Bianchi-I cosmology with scale dependent $G$ and $\Lambda$ in asymptotically safe gravity

We study anisotropic Bianchi-I cosmology, incorporating quantum gravitational corrections into the Einstein equation through the scale-dependent Newton coupling and cosmological term, as determined by the flow equation of the effective action for gravity. For the classical cosmological constant $\Lambda_0=0$, we derive the quantum mechanically corrected, or quantum-improved power-series solution for a general equation-of-state parameter $w$ in the range $-1<w\leq 1$ in the form of expansions in both inverse cosmic time and the anisotropy parameter. We give a general criterion, valid for any $\Lambda_0$, if the solution becomes isotropic in the late time, which indicates that the universe becomes isotropic for most cases of $-1<w<1$ except $w=1$. By numerical analysis, we show that quantum corrections lead to earlier isotropization compared to the classical case starting from an initially highly anisotropic state. In contrast, for $\Lambda_0 >0$, we obtain the inverse power-series solution in the exponential of the cosmic time. We find that the universe always becomes isotropic in the late time, in accordance with the cosmic no hair theorem, and the quantum corrections make the isotropization faster. We also briefly summarize the Kasner solution and its generalization with quantum corrections.