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 $Persistence$ $Signals$, an innovative method from recent literature that facilitates the embedding of persistence diagrams into vector spaces by re-conceptualizing them as signals in $\mathbb R^2_+$. Utilizing this methodology, we analyze three quintessential cosmic structures: clusters, filaments, and voids. A central discovery is the correlation between $Persistence$ $Energy$ and redshift values, linking persistent homology with cosmic evolution and providing insights into the dynamics of cosmic structures.