Cosmological Fluctuations on the Light Cone
  In studying temperature fluctuations in the cosmic microwave background
Weinberg has noted that some ease of calculation and insight can be achieved by
looking at the structure of the perturbed light cone on which the perturbed
photons propagate. In his approach Weinberg worked in a specific gauge and
specialized to fluctuations around the standard Robertson-Walker cosmological
model with vanishing spatial three-curvature. In this paper we generalize this
analysis by providing a gauge invariant treatment in which no choice of gauge
is made, and by considering geometries with non-vanishing spatial
three-curvature. By using the scalar, vector, tensor fluctuation basis we find
that the relevant gauge invariant combinations that appear in the light cone
temperature fluctuations have no explicit dependence on the spatial curvature
even if the spatial curvature of the background geometry is nonvanishing. We
find that a not previously considered, albeit not too consequential,
temperature fluctuation at the observer has to be included in order to enforce
gauge invariance. As well as working with comoving time we also work with
conformal time in which a background metric of any given spatial
three-curvature can be written as a time-dependent conformal factor (the
comoving time expansion radius as written in conformal time) times a static
Robertson-Walker geometry of the same spatial three-curvature. For temperature
fluctuations on the light cone this conformal factor drops out identically.
Thus the gauge invariant combinations that appear in the photon temperature
fluctuations have no explicit dependence on either the conformal factor or the
spatial three-curvature at all.
