The Information Content of Anisotropic Baryon Acoustic Oscillation Scale
  Measurements
  Anisotropic measurements of the Baryon Acoustic Oscillation (BAO) feature
within a galaxy survey enable joint inference about the Hubble parameter $H(z)$
and angular diameter distance $D_A(z)$. These measurements are typically
obtained from moments of the measured 2-point clustering statistics, with
respect to the cosine of the angle to the line of sight $\mu$. The position of
the BAO features in each moment depends on a combination of $D_A(z)$ and
$H(z)$, and measuring the positions in two or more moments breaks this
parameter degeneracy. We derive analytic formulae for the parameter
combinations measured from moments given by Legendre polynomials, power laws
and top-hat Wedges in $\mu$, showing explicitly what is being measured by each
in real-space for both the correlation function and power spectrum, and in
redshift-space for the power spectrum. The large volume covered by modern
galaxy samples means that the correlation function can be well approximated as
having no correlations at different $\mu$ on the BAO scale, and that the errors
on this scale are approximately independent of $\mu$. Using these
approximations, we derive the information content of various moments. We show
that measurements made using either the monopole and quadrupole, or the
monopole and $\mu^2$ power-law moment, are optimal for anisotropic BAO
measurements, in that they contain all of the available information using two
moments, the minimal number required to measure both $H(z)$ and $D_A(z)$. We
test our predictions using 600 mock galaxy samples, matched to the SDSS-III
Baryon Oscillation Spectroscopic Survey CMASS sample, finding a good match to
our analytic predictions. Our results should enable the optimal extraction of
information from future galaxy surveys such as eBOSS, DESI and Euclid.
