Silk Damping

Every acoustic peak in the cosmic microwave background stands on the same knife-edge: the photon–baryon plasma that rings before recombination is tightly coupled, but never perfectly. Photons scatter off free electrons roughly once per kiloparsec, and between scatterings they drift — hot to cold — carrying energy out of the very compressions they are meant to sustain. Joseph Silk saw this in 1968, years before any CMB anisotropy had been measured: the plasma is a leaky oscillator, and the leak has a scale — set not by the tiny mean free path but by the far larger distance a photon random-walks before recombination, the diffusion or Silk length. Because a random walk of $N$ steps covers a net distance $\sim\sqrt{N}\,\lambda_{\mathrm{mfp}}$ and $N$ is large, a sub-kiloparsec mean free path grows into a comoving diffusion length of several Mpc, and fluctuations shorter than this are ironed flat. In Fourier space each mode picks up $\exp[-(k/k_D)^2]$ on its amplitude and $\exp[-2(k/k_D)^2]$ on its power, with $k_D\approx0.14\,\mathrm{Mpc}^{-1}$ at last scattering. Projected onto the sky this cuts the CMB temperature spectrum off exponentially beyond $\ell\sim1000$ — the damping tail, the smooth slope under the third, fourth, and higher acoustic peaks. Silk damping is not a nuisance to be subtracted but a precision instrument: the tail's location depends on the baryon density (the mean free path), the expansion rate near recombination (how long diffusion runs), and the physics of hydrogen recombination. Measuring it with Planck, ACT, and SPT turns a leaky oscillator into one of cosmology's sharpest tools.

Key takeaways

Silk damping is the exponential suppression of small-scale CMB anisotropies caused by photons diffusing out of overdensities during the finite-duration epoch of recombination.
Photons random-walk through the coupled photon–baryon plasma, so their net diffusion length grows as $\lambda_D\sim\sqrt{N}\,\lambda_{\mathrm{mfp}}$ — a sub-kiloparsec step becomes a comoving scale of several Mpc.
Each mode is multiplied by $e^{-(k/k_D)^2}$ on the amplitude and $e^{-2(k/k_D)^2}$ on the power, with $k_D\approx0.14\,\mathrm{Mpc}^{-1}$ (comoving) at last scattering.
Projected on the sky this produces the CMB damping tail: the $TT$ spectrum falls off steeply for multipoles $\ell\gtrsim1000$ , becoming severe near $\ell_D\approx k_D D_A\approx1960$ .
Because $\lambda_D$ depends on $1/(n_e\sigma_T)$ and the duration of recombination, the tail is a clean probe of $\Omega_b h^2$ , the ionization history, and early departures from standard expansion (such as extra relativistic species).
Key idea: Photons random-walk out of small-scale overdensities during recombination; the net diffusion length $\lambda_D\propto\sqrt{N}\,\lambda_{\mathrm{mfp}}$ defines a wavenumber $k_D$ below which fluctuations survive and above which the power is cut off as $e^{-2(k/k_D)^2}$ .

Where this lives today

The damping tail is now everyday working data. Arcminute-resolution ground experiments — the South Pole Telescope and the Atacama Cosmology Telescope — measure temperature and polarization out to $\ell\approx3000$ – $4000$ , deep into the region where Silk damping has cut the power a hundredfold, while Planck maps the full sky to $\ell\approx2500$ . Fitting the fall-off does several jobs at once. It measures $\Omega_b h^2$ through the mean free path, independently of the peak heights, and its agreement with the deuterium abundance from Big Bang nucleosynthesis is one of cosmology's tightest consistency checks. It constrains the effective number of relativistic species $N_{\mathrm{eff}}$ : extra radiation speeds the pre-recombination expansion, shortens the diffusion time, and shifts the nearly geometry-free ratio $\ell_D/\ell_A$ . It tests recombination itself, since anything that changes how sharply $n_e$ drops — energy injection from decaying particles, a varying fine-structure constant, primordial magnetic fields — reshapes the tail, and proposed cures for the Hubble tension such as early dark energy are tested largely by how they would distort the tail relative to the peaks.

