Spectro-spatial evolution of the CMB I: discretisation of the thermalisation Green's function

Spectral distortions of the cosmic microwave background (CMB) have been recognized as an important future probe of the early Universe. Existing theoretical studies primarily focused on describing the evolution and creation of average distortions, ignoring spatial perturbations in the plasma. One of the main reasons for this choice is that a treatment of the spectro-spatial evolution of the photon field deep into the primordial Universe requires solving a radiative transfer problem for the distortion signals, which in full detail is computationally challenging. Here we provide the first crucial step towards tackling this problem by formulating a new spectral discretisation of the underlying average thermalisation Green's function. Our approach allows us to convert the high-dimensional partial differential equation system (~1,000-10,000 equations) into and set of ordinary differential equations of much lower dimension (~10 equations). We demonstrate the precision of the approach and highlight how it may be further improved in the future. We also clarify the link of the observable spectral distortion parameters (e.g., mu and y) to the computational spectral basis that we use in our frequency discretisation. This reveals how several basis-dependent ambiguities can be interpreted in future CMB analysis. Even if not exact, the new Green's function discretisation can be used to formulate a generalised photon Boltzmann-hierarchy, which can then be solved with methods that are familiar from theoretical studies of the CMB temperature and polarisation anisotropies. We will carry this program out in a series of companion papers, thereby opening the path to full spectro-spatial exploration of the CMB with future CMB imagers and spectrometers.