🍩 Database of Original & Non-Theoretical Uses of Topology
(found 9 matches in 0.002928s)
Loops Abound in the Cosmic Microwave Background: A \$4\sigma\$ Anomaly on Super-Horizon Scales (2021)Pratyush Pranav
AbstractWe present a topological analysis of the temperature fluctuation maps from the \emph\Planck 2020\ Data release 4 (DR4) based on the \texttt\NPIPE\ data processing pipeline. For comparison, we also present the topological characteristics of the maps from \emph\Planck 2018\ Data release 3 (DR3). We perform our analysis in terms of the homology characteristics of the maps, invoking relative homology to account for analysis in the presence of masks. We perform our analysis for a range of smoothing scales spanning sub- and super-horizon scales corresponding to \$FWHM = 5', 10', 20', 40', 80', 160', 320', 640'\$. Our main result indicates a significantly anomalous behavior of the loops in the observed maps compared to simulations that are modeled as isotopic and homogeneous Gaussian random fields. Specifically, we observe a \$4\sigma\$ deviation between the observation and simulations in the number of loops at \$FWHM = 320'\$ and \$FWHM = 640'\$, corresponding to super-horizon scales of \$5\$ degrees and larger. In addition, we also notice a mildly significant deviation at \$2\sigma\$ for all the topological descriptors for almost all the scales analyzed. Our results show a consistency across different data releases, and therefore, the anomalous behavior deserves a careful consideration regarding its origin and ramifications. Disregarding the unlikely source of the anomaly being instrumental systematics, the origin of the anomaly may be genuinely astrophysical -- perhaps due to a yet unresolved foreground, or truly primordial in nature. Given the nature of the topological descriptors, that potentially encodes information of all orders, non-Gaussianities, of either primordial or late-type nature, may be potential candidates. Alternate possibilities include the Universe admitting a non-trivial global topology, including effects induced by large-scale topological defects.
Persistent Homology in Cosmic Shear: Constraining Parameters With Topological Data Analysis (2021)Sven Heydenreich, Benjamin Brück, Joachim Harnois-Déraps
AbstractIn recent years, cosmic shear has emerged as a powerful tool for studying the statistical distribution of matter in our Universe. Apart from the standard two-point correlation functions, several alternative methods such as peak count statistics offer competitive results. Here we show that persistent homology, a tool from topological data analysis, can extract more cosmological information than previous methods from the same data set. For this, we use persistent Betti numbers to efficiently summarise the full topological structure of weak lensing aperture mass maps. This method can be seen as an extension of the peak count statistics, in which we additionally capture information about the environment surrounding the maxima. We first demonstrate the performance in a mock analysis of the KiDS+VIKING-450 data: We extract the Betti functions from a suite of \textlessi\textgreaterN\textlessi/\textgreater-body simulations and use these to train a Gaussian process emulator that provides rapid model predictions; we next run a Markov chain Monte Carlo analysis on independent mock data to infer the cosmological parameters and their uncertainties. When comparing our results, we recover the input cosmology and achieve a constraining power on that is 3% tighter than that on peak count statistics. Performing the same analysis on 100 deg\textlesssup\textgreater2\textlesssup/\textgreater of \textlessi\textgreaterEuclid\textlessi/\textgreater-like simulations, we are able to improve the constraints on \textlessi\textgreaterS\textlessi/\textgreater\textlesssub\textgreater8\textlesssub/\textgreater and Ω\textlesssub\textgreaterm\textlesssub/\textgreater by 19% and 12%, respectively, while breaking some of the degeneracy between \textlessi\textgreaterS\textlessi/\textgreater\textlesssub\textgreater8\textlesssub/\textgreater and the dark energy equation of state. To our knowledge, the methods presented here are the most powerful topological tools for constraining cosmological parameters with lensing data.
