The CMB encodes the state of the universe at $z\sim 1100$ with extraordinary fidelity. Temperature anisotropies and polarization patterns constrain geometry, baryon content, dark matter, neutrino physics, and the primordial fluctuation spectrum.

In the tight-coupling limit, the coupled photon-baryon system behaves like a driven oscillator. A simplified form is $$\ddot\delta_\gamma + c_s^2 k^2 \delta_\gamma = \text{driving terms},$$ where the sound speed is $$c_s^2 = \frac{1}{3(1+R)},\quad R\equiv \frac{3\rho_b}{4\rho_\gamma}.$$ Baryon loading shifts peak heights and phases.

Thomson scattering of a local quadrupole generates linear polarization. Scalar perturbations generate primarily E-modes, while B-modes can arise from primordial tensor modes or from gravitational lensing that remaps the polarization field.

Parameter estimation compares theoretical spectra to measured bandpowers using a likelihood function. In practice, one samples posterior distributions over $\Omega_b h^2$, $\Omega_c h^2$, $\theta_*$, $\tau$, $n_s$, and $A_s$, among others, using Bayesian or frequentist pipelines.