The physical picture

Picture a compression in the plasma — a hot spot — smaller than the distance photons can wander before recombination. Photons do not stay put; they scatter off electrons, drift, scatter again, and inch outward, carrying their energy to cooler surroundings and dragging the baryons through Coulomb coupling to the electron gas. By the time the electrons recombine and the photons free-stream away, the hot spot has been ironed flat. The mechanism is diffusion; the crucial multiplier is the random walk.

Figure

A CMB photon random-walks out of a small hot spot before recombination: N tiny steps of length λ_mfp ≈ 1.5 kpc net a displacement λ_D ≈ √N λ_mfp ≈ 7 Mpc comoving, so the temperature contrast of any hot spot smaller than λ_D is ironed flat

A single mean free path is minuscule — of order $1.5\,\mathrm{kpc}$ physical near recombination — but the photon scatters an enormous number of times, and $N$ uncorrelated steps net a displacement only $\sqrt{N}$ times one step. That $\sqrt{N}$ turns a microscopic mean free path into a macroscopic diffusion length: to order of magnitude the Silk length is the geometric mean of the horizon and the mean free path, landing at a few Mpc comoving, or $k_D\approx0.14\,\mathrm{Mpc}^{-1}$ . There are really two smoothings, and it is a common slip to remember only the first. One is the diffusion just described — shear viscosity plus heat conduction in the imperfectly-coupled fluid — giving $e^{-(k/k_D)^2}$ . The other is geometric: last scattering is a shell of finite comoving thickness $\sigma_{\mathrm{LS}}\sim19\,\mathrm{Mpc}$ , and averaging emission along the line of sight through it blurs small-scale features on a comparable scale. Both cut the same high- $\ell$ modes. And because a shorter mean free path (higher $\Omega_b h^2$ ) means less diffusion, raising the baryon density pushes $k_D$ higher and the tail to higher $\ell$ — the basis of the baryometer below.

Figure

Silk damping of the CMB TT power spectrum: the undamped acoustic peaks in grey are multiplied by the envelope exp(−2(ℓ/ℓ_D)²) with ℓ_D ≈ 1960, so the blue damped spectrum falls off steeply beyond ℓ ≈ 1000 while the low-ℓ peaks survive intact

The core equations

The whole phenomenon starts from one estimate: how far does a photon get by random-walking? A step is the Thomson mean free path, and $N$ steps net a $\sqrt{N}$ displacement.

\lambda_D \sim \sqrt{N}\,\lambda_{\mathrm{mfp}}, \qquad \lambda_{\mathrm{mfp}} = \frac{1}{n_e \sigma_T} \qquad (1)

where: $\lambda_D$ — photon diffusion (Silk) length; $N$ — number of scatterings before recombination; $\lambda_{\mathrm{mfp}}$ — photon mean free path; $n_e$ — free-electron number density; $\sigma_T$ — Thomson cross-section, $6.65\times10^{-29}\,\mathrm{m}^2$ . Reading it: because $N\gg1$ , the diffusion length is vastly larger than a single step, which is why the damping scale is macroscopic even though scattering is microscopic. A careful treatment integrates the per-mode dissipation over the plasma's history rather than using a single $N$ : the tight-coupling expansion to first order in $k\lambda_{\mathrm{mfp}}$ gives the comoving damping wavenumber as an integral over conformal time.

k_D^{-2}(\eta) = \int_0^{\eta} \frac{\mathrm{d}\eta'}{6(1+R)\,n_e\sigma_T a}\left[\frac{16}{15} + \frac{R^2}{1+R}\right] \qquad (2)

where: $k_D$ — comoving damping wavenumber; $\eta$ — conformal time; $a$ — scale factor; $R\equiv3\rho_b/4\rho_\gamma$ — baryon-to-photon momentum ratio. Reading it: the bracket collects the two dissipative channels — shear viscosity ( $16/15$ , enhanced by photon polarization) and heat conduction ( $R^2/(1+R)$ ) — and the $1/(n_e\sigma_T a)$ weight means the integral is dominated by the late, low-density epoch just before recombination. Once $k_D$ is known, the effect on any mode is a Gaussian cutoff. The amplitude carries one power of the damping factor, and the power spectrum — the amplitude squared — carries two.

