Journal
JOURNAL OF COSMOLOGY AND ASTROPARTICLE PHYSICS
Volume -, Issue 9, Pages -Publisher
IOP Publishing Ltd
DOI: 10.1088/1475-7516/2019/09/052
Keywords
non-gaussianity; CMBR theory
Funding
- Government of Canada through Industry Canada
- Province of Ontario through the Ministry of Economic Development, Job Creation and Trade
- John Templeton Foundation [FP5136-O]
- ERC advanced grant ARTHUS [740021]
- European Research Council (ERC) [740021] Funding Source: European Research Council (ERC)
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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 persistence diagrams. We, moreover, numerically demonstrate the sensitivity of the Betti numbers and persistence diagrams to the presence of non-Gaussianities.
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