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We propose a novel method to quantify the assembly histories of dark matterhalos with the redshift evolution of the mass-weighted spatial variance oftheir progenitor halos, i.e. the protohalo size history. We find that theprotohalo size history for each individual halo at z~0 can be described by adouble power-law function. The amplitude of the fitting function stronglycorrelates to the central-to-total stellar mass ratios of descendant halos. Thevariation of the amplitude of the protohalo size history can induce a stronghalo assembly bias effect for massive halos. This effect is detectable inobservation using the central-to-total stellar mass ratio as a proxy of theprotohalo size. The correlation to the descendant central-to-total stellar massratio and the halo assembly bias effect seen in the protohalo size are muchstronger than that seen in the commonly adopted half-mass formation timederived from the mass accretion history. This indicates that the informationloss caused by the compression of halo merger trees to mass accretion historiescan be captured by the protohalo size history. Protohalo size thus provides auseful quantity to connect protoclusters across cosmic time and to linkprotoclusters with their descendant clusters in observations.

Characterizing the assembly of dark matter halos with protohalo size  histories: I. Redshift evolution, relation to descendant halos, and halo  assembly bias