Patchy Reionization: How Ionized Bubbles Stamp the CMB

What Patchy Reionization Is and Why It Matters

After cosmological recombination at z ≈ 1100 the universe was a sea of neutral hydrogen — the cosmic dark ages. When the first stars, galaxies, and quasars ignited around z ≈ 15–20, their ultraviolet photons carved out expanding H II bubbles around each source. These ionized regions grew, overlapped, and finally percolated to fill the intergalactic medium (IGM) by z ≈ 5.5–6, the last redshift at which Lyman-α forest data still show residual neutral patches.

The word patchy captures the essential point: reionization was spatially inhomogeneous. Ionization did not rise uniformly everywhere but proceeded bubble by bubble, biased toward the densest regions where galaxies formed first. This spatial structure in the free-electron field δx_e(x) is exactly what leaves a CMB signature. A perfectly homogeneous reionization would only rescale the optical depth τ; the patchiness adds new, small-scale anisotropy that carries information about the sizes of the bubbles, the number and clustering of ionizing sources, and above all the duration of reionization — a quantity that global-averaged probes like τ cannot pin down on their own.

The Mechanism, Step by Step

Every CMB photon streaming from the last-scattering surface has a small chance of Thomson-scattering off a free electron produced during reionization; the total probability is the optical depth τ ≈ 0.054, meaning roughly 5% of photons rescatter. Three things happen when the electron field is patchy.

(1) Doppler / kinetic SZ. Electrons inside a bubble share the bulk peculiar velocity v of that gas. Scattering imprints a temperature shift ΔT/T = −∫ σ_T n_e (v·n̂/c) dl along each line of sight. Because both n_e (through δx_e) and v vary across the sky, the integral does not cancel, producing new anisotropy — the patchy kSZ effect.

(2) Screening. Where τ is locally higher, the primary CMB is more attenuated by a factor e^(−δτ), modulating the pattern already there.

(3) Polarization mixing. Rescattering of the CMB quadrupole by patchy electrons generates new polarization, and spatially varying τ converts primary E-modes into secondary B-modes. The kSZ term is first-order in velocity and dominates the temperature signal; screening and E→B are the subtler polarization channels.

Characteristic Numbers, Scales, and the Key Relation

The bubbles that matter span comoving radii of roughly a few to tens of Mpc, which at z ≈ 8 subtend arcminutes on the sky — hence the patchy kSZ power spectrum peaks near multipole ℓ ≈ 2000–5000. Its amplitude is conventionally quoted as D_ℓ = ℓ(ℓ+1)C_ℓ/2π at ℓ = 3000, where the patchy contribution is of order 1–3 μK², comparable to the homogeneous post-reionization kSZ (D₃₀₀₀ ≈ 1.76 μK² at σ₈ = 0.812).

The physically crucial scaling, established in semi-numerical models, is that the amplitude grows with how long reionization lasts: D₃₀₀₀^patchy ∝ z̄ · Δz^0.47, where z̄ is the midpoint redshift and Δz is the duration (often defined as the redshift interval over which the ionized fraction runs from 25% to 75%, or 5% to 95%). A more extended, drawn-out reionization means larger, longer-lived bubbles and more velocity coherence, boosting the signal. The optical-depth fluctuation itself has an rms of order δτ ~ 10⁻³ across the sky, roughly a few percent of the mean τ.

How It Is Observed and Detected

Patchy reionization lives at high multipole and low amplitude, so it demands arcminute-resolution, high-sensitivity ground-based CMB experiments. The South Pole Telescope (SPT-3G) and the Atacama Cosmology Telescope (ACT) measure the total kSZ power at ℓ ≈ 3000 from their multi-frequency (95/150/220 GHz) maps, separating it from the thermal SZ, CIB, and radio-source foregrounds. Because the patchy and homogeneous kSZ blend together, the reionization piece is extracted by subtracting a modeled late-time kSZ and by exploiting the higher-order kSZ statistics.

