Hierarchical Clustering in $Λ$CDM Cosmologies via Persistence Energy

In this research, we investigate the structural evolution of the cosmic web, employing advanced methodologies from Topological Data Analysis. Our approach involves leveraging LITE, an innovative method from recent literature that embeds persistence diagrams into elements of vector spaces. Utilizing this methodology, we analyze three quintessential cosmic structures: clusters, filaments, and voids. A central discovery is the correlation between \textit{Persistence Energy} and redshift values, linking persistent homology with cosmic evolution and providing insights into the dynamics of cosmic structures.