  Secondary halo bias, commonly known as 'assembly bias,' is the dependence of
halo clustering on a halo property other than mass. This prediction of the
Lambda-Cold Dark Matter cosmology is essential to modelling the galaxy
distribution to high precision and interpreting clustering measurements. As the
name suggests, different manifestations of secondary halo bias have been
thought to originate from halo assembly histories. We show conclusively that
this is incorrect for cluster-size haloes. We present an up-to-date summary of
secondary halo biases of high-mass haloes due to various halo properties
including concentration, spin, several proxies of assembly history, and subhalo
properties. While concentration, spin, and the abundance and radial
distribution of subhaloes exhibit significant secondary biases, properties that
directly quantify halo assembly history do not. In fact, the entire assembly
histories of haloes in pairs are nearly identical to those of isolated haloes.
In general, a global correlation between two halo properties does not predict
whether or not these two properties exhibit similar secondary biases. For
example, assembly history and concentration (or subhalo abundance) are
correlated for both paired and isolated haloes, but follow slightly different
conditional distributions in these two cases. This results in a secondary halo
bias due to concentration (or subhalo abundance), despite the lack of assembly
bias in the strict sense for cluster-size haloes. Due to this complexity,
caution must be exercised in using any one halo property as a proxy to study
the secondary bias due to another property.
