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

TL;DR

This paper demonstrates that CMB anisotropies depend only on sources within the observer's past light-cone and introduces the Fourier-Bessel expansion as an optimal basis for computing these anisotropies, capturing the relevant propagating information.

Contribution

It shows the causal dependence of CMB anisotropies on sources inside the past light-cone and proposes the Fourier-Bessel expansion as an optimal computational framework.

Findings

CMB anisotropies are regulated by spacetime window functions within the past light-cone.

Fourier-Bessel modes form a discrete set capturing source information propagating to the observer.

Angular power spectra are invariant under the choice of basis, confirming the Fourier-Bessel expansion's optimality.

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 happens order by order in a series expansion in powers of the visibility $\gamma=e^{-\mu}$, where $\mu$ is the optical depth to Thompson scattering. We show that 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. In that expansion, for each multipole $l$ there is a discrete tower of momenta $k_{i,l}$ (not a continuum) which can affect physical observables, with the smallest momenta being $k_{1,l} ~ l$. 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. (Abridged)