= Solution
Here $R(t)$ is the cosmological <scale factor (cosmology)>, $t$ is <cosmic time> and $H=\dot R/R$ is the <Hubble parameter>. With $R$ a length, $k=+1,0,-1$ labels positively curved, flat and negatively curved spatial sections. This $k$ is unrelated to Question 1's specific orbital <energy>. The density $\rho$ is mass-equivalent <energy density>: $\rho c^2$ has energy-per-volume units. In the displayed <Friedmann equation> $\Lambda$ has units $\mathrm{s^{-2}}$; the geometric <cosmological constant> is $\Lambda_{\rm geom}=\Lambda/c^2$. For a separately conserved fluid of constant <equation of state> $p=w\rho c^2$, the <cosmological perfect-fluid continuity equation> gives $\rho\propto R^{-3(1+w)}$, so $\epsilon=3w$: zero for <pressureless matter> and one for radiation. With several fluids, sum their densities in the expansion equation and evolve each separately.
Setting <cosmological spatial curvature> and the <cosmological constant> to zero gives the <mass-equivalent cosmological critical density>,
$$
\boxed{\rho_{\rm crit}(t)=\frac{3H(t)^2}{8\pi G}}.
$$
Its energy-density version is $\rho_{\rm crit}c^2$; keeping this distinction is necessary when dividing a <radiation energy density> by $c^2$ below.
In an <Einstein-de Sitter universe>, $H^2\propto R^{-3}$ integrates to $R\propto t^{2/3}$ and $H=2/(3t)$. With $1+z=R_0/R$,
$$
t(z)=\frac{2}{3H_0}(1+z)^{-3/2},\qquad
\boxed{t_2-t_1=\frac{2}{3H_0}\left[(1+z_2)^{-3/2}-(1+z_1)^{-3/2}\right]}.
$$
Here $z_1>z_2$ are the two epoch redshifts relative to today. This is elapsed <cosmic time>, not <comoving distance> divided by $c$.
For <collisionless preservation of a Planck spectrum>, use <frequency> redshift as well as photon-number conservation. Let $f=R/R_i$, so a <frequency> bin $d\nu$ now corresponds to $d\nu_i=f\,d\nu$ and $\nu_i=f\nu$ initially. Collisionless propagation conserves <photon> number in the corresponding comoving bin:
$$
R^3n_\nu(t)\,d\nu=R_i^3n_{f\nu}(t_i)\,f\,d\nu,
\qquad n_\nu(t)=f^{-2}n_{f\nu}(t_i).
$$
For an initial <blackbody spectrum>, insert its <Planck distribution> to obtain
$$
n_\nu(t)=\frac{8\pi\nu^2}{c^3\left[\exp\{h\nu/(k_B T_i/f)\}-1\right]},
\qquad\boxed{T_{\rm rad}=T_i/f=T_0(1+z)}.
$$
The factors $f^2$ from the numerator cancel the spectral dilution factor, while the exponent changes only the <temperature>. Integrating gives <photon number density> proportional to $T^3\propto R^{-3}$ and <radiation energy density> proportional to $T^4\propto R^{-4}$. Conservation of total <photon> number alone would not guarantee a <blackbody spectrum> shape; the frequency-bin mapping and absence of injection or spectral redistribution are essential.
For the numerical density comparisons, the supplied <Hubble constant> is $H_0=2.33312\times10^{-18}\,\mathrm{s^{-1}}$. Taking $G=6.67430\times10^{-11}\,\mathrm{m^3\,kg^{-1}\,s^{-2}}$ and $c=2.99792458\times10^8\,\mathrm{m\,s^{-1}}$ gives
$$
\rho_{\rm crit,0}=9.7353\times10^{-27}\,\mathrm{kg\,m^{-3}},\quad
\rho_{\gamma0}=\frac{aT_0^4}{c^2}=4.6760\times10^{-31}\,\mathrm{kg\,m^{-3}},\quad
\Omega_{\gamma0}=4.8032\times10^{-5}.
$$
The radiation specified here is <cosmic microwave background> <photons> only. Adding a relic <neutrino> density would change the <photon-only matter-radiation equality> result. The rounded present matter and vacuum fractions sum to one without this small <photon> contribution; the requested pairwise density comparisons do not require silently renormalizing them.
For the final unlabelled flight-time request, insert the case-(c) value $1+z_c=(0.7/\Omega_{\gamma0})^{1/4}=10.98735$ into the stipulated Einstein-de Sitter formula:
$$
\Delta t=\frac{2}{3H_0}\left[10.98735^{-3/2}-1101^{-3/2}\right]
=7.838\times10^{15}\,\mathrm s\simeq\boxed{2.48\times10^8\,\mathrm{yr}}.
$$
The two model ages are approximately $2.486\times10^8$ and $2.478\times10^5$ years. This uses exactly the instructed matter-only model with the same $H_0$. It is not an exact age difference in the matter-plus-vacuum cosmology; a high-redshift matter-era normalization using $H_0\sqrt{\Omega_{m0}}$ would be a different approximation.
\Image[/past-exam-of-the-relativistic-astrophysics-and-cosmology-course-of-the-university-of-cambridge/2012/paper-1-density-crossings.png]
{title=Matter, photon and vacuum density crossings}
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