Measuring the Moving Lens Effect with Wide-Separation Lensed Quasars

Abstract

The transverse motion of a gravitational potential induces frequency shifts in photons propagating through it, giving rise to the moving-lens effect and providing a probe of transverse peculiar velocities. Statistical evidence for this effect has recently been reported using cross-correlations between cosmic microwave background temperature maps and large-scale-structure tracers. Here, we investigate a complementary spectroscopic approach based on wide-separation lensed quasars, for which different lensed images probe distinct deflection geometries through a moving foreground lens. We focus on SDSS J1004+4112 and assess the differential redshift that could be measured from its broad emission-line spectra. Using an estimator that combines information across the line profiles, we perform Monte Carlo forecasts for C IV and C III] under an ELT-like configuration, considering photon noise only. For an assumed lens transverse velocity of $500\,{\rm km\,s^{-1}}$ and the orientation maximizing the signal, the expected differential redshift reaches $|\Delta z_{\rm ML}|\simeq6.7\times10^{-7}$. At an integration time of 40 hr, the joint photon-noise-limited precision is $\sigma_z\simeq2.5\times10^{-7}$. C IV dominates the constraint, while C III] provides only a modest additional improvement, and integrations of tens of hours reach the $10^{-7}$ precision regime. These results show that photon statistics alone can reach the characteristic moving-lens signal scale, although practical detectability will depend on controlling astrophysical and instrumental systematics.