Felix: A Topology based Framework for Visual Exploration of Cosmic Filaments

Abstract

The large-scale structure of the universe is comprised of virialized blob-like clusters, linear filaments, sheet-like walls and huge near empty three-dimensional voids. Characterizing the large scale universe is essential to our understanding of the formation and evolution of galaxies. The density range of clusters, walls and voids are relatively well separated, when compared to filaments, which span a relatively larger range. The large scale filamentary network thus forms an intricate part of the cosmic web. In this paper, we describe Felix, a topology based framework for visual exploration of filaments in the cosmic web. The filamentary structure is represented by the ascending manifold geometry of the 2-saddles in the Morse-Smale complex of the density field. We generate a hierarchy of Morse-Smale complexes and query for filaments based on the density ranges at the end points of the filaments. The query is processed efficiently over the entire hierarchical Morse-Smale complex, allowing for interactive visualization. We apply Felix to computer simulations based on the heuristic Voronoi kinematic model and the standard $Λ$CDM cosmology, and demonstrate its usefulness through two case studies. First, we extract cosmic filaments within and across cluster like regions in Voronoi kinematic simulation datasets. We demonstrate that we produce similar results to existing structure finders. Filaments that form the spine of the cosmic web, which exist in high density regions in the current epoch, are isolated using Felix. Also, filaments present in void-like regions are isolated and visualized. These filamentary structures are often over shadowed by higher density range filaments and are not easily characterizable and extractable using other filament extraction methodologies.

Figures (9)

• Figure 1: Ascending manifolds of a 2-saddle (yellow sphere). The scalar function is a sum of two 3D Gaussians centered on either side of the volume. The two arcs incident on the 2-saddle constitute the ascending manifold and terminate at the two maxima (red spheres) of the scalar function.
• Figure 2: Topological cancellation of a pair of critical points in a 2D Morse-Smale complex. (a) Morse-Smale complex of the function shown in Figure 1 restricted to a 2D slice. Maxima are denoted by $\odot$, saddles by $\oplus$, and minima by $\circledcirc$. A pair ($p_{i+1}, q_i$) of critical points connected by a single arc is scheduled to be canceled. (b) $D(p_{i+1})$ is the set of surviving index $i$ critical points connected to $p_{i+1}$ and $A(q_i)$ is the set of surviving index $i+1$ critical points connected to $q_i$. (c) Combinatorial realization: connect all critical points $D(p_{i+1})$ to those in $A(q_i)$. Geometric realization: merge the descending manifold of $p_{i+1}$ with those of critical points in $A(q_i)$. Merge the ascending manifold of $q_i$ with those of critical points in $D(p_{i+1})$.
• Figure 3: Hierarchical Morse-Smale complex. A family of Morse-Smale complexes generated by iteratively canceling pairs of critical points. $MSC_0$ is the Morse-Smale complex of a 2D equivalent of the function shown in Figure \ref{['fig:max-sad-concept']}. It is simplified to generate a coarser version, $MSC_1$, by canceling a pair of critical points (cyan) connected by a single arc and having least absolute difference in function value $t_1$. Successive versions $MSC_i$ are computed similarly by selecting arcs so that $t_0 = 0 \leq t_1 \leq \ldots \leq t_6$.
• Figure 4: \ref{['fig:2gauss_persplot']} A scatter plot of the function values of the canceled critical point pairs for the function shown in Figure \ref{['fig:max-sad-concept']}. A 2-saddle-maximum pair is the only pair that is far removed from the diagonal. This corresponds to cancellation of the 2-saddle with a maximum that represents one of the Gaussians in Figure \ref{['fig:max-sad-concept']}. Other pairs close to the diagonal represent insignificant features that manifest due to the added Gaussian noise as well as sampling noise. \ref{['fig:vk_new_3_persplot']} A scatter plot of the function values of the canceled critical point pairs for the Voronoi-Kinematic dataset B (see Section \ref{['subsec:voronoi-models']}). No discernible separation of points is seen, though there are many points that are far removed from the diagonal. Thus, no clear global simplification threshold may be used for filament extraction.
• Figure 5: (a) Filaments are modeled as the ascending paths of 2-saddles connecting two extrema. The 2-saddles are filtered based on the range constraints $[M_b,M_e]$ and $[S_b,S_e]$ on the highest and lowest values respectively along the ascending paths. The highest values along the 2-saddle's ascending manifold are at extrema and the lowest value is at the 2-saddle. The function along the paths needs to be simplified as it is rarely smooth. In the illustration, a simplification threshold of $t$ reveals a filament with appropriate density characteristics. However, imposing such a threshold uniformly will cause another filament (b) having the required density characteristics to be destroyed. It is therefore necessary to extract filaments by querying all Morse-Smale complexes within a given hierarchy.