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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 $n_e{\sigma }_{T}c$ 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 $T\simeq 3000\mathrm{K}$, 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 $T_0\simeq 2.725\mathrm{K}$, this occurs at $z\simeq 1100$. 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 areThe proper particle horizon is $a_{\ast }{r}_{\mathrm{hor},\ast }=3c{t}_{\ast }$ and the angular diameter distance is $D_A=a_{\ast }{\chi }_{\ast }$. Therefore the angular particle-horizon size in a power-law universe, taking one horizon radius as the transverse length, isThe 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.

As the temperature falls, electrons and protons form neutral hydrogen through cosmological recombination. The free electron fraction drops sharply, and so does the scattering rate ${\Gamma }_{\gamma }=n_e{\sigma }_{T}c$. When this rate becomes comparable to or smaller than the Hubble parameter, the plasma becomes transparent and photon decoupling occurs. The Thomson scattering cross-section itself does not abruptly vanish. The rapid change is in the abundance of free targets.

The Hydrogen binding energy is $13.6\mathrm{eV}$, but recombination occurs at a much smaller thermal energy, of order $0.3\mathrm{eV}$, or a few thousand kelvin. There are roughly a billion photons per baryon, so the high-energy tail can keep hydrogen ionized long after $k_{B}T$ drops below the binding energy. The falling exponential ionizing tail then makes the transition comparatively sharp. Last scattering follows when sufficiently few electrons remain free.^[ref]

To estimate a horizon, specify an expansion model as well as the redshift. The proper particle horizon at emission isAs a transparent analytic estimate, approximate a spatially flat matter-dominated universe by $a(t)=(t/t_{\mathrm{now}}{)}^{2/3}$ throughout. ThenTaking the illustrative present value $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, this model has $t_{\mathrm{now}}=2/(3H_0)\simeq 9.3\times 10^9\mathrm{yr}$. At $z_{\ast }=10^3$,This is a proper horizon radius at emission; its corresponding present comoving length is about $270\mathrm{Mpc}$.

For the same model the comoving distance to the Cosmic microwave background last-scattering surface and its angular diameter distance areTherefore the angular extent associated with one radial horizon length isThis is an order-one-degree estimate. Including the earlier radiation domination and late dark energy changes the coefficient; it does not make the horizon span the sky. A horizon patch diameter has twice this angle, whereas the acoustic scale uses a sound horizon and differs again. The redshift by itself, without an expansion model or normalization, is insufficient for a unique horizon distance.

The horizon problem is that the observed cosmic microwave background has nearly the same mean temperature across angular separations vastly greater than this scale. In the purely decelerating Hot Big Bang model, sufficiently separated last-scattering regions had neither time to exchange signals nor a common post-Big-Bang causal region that could establish their observed thermal uniformity. The puzzle concerns the missing causal explanation of those initial correlations; it is not an inconsistency of specifying uniform initial conditions mathematically. An earlier period of cosmic inflation can enlarge a formerly causally connected region beyond the later observable horizon.

The comoving distance to the Cosmic microwave background last-scattering surface is of order the present horizon, ${\chi }_{\ast }\sim 1.3\times 10^4\mathrm{Mpc}$. At recombination the particle horizon is much smaller, so replacing ${\chi }_{\ast }$ by today's horizon is adequate for the requested estimate. A transverse Fourier mode projects approximately to angular multipole $\ell \sim k{\chi }_{\ast }$. If an angular feature's width is estimated by $\theta \sim 1/\ell$, the resolution ${\theta }_{\min }\sim 10^{-3}$ givesUsing the common half-wavelength convention $\theta \sim \pi /\ell$ instead gives ${\ell }_{\max }\sim 3000$ and $k_{\max }\sim 0.2$--$0.3\mathrm{Mp}{\mathrm{c}}^{-1}$. These express the same order-of-magnitude resolution estimate with different angular-scale conventions. On the graph, a useful indicative band is therefore from about $10^{-4}\mathrm{Mp}{\mathrm{c}}^{-1}$ to order $10^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, potentially a few times $10^{-1}$; it is not a sharp instrumental boundary. The lowest useful multipoles are of order a few, with substantial cosmic variance.

