The intergalactic medium temperature and Compton y parameter

(Abridged) The precision modeling of the thermal Sunyaev Zeldovich (SZ) effect and future high accuracy measurements will provide a powerful way to constrain the thermal history of the universe. In this paper, we combine high resolution adiabatic hydro simulations and analytical models to provide a precision modeling of the gas density weighted temperature ${\bar{T}}_g$ and the mean SZ Compton $y$ parameter. Simulation artifacts are quantified and corrected and our analytical model passes tests of various simulations and is able to predict ${\bar{T}}_g$ and $\bar{y}$ to several percent accuracy. For low matter density $\mathrm{\Lambda }$CDM cosmology, the present ${\bar{T}}_g$ is $0.32 (\sigma_8/0.84)^{3.05-0.15\Omega_m}(\Omega_m/0.268)^{1.28-0.2\sigma_8} {\rm keV}$\mathrm{.}Themean$y$parameteris$2.6\times 10^{-6} (\sigma_8/0.84)^{4.1-2\Omega_m}(\Omega_m/0.268)^{1.28-0.2\sigma_8}$. The current upper limit of $y<1.5\times {10}^{-5}$ measured by FIRAS has already ruled out combinations of high ${\sigma }_8\text{ga}1.1$ and high ${\mathrm{\Omega }}_m\text{ga}0.5$.