  In this paper, we report a significant recovery of the linear baryonic
acoustic oscillation (BAO) signature by applying the isobaric reconstruction
algorithm to the non-linear matter density field. Assuming only the
longitudinal component of the displacement being cosmologically relevant, this
algorithm iteratively solves the coordinate transform between the Lagrangian
and Eulerian frames without requiring any specific knowledge of the dynamics.
For dark matter field, it produces the non-linear displacement potential with
very high fidelity. The reconstruction error at the pixel level is within a few
percent, and is caused only by the emergence of the transverse component after
the shell-crossing. As it circumvents the strongest non-linearity of the
density evolution, the reconstructed field is well-described by linear theory
and immune from the bulk-flow smearing of the BAO signature. Therefore this
algorithm could significantly improve the measurement accuracy of the sound
horizon scale. For a perfect large-scale structure survey at redshift zero
without Poisson or instrumental noise, the fractional error is reduced by a
factor of 2.7, very close to the ideal limit with linear power spectrum and
Gaussian covariance matrix.
