Nonlinear relativistic corrections to cosmological distances, redshift and gravitational lensing magnification: II. derivation

We present a derivation of the cosmological distance-redshift relation up to second order in perturbation theory. In addition, we find the observed redshift and the lensing magnification to second order. We do not require that the density contrast is small, but only that the metric potentials and peculiar velocities are small. Thus our results apply into the nonlinear regime, and can be used for most dark energy models. We present the results in a form that can be readily computed in an N-body simulation. This paper accompanies paper I (Umeh, Clarkson and Maartens 2014 Class. Quantum Grav. 31 202001), where the key results are summarized in a physically transparent form and applications are discussed.