Gauss–Bonnet effects in $$f(R,\Sigma ,T)$$ gravity

Abstract We investigate the cosmological dynamics of Parameterized Absolute Parallelism (PAP) geometry within the framework of modified $$f(R,\Sigma ,T)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Σ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> gravity. By generalizing the Gauss–Bonnet invariant $${\mathcal {G}}_b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> using the PAP curvature tensor $$B^{\alpha }{}_{\mu \nu \sigma },$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>B</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:mmultiscripts> <mml:mrow/> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> <mml:mi>σ</mml:mi> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> we incorporate both Riemannian curvature and torsion effects into a unified framework. The modified field equations are derived for a spatially flat FLRW background, and exact analytical solutions for the Hubble parameter, scale factor, energy density, and pressure are obtained under a constant equation-of-state parameter $$\omega .$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Our analysis shows that the PAP deformation parameter b and matter–torsion coupling $$\eta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> significantly modify cosmic expansion, the effective equation of state, and curvature diagnostics. Furthermore, we explore the role of $${\mathcal {G}}_b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> in torsion-driven Gauss–Bonnet inflation and demonstrate how PAP corrections alter the slow-roll dynamics, enabling sustained inflation even for flat potentials. A detailed stability analysis reveals the critical regions in the $$(b,\eta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>η</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> parameter space separating stable and unstable cosmological phases. This unified approach recovers General Relativity, teleparallel gravity, and minimally coupled PAP models as limiting cases, offering new insights into early- and late-time acceleration driven by torsion and higher-curvature effects.