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We argue that the lensing power spectrum of astrometric shift (lensing shiftpower spectrum) is a powerful tool of the clustering property of dark matter onsubgalactic scales. First we give the formalism to probe the nature of darkmatter by using the lensing shift power spectrum. Then, leveraging recentmeasurements of the lensing shift power spectrum on an angular scale ofapproximately $1~$arcsec towards the gravitationally lensed quasarMG$\,$J0414+0534 at the redshift of $z_S=2.639$, we place constraints on themass of warm dark matter (WDM) particles $m_{\rm WDM}$ and their fraction in amixed dark matter (MDM) model $r_{\rm WDM}$, in which WDM and cold dark mattercoexist. Although the constraint derived from the above single lensing systemis not as strong as the existing constraints, as we show in this paper, thelensing shift power spectrum has a great potential to obtain much tighterconstraints on WDM and MDM models through future observations, highlighting theimportance of well-controlled systematic error considerations for achievingenhanced precision.

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