  The location of the baryon acoustic oscillation (BAO) feature in the
two-point correlation function (2PCF) of matter produces a standard ruler that
is useful for the measurement of the expansion history of the Universe.
Inspired by the possibility of reconstructing the positions of protohalos in
the initial density field with a novel method rooted in optimal transport
theory, we revisit the BAO signal in the protohalo correlation function. Our
work examines the performance of a template 2PCF built on a tracer bias
relation that includes scale dependence -- a term that can be motivated by
peaks theory or a general bias expansion. Working in protohalos, halos, and the
linear combination of the protohalo and matter fields that is motivated by the
continuity equation, we demonstrate that this model accurately captures the
shape of the BAO feature and improves the precision of the BAO scale
measurement relative to a model that does not include scale-dependent bias by
47% in protohalos, 15% in halos, and 14% in the linear combination of the
protohalo and matter fields. Allowing for scale dependence does not appear to
introduce any shift in the BAO feature. The precision of the BAO distance scale
estimate is highest with the linear combination of the protohalo and matter
fields, which offers a factor of 3.5 improvement over Eulerian-space
measurements and a factor of 4-8 improvement over the estimate made with
protohalos alone.