SPT constraints on the total kSZ amplitude already translate into an upper bound on the duration, currently Δz ≲ 5 (95% ionization minus 5%), disfavoring very extended histories. Planck's large-angle EE 'reionization bump' fixes the global τ = 0.054 ± 0.007, anchoring the midpoint at z̄ ≈ 7.7. Future surveys — Simons Observatory and CMB-S4 — aim to detect the patchy signal directly, including the τ-fluctuation field reconstructed with lensing-like quadratic estimators and the patchy-screening B-modes.

Where It Operates and How It Differs from Related Effects

Patchy reionization is a phenomenon of the epoch of reionization, z ≈ 6–12, distinct from the smooth, fully ionized IGM at lower redshift. It is essential to distinguish its several CMB channels from look-alikes. The homogeneous (late-time) kSZ arises from the same Thomson-Doppler physics but in the already-ionized universe at z ≲ 6; it peaks at the same ℓ ≈ 3000 and must be modeled and subtracted to isolate the reionization part.

The thermal SZ effect, by contrast, comes from hot electrons in galaxy clusters and has a characteristic frequency dependence (a null near 217 GHz) that lets it be separated cleanly — patchy kSZ, being a pure Doppler shift, is frequency-independent (a blackbody distortion) and cannot be removed by spectral filtering. Gravitational lensing also generates B-modes and mimics some patchy-screening statistics, so reconstructions must jointly fit lensing and τ-modulation. Finally, the global optical depth τ measures only the integrated column of free electrons and is degenerate over reionization histories; only the patchy statistics break that degeneracy and reveal the morphology.

Open Questions and Significance

Patchy reionization is one of the few observational handles on the astrophysics of the first luminous sources. Its amplitude and shape encode which objects reionized the universe — faint dwarf galaxies versus rarer bright ones versus quasars — through the sizes and clustering of the bubbles they blew. The biggest open questions are the duration Δz and midpoint z̄ of reionization, whether the process was 'self-regulated' by feedback and recombinations, and the escape fraction of ionizing photons from early galaxies.

A robust measurement of the patchy kSZ amplitude would independently constrain Δz to complement the Planck τ and the growing z ≈ 5–9 Lyman-α, quasar-damping-wing, and 21-cm data. The patchy-screening B-modes are a double-edged prize: they are a signal in their own right, but also a contaminant for primordial gravitational-wave searches (the sought-after tensor B-modes with tensor-to-scalar ratio r), so cleanly separating patchy-reionization E→B leakage from inflationary B-modes is an active frontier for CMB-S4 and LiteBIRD. Robust theoretical predictions remain limited by our uncertain knowledge of the reionization history itself.

CMB secondary effects from reionization, compared by mechanism, angular scale, and characteristic amplitude

Effect	Physical mechanism	Peak angular scale	Characteristic amplitude
Homogeneous kSZ (post-reion.)	Doppler scattering off smooth ionized IGM bulk flows	ℓ ≈ 3000 (arcmin)	D₃₀₀₀ ≈ 1.5–2 μK² (≈1.76 μK² at σ₈=0.812)
Patchy kSZ (reionization)	Doppler scattering off ionized bubbles / velocity + δx_e	ℓ ≈ 2000–5000	≈ 1–3 μK² at ℓ=3000, scales ∝ Δz^0.47
Patchy screening (τ fluctuations)	Spatially varying Thomson optical depth attenuates primary	ℓ ≈ several thousand	δτ ~ 10⁻³ rms; sub-μK, reconstructed
Patchy E→B polarization	Modulation of primary E-mode by δτ makes secondary B	ℓ ≈ 100–1000	≈ few nK to ~0.1 μK (r-dependent)
Thomson optical depth (global)	Net scattering suppresses primary, boosts large-scale EE	ℓ ≲ 10 reionization bump	τ = 0.054 ± 0.007 (Planck)