Estimation of the temperature-density relation in the intergalactic
  medium at $\mathbf{z\sim 2-4}$ via Ly$\mathbf{\alpha}$ forest
  Quasar spectra provide a unique opportunity to investigate the intergalactic
medium at high redshifts. The spectral analysis of the Ly$\alpha$ forest allows
us to constrain the effective equation of state of the intergalactic medium and
its changes during the evolution of the Universe. Based on the Voigt profile
fitting of Ly$\alpha$ forest lines in 50 high-resolution QSO spectra obtained
at Keck telescope we present new measurements of the power-law index $\gamma$
of temperature-density relation in IGM for six redshift bins in the range
$z=1.6-3.7$. We find that the IGM state is close to isothermal ($\gamma\approx
1$) at $z\sim 3$ which may indicate that HeII reionization occurred at this
redshift.