\mathcal{D}(k) = \exp\!\left[-\left(\frac{k}{k_D}\right)^2\right], \qquad C_\ell \propto |\mathcal{D}(k)|^2 = e^{-2(k/k_D)^2} \qquad (3)

where: $\mathcal{D}(k)$ — damping factor on the perturbation amplitude; $C_\ell$ — angular power-spectrum coefficient. Reading it: this multiplies the acoustic spectrum $\cos^2(kr_s)$ , so peaks survive at low $k$ and are erased at high $k$ — the smooth envelope of the tail. The baryon loading $R$ appears in nearly every plasma quantity; here is a numerical handle near recombination.

R \equiv \frac{3\rho_b}{4\rho_\gamma} = \frac{3\Omega_b}{4\Omega_\gamma}\,a \approx 0.6\,\left(\frac{\Omega_b h^2}{0.02}\right)\left(\frac{a}{10^{-3}}\right) \qquad (4)

where: $\rho_b$ — baryon energy density; $\rho_\gamma$ — photon energy density; $\Omega_b h^2$ — physical baryon density parameter. Reading it: at last scattering the plasma is mildly baryon-loaded, $R\approx0.6$ , so viscosity and conduction terms in (2) contribute at comparable weight. Not all of the observed smoothing is diffusion: because we see photons emitted across a shell of finite thickness, a second, geometric Gaussian cutoff multiplies in.

\delta T_{\mathrm{obs}}(k) = \delta T_{\mathrm{intrinsic}}(k)\;e^{-(k/k_D)^2}\;e^{-(k\sigma_{\mathrm{LS}})^2/2} \qquad (5)

where: $\sigma_{\mathrm{LS}}$ — comoving width of the last-scattering surface ( $\sim19\,\mathrm{Mpc}$ ); $\delta T$ — temperature-fluctuation amplitude of a Fourier mode. Reading it: the two exponentials have comparable widths, so the line-of-sight blurring is not a small correction — it contributes on the same footing as diffusion. Finally, translating from comoving wavenumber to the sky requires projecting through the comoving distance to last scattering.

\ell_D \approx k_D\, D_A \approx 0.14 \times 14{,}000 \approx 1960 \qquad (6)

where: $\ell_D$ — multipole at which damping becomes severe; $D_A$ — comoving distance to last scattering, $\approx 14{,}000\,\mathrm{Mpc}$ . Reading it: damping is already noticeable by $\ell\sim1000$ and dominates past $\ell_D\approx1960$ , which is why resolving the higher peaks (ACT, SPT) demands arcminute resolution. The cosmological sensitivity is captured by $\lambda_D\propto\Omega_b^{-1/2}\,\Omega_m^{-1/4}$ near recombination: less baryon density means more diffusion, faster expansion means less time to diffuse, and because $\lambda_D$ and the sound horizon $r_s$ scale differently the ratio $\ell_D/\ell_A$ is a clean, largely geometry-independent probe.

Derivation ADiffusion length from a photon random walk

The goal is to get the Silk scale from almost nothing: treat a CMB photon as a drunkard taking steps of one Thomson mean free path, count the steps it manages before recombination, and use the random-walk law to turn that count into a net displacement — seeing explicitly why the answer is the geometric mean of the horizon and the mean free path, and why it lands at several comoving Mpc rather than at the tiny step itself.

Step 1. In a time $t$ a photon travels a path length $ct$ , broken into steps of length $\lambda_{\mathrm{mfp}}=1/(n_e\sigma_T)$ , so the number of steps is $N = ct/\lambda_{\mathrm{mfp}} = ct\,n_e\sigma_T$ . Why: the photon always moves at $c$ between scatters, so the total path is just $ct$ , and dividing by one step counts the scatters.

