  The cell cycle duration is a variable cellular phenotype that underlies
long-term population growth and age structures. By analyzing the stationary
solutions of a branching process with heritable cell division times, we
demonstrate existence of a phase transition, which can be continuous or
first-order, by which a non-zero fraction of the population becomes localized
at a minimal division time. Just below the transition, we demonstrate
coexistence of localized and delocalized age-structure phases, and power law
decay of correlation functions. Above it, we observe self-synchronization of
cell cycles, collective divisions, and slow 'aging' of population growth rates.
