The chromatic Point Spread Function of weak lensing measurement in Chinese Space Station survey Telescope

The light from distant objects is deflected by the gravitational potential of the massive objects along their path to us, which is referred to as gravitational lensing (e.g. Blandford et al. 1991; Bartelmann & Schneider 2001).In the limit of very weak deflection, i.e. without striking phenomena such as multiple images or arcs, lensing is referred to as ”weak lensing” (e.g. Kilbinger 2015). The images of distant galaxies are weakly distorted by the tidal effect of the gravitational potential, i.e. lensing shear. The resulting correlations in the shapes can be related directly to the statistical properties of the mass distribution in the universe, thus the weak lensing by large-scale structure, or cosmic shear, has been identified as a powerful tool for cosmology, and has been demonstrated by several observations, e.g. the COSMOS survey (Schrabback et al. 2010), the CFHTLenS survey (Heymans et al. 2012) etc. More recent studies yield competitive constraints on cosmological parameters (e.g. Abbott et al. 2018; Asgari et al. 2021; Hamana et al. 2020). Thus several ongoing or future missions are designed with weak lensing as a primary science driver, such as the Vera Rubin Observatory Legacy Survey of Space and Time (the LSST LSST Science Collaboration et al. 2009; Ivezić et al. 2019), the ESA space-borne telescope $Euclid$ (Laureijs et al. 2011), the Roman Space Telescope (WFIRST Spergel et al. 2015) and the Chinese Space Station Survey Telescope (CSST Zhan 2011, 2018).

The measurement of the cosmic shear requires stringent control of the systematic effects since the statistical error of large weak lensing surveys becomes less important (e.g. Mandelbaum 2018). In the past few decades, significant progress has been achieved on the shear measurements (e.g. Kaiser et al. 1995; Luppino & Kaiser 1997; Kuijken 1999; Bernstein & Jarvis 2002; Zhang 2011; Miller et al. 2013; Hoekstra 2021). One of the most dominant sources of measurement bias is the smearing of the images by the Point Spread Function (PSF). Precise modelling of the PSF is crucial and difficult. In reality, people usually estimate the PSF by measuring the shapes of star images (e.g. Hoekstra 2004; Jarvis et al. 2021), and construct the PSF model over the whole field of view. The implicit assumption is that the PSF affecting stars and galaxies is locally the same. However, it has been noticed that the PSF has a dependence on the observing wavelength. The measured star images over a wide band can provide an “effective” PSF, which is weighted by the Spectral Energy Distribution (SED) of the star. Once the SEDs of the galaxies are different from that of stars, the assumption that the effective PSF from a star is the same as that of a galaxy will be violated. Such a deviation is called colour bias. Cypriano et al. (2010) firstly proposed such an issue and discussed the impact on the shear measurements in a diffraction-limited telescope. They find such kind of a bias can be calibrated in cases of (i) the stars have the same colour as the galaxies; (ii) estimation of the galaxy SED using multiple colours and a PSF model of PSF using the optical design of the telescope. Meyers & Burchat (2015) study the impact of the colour bias on shape measurements of two atmospheric chromatic effects for ground-based surveys, the Dark Energy Survey (DES) and the LSST. Eriksen & Hoekstra (2018) explores various approaches to determine the effective PSF using broad-band data. They also study the correlations between photometric redshift and PSF estimates that arise from the use of the same photometry, considering the Euclid mission as a reference. Carlsten et al. (2018) measures the wavelength dependence of the PSF size in the Hyper Suprime-Cam (HSC) Subaru Strategic Program (SSP) survey, and constructs a PSF size model as a function of the wavelength. They proposed a power-law model and tried to calibrate the colour bias to fulfil the error budget proposed by Meyers & Burchat (2015). Plazas & Bernstein (2012) discuss how differential chromatic refraction can bias shear measurements in LSST by introducing a SED-dependent elongation of the PSF along the elevation vector.

