Warm dark matter chills out: constraints on the halo mass function and the free-streaming length of dark matter with eight quadruple-image strong gravitational lenses

The free-streaming length of dark matter depends on fundamental dark matter physics, and determines the abundance and concentration of dark matter haloes on sub-galactic scales. Using the image positions and flux ratios from eight quadruply imaged quasars, we constrain the free-streaming length of dark matter and the amplitude of the subhalo mass function (SHMF). We model both main deflector subhaloes and haloes along the line of sight, and account for warm dark matter free-streaming effects on the mass function and mass-concentration relation. By calibrating the scaling of the SHMF with host halo mass and redshift using a suite of simulated haloes, we infer a global normalization for the SHMF. We account for finite-size background sources, and marginalize over the mass profile of the main deflector. Parametrizing dark matter free-streaming through the half-mode mass m_hm, we constrain the thermal relic particle mass m_DM corresponding to m_hm. At 95 {per cent} CI: m_hm < 10^7.8 M_⊙ (m_{DM} > 5.2 {keV}). We disfavour m_{DM} = 4.0 {keV} and m_{DM} = 3.0 {keV} with likelihood ratios of 7:1 and 30:1, respectively, relative to the peak of the posterior distribution. Assuming cold dark matter, we constrain the projected mass in substructure between 10⁶ and 10⁹ M_⊙ near lensed images. At 68 {per cent} CI, we infer 2.0{-}6.1 × 10⁷ {M_{⊙ }} {kpc^{-2}}, corresponding to mean projected mass fraction \bar{f}_{sub} = 0.035_{-0.017}^{+0.021}. At 95 {per cent} CI, we obtain a lower bound on the projected mass of 0.6 × 10⁷ {M_{⊙ }} {kpc^{-2}}, corresponding to \bar{f}_{sub} > 0.005. These results agree with the predictions of cold dark matter.