  Light from `point sources' such as supernovae is observed with a beam width
of order of the sources' size - typically less than 1 AU. Such a beam probes
matter and curvature distributions that are very different from coarse-grained
representations in N-body simulations or perturbation theory, which are
smoothed on scales much larger than 1 AU. The beam typically travels through
unclustered dark matter and hydrogen with a mean density much less than the
cosmic mean, and through dark matter halos and hydrogen clouds. Using N-body
simulations, as well as a Press-Schechter approach, we quantify the density
probability distribution as a function of beam width and show that, even for
Gpc-length beams of 500 kpc diameter, most lines of sight are significantly
under-dense. From this we argue that modelling the probability distribution for
AU-diameter beams is absolutely critical. Standard analyses predict a huge
variance for such tiny beam sizes, and nonlinear corrections appear to be
non-trivial. It is not even clear whether under-dense regions lead to dimming
or brightening of sources, owing to the uncertainty in modelling the expansion
rate which we show is the dominant contribution. By considering different
reasonable approximations which yield very different cosmologies we argue that
modelling ultra-narrow beams accurately remains a critical problem for
precision cosmology. This could appear as a discordance between angular
diameter and luminosity distances when comparing SN observations to BAO or CMB
distances.
