In this work we present a new hybrid method to simulate the thermal effects of the reionization in cosmological hydrodynamical simulations. The method improves upon the standard approach used in simulations of the intergalactic medium (IGM) and galaxy formation without a significant increase of the computational cost allowing for efficient exploration of the parameter space. The method uses a small set of phenomenological input parameters and combines a semi-numerical reionization model to solve for the topology of reionization and an approximate model of how reionization heats the IGM, with the massively parallel texttt{Nyx} hydrodynamics code, specifically designed to solve for the structure of diffuse IGM gas. We have produced several large-scale high resolution cosmological hydrodynamical simulations ($2048^3$, $L_{rm box} = 40$ Mpc/h) with different instantaneous and inhomogeneous HI reionization models that use this new methodology. We study the IGM thermal properties of these models and find that large scale temperature fluctuations extend well beyond the end of reionization. Analyzing the 1D flux power spectrum of these models, we find up to $sim 50%$ differences in the large scale properties (low modes, $klesssim0.01$ s/km) of the post-reionization power spectrum due to the thermal fluctuations. We show that these differences could allow one to distinguish between different reionization scenarios already with existing Ly$alpha$ forest measurements. Finally, we explore the differences in the small-scale cutoff of the power spectrum and we find that, for the same heat input, models show very good agreement provided that the reionization redshift of the instantaneous reionization model happens at the midpoint of the inhomogeneous model.
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