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Context: Studies of extremely metal-poor stars indicate that chemicalabundance ratios [X/Fe] have an rms scatter as low as 0.05 dex (12 \%). Itremains unclear whether this reflects observational uncertainties or intrinsicastrophysical scatter arising from physical conditions in the ISM at earlytimes. Aims: Measure differential chemical abundance ratios in extremelymetal-poor stars to investigate the limits of precision and to understandwhether cosmic scatter or observational errors are dominant. Methods: We usedhigh resolution (R $\sim 95,000$) and high S/N (S/N $= 700$ at 5000$\AA$)HIRES/Keck spectra, to determine high precision differential abundances betweentwo extremely metal-poor stars through a line-by-line differential approach. Wedetermined stellar parameters for the star G64-37 with respect to the standardstar G64-12. We performed EW measurements for the two stars for the linesrecognized in both stars and performed spectral synthesis to study the carbonabundances. Results: The differential approach allowed us to obtain errors of$\sigma$(T$_{eff}$ ) $=$ 27 K, $\sigma$(log $g$) $=$ 0.06 dex, $\sigma$([Fe/H])$=$ 0.02 dex and $\sigma$(v$_{t}$ ) $=$ 0.06 kms$^{-1}$. We estimated relativechemical abundances with a precision as low as $\sigma$([X/Fe]) $\approx$ 0.01dex. The small uncertainties demonstrate that there are genuine abundancedifferences larger than the measurement errors. The observed Li difference cannot be explained by the difference in mass, because the less massive star hasmore Li. Conclusions: It is possible to achieve an abundance precision around$\approx$ 0.01-0.05 dex for extremely metal-poor stars, opening new windows onthe study of the early chemical evolution of the Galaxy.

The first high precision differential abundance analysis of Extremely Metal Poor stars