Cosmic microwave background anisotropy (CMB anisotropy)

Cosmic microwave background anisotropy is the direction dependence of the relic radiation's temperature and polarization. A temperature map can be written $T(\hat{\mathbf{n}})=\bar{T}[1+\Theta (\hat{\mathbf{n}})]$, separating a mean from its fractional angular variation. Comparing directions with a Differential Microwave Radiometer measures these variations without requiring the instrument to determine the absolute mean temperature.^[ref]

Intrinsic plasma fluctuations, motions and gravitational frequency shifts all contribute. The Cosmic microwave background power spectrum organizes the signal by angular scale; the Cosmic microwave background acoustic peaks record the evolution of the photon-baryon fluid. Frequency comparisons and instrumental checks help distinguish the relic signal from galactic foregrounds and noise before the map is used to infer cosmological parameters.

The dipole is the $\ell =1$ angular component of Cosmic microwave background anisotropy. Its leading observed contribution is a Doppler dipole from observer motion, producing one hotter and one cooler hemisphere. Interpreting the whole measured dipole as motion is a physical model; an intrinsic dipole is a separate contribution.

The Differential Microwave Radiometer measured differences in microwave power between pairs of sky directions to map Cosmic microwave background anisotropy. Comparing two directions suppresses a common absolute brightness offset; repeated comparisons during scanning are then combined into a sky map.

The COBE instrument used paired horn antennas at three frequencies, with independent channels that tested detector noise and frequency-dependent galactic foreground contamination. A differencing instrument does not by itself determine the absolute monopole temperature. Its measurement therefore complements the spectrum measured by the Far Infrared Absolute Spectrophotometer.^[ref]

In 1992 the COBE DMR team, including George F. Smoot and Charles L. Bennett, used radiometer antennas to compare antenna temperatures between sky directions $60^{\circ }$ apart at $31.5$, $53$ and $90\mathrm{GHz}$, with two nearly independent channels per frequency. Repeated sky coverage supported least-squares solution map reconstruction.^[ref]

The analysis fitted magnetic response and removed the assumed kinematic angular dipole and its second-order Doppler effect contribution. Channel sums retained sky structure; channel differences estimated noise. Cross-channel correlations, different frequency maps and Milky Way latitude cuts tested the interpretation. Residual large-angle structure supported Cosmic microwave background anisotropy; Galactic contamination particularly limited the cosmic microwave background quadrupole. Brightness fluctuations alone were not sufficient evidence of primordial structure.^[ref]

The two differences serve different purposes. Comparing two sky directions measures angular variation. Comparing two independently calibrated receiver maps observing the same sky tests noise and consistency. A common sky signal cancels in the latter comparison if their antenna beam patterns and response scales match; a systematic error shared by both channels need not cancel. Separating observer motion, local emission and intrinsic Cosmic microwave background anisotropy requires these additional tests.

An ideal differential measurement constrains two sky temperatures through $d_{ij}=T_i-T_j$. Represent every observed pair by a row of a matrix $A$, with $+1$ in column $i$, $-1$ in column $j$ and zeros elsewhere. Then $d=AT$ and $A1=0$: adding any constant to the entire map changes no measurement. This null space is the angular angular monopole ambiguity, explaining why a directional comparison alone cannot measure the absolute blackbody temperature established by FIRAS.

For an original three-pixel example choose $T=(30,-10,-20)\mu \mathrm{K}$ relative to an unspecified mean. Measurements $d_{12}=40\mu \mathrm{K}$ and $d_{23}=10\mu \mathrm{K}$ determine the third difference $d_{31}=-50\mu \mathrm{K}$. The zero-mean condition recovers the chosen map, but adding $100\mu \mathrm{K}$ to every pixel gives identical differences. The closure sum $d_{12}+d_{23}+d_{31}=0$ checks consistency without establishing the mean.

More generally, the graph joining measured pixel pairs must be a connected graph. If $AT=0$, each edge forces equal endpoint temperatures; a path propagates that equality throughout each connected component. Thus a connected observation graph leaves exactly one undetermined constant, whereas disconnected coverage leaves one per component. Fixing the mean makes an ideal connected reconstruction unique; real noise calls for least-squares solution fitting and uncertainty analysis.

