More than 90% baryons are likely in the form of intergalactic medium (IGM) and have been difficult to detect directly. To understand the IGM state, such as density, temperature, peculiar velocity and metalicity, stands as a major challenge to both observation and theory, and is crucial to understand the thermal history of the universe and galaxy formation.

In this thesis, I investigate various observable IGM tracers such as the thermal and kinetic Sunyaev Zel'dovich effect and the soft X-ray background. I build analytical models to calculate their statistics and their dependence on cosmological parameters, the IGM state and the thermal history of the universe. I then run and analyze hydro simulations to understand more detailed physics, test the analytical models and fix some free parameters in the analytical models. I then simulate upcoming observations using simulation data to suggest the optimal survey strategies and data analysis methods and forecast the sensitivity and accuracy of these observations. Applying these understandings to observation data, the IGM state and the thermal history of the universe is very likely to be extracted.