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The intergalactic medium (IGM) is the dominant reservoir of baryons at all cosmic epochs. We investigate the evolution of the IGM from z=2-0 in 48 Mpc/h, 110-million particle cosmological hydrodynamic simulations using three prescriptions for galactic outflows. We focus on the evolution of IGM physical properties, and how such properties are traced by Ly-alpha absorption as detectable using HST/COS. Our results broadly confirm the canonical picture that most Ly-alpha absorbers arise from highly ionized gas tracing filamentary large-scale structure. Growth of structure causes gas to move from the diffuse photoionized IGM into other cosmic phases, namely stars, cold and hot gas within galaxy halos, and the unbound and shock-heated warm-hot intergalactic medium (WHIM). By today, baryons are roughly equally divided between bound phases (35%), the diffuse IGM (41%), and the WHIM (24%). Here we (re)define the WHIM as gas with overdensities lower than that in halos and temperatures >10^5 K, in order to more closely align it with "missing baryons". When we tune our photoionizing background to match the observed evolution of the Ly-alpha mean flux decrement, we obtain a line count evolution that broadly agrees with available data. We predict a column density distribution slope of -1.70 for our favored momentum-driven wind model, in agreement with recent observations, and it becomes shallower with redshift. With improved statistics, the frequency of strong lines can be a valuable diagnostic of outflows, and our favored wind model matches existing data best among our models. The relationship between column density and physical density is fairly tight from z=2-0, and evolves as rho N_HI^0.74 10^(-0.37z) for diffuse absorbers. Linewidths only loosely reflect the temperature of the absorbing gas, which will hamper attempts to quantify the WHIM using broad Ly-alpha absorbers. [Abridged]


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