21 cm Signal from the Thermal Evolution of Lyman-$α$ during Cosmic Dawn

TL;DR

This work analyzes the thermal interplay between Lyman-$\alpha$ photons and neutral hydrogen during cosmic dawn by solving a coupled system that also includes X-ray heating, tracked over a 500 Myr span from $z\approx25$ to $z\approx8$. The authors derive and solve the evolution equations for the photon occupation number $J(x,t)$ and the spin/kinetic temperatures, highlighting the Wouthuysen-Field coupling and energy exchange between photons and gas. They show that at large Lyman-$\alpha$ photon densities, the gas temperature reaches a quasi-equilibrium largely set by the ratio of injected to continuum Lyman-$\alpha$ photons, effectively fixing the 21 cm signal's behavior during cosmic dawn and the crossover redshift $z_c$. When X-ray heating is included, the coupled dynamics implies that the global 21 cm signal can constrain the injected-to-continuum Lyman-$\alpha$ photon ratio, offering a path to interpret current and upcoming global signal measurements in the CD/EoR era.

Abstract

The Lyman-$α$ photons couple the spin temperature of neutral hydrogen (HI) to the kinetic temperature during the era of cosmic dawn. During this process, they also exchange energy with the medium, heating and cooling the HI. In addition, we expect X-ray photons to heat the mostly neutral gas during this era. We solve this coupled system (Lyman-$α$-HI system along with X-ray heating) for a period of 500 Myr (redshift range $8 <z < 25$). Our main results are: (a) Without X-ray heating, the temperature of the gas reaches an equilibrium which is nearly independent of photon intensity and only weakly dependent on the expansion of the universe. The main determinant of the quasi-static temperature is the ratio of injected and continuum Lyman-$α$ photons. (b) While X-ray photons provide an additional source of heating at initial times, for large enough Lyman-$α$ photon intensity, the system tends to reach the same quasi-static temperature as expected without additional heating. This limit is reached when the density of photons close to the Lyman-$α$ resonance far exceeds the HI number density. (c) We compute the global HI signal for these scenarios. In the limit of the large density of Lyman-$α$ photons, the spin temperature of the hyperfine line is fixed. This freezes the global HI signal from the era of cosmic dawn and the cross-over redshift from absorption to emission. This feature depends only on the ratio of injected to continuum Lyman-$α$ photons, and the global HI signal can help us determine this ratio.

Figures (5)

