  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.
