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  1. Abstract The density and temperature properties of the intergalactic medium (IGM) reflect the heating and ionization history during cosmological structure formation, and are primarily probed by the Ly α forest of neutral hydrogen absorption features in the observed spectra of background sources. We present the methodology and initial results from the Cholla IGM Photoheating Simulation (CHIPS) suite performed with the graphics process unit–accelerated Cholla code to study the IGM at high, uniform spatial resolution maintained over large volumes. In this first paper, we examine the IGM structure in CHIPS cosmological simulations that include IGM uniform photoheating and photoionization models wheremore »hydrogen reionization is completed early or by redshift z ∼ 6. Comparing with observations of the large- and small-scale Ly α transmitted flux power spectra P ( k ) at redshifts 2 ≲ z ≲ 5.5, the relative agreement of the models depends on scale, with the self-consistent Puchwein et al. IGM photoheating and photoionization model in good agreement with the flux P ( k ) at k ≳ 0.01 s km −1 at redshifts 2 ≲ z ≲ 3.5. On larger scales, the P ( k ) measurements increase in amplitude from z ∼ 4.6 to z ∼ 2.2, faster than the models, and lie in between the model predictions at 2.2 ≲ z ≲ 4.6 for k ≈ 0.002–0.01 s km −1 . We argue that the models could improve by changing the He ii photoheating rate associated with active galactic nuclei to reduce the IGM temperature at z ∼ 3. At higher redshifts, z ≳ 4.5, the observed flux P ( k ) amplitude increases at a rate intermediate between the models, and we argue that for models where hydrogen reionization is completed late ( z ∼ 5.5–6), resolving this disagreement will require inhomogeneous or “patchy” reionization. We then use an additional set of simulations to demonstrate that our results have numerically converged and are not strongly affected by varying cosmological parameters.« less