$\Lambda$CDM model against redshift-binned data: A mock analysis based on SNIa and Cosmic Chronometers

Despite the broad successes of the flat $\Lambda$CDM model and its fitness to the various cosmological observations, it confronts challenges stemming from anomalies in the measurements of the Hubble constant ($H_0$) and the amplitude of matter fluctuations ($\sigma_8$). These inconsistencies have necessitated a reassessment of the model parameters, with a particular focus on their potential dependence on redshift. This study pioneers a new investigation to probe this redshift dependency by generating mock data simulated from observational data of Type Ia supernovae (SNIa) and cosmic chronometers (CC), thereby increasing the data density in this field. By sorting the data into high-redshift and low-redshift bins, we aim to refine the cosmological constraints on the parameters of the $\Lambda$CDM model and determine whether the noted dependence on redshift is due to a lack of high-redshift observational data or if they signify intrinsic issues within the model itself. Our approach employs the Markov Chain Monte Carlo (MCMC) algorithm to minimize the $\chi^2$ function, thus tightening the cosmological constraints. Our findings within the mock analysis reveal discrepancies between the values of $\Omega_{m0}$ and $H_0$ derived from the mock data bins with high redshift and low redshift, indicating the potential deviation of the standard $\Lambda$ CDM cosmology from the high-redshift SNIa and CC data. If this deviation proposes a new physics beyond the standard model, then with better quality future data tracking the new physics, these discrepancies will be statistically significant.