Step 2. The mean-square net displacement of an $N$ -step uncorrelated walk is $N$ times the square of one step, so $\lambda_D^2 = N\,\lambda_{\mathrm{mfp}}^2 = (ct\,n_e\sigma_T)\times(n_e\sigma_T)^{-2}$ . Why: successive scattering directions are independent, so displacements add in quadrature, not linearly — the entire content of the $\sqrt{N}$ enhancement.

Step 3. The factors of $n_e\sigma_T$ collapse to give $\lambda_D^2 = ct/(n_e\sigma_T) = ct\,\lambda_{\mathrm{mfp}}$ , so $\lambda_D=\sqrt{ct\,\lambda_{\mathrm{mfp}}}$ is the geometric mean of the horizon $ct$ and the mean free path. Why: as a geometric mean the hierarchy is obvious — $\lambda_D$ sits logarithmically halfway between the tiny step and the huge horizon, macroscopic but sub-horizon.

Step 4. Numerically, with $\lambda_{\mathrm{mfp}}\approx1.5\,\mathrm{kpc}$ physical and a horizon of order $100\,\mathrm{kpc}$ physical near $z\approx1090$ , $\lambda_D$ is of order $10\,\mathrm{kpc}$ physical, and the $1/6$ and $(1+R)$ factors of the full integral bring the comoving value to $\lambda_D=1/k_D\approx7\,\mathrm{Mpc}$ , i.e. $k_D\approx0.14\,\mathrm{Mpc}^{-1}$ and $\ell_D\approx1960$ . Why: the crude estimate lands within a factor of a few of the rigorous result, confirming the physics really is a random walk. *Verified: reproduces the standard comoving damping scale $k_D\approx0.14\,\mathrm{Mpc}^{-1}$ ; order-of-magnitude consistent with the integrated tight-coupling result of equation (2).*

Go deeper · Derivation B · 1 minDamping factor from the tight-coupling expansion

Derivation BDamping factor from the tight-coupling expansion

The goal is to promote the hand-waving random walk into the exponential envelope $e^{-(k/k_D)^2}$ by treating each Fourier mode as a damped acoustic oscillator solved in the WKB approximation: add a small dissipative term to the photon–baryon oscillator, let the frequency go slightly complex, and show that integrating the imaginary part over conformal time gives exactly the Gaussian suppression, with the definition of $k_D$ absorbing the accumulated dissipation.

Step 1. In tight coupling the temperature monopole obeys a damped oscillator, $\ddot{\Theta}_0 + \Gamma_D\,\dot{\Theta}_0 + k^2 c_s^2\,\Theta_0 = 0$ , with $c_s$ the plasma sound speed and $\Gamma_D = k^2 D$ the dissipation rate from viscosity and conduction. Why: expanding the Boltzmann hierarchy to first order in $k\lambda_{\mathrm{mfp}}$ leaves precisely a friction term $\propto\dot\Theta_0$ — a nearly-ideal oscillator with a small drag.

Step 2. For slowly-varying coefficients try the WKB ansatz $\Theta_0\propto\exp[i\!\int\omega\,\mathrm{d}\eta]$ with $\omega = k c_s + i\,\tfrac{1}{2}\Gamma_D$ ; the real part is the acoustic frequency, the small imaginary part the drag. Why: when the damping per oscillation is weak, the frequency acquires a small imaginary correction and the two parts decouple into "oscillate" and "decay".

Step 3. The imaginary part accumulates as $\mathrm{Im}\!\int\omega\,\mathrm{d}\eta = \tfrac{1}{2}\int\Gamma_D\,\mathrm{d}\eta = \tfrac{1}{2}k^2\!\int D\,\mathrm{d}\eta \equiv (k/k_D)^2$ , precisely the definition of $k_D$ in equation (2) once $D$ is inserted. Why: this is where the integral-over-history enters — the damping is the summed dissipation along the whole trajectory of the mode, not an instantaneous rate.