Besides the colour bias, the galaxy shapes can differ significantly across filters. In other words, the SED of the galaxies varies spatially. Since in the weak lensing, the signature is achromatic, it has been proposed to measure cosmic shear from multiple filters (Jarvis & Jain 2008). However, a higher-order systematic bias in shear measurement can arise within a filter due to such an effect, which is called colour gradient bias (CG bias for short Voigt et al. 2012; Semboloni et al. 2013; Kamath et al. 2020). They estimate the CG bias with the chromatic PSF for the wide band images of Euclid and LSST, showing that the bias has to be taken into account for precise shear measurement. Er et al. (2018) performs analysis of CG bias using real data taken by the Hubble Space Telescope, and demonstrates that the CG bias can be calibrated by images from two narrower bands and presents its dependence on the galaxy properties, such as colour, galaxy size etc.

The stage-IV cosmological surveys target high-precision weak lensing measurements for over a billion galaxies. The shrinking statistical error causes all the systematic biases prominent. Thus, even the small, higher-order systematic bias, such as colour bias and CG bias need to be carefully estimated and controlled. In this work, we study the two biases due to the chromatic PSF effect in CSST weak lensing images. The basic formulae are given in Sect. 2. The estimates of colour bias and CG bias are given in the following Sects. 3 and 4. We discuss our results in the end.

We follow the conventional notations of gravitational lensing (e.g. Bartelmann & Schneider 2001). We introduce the angular coordinates ${\theta}=(\theta_{1},\theta_{2})$ on the lens plane, which is perpendicular to the line of sight. For an image of a galaxy or a star, we denote the photon brightness distribution of the image at each position $\theta$ and wavelength $\lambda$ by $I(\theta,\lambda)$. The resulting image observed with a PSF $P(\theta,\lambda)$ in a bandwidth $\Delta\lambda$ is given by \beI^obs(θ)=∫_Δλ dλ  I(θ,λ) * P(θ,λ), \eewhere $*$ denotes convolution, and $I(\theta;\lambda)$ is the brightness of the pre-convolution image, $I(\theta,\lambda)=\lambda S(\theta,\lambda)T(\lambda)$. $S(\theta,\lambda)$ is the SED of source at position $\theta$, and $T(\lambda)$ is the transmission function of the filter. The size of a star image usually is smaller than the pixel scale and can be considered as a delta function before the PSF smearing. Thus the observed star image can be used to estimate the PSF. The observed star image can be written as an integration of PSF at each wavelength and weighted by the star SED $S^{\rm star}(\lambda)$, \beP^star(θ) = ∫dλP(θ,λ) T(λ) S^star(λ) λ, \eewhich is also called the effective PSF in the shear measurements. Henceforth the PSF means effective PSF if we do not mention it. The analogous PSF which smears the galaxy and is weighted by the SED of the galaxy is named ”galaxy PSF” \beP^gal(θ) = ∫dλP(θ,λ) T(λ) S^gal(λ) λ. \ee

Apparently, $P^{\rm gal}$ cannot be measured directly. If the SED of the star has the same shape as that of the galaxy SED, then the PSF estimated from the star image can be used for the shear measurement. Otherwise, the difference between the effective PSF using star SED and that using galaxy SED will introduce a bias in the shear measurements, which is called ’colour bias’. We make following simplifications in our study of colour bias: (1) we ignore the difference of the PSF over the FOV, i.e. at the star position and that at the galaxy position. (2) We define the colour bias by the difference between the two effective PSFs, i.e. $P^{\rm star}$ and $P^{\rm gal}$. The measurement error in the cosmic shear is discussed later for only one particular case. (3) galaxy has a spatially uniform SED, i.e. the morphology of the galaxy is not taken into account in the analysis of colour bias. In the study of colour gradient bias (Sect.4), the spatial variation of SED or the morphology of the galaxy is considered.

