Thermal state of the intergalactic medium near to the optical limit for the Ly-$\alpha$ forest

In this article, the temperature-density relation of the intergalactic medium was studied in the region $1.6 \leq z < 2.0$ divided into two bins. For this purpose, the Ly-$\alpha$ forest decomposition into individual absorption profiles was used for the study of 35 publicly available quasar spectra obtained by the Ultraviolet and Visual Echelle Spectrograph (UVES) on the Very Large Telescope (ESO) and by the High Resolution Echelle Spectrometer (HIRES) on the Keck Telescope. For the determination of the thermal state sensitive cut-off position in the $b - N_{HI}$ distribution, the iterative fitting procedure was adopted. The measurements were calibrated using mock Ly-$\alpha$ forest data generated by 23 hydrodynamical simulations with different thermal histories. The value of the temperature at mean density corresponds to the decreasing trend predicted by various models at the lower redshifts. In the case of power law index, determined values are close to 1.6, which is expected after all reionization events in various models assuming the balance of photoheating with adiabatic cooling.