Running gravitational constant induced dark energy as a solution to $\sigma_8$ tension

We consider a modified gravity model with a running gravitational constant coupled to a varying dark energy fluid and test its imprint on the growth of structure in the universe. Using Redshift Space Distortion (RSD) measurement results, we show a tension at the $3 \sigma$ level between the best fit $\Lambda$CDM and the corresponding parameters obtained from the Planck data. Unlike many modified gravity-based solutions that overlook scale dependence and model-specific background evolution, we study this problem in the broadest possible context by incorporating both factors into our investigation. We performed a full perturbation analysis to demonstrate a scale dependence in the growth equation. Fixing the scale to $k = 0.1 h$ Mpc$^{-1}$ and introducing a phenomenological functional form for the varying Newton coupling $G$ with only one free parameter, we conduct a likelihood analysis of the RSD selected data. The analysis reveals that the model can bring the tension level within $1 \sigma$ while maintaining the deviation of $G$ from Newton's gravitational constant at the fifth order.