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 _ { hm } , we constrain the thermal relic particle mass m _ { DM } corresponding to m _ { hm } .
At 95 \% CI : { m _ { hm } } < 10 ^ { 7.8 } { M _ { \odot } } ( m _ { DM } > 5.2 keV ) .
We disfavor 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 10 ^ { 6 } -10 ^ { 9 } M _ { \odot } near lensed images .
At 68 \% CI , we infer 2.0 - 6.1 \times 10 ^ { 7 } { M _ { \odot } } { kpc ^ { -2 } } , corresponding to mean projected mass fraction \bar { f } _ { 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 } } { kpc ^ { -2 } } , corresponding to \bar { f } _ { sub } > 0.005 .
These results agree with the predictions of cold dark matter .