The comoving mass density of MgII from $z\sim2$ to $5.5$

We present the results of a survey for intervening MgII absorbers in the redshift range z $\simeq$2-6 in the foreground of four high redshift quasar spectra, 5.79$\le z_{em}\le$6.133, obtained with the ESO VLT X-shooter. We identify 24 absorbers at $\ge 5\sigma$ significance in the equivalent width range 0.117$\le W_{2796}\le 3.655\unicode{xC5}$ with the highest redshift absorber at $z=4.89031\pm4\times10^{-5}$. For weak ($W_{2796}<0.3\unicode{xC5}$) systems, we measure an incidence rate $dN/dz$=1.35$\pm$0.58 at <z>=2.34 and find that it almost doubles to $dN/dz$=2.58$\pm$0.67 by <z>=4.81. Weak absorbers exceeds the number expected from an exponential fit to stronger systems ($W_{2796}>0.3\unicode{xC5}$). We find that there must be significant evolution in the absorption halo properties of MgII absorbers with $W_{2796} >0.1\unicode{xC5}$ by <z>=4.77 and/or that they are associated with galaxies with luminosities beyond the limits of the current luminosity function at z $\sim$5. We find that the incidence rate of strong MgII absorbers ($W_{2796}>1.0\unicode{xC5}$) can be explained if they are associated with galaxies with $L\ge0.29L_{\ast}$ and/or their covering fraction increases. If they continue to only be associated with galaxies with $L\ge0.50L_{\ast}$ then their physical cross section ($\sigma_{phys}$) increases from 0.015 Mpc$^2$ at z=2.3 to 0.041 Mpc$^2$ at <z>=4.77. We measure $\Omega_{MgII}$=2.1$^{+6.3}_{-0.6}\times10^{-8}$, 1.9$^{+2.9}_{-0.2}\times10^{-8}$, 3.9$^{+7.1}_{-2.4}\times10^{-7}$ at <z>=2.48, 3.41, 4.77, respectively. At <z>=4.77, $\Omega_{MgII}$ exceeds the value expected from $\Omega_{HI}$ estimated from the global metallicity of DLAs at z $\simeq$4.85 by a factor of $\sim$44 suggesting that either MgII absorbers trace both ionised and neutral gas and/or are more metal rich than the average DLA at this redshift.