The intergalactic medium ( IGM ) is the dominant reservoir of baryons at all cosmic epochs . In this paper , we investigate the evolution of the IGM from z = 2 \rightarrow 0 in ( 48 h ^ { -1 } { Mpc } ) ^ { 3 } , 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 Hubble ’ s Cosmic Origins Spectrograph ( 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 comparably divided between bound phases ( 35 % in our favoured outflow model ) , the diffuse IGM ( 41 % ) , and the WHIM ( 24 % ) . Here we ( re ) define the WHIM as gas with overdensities lower than that in halos ( \rho / \bar { \rho } \la 100 today ) and temperatures T > 10 ^ { 5 } K , to more closely align it with the “ missing baryons ” that are not easily detectable in emission or Ly \alpha absorption . Strong galactic outflows can have a noticeable impact on the temperature of the IGM , though with our favoured momentum-driven wind scalings they do not . When we ( mildly ) tune our assumed photoionizing background to match the observed evolution of the Ly \alpha mean flux decrement , we obtain line count evolution statistics that broadly agree with available ( pre-COS ) observations . We predict a column density distribution slope of f ( N _ { HI } ) \propto N _ { HI } ^ { -1.70 } for our favoured wind model , in agreement with recent observational estimates , and it becomes shallower with redshift . Winds have a mostly minimal impact , but they do result in a shallower column density slope and more strong lines . With improved statistics , the frequency of strong lines can be a valuable diagnostic of outflows , and the momentum-driven wind model matches existing data significantly better than the two alternatives we consider . The relationship between column density and physical density broadens mildly from z = 2 \rightarrow 0 , and evolves as \rho \propto N _ { HI } ^ { 0.74 } 10 ^ { -0.37 z } for diffuse absorbers , consistent with previous studies . Linewidth distributions are quite sensitive to spectral resolution ; COS should yield significantly broader lines than higher-resolution data . Thermal contributions to linewidths are typically subdominant , so linewidths only loosely reflect the temperature of the absorbing gas . This will hamper attempts to quantify the WHIM using broad Ly \alpha absorbers , though it may still be possible to do so statistically . Together , COS data and simulations such as these will provide key insights into the physical conditions of the dominant reservoir of baryons over the majority of cosmic time .