Effect of non-equilibrium ionization on derived physical conditions of the high-$z$ intergalactic medium

Non-equilibrium ionization effects are important in cosmological hydrodynamical simulations but are computationally expensive. We study the effect of non-equilibrium ionization evolution and UV ionizing background (UVB) generated with different quasar spectral energy distribution (SED) on the derived physical conditions of the intergalactic medium (IGM) at $2\leq z \leq 6$ using our post-processing tool 'Code for Ionization and Temperature Evolution' (CITE). CITE produces results matching well with self-consistent simulations more efficiently. The HeII reionization progresses more rapidly in non-equilibrium model as compared to equilibrium models. The redshift of HeII reionization strongly depends on the quasar SED and occurs earlier for UVB models with flatter quasar SEDs. During this epoch the normalization of temperature-density relation, $T_0(z)$, has a maximum while the slope, $\gamma(z)$, has a minimum, but occurring at different redshifts. The $T_0$ is higher in non-equilibrium models using UVB obtained with flatter quasar SEDs. While our models produce the observed median HeII effective optical depth evolution and its scatter for equilibrium and non-equilibrium considerations, to explain the observed cumulative distributions we may need to consider fluctuating UVB. For a given UVB model, the redshift dependence of the HI photo-ionization rate derived from the observed HI effective optical depth ($\tau_{\rm eff,HI}$) for the equilibrium model is different from that for the non-equilibrium model. This may lead to different requirements on the evolution of ionizing emissivities of sources. We show that, in the absence of strong differential pressure smoothing effects, it is possible to recover the $T_0$ and $\gamma$ realised in non-equilibrium model from the equilibrium models generated by rescaling photo-heating rates while producing the same $\tau_{\rm eff,HI}$.