Toward gravitational lensing in modified theories of gravity

In this study, we investigate gravitational lensing within modified gravity frameworks, focusing on the Hu–Sawicki $f\left(R\right)$ and normal branch Dvali–Gabadadze–Porrati (nDGP) models, and we compare these results with those obtained from general relativity (GR). Our results reveal that both modified gravity models consistently enhance key lensing parameters relative to GR, including the Einstein radius, lensing optical depth, and time delays. Notably, we find that the Hu–Sawicki $f\left(R\right)$ and nDGP models yield significantly larger Einstein radii and higher lensing probabilities, especially at greater redshifts, indicating an increased likelihood of lensing events under modified gravity. Our analysis of time delays further shows that the broader mass distributions in these frameworks lead to pronounced differences in high-mass lens systems, providing potential observational markers of modified gravity. Additionally, we observe amplified magnification factors in wave optics regimes, highlighting the potential for gravitational wave (GW) lensing to differentiate modified gravity effects from GR predictions. Through these findings, we propose modified gravity theories as compelling alternatives to GR in explaining cosmic phenomena, with promising implications for future high-precision gravitational lensing surveys.