The gravitational constant and the cosmic microwave background

Abstract

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 ${\ell }_{\mathrm{eq}}$ and the damping scale ${\ell }_{\mathrm{damp}}$ 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 ${\lambda }_G\equiv \sqrt{G/G_{\mathrm{N}}}$, scanning ${\lambda }_G\in [0.25,2]$ 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 ${\ell }_{\mathrm{eq}}$ and ${\ell }_A$ are nearly independent of $G$, non-equilibrium recombination gives ${\ell }_{\mathrm{damp}}\propto {\lambda }_G^{-0.304}$. The dependence exists only above a critical value of $G$, set by fundamental and cosmological parameters, a factor of 49 below $G_{\mathrm{N}}$. 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, ${\alpha }_G\equiv G{m}_e\mu /\hslash c$, with $\mu$ 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 ${\ell }_{\mathrm{damp}}/{\ell }_A$ therefore isolates $G$ at recombination and, up to lensing and screening, compares it directly with the laboratory value.

Submitted to A&A