= Solution
\b[The <age of the universe> is a proper time along comoving cosmological observers, inferred within an expansion model; <stellar ages> provide partly independent consistency checks.] In an expanding FLRW model with a big-bang origin,
$$
t_0=\int_0^1\frac{\mathrm da}{aH(a)}
=\int_0^\infty\frac{\mathrm dz}{(1+z)H(z)},
$$
where $a_0=1$. The <Hubble time> $H_0^{-1}$ is a characteristic scale, not generally the age. For flat <radiation> and dust universes, $t_0=1/(2H_0)$ and $2/(3H_0)$, respectively.
For a flat matter-plus-vacuum model neglecting <radiation> in the age integral,
$$
\boxed{t_0=
\frac{2}{3H_0\sqrt{\Omega_\Lambda}}
\operatorname{arsinh}\sqrt{\frac{\Omega_\Lambda}{\Omega_m}},
\qquad\Omega_m+\Omega_\Lambda=1.}
$$
As an explicitly illustrative contemporary model, the Planck2015 temperature-plus-lensing estimates $H_0\simeq67.8\ {\rm km\,s^{-1}\,Mpc^{-1}}$ and $\Omega_m\simeq0.308$ give about $13.8\ {\rm Gyr}$. This is an inference under the assumed flat cosmology, not a model-independent clock reading or a numerical constant supplied in the examination.<Planck 2015 cosmological parameters by Planck Collaboration>[February2015 original-version abstract, temperature/lensing flat-LambdaCDM parameter values]{ref}
The lookback time to an object observed at <redshift> $z$ is
$$
t_L(z)=\int_0^z\frac{\mathrm dz'}{(1+z')H(z')},\qquad
t_{\rm universe}(z)=t_0-t_L(z).
$$
The object's age at emission cannot exceed this <cosmic time>, and its formation occurs later than the big bang. A stellar population observed at large <redshift> therefore constrains both <stellar evolution> and the cosmological time available, rather than simply carrying today's cosmic age.
A star cluster's <main-sequence turnoff> traces the most massive stars still burning core <hydrogen>. More massive stars generally consume their usable fuel faster; fitting a color-magnitude diagram with stellar-evolution tracks can infer the population's formation time. The estimate depends on distance, reddening, composition, helium abundance, mixing and the binary population. Old <globular clusters> give lower limits on the cosmic age, with a formation delay between the big bang and the cluster.
White-dwarf cooling sequences provide another clock: thermal energy and crystallization physics determine how <luminosity> decreases after the remnant forms. The faint-end population must be combined with progenitor lifetimes and the star-formation history. Radioactive abundance ratios supply <nucleocosmochronology>. If two isotopes with decay constants $\lambda_1,\lambda_2$ start at ratio $\mathcal Q_0$, their measured ratio satisfies
$$
\mathcal Q(t)=\mathcal Q_0e^{-(\lambda_1-\lambda_2)t}.
$$
The initial production ratio and subsequent chemical mixing can dominate the uncertainty; a measured abundance ratio alone is not an exact formation date.
Kinematic expansion ages $R/v$ can estimate young remnants but require a dynamical model because deceleration changes the relation between present speed and elapsed time. Agreement among cosmological, stellar, cooling and radioactive estimates is powerful precisely because their systematic errors differ. An apparently over-age object calls for examining both the astrophysical clock and the cosmological assumptions before claiming a contradiction.
Back to article page