Step 4. Assembling real and imaginary parts gives $\Theta_0(\eta,k)\propto\cos(kr_s)\,\exp[-(k/k_D)^2]$ : an acoustic oscillation in the sound-horizon phase $kr_s$ times the Gaussian envelope that suppresses high- $k$ peaks. Why: the observable is exactly this product — peaks from $\cos(kr_s)$ , tail from the envelope — so both great features of the CMB spectrum fall out of one line. *Verified: matches Hu and Sugiyama (1995) and the tight-coupling treatment in Dodelson, Modern Cosmology ch. 8.*

Limiting cases

The amplitude damping factor $\mathcal{D}(k)=e^{-(k/k_D)^2}$ and the rate $\Gamma_D$ have clean behaviour in the physical extremes:

Case	Result	Meaning
Large scales $k \ll k_D$	$\mathcal{D}(k)\to1$	Modes far larger than the diffusion length are undamped; the first and second acoustic peaks keep full amplitude and the low- $\ell$ spectrum is untouched by Silk damping.
Small scales $k \gg k_D$	$\mathcal{D}(k)\to0$ (exponentially)	Modes shorter than the diffusion length are erased; the $TT$ spectrum falls off steeply for $\ell\gtrsim1000$ , producing the damping tail.
Tight coupling $k\lambda_{\mathrm{mfp}}\ll1$	$\Gamma_D\simeq\dfrac{k^2}{6(1+R)}\dfrac{1}{n_e\sigma_T}\left[\dfrac{16}{15}+\dfrac{R^2}{1+R}\right]$	The first-order damping rate, valid while scattering is frequent; viscosity ( $16/15$ ) and conduction ( $R$ ) terms add.
Baryon-dominated $R\gg1$	$\Gamma_D\to\dfrac{k^2}{6}\dfrac{R}{n_e\sigma_T}$	Heavy baryon loading up-weights heat conduction; more baryons also shorten the mean free path, doubly suppressing diffusion.
Instant recombination $\sigma_{\mathrm{LS}}\to0$	second exponential in (5) $\to1$	Only the diffusion envelope survives; the finite-thickness blurring switches off, isolating the intrinsic Silk cut.

Common trapThe damping scale is not the mean free path

The single most common error is to identify the damping scale with the photon mean free path itself. It is not: the mean free path is minuscule ( $\sim1.5\,\mathrm{kpc}$ physical near recombination), while the damping scale is the far larger random-walk length $\lambda_D\sim\sqrt{N}\,\lambda_{\mathrm{mfp}}$ , because $N\gg1$ scatterings accumulate and the $\sqrt{N}$ factor is enormous. A related slip is to forget the second smoothing: the finite thickness of the last-scattering surface, $\sigma_{\mathrm{LS}}\sim19\,\mathrm{Mpc}$ comoving, blurs anisotropies on a comparable scale, so the tail is a product of two Gaussian cutoffs, not one. Finally, do not confuse Silk (collisional) diffusion damping with the collisionless free-streaming or Landau damping of neutrinos and dark matter — a different mechanism on different components.

Worked examples

1 · The diffusion scale and where the tail sits (fully worked)

Given the comoving damping wavenumber $k_D\approx0.14\,\mathrm{Mpc}^{-1}$ at last scattering and the comoving distance to the last-scattering surface $D_A\approx14{,}000\,\mathrm{Mpc}$ , find the comoving diffusion length, the damping multipole, and by how much the power is suppressed at $\ell=1000$ , $2000$ , and $3000$ .

Step 1. The comoving diffusion length is the reciprocal of the damping wavenumber, since the envelope is $e^{-(k/k_D)^2}=e^{-(k\lambda_D)^2}$ with $\lambda_D=1/k_D$ . So $\lambda_D = 1/0.14 = \mathbf{7.1\,\mathrm{Mpc}}$ comoving — thousands of times the kiloparsec mean free path, as the $\sqrt{N}$ argument demands.

Step 2. Project to the sky: $\ell_D \approx k_D D_A = 0.14\times14{,}000 = \mathbf{1960}$ — the multipole where $(\ell/\ell_D)^2$ reaches unity, damping significant well before it and overwhelming well after.

