Observational test for $f(Q)$ gravity with weak gravitational lensing

Abstract: In this article we confront a class of $f(Q)$ gravity models with observational data of galaxy-galaxy lensing. Specifically, we consider the $f(Q)$ gravity models containing a small quadratic correction when compared with General Relativity (GR), and quantify this correction by a model parameter $\alpha$. To derive the observational constraints, we start by extracting the spherically symmetric solutions which correspond to the deviations from the Schwarzschild solution that depends on the model parameter in a two-fold way, i.e., a renormalized mass and a new term proportional to $r^{-2}$. Then, we calculate the effective lensing potential, the deflection angle, the shear component, and the effective Excess Surface Density (ESD) profile. After that, we employ the group catalog and shape catalog from the SDSS DR7 for the lens and source samples respectively. Moreover, we handle the off-center radius as a free parameter and constrain it using the MCMC. Concerning the deviation parameter from GR we derive $\alpha=1.202^{+0.277}_{-0.179}\times 10^{-6} {\rm Mpc}^{-2}$ at 1 $\sigma$ confidence level, and then compare the fitting efficiency with the standard $\Lambda$CDM paradigm by applying the AIC and BIC information criteria. Our results indicate that the $f(Q)$ corrections alongside off-center effects yield a scenario that is slightly favored.