The VIMOS VLT Deep Survey: star formation rate density of Ly alpha emitters from a sample of 217 galaxies with spectroscopic redshifts 2 <= z <= 6.6

Aims. The aim of this work is to study the contribution of the Ly alpha emitters to the star formation rate density (SFRD) of the Universe in the interval 2 < z < 6.6. Methods. We assembled a sample of 217 Ly alpha emitters (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 similar to 1.5 x 10(-18) erg/s/cm(2). Of those Ly alpha emitters, 133 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 alpha, while the remaining 10% could be either [OII] 3727 or Ly alpha. We computed the luminosity function (LF) and derived the star-formation rate density using this sample of LAE. Results. The VVDS-LAE sample reaches faint line fluxes F(Ly alpha) = 1.5 x 10(-18) erg/s/cm(2) (corresponding to L(Ly alpha) similar to 10(41) erg/s at z similar to 3), allows the faint-end slope of the luminosity function to be constrained to alpha similar to -1.6 +/- 0.12 at redshift z similar to 2.5 and to alpha similar to -1.78(-0.12)(0.10) at redshift similar to 4, placing trends found in previous LAE studies on firm statistical grounds, and indicating that sub-L-* LAE (LLy-alpha less than or similar to 10(42.5) erg/s) contribute significantly to the SFRD. The projected number density and volume density of faint LAE in 2 <= z <= 6.6 with F > 1.5 x 10(-18) erg/s/cm(2) are 33 galaxies/arcmin(2) and similar to 4 x 10(-2) Mpc(-3), respectively. We find that the observed luminosity function (LF) of LAEs does not evolve from z = 2 to z = 6. This implies that, after correction for the redshift-dependent IGM absorption, the intrinsic luminosity function must have evolved significantly over 3 Gyr. The SFRD from LAE contributes around 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 similar to> 5-6, equivalent to Lyman Break galaxies.