Thermal state of the intergalactic medium at $z\sim2-4$
  We present a new method to infer parameters of the temperature-density
relation in the intergalactic medium in the post-reionization epoch at $z\sim
2-4$. This method is based on the analysis of the Ly$\alpha$ absorbers
distribution over column densities and Doppler parameters by the model joint
probability density function. This approach allows us to measure the power-law
index $\gamma$ of the temperature-density relation and a certain combination of
the temperature at the mean density $T_0$ and hydrogen photoionization rate
$\Gamma$. To estimate $T_0$ and $\Gamma$ separately, we employ measurements of
the Ly$\alpha$ forest effective opacity and the model gas probability density
function. We show that $\gamma$ tends to be lower than 1.6 and reaches 1.3 at
redshift $\sim3$. The inferred temperatures at the mean density are
$\sim(2\pm0.5)\times10^4$ K in the studied redshift range. Both these estimates
favour HeII reionization at $z\gtrsim3$. We find that the hydrogen
photoionization rate is $\sim0.6\times10^{-12}$ s$^{-1}$, which is consistent
with previous measurements.
