The luminosity function of Lyα emitters at 2.3 < z < 4.6 from integral-field spectroscopy

We have used Visible MultiObject Spectrograph Integral-Field Unit observations centred on a radio galaxy at z = 2.9 to search for Ly alpha emitters within a comoving volume of similar to 10(4) Mpc(3). We find 14 Ly alpha emitters with flux > 1.4 x 10(-20) W m(-2), yielding a comoving space density of 0.0018(-0.0005)(+0.0006) Mpc(-3). We fit a Schechter luminosity function that agrees well with previous studies both at similar redshift (z similar to 3.4) and higher redshift (z similar to 5.7). We therefore find no evidence for evolution in the properties of Ly alpha emitters between 3 < z < 6, although our sample is small. By summing the star-formation rates of the individual Ly alpha emitters we find a total cosmic star-formation rate density of rho(SFR) = 6.7 +/- 0.5 x 10(-3) M-circle dot yr(-1) Mpc(-3). Integrating over the luminosity function for the combined Ly alpha surveys at z similar to 3.4 and accounting for the difference in obscuration between the Ly alpha line and the ultraviolet-continuum yields an estimate of rho(SFR) similar to 2.2 x 10(-2) M-circle dot yr(-1) Mpc(-3), in line with previous multi-colour and narrow-band surveys of high-redshift star-forming galaxies.