Observational Constraints on the Dark Energy with a Quadratic Equation of State

In this study, we introduce a novel late-time effective dark energy model characterized by a quadratic equation of state (EoS) and rigorously examine its observational constraints. Initially, we delve into the background dynamics of this model, tracing the evolution of fluctuations in linear order. Our approach involves substituting the conventional cosmological constant with a dynamically effective dark energy fluid. Leveraging a diverse array of observational datasets encompassing the Planck 2018 Cosmic Microwave Background (CMB), Type Ia Supernovae (SNe), Baryon Acoustic Oscillations (BAO), and a prior on the Hubble constant $H_0$ (R21), we constrain the model parameters. We establish the model's consistency by comparing the Hubble parameter as a function of redshift against observational Hubble data (OHD), benchmarking its performance against the Standard $\Lambda$CDM model. Additionally, our investigation delves into studies of the model's dynamical behavior by computing cosmological parameters such as the deceleration parameter, relative Hubble parameter, and the evolution of the Hubble rate. Furthermore, employing Bayesian analysis, we determine the Bayesian Evidence for our proposed model compared to the reference $\Lambda$CDM model. While our analysis unveils the favorable behavior of the model in various observational tests, the well-known cosmological tensions persist when the full dataset combination is explored.