• Figure 1: We show the evolution of intensity profiles for Lyman-$\alpha$ continuum + 20% injected photons for 100 Myr. The left (right) panels correspond to $C=1\times 10^{-16}$$\rm cm^{-3} \, s^{-1} \, sr^{-1}$ ($C=1\times 10^{-14}$$\rm cm^{-3} \, s^{-1} \, sr^{-1}$). The following initial conditions are used: $z = 20$, $T = 10 \, \rm K$, fully neutral H$\,$i gas; the grid on the x-axis corresponds to the variable: $x_0=x(T=10\, \rm K)$. The evolution of the profile close to $x=0$ is shown in the inset. The bottom panels assume local approximation. In this approximation, the expansion rate and the number density of H$\,$i atoms are held fixed to their initial value for solving Eqs. (\ref{['eq:ts_def']}), (\ref{['eq:fineq2']}), (\ref{['eq:thermal']}), and (\ref{['eq:ene_exchange1']}). If this evolution is switched off, a quasi-static equilibrium is reached after nearly 50 Myr. The top panels, the realistic case in which all relevant quantities are allowed to evolve, show slow change of the photon profile even at late times.
• Figure 2: In the left panel, we show the thermal evolution of the gas for the photon injection model (Lyman-$\alpha$ continuum + 20% injected photons) and initial conditions as in Figure \ref{['fig:long_noz']} for 100 Myr with different photon injection rates, $C$. The solid curves include all the relevant physics, while for the dashed curves correspond to the aforementioned local approximation (Figure \ref{['fig:long_noz']}). In local approximation, a quasi-static thermal equilibrium is reached, while a slow evolution of the gas temperature is seen in the other case. For a very small value of $C$, the temperature initially decreases due to adiabatic cooling before the Lyman-$\alpha$ intensity builds up. In the right panel, we show the evolution of the coupling coefficient $y_\alpha$ for the same cases. $y_\alpha$ is proportional to $J(\nu_\alpha)$ and inversely proportional to $T_\alpha$ (Eq. (\ref{['eq:yalpha']})). Initially, $y_\alpha$ increases rapidly as $J(\nu_\alpha)$ (or equivalently $J(x=0)$) increases rapidly (Figure \ref{['fig:long_noz']}). If the temperature also increases sharply (left panel), $y_\alpha$ flattens. Over longer times, after the tilted profile is reached, $J(0)$ is nearly constant, but the temperature decreases slightly, leading to a small increase in $y_\alpha$. In local approximation, $y_\alpha$ reaches a constant as both $J(0)$ and the temperature reach steady state values (left panel and Figure \ref{['fig:long_noz']}).
• Figure 3: We show the temperature evolution of the gas including the impact of X-ray heating. The solid lines (dashed lines) correspond to cases when the thermal effect of Lyman-$\alpha$ photons is included (excluded). The model considered is the same as in Figure \ref{['fig:long_noz']}: continuum + 20% injected Lyman-$\alpha$ photons with initial gas temperature, $T = 10 \, \rm K$ at $z = 20$. In each panel, the excess heating is varied over four orders of magnitude, while the Lyman-$\alpha$ injection rates are: $C = 10^{-12}$$\rm cm^{-3} \, s^{-1} \, sr^{-1}$ (top left), $10^{-14}$$\rm cm^{-3} \, s^{-1} \, sr^{-1}$ (top right), $10^{-16}$$\rm cm^{-3} \, s^{-1} \, sr^{-1}$ (bottom left), and $10^{-18}$$\rm cm^{-3} \, s^{-1} \, sr^{-1}$ (bottom right). At initial times, both X-ray and Lyman-$\alpha$ photons act as heating sources. However, the net impact of Lyman-$\alpha$ photons is to drive the system toward an equilibrium temperature, in which the gas and Lyman-$\alpha$ colour temperature approach each other. This causes a fraction of the additional heat to be shared with Lyman-$\alpha$ photons, thereby preventing the X-ray photons from heating the medium. For most models shown in the four panels, this behaviour is seen, with the equilibrium reaching faster for larger $C$. Only for $C = 10^{-18}$ the impact of Lyman-$\alpha$ photons is negligible.
• Figure 4: We show the temperature evolution of the gas (left panels) and the global 21 cm signal (right panels) for the models parametrized by $N_{\rm heat}$ and $f_L$ (see the text for details). The Lyman-$\alpha$ model corresponds to continuum + 17% injected photons. In each panel, we fix $N_{\rm heat}$ and vary $f_L$ by several orders of magnitude: $N_{\rm heat} = 10$ (top panel), 1 (middle panel), and 0.1 (bottom panel). Left Panels: Thermal history is displayed with (solid curves) and without (dashed curve) the thermal effects of Lyman-$\alpha$ photons. These results are in agreement with thermal histories shown in Figure \ref{['fig:heat']}, even though the system is evolved for 500 Myr here. Right Panels: The global H$\,$i signal is displayed (Eq. (\ref{['overallnorm']})). There are two curves (solid and dashed) for each model (fixed $f_L$ and $N_{\rm heat}$). For dashed curves, Wouthuysen-Field coupling is included (Eq. (\ref{['eq:ts_def']}) and the right panel of Figure \ref{['fig:long_ns']}) but not the thermal impact of Lyman-$\alpha$ photons. Solid curves include all the relevant physics of Lyman-$\alpha$ photons. For small values of $f_L$ (small number density of Lyman-$\alpha$ photons), there is negligible difference between solid and dashed curves. For higher values of $f_L$, the inclusion of Lyman-$\alpha$ thermal effect causes the gas to heat faster initially. However, Lyman-$\alpha$ photons prevent the gas temperature from rising above the equilibrium temperature for the corresponding Lyman-$\alpha$ model (continuum + 17% injected photons), which is close to 40 K.
• Figure 5: The same as Figure \ref{['fig:g21']} but for Lyman-$\alpha$ model, continuum plus 10% injected photons. The equilibrium temperature in this case is $T \simeq 70 \, \rm K$, which gives the redshift of crossover of the global H$\,$i signal from absorption to emission, $z_c \simeq 17.5$.