On the robustness of the Hβ Lick index as a cosmic clock in passive early-type galaxies

We examine the H$\beta$ Lick index in a sample of $\sim 24000$ massive ($\rm log(M/M_\odot)>10.75$) and passive early-type galaxies extracted from SDSS at z<0.3, in order to assess the reliability of this index to constrain the epoch of formation and age evolution of these systems. We further investigate the possibility of exploiting this index as "cosmic chronometer", i.e. to derive the Hubble parameter from its differential evolution with redshift, hence constraining cosmological models independently of other probes. We find that the H$\beta$ strength increases with redshift as expected in passive evolution models, and shows at each redshift weaker values in more massive galaxies. However, a detailed comparison of the observed index with the predictions of stellar population synthesis models highlights a significant tension, with the observed index being systematically lower than expected. By analyzing the stacked spectra, we find a weak [NII]$\lambda6584$ emission line (not detectable in the single spectra) which anti-correlates with the mass, that can be interpreted as a hint of the presence of ionized gas. We estimated the correction of the H$\beta$ index by the residual emission component exploiting different approaches, but find it very uncertain and model-dependent. We conclude that, while the qualitative trends of the observed H$\beta$-z relations are consistent with the expected passive and downsizing scenario, the possible presence of ionized gas even in the most massive and passive galaxies prevents to use this index for a quantitative estimate of the age evolution and for cosmological applications.