  Observations of the microwave background fluctuations suggest a
scale-dependent amplitude asymmetry of roughly 2.5 sigma significance.
Inflationary explanations for this 'anomaly' require non-Gaussian fluctuations
which couple observable modes to those on much larger scales. In this Letter we
describe an analysis of such scenarios which significantly extends previous
treatments. We identify the non-Gaussian 'response function' which
characterizes the asymmetry, and show that it is non-trivial to construct a
model which yields a sufficient amplitude: many independent fine tunings are
required, often making such models appear less likely than the anomaly they
seek to explain. We present an explicit model satisfying observational
constraints and determine for the first time how large its bispectrum would
appear to a Planck-like experiment. Although this model is merely illustrative,
we expect it is a good proxy for the bispectrum in a sizeable class of models
which generate a scale-dependent response using a large eta parameter.
