  Heavy-ion experiments provide a new opportunity to gain a deeper
understanding of the structure of nuclei. To achieve this, it is crucial to
identify observables under circumstances that are minimally affected by the
process that leads to the initial state of heavy-ion collisions from nuclear
wavefunction. In this study, we demonstrate that when assuming
scale-invariance, the effect of this stage on the initial energy or entropy
density moments in ultra-central symmetric collisions is negligible for nucleon
sizes of approximately 0.7 fm or larger for large nuclei. By borrowing cluster
expansion method from statistical physics and using scale-invariance
assumption, we calculate the average ellipticity of initial density at the
presence of short-range correlation. We compare our calculations to Monte Carlo
studies and assess the accuracy of various methods of short-range correlation
sampling. Additionally, we find that the isobar ratio can constrain the initial
state parameters, in addition to deformation. Our study indicates that the
isobar ratios in ultra-central collisions are especially sensitive to the
fluctuation in the weight of the nuclei constituents and the two-body
correlation among nucleons. This insight is crucial for drawing conclusions
about nuclear deformations based on isobar ratios.
