Beyond the Linear-Order Relativistic Effect in Galaxy Clustering:
  Second-Order Gauge-Invariant Formalism
  We present the second-order general relativistic description of the observed
galaxy number density in a cosmological framework. The observed galaxy number
density is affected by the volume and the source effects, both of which arise
due to the mismatch between physical and observationally inferred quantities
such as the redshift, the angular position, the volume, and the luminosity of
the observed galaxies. These effects are computed to the second order in metric
perturbations without choosing a gauge condition or adopting any restrictions
on vector and tensor perturbations, extending the previous linear-order
calculations. Paying particular attention to the second-order gauge
transformation, we explicitly isolate unphysical gauge modes and construct
second-order gauge-invariant variables. Moreover, by constructing second-order
tetrads in the observer's rest frame, we clarify the relation between the
physical and the parametrized photon wavevectors. Our second-order relativistic
description will provide an essential tool for going beyond the power spectrum
in the era of precision measurements of galaxy clustering. We discuss potential
applications and extensions of the second-order relativistic description of
galaxy clustering.
