Testing Cotton gravity as dark matter substitute with weak lensing

Harada proposed a modified theory of gravity called Cotton gravity, and argued that it successfully explains the rotation curves of 84 galaxies without the need for dark matter. In this work, we use the galaxy-galaxy lensing technique to test whether the modification effect of Cotton gravity can indeed be a viable substitute for dark matter. Using the spherically symmetric solution of Cotton gravity, we obtain the deflection angle via the Gauss-Bonnet theorem and the weak lensing shear. We use five galaxy catalogs divided into 5 stellar mass bins from the Sloan Digital Sky Survey Data Release 7 (SDSS DR7), each further divided into blue star-forming galaxy and red passive galaxy sub-catalogs. We find that Cotton gravity on its own has a significant deviation from the measured galaxy-galaxy lensing signals, thus it cannot replace the role of dark matter. If we consider the combination of dark matter and Cotton gravity, the modification is tightly constrained. Our analysis also applies to other modified gravity theories whose an additional linear term appears in the Schwarzschild solution.