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Observations of the Lyman-$\alpha$ forest in distant quasar spectra withupcoming surveys are expected to provide significantly larger andhigher-quality datasets. To interpret these datasets, it is imperative todevelop efficient simulations. One such approach is based on the assumptionthat baryonic densities in the intergalactic medium (IGM) follow a lognormaldistribution. We extend our earlier work to assess the robustness of thelognormal model of the Lyman-$\alpha$ forest in recovering the parameterscharacterizing IGM state, namely, the mean-density IGM temperature ($T_0$), theslope of the temperature-density relation ($\gamma$), and the hydrogenphotoionization rate ($\Gamma_{12}$), by comparing with high-resolutionSherwood SPH simulations across the redshift range $2 \leq z \leq 2.7$. Theseparameters are estimated through a Markov Chain Monte Carlo technique, usingthe mean and power spectrum of the transmitted flux. We find that the usuallognormal distribution of IGM densities cannot recover the parameters of theSPH simulations. This limitation arises from the fact that the SPH baryonicdensity distribution cannot be described by a simple lognormal form. To addressthis, we extend the model by scaling the linear density contrast by a parameter$\nu$. While the resulting baryonic density is still lognormal, the additionalparameter gives us extra freedom in setting the variance of densityfluctuations. With this extension, values of $T_0$ and $\gamma$ implied in theSPH simulations are recovered at $\sim 1-\sigma$ ($\lesssim$ 10%) of the median(best-fit) values for most redshifts bins. However, this extended lognormalmodel cannot recover $\Gamma_{12}$ reliably, with the best-fit value discrepantby $\gtrsim 3-\sigma$ for $z > 2.2$. Despite this limitation in the recovery of$\Gamma_{12}$, we argue that the model remains useful for constrainingcosmological parameters.

A modified lognormal approximation of the Lyman-$\alpha$ forest: comparison with full hydrodynamic simulations at $2\leq z\leq 2.7$