The figure chooses a cosmic microwave background quadrupole sky $T(\theta )=Q{P}_2(\cos \theta )$, with $Q=30\mu \mathrm{K}$ and Legendre polynomial $P_2(u)=(3u^2-1)/2$. On a great-circle scan, with $\varphi$ measured from the chosen symmetry axis and ideal beams separated by $\alpha =60^{\circ }$,Expanding ${\cos }^2\varphi =(1+\cos 2\varphi )/2$ proves the formula; its peak magnitude is $45\sqrt{3}/2\simeq 38.9711\mu \mathrm{K}$. A constant sky gives zero. This deliberately restricted scan also misses some angular patterns, so rejecting the angular monopole is not a proof of complete sky reconstruction. Broad, overlapping coverage supplies the missing constraints; finite antenna beam patterns additionally smooth small angular features.

The cosmic microwave background quadrupole is the degree-two spherical harmonic part of the observed cosmic microwave background sky. It has five real coefficients. The axisymmetric example $Q{P}_2(\cos \theta )$ selects only one orientation and shape from this space; $Q$ is its chosen amplitude, not a normalization for every possible cosmic microwave background quadrupole.

A finite visibility width averages photon sources over emission time and radial position. For a Gaussian visibility profile, a plane-wave radial phase has the displayed damping factor; its power has the square of this amplitude factor. An acoustic source also varies with frequency $k{c}_s$, so its rapid oscillations are averaged. This averaging is distinct from Silk damping, which smooths the perturbations by photon diffusion before decoupling. Angular projection and source evolution prevent a universal isotropic damping coefficient based only on radial width.

For a statistically isotropic Cosmic microwave background anisotropy, rotational invariance organizes the angular three-point function using a Wigner 3j symbol:In the usual parity-even temperature convention, one instead factors out the Gaunt integralThe coefficient $b$ is the reduced CMB bispectrum: it removes the angular geometric factor and retains primordial and transfer-function information. A Sachs-Wolfe projection of a constant bispectrum is an analytic large-angle example.

Parity matters here. Since $Y_{\ell m}(-\mathbf{n})=(-1{)}^{\ell }{Y}_{\ell m}(\mathbf{n})$, the Gaunt integral vanishes when ${\ell }_1+{\ell }_2+{\ell }_3$ is odd. Statistical isotropy alone does not impose parity invariance. A parity-odd angular three-point function therefore needs its own geometric convention and cannot generally be represented by this parity-even Gaunt factor times a finite $b$.

For an isotropic, full-sky temperature map with a fixed bispectrum template $b=f_{\mathrm{NL}}{b}^1$, the inverse-variance-weighted cubic statistic has normalization $F=\frac{1}{6}\sum (b^1{)}^2{\mathcal{G}}^2/(C_1{C}_2{C}_3)$ over ordered triples. It is unbiased under the linear template relation and has Gaussian cosmic variance $1/F$. The factor $1/6$ counts the six cross-triple Wick contractions. A removed monopole cancels the nine internal-pairing terms; incomplete sky coverage or anisotropic noise generally require a linear correction.

In a full-sky cubic bispectrum estimator, pairwise Gaussian covariance within one triple gives a sum of Gaunt integrals with opposite $m$ indices. The spherical harmonic addition theorem turns their pair into the constant $(2\ell +1)/(4\pi )$; its integral against $Y_{LM}$ vanishes for $L>0$. A zero temperature monopole removes $L=0$. Thus only the six pairings connecting the two triples survive in the Gaussian variance; isotropic weights are essential for the cancellation.

Linear cosmological transfer maps the primordial bispectrum to the reduced CMB bispectrum. A Fourier representation of the momentum delta function and three Rayleigh plane-wave expansions separate the radial transfer integrals from a Gaunt integral. The six angular factors and three Fourier measures give $(4\pi {)}^6/(2\pi {)}^9=(2/\pi {)}^3$. The observer-position phase cancels by momentum conservation; the remaining radial position is an auxiliary integration variable.

For a constant primordial bispectrum and large-angle transfer function ${\Delta }_{\ell }(k)=j_{\ell }(kR)/5$, the spherical Bessel product integral reduces the radial projection to ${\int }_0^1{r}^{L+2}dr+{\int }_1^{\infty }{r}^{-L-1}dr$. For $L>0$ this equals $1/(L+3)+1/L$, giving the displayed reduced CMB bispectrum. All powers of the distance $R$ cancel. Under common large-multipole scaling it behaves as ${\ell }^{-4}$ at fixed shape; this is angular scale invariance, not a constant angular bispectrum. The all-monopole case $L=0$ has a logarithmically divergent radial tail and is excluded.

