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.
