Probing dark matter structure down to 10⁷ solar masses: flux ratio statistics in gravitational lenses with line-of-sight haloes

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 m_hm. Assuming cold dark matter (CDM), we forecast bounds on m_hm 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_hm < 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 percent 6 per cent}. With less severe tidal destruction we constrain m_hm 10⁷ 10^7.4)M⊙, or an 8.2 (6.2) keV thermal relic. If dark matter is warm, with m_hm= 10^7.7, M⊙ (5.1 keV), we would favour WDM with m_hm > 10^7.7 M⊙ over CDM with relative likelihoods of 22:1 and 8:1 with flux uncertainties of 2rm per cent and 6rm 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.