Shot noise and biased tracers: a new look at the halo model

Shot noise is an important ingredient to any measurement or theoretical modeling of discrete tracers of the large scale structure. Recent work has shown that the shot noise in the halo power spectrum becomes increasingly sub-Poissonian at high mass. Interestingly, while the halo model predicts a shot noise power spectrum in qualitative agreement with the data, it leads to an unphysical white noise in the cross halo-matter and matter power spectrum. In this work, we show that absorbing all the halo model sources of shot noise into the halo fluctuation field leads to meaningful predictions for the shot noise contributions to halo clustering statistics and remove the unphysical white noise from the cross halo-matter statistics. Our prescription straightforwardly maps onto the general bias expansion, so that the renormalized shot noise terms can be expressed as combinations of the halo model shot noises. Furthermore, we demonstrate that non-Poissonian contributions are related to volume integrals over correlation functions and their response to long-wavelength density perturbations. This leads to a new class of consistency relations for discrete tracers, which appear to be satisfied by our reformulation of the halo model. We test our theoretical predictions against measurements of halo shot noise bispectra extracted from a large suite of numerical simulations. Our model reproduces qualitatively the observed sub-Poissonian noise, although it underestimates the magnitude of this effect.