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Two stages of the thermal history. Big Bang nucleosynthesis is nuclear assembly in an almost fully ionized, radiation-dominated plasma. Around $t\sim 1\mathrm{s}$, at temperatures of order $1\mathrm{MeV}/k_B$, the weak neutron–proton conversion rate falls below the expansion rate. The neutron-to-proton abundance freezes out at order $1/5$ and then decreases by neutron decay. The deuterium bottleneck delays effective nuclear assembly until $k_{B}T\sim 0.07$–$0.1\mathrm{MeV}$, at ages of a few minutes: the high photon-to-baryon ratio permits photodissociation despite the deuteron binding energy exceeding the mean photon energy. Once deuterium survives, reactions rapidly put most remaining neutrons into helium-4. A neutron-to-proton ratio near $1/7$ gives a helium mass fraction $Y_p\simeq 2(n/p)/(1+n/p)\simeq 1/4$, with hydrogen dominant by number and small residual deuterium, helium-3 and lithium-7 abundances. Expansion and low density prevent substantial heavy-element production. Nuclear binding is involved; neutral atoms do not yet survive.

Photon decoupling occurs much later, near a few thousand kelvin and a few hundred thousand years, after matter has overtaken radiation in the expansion dynamics. Electrons and nuclei form neutral atoms during cosmological recombination. The resulting decline of free-electron density reduces Thomson opacity, and photons begin to free stream over cosmological distances. Those photons form the observed cosmic microwave background. Atomic binding and scattering opacity, rather than new nuclear reactions, control this stage. Helium forms neutral atoms earlier than hydrogen. Recombination, photon last scattering, baryon drag release and the later thermal decoupling of the gas are related but not identical events.

The temperature falls through expansion: after the relevant entropy releases have ended, $T_{\gamma }\propto a^{-1}$ and number densities scale as $a^{-3}$. Big Bang nucleosynthesis lies in the radiation-dominated universe; photon decoupling lies after matter–radiation equality, when a matter-dominated approximation is useful but radiation is not exactly negligible.

Photon density and baryon abundance. The cosmic microwave background has a black-body radiation spectrum obeying Planck's law:where $u_{\nu }$ is spectral energy density, not a photon-number density. The measured temperature is about $2.72548\mathrm{K}$.^[ref] The mean photon energy in a black body isUsing the supplied rounded energy density givesThe exact black-body number integral at $2.725\mathrm{K}$ gives $4.1\times 10^8{\mathrm{m}}^{-3}$; the difference reflects rounding of the stated energy density. Either estimate is adequate.

A numerical photon-to-baryon ratio additionally needs a baryon density, which is not printed in this question. Make that observational input explicit rather than infer baryons from radiation alone. For illustration take the standard rounded physical density ${\Omega }_b{h}_H^2\simeq 0.022$, with $H_0=100h_H\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$. This is consistent with the cosmological parameter fits available around the paper's date.^[ref] It givesCounting baryons, including those bound in helium, rather than counting nuclei then givesBoth populations dilute as $a^{-3}$ after electron–positron annihilation, so this ratio also applies at hydrogen recombination in the standard adiabatic history.

Estimating decoupling. The large photon excess explains why setting $k_{B}T$ equal to the Hydrogen binding energy ${\chi }_H=13.6\mathrm{eV}$ is wrong. Even a small ionizing tail can oppose neutral-atom formation. The crude exponential balance $e^{-{\chi }_H/(k_{B}T)}\sim \eta$ gives ${\chi }_H/(k_{B}T)\sim \ln (1/\eta )\sim 21$, or $T\sim 7\times 10^3\mathrm{K}$. This only establishes the scale and the large suppression below ${\chi }_H/k_B$; it neglects phase-space factors and the opacity criterion.

A more controlled elementary Saha-Thomson decoupling estimate uses the hydrogen-only Saha equation. With $X_e=n_e/n_H$ and ground-state statistical weights combined in the usual way,The electron translational phase-space factor is significant inside the logarithm. This equilibrium relation places the sharp neutralization interval around $T\sim (3\text{–}4)\times 10^3\mathrm{K}$. Combine it with the approximate Thomson decoupling rate criterionFor $Y_p\simeq 0.25$, $n_{b0}\simeq 0.25{\mathrm{m}}^{-3}$, $H_0\simeq 68\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, and ${\Omega }_m\simeq 0.31$, solving these two approximate relations gives $T\simeq 3.1\times 10^3\mathrm{K}$, with equilibrium $X_e$ of order $10^{-2}$. Thus the appropriate estimate isThis is not a precision recombination calculation. Hydrogen-only equilibrium omits the kinetic bottleneck in escaping Lyman photons, two-photon processes, the detailed helium history and freeze-out; $\Gamma /H\sim 1$ is not identical to the maximum of the last-scattering visibility function. These approximations must be improved to predict an accurate free-electron fraction or a percent-level last-scattering redshift. The high photon-to-baryon ratio alone does not fix that detailed history.

The electrons are highly nonrelativistic: $k_{B}T/(m_e{c}^2)\sim 5\times 10^{-7}$, and their thermal root-mean-square speed is only about $10^{-3}c$. Photon coupling does not make their kinetic energies relativistic.

Age–temperature relation. During a nearly flat pressureless matter-dominated phase, with radiation, curvature and vacuum energy neglected in the expansion equation,Here the Big Bang time is chosen as $t=0$. Conserved photon entropy gives $T\propto a^{-1}$, soTo normalize this relation, use the matter-era expansion rate expressed in present-day measured parameters, rather than assume the late universe is still matter dominated:The same illustrative parameters give $t_{\mathrm{dec}}\sim 4.5\text{–}5\times 10^5\mathrm{yr}$ for $T_{\mathrm{dec}}\sim 3.0\text{–}3.1\times 10^3\mathrm{K}$. Radiation was still a meaningful correction; retaining it reduces the age, and a more complete recombination calculation gives the familiar scale of roughly $4\times 10^5\mathrm{yr}$.^[ref] A numerical age cannot follow from a proportionality $t\propto T^{-3/2}$ alone: an expansion or density normalization is essential. The small amount of dark energy at decoupling can be neglected even though it affects the present age.

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.