The CMB lensing potential is the line-of-sight projection of gravitational potential that remaps observed CMB directions by the deflection angle $\nabla \psi$.

The CMB angular power spectrum decomposes the variance of temperature or polarization anisotropy by angular multipole. Its large-angle plateau and acoustic peaks encode primordial perturbations and photon-baryon evolution.

For a scale-invariant potential spectrum with ${\mathcal{A}}_{\Phi }=k^3{P}_{\Phi }/(2{\pi }^2)$ constant, the matter-era Sachs-Wolfe effect gives $C_{\ell }=(4\pi {\mathcal{A}}_{\Phi }/9)\int j_{\ell }(x{)}^{2}dx/x$. The logarithmic spherical Bessel square integral then gives $\ell (\ell +1)C_{\ell }=2\pi {\mathcal{A}}_{\Phi }/9$. This plateau applies to large-angle temperature anisotropies, neglecting acoustic evolution and late evolving potentials; it is not a constant $C_{\ell }$ spectrum.

Cosmic variance is the sampling uncertainty caused by observing only one realization of a cosmological random field. For a Gaussian isotropic full-sky temperature field and an ideal noiseless estimate, $\mathrm{Var}({\hat{C}}_{\ell })=2C_{\ell }^2/(2\ell +1)$ because there are only $2\ell +1$ independent real angular degrees of freedom.

Thomson scattering of a local radiation quadrupole generates linear cosmic microwave background polarization. Its acoustic phase follows the photon velocity and is shifted by one quarter-period relative to the monopole temperature oscillation.

Instrumental conversion of a larger temperature-intensity signal into apparent sky polarization, for example through unequal channel gains or mismatched beam responses. Relative calibration, beam characterization, scan-angle coverage and map-level null tests are needed to control it. Multifrequency foreground removal alone does not establish that instrumental leakage is negligible.

The parity-odd part of linear sky polarization is represented by the pseudoscalar potential $P_B$. At linear order an axisymmetric scalar cosmological perturbation gives no physical B mode. Tensor or vector sources and lensing conversion of E-mode polarization are outside that scalar-source statement.

The parity-even part of linear sky polarization can be represented by a scalar potential $P_E$ through $Q\pm iU={ð}_{\pm }^2(P_E\pm i{P}_B)$. It is naturally separated from B-mode polarization by the parity of its spherical multipoles. Linear scalar cosmological perturbations generate E modes; the potential has an unobservable low-multipole kernel.

For a scalar photon quadrupole emitted at distance ${\chi }_{\ast }$, the meridian-basis source is $Q\pm iU\propto {\Theta }_2{e}^{-ik{\chi }_{\ast }\mu }(1-{\mu }^2)$. Integrating its scalar-potential equation and removing the low-multipole kernel of polarization potentials yields $P_E\propto {\sum }_{\ell \ge 2}(-i{)}^{\ell }(2\ell +1){\Theta }_2{j}_{\ell }(k{\chi }_{\ast })P_{\ell }(\mu )/(k{\chi }_{\ast }{)}^2$, with no scalar $P_B$. The apparent $k^{-2}$ divergence lies in discarded low multipoles; $j_{\ell }(x)=O(x^{\ell })$ makes the physical small-$x$ limit regular.

Twice-applied spin-raising or lowering annihilates the ordinary spherical-harmonic multipoles $\ell =0,1$. Polarization potentials therefore have a four-dimensional monopole/dipole ambiguity on the sphere. For an axisymmetric mode it is $A+B\cos \theta$. Setting these components to zero defines the observable $\ell \ge 2$ potentials without changing $Q$ or $U$.

For Fourier wavevector $\mathbf{k}$ and photon direction $\hat{\mathbf{p}}$, the temperature perturbation is expanded as

For a fixed angular order $m$, write $\Theta ={\sum }_{\ell }{C}_{\ell }{q}_{\ell }{Y}_{\ell m}$. Streaming by $ik\mu$ contributes $ik(C_{\ell +1}/C_{\ell })a_{\ell +1,m}{q}_{\ell +1}+ik(C_{\ell -1}/C_{\ell })a_{\ell ,m}{q}_{\ell -1}$. Thus rescaling the expansion coefficients also rescales the hierarchy couplings and source amplitudes. Replacing $C_{\ell }=(-i{)}^{\ell }\sqrt{(2\ell +1)/(8\pi )}$ by ${\tilde{C}}_{\ell }=(-i{)}^{\ell }\sqrt{2\pi /(2\ell +1)}$ requires ${\tilde{q}}_{\ell }=(2\ell +1)q_{\ell }/(4\pi )$; the temperature field is unchanged. This prevents mixing incompatible photon temperature multipole conventions.

