Evolve angular radiation moments with gravity and collisions

For a scalar Fourier mode, let μ be the cosine between photon direction and k, and Θ its fractional temperature disturbance. Expand Θ in Legendre polynomials with coefficients (−i)ℓ(2ℓ+1). That phase convention determines the signs in the moment equations. In a collisionless test problem without metric driving, Θ′+ikμΘ=0. The Legendre recurrence then couples each multipole only to its two neighbors, transferring structure into progressively finer angular variation.

The monopole describes the angular average, the dipole relates to bulk motion, and the quadrupole contributes to anisotropic stress. Factors depend on whether one expands temperature, intensity, or a momentum-dependent distribution. In particular, a photon intensity perturbation is four times the temperature perturbation at linear order. Massive neutrinos require momentum dependence because their speed changes with momentum and expansion; replacing them by one massless temperature hierarchy would discard essential information.

Thomson scattering couples photons to free electrons and hence baryons. Define the positive conformal scattering rate Γ=a neσT, rather than silently using a signed optical-depth derivative. Tight coupling requires Γ much larger than both the relevant wave number and conformal expansion rate, together with slowly varying coefficients for the expansion used. The nearly common photon–baryon velocity then supports acoustic oscillations with a sound speed reduced by baryon inertia.

Finite mean free path generates slip, higher moments, polarization, and diffusion damping. A controlled calculation expands in the small scattering time and switches treatment when that expansion loses accuracy. Recombination changes the free-electron density and therefore the collision history; it is not simply an abrupt deletion of every scattering term at one fixed time. Free-streaming neutrino anisotropic stress also feeds metric potentials, so setting Φ=Ψ throughout radiation evolution would generally remove a physical source.

The full linear system couples radiation moments, species densities and velocities, and Einstein constraints. Its solution at each k supplies transfer functions projected onto observed sky multipoles. For a single adiabatic scalar initial mode, the sky power integrates primordial power times the squared radiation transfer. Other initial modes and correlations add cross terms. An internal hierarchy multipole at one time is not automatically the observed sky coefficient; propagation and projection connect them. Artificially setting all higher moments to zero can reflect errors back toward lower multipoles as angular structure reaches the cutoff. Increase the maximum multipole and test the closure and time integration; apparent convergence of one observable at one k does not certify every scale.

The spectral lab offers a separate cutoff comparison. Reset and set mode count 2,decay exponent 1; at time 0 predict sum at x=π/2 equal to 1 and smallest wavelengthπ. Reset and set count 4,exponent 1; predict sum 2/3 and smallest wavelengthπ/2. Its finite traveling sine sum is prescribed, not a Legendre hierarchy. It contains no gravitational sources, collisions, polarization, or line-of-sight projection; changing its cutoff cannot validate a cosmological hierarchy closure.