= Solution
The <cosmic microwave background> consists of relic <photons> from the early hot, nearly thermal <photon-baryon fluid>. Before <cosmological recombination>, abundant free <electrons> repeatedly scatter <photons> by <Thomson scattering>. Their <mean free path> $\lambda_\gamma=(n_e\sigma_T)^{-1}$ is short and their <Thomson scattering> rate $n_e\sigma_Tc$ is much greater than the <Hubble parameter>. An earlier <photon> therefore cannot stream directly to us: subsequent scattering repeatedly changes its direction.
As <cosmic expansion> cools the plasma to about $T\simeq3000\,\mathrm K$, <electrons> and <protons> combine into neutral <hydrogen>. The free-electron abundance drops rapidly, the scattering <optical depth> to the observer becomes of order one, and <photon decoupling> occurs. Since the <cosmic microwave background> <temperature> scales as $T=T_0(1+z)$, with $T_0\simeq2.725\,\mathrm K$, this occurs at $z\simeq1100$. The small <baryon-to-photon ratio> delays neutralization to well below $13.6\,\mathrm{eV}/k_B$: the high-energy tail contains enough <photons> to ionize <hydrogen> until the <temperature> is much lower than the <Hydrogen binding energy> expressed as a <temperature>. The observed <cosmic microwave background> is primarily this pre-existing radiation undergoing its last scattering; its nearly <blackbody spectrum> is then preserved by <cosmological redshift>.
For the requested estimate take a flat <matter-dominated universe>, $a(t)=(t/t_0)^{2/3}$. Restore $c$ in distances. The <comoving particle horizon> at <cosmological recombination> and the <comoving radial distance> to the <Cosmic microwave background last-scattering surface> are
$$
r_{\mathrm{hor},*}=c\int_0^{t_*}\frac{dt}{a(t)}=3ct_0\sqrt{a_*},\qquad
\chi_*=c\int_{t_*}^{t_0}\frac{dt}{a(t)}=3ct_0(1-\sqrt{a_*}).
$$
The proper <particle horizon> is $a_*r_{\mathrm{hor},*}=3ct_*$ and the <angular diameter distance> is $D_A=a_*\chi_*$. Therefore the <angular particle-horizon size in a power-law universe>, taking one horizon radius as the transverse length, is
$$
\boxed{\theta_{\mathrm{hor}}\simeq\frac{r_{\mathrm{hor},*}}{\chi_*}=\frac1{\sqrt{1+z_*}-1}\simeq0.0311\,\mathrm{rad}\simeq1.78^\circ.}
$$
The angular diameter of a patch of that radius is about $3.56^\circ$. The factor of two is a radius-versus-diameter convention; either gives a horizon scale of only a few degrees. This is a <particle horizon>, rather than the smaller <sound horizon>. Including early <radiation in cosmology> changes the numerical estimate but preserves its order of magnitude.
The <horizon problem> is that the <cosmic microwave background> has almost the same <temperature> across the whole sky, including pairs of regions whose past light cones do not overlap in this noninflating model. Local thermal equilibration cannot explain that observation within this history. A sufficiently long period of <cosmic inflation> makes the <comoving Hubble radius> decrease: a region once small enough to share a causal past can expand to encompass our entire observed <Cosmic microwave background last-scattering surface>. After <reheating>, it can be homogeneous across many later <particle horizon> patches. The matter-only estimate integrates from a decelerating <Big Bang>; extending the earlier history by <cosmic inflation> changes that causal conclusion.
Back to article page
