🍩 Database of Original & NonTheoretical Uses of Topology
(found 9 matches in 0.002295s)


Persistent Homology in Cosmic Shear: Constraining Parameters With Topological Data Analysis (2021)
Sven Heydenreich, Benjamin Brück, Joachim HarnoisDérapsAbstract
In recent years, cosmic shear has emerged as a powerful tool for studying the statistical distribution of matter in our Universe. Apart from the standard twopoint 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+VIKING450 data: We extract the Betti functions from a suite of \textlessi\textgreaterN\textlessi/\textgreaterbody 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/\textgreaterlike 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. 
Topological Echoes of Primordial Physics in the Universe at Large Scales (2020)
Alex Cole, Matteo Biagetti, Gary ShiuAbstract
We 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 nonGaussianities (NG) of primordial density perturbations. We compute persistence diagrams and derived statistics for simulations of dark matter halos with Gaussian and nonGaussian initial conditions. For computational reasons and to make contact with experimental observations, our pipeline computes persistence in subboxes 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 firstprinciples calculations and make new predictions. 
The Persistence of Large Scale Structures I: Primordial NonGaussianity (2020)
Matteo Biagetti, Alex Cole, Gary ShiuAbstract
We develop an analysis pipeline for characterizing the topology of large scale structure and extracting cosmological constraints based on persistent homology. Persistent homology is a technique from topological data analysis that quantifies the multiscale topology of a data set, in our context unifying the contributions of clusters, filament loops, and cosmic voids to cosmological constraints. We describe how this method captures the imprint of primordial local nonGaussianity on the latetime distribution of dark matter halos, using a set of Nbody simulations as a proxy for real data analysis. For our best single statistic, running the pipeline on several cubic volumes of size \$40~(\rm\Gpc/h\)\textasciicircum\3\\$, we detect \$f_\\rm NL\\textasciicircum\\rm loc\=10\$ at \$97.5\%\$ confidence on \$\sim 85\%\$ of the volumes. Additionally we test our ability to resolve degeneracies between the topological signature of \$f_\\rm NL\\textasciicircum\\rm loc\\$ and variation of \$\sigma_8\$ and argue that correctly identifying nonzero \$f_\\rm NL\\textasciicircum\\rm loc\\$ in this case is possible via an optimal template method. Our method relies on information living at \$\mathcal\O\(10)\$ Mpc/h, a complementary scale with respect to commonly used methods such as the scaledependent bias in the halo/galaxy power spectrum. Therefore, while still requiring a large volume, our method does not require sampling longwavelength modes to constrain primordial nonGaussianity. Moreover, our statistics are interpretable: we are able to reproduce previous results in certain limits and we make new predictions for unexplored observables, such as filament loops formed by dark matter halos in a simulation box. 
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 WeygaertAbstract
We 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 multiscale, 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/\textgreatertest shows differences between observations and simulations, yielding \textlessi\textgreaterp\textlessi/\textgreatervalues 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. Nonparametric tests show even stronger differences at almost all scales. Crucially, Gaussian simulations based on powerspectrum 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 latetime 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 superhorizon scales involved, may motivate the study of primordial nonGaussianity. Alternative scenarios worth exploring may be models with nontrivial topology, including topological defect models. 
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 WintraeckenAbstract
Abstract. We introduce a multiscale topological description of the Megaparsec weblike cosmic matter distribution. Betti numbers and topological persistence of 
Cosmic Web Reconstruction Through Density Ridges: Method and Algorithm (2015)
YenChi Chen, Shirley Ho, Peter E. Freeman, Christopher R. Genovese, Larry WassermanAbstract
The detection and characterization of filamentary structures in the cosmic web allows cosmologists to constrain parameters that dictate the evolution of the Universe. While many filament estimators have been proposed, they generally lack estimates of uncertainty, reducing their inferential power. In this paper, we demonstrate how one may apply the subspace constrained mean shift (SCMS) algorithm (Ozertem & Erdogmus 2011; Genovese et al. 2014) to uncover filamentary structure in galaxydata. The SCMS algorithm is a gradient ascent method that models filaments as density ridges, onedimensional smooth curves that trace highdensity regions within the point cloud. We also demonstrate how augmenting the SCMS algorithm with bootstrapbased methods of uncertainty estimation allows one to place uncertainty bands around putative filaments. We apply the SCMS first to the data set generated from the Voronoi model. The density ridges show strong agreement with the filaments from Voronoi method. We then apply the SCMS method data sets sampled from a P3M Nbody simulation, with galaxy number densities consistent with SDSS and WFIRSTAFTA, and to LOWZ and CMASS data from the Baryon Oscillation Spectroscopic Survey (BOSS). To further assess the efficacy of SCMS, we compare the relative locations of BOSS filaments with galaxy clusters in the redMaPPer catalogue, and find that redMaPPer clusters are significantly closer (with pvalues \textless10−9) to SCMSdetected filaments than to randomly selected galaxies. 
Alpha, Betti and the Megaparsec Universe: On the Topology of the Cosmic Web (2011)
Rien Van De Weygaert, Gert Vegter, Herbert Edelsbrunner, Bernard J. T. Jones, Pratyush Pranav, Changbom Park, Wojciech A. Hellwing, Bob Eldering, Nico Kruithof, E. G. P. Bos, Johan Hidding, Job Feldbrugge, Eline Ten Have, Matti Van Engelen, Manuel Caroli, Monique TeillaudAbstract
We study the topology of the Megaparsec Cosmic Web in terms of the scaledependent Betti numbers, which formalize the topological information content of... 
The Persistent Cosmic Web and Its Filamentary Structure – II. Illustrations (2011)
T. Sousbie, C. Pichon, H. KawaharaAbstract
Abstract. The recently introduced discrete persistent structure extractor (DisPerSE, Sousbie, Paper I) is implemented on realistic 3D cosmological simulations