Improving the Line of Sight for the Anisotropic 3-Point Correlation
  Function of Galaxies: Centroid and Unit-Vector-Average Methods Scaling as
  $\mathcal{O}(N^2)$
  The 3-Point Correlation Function (3PCF), which measures correlations between
triplets of galaxies encodes information about peculiar velocities, which
distort the observed positions of galaxies along the line of sight away from
their true positions. To access this information, we must track the 3PCF's
dependence not only on each triangle's shape, but also on its orientation with
respect to the line of sight. Consequently, different choices for the line of
sight will affect the measured 3PCF. Up to now, the line of sight has been
taken as the direction to a single triplet member (STM), but which triplet
member is used impacts the 3PCF by ~20% of the statistical error for a
BOSS-like survey. For DESI (2019-24), which is 5X more precise, this would
translate to 100% of the statistical error, increasing the total error bar by
40%. We here propose a new method that is fully symmetric between the triplet
members, and uses either the average of the three galaxy position vectors
(which we show points to the triangle centroid), or the average of their unit
(direction) vectors. Naively, these approaches would seem to require triplet
counting, scaling as $N^3$, with $N$ the number of objects in the survey. By
harnessing the solid harmonic shift theorem, we here show how these methods can
be evaluated scaling as $N^2$. We expect that they can be used to make a
robust, systematics-free measurement of the anisotropic 3PCF of upcoming
redshift surveys such as DESI. So doing will in turn open an additional channel
to constrain the growth rate of structure and thereby learn the matter density
as well as test the theory of gravity.