Step 3. The power carries two factors of the envelope, $C_\ell\propto e^{-2(\ell/\ell_D)^2}$ . Evaluate at three multipoles: at $\ell=1000$ , $e^{-2(1000/1960)^2}=e^{-0.52}=\mathbf{0.59}$ ; at $\ell=2000$ , $e^{-2(2000/1960)^2}=e^{-2.08}=\mathbf{0.12}$ ; at $\ell=3000$ , $e^{-2(3000/1960)^2}=e^{-4.68}=\mathbf{0.0093}$ .

Step 4. Read the numbers: near the third peak the tail has lost about $40\%$ of its power, by $\ell\approx2000$ roughly $88\%$ , and by $\ell\approx3000$ more than $99\%$ . A comoving diffusion length of $7\,\mathrm{Mpc}$ becomes a sky cutoff at $\ell_D\approx1960$ , suppressing the power a hundredfold by $\ell\approx3000$ .

2 · Baryon loading and the mean free path at recombination

Take $\Omega_b h^2 = 0.022$ and last scattering at $z\approx1090$ (so $a\approx1/1090$ ). Find the baryon loading $R$ and estimate the physical Thomson mean free path, assuming the plasma is still nearly fully ionized just before recombination.

Step 1. Baryon loading from equation (4): $R\approx0.6\,(\Omega_b h^2/0.02)(a/10^{-3}) = 0.6\times1.1\times0.917 = \mathbf{0.61}$ — mildly baryon-loaded, so both dissipative terms in the diffusion coefficient contribute at similar weight.

Step 2. Baryon number density today is $n_{b,0}\approx1.12\times10^{-5}\,\Omega_b h^2\,\mathrm{cm^{-3}}=2.5\times10^{-7}\,\mathrm{cm^{-3}}=0.25\,\mathrm{m^{-3}}$ . Scale up as $(1+z)^3$ : $n_b(z{=}1090)=0.25\times(1091)^3\approx\mathbf{3.2\times10^{8}\,\mathrm{m^{-3}}}$ . Just before recombination the ionization fraction is near unity, so $n_e\approx n_b$ .

Step 3. The mean free path is $\lambda_{\mathrm{mfp}}=1/(n_e\sigma_T)=1/(3.2\times10^{8}\times6.65\times10^{-29})=4.7\times10^{19}\,\mathrm{m}$ . Converting with $1\,\mathrm{Mpc}=3.09\times10^{22}\,\mathrm{m}$ gives $\lambda_{\mathrm{mfp}}\approx1.5\times10^{-3}\,\mathrm{Mpc}$ physical, i.e. $\lambda_{\mathrm{mfp}}\approx1.5\,\mathrm{kpc}$ physical — a stride thousands of times shorter than the $7\,\mathrm{Mpc}$ diffusion length, confirming that only the accumulated random walk, not a single step, sets the Silk scale.

3 · The tail as a baryometer (scaling estimate)

Using $\lambda_D\propto\Omega_b^{-1/2}$ , estimate how the damping multipole $\ell_D$ shifts if the baryon density is raised by $10\%$ , at fixed geometry.

Step 1. Since $\ell_D\approx k_D D_A = D_A/\lambda_D$ with $\lambda_D\propto\Omega_b^{-1/2}$ , we have $\ell_D\propto\Omega_b^{+1/2}$ ; a $10\%$ rise gives a factor $(1.1)^{1/2}=1.049$ , about $\mathbf{+5\%}$ , shifting $\ell_D$ from $\approx1960$ to $\approx2060$ .

Step 2. More baryons mean a shorter mean free path and less diffusion, so the tail moves to higher $\ell$ . This baryon sensitivity is distinct from the odd/even peak-height asymmetry, giving the tail an independent handle on $\Omega_b h^2$ and a cross-check on Big Bang nucleosynthesis.

Physics of the portrait

Silk damping is one of the rare predictions that preceded its data by a full generation — when Joseph Silk wrote it down in 1968, CMB anisotropies would not be detected for another two and a half decades. The portrait is of the man who read the leak in the oscillator before anyone had heard it ring.

