Probing the optical depth with galaxy number counts

We study the prospects for measuring the cosmological distribution and abundance of ionized electrons in the intergalactic medium using galaxy surveys. Optical light from distant galaxies is subject to Thomson screening by intervening electrons which distorts the observed galaxy number density, similar to the effect of weak gravitational lensing magnification. We construct an estimator for the optical-depth fluctuations from the statistical anisotropies of galaxy number counts induced by the spatially-varying optical-depth field. We find near-future galaxy surveys can detect this signal at signal-to-noise above $\sim10$ depending on galaxy survey specifications. We highlight various science cases for the measurement of optical-depth fluctuations.