= Solution
The <age of the universe> is a model-dependent elapsed <cosmic time>, not simply the inverse present <Hubble constant>. For a model with a hot <Big Bang> origin,
$$
\boxed{t_0=\int_0^\infty\frac{dz}{(1+z)H(z)}}.
$$
An <Einstein-de Sitter universe> gives $t_0=2/(3H_0)$. The <age of a flat matter-Lambda universe>, neglecting early <radiation>, is
$$
t_0=\frac{2}{3H_0\sqrt{\Omega_{\Lambda0}}}
\operatorname{arsinh}\sqrt{\frac{\Omega_{\Lambda0}}{\Omega_{m0}}},\qquad
\Omega_{m0}=1-\Omega_{\Lambda0}.
$$
This follows by integrating $dt=da/(aH)$ and substituting $x=\sqrt{\Omega_{\Lambda0}/\Omega_{m0}}\,(a/a_0)^{3/2}$. It tends to $2/(3H_0)$ as the vacuum fraction tends to zero. Pure <de Sitter spacetime> instead has constant $H$ and no finite dust <Big Bang> age obtained from this integral. <cosmic microwave background> ages therefore depend on the assumed expansion model and fitted parameters; the <WMAP> nine-year combined-data <Lambda-CDM model> inference was approximately 13.8 billion years, not a direct clock reading.<Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results by C. L. Bennett and others>[Version 1 abstract, inferred cosmic age]{ref}
Old <globular clusters> provide an independent lower bound. Their <main-sequence turnoff> identifies stars just exhausting core hydrogen; the nuclear fuel divided by <luminosity> sets a mass-dependent lifetime, so a fitted <colour-magnitude diagram> constrains the common <stellar age>. Distance, extinction, <metallicity>, helium abundance, diffusion and stellar-model physics introduce uncertainties. The elapsed time before cluster formation must be added when comparing with the cosmic age.<New Globular Cluster Age Estimates and Constraints on the Cosmic Equation of State and The Matter Density of the Universe by Lawrence M. Krauss and Brian Chaboyer>[Abstract and joint stellar-age/cosmological interpretation]{ref}
<white-dwarf cooling ages> offer another clock: faint remnants have lost more stored thermal <energy>, but progenitor lifetimes must be added and <crystallization>, envelope properties and sample completeness affect the inference. <nucleocosmochronology> uses long-lived radioactive abundance ratios, with $Q/Q_0=e^{-(\lambda_1-\lambda_2)t}$ for two radioactive species, or a stable reference with $\lambda_2=0$. Uncertain initial production ratios and subsequent enrichment limit precision. \b[Every reliably dated object must be younger than the universe by at least its formation delay]; consistency among distinct clocks tests both cosmology and their astrophysical calibrations.
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