A 5D Scalar Field Integrated with General Relativity and Photonic Contributions: A Dark Matter-Free Explanation for Gravitational Lensing and Cosmological Observations

We propose and extend a 5-dimensional (5D) scalar field model integrated with General Relativity (GR) to explain gravitational lensing and cosmological observations without invoking dark matter. The scalar field $\stackrel{？}{\tau }\left(x^{\mu },\chi \right)$ , with $\chi \in S^1$ as a compactified extra dimension, generates a 4D time-delay potential ${\tau }_0\left(x\right)$ that contributes to spacetime curvature alongside baryonic matter. We further incorporate an explicit photonic term, ${\tau }_{\gamma }$ , to the time-delay field $\tau \left(x,t\right)$ , accounting for the energy-momentum carried by photons, which produces an additional observable delay component. By combining real (baryonic), relativistic (scalar field), and photonic mass contributions, we derive a deflection angle $\alpha =\frac{4GM}{c^{2}b}$ , consistent with GR, using Fermat’s principle. The model reproduces the angular power spectrum $C_{？}^{\left(\tau \right)}$ , matching DES Y3 weak lensing data within $<0.01\sigma$ for $30<？<3000$ . Preliminary tests against strong lensing (CLASH), galaxy