Let $⟨F⟩=(4\pi {)}^{-1}\int Fd\Omega$ and $f=\bar{f}(ϵ)-ϵ{\bar{f}}^{\prime }(ϵ)\Theta (\mathbf{e})$ with comoving energy $ϵ=aE$. If ${ϵ}^4\bar{f}$ vanishes at both integration endpoints, integration by parts gives $-\int {ϵ}^4{\bar{f}}^{\prime }dϵ=4\int {ϵ}^3\bar{f}dϵ$. The kinetic stress-energy tensor then gives ${\delta }_{\gamma }=4⟨\Theta ⟩$, $v_{\gamma }^i=3⟨\Theta e^i⟩$, $\delta P=\bar{\rho }{\delta }_{\gamma }/3$, and ${\pi }^{ij}=4\bar{\rho }⟨\Theta (e^i{e}^j-{\delta }^{ij}/3)⟩$. If $\Pi =-\pi$ is used, the last expression changes sign. Frequency independence of $\Theta$ is essential to this common temperature description.

A direction-dependent fractional photon temperature change. For a thermal spectrum, $f(ϵ)=\bar{f}(ϵ)-\Theta ϵ{\bar{f}}^{\prime }(ϵ)$ is its linear expansion at fixed comoving energy. For a nonthermal spectrum the same ansatz is a brightness dilation, rather than necessarily a thermodynamic temperature. In unweighted Legendre polynomial multipoles, $\Theta ={\sum }_{\ell }(-i{)}^{\ell }{\Theta }_{\ell }{P}_{\ell }$, the photon density, velocity and anisotropic stress are ${\delta }_{\gamma }=4{\Theta }_0$, $v_{\gamma }=-{\Theta }_1$ and ${\Pi }_{\gamma }=-3{\Theta }_2/5$.

The photon monopole is the direction-averaged fractional temperature perturbation. In Newtonian gauge its gravitationally observable effective temperature is the Sachs-Wolfe combination ${\Theta }_0+\Psi$.

In Newtonian gauge, ${\Theta }_0+\Psi$ combines the intrinsic photon temperature perturbation with the gravitational redshift at emission. With constant potentials and negligible baryon loading it oscillates as $\cos (k{r}_s)$ for adiabatic initial conditions.

For a constant growing adiabatic mode with curvature convention $\mathcal{R}=-\varphi -\mathcal{H}({\varphi }^{\prime }+\mathcal{H}\varphi )/[4\pi G{a}^2(\bar{\rho }+\bar{P})]$, ${\varphi }_{\mathrm{rad}}=-2\mathcal{R}/3$ and ${\varphi }_{\mathrm{mat}}=-3\mathcal{R}/5$. The superhorizon photon continuity equation conserves ${\Theta }_0-\varphi$. With radiation initial value ${\Theta }_0=\mathcal{R}/3$, it gives ${\Theta }_{0,\mathrm{mat}}=2\mathcal{R}/5$ and $({\Theta }_0+\psi {)}_{\mathrm{mat}}=-\mathcal{R}/5$ when $\psi =\varphi$.

The Sachs-Wolfe effect is the large-angle CMB temperature anisotropy produced by intrinsic photon-temperature perturbations and gravitational redshift at last scattering. For an adiabatic mode during matter domination, ${\Theta }_0+\Psi =\Psi /3$ on superhorizon scales.

The Doppler CMB anisotropy is the line-of-sight velocity contribution from the last-scattering plasma. With the convention used here it contributes $-\hat{\mathbf{n}}\cdot {\mathbf{v}}_e$ to the observed fractional temperature perturbation.

The integrated Sachs-Wolfe effect is the CMB temperature shift produced when the gravitational potentials traversed by a photon evolve with time. Its line-of-sight source is proportional to ${\Phi }^{\prime }+{\Psi }^{\prime }$.

