Testing Lambert$W$ equation of state with observational Hubble parameter data

In this paper, we investigate the possibility that the Universe is driven by a single dark fluid described by a Lambert $W$ equation of state parameter, $w_{eff}$, which is essentially dependent on two parameters $\vartheta_{1}$ and $\vartheta_{2}$ which need to be fixed from observations. We obtain the constraints on these parameters using the latest 51 data points of $H(z)$ measurements, spanning the redshift range $0.07\leq z \leq 2.36$. The present study shows that the Universe is indeed undergoing an accelerated expansion phase following the decelerated one at the transition redshift, $z_{t}=0.77\pm0.03$ ($1\sigma$) and is well consistent with the recent observations. We also find that at low redshifts, $w_{eff}$ evolves only in the quintessence regime ($-1<w_{eff}<-\frac{1}{3}$) within $1\sigma$ confidence level. Its present value is found to be $-0.96\pm0.02$ ($1\sigma$). The fact that the present value of $w_{eff}$ is very close to the Cosmological Constant $\Lambda$ implies that our proposed equation of state parameter might serve as a unification of dark matter and dark energy. Furthermore, we compare the evolution of $H(z)$ for the model under consideration with that of the $\Lambda$CDM model. Finally, we observe that for the best-fit case, the differences between the two models are negligible at $z\sim 0.67$.