Dynamical system and statefinder analysis of cosmological models in f(T, B) gravity

TL;DR

The paper investigates cosmic acceleration within $f(T,B)$ gravity in a flat FLRW background using a dynamical-systems framework and statefinder diagnostics. It analyzes two prototypical forms, a multiplicative $f(T,B)=c_1 T^{\alpha} B^{\beta}$ and an additive $f(T,B)=c_2 T^{\alpha}+c_3 B^{\beta}$, and shows both yield de Sitter attractors as late-time solutions, with distinct transitional behavior: the multiplicative case follows a Chaplygin-gas-like path to a $\Λ$CDM-like state, while the additive case exhibits richer dynamics including damped oscillations and potential spiral approaches to de Sitter. Statefinder trajectories in the $r$-$q$ and $r$-$s$ planes discriminate between the two couplings and from $\Λ$CDM, revealing observationally testable fingerprints of the underlying geometric construction. Overall, $f(T,B)$ gravity provides a consistent geometric framework for late-time acceleration with diagnostic features that can help distinguish it from standard dark energy scenarios.

Abstract

This study systematically investigates the cosmological dynamics of two well-motivated functional forms in $f(T,B)$ gravity within a flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe. Here $T$ denotes the torsion scalar and $B$ the boundary term, with the special choice $f(T,B) = - T + B$ recovering General Relativity. We focus on a multiplicative power-law model $f(T,B) = c_1 T^αB^β$ and an additive mixed power-law model $f(T,B) = c_2 T^α+ c_3 B^β$. Using dynamical system techniques, we construct autonomous systems and identify de Sitter attractors that naturally explain late-time cosmic acceleration. Analytical stability conditions for these fixed points are derived, and numerical simulations reveal characteristic evolutionary patterns, such as spiral trajectories and damped oscillations in the additive mixed power-law model. Furthermore, statefinder diagnostics are applied to quantitatively distinguish these models from the standard $Λ$CDM paradigm and other dark energy scenarios. The results indicate that $f(T,B)$ gravity offers a theoretically consistent and observationally distinguishable geometric framework for explaining cosmic acceleration, presenting a compelling alternative to conventional dark energy models.

Figures (6)

• Figure 1: Phase space flow of the model $f(T, B)=c_1 T^\alpha B^\beta$ for $(\alpha_1, \beta_1)=(3,-4)$.
• Figure 2: Evolution of cosmological parameters for $(\alpha_1, \beta_1)=(3,-4)$ and initial conditions $(\Omega_{m_0}, x_0, y_0)=(0.3, 0.3, 2.5)$ .
• Figure 3: Phase space flow of the mixed power-law model under different parameters: (a), (c): $(\alpha_2, \beta_2, v_2)=(5,-1000, 0.00125)$; (b): $(\alpha_3, \beta_3, v_3)=(5, 2/3, 1)$; (d): $(\alpha_3, \beta_3)=(5, 2/3)$.
• Figure 4: Evolution of cosmological parameters for the mixed power-law model from initial conditions $(\Omega_{m_0}, u_0, y_0, v_0)=(0.3, 1, 2.5, 1)$. (a): $(\alpha_2, \beta_2)=(5,-1000)$; (b): $(\alpha_3, \beta_3)=(5,2/3)$.
• Figure 5: Evolution of statefinder parameters in the $r$-$q$ and $r$-$s$ planes for the model $f(T, B) = c_1 T^\alpha B^\beta$, with initial values $(\Omega_{m_0}, x_0, y_0) = (0.3, 0.3, 2.5)$ and parameter choice $(\alpha_1, \beta_1) = (3, -4)$.