Cross-correlating a cosmic microwave background temperature map with a low-redshift matter tracer can isolate a late gravitational contribution from the much larger primordial anisotropy. Decaying potential wells generally produce a positive correlation of matter overdensity with temperature. Tracer bias, overlapping redshift distributions, foreground emission, masks and chance correlations limit the inference. Potential evolution also depends on curvature and gravity assumptions, so this is not a dark-energy measurement independent of a cosmological model.

The Rees-Sciama effect is the microwave temperature shift produced by time-dependent gravitational potentials in nonlinear structure. A photon gains energy entering a potential well and loses it leaving; the shifts do not cancel when the well changes during transit. To leading order in its effect on the photon spectrum, this is a frequency-independent thermodynamic temperature shift, unlike the thermal Sunyaev-Zeldovich effect. A nonlinear potential can evolve even during matter domination, when linear growing-mode potentials are constant.

The photon dipole represents the bulk velocity of the radiation. In tight coupling to baryons, $v_b=-3{\Theta }_1$ under the convention used here.

The photon quadrupole is the leading anisotropy that sources linear polarization through Thomson scattering.

Before recombination, frequent Thomson scattering couples photons, electrons, and baryons into an acoustic fluid.

For the velocity convention $v_{\gamma }=-{\Theta }_1$ and $\dot{\tau }<0$, momentum-conserving Thomson scattering gives ${\dot{v}}_b+\mathcal{H}{v}_b+k\psi =(\dot{\tau }/R)(v_b-v_{\gamma })$. Here $R$ is the baryon loading parameter. Its collision force is opposite to the photon force when both are weighted by their enthalpies.

The inertia ratio of the photon-baryon fluid is $R=3{\bar{\rho }}_b/(4{\bar{\rho }}_{\gamma })$. Photon momentum is weighted by the radiation enthalpy $4{\bar{\rho }}_{\gamma }/3$, so this ratio controls momentum exchange, the common sound speed and the offset of the acoustic oscillator. Background conservation gives $R\propto a$ and $\dot{R}=\mathcal{H}R$.

Thomson scattering is the low-energy elastic scattering of electromagnetic radiation by a free charged particle. Before cosmological recombination, repeated photon-electron scattering couples photons to the baryon velocity and damps higher photon multipoles.

For a dilute ionized plasma with nonrelativistic electrons and photon energies well below $m_e{c}^2$, independent Thomson scattering events give photon survival probability $\exp (-n_e{\sigma }_{T}s)$ over proper distance $s$. Its mean gives the displayed mean free path. The scattering rate is ${\Gamma }_{\gamma }=n_e{\sigma }_{T}c$. Photons are tightly coupled when this rate greatly exceeds the Hubble parameter; photon decoupling becomes possible as the free-electron fraction falls. Frequent elastic scattering does not by itself imply chemical equilibration of the photon number.

Coherent Thomson scattering by cold electrons changes a photon's direction without changing its frequency in the electron rest frame. A monochromatic incident radiation field therefore remains monochromatic after arbitrarily many such scatterings. Large scattering optical depth can isotropize radiation but does not imply a Planck photon distribution. True absorption and emission, or suitable additional inelastic processes and photon production, are needed to establish a blackbody spectrum independently of the initial radiation field.

A free nonrelativistic charge driven by a weak plane electromagnetic wave has peak acceleration $|q|E_0/m$. Its time-averaged Larmor formula power is $q^4{E}_0^2/(12\pi {ϵ}_0{m}^2{c}^3)$. Divide by incident light intensity ${ϵ}_{0}c{E}_0^2/2$ to obtain the displayed cross section, independent of frequency. For an electron, it is $8\pi r_e^2/3$. The approximation assumes negligible Compton recoil kinematics and small driven velocity compared with the speed of light.

The tight-coupling approximation expands the photon-baryon Boltzmann equations in powers of $k/\Gamma$ and $\mathcal{H}/\Gamma$, where $\Gamma$ is the Thomson scattering rate.

Let $\mathbf{v}=i\mathbf{k}\theta$ for a nonzero Fourier mode, so its velocity divergence is $-k^2\theta$. In units $c=1$, the conformal Thomson scattering rate is $q=a{n}_e{\sigma }_T$, with proper electron number density $n_e$ and scale factor $a$. The photon drag term is $-q({\mathbf{v}}_{\gamma }-{\mathbf{v}}_b)$. Taking a divergence and dividing the entire equation by $-k^2$ therefore givesNo additional $k^{-2}$ survives. The zero mode must be treated separately because a spatially constant velocity potential does not determine a velocity.

