Cosmic Microwave Background Physics

1. The Boltzmann Hierarchy

The photon distribution function is expanded in Legendre multipoles. Defining the brightness perturbation $\Theta(\mathbf{k}, \hat{n}, \tau) = \delta T/T$ and expanding in multipoles:

$$\Theta(\mathbf{k}, \hat{n}, \tau) = \sum_\ell (-i)^\ell (2\ell+1)\,\Theta_\ell(k,\tau)\,P_\ell(\hat{k}\cdot\hat{n})\,,$$

the collisional Boltzmann equation in Fourier space generates the hierarchy:

$$\dot\Theta_0 = -k\Theta_1 - \dot\Phi\,,$$

$$\dot\Theta_1 = \frac{k}{3}(\Theta_0 - 2\Theta_2 + \Psi) + \dot\tau_c(\Theta_1 - v_b/3)\,,$$

$$\dot\Theta_\ell = \frac{k}{2\ell+1}\left[\ell\,\Theta_{\ell-1} - (\ell+1)\Theta_{\ell+1}\right] - \dot\tau_c\,\Theta_\ell\,,\quad \ell \geq 2\,,$$

where $\Phi$ and $\Psi$ are the Bardeen potentials,$\dot\tau_c = n_e\sigma_T a$ is the Thomson scattering rate, and$v_b$ is the baryon velocity. The monopole $\Theta_0$ represents the temperature perturbation, the dipole $\Theta_1$ the bulk velocity, and$\Theta_2$ the quadrupole (source of polarization).

2. Tight-Coupling Approximation

Before recombination, Thomson scattering is rapid ($\dot\tau_c \gg H$), and multipoles with $\ell \geq 2$ are suppressed. The photon-baryon fluid then satisfies a forced harmonic oscillator equation:

$$\ddot\Theta_0 + \frac{\dot R}{1+R}\dot\Theta_0 + k^2 c_s^2\,\Theta_0 = -\frac{k^2}{3}\Psi - \frac{\dot R}{1+R}\dot\Phi - \ddot\Phi\,,$$

where $R = 3\rho_b/(4\rho_\gamma)$ is the baryon loading parameter and the sound speed is

$$c_s = \frac{1}{\sqrt{3(1+R)}}\,.$$

The general solution is an oscillation with a zero-point shift from the gravitational potential:

$$\Theta_0(\tau) + \Psi = A\cos(k r_s) + B\sin(k r_s)\,,$$

where $r_s(\tau) = \int_0^\tau c_s\,d\tau'$ is the comoving sound horizon.

3. Angular Power Spectrum

The CMB anisotropy is decomposed into spherical harmonics:$\delta T/T(\hat{n}) = \sum_{\ell m} a_{\ell m}\,Y_{\ell m}(\hat{n})$. Statistical isotropy implies $\langle a_{\ell m}\,a_{\ell'm'}^*\rangle = C_\ell\,\delta_{\ell\ell'}\delta_{mm'}$. The power spectrum is related to the primordial spectrum by

$$C_\ell = \frac{2}{\pi}\int_0^\infty k^2\,P(k)\,|\Delta_\ell(k)|^2\,dk\,,$$

where $\Delta_\ell(k)$ is the radiation transfer function that encodes all the physics of photon evolution from the primordial perturbation to the last-scattering surface. The transfer function includes contributions from the intrinsic temperature perturbation, the Doppler effect, and gravitational redshifts.

4. The Sachs-Wolfe Effect

On scales larger than the sound horizon at recombination ($\ell \lesssim 100$), perturbations have not had time to oscillate. The observed temperature perturbation is set by the gravitational redshift from the potential well plus the intrinsic temperature fluctuation. For adiabatic initial conditions in a matter-dominated universe:

$$\frac{\delta T}{T}\bigg|_{\rm SW} = \frac{\Phi}{3}\,,$$

where the factor of 1/3 arises because the intrinsic temperature perturbation is$\Theta_0 = -2\Phi/3$ (adiabatic condition) and the gravitational blueshift from climbing out of the well is $\Psi = \Phi$. This gives the large-angle plateau:

$$\frac{\ell(\ell+1)C_\ell}{2\pi}\bigg|_{\rm SW} = \frac{A_s}{9}\quad (\text{approximately scale-invariant})\,.$$

