Neglecting the second order corrections in weak lensing measurements can lead to a few percent uncertainties on cosmic shears, and becomes more important for cluster lensing mass reconstructions. Existing methods which claim to measure the reduced shears are not necessarily accurate to the second order when a point spread function (PSF) is present. We show that the method of Zhang (2008) exactly measures the reduced shears at the second order level in the presence of PSF. A simple theorem is provided for further confirming our calculation, and for judging the accuracy of any shear measurement method at the second order based on its properties at the first order. The method of Zhang (2008) is well defined mathematically. It does not require assumptions on the morphologies of galaxies and the PSF. To reach a sub-percent level accuracy, the CCD pixel size is required to be not larger than 1/3 of the Full Width at Half Maximum (FWHM) of the PSF. Using a large ensemble (> 10^7) of mock galaxies of unrestricted morphologies, we find that contaminations to the shear signals from the noise of background photons can be removed in a well defined way because they are not correlated with the source shapes. The residual shear measurement errors due to background noise are consistent with zero at the sub-percent level even when the amplitude of such noise reaches about 1/10 of the source flux within the half-light radius of the source. This limit can in principle be extended further with a larger galaxy ensemble in our simulations. On the other hand, the source Poisson noise remains to be a cause of systematic errors. For a sub-percent level accuracy, our method requires the amplitude of the source Poisson noise to be less than 1/80 ~ 1/100 of the source flux within the half-light radius of the source, corresponding to collecting roughly 10^4 source photons.
Hyper Suprime-Cam Y3 results: photo-$z$ bias calibration with lensing shear ratios and cosmological constraints from cosmic shear
Leiden Observatory, Leiden University（莱顿天文台，莱顿大学）；Inter University Centre for Astronomy and Astrophysics（大学天文学与天体物理中心）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo（东京大学 Kavli宇宙物理学与数学研究所（WPI））；Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI), Nagoya University（名古屋大学 Kobayashi-Maskawa粒子宇宙起源研究所（KMI））；Institute for Advanced Research, Nagoya University（名古屋大学高等研究院）；Department of Physics and Astronomy, University of Pennsylvania（宾夕法尼亚大学物理与天文学系）；Department of Physics and Astronomy and PITT PACC, University of Pittsburgh（匹兹堡大学物理与天文学系及PITT PACC）；National Astronomical Observatory of Japan (NAOJ), National Institutes of Natural Science（日本国立天文台（NAOJ），国立自然科学研究院）；The Institute of Statistical Mathematics（统计科学研究所）
