The Lumina Project: CMB Optical Depth Fluctuations from Patchy Reionization

Patchy reionization couples the ionized-bubble morphology to the underlying density field, making the CMB Thomson optical depth sensitive to both the global ionization history and anisotropic fluctuations on the sky. Using the large-volume radiation-hydrodynamical Lumina simulation, we compute $\tau_{\rm CMB}$ in two ways: (i) from global volume- and mass-weighted ionization histories, and (ii) from explicit line-of-sight integrations through on-the-fly light cones. We find that the sightline-averaged optical depth in the light cone, $\langle \tau_{\rm LOS} \rangle = 0.0550$, exceeds the value inferred from a global volume-weighted history, $\tau_{{\rm CMB},V} = 0.0515$, by $\approx 7\%$. This enhancement is largely captured by the global mass-weighted prediction, $\tau_{{\rm CMB},m} = 0.0544$, indicating that precision comparisons to CMB optical-depth constraints should use mass-weighted electron fractions or explicit light-cone integration rather than volume-weighted ionized fractions alone. The excess optical depth accumulates primarily near $z_{\rm LOS} = 8.0^{+1.9}_{-1.3}$, where the combination of high physical density and strong ionization-field patchiness is greatest. The resulting $\tau_{\rm LOS}$ field is non-Gaussian and exhibits $\gtrsim 5\%$ sightline-to-sightline scatter, with fluctuations tracing rare early-ionized overdensities and large-scale structure. Coarse-graining experiments show that smoothing the ionization field on $\gtrsim 3 {\rm cMpc}$ scales suppresses the density-ionization correlation and biases $\tau_{\rm CMB}$ low relative to the resolved calculation. Finally, angular power spectra and real-space correlation functions decomposed into HII, HeII, and HeIII auto- and cross-contributions reveal scale-dependent departures from simple hydrogen-helium co-tracing and evolving characteristic scales with redshift.