Limits on the size of electron density fluctuations in the early universe and implications for the Hubble tension

Abstract

Little is known about the early universe on scales smaller than those resolved by the cosmic microwave background (CMB), where electron density fluctuations on kiloparsec to megaparsec scales are expected in many extensions of LCDM. These fluctuations nonetheless leave two imprints on the CMB. They accelerate recombination, the effect behind previous constraints and proposals to raise the inferred Hubble constant, $H_0$. Through correlations along the line of sight, they also reduce the effective Thomson scattering rate, enhancing diffusion damping by an amount that grows with their coherence length. We implement both effects in a modified Boltzmann code, classito, and use CMB data from Planck, ACT DR6, and SPT-3G D1 to place the first limits on such fluctuations that include this modified photon transport. We find no evidence for them: the variance of the electron density contrast is below $0.21$ ($95\%$ CL), falling to $0.033$ for a coherence length of $2\,{\rm Mpc}$, and we present these limits in terms of the electron power spectrum at $1\,{\rm Mpc}^{-1}\lesssim k\lesssim10^2\,{\rm Mpc}^{-1}$. We also reassess small-scale electron density fluctuations as a resolution of the Hubble tension. In Planck data the transport effect is partially degenerate with accelerated recombination, widening the range of $H_0$ allowed: combined with a distance-ladder prior, Planck gives $H_0 = 71.1\pm0.7\,{\rm km\,s^{-1}\,Mpc^{-1}}$ and an improved fit relative to LCDM. ACT and SPT-3G break this degeneracy, chiefly through the temperature--$E$-mode correlation in the damping tail, and with the same prior they give only $H_0 = 69.3\pm0.5\,{\rm km\,s^{-1}\,Mpc^{-1}}$ and no preference over LCDM. With our treatment and publicly available code, models that generate small-scale electron density fluctuations can be directly confronted with CMB data through the variance and coherence length that they predict.