What the small angle CMB really tells us about the curvature of the Universe

It is well known that observations of the cosmic microwave background (CMB) are highly sensitive to the spatial curvature of the Universe, k. Here we find that what is in fact being tightly constrained by small angle fluctuations is spatial curvature near the surface of last scattering, and that if we allow k to be a function of position, rather than taking a constant value everywhere, then considerable spatial curvature is permissible within our own locale. This result is of interest for the giant void models that attempt to explain the supernovae observations without Dark Energy. We find such voids to be compatible with the observed small angle CMB, but they must be either very deep (and unnaturally empty) or exist in a positively curved Universe