Cosmic age, Statefinder and $Om$ diagnostics in the decaying vacuum
  cosmology
  As an extension of $\Lambda$CDM, the decaying vacuum model (DV) describes the
dark energy as a varying vacuum whose energy density decays linearly with the
Hubble parameter in the late-times, $\rho_\Lambda(t) \propto H(t)$, and
produces the matter component. We examine the high-$z$ cosmic age problem in
the DV model, and compare it with $\Lambda$CDM and the Yang-Mills condensate
(YMC) dark energy model. Without employing a dynamical scalar field for dark
energy, these three models share a similar behavior of late-time evolution. It
is found that the DV model, like YMC, can accommodate the high-$z$ quasar APM
08279+5255, thus greatly alleviates the high-$z$ cosmic age problem. We also
calculate the Statefinder $(r,s)$ and the {\it Om} diagnostics in the model. It
is found that the evolutionary trajectories of $r(z)$ and $s(z)$ in the DV
model are similar to those in the kinessence model, but are distinguished from
those in $\Lambda$CDM and YMC. The ${\it Om}(z)$ in DV has a negative slope and
its height depends on the matter fraction, while YMC has a rather flat ${\it
Om}(z)$, whose magnitude depends sensitively on the coupling.
