Some New Types of Well-Behaved Polynomial Redshift Parametrization of Dark Energy Equation of State

Abstract: In this paper, we explore a new type of smooth and well-behaved polynomial redshift function that can avoid a future singularity. Using this function, we have proposed different redshift parametrizations of the dark energy equation of state, drawing motivation from different polynomial functions like conventional polynomial, Legendre polynomial, Laguerre polynomial, Chebyshev polynomial and Fibonacci polynomial. The main feature of these parametrizations is their well-behaved nature throughout the evolution of the universe, which was a matter of concern in most of the previous polynomial parametrizations of the dark energy equation of state (EoS). This form of parametrization may be considered as an extension of those forms with no divergence at any redshift value. A comprehensive observational data analysis is performed with the Hubble, BAO and DESI datasets to constrain the parameter space of the models. Confidence contours showing joint and marginalized posterior distribution with different combinations of datasets are generated using a Markov Chain Monte Carlo approach. We see that our improved parametrizations enable us to derive more stringent restrictions on the current dark energy EoS and its derivative, which improves performance. Finally, a machine learning analysis is performed using some suitable algorithms like ELR, PILR, ANN, SVR, ERFR and GBR to compare the models. Among all the tested polynomial bases, the Legendre basis demonstrated superior performance with the lowest test RMSE and reduced $\chi^{2}$ value under the Modified Differential Evolution theoretical model, indicating exceptional physical accuracy and numerical stability.