Several methods have been proposed and adopted in the shear measurements. First one is based on the brightness moments of the galaxy images,(e.g. Kaiser et al. 1995; Luppino & Kaiser 1997; Hoekstra et al. 1998). Second one makes use of the model fitting of the galaxy shape (e.g. Kuijken 1999; Bernstein & Jarvis 2002; Refregier et al. 2002; Voigt & Bridle 2010). Some disadvantages have been found in these methods(e.g. Zhang & Komatsu 2011; Sheldon & Huff 2017; Mandelbaum 2018), and more new measurement methods have been proposed (e.g. Zhang et al. 2015; Eckert & Bernstein 2019; Springer et al. 2020; Theobald et al. 2021). Each method will suffer different colour and colour gradient biases. To simplify the calculation, we adopt the brightness moment method to quantify the shape of PSF and galaxy throughout this paper. Other kinds of measurement noise or bias are not considered either. Therefore, we define the difference of second order brightness moment between $P^{\rm star}$ and $P^{\rm gal}$ as the colour bias. The second order brightness moment is written as \beQ_ij ≡∫d2θ θiθjW(θ) ∫dλ I(θ;λ) * P(θ;λ)∫d2θ  W(θ)∫dλ I(θ;λ) * P(θ;λ), \eewhere $W(\theta)$ is the weight function to reduce the noise in real measurement, $i,j$ indicates two directions on the sky. $I(\theta;\lambda)$ is the delta function when we calculate the moments of the stellar image. Then the size and the ellipticity of the image can be calculated from

where $R$ represent the size, $\epsilon_{1}$ and $\epsilon_{2}$ are two components of ellipticity $\epsilon=(\epsilon_{1}^{2}+\epsilon_{2}^{2})^{1/2}$. We estimate the colour bias for the size $R$ and the module of the ellipticity $\epsilon$ separately, which is defined as \beb_c≡Rgal-RstarRstar,  for  size  of  PSF;   b_c≡ϵgal-ϵstarϵstar  for  ellipticity  of  PSF, \eewhere the superscript ’gal’ or ’star’ indicates that the quantities are calculated from the effective PSF using the SED of galaxy or star respectively.

Moreover, the shape of the galaxy varies over the wavelength (or the spectral energy distribution of galaxy varies spatially), i.e. the colour gradient. The shape measurement without taking into account such a kind of effect can induce another bias as well, which is called ’colour gradient bias’ (CG bias for short). The multiplicative bias induced by the colour gradient in shear measurement is defined by \beb_CG = ^γi-γiγi, i=1,2, \eewhere the subscript ’$i$’ indicates one of two components of the shear. $\gamma_{i}$ is the true shear and $\hat{\gamma_{i}}$ is the estimate of shear without taking into account of colour gradient. One can find more details on the CG bias inSemboloni et al. (2013); Er et al. (2018).

The SED of a galaxy varies spatially, generating colour gradient. We estimate the shear measurement bias when the colour gradient of the galaxy is ignored, i.e. the CG bias. It has been shown that the CG bias depends on several factors, such as the measurement method, the relative size of the galaxy with respect to the size of the PSF, the pixel scale of the CCD, the transmission function of the filters etc (e.g. Voigt et al. 2012; Semboloni et al. 2013). On top of that is the colour gradient of the galaxy. For example, if there is a great difference in the SED between the bulge and disk of the galaxy, one would expect a significant CG bias. More detailed discussions can be found in Semboloni et al. (2013). We evaluate the CG bias for CSST four filters following a similar approach in Er et al. (2018), and only introduce basic steps here (one can find more details from Fig. 1 in Er et al. (2018)). Following the step in the Figure, one can generate the galaxy images without CG bias from the top flowchart: 1) convolve the image with PSF at each wavelength; 2) integrate the images over the band of wavelength; 3) deconvolve the image by the effective PSF; 4) shear the deconvolved image. In the bottom flowchart, one follows the real image process and can simulate the image with CG. The only difference is that the shear step is at the beginning. In reality, there are convolution with the effective PSF in the end for both top and bottom flow. Since we don’t employ any particular PSF correction method, a direct deconvolution is used, and cancel out the last convolution. Then we use the second-order brightness moment and Gaussian weighting function to measure the shear from these two kinds of images.

