Non linear effects in the observed redhisft: how to obtain gauge invariance for non linear observables

I present the second-order expression for the observed redshift, accounting for all the relativistic effects from the light propagation and from the frame change at the observer and the source positions. Moreover I derive the generic gauge-transformation law that any observable quantities should satisfy and verify this second-order expression for the observed redshift by explicitly checking its expected gauge transformation property. After this, I discuss the importance of expressing all the physical observables in terms of the observed redshift, in order to get gauge invariant quantities up to non linear order. All the derivation is based on the Geodesic Light Cone (GLC) gauge. Along the talk, I will clarify how to exploit the "residual gauge freedom" allowed within the GLC itself in order to get correct result.