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The growth history of the inner core is inherently linked to the thermal history of the Earth. The crystallization of the inner core may have been delayed by supercooling and gone through an initial phase of fast growth after the nucleation barrier has been passed, but so far no evidence or constraint has been proposed to time this possible event. With two-phase flow dynamics, we explore the effect of different growth scenarios for the inner core to determine their effects on its liquid fraction structure. Seismic observations on the melt fraction inside the inner core at present limit the porosity to a maximum of 10% of liquid. Our model provides constraints for the delay in nucleation compared to the time where the first crystal may have started to nucleate, and we find that the supercooling cannot have exceeded 100 K.