The discrepancy between cosmological Li abundance inferred from Population II dwarf stars and that derived from WMAP/BBNS is still far from being solved.We investigated, as an alternative route, the use of Li abundances in Population II lower RGB stars as empirical diagnostic of the cosmological Li. Both theory and observations suggest that the surface A(Li) in red giants after the completion of the first dredge-up and before the RGB bump, are significantly less sensitive to the efficiency of atomic diffusion, compared with dwarf stars. Standard stellar models computed under different physical assumptions show that the inclusion of the atomic diffusion has an impact of 0.07dex in the determination of A(Li)0 (much smaller than the case of MS stars) and it is basically unaffected by reasonable variations of other parameters (overshooting, age,initial Y, mixing length). We have determined the surface Li content of 17 Halo lower RGB stars,in the metallicity range [Fe/H]=-3.4 /-1.4 dex. The initial Li has then been inferred by accounting for the difference between initial and post-dredge up A(Li) in the appropriate stellar models. It depends mainly on the used T(eff) scale and is only weakly sensitive to the efficiency of atomic diffusion,so long as one neglects Li destruction caused by the process competing with atomic diffusion. Final A(Li)0 span a relatively narrow range (2.28 /2.46 dex), and is 0.3-0.4 dex lower the WMAP/BBNS predictions. These values of A(Li)0 are corroborated by the analysis of the GCs NGC6397, NGC6752 and M4. Our result provides an independent quantitative estimate of the difference with the Big Bang value and sets a very robust constraint for the physical processes invoked to resolve this discrepancy.
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