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.