Bianchi-I cosmology with scale dependent $G$ and $Λ$ in asymptotically safe gravity
Chiang-Mei Chen, Akihiro Ishibashi, Rituparna Mandal, Nobuyoshi Ohta
TL;DR
This work embeds quantum gravity effects from asymptotically safe gravity into the Einstein equations for a Bianchi-I universe by promoting $G$ and $\Lambda$ to scale-dependent functions $G(k)$ and $\Lambda(k)$ and identifying the RG scale with cosmological quantities. It derives quantum-improved BI dynamics and produces both analytical power-series and numerical solutions for the volume evolution across the two FRG branches: $\Lambda_0=0$ and $\Lambda_0>0$. The key findings are that quantum corrections accelerate isotropization relative to the classical BI model for $\Lambda_0=0$, while for $\Lambda_0>0$ the universe always isotropizes, consistent with the cosmic no-hair theorem, with corrections enhancing the approach to de Sitter. These results extend Kasner-type analyses to the FRG-improved regime and offer insights into late-time anisotropy, potential observational signatures, and the robustness of isotropization in quantum-corrected cosmologies.
Abstract
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 $Λ_0=0$, 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 $Λ_0$, 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 $Λ_0 >0$, 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.