Origin and full characterization of the secondary (assembly) halo bias

Not only are halos of different masses differently clustered, the so-called primary bias, but halos with distinct internal properties also are, which is known as secondary bias. Contrarily to the former bias, the latter is not understood within the Press-Schechter (PS) and excursion set (ES) formalisms of structure formation, where protohalos are fully characterised by their height and scale. As halos in different backgrounds undergo mergers at different rates, the secondary bias was suggested not to be innate but to arise from the distinct assembly history of halos. However, it has recently been proven that mergers leave no imprint in halo properties. But, in the peak model of structure formation, the secondary bias could still be innate and arise from the different typical curvature of peaks lying in different backgrounds. Here we show that this is the case, indeed. Using the CUSP formalism, we find that peaks lying in different backgrounds with different typical curvatures evolve in halos with different density profiles which in turn lead to many other distinct properties. The dependence we find of halo bias with those properties reproduces the results of simulations.