We characterize the importance of metallicity on the presence of molecular hydrogen in damped Lyman-alpha (DLA) systems. We construct a representative sample of 18 DLA/sub-DLA systems with log N(HI)>19.5 at high redshift (zabs>1.8) with metallicities relative to solar [X/H]>-1.3(with[X/H]= logN(X)/N(H)-log(X/H)solar and X either Zn, S or Si). We gather data covering the expected wavelength range of redshifted H2 absorption lines on all systems in the sample from either the literature (10 DLAs), the UVES-archive or new VLT-UVES observations for four of them. The sample is large enough to discuss for the first time the importance of metallicity as a criterion for the presence of molecular hydrogen in the neutral phase at high-z. From the new observations, we report two new detections of molecular hydrogen in the systems at zabs=2.431 toward Q2343+125 and zabs=2.426 toward Q2348-011. We compare the H2 detection fraction in the high-metallicity sample with the detection fraction in the overall sample from Ledoux et al. (2003). We show that the fraction of DLA systems with logf=log 2N(H2)/(2N(H2)+N(HI))>-4 is as large as 50% for [X/H]>-0.7 when it is only about 5% for [X/H]<-1.3 and about 15% in the overall sample (with -2.5<[X/H]<-0.3). This demonstrates that the presence of molecular hydrogen at high redshift is strongly correlated with metallicity.
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