taken when transforming galaxy ellipticity and including multiplicative bias.
The code used for add multiplicative bias to the HSC three-year mock shape catalog can be found here.
The first thing is to understand how ellipticity transform under shear distortion. We start from the transform equations of ellipticity under shear distortion (Bernstein & Jarvis 2002), which is summarized in equation (24) of Shirasaki et. al (2019). Here we assume \kappa=0 and \bm{\delta}=2\bm{\gamma}:
\begin{aligned}\tag{1} e_1^{\text{mock}}&=\frac{e_1^{\text{int}}+2\gamma_1+ (\gamma_2/\gamma^2)[1-(1-4\gamma^2)^{1/2}] (\gamma_1 e_2^{\text{int}}-\gamma_2 e_1^{\text{int}})}{1+2\bm{\gamma} \cdot \bm{e^{\text{int}}}} + e_1^{\text{mea}}\,,\\ e_2^{\text{mock}}&=\frac{e_2^{\text{int}}+2\gamma_2+ (\gamma_1/\gamma^2)[1-(1-4\gamma^2)^{1/2}] (\gamma_2 e_1^{\text{int}}-\gamma_1 e_2^{\text{int}})}{1+2\bm{\gamma} \cdot \bm{e^{\text{int}}}} + e_2^{\text{mea}}\,. \end{aligned} Here \bm{e^{\text{int}}}=e_1^{\text{int}}+ie_2^{\text{int}} and \bm{\gamma}=\gamma_1+i\gamma_2 are spin-2 tensors, which can be written into complex numbers. \gamma refers to the l^2 norm of the tensor. e^{\text{int}} and e^{\text{mea}} are intrinsic galaxy shape and measurement error, respectively.
Since \gamma \sim 0.03 on average, it is reasonable to Taylor expand equation (1) as a function of \gamma_{1,2} and only keep the first order of \gamma_{1,2}: \tag{2} \begin{aligned} e_1^{\text{mock}}&= e_{1}^{\text{noi}}+ 2\gamma_1 \left(1-(e_1^{\text{int}})^2\right)-2\gamma_2 e_1^{\text{int}} e_2^{\text{int}}+ \mathcal{O}{(\gamma^2)} \,,\\ e_2^{\text{mock}}&= e_{2}^{\text{noi}}+ 2\gamma_2 \left(1-(e_2^{\text{int}})^2\right)-2\gamma_1 e_1^{\text{int}} e_2^{\text{int}}+ \mathcal{O}{(\gamma^2)} \,,\\ \end{aligned} where e_{1,2}^{\text{noi}}=e_{1,2}^{\text{int}}+ e_{1,2}^{\text{mea}} is the total noise in ellipticity.
Let’s recall the definition of shear responsivity: \mathcal{R}=1-\left\langle \frac{(e^{\text{int}}_1)^2 + (e^{\text{int}}_2)^2}{2} \right\rangle\,, and we define the second-order shape noise: \begin{aligned} \Delta_{11}&=(e^{\text{int}}_1)^2-\left\langle \frac{(e^{\text{int}}_1)^2 + (e^{\text{int}}_2)^2}{2} \right\rangle\,,\\ \Delta_{22}&=(e^{\text{int}}_2)^2-\left\langle \frac{(e^{\text{int}}_1)^2 + (e^{\text{int}}_2)^2}{2} \right\rangle\,,\\ \Delta_{12}&=e^{\text{int}}_1e^{\text{int}}_2\,. \end{aligned} Then we substitute shear responsivity and second-order shape noise into equation (2) — \begin{aligned}\tag{3} e_1^{\text{mock}}&=e_1^{\text{noi}}+ 2\gamma_1 (\mathcal{R}-\Delta_{11})-2\gamma_2 \Delta_{12}+ \mathcal{O}{(\gamma^2)} \,,\\ e_2^{\text{mock}}&=e_2^{\text{noi}}+ 2\gamma_2 (\mathcal{R}-\Delta_{22})-2\gamma_1 \Delta_{12}+ \mathcal{O}{(\gamma^2)} \,.\\ \end{aligned}
If we further neglect the terms with the second order of intrinsic shape noise in equation (3), (the amplitude of intrinsic shape is at the level of 0.25) we have \begin{aligned}\tag{4} e_{1}^{\text{mock}}&= e_1^{\text{noi}}+ 2\gamma_1 \mathcal{R}+\mathcal{O}{(\gamma^2)}+\mathcal{O}(\Delta)\,,\\ e_{2}^{\text{mock}}&= e_2^{\text{noi}}+ 2\gamma_2 \mathcal{R}+\mathcal{O}{(\gamma^2)}+\mathcal{O}(\Delta)\,. \end{aligned} After taking these approximation, we are able separate the cosmic covariance into shear-shear covariance, noise-noise covariance and shear-noise covariance following Masato Shirasaki’s slides.
We have three equations describing galaxy shape transform under shear distortion:
It is important to note that those three choices have different level of accuracy when estimating covariance. Although when estimating shear, equation (2) and equation (4) have the same level of accuracy — they both accurate to the second order of shear since terms with second order shape noise and second order shear averages to zero.
In the last section, we use the ellipticity transform equations (equation (1)) that are accurate for isolated galaxies. However, as we discovered with image simulation, the transform equations are biased due to blendings and selection effects etc. Here we apply the multiplicative bias (denoted as m) correction derived from image simulation to rescale the responsivity from \mathcal{R} to (1+m)\mathcal{R} in the shear transform equations.
Since we have ensured in the catalog paper that the root-mean-square of e_{1}^{\text{noi}}+e_{2}^{\text{noi}} equals the root-mean-square of the observed ellipticity; we take the assumption that e_{1,2}^{\text{noi}} is draw from an unbiased distribution.
To simplify our discussion here, we take equation (4) as our ellipticity transform equaiton, but the conclusion shall be also valid to equation (2) if we assume the components of response matrix is uniformly scaled by the bias. After scaling the responsivity by (1+m) for bias correction, equation (4) changes to \begin{aligned} e_{1}^{\text{mock}}&= e_1^{\text{noi}}+ 2\gamma_1 (1+m) \mathcal{R}\,,\\ e_{2}^{\text{mock}}&= e_2^{\text{noi}}+ 2\gamma_2 (1+m) \mathcal{R}\,. \end{aligned} We shall use te rescaled responsivity for accurate shear estimation, and shear at single galaxy level is \hat{\gamma}_{1,2}=\frac{e^{\text{mock}}_{1,2}}{2\mathcal{R}(1+m)}= \frac{e_{1,2}^{\text{noi}}}{2(1+m)\mathcal{R}}+ \gamma_{1,2}\,. Note that the noise term is scaled by 1/(1+m)\,.