The PDF perspective on the tracer-matter connection: Lagrangian bias and\n non-Poissonian shot noise

Abstract

We study the connection of matter density and its tracers from the PDF\nperspective. One aspect of this connection is the conditional expectation value\n$\\langle \\delta_{\\mathrm{tracer}}|\\delta_m\\rangle$ when averaging both tracer\nand matter density over some scale. We present a new way to incorporate a\nLagrangian bias expansion of this expectation value into standard frameworks\nfor modelling the PDF of density fluctuations and counts-in-cells statistics.\nUsing N-body simulations and mock galaxy catalogs we confirm the accuracy of\nthis expansion and compare it to the more commonly used Eulerian\nparametrization. For halos hosting typical luminous red galaxies, the\nLagrangian model provides a significantly better description of $\\langle\n\\delta_{\\mathrm{tracer}}|\\delta_m\\rangle$ at second order in perturbations. A\nsecond aspect of the matter-tracer connection is shot-noise, \\ie the scatter of\ntracer density around $\\langle \\delta_{\\mathrm{tracer}}|\\delta_m\\rangle$. It is\nwell known that this noise can be significantly non-Poissonian and we validate\nthe performance of a more general, two-parameter shot-noise model for different\ntracers and simulations. Both parts of our analysis are meant to pave the way\nfor forthcoming applications to survey data.\n