Equivalence of the Fermat potential and the lensing potential approaches to computing the integrated Sachs-Wolfe effect

TL;DR

This paper demonstrates that the Fermat potential approach for calculating the ISW effect is mathematically equivalent to the traditional lensing potential method, offering a simpler computational alternative for flat cosmologies.

Contribution

It establishes the equivalence between the Fermat potential and lensing potential methods for the ISW effect, simplifying calculations in flat background cosmologies.

Findings

Fermat potential approach is equivalent to the conventional method.

The new approach simplifies the calculation process.

Applicability confirmed for flat background cosmologies.

Abstract

We show in detail that the recently derived expression for evaluating the integrated Sachs-Wolfe (ISW) temperature shift in the cosmic microwave background (CMB) caused by individual embedded (compensated) lenses is equivalent to the conventional approach for flat background cosmologies. The conventional approach requires evaluating an integral of the time derivative of the lensing potential, whereas the new Fermat potential approach is simpler and only requires taking a derivative of the potential part of the time delay.