Photon-baryon fluid

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.

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.

With Fourier convention $\mathbf{v}=i\mathbf{k}\theta$, the velocity divergence is $-k^2\theta$. Thomson drag in the velocity equation is the conformal scattering rate $q=a{n}_e{\sigma }_T$ times the velocity difference. Taking its divergence and dividing the whole divergence equation by $-k^2$ leaves the same rate multiplying the difference of velocity potentials; an additional $k^{-2}$ is not retained. Weighting the photon Euler equation by $R=4{\bar{\rho }}_{\gamma }/(3{\bar{\rho }}_b)$ and adding the baryon equation cancels the drag. In tight coupling the remaining equation is $(1+R){\theta }^{\prime }+\mathcal{H}\theta +R({\delta }_{\gamma }/4-{\sigma }_{\gamma })=0$, with sound speed $R/[3(1+R)]$.

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 is the suppression of small-angular-scale anisotropy as photons random-walk out of overdense regions during the finite-width recombination epoch. It produces an approximately exponential damping tail in the Cosmic microwave background power spectrum at multipoles $\ell \gtrsim 1000$.