Aims. The aim of this work is to study the contribution of the Ly-a emitters (LAE) to the star formation rate density (SFRD) of the Universe in the interval 2<z<6.6. Methods. We assembled a sample of 217 LAE from the Vimos-VLT Deep Survey (VVDS) with secure spectroscopic redshifts in the redshift range 2 < z < 6.62 and fluxes down to F=1.5x10^18 erg/s/cm^2. 133 LAE are serendipitous identifications in the 22 arcmin^2 total slit area surveyed with the VVDS-Deep and the 3.3 arcmin^2 from the VVDS Ultra-Deep survey, and 84 are targeted identifications in the 0.62 deg^2 surveyed with the VVDS-DEEP and 0.16 deg^2 from the Ultra-Deep survey. Among the serendipitous targets we estimate that 90% of the emission lines are most probably Ly-a, while the remaining 10% could be either [OII]3727 or Ly-a. We computed the LF and derived the SFRD from LAE at these redshifts. Results. The VVDS-LAE sample reaches faint line fluxes F(Lya) = 1.5x1^18 erg/s/cm^2 (corresponding to L(Lya)=10^41 erg/s at z~3) enabling the faint end slope of the luminosity function to be constrained to a=-1.6+-0.12 at redshift z~2.5 and to a=-1.78+0.1-0.12 at z=4, placing on firm statistical grounds trends found in previous LAE studies, and indicating that sub-L* LAE contribute significantly to the SFRD. The projected number density and volume density of faint LAE in 2<z<6.6 with F>1.5x10^18 erg/s/cm^2 are 33 galaxies/arcmin^2 and 4x10^-2 Mpc^-3, respectively. We find that the the observed luminosity function of LAE does not evolve from z=2 to z=6. This implies that, after correction for the redshift-dependent IGM absorption, the intrinsic LF must have evolved significantly over 3 Gyr. The SFRD from LAE contributes to about 20% of the SFRD at z =2-3, while the LAE appear to be the dominant source of star formation producing ionizing photons in the early universe z>5-6, becoming equivalent to that of Lyman Break galaxies.
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