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The effect of AGN feedback on the halo mass function

[Abridged.] We investigate baryon effects on the halo mass function (HMF), with emphasis on the role played by AGN feedback. Halos are identified with both Friends-of-Friends (FoF) and Spherical Overdensity (SO) algorithms. We embed the standard SO algorithm into a memory-controlled frame program and present the {\bf P}ython spher{\bf I}c{\bf A}l {\bf O}verdensity code --- {\small PIAO}. For both FoF and SO halos, the effect of AGN feedback is that of suppressing the HMFs to a level even below that of Dark Matter simulations. The ratio between the HMFs in the AGN and in the DM simulations is $\sim 0.8$ at overdensity $Δ_c=500$, a difference that increases at higher overdensity $Δ_c=2500$, with no significant redshift and mass dependence. A decrease of the halo masses ratio with respect to the DM case induces the decrease of the HMF in the AGN simulation. The shallower inner density profiles of halos in the AGN simulation witnesses that mass reduction is induced by the sudden displacement of gas induced by thermal AGN feedback. We provide fitting functions to describe halo mass variations at different overdensities, which can recover the HMFs with a residual random scatter $\lt 5$ per cent for halo masses larger than $10^{13} ~h^{-1}{\rm M_\odot}$.