http://arxiv.org/abs/2111.15427

We present a multi-scale topological analysis of the temperature fluctuation maps from the NPIPE and FFP10 datasets, invoking relative homology to account for the analysis in the presence of masks. For the topological components, we detect a $2.96\sigma$ deviation between the observations and simulations at $N = 128, FWHM = 80’$, for the FFP10 dataset. For the topological loops, we observe a high deviation between the observation and simulations in the number of loops at $FWHM = 320’$, at a low dimensionless threshold $\nu = -2.5$, for the NPIPE dataset. Under a Gaussian assumption, this would amount to a deviation of $\sim 4\sigma$ . However, the distribution in this bin is manifestly non-Gaussian and does not obey Poisson statistics either. In the absence of a true theoretical understanding, we simply note that the significance is higher than what may be resolved by $600$ simulations. The FFP10 dataset, indicates a $2.77\sigma$ deviation at this resolution and threshold. The Euler characteristic reflects the deviations in the components and loops. To assess the significance of combined levels for a given scale, we employed the empirical and theoretical versions of the $\chi^2$ test as well as the nonparametric Tukey depth test. Although all statistics exhibit a stable distribution, we favor the empirical version of the $\chi^2$ test in the final interpretation, as it indicates the most conservative differences. Even though both datasets exhibit mild to significant discrepancies, they also exhibit contrasting behaviors at various instances. Therefore, we do not find it feasible to convincingly accept or reject the null hypothesis. Disregarding the large-scale anomalies that persist at similar scales in WMAP and Planck, observations of the cosmic microwave background are largely consistent with the standard cosmological model within $2\sigma$.

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P. Pranav
Wed, 1 Dec 21
53/110

Comments: 16 figures, 2 tables. Matches accepted version in A&A. arXiv admin note: substantial text overlap with arXiv:2101.02237

http://arxiv.org/abs/2109.08721

This paper is second in the series, following Pranav et al. (2019), focused on the characterization of geometric and topological properties of 3D Gaussian random fields. We focus on the formalism of persistent homology, the mainstay of Topological Data Analysis (TDA), in the context of excursion set formalism. We also focus on the structure of critical points of stochastic fields, and their relationship with formation and evolution of structures in the universe.
The topological background is accompanied by an investigation of Gaussian field simulations based on the LCDM spectrum, as well as power-law spectra with varying spectral indices. We present the statistical properties in terms of the intensity and difference maps constructed from the persistence diagrams, as well as their distribution functions. We demonstrate that the intensity maps encapsulate information about the distribution of power across the hierarchies of structures in more detailed than the Betti numbers or the Euler characteristic. In particular, the white noise ($n = 0$) case with flat spectrum stands out as the divide between models with positive and negative spectral index. It has the highest proportion of low significance features. This level of information is not available from the geometric Minkowski functionals or the topological Euler characteristic, or even the Betti numbers, and demonstrates the usefulness of hierarchical topological methods. Another important result is the observation that topological characteristics of Gaussian fields depend on the power spectrum, as opposed to the geometric measures that are insensitive to the power spectrum characteristics.

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P. Pranav
Tue, 21 Sep 21
7/85

Comments: 29 pages, 17 figures, 2 tables, comments welcome

http://arxiv.org/abs/2101.02237

We 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.

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P. Pranav
Fri, 8 Jan 21
23/48

Comments: 21 pages, 8 figures, 2 tables

http://arxiv.org/abs/1908.01619

The topology and geometry of random fields – in terms of the Euler characteristic and the Minkowski functionals – has received a lot of attention in the context of the Cosmic Microwave Background (CMB), as the detection of primordial non-Gaussianities would form a valuable clue on the physics of the early Universe. The virtue of both the Euler characteristic and the Minkowski functionals in general, lies in the fact that there exist closed form expressions for their expectation values for Gaussian random fields. However, the Euler characteristic and Minkowski functionals are summarizing characteristics of topology and geometry. Considerably more topological information is contained in the homology of the random field, as it completely describes the creation, merging and disappearance of topological features in superlevel set filtrations.
In the present study we extend the topological analysis of the superlevel set filtrations of two-dimensional Gaussian random fields by analysing the statistical properties of the Betti numbers – counting the number of connected components and loops – and the persistence diagrams – describing the creation and mergers of homological features. Using the link between homology and the critical points of a function – as illustrated by the Morse-Smale complex – we derive a one-parameter fitting formula for the expectation value of the Betti numbers and forward this formalism to the persistent diagrams. We, moreover, numerically demonstrate the sensitivity of the Betti numbers and persistence diagrams to the presence of non-Gaussianities.

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J. Feldbrugge, M. Engelen, R. Weygaert, et. al.
Tue, 6 Aug 19
6/76

Comments: N/A