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Abstract Strong lensing provides a powerful means of investigating the nature of dark matter as it probes dark matter structure on sub-galactic scales. We present an extension of a forward modelling framework that uses flux ratios from quadruply imaged quasars (quads) to measure the shape and amplitude of the halo mass function, including line-of-sight (LOS) haloes and main deflector subhaloes. We apply this machinery to 50 mock lenses – roughly the number of known quads – with warm dark matter (WDM) mass functions exhibiting free-streaming cut-offs parametrized by the half-mode mass mhm. Assuming cold dark matter (CDM), we forecast bounds on mhm and the corresponding thermal relic particle masses over a range of tidal destruction severity, assuming a particular WDM mass function and mass–concentration relation. With significant tidal destruction, at 2σ we constrain m ₇₌ 10^7. 9 (10^8. 4) \, M, or a 4. 4 (3. 1) keV thermal relic, with image flux uncertainties from measurements and lens modelling of 2{\ per\ cent} (6{\ per\ cent}). With less severe tidal destruction we constrain m ₇₌ 10^7 (10^7. 4) \, M, or an 8. 2 (6. 2) keV thermal relic. If dark matter is warm, with m ₇₌ = 10^7. 7 \, M (5. 1 keV), we would favour WDM with m ₇₌ 10^7. 7 \, M over CDM with relative likelihoods of 22: 1 and 8: 1 with flux uncertainties of 2{\ per\ cent} and 6{\ per\ cent}, respectively. These bounds improve over those obtained by modelling only main deflector subhaloes because LOS objects produce additional flux perturbations, especially for high-redshift systems. These results indicate that ∼50 quads can conclusively differentiate between WDM and CDM.
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