Evaluating a sigmoid dark energy model to explain the Hubble tension

In this study, we analyze Type Ia supernovae (SNe Ia) data sourced from the Pantheon+ compilation to investigate late-time physics effects influencing the expansion history, $H\left(z\right)$, at redshifts . Our focus centers on a time-varying dark energy (DE) model that introduces a rapid transition in the equation of state, at a specific redshift, $z_a$, from the baseline, $w_{\Lambda }=-1$, value to the present value, $w_0$. The change in the equation of state is implemented as a transition in the DE density scale factor driven by a sigmoid function. The constraints obtained for the DE sigmoid phenomenological parametrization have broad applicability for dynamic DE models that invoke late-time physics. Our analysis indicates that the sigmoid model provides a slightly better, though not statistically significant, fit to the SNe Pantheon+ data compared to the standard $\Lambda$ cold dark matter ($\mathrm{\Lambda CDM}$) model. The fit results, assuming a flat geometry and maintaining ${\Omega }_m$ constant at the 2018-Planck value of $0.3153$, are as follows: $H_0{=73.3}_{-0.6}^{+0.2}$ km s${\hspace{0.17em}}^{-1}$ Mpc${\hspace{0.17em}}^{-1}$, $w_0=-{0.95}_{-0.02}^{+0.15}$, $z_a=0.8\pm 0.46$. The errors represent statistical uncertainties only. The available SN dataset lacks sufficient statistical power to distinguish between the baseline $\mathrm{\Lambda CDM}$ model and the alternative sigmoid models. A feature of interest offered by the sigmoid model is that it identifies a specific redshift, $z_a=0.8$, where a potential transition in the equation of state could have occurred. The sigmoid model does not favor a DE in the phantom region (). Further constraints to the dynamic DE model have been obtained using CMB data to compute the distance to the last scattering surface. While the sigmoid DE model does not completely resolve the $H_0$ tension, it offers a transition mechanism that can still play a role alongside other potential solutions.