= Solution
The metric uses signature $(+---)$ and a time-valued interval $ds$: a comoving clock has $ds=dt$. Here $t$ is cosmic <proper time>, $R(t)$ a <scale factor> with dimensions of <length>, $c$ the <speed of light>, $\chi$ a <dimensionless> comoving radial coordinate, and $\theta,\phi$ the usual polar and azimuthal angles. With curvature normalized to $k=+1,0,-1$, $S(\chi)=\sin\chi,\chi,\sinh\chi$, respectively. Homogeneity and <isotropy> apply to each constant-$t$ spatial slice. A comoving object's fixed coordinates do not imply a constant proper separation: spatial <lengths> scale with $R$.
Radial null propagation obeys $d\chi=c\,dt/R(t)$ in magnitude. Two neighbouring wave crests between the same comoving emitter and observer have the same radial integral, giving $\delta t_0/R_0=\delta t_e/R_e$. The <wavelength> <redshift> is consequently
$$
\boxed{1+z=\frac{\lambda_0}{\lambda_e}=\frac{\delta t_0}{\delta t_e}=\frac{R_0}{R_e}.}
$$
Peculiar Doppler shifts are excluded by the comoving assumption. Define <angular diameter distance> by $d_\theta=\ell_\perp/\delta\theta$, where $\ell_\perp$ is proper transverse size at emission. The transverse metric gives $\ell_\perp=R_eS(\chi)\delta\theta$, hence
$$
\boxed{d_\theta=R_eS(\chi)=\frac{R_0S(\chi)}{1+z}.}
$$
This distance is neither the <photon> travel <time> <times> $c$ nor the object's present proper distance.
The <comoving particle horizon> at $t_1$, with the initial boundary at the <Big Bang>, is $\chi_p(t_1)=\int_0^{t_1}c\,dt/R(t)$. Its proper radial distance is $D_p=R(t_1)\chi_p$; for a power-law <scale factor> $R=At^p$, $0<p<1$,
$$
\boxed{\chi_p=\frac{c\,t_1^{1-p}}{A(1-p)},\qquad D_p=\frac{ct_1}{1-p}.}
$$
Thus a matter-only flat <Einstein-de Sitter universe>, $p=2/3$, has $D_p=3ct_1$, and the flat radiation-only case, $p=1/2$, has $D_p=2ct_1$. Strictly, Einstein-de Sitter denotes the <pressureless matter> model; the radiation calculation uses the flat zero-$\Lambda$ radiation analogue requested here.
For the source seen at <time> $t_1=t_{\rm rec}$, the radial coordinate is the integral from <recombination> to today, not the earlier horizon integral. In these flat models,
$$
\chi_s=\frac{c}{A(1-p)}(t_0^{1-p}-t_1^{1-p}),\qquad
d_\theta=\frac{ct_1^p}{1-p}(t_0^{1-p}-t_1^{1-p}).
$$
Taking the linear horizon scale in the question to be $D_p$, its small angular extent is
$$
\boxed{\theta_p=\frac{D_p}{d_\theta}
=\frac{(t_1/t_0)^{1-p}}{1-(t_1/t_0)^{1-p}}.}
$$
Therefore
$$
\boxed{\theta_{p,m}=\frac{(t_{\rm rec}/t_0)^{1/3}}{1-(t_{\rm rec}/t_0)^{1/3}},\qquad
\theta_{p,r}=\frac{(t_{\rm rec}/t_0)^{1/2}}{1-(t_{\rm rec}/t_0)^{1/2}}.}
$$
Using the supplied <times> gives $\theta_{p,m}=0.02684\,\mathrm{rad}=1.54^\circ$ and $\theta_{p,r}=0.004244\,\mathrm{rad}=0.243^\circ$. There is a genuine radius/diameter convention to state: the supplied $3ct_1$ is a radial horizon distance. If angular diameter <means> the full diameter of a spherical horizon patch of radius $D_p$, the two answers are instead \b[$2\theta_{p,m}=3.08^\circ$ and $2\theta_{p,r}=0.486^\circ$]. Both conventions follow from the same distance calculation; a factor of two should not be hidden in the word horizon.
Matter-only evolution is the better of these two approximations because the universe is already matter dominated around <recombination> and evolves mostly through a subsequent matter era, rather than remaining radiation dominated to the present. Neither is exact: pre-recombination radiation changes the <particle horizon>, and late-time <dark energy> changes the distance to the last-scattering surface. No supplied ages have been replaced by later best-fit values.
The <cosmic microwave background> originates near last scattering and samples many horizon-sized regions. In an always-decelerating hot-Big-Bang model, widely separated last-scattering patches lack overlapping early causal histories, motivating the <horizon problem>. The measured acoustic angular scale uses the smaller <sound horizon>, not automatically this light-travel <particle horizon>; baryon-photon pressure changes the propagation speed. Inflation supplies an earlier accelerated expansion during which the <comoving Hubble radius> shrinks, allowing a patch with a common earlier causal history to expand to the size of the observable universe. Quantum perturbations can then seed later acoustic and density fluctuations. This does not require local <signals> to propagate faster than light, and the <Hubble radius> must not be equated indiscriminately with the <particle horizon>. CMB <temperature> and lensing spectra test the resulting perturbation and expansion model.<Planck 2013 results. XVI. Cosmological parameters by Planck Collaboration>[abstract; temperature/lensing spectra and the spatially flat standard cosmological model]{ref}
\Image[/past-exam-of-the-relativistic-astrophysics-and-cosmology-course-of-the-university-of-cambridge/2014/paper-1-horizon-angles.png]
{title=Original horizon-length angular extents in the two stated power-law cosmologies; full patch diameters are twice the plotted values}
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