On the robustness of the H$\beta$ 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.
