Optical depth to reionization from perturbative 21 cm clustering

The optical depth $\tau$ is the least well determined parameter in standard model of cosmology, and one whose precise value important for both understanding reionization inferring fundamental physics from cosmological measurements. We forecast how future epoch experiments could constraint using a symmetries-based bias expansion that highlights special role played by anisotropies power spectrum on large scales. Given parametric ionization evolution inspired physical behavior more detailed simulations, we find 21cm place tight constraints timing duration hence are competitive with proposed, space-based CMB missions provided they can measure $k\approx 0.1\,h\,\text{Mpc}^{-1}$ clean foreground wedge across redshifts spanning most active periods reionization, corresponding to fractions $0.2 \lesssim x 0.8$. Significantly improving upon existing CMB-based measurements next-generation surveys would require substantially longer observations ($\sim5$ years) than $\mathcal{O}(1000 \,\,\text{hour})$ integration times. Precise smaller scales will not improve until better astrophysics achieved. In presence noise foregrounds even struggle constrain if deviates significantly simple forms.