5. Acoustic Peak Structure

The peaks in $C_\ell$ correspond to modes that have completed a half-integer number of oscillations by recombination. The $n$-th peak occurs at

$$\ell_n \approx n\,\frac{\pi\,d_A(z_*)}{r_s(z_*)}\,,$$

where $d_A(z_*)$ is the angular diameter distance to recombination and$r_s(z_*)$ is the sound horizon at recombination. Each cosmological parameter leaves a distinctive signature:

6. Silk Damping

Photon diffusion during recombination erases anisotropies on small scales. The damping scale is set by the photon mean free path integrated over the recombination epoch:

$$k_D^{-2} = \int_0^{\tau_*} \frac{d\tau}{6(1+R)\dot\tau_c}\left[\frac{R^2+16(1+R)/15}{1+R}\right]\,.$$

The damping suppresses the power spectrum as $C_\ell \propto e^{-(\ell/\ell_D)^2}$with $\ell_D \sim 1500$. This exponential tail at high multipoles is sensitive to the baryon density and the number of relativistic species through $N_{\rm eff}$.

7. Constraints on Relativistic Species

The radiation energy density before recombination is parametrized as

$$\rho_{\rm rad} = \left[1 + \frac{7}{8}\left(\frac{4}{11}\right)^{4/3}N_{\rm eff}\right]\rho_\gamma\,.$$

The SM predicts $N_{\rm eff}^{\rm SM} = 3.044$ (including finite-temperature QED corrections). Increasing $N_{\rm eff}$ increases the expansion rate before recombination, shifting the epoch of matter-radiation equality and modifying the damping scale. Planck measures

$$N_{\rm eff} = 2.99 \pm 0.17\quad (95\%\;\text{CL})\,,$$

consistent with three neutrino species and constraining additional light relics (sterile neutrinos, axions thermalized before neutrino decoupling, dark radiation) to contribute$\Delta N_{\rm eff} \lesssim 0.3$ at 95% CL.

8. CMB Polarization

Thomson scattering of a quadrupole anisotropy generates linear polarization. The polarization is decomposed into E-modes (curl-free) and B-modes (divergence-free). E-modes are produced by both scalar and tensor perturbations:

$$C_\ell^{EE} \propto \int k^2\,P(k)\,|\Delta_\ell^E(k)|^2\,dk\,,$$

while primordial B-modes are produced only by tensor perturbations (gravitational waves from inflation) with amplitude parametrized by the tensor-to-scalar ratio:

$$r = \frac{\Delta_t^2}{\Delta_s^2} = 16\epsilon\,,$$

where $\epsilon$ is the first slow-roll parameter. Current limits from BICEP/Keck give $r < 0.036$ (95% CL), constraining the energy scale of inflation:

$$V^{1/4} = 1.06\times10^{16}\;\text{GeV}\left(\frac{r}{0.01}\right)^{1/4}\,.$$

Gravitational lensing of E-modes by large-scale structure produces a secondary B-mode signal that must be “delensed” to access the primordial signal. CMB-S4 aims to reach$\sigma(r) \sim 5\times10^{-4}$.

9. Secondary Anisotropies

After recombination, several effects generate secondary CMB anisotropies. The integrated Sachs-Wolfe (ISW) effect arises from time-varying potentials during dark energy domination:

$$\frac{\delta T}{T}\bigg|_{\rm ISW} = 2\int_0^{\tau_0}\dot\Phi\,e^{-\tau}\,d\tau\,.$$

The thermal Sunyaev-Zel’dovich (tSZ) effect from hot gas in galaxy clusters shifts the CMB spectrum: $\Delta T/T = -2y$ in the Rayleigh-Jeans regime, where the Compton $y$-parameter is

$$y = \int \frac{k_B T_e}{m_e c^2}\,n_e\,\sigma_T\,d\ell\,.$$

The kinematic SZ effect from the peculiar velocity of clusters provides$\Delta T/T = -\tau_T(v_r/c)$, a direct probe of the velocity field that tests the growth of structure.