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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,An Einstein-de Sitter universe gives $t_0=2/(3H_0)$. The age of a flat matter-Lambda universe, neglecting early radiation, isThis follows by integrating $dt=da/(aH)$ and substituting $x=\sqrt{{\Omega }_{\Lambda 0}/{\Omega }_{m0}}(a/a_0{)}^{3/2}$. 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.^[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.^[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. 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.

The geodesic with nonzero conserved spatial momentum reaches $t=-\infty$ after the finite past proper timeThis is incompleteness of the expanding coordinate patch, whose geodesics extend smoothly in global de Sitter spacetime; it is not a curvature singularity.

This region $0\le r<R$ of de Sitter spacetime is static for the central observer, with a cosmological Killing horizon at $r=R$. The central proper time fixes the normalization of ${\partial }_t$. Static observers have acceleration $r/(R^2\sqrt{1-r^2/R^2})$. Horizon-crossing null coordinates remove the coordinate singularity, while the global extension has compact spherical spatial slices and spacelike past/future conformal infinity.

In the flat slicing of de Sitter spacetime, $a\propto -1/\eta$, $\eta <0$, a Fourier mode of a minimally coupled massless scalar field obeys $f_k^{\prime \prime }-2f_k^{\prime }/\eta +c^2{k}^2{f}_k=0$. Substitution verifies the displayed positive-frequency branch. The Canonically rescaled de Sitter scalar mode is $u_k=a{f}_k$, a different variable satisfying $u_k^{\prime \prime }+(c^2{k}^2-2/{\eta }^2)u_k=0$. For nonzero $k$, the opposite-frequency branch completes the time solution; at $k=0$ the solutions are $1$ and ${\eta }^3$. This separation avoids losing the homogeneous decaying mode by taking a restricted ansatz literally at zero wave number.

The equation of state $p=-\varrho$ makes the cosmological continuity equation imply constant energy density ${\varrho }_0$. For ${\varrho }_0>0$, put $H_{\Lambda }^2=8\pi G{\varrho }_0/3$. Solving both the Friedmann equation and Friedmann acceleration equation givesUp to shifts of cosmic time and time reversal, the positive scale factors are $A{e}^{H_{\Lambda }t}$ for $k=0$, $H_{\Lambda }^{-1}\cosh (H_{\Lambda }t)$ for $k=1$, and $H_{\Lambda }^{-1}\sinh (H_{\Lambda }t)$ on $t>0$ for $k=-1$. They are the flat, closed and open slicings of de Sitter spacetime. The open zero is a coordinate boundary with finite Ricci scalar $12H_{\Lambda }^2$ and finite Kretschmann scalar $24H_{\Lambda }^4$.

For the expanding flat branch, the cosmological event horizon has comoving radial distance ${\int }_t^{\infty }ds/a(s)=1/[H_{\Lambda }a(t)]$ and proper distance in cosmology $H_{\Lambda }^{-1}$. An event on the horizon reaches the central comoving observer only at infinite reception time; events beyond it never do.