The 21cm angular-power spectrum from the dark ages

TL;DR

The paper develops a comprehensive linear-theory treatment of the 21cm absorption signal from the dark ages, deriving the full angular-power spectrum $C_\\ell(z,z')$ via a Boltzmann equation in linear GR and including a wide set of sources beyond density and redshift-distortion. It shows that while monopole and redshift-distortion terms dominate, percent-level contributions arise from optical-depth perturbations, ionization fraction perturbations, and post-Newtonian velocity/potential terms that matter on large angular scales, with additional small-scale non-linear evolution and gravitational lensing. The authors provide analytic approximations for intuition (super- and sub-horizon regimes, Limber limit) and perform detailed numerical calculations across redshifts, discussing the impact of frequency-window width and cross-redshift correlations. They also explore polarization signatures and non-linear effects, concluding that although observational challenges are formidable, high-redshift 21cm data could offer rich cosmological information if modeling and measurements reach the necessary precision, with public code available to the community.

Abstract

At redshifts z >~ 30 neutral hydrogen gas absorbs CMB radiation at the 21cm spin-flip frequency. In principle this is observable and a high-precision probe of cosmology. We calculate the linear-theory angular power spectrum of this signal and cross-correlation between redshifts on scales much larger than the line width. In addition to the well known redshift-distortion and density perturbation sources a full linear analysis gives additional contributions to the power spectrum. On small scales there is a percent-level linear effect due to perturbations in the 21cm optical depth, and perturbed recombination modifies the gas temperature perturbation evolution (and hence spin temperature and 21cm power spectrum). On large scales there are several post-Newtonian and velocity effects; although negligible on small scales, these additional terms can be significant at l <~ 100 and can be non-zero even when there is no background signal. We also discuss the linear effect of reionization re-scattering, which damps the entire spectrum and gives a very small polarization signal on large scales. On small scales we also model the significant non-linear effects of evolution and gravitational lensing. We include full results for numerical calculation and also various approximate analytic results for the power spectrum and evolution of small scale perturbations.

Figures (12)

• Figure 1: Evolution of the interaction times for H-H, H-e, H-p and H-photon spin-coupling processes, and how this influences the spin temperature $T_s$ relative to the background CMB and gas temperatures. At high temperatures the H-H collision time is short and collisions couple $T_s$ to the gas temperature $T_g$; at lower redshifts the gas is diffuse and CMB photon interactions drive $T_s$ to the CMB temperature $T_\gamma$. This figure assumes purely linear evolution and no Lyman-alpha coupling; in reality non-linear effects are likely to change the result at $z\alt 30$.
• Figure 2: The background 21cm brightness $T_b$, optical depth $\tau_\epsilon$, and $[l(l+1)C_l/2\pi]^{1/2}$ at $l=10^4$ as a function of source redshift. The dashed line shows the result for $T_b$ neglecting the second term in Eq. \ref{['Ts_eq']} due to the effect of absorption on the ambient blackbody spectrum.
• Figure 3: Evolution of the fractional baryon, matter and spin temperature perturbations as a fraction of the CDM density perturbation. The left figure is for a $k=0.1 \text{Mpc}^{-1}$ mode, the right figure shows the effect of baryon pressure at $k=500 \text{Mpc}^{-1}$. The dotted lines show the equivalent results neglecting ionization fraction perturbations. In both cases the baryon perturbation is significantly less than the CDM perturbation at all relevant redshifts. There is no reionization.
• Figure 4: Transfer function for monopole source at redshift $z=50$ given unit initial curvature perturbation, compared to other relevant perturbations. The perturbations are numerically evaluated in the synchronous gauge. The Newtonian-gauge fluctuations equal those in the synchronous gauge well inside the horizon ($k \gg 10^{-3}\, \text{Mpc}^{-1}$). On large scales, the Newtonian-gauge $\Delta_b$ flattens out at $\approx 6/5$ times the primordial curvature perturbation, as shown by the dotted curve. The spin temperature perturbation is negative at $z=50$ (see Fig. \ref{['evolve']}).
• Figure 5: The effect of perturbations to the optical depth and ionization fraction on the 21cm power spectrum at $z=50$ with a sharp window function. The solid line shows our main result, the dashed-dotted line is the larger result using the a zeroth-order expansion in $\tau_\epsilon$, the dashed line is the lower result if the effect of ionization fraction perturbations on the gas temperature evolution is neglected. The fractional change in the spectrum is roughly the same on all scales where baryon pressure is negligible.