The cosmic microwave background constrains the matter power spectrum indirectly through the primordial spectrum and radiation/matter cosmological transfer functions. Beam resolution, Cosmic microwave background diffusion damping, foreground emission, and projection of three-dimensional fluctuations into angular anisotropy limit the recoverable small-scale information. Resolving a given angle alone does not guarantee a precise matter-spectrum measurement there.

Expand the fractional CMB temperature fluctuation as $\Theta (\mathbf{n})={\sum }_{\ell m}{a}_{\ell m}{Y}_{\ell m}(\mathbf{n})$. Under statistical isotropy, its CMB angular power spectrum is defined byThe normalization is for dimensionless temperature, rather than temperature in kelvin. The spherical harmonics separate angular scales, with large $\ell$ corresponding to small angles.

For approximately constant $R,\varphi ,\psi$, a useful solution of the photon-baryon acoustic oscillator isHere $r_s$ is the comoving sound horizon. With slowly varying $R$, the oscillatory phase is still $k{r}_s$ and its homogeneous amplitude varies approximately as $(1+R{)}^{-1/4}$ under the short-period approximation. Evolving potentials add gravitational driving rather than erase this phase coherence.

The regular growing adiabatic initial conditions produce an initial density displacement and negligible initial acoustic velocity outside the horizon. They therefore select the cosine phase, with $B$ approximately zero, for all modes. At last scattering, extrema of compression and rarefaction occur at $k{r}_s({\eta }_{\ast })\simeq n\pi$. The observed monopole-plus-potential Sachs-Wolfe combination is ${\Theta }_0+\psi ={\delta }_{\gamma }/4+\psi$, so baryon loading shifts its equilibrium and makes alternating peaks unequal. The velocity produces a phase-shifted Doppler CMB anisotropy.

Projection from the Cosmic microwave background last-scattering surface places a mode near $\ell \simeq k{\chi }_{\ast }$, where ${\chi }_{\ast }={\eta }_0-{\eta }_{\ast }$ is its comoving radial distance in a flat universe. The resulting adiabatic acoustic-peak spacing isCoherent acoustic phases turn successive compressions and rarefactions into the CMB peak series. Finite-width last scattering, Doppler projection, evolving potentials and diffusion shift or broaden actual peaks, so this is the requested approximate spacing, not an exact prediction for every peak.

Put $\Gamma (\eta )=a{\bar{n}}_e{\sigma }_T\ge 0$, the Thomson scattering rate per unit conformal time. The cosmological optical depth is the integrated scattering rate from emission time to observation, so a Poisson scattering process has survival probabilitySince $\dot{\tau }=-\Gamma$, the cosmological visibility function isA photon scatters in an interval $d\eta$ with rate factor $\Gamma d\eta$ and then reaches us without another scattering with probability $E$, so $gd\eta$ is its probability of last scattering in that interval. For an optically thick initial epoch,If the initial optical depth is finite, the missing weight represents photons that have already stopped scattering before ${\eta }_i$.

Before cosmological recombination, the optical depth to the present is enormous: $E\simeq 0$, and $g$ is also small because almost every scattering is followed by another. Through photon decoupling, $E$ rises rapidly and $g$ has its dominant positive peak. After decoupling, without reionization, $E$ is close to one and reaches exactly one at ${\eta }_0$, while $g$ is small because the remaining electron density and scattering rate are small. Residual ionization can give a weak tail; there is no second reionization peak. The peak marks the Cosmic microwave background last-scattering surface.

The Cosmic neutrino background began free streaming at neutrino decoupling, long before cosmological recombination, so its directional flux can retain information about density fluctuations from epochs inaccessible to the cosmic microwave background. Earlier decoupling does not by itself guarantee a larger comoving radial distance. A sufficiently massive neutrino eventually becomes nonrelativistic, and its travelled distance iswhich may be smaller than the photon distance because $v(a)<c$. Hence the cosmic neutrino background last-scattering surface need not lie beyond the Cosmic microwave background last-scattering surface; it generally does for neutrinos that remain sufficiently relativistic for sufficiently long.