Geometry and topology of spin random fields

Abstract

Spin (spherical) random fields are very important in many physical applications, in particular they play a key role in Cosmology, especially in connection with the analysis of the Cosmic Microwave Background radiation. These objects can be viewed as random sections of the s-th complex tensor power of the tangent bundle of the 2-sphere. In this paper, we discuss how to characterize their expected geometry and topology. In particular, we investigate the asymptotic behaviour, under scaling assumptions, of general classes of geometric and topological functionals including Lipschitz-Killing Curvatures and Betti numbers for (properly defined) excursion sets; we cover both the cases of fixed and diverging spin parameters s. In the special case of monochromatic fields (i.e., spin random eigenfunctions) our results are particularly explicit; we show how their asymptotic behaviour is non-universal and we can obtain in particular complex versions of Berry's random waves and of Bargmann-Fock's models as subcases of a new generalized model, depending on the rate of divergence of the spin parameter s.

Minkowski Functionals in 𝖲𝖮(3) for the spin-2 CMB polarisation field

“…To overcome this issue, we lift the field to a higher-dimensional space where it can be seen as a scalar field, following the framework introduced in [32]. See also [33] for further mathematical discussion on spin random fields.…”

“…This cannot be the case for fields produced by lifting a spin field, such as CMB polarisation, since all multipoles are JCAP01(2024)039 constituted only by s = ±2. The details of this construction and the consequence on random fields can be found in [32]; some statistical properties of fields where the spin increases with the multipole can be found in [33].…”