Constraining the H${}_2$ column density distribution at z$\sim$3 from composite DLA spectra

We present the detection of the average H${}_2$ absorption signal in the overall population of neutral gas absorption systems at $z\sim 3$ using composite absorption spectra built from the Sloan Digital Sky Survey-III damped Lyman-$\alpha$ catalogue. We present a new technique to directly measure the H${}_2$ column density distribution function $f_{{\mathrm{H}}_2}(N)$ from the average H${}_2$ absorption signal. Assuming a power-law column density distribution, we obtain a slope $\beta =-1.29\pm 0.06(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t})\pm 0.10(\mathrm{s}\mathrm{y}\mathrm{s})$ and an incidence rate of strong H${}_2$ absorptions (with $N$(H$_2)\gtrsim 10^{18},$cm$^{-2}$)tobe$4.0 \pm 0.5(\rm stat) \pm 1.0 (\rm sys),%$ in H$$I absorption systems with $N($H$$I)$\ge {10}^{20}$cm${}^{-2}$. Assuming the same inflexion point where $f_{{\mathrm{H}}_2}(N)$ steepens as at $z=0$, we estimate that the cosmological density of H${}_2$ in the column density range $\log N({\mathrm{H}}_2)$(cm${}^{-2})=18-22$ is $\sim 15\mathrm{\%}$ of the total. We find one order of magnitude higher H${}_2$ incident rate in a sub-sample of extremely strong DLAs ($\log N($H$$I)(cm${}^{-2})\ge 21.7$), which, together with the the derived shape of $f_{{\mathrm{H}}_2}(N)$, suggests that the typical H$$I-H${}_2$ transition column density in DLAs is $\log N(\mathrm{H})$(cm${}^{-2})\gtrsim 22.3$ in agreement with theoretical expectations for the average (low) metallicity of DLAs at high-$z$.