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The age is the proper cosmic time elapsed from the scale-factor singularity $a=0$ to the present epoch.

The age of the universe is the cosmological quantity; an FLRW model computes it by integrating proper cosmic time from its initial scale-factor singularity.

The age of the universe is elapsed cosmic time since the early hot Big Bang phase, inferred within a cosmological model from expansion and matter-content measurements. The age of an FLRW universe is the corresponding model computation from its scale-factor evolution.

virial pressure-radius curve with uniform-sphere binding energy. Put $c_s^2=k_{B}T/\mu$. Isothermality makes $\int (p/\rho )dM=c_s^2{M}_0$, so the virial theorem givesThe curve crosses zero at $r_z=B/A=G{M}_0/(5c_s^2)$, rises to its maximum atthen decreases toward zero at large radius. Since $p_0^{\prime }=(4B-3A{r}_0)/r_0^5$, a cloud displaced outward at fixed external pressure has a restoring pressure deficit only on the descending branch. Thus the homologous, isothermal radial mode is stable for $r_0>r_c$, unstable for $r_z<r_0<r_c$, and marginal at $r_c$. The negative-pressure branch is not a physical equilibrium with a positive confining pressure. This conclusion uses the stipulated gravitational-energy approximation, not the exact isothermal density profile.

critical mass from a virial pressure turning point. At the turning point, $B=3A{r}_c/4$; henceUsing the cloud's mean density $\bar{\rho }=3M_0/(4\pi r_0^3)$, the condition $r_0>4G{M}_0/(15c_s^2)$ becomesUsing the boundary density instead changes the numerical coefficient, not this scaling. The coefficient should not be identified with the exact Bonnor-Ebert sphere threshold because the assumed potential energy is different.

mass and luminosity weighting of a stellar initial mass function. Impose finite birth-mass cutoffs $m_l,m_u$. The stellar birth mass and main-sequence luminosity scale asFor a broad mass interval, the low-mass end controls the first integral and the high-mass end the second. Per logarithmic mass interval the weights are respectively $m^2\xi \propto m^{-0.35}$ and $mL\xi \propto m^{2.65}$. Finite lifetimes remove massive stars from an aging population, so the luminosity conclusion applies while those stars remain on the main sequence.

exponential closed-box gas consumption. The gross formation rate is $\Psi =ϵM_g$, but the permanent locking rate is $\alpha \Psi$. The instantaneous recycling approximation and mass conservation give ${\dot{M}}_g=-\alpha ϵM_g$, ${\dot{M}}_{\ast }=\alpha ϵM_g$. ThereforeHere $M_{\ast }$ means surviving locked stellar mass in the galactic gas fraction, not the cumulative gross mass that has passed through star formation.

locked-mass and returned-mass yield conventions. Pre-existing metals are locked at rate $\alpha Z\Psi$; the stipulated newly produced metal source is $b(1-\alpha )\Psi$. ThusFor $\alpha >0$ and initially pristine gas,The effective yield $y$ is per unit permanently locked mass; it is not the source parameter $b$. For $\alpha =0$, $M_g=M_0$ and the direct time equation gives $Z=bϵt$. For $\alpha =1$, the explicitly stipulated metal source vanishes, so this particular enrichment model gives $Z=0$. Constant yield is a trace-enrichment approximation; the formal logarithmic divergence as gas is exhausted must not be interpreted as a physical metal fraction exceeding one.

backward extrapolation of exponential closed-box star formation. Denote the present age of the universe by $t_{\mathrm{now}}$. With the last paragraph's $\alpha =1$,Adopting a present age of the universe $t_{\mathrm{now}}=13.8\mathrm{Gyr}$ for the estimate gives about $365M_{\odot }\mathrm{y}{\mathrm{r}}^{-1}$, requiring $7.3\times 10^{11}{M}_{\odot }$ of gas at that epoch in the same model. The boxed expression makes the dependence on the adopted present age explicit. This is an extreme history for a present Milky-Way-like system, although intense high-redshift star formation by itself is not forbidden.

The extrapolation forces a fixed gas reservoir to decline exponentially over many gas-depletion times. Gas accretion, outflows, evolving gas density and star-formation efficiency, delayed recycling, and galaxy assembly break that assumption. The empirical disk-averaged relation is not a universal constant-efficiency law for all gas: the original measurements combined Balmer series H-alpha, atomic gas, carbon monoxide and infrared data, with tracer calibration and dust corrections entering the inference.^[ref] Cosmological calculations explicitly support continuing cold-gas supply to rapidly forming high-redshift galaxies, providing a physical alternative to this closed-box backward extrapolation.^[ref]

For monotonically expanding scale factor $R$, $dt=dR/(RH)$, so the elapsed cosmic time isThe second expression uses cosmological redshift $1+z=R_0/R$; the age of the universe is the limit to its earliest specified state, if that limit is finite.

An age comparison for an initial vacuum-dominated phase needs a common normalization. Match the two histories at the end of S1, with $R=R_1$ and $H=H_1>0$, and give them identical subsequent S2 and S3 evolution, including transition radii and today's state. Below $R_1$, the history with S1 has constant $H=H_1$. The comparison history extends S2 backwards, so $\rho \propto R^{-4}$ and $H=H_1(R_1/R{)}^2$. Starting both at $R_i=ϵR_1$, where $0<ϵ<1$, their early durations areThe difference is positive becauseThe common later evolution adds the same time to both histories. Thus S1 makes this matched universe older. More strongly, as $R_i\to 0$, the exact S1 duration diverges logarithmically, whereas a radiation beginning has finite duration $1/(2H_1)$ to the match. The exact de Sitter spacetime expanding patch reaches $R=0$ only as $t\to -\infty$ for its comoving observers; a finite “age since $R=0$” does not exist in this ideal model. This does not establish past geodesic completeness of the patch.

The matching assumption is essential for a finite cutoff or realistic finite period of cosmic inflation: arbitrary changes of the present Hubble parameter or transition densities prevent a universal age comparison. The literal zero-size limit in this question is instead the infinite-duration idealization just derived.

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 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,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,As an explicitly illustrative contemporary model, the Planck2015 temperature-plus-lensing estimates $H_0\simeq 67.8\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ and ${\Omega }_m\simeq 0.308$ give about $13.8\mathrm{Gyr}$. This is an inference under the assumed flat cosmology, not a model-independent clock reading or a numerical constant supplied in the examination.^[ref]

The lookback time to an object observed at redshift $z$ isThe 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 satisfiesThe 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.