TDCOSMO XXXI: New techniques in line-of-sight studies of time delay lenses

The distribution of matter along the same line of sight but external to the main lens, quantified by κ ext, is a key source of uncertainty in time delay cosmography and other applications of strong lensing, and must be independently estimated to avoid biasing the inferred value of the Hubble constant (H 0). We present advancements and standardisations in the weighted number counts techniques used to constrain κ ext, in particular the use of the Euclid Flagship Simulation and a breakdown of the line-of-sight contributions between the observer, lens and source. As part of the TDCOSMO 2026 milestone analysis, we apply this updated method to the sample of 11 time delay lenses used in that study. Our estimates for certain systems are sensitive to these methodological changes, but are nonetheless consistent within 1σ for all but one of the systems which had been studied previously, with the median κ ext across those systems changing from-0.002 to-0.006 following this new analysis. This work represents the first estimate of κ ext values which includes the contribution of the observer-lens and lens-source terms, and largest standardised analysis of time delay lens environments to date.