Semi-analytical approach to Ly$α$ multiple-scattering in 21-cm signal simulations

TL;DR

This work provides a fast, semi-analytical treatment of Lyα multiple scattering in cosmic dawn 21-cm signal simulations. By deriving an analytic MS window function from beta-distributed radial distance distributions—characterized by a single parameter $x_{ m em}$—the authors embed MS effects into 21cmFAST via SPαRTA, a gridless Monte Carlo tool that includes peculiar velocities, finite temperature, anisotropic scattering, and recoil. They demonstrate that MS enhances Lyα flux fluctuations and can boost the 21-cm power spectrum by about 50% at high redshift before strong Lyα coupling, but has negligible impact on Lyα heating and overall heating in realistic X-ray scenarios. The framework connects MS physics to observable 21-cm statistics, offering a practically efficient avenue for parameter studies during cosmic dawn.

Abstract

A crucial physical quantity in determining the 21-cm signal during cosmic dawn is the inhomogeneous background of Ly$α$ photons originating from the first galaxies. As these photons travel through the intergalactic medium, their scattering cross-section is often approximated as a delta function at resonance due to computational cost. That is, photons with emitted wavelengths between Ly$α$ and Ly$β$ are assumed to travel in straight lines until they redshift into the Ly$α$ resonance. However, due to the damping wing in the Ly$α$ cross-section, this approximation fails as the frequency of the photon approaches the resonant frequency, resulting in multiple scatterings events that could be separated by non-negligible distances. Some previous works studied this effect of Ly$α$ multiple scattering by running computationally heavy radiative-transfer simulations. However, robustly interpreting the cosmic 21cm signal requires exploring a large parameter space of astrophysical uncertainties, motivating more computationally-efficient approaches. Here we incorporate Ly$α$ multiple scatterings in the public, semi-numerical simulation 21cmFAST. We employ Monte Carlo simulations to study the trajectories of Ly$α$ photons on different scales. We find that the distance distributions of Ly$α$ photons with respect to the absorption point can be modeled as analytical functions that are governed by a single parameter. Upon implementing the distance distributions in 21cmFAST, we find that the multiple scattering effect is important (about 50% difference in the 21-cm power spectrum) only at high redshifts before the spin temperature is fully coupled to the kinetic temperature. Furthermore, we find that Ly$α$ multiple scattering does not enhance Ly$α$ heating, and that the combined effect is negligible, especially under realistic X-ray heating scenarios.

Figures (10)

• Figure 3: Radial distributions $f_\mathrm{MS}\left(y;z_\mathrm{abs}\right)$ of the normalized distance variable, $y\equiv r/R_\mathrm{SL}$, for different values of $x_\mathrm{em}\equiv R_\mathrm{SL}\left(z_\mathrm{abs},z_\mathrm{em}\right)/R_*\left(z_\mathrm{abs}\right)$ (for better interpretation of that quantity, we report also the corresponding $\Delta z = z_\mathrm{em}-z_\mathrm{abs}$). All curves shown here were obtained by fitting the numerical distributions from SP$\alpha$RTA to a beta function. Solid (dashed) curves correspond to having turned on (off) all the physical features listed in Sec. \ref{['sec: SPaRTA']}. All distributions correspond to $z_\mathrm{abs}=10$, $T_k=10^4\,\mathrm{K}$ and $x_\mathrm{HI}=1$.
• Figure 4: Left panel: distribution of the relative peculiar velocity between the absorber at $z_\mathrm{abs}=10$ and the emitter (whose redshift is implied by the $x_\mathrm{em}$ value), as extracted from the SP$\alpha$RTA simulation. All curves are cut at the highest and lowest values obtained in the simulation. We also show, by the dashed black curve, the Gaussian distribution that corresponds to $|\mathbf v|/c$ at $z=0$ (computed in Newtonian gauge), smoothed on a scale of $10\,\mathrm{Mpc}$ (this curve is much wider than all other curves since the 3D velocity field can get much higher values compared to the 1D relative velocity field). Right panel: the apparent Voigt cross-section $\sigma_\alpha\left(\nu,T_k\right)$ (Eq. \ref{['eq: 14']}) as modified by relative peculiar velocities due to Doppler shift. In all curves (except the black curve) $T_k=10^4\,\mathrm{K}$. The black curve corresponds to $T_k=0$ and zero relative peculiar velocity. The vertical dotted line corresponds to the mean initial frequency in the SP$\alpha$RTA simulation (c.f. right panel of Fig. \ref{['fig: 2']}).
• Figure 5: Similarly as Fig. \ref{['fig: 3']}, but here solid curves correspond to accounting for correlations in the peculiar velocity field, while the dashed curves correspond to zeroing the Pearson correlation coefficient $\rho$ (see exact definition at Appendix \ref{['sec: Conditional Gaussian random variables']}), meaning that all velocity samples in the simulation are completely uncorrelated.
• Figure 6: Similarly as Fig. \ref{['fig: 3']}, but here solid, dashed and dotted curves correspond to $z_\mathrm{abs}=10$, $z_\mathrm{abs}=20$ and $z_\mathrm{abs}=30$, respectively. Some high $x_\mathrm{em}$ values are not presented for some of the curves, as these $x_\mathrm{em}$ values correspond to scales where the apparent frequency of the photon is higher than Ly$\beta$ (since $x_\mathrm{em}\equiv R_\mathrm{SL}/R_*\left(z_\mathrm{abs}\right)$ and $R_*\left(z_\mathrm{abs}\right)\propto1+z_\mathrm{abs}$, $R_\mathrm{SL}$ ought to be increased if $z_\mathrm{abs}$ is increased while $x_\mathrm{em}$ is fixed, and this consequently increases the frequency of the photon at $z_\mathrm{em}$).
• Figure 7: The MS window function, as given by Eq. \ref{['eq: 32']} and the fit of Eqs. \ref{['eq: 28']}-\ref{['eq: 30']}, for different $x_\mathrm{em}$ values. For comparison, we show the SL window function, Eq. \ref{['eq: 13']}, by the black dashed curve. For better visualization, the inset shows how the window functions behave at high $kR$ values.