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Invariant random fields in vector bundles and application to cosmology. (English. French summary) Zbl 1268.60072
The author provides various expansions of invariant random fields. Namely, the spectral decomposition of an invariant random field in a homogeneous vector bundle generated by an induced representation of a compact connected Lie group \(G\) is obtained. More details could be found in the recent book by the author [Invariant random fields on spaces with a group action. Berlin: Springer (2013; Zbl 1268.60006)]. The important case \(G=\mathrm{SO}(3)\) is considered as it is related with an analysis of the cosmic microwave background radiation. Various approaches to the study of relic radiation are compared. The role of the expansion coefficients, called electric multipoles and magnetic multipoles, is clarified.

MSC:
60G60 Random fields
Citations:
Zbl 1268.60006
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[1] P. Baldi, D. Marinucci and V. S. Varadarajan. On the characterization of isotropic Gaussian fields on homogeneous spaces of compact groups. Electron. Comm. Probab. 12 (2007) 291-302. · Zbl 1128.60039
[2] A. O. Barut and R. Rączka. Theory of Group Representations and Applications , 2nd edition. World Scientific, Singapore, 1986. · Zbl 0644.22011
[3] P. Cabella and M. Kamionkowski. Theory of cosmic microwave background polarization, 2005. Available at .
[4] R. Camporesi. The Helgason Fourier transform for homogeneous vector bundles over compact Riemannian symmetric spaces - The local theory. J. Funct. Anal. 220 (2005) 97-117. · Zbl 1060.43004
[5] A. Challinor. Anisotropies in the cosmic microwave background, 2004. Available at . · Zbl 1068.85501
[6] A. Challinor. Cosmic microwave background polarisation analysis. In Data Analysis in Cosmology 121-158. V. J. Martinez, E. Saar, E. Martínez-González and M.-J. Pons-Borderia (Eds). Lecture Notes in Phys. 665 . Springer, Berlin, 2009. · Zbl 1179.85027
[7] A. Challinor and H. Peiris. Lecture notes on the physics of cosmic microwave background anisotropies. In Cosmology and Gravitation: XIII Brazilian School on Cosmology and Gravitation (XIII BSCG), Rio de Janeiro (Brazil), 20 July-2 August 2008 86-140. M. Novello and S. Perez (Eds). AIP Conf. Proc. 1132 . American Institute of Physics, Melville, NY, 2008.
[8] R. Durrer. The Cosmic Microwave Background . Cambridge Univ. Press, Cambridge, 2008. · Zbl 0875.83076
[9] I. M. Gelfand and Z. Ya. Shapiro. Representations of the group of rotations in three-dimensional space and their applications. Uspehi Mat. Nauk 7 (1952) 3-117 (in Russian).
[10] D. Geller, X. Lan and D. Marinucci. Spin needlets spectral estimation. Electron. J. Stat. 3 (2009) 1497-1530. · Zbl 1326.62195
[11] D. Geller and D. Marinucci. Spin wavelets on the sphere. J. Fourier Anal. Appl. 16 (2010) 840-884. · Zbl 1206.42039
[12] A. H. Jaffe. Bayesian analysis of cosmic microwave background data. In Bayesian Methods in Cosmology 229-244. M. P. Hobson, A. H. Jaffe, A. R. Liddle, P. Mukherjee and D. Parkinson (Eds). Cambridge Univ. Press, Cambridge, 2009.
[13] M. Kamionkowski, A. Kosowsky and A. Stebbins. Statistics of cosmic microwave background polarization. Phys. Rev. D 55 (1997) 7368-7388.
[14] N. Leonenko and L. Sakhno. On spectral representations of tensor random fields on the sphere, 2009. Available at . · Zbl 1239.60038
[15] Y.-T. Lin and B. D. Wandelt. A beginner’s guide to the theory of CMB temperature and polarization power spectra in the line-of-sight formalism. Astroparticle Physics 25 (2006) 151-166.
[16] D. Marinucci and G. Peccati. High-frequency asymptotics for subordinated stationary fields on an Abelian compact group. Stochastic Process. Appl. 118 (2008) 585-613. · Zbl 1143.60007
[17] D. Marinucci and G. Peccati. Group representations and high-resolution central limit theorems for subordinated spherical random fields. Bernoulli 16 (2010) 798-824. · Zbl 1284.60099
[18] D. Marinucci and G. Peccati. Representations of SO (3) and angular polyspectra. J. Multivariate Anal. 101 (2010) 77-100. · Zbl 1216.60027
[19] M. A. Naĭmark and A. I. Ŝtern. Theory of Group Representations . Springer, New York, 1982.
[20] E. T. Newman and R. Penrose. Note on the Bondi-Metzner-Sachs group. J. Math. Phys. 7 (1966) 863-870.
[21] A. M. Obukhov. Statistically homogeneous random fields on a sphere. Uspehi Mat. Nauk 2 (1947) 196-198.
[22] Yu. A. Rozanov. Spectral theory of n -dimensional stationary stochastic processes with discrete time. Uspehi Mat. Nauk 13 (1958) 93-142 (in Russian).
[23] K. S. Thorne. Multipole expansions of gravitational radiation. Rev. Modern Phys. 52 (1980) 299-339.
[24] N. Ya. Vilenkin. Special Functions and the Theory of Group Representations. Translations of Mathematical Monographs 22 . American Mathematical Society, Providence, RI, 1968. · Zbl 0172.18404
[25] S. Weinberg. Cosmology . Oxford Univ. Press, Oxford, 2008.
[26] A. M. Yaglom. Second-order homogeneous random fields. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob. , Vol. II 593-622. Univ. California Press, Berkeley, CA, 1961. · Zbl 0123.35001
[27] M. Zaldarriaga and U. Seljak. An all-sky analysis of polarisation in the microwave background. Phys. Rev. D 55 (1997) 1830-1840.
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