Non-Gaussianities due to Relativistic Corrections to the Observed Galaxy
  Bispectrum
  High-precision constraints on primordial non-Gaussianity (PNG) will
significantly improve our understanding of the physics of the early universe.
Among all the subtleties in using large scale structure observables to
constrain PNG, accounting for relativistic corrections to the clustering
statistics is particularly important for the upcoming galaxy surveys covering
progressively larger fraction of the sky. We focus on relativistic projection
effects due to the fact that we observe the galaxies through the light that
reaches the telescope on perturbed geodesics. These projection effects can give
rise to an effective $f_{\rm NL}$ that can be misinterpreted as the primordial
non-Gaussianity signal and hence is a systematic to be carefully computed and
accounted for in modelling of the bispectrum. We develop the technique to
properly account for relativistic effects in terms of purely observable
quantities, namely angles and redshifts. We give some examples by applying this
approach to a subset of the contributions to the tree-level bispectrum of the
observed galaxy number counts calculated within perturbation theory and
estimate the corresponding non-Gaussianity parameter, $f_{\rm NL}$, for the
local, equilateral and orthogonal shapes. For the local shape, we also compute
the local non-Gaussianity resulting from terms obtained using the consistency
relation for observed number counts. Our goal here is not to give a precise
estimate of $f_{\rm NL}$ for each shape but rather we aim to provide a scheme
to compute the non-Gaussian contamination due to relativistic projection
effects. For the terms considered in this work, we obtain contamination of
$f_{\rm NL}^{\rm loc} \sim {\mathcal O}(1)$.
