Running gravitational constant induced dark energy as a solution to $$\sigma _8$$ tension

Abstract 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 $$3 \sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> level between the best fit $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> 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 $$k = 0.1 h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.1</mml:mn> <mml:mi>h</mml:mi> </mml:mrow> </mml:math> Mpc $$^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mmultiscripts> </mml:math> 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 $$1 \sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> while maintaining the deviation of G from Newton’s gravitational constant at the fifth order.