  We present a new proxy for the overdensity peak height for which the
large-scale clustering of haloes of a given mass does not vary significantly
with the assembly history. The peak height, usually taken to be well
represented by the virial mass, can instead be approximated by the mass inside
spheres of different radii, which in some cases can be larger than the virial
radius and therefore include mass outside the individual host halo. The sphere
radii are defined as r = $a$ delta_t + $b$ log_10(M_vir/M_nl), where delta_t is
the age relative to the typical age of galaxies hosted by haloes with virial
mass M_vir, M_nl is the non-linear mass, and $a$=0.2 and $b$=-0.02 are the free
parameters adjusted to trace the assembly bias effect. Note that $r$ depends on
both halo mass and age. In this new approach, some of the objects which were
initially considered low-mass peaks belong to regions with higher
overdensities. At large scales, i.e. in the two-halo regime, this model
properly recovers the simple prescription where the bias responds to the height
of the mass peak alone, in contrast to the usual definition (virial mass) that
shows a strong dependence on additional halo properties such as formation time.
The dependence on the age in the one-halo term is also remarkably reduced. The
population of galaxies whose "peak height" changes with this new definition
consists mainly of old stellar populations and are preferentially hosted by
low-mass haloes located near more massive objects. The latter is in agreement
with recent results which indicate that old, low-mass haloes would suffer
truncation of mass accretion by nearby larger haloes or simply due to the high
density of their surroundings, thus showing an assembly bias effect. The change
in mass is small enough that the Sheth et al. (2001) mass function is still a
good fit to the resulting distribution of new masses.
