Inflationary dynamics in combined f(φ )T and Gauss-Bonnet gravity with three representative coupling schemes

We analyze inflationary dynamics in combined \(f(φ )T\) and Gauss-Bonnet gravity, considering three representative coupling schemes: (i) exponential \(V(φ )=V_0 e-λ φ\) , \(f(φ )=f_0 e-λ φ\) , and linear \(ξ (φ )=ξ _0φ \) ; (ii) power-law \(V(φ )=V_0φ ^2\) , inverse power-law \(ξ (φ )=ξ _0/φ ^2\) , and constant \(f(φ )=f_0\) ; (iii) dilaton-like \(V(φ )=V_0(1-φ )⁻²\) , \(f(φ )=f_0(1-φ )⁻¹\) , \(ξ (φ )=ξ _0(1-φ )^2\) . For each case we derive slow-roll parameters and compute the scalar spectral index \(n_s\) and tensor-to-scalar ratio r , comparing with Planck 2018+BK18 ( \(n_s=0.9665± 0.0038\) , \(r<0.065\) at \(95%\) CL) and ACT DR6 ( \(n_s=0.969± 0.004\) , \(r<0.021\) at \(95%\) CL). Case 1 yields \(r~ 10⁻⁴\) – \(10⁻³\) with \(n_s\) within Planck \(68%\) CL for \(Y=β f_0~ 8.3\) –8.7. Case 2 gives \(r~ 0.03\) –0.07, consistent with Planck+BK18 but disfavored by ACT DR6. Case 3 produces \(r~ 10⁻⁴\) with \(n_s\) matching Planck central values for \(X≡ ξ _0 V_0~ 0.20\) . This comparative analysis provides testable signatures for future CMB experiments including LiteBIRD and CMB-S4.