Not yet reviewed by Pith; the record is open.

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.