Size evolution of star-forming galaxies with $2<z<4.5$ in the VIMOS Ultra-Deep Survey

We measure galaxy sizes on a sample of $\sim1200$ galaxies with confirmed spectroscopic redshifts $2 \leq z_{spec} \leq 4.5$ in the VIMOS Ultra Deep Survey (VUDS), representative of star-forming galaxies with $i_\mathrm{AB} \leq 25$. We first derive galaxy sizes applying a classical parametric profile fitting method using GALFIT. We then measure the total pixel area covered by a galaxy above a given surface brightness threshold, which overcomes the difficulty of measuring sizes of galaxies with irregular shapes. We then compare the results obtained for the equivalent circularized radius enclosing 100\% of the measured galaxy light $r_T^{100}$ to those obtained with the effective radius $r_{e,\mathrm{circ}}$ measured with GALFIT. We find that the sizes of galaxies computed with our non-parametric approach span a large range but remain roughly constant on average with a median value $r_T^{100}\sim2.2$ kpc for galaxies with $2<z<4.5$. This is in stark contrast with the strong downward evolution of $r_e$ with increasing redshift, down to sizes of $<1$ kpc at $z\sim4.5$. We analyze the difference and find that parametric fitting of complex, asymmetric, multi-component galaxies is severely underestimating their sizes. By comparing $r_T^{100}$ with physical parameters obtained through SED fitting we find that the star-forming galaxies that are the largest at any redshift are, on average, more massive and more star-forming. We discover that galaxies present more concentrated light profiles as we move towards higher redshifts. We interpret these results as the signature of several, possibly different, evolutionary paths of galaxies in their early stages of assembly, including major and minor merging or star-formation in multiple bright regions. (abridged)