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

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 lightcone 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 = e −µ , where µ is the optical depth to Thompson scattering. We show that the contributions of order γ N to the 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 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 = 0 and time t0. Hence, for each multipole there is a discrete tower of momenta k i (not a continuum) which can affect physical observables, with the smallest momenta being k 1 ∼ . 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.

CMB and random flights: temperature and polarization in position space

“…Due to the general vanishing property of the probability density functions for the extended random flight if intermediate displacements do not form a polygon, and the decreasing of the visibility function for z >> 10 3 , we can in practice take all the sources to vanish outside of a sphere of radius R sufficiently large, and calculate the temperature and polarization corrections using Fourier-Bessel expansions, as shown in ref. [1]. In Fourier-Bessel basis only a discretized tower of modes contribute to each observable at each multipole, and the computational advantages of this approach are described in ref.…”

“…If l ≥ L (which is always the case in our iterative solutions), then the product of two spherical Bessel functions of order l can be written in terms of a single spherical Bessel function of order L. This is a consequence of Gegenbauer's relation [17,18] and of the orthogonality of associated Legendre polynomials -see [1] for a derivation:…”

“…(7.1) by taking P (k, η) to be 1 2 Θ 2 (k, η) -i. e., only the temperature quadrupole is taken as a source for the temperature fluctuations, since the polarization (which is of higher order, since its source is the temperature quadrupole) is at least a second-order contribution to the temperature. The first order contribution to the temperature, θ (1) lm , is therefore computed as:…”

“…The argument which leads to these conclusions will take us through a treatment of the CMB in position space and, in many senses, the present work is a further development of [1]. When performing the conversion from Fourier space to position space, it is unavoidable that a certain class of integrals over products of spherical Bessel functions appear in the description.…”

3DEX: a code for fast spherical Fourier-Bessel decomposition of 3D surveys

“…. , N (where N is the number of galaxies in the latter example), the survey may be considered as a superposition of 3D Dirac deltas and each coefficient f mn can simply be estimated with a sum (Heavens & Taylor 1995;Fisher et al 1995;Erdogdu et al 2006b;Abramo et al 2010)…”