Maximal X-ray feedback in the pre-reionization universe

X-ray feedback in the pre-reionization Universe provided one of the major energy sources for reionization and the thermal evolution of the early intergalactic medium. However, X-ray sources at high redshift have remained largely inaccessible to observations. One alternative approach to study the overall effect of X-ray feedback in the early Universe is a full cosmological simulation. Toward this goal, in this paper we create an analytic model of X-ray feedback from accretion onto supermassive black holes (SMBHs), to be used as a sub-grid model in future cosmological simulations. Our analytic model provides a relation between the mass of a dark matter halo and the SMBH it hosts, where the efficiency is governed by an energy balance argument between thermal feedback and the confining gravitational potential of the halo. To calibrate the model, we couple the halo-level recipe with the Press-Schechter halo mass function and derive global mass and energy densities. We then compare our model to various observational constraints, such as the resulting soft X-ray and IR cosmic radiation backgrounds, to test our choice of model parameters. We in particular derive model parameters that do not violate any constraints, while providing maximal X-ray feedback prior to reionization. In addition, we consider the contribution of SMBH X-ray sources to reionization and the global 21 cm absorption signal.