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Warm dark matter chills out: constraints on the halo mass function and the free-streaming length of dark matter with 8 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 halos 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 subhalos and halos along the line of sight, and account for warm dark matter (WDM) 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 halos, 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. Parameterizing dark matter free-streaming through the half-mode mass $m_{\rm{hm}}$, we constrain the thermal relic particle mass $m_{\rm{DM}}$ corresponding to $m_{\rm{hm}}$. At $95 \%$ CI: $m_{\rm{hm}} < 10^{7.8} M_{\odot}$ ($m_{\rm{DM}} > 5.2 \ \rm{keV}$). We disfavor $m_{\rm{DM}} = 4.0 \rm{keV}$ and $ m_{\rm{DM}} = 3.0 \rm{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 $10^6 - 10^{9} M_{\odot}$ near lensed images. At $68 \%$ CI, we infer $2.0 - 6.1 \times 10^{7} M_{\odot} \rm{kpc^{-2}}$, corresponding to mean projected mass fraction $\bar{f}_{\rm{sub}} = 0.035_{-0.017}^{+0.021}$. At $95 \%$ CI, we obtain a lower bound on the projected mass of $0.6 \times 10^{7} M_{\odot} \rm{kpc^{-2}}$, corresponding to $\bar{f}_{\rm{sub}} > 0.005$. These results agree with the predictions of cold dark matter.