To display momentum conservation, define $R_{\gamma }=4{\bar{\rho }}_{\gamma }/(3{\bar{\rho }}_b)$, the inverse of the usual baryon loading parameter. Neglect baryon thermal pressure. With Newtonian potential $\Psi$ and photon shear convention ${\sigma }_{\gamma }$, the two Euler equations can be writtenWeighting the photon equation by $R_{\gamma }$ and adding the baryon equation cancels the drag. In the tight-coupling approximation, ${\theta }_{\gamma }\simeq {\theta }_b=\theta$, henceFor adiabatic acoustic fluctuations the squared sound speed is $c_s^2=R_{\gamma }/[3(1+R_{\gamma })]$, so $c_s=\sqrt{R_{\gamma }/[3(1+R_{\gamma })]}$. This tends to $1/\sqrt{3}$ in the negligible-baryon limit. Calling the squared expression the sound speed loses the square root.

For $\Theta (\mu )={\sum }_{\ell }(-i{)}^{\ell }{\Theta }_{\ell }{P}_{\ell }(\mu )$ and a Thomson collision operator omitting polarization feedback, the quadrupole relaxes with rate $(9/10){\Gamma }_T$, where ${\Gamma }_T=-{\tau }^{\prime }>0$. Balancing its leading streaming source gives ${\Theta }_2=20k{\Theta }_1/(27{\Gamma }_T)$. This differs from photon quadrupole in tight coupling with polarization; the two formulas expand different collision operators.

With $v_b=v_{\gamma }$ at leading order and $c_s^2=1/[3(1+R)]$, the photon continuity equation and total Euler equation give ${\ddot{\delta }}_{\gamma }+\dot{R}{\dot{\delta }}_{\gamma }/(1+R)+k^2{\delta }_{\gamma }/[3(1+R)]=4\ddot{\varphi }+4\dot{R}\dot{\varphi }/(1+R)-4k^2\psi /3$. The baryon loading parameter changes inertia and shifts the gravitational equilibrium. The leading equations omit diffusion damping.

A damped mode $\delta =e^{-k^2/k_D^2}(A\cos q+B\sin q)$ does not alone imply oscillations in its ensemble power spectrum. Averaging its modulus squared gives the displayed formula. Equal independent initial variances, $P_A=P_B$ and $P_{AB}=0$, erase the oscillatory variance through ${\cos }^{2}q+{\sin }^{2}q=1$. A coherent initial phase, as supplied by a regular growing adiabatic mode, selects one common oscillator phase and retains acoustic peaks. Its leading density and velocity oscillations are a quarter-period apart, and angular projection also preserves their correlated contribution.

For baryon-loading ratio $R=3{\rho }_b/(4{\rho }_{\gamma })$, the photon-baryon sound speed is $c_s^2=1/[3(1+R)]$.

The velocity slip is the difference between the bulk velocities of photons and baryons. Thomson scattering suppresses it at a rate $\Gamma (1+R^{-1})$, where $R=3{\rho }_b/(4{\rho }_{\gamma })$ and $\Gamma =-\dot{\tau }>0$. Its first correction in the tight-coupling approximation produces heat-conduction damping of acoustic waves.

For the expansion $\Theta (\mu )={\sum }_{\ell }(-i{)}^{\ell }{\Theta }_{\ell }{P}_{\ell }(\mu )$ without a $(2\ell +1)$ factor, the leading polarization relation is $E_2=-\sqrt{6}{\Theta }_2/4$. The photon Boltzmann hierarchy then gives ${\Theta }_2=-8k{\Theta }_1/(9\dot{\tau })$. This small photon quadrupole supplies the shear-viscosity part of Silk damping.

When gravity and expansion can be neglected and $R$ is constant, the photon monopole satisfiesThe first damping contribution comes from photon-baryon velocity slip; the second comes from the photon quadrupole. This is a local form of Silk damping.

For the simplified quadrupole equation ${\sigma }_{\gamma }^{\prime }+4k^2{\theta }_{\gamma }/15=-{\sigma }_{\gamma }/{\tau }_c$, neglecting its time derivative and gravitational driving gives ${\sigma }_{\gamma }=-{\tau }_c{\delta }_{\gamma }^{\prime }/5$. A radiation-dominated inertia then obeys ${\delta }_{\gamma }^{\prime \prime }+4{\tau }_c{k}^2{\delta }_{\gamma }^{\prime }/15+k^2{\delta }_{\gamma }/3=0$. On slowly varying subhorizon scales the amplitude is damped by $\exp [-(2/15)k^2\int {\tau }_{c}d\tau ]$. This coefficient belongs to that truncated collision model, rather than the polarization-inclusive Silk damping equation.

