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 $z \sim 4.5$.