Dynamical properties of z $\sim 4.5$ dusty star-forming galaxies and
  their connection with local early type galaxies
  There is a large consensus that gas in high-$z$ galaxies is highly turbulent,
because of a combination of stellar feedback processes and gravitational
instabilities driven by mergers and gas accretion. In this paper, we present
the analysis of a sample of five Dusty Star Forming Galaxies (DSFGs) at $4
\lesssim z\lesssim 5$. Taking advantage of the magnifying power of strong
gravitational lensing, we quantified their kinematic and dynamical properties
from ALMA observations of their [CII] emission line. We combined the dynamical
measurements obtained for these galaxies with those obtained from previous
studies to build the largest sample of $z \sim 4.5$ galaxies with high-quality
data and sub-kpc spatial resolutions, so far. We found that all galaxies in the
sample are dynamically cold, with rotation-to-random motion ratios, $V/\sigma$,
between 7 to 15. The relation between their velocity dispersions and their
star-formation rates indicates that stellar feedback is sufficient to sustain
the turbulence within these galaxies and no further mechanisms are needed. In
addition, we performed a rotation curve decomposition to infer the relative
contribution of the baryonic (gas, stars) and dark matter components to the
total gravitational potentials. This analysis allowed us to compare the
structural properties of the studied DSFGs with those of their descendants, the
local early type galaxies. In particular, we found that five out of six
galaxies of the sample show the dynamical signature of a bulge, indicating that
the spheroidal component is already in place at $z \sim 4.5$.
