CMB in a box: Causal structure and the Fourier-Bessel expansion

L. Raul Abramo, Paulo H. Reimberg, and Henrique S. Xavier
Phys. Rev. D 82, 043510 – Published 6 August 2010

Abstract

This paper makes two points. First, we show that the line-of-sight solution to cosmic microwave anisotropies in Fourier space, even though formally defined for arbitrarily large wavelengths, leads to position-space solutions which only depend on the sources of anisotropies inside the past light cone of the observer. This foretold manifestation of causality in position (real) space happens order by order in a series expansion in powers of the visibility γ=eμ, where μ is the optical depth to Thomson scattering. We show that the contributions of order γN to the cosmic microwave background (CMB) anisotropies are regulated by spacetime window functions which have support only inside the past light cone of the point of observation. Second, we show that the Fourier-Bessel expansion of the physical fields (including the temperature and polarization momenta) is an alternative to the usual Fourier basis as a framework to compute the anisotropies. The viability of the Fourier-Bessel series for treating the CMB is a consequence of the fact that the visibility function becomes exponentially small at redshifts z103, effectively cutting off the past light cone and introducing a finite radius inside which initial conditions can affect physical observables measured at our position x=0 and time t0. Hence, for each multipole there is a discrete tower of momenta ki (not a continuum) which can affect physical observables, with the smallest momenta being k1. The Fourier-Bessel modes take into account precisely the information from the sources of anisotropies that propagates from the initial value surface to the point of observation—no more, no less. We also show that the physical observables (the temperature and polarization maps), and hence the angular power spectra, are unaffected by that choice of basis. This implies that the Fourier-Bessel expansion is the optimal scheme with which one can compute CMB anisotropies.

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  • Received 10 May 2010

DOI:https://doi.org/10.1103/PhysRevD.82.043510

© 2010 The American Physical Society

Authors & Affiliations

L. Raul Abramo*

  • Instituto de Física, Universidade de São Paulo, CP 66318, 05314-970, São Paulo, Brazil and Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, New Jersey 08544, USA

Paulo H. Reimberg and Henrique S. Xavier

  • Instituto de Física, Universidade de São Paulo, CP 66318, 05314-970, São Paulo, Brazil

  • *abramo@fma.if.usp.br

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Vol. 82, Iss. 4 — 15 August 2010

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