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