Sachs-Wolfe effect: Gauge independence and a general expression

We address two points concerning the Sachs-Wolfe effect: (i) the gauge independence of the observable temperature anisotropy and (ii) a gauge-invariant expression of the effect considering the most general situation of hydrodynamic perturbations. The first result follows because the gauge transformation of the temperature fluctuation at the observation event only contributes to the isotropic temperature change which, in practice, is absorbed into the definition of the background temperature. Thus, we proceed without fixing the gauge condition, and express the Sachs-Wolfe effect using the gauge-invariant variables.