We combine a bulge and a disk to mimic the spatial variation of synthetic galaxy SED. Again we use an Ell galaxy SED for the bulge and an Irr galaxy SED for the disk. The image profiles are described by the Sersic profile at each wavelength \beI_s(x,y) = I_0 e ^-κ(θrh)^1/n,  with    θ=x^2/q+y^2q, \eewhere $I_{0}$ is the central intensity, $\kappa=1.9992n-0.3271$, $n$ is the Sersic index, and $q$ is the axis ratio, $r_{h}$ is the half-light radius. We take $n=1.5$ for the bulge and $n=1$ for the disk. The source galaxies are assumed to be circular initially. Two sizes of galaxies are considered in the first test. The size for the big galaxy and the small galaxy is given by the half-light radius of its bulge and disk: $r_{h}^{bulge}=0.17,r_{h}^{disk}=1.2$ arcsec and $r_{h}^{bulge}=0.09,r_{h}^{disk}=0.6$ arcsec respectively. The total flux of synthetic galaxy is normalised at the wavelength of $550$ nm where the bulge and disk contain 25 and 75 percent of the flux respectively. A shear value ($\gamma_{1}=\gamma_{2}=0.05$) is uniformly used in all the tests. The PYTHON-based GalSim package (Rowe et al. 2015) is used to simulate the galaxy image, which has been widely adopted in weak lensing studies (e.g. Hoekstra et al. 2016). The stamp size of each galaxy image is $512\times 512$ pixels with pixel size $0.074$ arcsec. The wavelength sampling interval is $1$ nm. The PSF model is simply simulated by the Airy function, which shows a similar shape to the PSF in the CSST $i$ band (Shen et al. 2022). The profile of Airy disk is expressed as \beP(x)=I0(1-κ2)2 (2J1(x)x-2κJ1(κx)κx2)^2, \eewhere $I_{0}$ is the maximum intensity at the centre, $\kappa$ is the aperture obscuration ratio, and $J_{1}(x)$ is the first kind of Bessel function of order one; $x$ is defined as $x=\frac{\pi\theta}{\lambda{D}}$. We take the CSST aperture diameter $D=2$ m and obscuration $\kappa=0.1$ in our simulation. In this study, we do not consider the evolution of the galaxies over the redshift, i.e. we use the same angular half-light radius of the bulge and the disk for galaxies at different redshifts. But the size of synthetic galaxies vary due to the SEDs, i.e. the ratio of the bulge-to-disk can change with redshift. The spatial variation of CG bias over the FOV has not been taken into account.

We calculate the “shear” from the simulated images, and obtain the bias $b_{CG}$. In Fig. 9 and Fig. 10, we show the $b_{CG}$ vs. redshift in each band with step $\Delta z=0.1$. Overall, the bias is small, especially in the big galaxy ( $<1.8\times 10^{-4}$ ). It is slightly larger in the small galaxy ( $<1.7\times 10^{-3}$ ). For both galaxies, the bias decreases with redshift, which agrees with the previous study Er et al. (2018). The two peaks in the CG bias curves correspond to the two strong emission lines of Irr SED. We estimate the colour gradient of our synthetic galaxy for comparison. In order to calculate the ’colour’, we split each band into two at the central wavelength of the bandwidth. The flux ratio between the two sub-band of the galaxy image is taken as the colour. The colour difference between the central region ($r<r^{bulge}_{h}$) and outer region ($r^{bulge}_{h}<r<r^{disk}_{h}$) of the galaxy image is defined as the colour gradient. We calculate the colour gradient as a function of the redshift and find this relation shows a similar trend as that of CG bias.

In the $i$ band and $z$ band, there are unusually large biases in the case of a small galaxy at low redshift. The main reason is that the relative size of the galaxy with respect to that of PSF becomes small and critical. From the relative strength of SED between the Ell and Irr galaxies (Fig. 1), one can see that the synthetic galaxy is bulge dominated and compacted at low redshift, and becomes disk dominated and extended at high redshift. In additional tests with smaller PSF sizes, the CG bias shows similar behaviour as that using large galaxies.

Moreover, we take different combinations between bulge size and disk size into account. Only CSST $i$ band is performed in this part. Fig. 11 shows the relation of CG bias with the size of two components of the synthetic galaxy. We use the bulge size from $0.17$ arcsec to $0.09$ arcsec and the disk size from $1.2$ arcsec to $0.6$ arcsec in each panel. Three redshifts ($z=0.0,1.0,2.0$) are used here. We can see that the CG bias is more sensitive to the bulge size than the disk size. There is rapid growth when the galaxy becomes small.

