ALMA Measurement of 10 kpc-scale Lensing Power Spectra towards the Lensed Quasar MG J0414+0534

We report the first measurement of lensing power spectra for gravitational potential, astrometric shifts, and convergence perturbations towards the anomalous quadruply lensed quasar MG$\,$J0414+0534. To obtain the spectra, we conducted observations of MG$\,$J0414+0534 using the Atacama Large Millimeter/submillimeter Array (ALMA) with high angular resolution (0.02-0.05 arcsec). We developed a new method in which Fourier coefficients of potential perturbation are adjusted to minimise the difference between linear combinations of weighted mean de-lensed images. Noise cancellation due to synthesised de-lensed images makes our method suitable for systems with extended images obtained from interferometres. Using positions of radio jet components, extended dust emission on scales $>1\,$kpc, and mid-infrared flux ratios, which are microlensing free, our new multi-wavelength method provides us with a very effective tool for probing cosmological matter fluctuations on scales $\lesssim 10\,$kpc. Assuming that contributions from structures on angular scales $\lesssim 1.0$ arcsec are negligible, on an angular scale of $\sim 1.3$ arcsec (corresponding to an angular wave number of $l \sim 1.1\times 10^6$ or $\sim 8\,$kpc in the lens plane), the measured convergence, astrometric shift, and potential powers are $\varDelta_\kappa=0.02-0.025$, $\varDelta_\alpha =7-8\,$mas, and $\varDelta_\psi=1.1-1.5\,$$\textrm{mas}^2$, respectively. Our result is consistent with the predicted abundance of haloes in the line of sight and subhaloes in cold dark matter models. Our lens models suggest a presence of a clump in the vicinity of object Y (inoue2017), a possible dusty dwarf galaxy. Although much fainter than the previous report, we confirmed weak continuum emission from object Y with a peak flux of $\sim 100\,\mu \textrm{Jy}\, \textrm{beam} ^{-1}$ at the $\sim 4\,\sigma$ level.