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.