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} \left(10^{8.4}\right) M_{\odot }, or a 4.4 (3.1) keV thermal relic, with image flux uncertainties from measurements and lens modelling of 2{{ per cent}} \left(6{{ per cent}}\right). With less severe tidal destruction we constrain m_{hm}< 10⁷ \left(10^{7.4}\right) M_{\odot }, or an 8.2 (6.2) keV thermal relic. If dark matter is warm, with m_{hm} = 10^{7.7} M_{\odot } (5.1 keV), we would favour WDM with m_{hm} > 10^{7.7} M_{\odot } 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.