Effective reheating in Gauss–Bonnet inflation with $\mu (\phi ,X)$ coupling

We study effective reheating in a scalar–Gauss–Bonnet inflationary model with a phase-space-dependent coupling $$\mu (\phi ,X),$$ μ ( ϕ , X ) , in which a compact field-space feature is combined with a bounded kinetic gate. The modified inflationary background determines the pivot-scale quantities and the effective energy density at the end of inflation. These quantities are then used to derive the reheating duration $$N_{\textrm{re}}$$ N re and temperature $$T_{\textrm{re}}$$ T re through the thermal-history matching relation. We first perform two fixed-pivot reference scans by varying the overall Gauss–Bonnet strength $$\lambda _{\textrm{GB}}$$ λ GB and the kinetic-gate amplitude $$g_{_X}$$ g X separately. For the reference parameter choices, increasing either parameter increases $$N_{\textrm{re}}$$ N re and decreases $$T_{\textrm{re}}$$ T re for the selected reheating equations of state. Additional benchmark calculations clarify how these variations depend on the dynamical regime of the model. In the $$\lambda _{\textrm{GB}}$$ λ GB scan, the increase in $$N_{\textrm{re}}$$ N re and the decrease in $$T_{\textrm{re}}$$ T re persist, although both variations become strongly suppressed when the coupling is more localized or when the end of inflation is controlled more strongly by the E-model potential. In the $$g_{_X}$$ g X scan, stronger field-space localization and kinetic saturation can instead lead to a slight decrease in $$N_{\textrm{re}}$$ N re and an increase in $$T_{\textrm{re}}$$ T re as $$g_{_X}$$ g X is increased. When the bounded kinetic contribution is considered together with a weaker overall Gauss–Bonnet interaction, the resulting changes in the reheating quantities become nearly negligible. The fixed-pivot predictions of the representative and alternative benchmarks are compared with CMB constraints. These reheating constraints are then discussed for four representative values of the effective equation-of-state parameter, $$\overline{w}_{\textrm{re}}=-1/3,0,2/3,$$ w ¯ re = - 1 / 3 , 0 , 2 / 3 , and 1. Finally, the analysis is generalized to a time-dependent reheating equation of state $$w_{\textrm{re}}(N),$$ w re ( N ) , showing that the thermal-history matching relations depend only on its e-fold average $$\overline{w}_{\textrm{re}},$$ w ¯ re , while the detailed evolution of $$w_{\textrm{re}}(N)$$ w re ( N ) remains degenerate within the effective framework adopted here.