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 ℓA, the equality scale ℓeq and the damping scale ℓdamp 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 λG ≡ √(G/GN), scanning λG ∈ [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 ℓeq and ℓA are nearly independent of G, non-equilibrium recombination gives ℓdamp ∝ λG−0.304. The dependence exists only above a critical value of G, set by fundamental and cosmological parameters, a factor of 49 below GN. 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, αG ≡ G me μ/ℏ 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 ℓdamp/ℓA therefore isolates G at recombination and, up to lensing and screening, compares it directly with the laboratory value.
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BibTeX
@misc{lamine2026gravitational,
title = {{The gravitational constant and the cosmic microwave background}},
author = {Brahim Lamine and Sébastien Beauchamp and Alain Blanchard and Jehanne Delhomelle and Raphaël Kou and James Rich},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22128},
eprint = {2610.22128},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22128},
}
RIS
TY - UNPB
TI - The gravitational constant and the cosmic microwave background
AU - Brahim Lamine
AU - Sébastien Beauchamp
AU - Alain Blanchard
AU - Jehanne Delhomelle
AU - Raphaël Kou
AU - James Rich
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22128
UR - https://arxiv.org/abs/2610.22128
AB - 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 ℓ_A, the equality scale ℓ_eq and the damping scale ℓ_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 ≡ √(G/G_N), scanning λ_G ∈ [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 ℓ_eq and ℓ_A are nearly independent of G, non-equilibrium recombination gives ℓ_damp ∝ 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_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 ≡ G m_e μ/ℏ 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 ℓ_damp/ℓ_A therefore isolates G at recombination and, up to lensing and screening, compares it directly with the laboratory value.
ER -