Ambiguities in gravitational lens models: impact on time delays of the source position transformation

TL;DR

This paper investigates how the source position transformation (SPT), a generalization of the mass-sheet transformation, affects gravitational lens time delay predictions, finding that its impact is generally small and may not significantly bias cosmological measurements.

Contribution

It introduces a comprehensive analysis of the SPT's effect on time delays using the new pySPT tool, showing that SPT-induced deviations are typically minor.

Findings

Time delay ratios scale with source position ratios for axisymmetric lenses.

In the quadruple image regime, time delay ratios are not conserved under SPT.

Impact of SPT on time delays is generally within a few percent, less critical than previously thought.

Abstract

The central ambition of the modern time delay cosmography consists in determining the Hubble constant $H_0$ with a competitive precision. However, the tension with $H_0$ obtained from the Planck satellite for a spatially-flat $\Lambda$CDM cosmology suggests that systematic errors may have been underestimated. The most critical one probably comes from the degeneracy existing between lens models that was first formalized by the well-known mass-sheet transformation (MST). In this paper, we assess to what extent the source position transformation (SPT), a more general invariance transformation which contains the MST as a special case, may affect the time delays predicted by a model. To this aim we use pySPT, a new open-source python package fully dedicated to the SPT that we present in a companion paper. For axisymmetric lenses, we find that the time delay ratios between a model and its SPT-modified counterpart simply scale like the corresponding source position ratios, $\Delta \hat{t}/ \Delta t \approx \hat{\beta}/\beta$, regardless of the mass profile and the isotropic SPT. Similar behavior (almost) holds for non-axisymmetric lenses in the double image regime and for opposite image pairs in the quadruple image regime. In the latter regime, we also confirm that the time delay ratios are not conserved. In addition to the MST effects, the SPT-modified time delays deviate in general no more than a few percent for particular image pairs, suggesting that its impact on time-delay cosmography seems not be as crucial as initially suspected. We also reflected upon the relevance of the SPT validity criterion and present arguments suggesting that it should be reconsidered. Even though a new validity criterion would affect the time delays in a different way, we expect from numerical simulations that our conclusions will remain unchanged.