Diffuse baryonic matter beyond 2020

The hot, diffuse gas that fills the largest overdense structures in the Universe -- clusters of galaxies and a web of giant filaments connecting them -- provides us with tools to address a wide array of fundamental astrophysical and cosmological questions via observations in the X-ray band. Clusters are sensitive cosmological probes. To utilize their full potential for precision cosmology in the following decades, we must precisely understand their physics -- from their cool cores stirred by jets produced by the central supermassive black hole (itself fed by inflow of intracluster gas), to their outskirts, where the infall of intergalactic medium (IGM) drives shocks and accelerates cosmic rays. Beyond the cluster confines lies the virtually unexplored warm IGM, believed to contain most of the baryonic matter in the present-day Universe. As a depository of all the matter ever ejected from galaxies, it carries unique information on the history of energy and metal production in the Universe. Currently planned major observatories, such as Astro-H and IXO, will make deep inroads into these areas, but to see the most interesting parts of the picture will require an almost science-fiction-grade facility with tens of m^2 of effective area, subarcsecond angular resolution, a matching imaging calorimeter and a super high-dispersion spectrograph, such as Generation-X.