Recently it has been claimed that the warm dark matter (WDM) model cannot at the same time reproduce the observed Lyman-$\alpha$ forests in distant quasar spectra and solve the small-scale issues in the cold dark matter (CDM) model. As an alternative candidate, it was shown that the mixed dark matter (MDM) model that consists of WDM and CDM can satisfy the constraint from Lyman-$\alpha$ forests and account for the “missing satellite problem” as well as the reported 3.5 keV anomalous x-ray line. We investigate observational constraints on the MDM model using strong gravitational lenses. We first develop a fitting formula for the nonlinear power spectra in the MDM model by performing $N$-body simulations and estimate the expected perturbations caused by line-of-sight structures in four quadruply lensed quasars that show anomaly in the flux ratios. Our analysis indicates that the MDM model is compatible with the observed anomaly if the mass fraction of the warm component is smaller than 0.47 at the 95% confidence level. The MDM explanation to the anomalous x-ray line and the small-scale issues is still viable even after this constraint is taken into account.

• Received 12 April 2016

DOI:https://doi.org/10.1103/PhysRevD.94.023522