Expected Lipschitz-Killing curvatures for spin random fields and other non-isotropic fields

Spherical spin random fields are used to model the Cosmic Microwave Background polarization, the study of which is at the heart of modern Cosmology and will be the subject of the LITEBIRD mission, in the 2030s. Its scope is to collect datas to test the theoretical predictions of the Cosmic Inflation model. In particular, the Minkowski functionals, or the Lipschitz-Killing curvatures, of excursion sets can be used to detect deviations from Gaussianity and anisotropies of random fields, being fine descriptors of their geometry and topology. In this paper we give an explicit, non-asymptotic, formula for the expectation of the Lipshitz-Killing curvatures of the excursion set of the real part of an arbitrary left-invariant Gaussian spin spherical random field, seen as a field on $SO(3)$. Our findings are coherent with the asymptotic ones presented in Carrón Duque, J. et al. “Minkowski Functionals in $SO(3)$ for the spin-2 CMB polarisation field”, Journal of Cosmology and Astroparticle Physics (2024). We also give explicit expressions for the Adler-Taylor metric, and its curvature. We obtain such result as an application of a general formula that applies to any nondegenerate Gaussian random field defined on an arbitrary three dimensional compact Riemannian manifold. The novelty is that the Lipschitz-Killing curvatures are computed with respect to an arbitrary metric, possibly different than the Adler-Taylor metric of the field.

💡 Research Summary

The paper addresses a fundamental problem in modern cosmology: how to quantify the geometry and topology of excursion sets of spin‑random fields, which model the polarization of the Cosmic Microwave Background (CMB). While the scalar temperature field has been extensively studied, the polarization field is naturally represented by a complex spin‑2 field on the sphere. By taking the real part of this field one obtains a real‑valued random field on the rotation group SO(3). The authors develop a fully non‑asymptotic, exact formula for the expected Lipschitz‑Killing curvatures (LKCs) of the excursion set ({p\in SO(3): f(p)\ge u}) for any deterministic threshold (u).

The novelty lies in two directions. First, the classical Adler‑Taylor theory provides expectation formulas for LKCs under the assumption that the field’s induced metric (the Adler‑Taylor metric) is proportional to the underlying manifold metric, i.e., the field is isotropic. Spin fields on SO(3) violate this condition, because the metric induced by the field is generally anisotropic. The authors therefore generalize the Adler‑Taylor framework to arbitrary three‑dimensional compact Riemannian manifolds ((M,g)) and arbitrary non‑degenerate Gaussian fields (f). They introduce the eigenvalues (a_1(p),a_2(p),a_3(p)) of the Adler‑Taylor metric (g_f) with respect to the background metric (g) and define two auxiliary functions (E_1) and (E_2) (expectations of rational functions of independent standard normals weighted by the eigenvalues). The main result, Theorem 1.2, expresses the expected LKCs as integrals over (M) of simple functions of (u) multiplied by (E_1) or (E_2), plus terms involving the volume, the Euler characteristic, and the scalar curvature of (g_f). In particular, the formulas for the expected surface area ((L_2)) and mean curvature ((L_1)) are new; they cannot be recovered from the classical theorem unless all eigenvalues are equal to one.

Second, the authors specialize the general theorem to the concrete case of spin‑(s) Gaussian fields on SO(3). Such fields are represented by a series expansion in Wigner‑(D) functions with coefficients (c_\ell) and i.i.d. complex Gaussian variables (\gamma_{\ell m,s}). The covariance function is (k(\theta)=\sum_{\ell\ge|s|}c_\ell^2 d_{\ell,s,s}(\theta)). By evaluating (k(0)) and (-k’’(0)) they define two scalar parameters (\xi^2) and (\xi_{\text{der}}^2) that capture the variance and the “derivative variance” of the field. The Adler‑Taylor metric then has a diagonal Gram matrix with eigenvalues (\xi^2,\xi^2,s^2) in the Euler‑angle coordinates. The scalar curvature of this metric turns out to be constant, (\operatorname{Scal}_f= (2\xi^2-s^2)/(2\xi^4)).

With these explicit eigenvalues the auxiliary functions (E_1) and (E_2) can be computed in closed form (formulas (1.14) and (1.15)), involving elementary logarithms and arcsine functions. Substituting them into Theorem 1.2 yields Theorem 1.1, which gives the expected LKCs for the excursion set of the real part of any left‑invariant spin field on SO(3). The result is presented as a linear combination of the intrinsic LKCs of SO(3) (the volume, surface area, mean curvature, Euler characteristic) multiplied by threshold‑dependent functions (\Xi_i(u)). The (\Xi_i) are expressed through the Gaussian tail (1-\Phi(u)) and the Gaussian density (e^{-u^2/2}) multiplied by constants (d_i(\xi,s)) that depend only on the spin weight and the covariance parameters.

The paper also includes detailed derivations of the Adler‑Taylor metric, its curvature tensors, and the explicit computation of the LKCs for SO(3) under this metric (Appendix A). Appendices B and C contain the proofs of the closed‑form expressions for (E_1) and (E_2) and a thorough exposition of the geometric preliminaries needed for the Lipschitz‑Killing curvature formulas.

From a practical standpoint, the formulas enable a direct, non‑asymptotic statistical analysis of CMB polarization data. By plugging in the empirically estimated angular power spectrum ({c_\ell}) one can compute the expected Minkowski functionals (the physical counterpart of LKCs) for any threshold, without resorting to large‑scale simulations or high‑order approximations. This provides a powerful diagnostic for detecting non‑Gaussianity or statistical anisotropy in upcoming missions such as LiteBIRD.

In summary, the authors have (i) extended the Adler‑Taylor theory to fully anisotropic Gaussian fields on three‑dimensional manifolds, (ii) derived explicit, non‑asymptotic expectation formulas for all four Lipschitz‑Killing curvatures on SO(3), and (iii) demonstrated how these results apply to spin‑random fields relevant for CMB polarization studies. The work opens the door to rigorous geometric‑topological inference on a broad class of non‑isotropic random fields in cosmology and beyond.