no. 4
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.

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.

DOI : 10.1214/10-AIHP409
Classification : 60G60
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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1214/10-AIHP409/

[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. | MR | Zbl

[2] A. O. Barut and R. Rączka. Theory of Group Representations and Applications, 2nd edition. World Scientific, Singapore, 1986. | MR | Zbl

[3] P. Cabella and M. Kamionkowski. Theory of cosmic microwave background polarization, 2005. Available at arXiv:astro-ph/0403392v2.

[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. | MR | Zbl

[5] A. Challinor. Anisotropies in the cosmic microwave background, 2004. Available at arXiv:astro-ph/0403344v1.

[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

[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.

[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). | MR | Zbl

[10] D. Geller, X. Lan and D. Marinucci. Spin needlets spectral estimation. Electron. J. Stat. 3 (2009) 1497-1530. | MR

[11] D. Geller and D. Marinucci. Spin wavelets on the sphere. J. Fourier Anal. Appl. 16 (2010) 840-884. | MR | Zbl

[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 arXiv:0912.3389v1.

[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. | MR | Zbl

[17] D. Marinucci and G. Peccati. Group representations and high-resolution central limit theorems for subordinated spherical random fields. Bernoulli 16 (2010) 798-824. | MR

[18] D. Marinucci and G. Peccati. Representations of SO(3) and angular polyspectra. J. Multivariate Anal. 101 (2010) 77-100. | MR | Zbl

[19] M. A. Naĭmark and A. I. Ŝtern. Theory of Group Representations. Springer, New York, 1982. | MR | Zbl

[20] E. T. Newman and R. Penrose. Note on the Bondi-Metzner-Sachs group. J. Math. Phys. 7 (1966) 863-870. | MR

[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). | MR | Zbl

[23] K. S. Thorne. Multipole expansions of gravitational radiation. Rev. Modern Phys. 52 (1980) 299-339. | MR

[24] N. Ya. Vilenkin. Special Functions and the Theory of Group Representations. Translations of Mathematical Monographs 22. American Mathematical Society, Providence, RI, 1968. | MR | Zbl

[25] S. Weinberg. Cosmology. Oxford Univ. Press, Oxford, 2008. | MR | Zbl

[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. | MR | Zbl

[27] M. Zaldarriaga and U. Seljak. An all-sky analysis of polarisation in the microwave background. Phys. Rev. D 55 (1997) 1830-1840.

Cité par Sources :