Statistical and Observation Comparison of Weyl-Type $f(Q,T)$ Models with the $\Lambda$CDM Paradigm

We study the $f(Q,T)$ gravity in the framework of Weyl geometry (known as Weyl-type $f(Q,T)$ gravity), where $Q$ denotes the non-metricity scalar, and $T$ denotes the energy-momentum tensor trace. In this work, we consider the $f(Q,T)$ model, which is defined as $f(Q,T)=\alpha Q^{m+1}+\frac{\beta}{6\kappa^2}T$ and investigating two scenarios: $(I)$ $m=0$ (linear model) and $(II)$ $m\neq 0$ (nonlinear model). For both scenarios, we find the explicit solution for the field equations by using the barotropic equation of state as $p=w\rho$, where $w$ is the equation-of-state (EoS) parameter. Further, we study the obtained solutions statistically using the $Pantheon^+$ (Without SHOES Calibrated) dataset with 1701 data points. For both models, the best-fit values of model parameters for $1-\sigma$ and $2-\sigma$ confidence level. The higher Hubble constant values in both models emphasize the presence of Tension. We statistically compare our models to the $\Lambda$CDM model using ${{\protect\chi}^2_{min}}$, ${{\protect\chi}^2_{red}}$, $AIC$, $\Delta AIC$, $BIC$ and $\Delta BIC$. We also examine cosmological parameters such as deceleration and EoS parameters to determine the current acceleration expansion of the Universe. Furthermore, we test our model using $Om$ diagnostic and compare it to the $\Lambda$CDM model to determine its dark energy profile. Finally, we draw the conclusion that statistically speaking, both linear and nonlinear models show good compatibility with the $\Lambda$CDM model.