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

Understand

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.