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We study the average Ly$\alpha$ emission associated with high-$z$ strong (log$N$(H I) $\ge$ 21) damped Ly$\alpha$ systems (DLAs). We report Ly$\alpha$luminosities ($L_{\rm Ly\alpha}$) for the full as well as various sub-samplesbased on $N$(H I), $z$, $(r-i)$ colours of QSOs and rest equivalent width of SiII$\lambda$1526 line (i.e., $W_{1526}$). For the full sample, we find $L_{\rmLy\alpha}$$< 10^{41} (3\sigma)\ \rm erg\ s^{-1}$ with a $2.8\sigma$ leveldetection of Ly$\alpha$ emission in the red part of the DLA trough. The $L_{\rmLy\alpha}$ is found to be higher for systems with higher $W_{1526}$ with itspeak, detected at $\geq 3\sigma$, redshifted by about 300-400 $\rm km\ s^{-1}$with respect to the systemic absorption redshift, as seen in Lyman BreakGalaxies (LBGs) and Ly$\alpha$ emitters. A clear signature of a double-humpLy$\alpha$ profile is seen when we consider $W_{1526} \ge 0.4$ \AA\ and $(r-i)< 0.05$. Based on the known correlation between metallicity and $W_{1526}$, weinterpret our results in terms of star formation rate (SFR) being higher inhigh metallicity (mass) galaxies with high velocity fields that facilitateseasy Ly$\alpha$ escape. The measured Ly$\alpha$ surface brightness requireslocal ionizing radiation that is 4 to 10 times stronger than the metagalacticUV background at these redshifts. The relationship between the SFR and surfacemass density of atomic gas seen in DLAs is similar to that of local dwarf andmetal poor galaxies. We show that the low luminosity galaxies will contributeappreciably to the stacked spectrum if the size-luminosity relation seen for HI at low-$z$ is also present at high-$z$. Alternatively, large Ly$\alpha$ halosseen around LBGs could also explain our measurements.

The Ly$\alpha$ emission from high-$z$ galaxies hosting strong Damped Ly$\alpha$ systems