The gravitational constant and the cosmic microwave background
The CMB constrains the gravitational constant $G$ at recombination to a few percent of its laboratory value, but the effect of $G$ on recombination itself is usually absorbed into a global fit rather than made explicit. We return to the semi-analytic description of the acoustic spectrum to ask how the acoustic scale $\ell_A$, the equality scale $\ell_{\rm eq}$ and the damping scale $\ell_{\rm damp}$ respond to a variation of $G$ during recombination, and where $G$ enters the residual ionization of the dark ages. We use a modified version of CLASS in which $G$ is varied through $λ_G \equiv \sqrt{G/G_{\rm N}}$, scanning $λ_G \in [0.25,2]$ at fixed physical densities and CMB temperature, with helium recomputed for each model. Analytic rate arguments are confronted with the numerical ionization histories, the visibility function, the Hu & White damping integral and the TT spectra. While $\ell_{\rm eq}$ and $\ell_A$ are nearly independent of $G$, non-equilibrium recombination gives $\ell_{\rm damp} \propto λ_G^{-0.304}$. The dependence exists only above a critical value of $G$, set by fundamental and cosmological parameters, a factor of 49 below $G_{\rm N}$. The dark-ages optical depth varies by only 16% for $G$ multiplied by 64, a negligible channel. Because $G$ enters only through the expansion rate, the imprint comes entirely from out-of-equilibrium recombination, and the competition between the atomic rates and the expansion is governed by one dimensionless combination, $α_G \equiv G m_e μ/\hbar c$, with $μ$ the mean mass of matter per photon. Each scale is the distance to last scattering divided by a length imprinted before it, so a late-time change of gravity rescales all three alike. The ratio $\ell_{\rm damp}/\ell_A$ therefore isolates $G$ at recombination and, up to lensing and screening, compares it directly with the laboratory value.