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.