The Challenge of Observing Patchy Reionization with CMB Optical-Depth Fluctuations

Abstract: Spatial fluctuations in the Thomson optical depth encode information about the inhomogeneous nature of cosmic reionization. We compute the optical-depth angular power spectrum, $C_\ell<sup>{ττ}$, using past lightcones constructed from five Cosmic Reionization on Computers (CROC) radiation-hydrodynamical simulations. By decomposing the electron-density field into patchy and density components, we quantify the separate contributions of ionization-fraction and baryon-density fluctuations to the optical-depth anisotropy. Because the simulations end at $z\approx5$, we supplement the reionization-era signal with an analytic estimate of the fully ionized low-redshift contribution. We find that baryon-density fluctuations dominate the high-redshift signal over most angular scales, while the accumulated low-redshift contribution exceeds the high-redshift signal across the full multipole range considered. Our results demonstrate that a significant fraction of the optical-depth power is not uniquely associated with reionization morphology, implying that future interpretations of $C_\ell<sup>{ττ}$ must account for the density contribution in addition to patchy ionization.

• The paper systematically evaluates how well CMB optical-depth fluctuations can probe patchy reionization, finding that baryon density fluctuations dominate the optical-depth signal.
• The study uses radiation-hydrodynamical simulations from the CROC suite to demonstrate that density fluctuations, not ionization fluctuations, predominantly determine the power spectrum of cosmological optical depth both during and after reionization.
• The low redshift density contribution significantly outweighs the reionization era signal: This finding implies that efforts to observe patchy reionization must address density fluctuation models or suppression methods designed to ignore the density term.

Overview and motivation

Fluctuations in the Thomson optical depth $\tau(\hat{\mathbf n})$ are widely treated as a direct probe of the morphology of cosmic reionization: lines of sight that pass through regions ionized earlier accumulate more free electrons, so $\delta\tau$ should encode the topology and timing of ionized bubbles (2608.19307). The paper by Takoudes, Zhu, and Gnedin subjects this assumption to a quantitative test using radiation-hydrodynamical simulations from the Cosmic Reionization on Computers (CROC) suite. Its central result is a cautionary one: when the electron-density field $n_e = x_e n_b$ is decomposed into a patchy contribution (spatial variations in the electron-per-baryon fraction $x_e$) and a density contribution (variations in baryon density at fixed mean ionization state), the density term dominates the optical-depth power spectrum both during reionization itself and, overwhelmingly, once the fully ionized low-redshift universe is included.

Simulation methodology

The analysis uses five CROC volumes of side $L_{\rm box}=80\,h^{-1}\,{\rm cMpc}$ with $1024^3$ cells. Three are independent Gaussian realizations (A, B, C); two are variants of C with manually imposed DC modes sampling underdense (${\rm DC}=-1$) and overdense (${\rm DC}=+1$) large-scale environments. This design lets the authors separate realization-to-realization scatter from environmental dependence of the signal.

Past lightcones are constructed by projecting simulation snapshots onto angular maps, with two independent schemes — "tiles" (boxes tiled along the radial axis with time interpolation) and "slabs" (box portions centered exactly at snapshot times). Both reuse the finite periodic volume, mitigated by random rotations, translations, and reflections applied per element. The difference between tile- and slab-based spectra serves as an empirical estimate of systematic uncertainty from lightcone construction. A reference angular grid defined by $a_{\rm ref}=0.1667$ maps all redshifts onto a common pixelization; an appendix shows results are robust to this choice over intermediate multipoles, with deviations confined to the lowest and highest $\ell$ where mode sampling and resampling effects matter most. Power spectra are estimated with a flat-sky FFT estimator truncated at the 1D Nyquist multipole $\delta\tau$0.

Decomposition of optical-depth fluctuations

At each snapshot the electron-density fluctuation is split exactly as

$\delta\tau$1

and each term is projected to its own optical-depth map. Because $\delta\tau$2, the spectra satisfy $\delta\tau$3, which is used as a normalization consistency check. Since CROC outputs terminate at $\delta\tau$4, the authors add an analytic low-redshift density contribution for $\delta\tau$5 assuming spatially uniform ionization ($\delta\tau$6 above $\delta\tau$7, $\delta\tau$8 below), evaluated via the Limber approximation with the nonlinear CAMB matter power spectrum extrapolated to $\delta\tau$9. The sim–low-$n_e = x_e n_b$0 cross term vanishes under Limber for non-overlapping radial intervals and is neglected.

Results

Three findings stand out:

The high-redshift signal is already density dominated. In every simulation, the density component exceeds the patchy component at essentially all multipoles shown, typically by a factor of a few. Even during the partially neutral epoch, then, most of the optical-depth power traces the underlying gas distribution rather than reionization morphology. The only exception is the underdense DC-mode run at the lowest multipoles, where patchy power is largest among all five boxes — plausibly because late reionization preserves large-scale ionization structure longer, though the spectra do not disentangle this geometric effect (later reionization places structure at smaller comoving distance) from genuine morphological differences.

A negative patchy–density cross-correlation emerges at high multipoles. At $n_e = x_e n_b$1, the total high-redshift spectrum falls below the density component alone, requiring $n_e = x_e n_b$2. The authors attribute this plausibly to enhanced recombination and self-shielding in overdense gas, which suppresses $n_e = x_e n_b$3 where $n_e = x_e n_b$4 is large. The physical origin is left as an open question rather than demonstrated directly.

The low-redshift contribution exceeds everything else. The analytic $n_e = x_e n_b$5 density term — driven by the long path length through the fully ionized IGM and late-time nonlinear growth — dominates the total $n_e = x_e n_b$6 across the full multipole range considered. Consequently, the total optical-depth power cannot be interpreted as a measurement of patchy reionization without explicit modeling and subtraction of the density-sourced component. Forecasts built solely on projected ionization-fraction power spectra systematically misattribute this power; the implication for experiments targeting reionization morphology through $n_e = x_e n_b$7 fluctuations is that either density modeling or statistics designed to suppress the density term are required.

Limitations and open questions

The paper concedes several limitations plainly. The five-volume ensemble limits quantification of sample variance and DC-mode environmental scatter in $n_e = x_e n_b$8, particularly at low $n_e = x_e n_b$9 where few independent modes fit in the box. The low-redshift treatment is analytic rather than simulated, resting on the assumptions of uniform ionization, $x_e$0, Limber validity, and power-spectrum extrapolation beyond $x_e$1. The sign and magnitude of the negative patchy–density correlation are inferred indirectly from the decomposition identity rather than measured independently. Finally, the scale-dependent reversal of DC-mode ordering in the patchy component conflates geometric projection with ionization-field morphology.

Conclusion

Using CROC past lightcones, this work demonstrates that baryon-density fluctuations, not ionization-fraction fluctuations, supply most of the optical-depth anisotropy power even at $x_e$2, and that the accumulated post-reionization contribution exceeds the reionization-era signal entirely. The practical conclusion is methodological rather than celebratory: extracting reionization morphology from $x_e$3 requires explicitly separating or suppressing the density contribution, and future work must enlarge the simulation ensemble and resolve the origin of the negative patchy–density cross-correlation before optical-depth fluctuations can serve as a clean probe of reionization topology.