Model-Independent Reionization Observables in the CMB

TL;DR

The paper addresses biases in inferring the reionization history from CMB polarization by introducing a model-independent framework: the ionization fraction $x(z)$ is treated as a free function of redshift and projected onto a basis of delta-function redshift modes. A Fisher-based PCA reveals that five eigenmodes capture essentially all observable information in the $E$-mode power spectrum $C_^{EE}$, with the first few modes constraining the total optical depth $\tau$ and the higher modes mainly shaping spectral ringing. In an ideal, cosmic-variance-limited experiment, the total optical depth can be measured to about $\sigma_\tau \approx 0.01$, and the best-constrained single mode yields $\sigma_\tau \approx 0.0026$, enabling precise normalization of initial fluctuations independent of the detailed reionization history. This approach provides a practical, model-independent tool for data analysis and model testing, reducing biases in $\tau$ while preserving sensitivity to the entire reionization history through a compact set of observable modes.

Abstract

We represent the reionization history of the universe as a free function in redshift and study the potential for its extraction from CMB polarization spectra. From a principal component analysis, we show that the ionization history information is contained in 5 modes, resembling low-order Fourier modes in redshift space. The amplitude of these modes represent a compact description of the observable properties of reionization in the CMB, easily predicted given a model for the ionization fraction. Measurement of these modes can ultimately constrain the total optical depth, or equivalently the initial amplitude of fluctuations to the 1% level regardless of the true model for reionization.

Figures (5)

• Figure 1: Hydrogen ionization fraction $x$ as a function of redshift $z$ in the fiducial model (thick), traditional step function ionization (dashed) and delta-function perturbations (thin).
• Figure 2: Top: $E$-mode polarization power spectrum for: the fiducial model of Fig. \ref{['fig:xe']} (thick); the step function model (thin); the step function model with deviations transferred onto the fiducial model (dashed); instrumental noise $w_P^{-1/2}$ (denoted in $\mu$K-arcmin) that roughly brackets expectations from WMAP and Planck (long dashed). Bottom: the transfer function or fractional power spectrum response to a delta function perturbation of unit amplitude at $8\le z_i \le 25$.
• Figure 3: Eigenmodes. (a) 5 best (decreasing eigenvalue thick to thin) constrained eigenmodes or linear combinations of ionization history. (b) 5 worst constrained eigenmodes.
• Figure 4: Eigenmode statistics. Top curve: rms error $\sigma_\mu$ on mode amplitude; dashed line represents a physicality prior on $x$; only the first 5 modes contain interesting information. Bottom curve: optical depth per unit-amplitude mode $\tau_\mu$. Middle curve: rms error on total optical depth shown as the cumulative contribution from modes $\le \mu$; dashed line represents the physicality prior on $x$.
• Figure 5: Representation of an arbitrary ionization history with the first 5 eigenmodes. Inset: ionization history in the true model (thick dashed) compared with representation with 1 to 5 eigenmodes (solid, increasing thickness) away from the fiducial model (thin dashed). Main panel: resulting prediction for the power spectrum. With three or more modes the prediction is indistinguishable from the true model.