The Lambert $W$ equation of state in light of DESI BAO

We investigate the hypothesis that the evolution of the Universe can be described by a single dark fluid whose effective equation of state (EoS), ${\omega }_{\mathrm{e}\mathrm{f}\mathrm{f}}$, is a linear combination of a logarithmic term and a power law term, both involving the Lambert $W$ function. This particular form of EoS was first proposed by S. Saha and K. Bamba in 2019 and has two parameters, ${\theta }_1$ and ${\theta }_2$, which must be determined from observations. To this end, we place limits on these parameters by combining recent baryon acoustic oscillation (BAO) data – including measurements from the Dark Energy Spectroscopic Instrument (DESI) – with Type Ia supernova observations from the Pantheon+ compilation, along with direct determinations of the Hubble parameter. From this combined analysis, we obtain a best-fit value for the Hubble parameter, $H_0=67.4\pm 1.2{\text{km}\text{s}}^{-1}{\text{Mpc}}^{-1}$, while current measurements of the sound horizon at the baryon drag epoch yield $r_d=146\pm 2.5$~Mpc. Furthermore, we study the evolution of the deceleration parameter, the effective EoS, and the jerk parameter, and support our findings using the $Om(z)$ diagnostic. The model exhibits noticeable deviations from the predictions of the concordance $\mathrm{\Lambda }$CDM model. Despite these differences, our results indicate that the model provides a coherent description of late-time cosmic evolution and the observed accelerated expansion of the Universe. Finally, we assess the observational viability of the model using information criteria, particularly the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), and compare these results with those obtained for the $\mathrm{\Lambda }$CDM model, which serves as our reference.