Constraining the Temperature-Density Relation of the Inter-Galactic Medium from Analytically Modeling Lyman-alpha Forest Absorbers

The absorption by neutral hydrogen in the intergalactic medium (IGM) produces the Ly$\alpha$ forest in the spectra of quasars. The Ly$\alpha$ forest absorbers have a broad distribution of neutral hydrogen column density $N_{\rm HI}$ and Doppler $b$ parameter. The narrowest Ly$\alpha$ absorption lines (of lowest $b$) with neutral hydrogen column density above $\sim 10^{13}{\rm cm^{-2}}$ are dominated by thermal broadening, which can be used to constrain the thermal state of the IGM. Here we constrain the temperature-density relation $T=T_0(\rho/\bar{\rho})^{\gamma-1}$ of the IGM at $1.6<z<3.6$ by using $N_{\rm HI}$ and $b$ parameters measured from 24 high-resolution and high-signal-to-noise quasar spectra and by employing an analytic model to model the $N_{\rm HI}$-dependent low-$b$ cutoff in the $b$ distribution. In each $N_{\rm HI}$ bin, the $b$ cutoff is estimated using two methods, one non-parametric method from computing the cumulative $b$ distribution and a parametric method from fitting the full $b$ distribution. We find that the IGM temperature $T_0$ at the mean gas density $\bar{\rho}$ shows a peak of $\sim 1.5\times 10^4$K at $z\sim$2.7-2.9. At redshift higher than this, the index $\gamma$ approximately remains constant, and it starts to increase toward lower redshifts. The evolution in both parameters is in good agreement with constraints from completely different approaches, which signals that He II reionization completes around $z\sim 3$.