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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, $H(z)$, at redshifts $(z < 2)$. 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, $\Lambda = -1$, value to the present value, $w_0$, through the implementation of 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 LCDM alone. The fit results, assuming a flat geometry and maintaining $\Omega_m$ constant at the 2018-Planck value of $0.3153$, are as follows: $H_0 = 73.3^{+0.2}_{-0.6}$ km s$^{-1}$ Mpc$^{-1}$, $w_{0} = -0.95^{+0.15}_{-0.02}$, $z_a = 0.8 \pm 0.46$. The errors represent statistical uncertainties only. The available SN dataset lacks sufficient statistical power to distinguish between the baseline LCDM 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 ($w_0 < -1$). 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., Comment: 9 pages, 5 figures

Measurements of the current expansion rate of the Universe, $H_0$, using standard candles, disagree with those derived from observations of the Cosmic Microwave Background (CMB). This discrepancy, known as the Hubble tension, is substantial and suggests the possibility of revisions to the standard cosmological model (Cosmological constant $\Lambda$ and cold dark matter - $\Lambda$CDM). Dynamic dark energy (DE) models that introduce deviations in the expansion history relative to $\Lambda$CDM could potentially explain this tension. We used Type Ia supernovae (SNe) data to test a dynamic DE model consisting of an equation of state that varies linearly with the cosmological scale factor $a$. To evaluate this model, we developed a new statistic (the $T_{\alpha}$ statistic) used in conjunction with an optimization code that minimizes its value to obtain model parameters. The $T_{\alpha}$ statistic reduces bias errors (in comparison to the $\chi^2$ statistic) because it retains the sign of the residuals, which is meaningful in testing the dynamic DE model as the deviations in the expansion history introduced by this model act asymmetrically in redshift space. The DE model fits the SNe data reasonably well, but the available SNe data lacks the statistical power to discriminate between $\Lambda$CDM and alternative models. To further assess the model using CMB data, we computed the distance to the last scattering surface and compared the results with that derived from the Planck observations. Although the simple dynamic DE model tested does not completely resolve the tension, it is not ruled out by the data and could still play a role alongside other physical effects., Comment: 21 pages, 9 figures