Modelling the Galactic Foreground and Beam Chromaticity for Global 21-cm
  Cosmology
  In order to characterize and model the beam-weighted foreground for global
21-cm signal experiments, we present a methodology for generating basis
eigenvectors that combines analytical and observational models of both the
galactic spectral index and sky brightness temperature with simulations of
beams having various angular and spectral dependencies and pointings. Each
combination creates a unique beam-weighted foreground. By generating
eigenvectors to fit each foreground model using Singular Value Decomposition
(SVD), we examine the effects of varying the components of the beam-weighted
foreground. We find that the eigenvectors for modelling an achromatic,
isotropic beam -- the ideal case -- are nearly identical regardless of the
unweighted foreground model used, and are practicably indistinguishable from
polynomial-based models. When anisotropic, chromatic beams weight the
foreground, however, a coupling is introduced between the spatial and spectral
structure of the foreground which distorts the eigenvectors away from the
polynomial models and induces a dependence of the basis upon the exact features
of the beam (chromaticity, pattern, pointing) and foreground (spectral index,
sky brightness temperature map). We find that the beam has a greater impact
upon the eigenvectors than foreground models. Any model which does not account
for its distortion may produce RMS uncertainties on the order of $\sim 10$ -
$10^3$ Kelvin for six-parameter, single spectrum fits. If the beam is
incorporated directly using SVD and training sets, however, the resultant
eigenvectors yield milli-Kelvin level uncertainties. Given a sufficiently
detailed description of the sky, our methodology can be applied to any
particular experiment with a suitably characterized beam for the purpose of
generating accurate beam-weighted foreground models.
