CHAPTER 8: Anisotropic Model in Scale-Dependent Gravity

Abstract

We investigate the cosmological dynamics of a scale-dependent (SD) gravitational framework in which both the Newton couplingGk and the cosmological termΛk evolve with cosmic time. The analysis is carried out in a spatially homogeneous but anisotropic Bianchi I Universe, allowing us to probe the interplay between anisotropic expansion and running gravitational couplings. By employing the dimensionless variables h = H/H0, g = G/G0, and the density parameters Ωm, Ωr, and Ωσ, the modified field equations are recast into a closed system of firstorder differential equations in redshift space and numerically integrated using a fourth-order Runge–Kutta method with initial conditions qualitatively consistent with current cosmological constraints. The obtained solutions reproduce the expected early-time behavior dominated by matter and radiation while yielding a late-time accelerating expansion driven by the dynamical running of Gk and Λk. The effective equation of state approaches a quintessence-like regime with weff(0) ≃ −0.70, and the present deceleration parameter q0 ≈ −0.60 lies within the range inferred from observational data. The negative slope of the Om(z) diagnostic further confirms that the model behaves as a quintessence-like dark-energy scenario rather than a cosmological-constant model. Altogether, the results demonstrate that scale-dependent gravitational couplings in an anisotropic background not only reproduce the essential phenomenology of ΛCDM at low redshift, but also introduce controlled and observationally admissible deviations that provide a compelling alternative description of late-time cosmic acceleration.