The ionizing photon production efficiency of bright z$\sim$2-5 galaxies

We investigate the production efficiency of ionizing photons ($\xi_{ion}^*$) of 1174 galaxies with secure redshift at z=2-5 from the VANDELS survey to determine the relation between ionizing emission and physical properties of bright and massive sources. We constrain $\xi_{ion}^*$ and galaxy physical parameters by means of spectro-photometric fits performed with the BEAGLE code. The analysis exploits the multi-band photometry in the VANDELS fields, and the measurement of UV rest-frame emission lines (CIII]$\lambda 1909$, HeII$\lambda 1640$, OIII]$\lambda 1666$) from deep VIMOS spectra. We find no clear evolution of $\xi_{ion}^*$ with redshift within the probed range. The ionizing efficiency slightly increases at fainter $M_{UV}$, and bluer UV slopes, but these trends are less evident when restricting the analysis to a complete subsample at log(M$_{star}$/M$_{\odot}$)$>$9.5. We find a significant trend of increasing $\xi_{ion}^*$ with increasing EW(Ly$\alpha$), with an average log($\xi_{ion}^*$/Hz erg$^{-1}$)$>$25 at EW$>$50\AA, and a higher ionizing efficiency for high-EW CIII]$\lambda 1909$ and OIII]$\lambda 1666$ emitters. The most significant correlations are found with respect to stellar mass, specific star-formation rate (sSFR) and SFR surface density ($\Sigma_{SFR}$). The relation between $\xi_{ion}^*$ and sSFR shows a monotonic increase from log($\xi_{ion}^*$/Hz erg$^{-1}$) $\sim$24.5 at log(sSFR)$\sim$-9.5$yr^{-1}$ to $\sim$25.5 at log(sSFR)$\sim$-7.5$yr^{-1}$, a low scatter and little dependence on mass. The objects above the main-sequence of star-formation consistently have higher-than-average $\xi_{ion}^*$. A clear increase of $\xi_{ion}^*$ with $\Sigma_{SFR}$ is also found, with log($\xi_{ion}^*$/Hz erg$^{-1}$)$>$25 for objects at $\Sigma_{SFR}>$10 M$_{\odot}/yr/kpc^2$.(Abridged)