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We obtain the non-linear generalization of the Sachs-Wolfe + integratedSachs-Wolfe (ISW) formula describing the CMB temperature anisotropies. Ourformula is valid at all orders in perturbation theory, is also valid in allgauges and includes scalar, vector and tensor modes. A direct consequence ofour results is that the maps of the logarithmic temperature anisotropies aremuch cleaner than the usual CMB maps, because they automatically remove manysecondary anisotropies. This can for instance, facilitate the search forprimordial non-Gaussianity in future works. It also disentangles the non-linearISW from other effects. Finally, we provide a method which can iteratively beused to obtain the lensing solution at the desired order.

CMB anisotropies at all orders: the non-linear Sachs-Wolfe formula