Persistent Homology of the Cosmic Web. I: Hierarchical Topology in \$\Lambda\$CDM Cosmologies (2021)Georg Wilding, Keimpe Nevenzeel, Rien van de Weygaert, Gert Vegter, Pratyush Pranav, Bernard J. T. Jones, Konstantinos Efstathiou, Job Feldbrugge
AbstractUsing a set of \$\Lambda\$CDM simulations of cosmic structure formation, we study the evolving connectivity and changing topological structure of the cosmic web using state-of-the-art tools of multiscale topological data analysis (TDA). We follow the development of the cosmic web topology in terms of the evolution of Betti number curves and feature persistence diagrams of the three (topological) classes of structural features: matter concentrations, filaments and tunnels, and voids. The Betti curves specify the prominence of features as a function of density level, and their evolution with cosmic epoch reflects the changing network connections between these structural features. The persistence diagrams quantify the longevity and stability of topological features. In this study we establish, for the first time, the link between persistence diagrams, the features they show, and the gravitationally driven cosmic structure formation process. By following the diagrams' development over cosmic time, the link between the multiscale topology of the cosmic web and the hierarchical buildup of cosmic structure is established. The sharp apexes in the diagrams are intimately related to key transitions in the structure formation process. The apex in the matter concentration diagrams coincides with the density level at which, typically, they detach from the Hubble expansion and begin to collapse. At that level many individual islands merge to form the network of the cosmic web and a large number of filaments and tunnels emerge to establish its connecting bridges. The location trends of the apex possess a self-similar character that can be related to the cosmic web's hierarchical buildup. We find that persistence diagrams provide a significantly higher and more profound level of information on the structure formation process than more global summary statistics like Euler characteristic or Betti numbers.
Topological Echoes of Primordial Physics in the Universe at Large Scales (2020)Alex Cole, Matteo Biagetti, Gary Shiu
AbstractWe present a pipeline for characterizing and constraining initial conditions in cosmology via persistent homology. The cosmological observable of interest is the cosmic web of large scale structure, and the initial conditions in question are non-Gaussianities (NG) of primordial density perturbations. We compute persistence diagrams and derived statistics for simulations of dark matter halos with Gaussian and non-Gaussian initial conditions. For computational reasons and to make contact with experimental observations, our pipeline computes persistence in sub-boxes of full simulations and simulations are subsampled to uniform halo number. We use simulations with large NG (\$f_\\rm NL\\textasciicircum\\rm loc\=250\$) as templates for identifying data with mild NG (\$f_\\rm NL\\textasciicircum\\rm loc\=10\$), and running the pipeline on several cubic volumes of size \$40~(\textrm\Gpc/h\)\textasciicircum\3\\$, we detect \$f_\\rm NL\\textasciicircum\\rm loc\=10\$ at \$97.5\%\$ confidence on \$\sim 85\%\$ of the volumes for our best single statistic. Throughout we benefit from the interpretability of topological features as input for statistical inference, which allows us to make contact with previous first-principles calculations and make new predictions.
Unexpected Topology of the Temperature Fluctuations in the Cosmic Microwave Background (2019)Pratyush Pranav, Robert J. Adler, Thomas Buchert, Herbert Edelsbrunner, Bernard J. T. Jones, Armin Schwartzman, Hubert Wagner, Rien van de Weygaert
AbstractWe study the topology generated by the temperature fluctuations of the cosmic microwave background (CMB) radiation, as quantified by the number of components and holes, formally given by the Betti numbers, in the growing excursion sets. We compare CMB maps observed by the \textlessi\textgreaterPlanck\textlessi/\textgreater satellite with a thousand simulated maps generated according to the ΛCDM paradigm with Gaussian distributed fluctuations. The comparison is multi-scale, being performed on a sequence of degraded maps with mean pixel separation ranging from 0.05 to 7.33°. The survey of the CMB over 𝕊\textlesssup\textgreater2\textlesssup/\textgreater is incomplete due to obfuscation effects by bright point sources and other extended foreground objects like our own galaxy. To deal with such situations, where analysis in the presence of “masks” is of importance, we introduce the concept of relative homology. The parametric \textlessi\textgreaterχ\textlessi/\textgreater\textlesssup\textgreater2\textlesssup/\textgreater-test shows differences between observations and simulations, yielding \textlessi\textgreaterp\textlessi/\textgreater-values at percent to less than permil levels roughly between 2 and 7°, with the difference in the number of components and holes peaking at more than 3\textlessi\textgreaterσ\textlessi/\textgreater sporadically at these scales. The highest observed deviation between the observations and simulations for \textlessi\textgreaterb\textlessi/\textgreater\textlesssub\textgreater0\textlesssub/\textgreater and \textlessi\textgreaterb\textlessi/\textgreater\textlesssub\textgreater1\textlesssub/\textgreater is approximately between 3\textlessi\textgreaterσ\textlessi/\textgreater and 4\textlessi\textgreaterσ\textlessi/\textgreater at scales of 3–7°. There are reports of mildly unusual behaviour of the Euler characteristic at 3.66° in the literature, computed from independent measurements of the CMB temperature fluctuations by \textlessi\textgreaterPlanck\textlessi/\textgreater’s predecessor, the \textlessi\textgreaterWilkinson\textlessi/\textgreater Microwave Anisotropy Probe (WMAP) satellite. The mildly anomalous behaviour of the Euler characteristic is phenomenologically related to the strongly anomalous behaviour of components and holes, or the zeroth and first Betti numbers, respectively. Further, since these topological descriptors show consistent anomalous behaviour over independent measurements of \textlessi\textgreaterPlanck\textlessi/\textgreater and WMAP, instrumental and systematic errors may be an unlikely source. These are also the scales at which the observed maps exhibit low variance compared to the simulations, and approximately the range of scales at which the power spectrum exhibits a dip with respect to the theoretical model. Non-parametric tests show even stronger differences at almost all scales. Crucially, Gaussian simulations based on power-spectrum matching the characteristics of the observed dipped power spectrum are not able to resolve the anomaly. Understanding the origin of the anomalies in the CMB, whether cosmological in nature or arising due to late-time effects, is an extremely challenging task. Regardless, beyond the trivial possibility that this may still be a manifestation of an extreme Gaussian case, these observations, along with the super-horizon scales involved, may motivate the study of primordial non-Gaussianity. Alternative scenarios worth exploring may be models with non-trivial topology, including topological defect models.
Topological Data Analysis and Diagnostics of Compressible Magnetohydrodynamic Turbulence (2018)Irina Makarenko, Paul Bushby, Andrew Fletcher, Robin Henderson, Nikolay Makarenko, Anvar Shukurov
AbstractThe predictions of mean-field electrodynamics can now be probed using direct numerical simulations of random flows and magnetic fields. When modelling astrophysical magnetohydrodynamics, it is important to verify that such simulations are in agreement with observations. One of the main challenges in this area is to identify robust quantitative measures to compare structures found in simulations with those inferred from astrophysical observations. A similar challenge is to compare quantitatively results from different simulations. Topological data analysis offers a range of techniques, including the Betti numbers and persistence diagrams, that can be used to facilitate such a comparison. After describing these tools, we first apply them to synthetic random fields and demonstrate that, when the data are standardized in a straightforward manner, some topological measures are insensitive to either large-scale trends or the resolution of the data. Focusing upon one particular astrophysical example, we apply topological data analysis to H i observations of the turbulent interstellar medium (ISM) in the Milky Way and to recent magnetohydrodynamic simulations of the random, strongly compressible ISM. We stress that these topological techniques are generic and could be applied to any complex, multi-dimensional random field.
The Topology of the Cosmic Web in Terms of Persistent Betti Numbers (2017)Pratyush Pranav, Herbert Edelsbrunner, Rien van de Weygaert, Gert Vegter, Michael Kerber, Bernard J. T. Jones, Mathijs Wintraecken
AbstractAbstract. We introduce a multiscale topological description of the Megaparsec web-like cosmic matter distribution. Betti numbers and topological persistence of
Felix: A Topology Based Framework for Visual Exploration of Cosmic Filaments (2016)Nithin Shivshankar, Pratyush Pranav, Vijay Natarajan, Rien van de Weygaert, E. G. Patrick Bos, Steven Rieder
AbstractThe 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 \$\Lambda\$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.