= Solution
\b[Two stages of the thermal history.] <Big Bang nucleosynthesis> is nuclear assembly in an almost fully ionized, radiation-dominated plasma. Around $t\sim1\ \mathrm s$, 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 $k_BT\sim0.07$–$0.1\ \mathrm{MeV}$, 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 $Y_p\simeq2(n/p)/(1+n/p)\simeq1/4$, 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.
\b[<Photon> density and <baryon> abundance.] The <cosmic microwave background> has a <black-body radiation> spectrum obeying <Planck's law>:
$$
u_\nu=\frac{8\pi h\nu^3}{c^3}\frac1{e^{h\nu/(k_BT)}-1},\qquad T_0\simeq2.725\ \mathrm K,
$$
where $u_\nu$ is spectral energy density, not a photon-number density. The measured <temperature> is about $2.72548\ \mathrm K$.<The Temperature of the Cosmic Microwave Background by Dale J. Fixsen>[Abstract: literature-combined black-body <temperature>, 2.72548 ± 0.00057 K]{ref} The mean <photon energy> in a black body is
$$
\langle E_\gamma\rangle=\frac{\pi^4}{30\zeta(3)}k_BT\simeq2.701k_BT.
$$
Using the supplied rounded energy density gives
$$
\boxed{n_{\gamma0}\simeq\frac{4\times10^{-14}}{2.701k_BT_0}\simeq3.9\times10^8\ \mathrm{m^{-3}}}.
$$
The exact black-body number integral at $2.725\ \mathrm K$ gives $4.1\times10^8\ \mathrm{m^{-3}}$; 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 $\Omega_bh_H^2\simeq0.022$, with $H_0=100h_H\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$. This is consistent with the cosmological parameter fits available around the paper's date.<Planck 2015 cosmological parameters by Planck Collaboration>[Base ΛCDM parameter tables: physical <baryon> density near 0.022; illustrative expansion normalization H0 about 68 km s−1 Mpc−1 and matter fraction about 0.31]{ref} It gives
$$
\rho_{b0}=(\Omega_bh_H^2)\frac{3(100\ \mathrm{km\,s^{-1}\,Mpc^{-1}})^2}{8\pi G}\simeq4.1\times10^{-28}\ \mathrm{kg\,m^{-3}},\qquad n_{b0}\simeq\frac{\rho_{b0}}{m_p}\simeq0.25\ \mathrm{m^{-3}}.
$$
Counting <baryons>, including those bound in <helium>, rather than counting nuclei then gives
$$
\boxed{\frac{n_\gamma}{n_b}\sim1.6\times10^9,\qquad \eta\equiv\frac{n_b}{n_\gamma}\sim6\times10^{-10}}.
$$
Both populations dilute as $a^{-3}$ after <electron>–positron annihilation, so this ratio also applies at <hydrogen> recombination in the standard adiabatic history.
\b[Estimating decoupling.] The large <photon> excess explains why setting $k_BT$ 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 $e^{-\chi_H/(k_BT)}\sim\eta$ gives $\chi_H/(k_BT)\sim\ln(1/\eta)\sim21$, or $T\sim7\times10^3\ \mathrm K$. 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,
$$
\frac{X_e^2}{1-X_e}=\frac1{n_H}\left(\frac{2\pi m_ek_BT}{h^2}\right)^{3/2}e^{-\chi_H/(k_BT)},\qquad
n_H\simeq(1-Y_p)n_{b0}\left(\frac T{T_0}\right)^3.
$$
The <electron> translational phase-space factor is significant inside the logarithm. This equilibrium relation places the sharp neutralization interval around $T\sim(3\text{–}4)\times10^3\ \mathrm K$. Combine it with the approximate <Thomson decoupling rate criterion>
$$
\Gamma_\gamma=n_e\sigma_Tc\sim H,\qquad
H\simeq H_0\sqrt{\Omega_m}\left(\frac T{T_0}\right)^{3/2}.
$$
For $Y_p\simeq0.25$, $n_{b0}\simeq0.25\ \mathrm{m^{-3}}$, $H_0\simeq68\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$, and $\Omega_m\simeq0.31$, solving these two approximate relations gives $T\simeq3.1\times10^3\ \mathrm K$, with equilibrium $X_e$ of order $10^{-2}$. Thus the appropriate estimate is
$$
\boxed{T_{\rm dec}\sim3\times10^3\ \mathrm K,\qquad k_BT_{\rm dec}\sim0.26\ \mathrm{eV},\qquad z_{\rm dec}\sim1100}.
$$
This 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; $\Gamma/H\sim1$ 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: $k_BT/(m_ec^2)\sim5\times10^{-7}$, and their thermal root-mean-square speed is only about $10^{-3}c$. <Photon> coupling does not make their kinetic energies relativistic.
\b[Age–<temperature> relation.] During a nearly flat pressureless matter-dominated phase, with radiation, curvature and vacuum energy neglected in the expansion equation,
$$
\rho_m\propto a^{-3},\qquad \left(\frac{\dot a}a\right)^2\propto a^{-3},\qquad \boxed{a(t)\propto t^{2/3}}.
$$
Here the Big Bang time is chosen as $t=0$. Conserved <photon> entropy gives $T\propto a^{-1}$, so
$$
\boxed{T\propto t^{-2/3},\qquad t\propto T^{-3/2}}.
$$
To 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:
$$
\boxed{t(T)\simeq\frac{2}{3H_0\sqrt{\Omega_m}}\left(\frac{T_0}T\right)^{3/2}}.
$$
The same illustrative parameters give $t_{\rm dec}\sim4.5\text{–}5\times10^5\ \mathrm{yr}$ for $T_{\rm dec}\sim3.0\text{–}3.1\times10^3\ \mathrm K$. Radiation was still a meaningful correction; retaining it reduces the age, and a more complete recombination calculation gives the familiar scale of roughly $4\times10^5\ \mathrm{yr}$.<Planck 2015 cosmological parameters by Planck Collaboration>[§6.7: recombination and <photon decoupling> near redshift 10^3, at an age around 400000 years; distinction from detailed kinetic ionization-history calculations]{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.
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