Past and future isotropization in f ( R ) dark energy models

We present a dynamical system analysis for the homogeneous and anisotropic Bianchi I cosmology in the context of Hu-Sawicki $f(R)$ gravity with parameters ${n,{C}_{1}}={1,1}$. Deriving the corresponding four-dimensional dynamical system from the field equations allows us to find the fixed points of the system in both the vacuum and matter cases alongside their stability. This approach also provides analytical expressions for both the cosmic expansion and shear of each fixed point. More importantly, it allows us to address one of the least satisfying aspects of the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ picture: the apparent fine-tuning of the initial state. We show that, for the chosen parameters, Hu-Sawicki gravity allows for isotropization both in the past and in the future for both vacuum and matter, demonstrating that Hu-Sawicki gravity may alleviate the anisotropy-related initial-condition problem for a subsequent inflationary phase. For the matter case, the analysis was subdivided into the dust and radiation cases, and we show that both cases exhibit past and future isotropization. The obtained results also revealed interesting aspects of the phase space such as the existence and phenomenology of lines and two-dimensional sheets of fixed points.