Removing the first derivative from the truncated photon diffusion equation gives the displayed transformed oscillator. Constant mean free time changes its oscillation frequency only at second order. For a slowly varying mean free time and subhorizon wavelengths, derivative corrections to the phase and prefactor are small; the leading solution is an acoustic sine/cosine multiplied by $\exp [-k^2/k_D^2]$, where $k_D^{-2}=2\int {\tau }_{c}d\tau /15$. Dropping the derivative correction requires slow variation as well as a small mean free time; it is not an exact solution for arbitrary time dependence.

The comoving sound horizon is the distance traveled by an acoustic wave,

For last scattering well inside matter domination, $c_s=c/\sqrt{3}$ gives physical sound radius $R_s=2c{a}_{\ast }^{3/2}/(\sqrt{3}{H}_0\sqrt{{\Omega }_m})$. Neglecting the early conformal time in the observer distance, an open dust cosmology has transverse comoving last-scattering distance $r_{\ast }=2c/(H_0{\Omega }_m)$. Thus ${\theta }_s=R_s/(a_{\ast }{r}_{\ast })=\sqrt{{\Omega }_m{a}_{\ast }/3}$ and $\ell \sim \pi /{\theta }_s$. Radiation, baryon sound-speed evolution and acoustic phase shifts refine this estimate.

// Target: cosmology.bigb

Cosmic microwave background diffusion damping, or Silk damping, damps the photon-baryon acoustic oscillator on small scales because Thomson scattering gives photons a finite mean free path. Repeated scatterings mix hotter and colder regions. Departures from perfect tight coupling also appear as photon-baryon velocity slip and a photon quadrupole, producing heat-conduction and shear-viscosity damping.

In natural units, the positive conformal scattering rate is ${\Gamma }_T=a{n}_e{\sigma }_T$, with proper free-electron number density $n_e$. A random-walk estimate gives squared comoving diffusion distance of order $\int d\eta /{\Gamma }_T$. More precisely, the leading polarization-inclusive tight-coupling amplitude containswhere $R=3{\rho }_b/(4{\rho }_{\gamma })$ is the baryon loading parameter. The corresponding power spectrum suppression is $e^{-2k^2/k_D^2}$. The approximation applies while the expansion in scattering time is controlled; it cannot be extended through a transparent universe by simply integrating $1/{\Gamma }_T$ indefinitely.^[ref]

Diffusion accumulates before last scattering and becomes especially important as the free-electron population falls. The finite width of the cosmological visibility function can also average oscillating line-of-sight sources; that averaging is a distinct mechanism acting in the same epoch. Both effects help shape the small-angular-scale damping tail, whose scale probes the ionization and baryon history.

Adiabatic initial conditions perturb all relevant local physical quantities along the same background time trajectory. For quantities with nonzero background time derivatives, their perturbations satisfy $\delta X/{\bar{X}}^{\prime }=\delta Y/{\bar{Y}}^{\prime }$. For separately conserved barotropic fluids, cosmological energy conservation then givesFor example, pressureless matter and radiation in cosmology obey ${\delta }_m=3{\delta }_{\gamma }/4$. Their relative particle-number ratio is unperturbed at leading order.

The density condition tests relative entropy between components. In a general nonbarotropic component it does not by itself eliminate intrinsic entropy, measured by $\delta P_i-({\bar{P}}_i^{\prime }/{\bar{\rho }}_i^{\prime })\delta {\rho }_i$. The stronger statement that every physical variable follows the same time shift includes the pressure and any independent field-velocity data. A single scalar species alone does not make the density-ratio test sufficient, since that test is then vacuous.

For vanishing total non-adiabatic pressure perturbation and negligible regular gradient terms, uniform-density curvature is conserved on superhorizon scales.^[ref] Relating this to conserved comoving curvature perturbation also requires the appropriate regular adiabatic mode; a non-attractor phase such as ultra-slow-roll inflation needs separate analysis. The ordinary growing adiabatic photon mode selects coherent acoustic phases, explaining the adiabatic acoustic-peak spacing.

