Evolution of the 21 cm signal throughout cosmic history

The potential use of the redshifted 21 cm line from neutral hydrogen for probing the epoch of reionization is motivating the construction of several low-frequency interferometers. There is also much interest in the possibility of constraining the initial conditions from inflation and the nature of the dark matter and dark energy by probing the power spectrum of density perturbations in three dimensions and on smaller scales than probed by the microwave background anisotropies. Theoretical understanding of the 21 cm signal has been fragmented into different regimes of physical interest. In this paper, we make the first attempt to describe the full redshift evolution of the 21 cm signal between $0<z<300$. We include contributions to the 21 cm signal from fluctuations in the gas density, temperature, and neutral fraction, as well as the $\mathrm{Ly}\alpha$ flux, and allow for a post-reionization signal from damped $\mathrm{Ly}\alpha$ systems. Our comprehensive analysis provides a useful foundation for optimizing the design of future arrays whose goal is to separate the particle physics from the astrophysics, either by probing the peculiar velocity distortion of the 21 cm power spectrum, or by extending the 21 cm horizon to $z\gtrsim 25$ before the first galaxies had formed, or to $z\lesssim 6$ when the residual pockets of hydrogen trace large-scale structure.

Figure 1

Top panel: Evolution of the CMB temperature $T_{\mathrm{CMB}}$ (dotted curve), the gas kinetic temperature $T_K$ (dashed curve), and the spin temperature $T_S$ (solid curve). Middle panel: Evolution of the gas fraction in ionized regions $x_i$ (solid curve) and the ionized fraction outside these regions (due to diffuse x-rays) $x_e$ (dotted curve). Bottom panel: Evolution of mean 21 cm brightness temperature $T_b$. In each panel we plot curves for Model A (thin curves), Model B (medium curves), and Model C (thick curves).

Figure 2

Redshift evolution of the angle-averaged 21 cm power spectrum ${\overline{\Delta }}_{T_b}$ for Model A at $k=0.01$ (solid curve), 0.1 (dotted curve), 1.0 (short-dashed curve), and 10.0 (long-dashed curve) ${\mathrm{Mpc}}^{-1}$. Reionization at $z=6.5$.

Figure 3

Redshift evolution of the angle-averaged 21 cm power spectrum ${\overline{\Delta }}_{T_b}$ for Model B. Reionization at $z=9.8$. Same line conventions as Fig. 2.

Figure 4

Redshift evolution of the angle-averaged 21 cm power spectrum ${\overline{\Delta }}_{T_b}$ for Model C. Reionization at $z=11.8$. Same line conventions as Fig. 2.

Figure 5

Angular decomposition of 21 cm power spectrum for model A at wave number $k=0.1{\mathrm{Mpc}}^{-1}$. Top panel: ${\Delta }_{{\mu }^0}(k)$ (solid curve). We also show the separate contribution from the $bb$ (dotted curve), $\alpha \alpha$ (long dashed curve), TT (short-dashed curve), and XX (dotted-dashed curve) terms. Note that there are additional cross terms, many of which contribute to the power spectrum with negative sign, not shown here. Middle panel: ${\Delta }_{{\mu }^2}(k)$ (solid curve). We also plot the contribution from the bV (dotted curve), $\alpha \mathrm{V}$ (long-dashed curve), TV (short-dashed curve), and XV (dotted-dashed curve) terms, indicating the sign of the contribution to the power spectrum as positive (thick curves) or negative (thin curves). Bottom panel: ${\Delta }_{{\mu }^4}(k)$ (solid curve).

Figure 6

Evolution of $z_{\mathrm{trans},\alpha }$ with $f_{\alpha }$ (thin curves) and $f_X$ (thick curves) for $k=0.01$ (dotted curves), 0.1 (short dashed curve), and $1{\mathrm{Mpc}}^{-1}$ (long-dashed curve). Also plotted is the redshift at which $x_{\alpha }=x_c$ (solid curves).

Figure 7

Evolution of $z_{\mathrm{trans},T}$ with the same line conventions as Fig. 6. Also plotted is the redshift at which $T_K=T_{\mathrm{CMB}}$ (solid curves).

Figure 8

Redshift evolution of angular scales. $\Theta =(2\pi /k)/d_A(z)$, which we map to angular moments using $l\approx \pi /\Theta$. This is meant to be an approximate guide, rather than an exact conversion.

Figure 9

Errors on the angle-averaged power spectrum ${\overline{\Delta }}_{T_b}$ (solid curves) for our optimized experiments MWA (dotted curve), SKA (short dashed curve), and LA (long-dashed curve), as functions of wavenumber $k$ at redshifts $z=7.9$ (top panel), 15.7 (middle panel), and 30.2 (bottom panel).

Figure 10

Signal-to-noise ratio as a function of redshift for our optimized experiments. Curves indicate $k=0.1$ (dotted curve), 1.0 (short-dashed curve), and $10{\mathrm{Mpc}}^{-1}$ (long-dashed curve). Top panel: MWA. Middle panel: SKA. Bottom panel: LA.

Figure 11

Experimental sensitivity to the separation of powers. Plotted is the redshift evolution of the power spectrum (thick solid curve) at $k=0.1{\mathrm{Mpc}}^{-1}$ and the corresponding sensitivity for MWA (thin-solid curve), SKA (thin-dotted curve), and LA (thin-dashed curve). We also plot sensitivity for LA with its observing time split between 16 separate fields (thin dotted-dashed curve). Top panel: ${\Delta }_{{\mu }^0}$. Middle panel: ${\Delta }_{{\mu }^2}$. Bottom panel: ${\Delta }_{{\mu }^4}$. Calculations are for Model A.