Second-order gauge-invariant formalism for the cosmological observables:
  Complete verification of their gauge-invariance
  Accounting for all the relativistic effects, we have developed the fully
nonlinear gauge-invariant formalism for describing the cosmological observables
and presented the second-order perturbative expressions associated with light
propagation and observations without choosing a gauge condition. For the first
time, we have performed a complete verification of the validity of our
second-order expressions by comparing their gauge-transformation properties
from two independent methods: one directly obtained from their expressions in
terms of metric perturbations and the other expected from their nonlinear
relations. The expressions for the cosmological observables such as galaxy
clustering and the luminosity distance are invariant under diffeomorphism and
gauge-invariant at the observed position. We compare our results to the
previous work and discuss the differences in the perturbative expressions. Our
second-order gauge-invariant formalism constitutes a major step forward in the
era of precision cosmology and its applications in the future will play a
crucial role for going beyond the power spectrum and probing the early
universe.
