We use galaxy-galaxy weak lensing data to perform a novel test on general relativity (GR) and $f(T)$ torsional gravity. In particular, we impose strong constraints using the torsional (teleparallel) formulation of gravity in which the deviation from GR is quantified by a single parameter $\alpha$, an approximation which is always valid at low-redshift Universe and weak gravitational fields. We calculate the difference in the deflection angle and eventually derive the modified excess surface density profile, which is mainly affected at small scales. Confronting the predictions with weak lensing data from the Sloan Digital Sky Survey Data Release 7, we obtain the upper bound on the deviation parameter, which, expressed via the dimensionless percentage in the Universe energy content, reads as ${\log }_{10}{\mathrm{\Omega }}_{\alpha }\le {-18.52}_{-0.42}^{+0.80}(\mathrm{stat}{)}_{-0.37}^{+1.50}(\mathrm{sys})[R_c/0.015R_{200}]$ with systematics mainly arising from the modeling that includes astrophysics effects, upon a reasonable choice of cutoff radius for $f(T)$ gravity in form of $T+\alpha T^2$. To our knowledge, this is the first time that GR is verified at such an accuracy at the corresponding scales.

• Received 7 August 2019
• Revised 17 October 2019
• Accepted 14 October 2020

DOI:https://doi.org/10.1103/PhysRevD.102.104044