  Stringent observational constraints on the sub-galactic matter power spectrum
would allow one to distinguish between the concordance $\Lambda$CDM and the
various alternative dark-matter models that predict significantly different
properties of mass structure in galactic haloes. Galaxy-galaxy strong
gravitational lensing provides a unique opportunity to probe the sub-galactic
mass structure in lens galaxies beyond the Local Group. Here, we demonstrate
the first application of a novel methodology to observationally constrain the
sub-galactic matter power spectrum in the inner regions of massive elliptical
lens galaxies on 1-10 kpc scales from the power spectrum of surface-brightness
anomalies in highly magnified galaxy-scale Einstein rings and gravitational
arcs. The pilot application of our approach to Hubble Space Telescope
(HST/WFC3/F390W) observations of the SLACS lens system SDSS J0252+0039 allows
us to place the following observational constraints (at the 99 per cent
confidence level) on the dimensionless convergence power spectrum
$\Delta^{2}_{\delta\kappa}$ and the standard deviation in the aperture mass
$\sigma_{\rm AM}$: $\Delta^{2}_{\delta\kappa}<1$ ($\sigma_{\rm AM}< 0.8 \times
10^8 M_\odot$) on 0.5-kpc scale, $\Delta^{2}_{\delta\kappa}<0.1$ ($\sigma_{\rm
AM}< 1 \times 10^8 M_\odot$) on 1-kpc scale and
$\Delta^{2}_{\delta\kappa}<0.01$ ($\sigma_{\rm AM}< 3 \times 10^8 M_\odot$) on
3-kpc scale. These first upper-limit constraints still considerably exceed the
estimated effect of CDM subhaloes. However, future analysis of a larger sample
of galaxy-galaxy strong lens systems can substantially narrow down these limits
and possibly rule out dark-matter models that predict a significantly higher
level of density fluctuations on the critical sub-galactic scales.