Origins · 1 minJoseph Silk and how this was discovered

Joseph Silk

b. 1942 · diffusion damping of cosmological fluctuations

Joseph Silk was born in London in 1942 and educated at Cambridge before taking his PhD at Harvard in 1968 — the same year as the paper that carries his name. In Cosmic Black-Body Radiation and Galaxy Formation (ApJ, 1968) he asked what imperfect photon-matter coupling does to primordial density fluctuations, and answered that photon diffusion must wipe out perturbations below a characteristic mass and length — now the Silk mass and the Silk length. It was remarkably prescient: the fluctuations he was damping would not be observed until COBE in 1992, yet his diffusion scale is today read directly off the CMB damping tail by Planck, ACT, and SPT. Silk spent much of his career at Berkeley, became the Savilian Professor of Astronomy at Oxford in 2005, and later worked at the Institut d'Astrophysique de Paris and Johns Hopkins. Extraordinarily prolific across galaxy formation, dark matter, and cosmology, he made the 1968 result the cleanest example of his instinct for the physically decisive effect — a mechanism he identified with a one-page random-walk argument and that is completed today as the Gaussian envelope $e^{-(k/k_D)^2}$ built from the tight-coupling transport coefficients.

History

1968

— Joseph Silk predicts that photon diffusion damps small-scale fluctuations in the pre-recombination plasma

1970

— Peebles and Yu, and independently Sunyaev and Zeldovich, compute the full acoustic and damping spectrum of the photon-baryon fluid

1983

— Kaiser shows photon polarization enhances the shear viscosity, raising the damping coefficient toward the 16/15 factor

1995

— Hu and Sugiyama derive the analytic tight-coupling damping envelope used in modern CMB codes

2013

— SPT, ACT, and Planck resolve the damping tail out to multipoles near 3000, pinning the baryon density and N_eff

Going further

Texts. Dodelson, Modern Cosmology, 2nd ed., ch. 8 (photon diffusion and the damping tail) · Mo, van den Bosch and White, Galaxy Formation and Evolution, §6.5 · Ryden, Introduction to Cosmology, 2nd ed., ch. 9 · Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 30 · Binney and Tremaine, Galactic Dynamics, 2nd ed. (for collisionless damping, to contrast).

Papers. Silk, J. (1968), Cosmic Black-Body Radiation and Galaxy Formation, ApJ 151, 459 · Kaiser, N. (1983), Small-angle anisotropy of the microwave background, MNRAS 202, 1169 (polarization and the $16/15$ viscosity factor) · Hu, W. and Sugiyama, N. (1995), Anisotropies in the Cosmic Microwave Background, ApJ 444, 489.

Provenance. Equations follow the tight-coupling treatment in Dodelson ch. 8 and Hu and Sugiyama (1995). A verifier should reconfirm: (1) the random-walk scaling $\lambda_D^2=ct/(n_e\sigma_T)$ ; (2) the diffusion coefficient of equation (2) with the $16/15$ polarization-enhanced viscosity and $R^2/(1+R)$ conduction terms; (3) $\sigma_T=6.65\times10^{-29}\,\mathrm{m}^2$ ; (4) the damping scale $k_D\approx0.14\,\mathrm{Mpc}^{-1}$ , $\lambda_D=1/k_D\approx7\,\mathrm{Mpc}$ comoving, and $\ell_D\approx k_D D_A\approx1960$ (this corrects an inconsistent $\ell_D\sim1400$ , which does not satisfy $\ell_D=k_D D_A$ for $D_A\approx14{,}000\,\mathrm{Mpc}$ ); (5) $R\approx0.61$ and $\lambda_{\mathrm{mfp}}\approx1.5\,\mathrm{kpc}$ physical ( $n_e\approx3.2\times10^{8}\,\mathrm{m^{-3}}$ ) at last scattering.

Prerequisites. Acoustic Oscillations in the Photon–Baryon Fluid · Density Perturbations and the Density Contrast · The Sound Horizon and the Standard Ruler. Leads to. The CMB Anisotropy Power Spectrum · The ΛCDM Concordance Model · Baryon Acoustic Oscillations.

Ask the tutor: Ask me why the $16/15$ viscosity factor needs photon polarization, or how the ratio $\ell_D/\ell_A$ turns the damping tail into a measurement of extra relativistic species.

