  We revisit the uncertainty in baryon acoustic oscillation (BAO) forecasts and
data analyses. In particular, we study how much the uncertainties on both the
measured mean dilation scale and the associated error bar are affected by the
non-Gaussianity of the non-linear density field. We examine two possible
impacts of non-Gaussian analysis: (1) we derive the distance estimators from
Gaussian theory, but use 1000 N-Body simulations to measure the actual errors,
and compare this to the Gaussian prediction, and (2) we compute new optimal
estimators, which requires the inverse of the non-Gaussian covariance matrix of
the matter power spectrum. Obtaining an accurate and precise inversion is
challenging, and we opted for a noise reduction technique applied on the
covariance matrices. By measuring the bootstrap error on the inverted matrix,
this work quantifies for the first time the significance of the non-Gaussian
error corrections on the BAO dilation scale. We find that the variance (error
squared) on distance measurements can deviate by up to 12% between both
estimators, an effect that requires a large number of simulations to be
resolved. We next apply a reconstruction algorithm to recover some of the BAO
signal that had been smeared by non-linear evolution, and we rerun the
analysis. We find that after reconstruction, the rms error on the distance
measurement improves by a factor of ~1.7 at low redshift (consistent with
previous results), and the variance ({\sigma}^2) shows a change of up to 18%
between optimal and sub-optimal cases (note, however, that these discrepancies
may depend in detail on the procedure used to isolate the BAO signal). We
finally discuss the impact of this work on current data analyses.
