Stability Analysis of Four $f(Q)$ Gravity Models : A Cosmological Review in the Background of Bianchi-I Anisotropy

TL;DR

This paper investigates the stability of four representative f(Q) gravity models in a Bianchi-I anisotropic cosmology to understand their ability to realize a transition from early inflation to late-time acceleration while isotropizing the universe. Employing a dynamical-systems framework with dimensionless variables $x$, $y$, and $z$ (satisfying $x+y+z=1$) and the connection function $oldsymbol{ extGamma}(Q)=rac{f_Q}{Qf_{QQ}}$, the authors derive autonomous equations and perform fixed-point analyses, including centre-manifold reductions for non-hyperbolic points. They assess viability through tensor and scalar perturbations via $c_T^2$ and $c_s^2$, and summarize results for four models: a power-law, an exponential, a quadratic-plus-log, and a mixed square-root-log form; the exponential model emerges as the most robust across stability and perturbative criteria, while the others require parameter regions close to General Relativity to maintain viability. The study demonstrates that purely geometric non-metricity-driven dynamics in anisotropic backgrounds can yield de Sitter attractors and natural isotropization, offering a framework for testing$f(Q)$ theories against both background evolution and perturbative consistency. These insights guide future work toward observationally viable anisotropic cosmologies within symmetric teleparallel gravity and motivate more complete anisotropic perturbation analyses.

Abstract

With the non-metricity scalar $Q$ as the functional argument, several $f(Q)$ gravity models are found to be proposed which are perfectly able to mimic the late-time accelerated expansion as pointed out by the type Ia supernovae observations. Temperature fluctuation differences for two celestial hemispheres, Hubble tension, voids, dipole modulation, anisotropic inflation, etc. motivates us to think beyond the $Λ$CDM model and the cosmological principle. Bianchi-I model portrays an anisotropic universe imposing shear. $f(Q)$ model also enables us to produce early inflation to late de Sitter universe without the requirement of $Λ$CDM. Ambiguities regarding fine-tuning or coincidences can be avoided alongwith. So, this article finds different stationary points of cosmic evolution with $f(Q)$ models habilitating in Bianchi-I anisotropic universe. Depending on models' nature, fixed points with different categories are found. Perturbations are followed wherever are applicable. While pursuing cosmological implications towards these fixed points, some are found to be formed only for the consideration of $f(Q)$ gravity and Bianchi-I both. Besides different prediction towards early inflation to late-time expansion which are available in existing literature of dynamical system studies, occurances of ultra slow roll inflation is predicted. For particular $f(Q)$ model, shear is predicted to decay leaving behind a constant valued residue. This models a universe that gradually turns more homogeneous. In some other models, depending on initial conditions, a final isotropic leftover is marked as the future fate of anisotropic world. More than one stable points are marked for special cases and are cosmologically interpreted.

Figures (4)

• Figure 1: Figure $(a)-(d)$ are $z$ vs $x$ phase diagrams for the model $f(Q) = mQ^n$ in Bianchi-I cosmology at different values of $n$ and $\omega$. A red dot indicates a stable point, a green dot indicates a saddle point and a blue dot indicates an unstable point.
• Figure 2: Figure $(a)-(d)$ are $z$ vs $x$ phase diagrams for the model $f(Q)= exp \{nQ\}$ at different values of $\omega$. A red dot indicates a stable point, a green dot indicates a saddle point and a blue dot indicates an unstable point.
• Figure 3: Figure $(a)-(d)$ are $z$ vs $x$ phase diagrams for the model $f(Q)=\alpha Q^2+ v Q^2 \log (Q)$ at different values of $\omega$. A red dot indicates a stable point, a green dot indicates a saddle point and a blue dot indicates an unstable point.
• Figure 4: Figure $(a)-(d)$ are $z$ vs $x$ phase diagrams for the model $f(Q)=\eta Q_0 \sqrt{\frac{Q}{Q_0}} \log\left(\frac{\lambda Q_0}{Q}\right)$ at different values of $\omega$. A red dot indicates a stable point, a green dot indicates a saddle point and a blue dot indicates an unstable point.