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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.