We study structure formation in a set of cosmological simulations to uncover the scales in the initial density field that gave rise to the formation of present-day structures. Our simulations share a common primordial power spectrum (here Lambda cold dark matter, Lambda CDM), but the introduction of hierarchical variations of the phase information allows us to systematically study the scales that determine the formation of structure at later times. We consider the variance in z = 0 statistics such as the matter power spectrum and halo mass function. We also define a criterion for the existence of individual haloes across simulations, and determine what scales in the initial density field contain sufficient information for the non-linear formation of unique haloes. We study how the characteristics of individual haloes such as the mass and concentration, as well as the position and velocity, are affected by variations on different scales, and give scaling relations for haloes of different mass. Finally, we use the example of a cluster-mass halo to show how our hierarchical parametrization of the initial density field can be used to create variants of particular objects. With properties such as mass, concentration, kinematics, and substructure of haloes set on distinct and well-determined scales, and its unique ability to introduce variations localized in real space, our method is a powerful tool to study structure formation in cosmological simulations.