  There are well-observed differences between cool-core (CC) and non-cool-core
(NCC) clusters, but the origin of this distinction is still largely unknown.
Competing theories can be divided into internal (inside-out), in which internal
physical processes transform or maintain the NCC phase, and external
(outside-in), in which the cluster type is determined by its initial
conditions, which in turn lead to different formation histories (i.e., assembly
bias). We propose a new method that uses the relative assembly bias of CC to
NCC clusters, as determined via the two-point cluster-galaxy cross-correlation
function (CCF), to test whether formation history plays a role in determining
their nature. We apply our method to 48 ACCEPT clusters, which have well
resolved central entropies, and cross-correlate with the SDSS-III/BOSS LOWZ
galaxy catalog. We find that the relative bias of NCC over CC clusters is $b =
1.42 \pm 0.35$ ($1.6\sigma$ different from unity). Our measurement is limited
by the small number of clusters with core entropy information within the BOSS
footprint, 14 CC and 34 NCC. Future compilations of X-ray cluster samples,
combined with deep all-sky redshift surveys, will be able to better constrain
the relative assembly bias of CC and NCC clusters and determine the origin of
the bimodality.