For separately conserved barotropic components, an adiabatic cosmological perturbation has equal ${\delta }_N/(1+w_N)$. Pressureless matter and a $w=-1/3$ string component therefore have ${\delta }_s=2{\delta }_c/3$ on superhorizon scales. In the supplied perfect-fluid synchronous equations, neglecting spatial gradients gives $S^{\prime \prime }+2\mathcal{H}{S}^{\prime }=0$ for $S={\delta }_s-2{\delta }_c/3$. Both $S$ and its independent decaying-mode amplitude must vanish to select the growing adiabatic solution. At finite wavenumber, pressure gradients generate departures from this leading long-wavelength relation.

Cosmic microwave background acoustic peaks arise from photon-baryon oscillations caught at successive extrema at recombination. Their approximate phase is set by $k{r}_s=n\pi$ and their angular positions measure the sound horizon relative to the distance to last scattering.

For an outward sightline, Fourier velocity $v_b^i=i{k}^i{\theta }_b$, and sudden last scattering, intrinsic photon temperature projects with a Spherical Bessel function, while the Doppler term projects with its derivative. Density and velocity acoustic sources are approximately a quarter cycle out of phase. Their amplitude is suppressed by photon diffusion, giving a power damping envelope $e^{-2k^2/k_D^2}$. Projection maps the sound scale to peak spacing approximately $\pi D_{\ast }/r_s$ and the damping scale to angular multipoles of order $k_D{D}_{\ast }$.

Adiabatic initial conditions select coherent cosine phases of the photon-baryon acoustic oscillator. Its extrema at last scattering satisfy $k{r}_s\simeq n\pi$ and project to $\ell \simeq k{\chi }_{\ast }$. Thus ${\ell }_n\simeq n\pi {\chi }_{\ast }/r_s({\eta }_{\ast })$, with comoving sound horizon $r_s=\int d\eta /\sqrt{3(1+R)}$. Projection, driving and finite last-scattering width modify the exact peak locations.

In the absence of collisions, a photon phase-space density is constant along its geodesic. Linearizing this statement gives a first-order transport equation for the direction-dependent temperature perturbation.

The photon brightness perturbation is the fractional frequency-integrated directional energy perturbation. For a small blackbody temperature shift $\Theta =\delta T/T$, $f_1=-q{f}_0^{\prime }(q)\Theta$. Integration by parts gives $\Delta =4\Theta$. Its direction average is the photon density contrast. This temperature identification assumes a perturbation of blackbody form; the integrated brightness definition itself is more general.

// Target: cosmology.bigb

For collisionless photons in synchronous gauge, their perturbed momentum obeys $q^{\prime }=-q{h}_{ij}^{\prime }{n}^i{n}^j/2$. Terms in $q^{\prime }{\partial }_q{f}_1$ and direction deflection times angular derivatives of $f_1$ are second order. Integrating the remaining linear Collisionless Boltzmann equation over $q^{3}dq$ gives the displayed equation, because $\int q^4{f}_0^{\prime }dq=-4\int q^3{f}_{0}dq$.

// Target: cosmology.bigb

For synchronous gauge metric $d{s}^2=a^2[-d{\tau }^2+({\delta }_{ij}-h_{ij})d{x}^{i}d{x}^j]$, conserved-background comoving momentum obeys $q^{\prime }=q{h}_{ij}^{\prime }{n}^i{n}^j/2$. In the linear Collisionless Boltzmann equation, insert $\Delta =-4f_1/(q{f}_0^{\prime })$ to obtain the displayed source. Using ${\delta }_{ij}+h_{ij}$ instead reverses both signs; it does not change the physics. The scalar contraction is $h_{ij}^{\prime }{n}^i{n}^j=(h^{\prime }-h_s^{\prime })/3+{\mu }^2{h}_s^{\prime }$.

For a photon in $d{s}^2=a^2[-d{\tau }^2+({\delta }_{ij}+h_{ij})d{x}^{i}d{x}^j]$, the null geodesic condition cancels the homogeneous $2\mathcal{H}{p}^0$ term in the derivative of $q=a^2{p}^0$. The remaining first-order geodesic equation gives the displayed momentum change. Its direction is constant at zeroth order, so first-order redshift and brightness sources can be evaluated on the unperturbed ray; direction deflection multiplied by an already first-order anisotropy is second order.