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

## TL;DR

This paper derives a comprehensive non-linear formula for CMB temperature anisotropies, valid at all perturbation orders and gauges, simplifying the analysis of primordial signals and secondary effects.

## Contribution

It presents the first all-orders, gauge-invariant non-linear Sachs-Wolfe and ISW formula including scalar, vector, and tensor modes.

## Key findings

- Logarithmic temperature maps are cleaner and better suited for non-Gaussianity searches.
- The formula disentangles non-linear ISW effects from other secondary anisotropies.
- Provides an iterative method for calculating lensing effects at any order.

## Abstract

We obtain the non-linear generalization of the Sachs-Wolfe + integrated Sachs-Wolfe (ISW) formula describing the CMB temperature anisotropies. Our formula is valid at all orders in perturbation theory, is also valid in all gauges and includes scalar, vector and tensor modes. A direct consequence of our results is that the maps of the logarithmic temperature anisotropies are much cleaner than the usual CMB maps, because they automatically remove many secondary anisotropies. This can for instance, facilitate the search for primordial non-Gaussianity in future works. It also disentangles the non-linear ISW from other effects. Finally, we provide a method which can iteratively be used to obtain the lensing solution at the desired order.