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

Journal Article · Sun Aug 15 04:00:00 EDT 2010 · Physical Review. D, Particles Fields

DOI:https://doi.org/10.1103/PHYSREVD.82.043510· OSTI ID:21432924

Abramo, L Raul; Reimberg, Paulo H; Xavier, Henrique S ^[1]

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1. Instituto de Fisica, Universidade de Sao Paulo, CP 66318, 05314-970, Sao Paulo, Brazil and Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, New Jersey 08544 (United States)

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 {gamma}=e{sup -}{mu}, where {mu} is the optical depth to Thomson scattering. We show that the contributions of order {gamma}{sup N} 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 z>>10{sup 3}, 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-vector=0 and time t{sub 0}. Hence, for each multipole l there is a discrete tower of momenta k{sub il} (not a continuum) which can affect physical observables, with the smallest momenta being k{sub 1l{approx}}l. 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.

• OSTI ID: 21432924
• Journal Information: Physical Review. D, Particles Fields, Journal Name: Physical Review. D, Particles Fields Journal Issue: 4 Vol. 82; ISSN PRVDAQ; ISSN 0556-2821
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANISOTROPY
CAUSALITY
ELECTROMAGNETIC RADIATION
ELECTRONS
ELEMENTARY PARTICLES
FERMIONS
INELASTIC SCATTERING
LEPTONS
LIGHT CONE
MATHEMATICAL SOLUTIONS
MICROWAVE RADIATION
POLARIZATION
RADIATIONS
RED SHIFT
RELICT RADIATION
SCATTERING
SERIES EXPANSION
SIMULATION
SPACE
SPACE-TIME
SPECTRA
SURFACES
THOMSON SCATTERING
WAVELENGTHS