Running gravitational constant induced dark energy as a solution to σ 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 σ 3 σ level between the best fit Λ Λ 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 $$k = 0.1 h$$ k = 0.1 h Mpc ⁻¹ - 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 σ 1 σ while maintaining the deviation of G from Newton’s gravitational constant at the fifth order.