Angular-scale limits of instantaneous photon decoupling

Approximating decoupling by a single time and retaining only a photon monopole and Doppler dipole omits the finite cosmological visibility function width, diffusion or Silk damping, quadrupole anisotropy and polarization generated as tight coupling fails. The photon-baryon acoustic oscillator also make the emission density and velocity strongly scale dependent. They can be included in the initial monopole and dipole, so their existence alone does not invalidate the subsequent collisionless line-of-sight solution for free-streaming photons. Smaller angles correspond to larger angular multipole $\ell$.