Weak gravitational lensing is one of the most powerful tools in modern cosmology. The next generation weak lensing surveys, such as $Euclid$, CSST, LSST and WFIRST, will measure the weak lensing signal with unprecedented precision. The shape measurements benefit from the compact diffraction-limited PSF of the space-borne telescope. Thus small instrumental effects can become important and need to be carefully investigated. One of the effects is the wavelength dependence of the PSF. In this paper, we study two biases due to the chromatic PSF in the CSST weak lensing measurement. The first one is the colour bias. The difference between the star SED and galaxy SED leads to the difference between PSF obtained from star images and the PSF that smears the galaxy. The other one is the colour gradient bias. It arises due to the spatial variations in the colours of galaxies. In our simulations, we find both of these two biases are small in most cases of the weak lensing measurement in the four optical filters of CSST.

For the colour bias, two types of SEDs are taken as reference for the galaxy. One is an irregular galaxy, and the other one is an elliptical galaxy. Three types of SEDs of the star (O5 V, G5 II, K2 I) and two types of SEDs of QSO (type I and type II) are used since QSO has point like image and can be misclassified as the star. We study the colour bias of PSF size and ellipticity for the CSST. We find that the colour bias of size is one order of magnitude smaller than that of ellipticity in general. The order of magnitude distributions of the colour bias of ellipticity show similarity in four bands, and cover a wide range. One of the reasons is that the ellipticity of the PSF is sensitive to the position on the field of view. While the distribution of the colour bias of size is centrally distributed in all bands and has widely separated among the filters. Two other properties can be found: 1) in the shorter wavelength filter, the colour bias will be larger; 2) the strong emission line of the source galaxy has a big impact on the colour bias.

We study the dependence of colour bias on some other factors. By generating a synthetic galaxy SED library, we find that there is a positive correlation between the colour bias of size and the colour of the galaxy. While the relation between the bias of ellipticity and galaxy colour is not clear. In the test of colour bias for the galaxy at different redshifts, we do not find a large difference between the elliptical galaxy and the irregular galaxy. We also calculate the shear measurement error due to colour bias, and find in all four bands, the shear measurement errors are small ($<1\%$). In the redshift combination of galaxy and QSO, we find the colour bias is significantly affected by the strong emission lines in the SED of galaxy and QSO. We perform calibration to the colour bias by reconstructing the SED of the galaxy. We use the brightness of two neighbouring bands and linearly interpolate the SED of the galaxy. The PSF based on the reconstructed SED provides a better estimate of PSF which is close to one that smears the galaxy.

For the colour gradient bias, only the multiplicative bias is studied. We adopt the Airy profile to simulate the PSF and estimate the CG bias at different redshifts for four CSST bands. For galaxies at different redshifts, we only consider the changing of the SED, but the evolution of the galaxies. The CG bias is also small $\sim 10^{-4}$ and decreases with redshift in general. Similar to that in the colour bias, a strong emission line of the source galaxy can increase the CG bias. When the size of the galaxy becomes small, the CG bias also increases rapidly. Particular in the CSST $i$ band and $z$ band, the bias can be up to $1.7\times 10^{-3}$ and $1.2\times 10^{-3}$ respectively. The main reason is that the ratio between the PSF size and galaxy size is relatively large in these bands. Our results show that the CG bias in CSST shear measurement is a subdominant effect for relatively big galaxies, but for small galaxies, in CSST $i$ and $z$ band CG bias can cause non-negligible systematic bias in the Stage-IV weak lensing survey.

In this study, we show that the chromatic effect in PSF can be small in most cases of shear measurements. However, some exceptions one has to be careful of. For example, the misclassification of QSO as the star. Measuring shear from small-size galaxies. Moreover, there are some simplifications in our study. First, in our simulation, we only consider the optics of the CSST. Other instrumental effects, such as those from the detector need to be included as well. The linear reconstruction method to calibrate the colour bias needs to be processed for every galaxy, which will cost massive computations. The relation between the colour and the colour bias can provide a neat estimate of the bias. The evaluation of the colour gradient bias in this paper is based on the Airy profile. An estimate with realistic noise and calibration to other wider band filters (Liu et al. 2023) will be important to the weak lensing measurement as well.