The Shapes of the Rotation Curves of Star-forming Galaxies Over the Last
  $\approx$10 Gyr
  We analyse maps of the spatially-resolved nebular emission of $\approx$1500
star-forming galaxies at $z\approx0.6$-$2.2$ from deep KMOS and MUSE
observations to measure the average shape of their rotation curves. We use
these to test claims for declining rotation curves at large radii in galaxies
at $z\approx1$-$2$ that have been interpreted as evidence for an absence of
dark matter. We show that the shape of the average rotation curves, and the
extent to which they decline beyond their peak velocities, depends upon the
normalisation prescription used to construct the average curve. Normalising in
size by the galaxy stellar disk-scale length after accounting for seeing
effects ($R_{\rm{d}}^{\prime}$), we construct stacked position-velocity
diagrams that trace the average galaxy rotation curve out to
$6R_{\rm{d}}^{\prime}$ ($\approx$13 kpc, on average). Combining these curves
with average HI rotation curves for local systems, we investigate how the
shapes of galaxy rotation curves evolve over $\approx$10 Gyr. The average
rotation curve for galaxies binned in stellar mass, stellar surface mass
density and/or redshift is approximately flat, or continues to rise, out to at
least $6R_{\rm{d}}^{\prime}$. We find a trend between the outer slopes of
galaxies' rotation curves and their stellar mass surface densities, with the
higher surface density systems exhibiting flatter rotation curves. Drawing
comparisons with hydrodynamical simulations, we show that the average shapes of
the rotation curves for our sample of massive, star-forming galaxies at
$z\approx0$-$2.2$ are consistent with those expected from $\Lambda$CDM theory
and imply dark matter fractions within $6R_{\rm{d}}$ of at least $\approx60$
percent.
