Application and development of advanced mathematical tools for population and time series analysis in pulsar astrophysics

TL;DR

This work develops and applies a cohesive mathematical toolkit to high-energy astrophysical populations, extending beyond traditional $P$–$\dot P$ representations. By integrating Principal Component Analysis, minimum spanning trees (Pulsar Tree), and betweenness centrality, it uncovers multidimensional structure in pulsar populations, MSP binaries, and FRB repeats, including the Pulsar Tree web for interactive exploration. It further introduces Dynamic Time Warping to quantify gamma-ray light-curve morphology, enabling objective clustering of pulsar light curves and revealing geometry-dominated emission patterns. The combined framework yields new classifications, predictive insights for binary MSPs, FRB repeater candidate identification, and a novel perspective on population evolution and evolution tracks, offering meaningful alternatives to conventional analyses in high-energy astrophysics.

Abstract

In this thesis, we introduce novel methods for analyzing pulsar populations using a variety of mathematical techniques. These tools-particularly graph theory-have been thoroughly validated in advanced mathematics, enabling us to overcome some of the constraints (even dimensional) inherent in conventional visualization approaches. This exploration benefits from dimensionality reduction techniques, which not only lessen computational demands but also highlight potential for describing physical characteristics. The resulting structures encode information about pulsar similarities that extend beyond standard spin parameters, revealing relationships that are not readily apparent in traditional diagrams. With a physically motivated topological perspective, we leverage the strengths of these methods and present results that span from prospective source classification and the emergence of new classes to catalog comparison, among other applications. This new approach enables fresh interpretations of longstanding problems, laying a new foundation for visualizing the pulsar population and categorizing sources. Building on this, we identify several sources as likely members of specific binary subclasses and investigate the potential transitional nature of others. Furthermore, we extend the use of graph theory to the boundary of machine learning, demonstrating its capability for binary separation in an unsupervised context. Finally, we introduce and apply an innovative, flexible time-series alignment technique to the field of gamma-ray astrophysics. The method identifies notable similarities among the light curves of gamma-ray pulsars. The results presented here are promising, offering a refreshing direction for the field and new pathways for rigorous mathematical analysis, ultimately providing meaningful alternatives to traditional approaches in high-energy astrophysics.

Figures (60)

• Figure 1: First column: shows the actual first detection charts taken from Jocelyn Bell (figure adopted from https://www.cam.ac.uk/stories/journeysofdiscovery-pulsars). The first panel displays the initial record of CP1919, dated August 6, 1967, in which the anomalous regular pulse was observed. The second panel presents the record dated November 28, 1967, which consolidated the previous observations and, as a result, confirmed the detection of the first pulsar: PSR B1919+21. Second column: Stacked plot on CP1919, also known as a plot illustrating successive mean pulse spectra, as referenced in Harold Dumont Craft Jr, who titled it as "Many consecutive pulses from CP1919" seen in his PhD dissertation "Radio observations of the pulse profiles and dispersion measures of twelve pulsars", Cornell University, 1970, p. 214 (figure adopted from Turchetti_2023, see this publication for more information).
• Figure 2: The Crab Nebula as seen by the James Webb Space Telescope in infrared light. At the center of this ring-shaped structure is a bright white spot: the Crab Pulsar (Credits: NASA, ESA, CSA, STScI, Jeff Hester (ASU), Allison Loll (ASU), Tea Temim (Princeton University)).
• Figure 3: Scheme of the pulsar "lighthouse" model showing its emission geometry (not drawn to scale). The rotation axis (blue) is inclined concerning the magnetic axis (red), producing a beam (yellow) of radiation that sweeps across the observer's line of sight. The description of the rest of the scheme can be found in Section \ref{['chapter1: magnetosphere']} (figure adopted from Lorimer2012).
• Figure 4: The $P\dot P$ diagram based on the the ATNF v2.6.1 restricted to $\dot P>0$ resulting in 2752 pulsars. Lines of constant magnetic field at the surface ($B_s$) in dashed, characteristic age ($\tau_c$) in dotted, and spin-down energy loss rate ($\dot{E}_{sd}$) in dash-dotted are also shown.
• Figure 5: Representation of a graph composed of nodes (green) and edges (black). An edge indicates a direct relationship between two nodes. This relationship can be quantified, for example, by assigning a weight to the corresponding edge, which represents a distance or similarity.