Why is the damping scale so much larger than the photon mean free path?

Because damping is set by the random-walk diffusion length, not a single step. A photon takes an enormous number $N$ of tiny steps of length $\lambda_{\mathrm{mfp}}$ before recombination, and the net displacement is $\sqrt{N}\,\lambda_{\mathrm{mfp}}$ , not $\lambda_{\mathrm{mfp}}$ . Near recombination $\lambda_{\mathrm{mfp}}\approx1.5\,\mathrm{kpc}$ physical, but the accumulated walk reaches a comoving $\lambda_D\approx7\,\mathrm{Mpc}$ — a factor of thousands. The Silk length is essentially the geometric mean of the horizon and the mean free path, sitting logarithmically midway between step and horizon, which is why the tail is a large-angular-scale effect even though scattering is microscopic.

Does Silk damping erase the fluctuations permanently, or only the photons?

It erases the baryon-photon fluctuations on small scales, but not the dark matter. Photons diffuse and drag the baryons through Coulomb coupling to electrons, so both the photon temperature contrast and the baryon overdensity are smoothed below the diffusion scale. Cold dark matter does not scatter and is untouched — its small-scale perturbations survive. After recombination the baryons, no longer pressure-supported, fall back into the dark-matter wells, so the small-scale matter power is partly restored even though the CMB tail keeps a permanent record. Silk damping is thus a permanent feature of the CMB but only a temporary suppression of the total matter perturbation.

What is the difference between diffusion damping and last-scattering-thickness smoothing?

Two independent smoothings on similar scales. Diffusion damping is dynamical: while the plasma is coupled, photons random-walk out of small-scale fluctuations, giving the intrinsic $e^{-(k/k_D)^2}$ . Last-scattering-thickness smoothing is geometric: recombination is not instantaneous, so we observe photons from a shell of comoving thickness $\sigma_{\mathrm{LS}}\sim19\,\mathrm{Mpc}$ , and averaging along the line of sight adds $e^{-(k\sigma_{\mathrm{LS}})^2/2}$ . Because the widths are comparable both matter, and the observed damping is the product of the two Gaussians.

How does the damping factor act on the power spectrum versus the amplitude?

The amplitude is multiplied by $\mathcal{D}(k)=e^{-(k/k_D)^2}$ ; the power spectrum, being the amplitude squared, is multiplied by $\mathcal{D}(k)^2=e^{-2(k/k_D)^2}$ — twice the exponent. The factor of two matters when reading off the CMB $C_\ell$ : at $\ell=\ell_D$ the amplitude is down by $e^{-1}=0.37$ but the power by $e^{-2}=0.14$ . A frequent slip is to apply the amplitude factor to the power spectrum, underestimating the suppression by a square root.

Why does the damping tail help constrain the number of relativistic species?

Extra relativistic species — a larger $N_{\mathrm{eff}}$ — increase the radiation density and speed the expansion just before recombination. Faster expansion leaves less conformal time to diffuse, changing $k_D$ relative to the acoustic scale set by the sound horizon $r_s$ . Because both scales project through the same geometry, the ratio $\ell_D/\ell_A$ is nearly independent of the distance to last scattering and of curvature, isolating the pre-recombination physics. That ratio therefore pins $N_{\mathrm{eff}}$ and bounds early dark energy and other exotic contributions to the early expansion rate.

Is Silk damping the same as free-streaming or Landau damping?

No — different mechanisms on different components. Silk damping is collisional: in the tightly but imperfectly coupled photon-baryon plasma, frequent Thomson scattering makes photons diffuse and drag baryons, erasing small-scale plasma fluctuations. Free-streaming (collisionless Landau damping) acts on particles that are not scattering — neutrinos and hot or warm dark matter — which stream out of overdensities at their thermal speeds, washing out structure below their free-streaming length. Both suppress small-scale power, but Silk damping is tied to the recombination-epoch baryon-photon fluid and shows up in the CMB tail, while free-streaming is tied to a collisionless species' thermal history and shows up as a cutoff in the matter power spectrum.