SAM for different redshifts (as marked). The dashed horizontal lines indicate the number densities of the samples used in this work.[]{data-label="Fig:AMF"}](Fig1a.pdf "fig:"){width="50.00000%"} ![The cumulative stellar mass function (top panel) and the cumulative SFR function (bottom panel) predicted by the G13 SAM for different redshifts (as marked). The dashed horizontal lines indicate the number densities of the samples used in this work.[]{data-label="Fig:AMF"}](Fig1b.pdf "fig:"){width="48.00000%"}

### Classification by halo properties

To investigate assembly bias we define subsets of the fixed number density galaxy samples by selecting haloes using two different intrinsic or secondary properties: formation time (age) and concentration.

We define the formation time of a halo as the redshift when its main progenitor reaches half of the halo’s present-day mass for the first time. This definition is commonly used in the study of assembly bias (e.g., @Gao:2004 [@Gao:2005; @Croton:2007], Z18). We note that the formation time of a halo is calculated at each redshift independently. We calculate the formation time using the merger trees available in the database and linearly interpolate the mass of the haloes between snapshots.

The other halo property we consider is the concentration. The halo concentration characterizes the density profile. It is canonically defined as $c_{\rm vir} = 