Scalar-Tensor Symmetric Teleparallel Gravity: Reconstruct the Cosmological History with a Steep Potential

TL;DR

The paper develops a dynamical-systems analysis of a scalar–nonmetricity gravity model with a steep potential $U(\phi)$ and a power-law coupling $f(\phi)$ across three symmetric teleparallel connection branches in a flat FLRW setting. By recasting the field equations into autonomous systems and applying center-manifold theory to non-hyperbolic points, it identifies finite and infinite (asymptotic) critical points corresponding to matter-dominated, stiff-fluid, and de Sitter epochs, as well as Big Rip/Big Crunch singularities. The results show that, under specific parametric conditions (e.g., $m\ge3$, $n>1$, appropriate $h_0$), ghost-free de Sitter attractors can unify early and late cosmic acceleration across all connections, though a full observational viability assessment remains to be done. The work also highlights the sensitivity of late-time dynamics to the coupling power and potential steepness, and it outlines directions for future study, including conformal-frame considerations.

Abstract

Within the framework of scalar-non-metricity gravity, we introduce a steep potential together with a power-law coupling function and investigate whether the acceleration phases of the universe can be consistently described by this model. In the symmetric teleparallel formulation, and under a Friedmann--Lemaître--Robertson--Walker background, three distinct branches of the connection arise, leading to three different cosmological scenarios. We perform a detailed dynamical analysis of these models by examining the phase space and determining the asymptotic cosmological solutions. The analysis reveals a rich hierarchy of critical points, including matter-dominated epochs, kinetic-dominated stiff-fluid regimes, and steep potential-dominated de Sitter solutions, along with asymptotic trajectories that approach Big Crunch or Big Rip singularities, as well as transient, unstable matter-dominated eras. The stability of the steep potential-dominated de Sitter points is further studied using Center Manifold Theory, showing that, under specific parametric conditions, the model can provide a unified description of both the early and late-time acceleration phases of the universe.

Figures (6)

• Figure 1:
• Figure 2: Qualitative evolution of the deceleration parameter of the dynamical system (\ref{['Xeq_conl']})-(\ref{['Zetaeq_conl']}) for different values of $n$ and $m$, with initial conditions ($X[0]=0.8,~Y[0]=0.3 ,~\lambda[0]=0.9,~\zeta[0]=0.4$) with parameters values are $h_0=0.5,~\beta=2,~\alpha=1$.
• Figure 3: 3D phase portraits for the dynamical system given in Eqs. (\ref{['Xeq_conll']})-(\ref{['Zetaeq_conll']}).
• Figure 4: Qualitative evolution of the deceleration parameter of the dynamical system (\ref{['Xeq_conll']})-(\ref{['Zetaeq_conll']}) for different values of $n$ and $m$, with initial conditions ($X[0]=0.2,~Y[0]=0.7,~Z[0]=0.1 ,~\lambda[0]=0.9,~\zeta[0]=0.1$) with parameters values are $h_0=0.5,~\beta=2,~\alpha=1$.
• Figure 5: 3D phase portraits for the dynamical system given in Eqs. (\ref{['Xeq_conlll']})-(\ref{['Zetaeq_conlll']}).