Nous développons la théorie des champs aléatoires invariants dans les fibrés vectoriels. Nous obtenons la décomposition spectrale d'un champ aléatoire invariant dans un fibré vectoriel homogène engendré par une représentation induite par un groupe de Lie compact et connexe. Nous discutons une application à la théorie du rayonnement fossile, où G = SO(3). Un théorème sur l'équivalence de deux groupes d'hypothèses cosmologiques est aussi démontré.
We develop the theory of invariant random fields in vector bundles. 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. We discuss an application to the theory of relic radiation, where G = SO(3). A theorem about equivalence of two different groups of assumptions in cosmological theories is proved.
Mots clés : invariant random field, vector bundle, cosmic microwave background
@article{AIHPB_2011__47_4_1068_0, author = {Malyarenko, Anatoliy}, title = {Invariant random fields in vector bundles and application to cosmology}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {1068--1095}, publisher = {Gauthier-Villars}, volume = {47}, number = {4}, year = {2011}, doi = {10.1214/10-AIHP409}, zbl = {1268.60072}, language = {en}, url = {http://www.numdam.org/articles/10.1214/10-AIHP409/} }
TY - JOUR AU - Malyarenko, Anatoliy TI - Invariant random fields in vector bundles and application to cosmology JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2011 SP - 1068 EP - 1095 VL - 47 IS - 4 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/10-AIHP409/ DO - 10.1214/10-AIHP409 LA - en ID - AIHPB_2011__47_4_1068_0 ER -
%0 Journal Article %A Malyarenko, Anatoliy %T Invariant random fields in vector bundles and application to cosmology %J Annales de l'I.H.P. Probabilités et statistiques %D 2011 %P 1068-1095 %V 47 %N 4 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/10-AIHP409/ %R 10.1214/10-AIHP409 %G en %F AIHPB_2011__47_4_1068_0
Malyarenko, Anatoliy. Invariant random fields in vector bundles and application to cosmology. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 1068-1095. doi : 10.1214/10-AIHP409. http://www.numdam.org/articles/10.1214/10-AIHP409/
[1] On the characterization of isotropic Gaussian fields on homogeneous spaces of compact groups. Electron. Comm. Probab. 12 (2007) 291-302. | MR | Zbl
, and .[2] Theory of Group Representations and Applications, 2nd edition. World Scientific, Singapore, 1986. | MR | Zbl
and .[3] Theory of cosmic microwave background polarization, 2005. Available at arXiv:astro-ph/0403392v2.
and .[4] The Helgason Fourier transform for homogeneous vector bundles over compact Riemannian symmetric spaces - The local theory. J. Funct. Anal. 220 (2005) 97-117. | MR | Zbl
.[5] Anisotropies in the cosmic microwave background, 2004. Available at arXiv:astro-ph/0403344v1.
.[6] 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
.[7] 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.
and .[8] The Cosmic Microwave Background. Cambridge Univ. Press, Cambridge, 2008.
.[9] Representations of the group of rotations in three-dimensional space and their applications. Uspehi Mat. Nauk 7 (1952) 3-117 (in Russian). | MR | Zbl
and .[10] Spin needlets spectral estimation. Electron. J. Stat. 3 (2009) 1497-1530. | MR
, and .[11] Spin wavelets on the sphere. J. Fourier Anal. Appl. 16 (2010) 840-884. | MR | Zbl
and .[12] 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] Statistics of cosmic microwave background polarization. Phys. Rev. D 55 (1997) 7368-7388.
, and .[14] On spectral representations of tensor random fields on the sphere, 2009. Available at arXiv:0912.3389v1.
and .[15] 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.
and .[16] High-frequency asymptotics for subordinated stationary fields on an Abelian compact group. Stochastic Process. Appl. 118 (2008) 585-613. | MR | Zbl
and .[17] Group representations and high-resolution central limit theorems for subordinated spherical random fields. Bernoulli 16 (2010) 798-824. | MR
and .[18] Representations of SO(3) and angular polyspectra. J. Multivariate Anal. 101 (2010) 77-100. | MR | Zbl
and .[19] Theory of Group Representations. Springer, New York, 1982. | MR | Zbl
and .[20] Note on the Bondi-Metzner-Sachs group. J. Math. Phys. 7 (1966) 863-870. | MR
and .[21] Statistically homogeneous random fields on a sphere. Uspehi Mat. Nauk 2 (1947) 196-198.
.[22] Spectral theory of n-dimensional stationary stochastic processes with discrete time. Uspehi Mat. Nauk 13 (1958) 93-142 (in Russian). | MR | Zbl
.[23] Multipole expansions of gravitational radiation. Rev. Modern Phys. 52 (1980) 299-339. | MR
.[24] Special Functions and the Theory of Group Representations. Translations of Mathematical Monographs 22. American Mathematical Society, Providence, RI, 1968. | MR | Zbl
.[25] Cosmology. Oxford Univ. Press, Oxford, 2008. | MR | Zbl
.[26] Second-order homogeneous random fields. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II 593-622. Univ. California Press, Berkeley, CA, 1961. | MR | Zbl
.[27] An all-sky analysis of polarisation in the microwave background. Phys. Rev. D 55 (1997) 1830-1840.
and .Cité par Sources :