Abstract
The fluctuations in the temperature and polarization of the cosmic microwave background are described by a hierarchy of Boltzmann equations. In its integral form, this Boltzmann hierarchy can be converted from the usual Fourier-space base into a position-space and causal description. We shall here restrict ourselves to a particular case when the evolution of the temperature is uncoupled to the polarization, and show that probability densities for random flights play a key role in this description. The integral system can be treated as a perturbative series in the number of steps of the random flights, and the properties of random flight probabilities impose constraints on the domains of dependence.
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References
Abramo, L.R., Reimberg, P.H., Xavier, H.S.: CMB in a box: causal structure and the Fourier-Bessel expansion. Phys. Rev D 82, 043510 (2010)
Bennett, C.L., Larson, D., Weiland, J.L., Jarosik, N., Hinshaw, G., Odegard, N., Smith, K.M., Hill, R.S., Gold, B., Halpern, M., Komatsu, E., Nolta, M.R., Page, L., Spergel, D.N., Wollack, E., Dunkley, J., Kogut, A., Limon, M., Meyer, S.S., Tucker, G.S., Wright, E.L.: Nine-year Wilkinson microwave anisotropy probe (WMAP) observations: final maps and results. ArXiv e-prints (2012). arXiv:1212.5225
Chandrasekhar, S.: Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15 (1), 1–89 (1943)
Dutka, J.: On the problem of random flights. Arch. Hist. Exact Sci. 32 (3), 351–375 (1985)
Leistedt, B., Rassat, A., Réfrégier, A., Starck, J.-L.: 3DEX: a code for fast spherical Fourier-Bessel decomposition of 3D surveys. Astron. Astrophys. 540, A60 (2012)
Planck Collaboration, Ade, P.A.R., Aghanim, N., Armitage-Caplan, C., Arnaud, M., Ashdown, M., Atrio-Barandela, F., Aumont, J., Baccigalupi, C., Banday, A.J., et al.: Planck 2013 results. I. Overview of products and scientific results. Astron. Astrophys. 571, A1 (2014). arXiv:1303. 5062
Reimberg, P.H., Abramo, L.R.: J. Cosmol. Astropart. Phys. 06, 043 (2013)
Seljak, U., Zaldarriaga, M.: A line of sight approach to cosmic microwave background anisotropies. Astrophys. J. 469, 437–444 (1996). arXiv: astro-ph/9603033
Smoot, G.F., et al.: Structure in the COBE differential microwave radiometer first year maps. Astrophys. J. 396, L1–L5 (1992)
Straumann, N.: From primordial quantum fluctuations to the anisotropies of the cosmic microwave background radiation. Ann. Phys. 15, 701–845 (2006)
Talman, J.D.: Special Functions: A Group Theoretic Approach. W. A. Benjamin, New York (1968)
Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944)
Acknowledgements
The authors thank the organizers of the II JPB School of Cosmology, and FAPESP for financial support.
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Reimberg, P.H. (2016). A Path-Integral Approach to CMB. In: Fabris, J., Piattella, O., Rodrigues, D., Velten, H., Zimdahl, W. (eds) The Cosmic Microwave Background. Astrophysics and Space Science Proceedings, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-319-44769-8_12
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DOI: https://doi.org/10.1007/978-3-319-44769-8_12
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