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 $f_{\rm H_2}(N)$ from the average H$_2$ absorption signal. Assuming a power-law column density distribution, we obtain a slope $\beta = -1.29 \pm 0.06(\rm stat) \pm 0.10 (\rm sys)$ and an incidence rate of strong H$_2$ absorptions (with $N$(H$_2)\gtrsim 10^{18}\,$cm$^{-2}$) to be $4.0 \pm 0.5(\rm stat) \pm 1.0 (\rm sys)\,\%$ in H$\,$I absorption systems with $N($H$\,$I)$\ge 10^{20}\,$cm$^{-2}$. Assuming the same inflexion point where $f_{\rm H_2}(N)$ steepens as at $z=0$, we estimate that the cosmological density of H$_2$ in the column density range $\log N(\rm H_2)$(cm$^{-2})= 18-22$ is $\sim 15\%$ 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 $f_{\rm H_2}(N)$, suggests that the typical H$\,$I-H$_2$ transition column density in DLAs is $\log N({\rm H})$(cm$^{-2}) \gtrsim 22.3$ in agreement with theoretical expectations for the average (low) metallicity of DLAs at high-$z$.