Probing warm and mixed dark matter models using lensing shift power spectrum

We argue that the lensing power spectrum of astrometric shift (lensing shift power spectrum) is a powerful tool of the clustering property of dark matter on subgalactic scales. First we give the formalism to probe the nature of dark matter by using the lensing shift power spectrum. Then, leveraging recent measurements of the lensing shift power spectrum on an angular scale of approximately $1~$arcsec towards the gravitationally lensed quasar MG$\,$J0414+0534 at the redshift of $z_S=2.639$, we place constraints on the mass of warm dark matter (WDM) particles $m_{\rm WDM}$ and their fraction in a mixed dark matter (MDM) model $r_{\rm WDM}$, in which WDM and cold dark matter coexist. Although the constraint derived from the above single lensing system is not as strong as the existing constraints, as we show in this paper, the lensing shift power spectrum has a great potential to obtain much tighter constraints on WDM and MDM models through future observations, highlighting the importance of well-controlled systematic error considerations for achieving enhanced precision.