The effect of a short mean free path on HII regions and 21-cm tomography during reionization

TL;DR

The authors address how a short mean free path (MFP) of ionizing photons alters the topology of reionization and the prospects for 21-cm tomography. They formulate an analytic excursion-set model with the MFP as a free parameter, deriving a MFP-inclusive ionization barrier and computing the first-crossing distribution to obtain the HII-region size distribution and, crucially, the one-point PDF of the ionization field and the 21-cm signal. They show that a shorter MFP dramatically reduces bubble sizes toward the end of reionization and yields a 21-cm contrast of order $O(1\,\mathrm{mK})$ at relevant resolutions, much smaller than naive estimates. By comparing to noise expectations for HERA- and SKA-like arrays, they find SKA has potential for direct imaging while HERA-like arrays face substantial challenges, highlighting the need for more detailed sensitivity studies in short-MFP scenarios. Overall, the work provides a transparent, efficient framework to explore MFP effects on reionization observables and informs interpretation and planning for upcoming 21-cm tomography experiments.

Abstract

Recent measurements of the mean free path (MFP) of ionizing photons at $z=6$ find that it is significantly shorter than extrapolations from lower $z$. This has a substantial impact on the topology of reionization and thus the prospects of tomography of the 21-cm signal from upcoming radio interferometers. In this work we develop the first analytic model of reionization which explicitly incorporates the MFP as a free parameter, allowing us to transparently explore its effect on the process. Our model is based on the excursion set formalism with an ionization condition which accounts for absorptions parameterized through the MFP. With the goal of observational comparison, we include additional modifications which make our model particularly suitable for predicting one-point statistics of the ionization field (and 21-cm signal), which are among the fundamental quantities for tomography. We find that the effect of the MFP is much more significant during the later stages of reionization, and that including a shorter MFP reduces the size of HII regions by around an order of magnitude towards the end of reionization compared with analytic models which do not account for the MFP. We find that the reported MFP value produces a contrast in the 21-cm signal of $\mathcal{O}$(1 mK) or less at resolutions $θ\sim $ 15--35 arcmin, an order of magnitude below naive estimates and up to a factor of several smaller than when using a larger MFP value extrapolated from low $z$, requiring significantly more sensitivity for imaging. We compare the contrast to noise estimates for arrays similar in size to HERA and SKA-Low and find that SKA has sufficient sensitivity for direct imaging (at the largest scales considered), while the predicted signal will be challenging for arrays similar in size to HERA. Our model indicates that more detailed sensitivity estimates are warranted in the context of a short MFP.

Figures (12)

• Figure 1: A short MFP limits the sizes of fully ionized regions by enforcing larger density requirements for a fully ionized region at larger radii. Lines show the barrier used in the excursion set formalism for the case where no MFP is included (red line) and for the ionization criterion which includes the MFP (black lines). The thin dark line assumes a constant ionizing efficiency with no mass dependence ($\xi=0$), while the two thicker lines assume a power law on the mass-dependence of $\xi = 2/3$ which reflects energy-regulated stellar feedback (see section \ref{['ss:mass-depend']}). The lighter of these two lines assumes a smooth star formation rate across the entire halo mass range, while the darker line assumes bursty star formation. All cases use $\lambda = 5$ cMpc and are normalized to have the same global neutral fraction of $x_{\rm HI} = 0.1$ at $z=6$.
• Figure 2: A smaller MFP value results in a significant change in the ionization field. Each panel shows a slice from an ionization box output by 21cmFASTv1 using the ionization criterion which accounts for the MFP, where each box is 512 cMpc on a side and 1 cMpc thick. Columns show different redshifts ($z = 8,7,6$) at fixed global neutral fractions ($\left< x_{\rm HI} \right> = 0.9, 0.5, 0.1$) and rows show different values of the MFP ($\lambda = 2,10,20$ cMpc), corresponding to the limits measured by B21 (2 and 10 cMpc) and the value used in many of the results in DF22 (20 cMpc), before the shorter MFP value was reported. The colorbar shows the ionized fraction of each pixel, where white pixels are fully ionized. For comparison, green circles are shown with radii $50,25$, and 15 cMpc, corresponding to smoothing scales used throughout this work. Slices of the ionization field clearly show several trends. Firstly, at the lowest global neutral fraction, smaller MFP values result in a striking difference in the ionization field, with neutral islands appearing smaller and more numerous and with less extreme neutral fractions, resulting in a more homogeneous ionization field. Secondly, the MFP has a much more significant effect later in reionization; panels at $z=8$ ($\left< x_{\rm HI} \right> = 0.9$) show very little difference between MFP values, while differences are visually clear by $\left< x_{\rm HI} \right> = 0.5$ and particularly by $\left< x_{\rm HI} \right> = 0.1$. In all cases the ionizing efficiency is assumed to be constant with mass ($\xi=0$; see section \ref{['ss:mass-depend']}).
• Figure 3: A shorter MFP may delay the onset of the percolation threshold during reionization. Each panel shows a hemisphere from the ionization boxes shown in figure \ref{['fig:21cmFASTslices']}. Each sphere has a radius of 25 cMpc and corresponds to the middle circle of each panel in figure \ref{['fig:21cmFASTslices']}, where the slice of each sphere is on the same plane as the slice in figure \ref{['fig:21cmFASTslices']}. Voxels outlined in green are those on the slice which are not fully ionized (i.e., those visible in figure \ref{['fig:21cmFASTslices']}). The colorbar shows the ionized fraction of each voxel, where fully ionized voxels are transparent. The same general trends are apparent as in the 2D slices, but the 3D representations reveal more nuance; for example, while neutral regions appear smaller for smaller MFP values in a 2D slice, it is clear from the 3D images that these smaller neutral regions are in fact connected by narrow channels, whereas the neutral regions for larger MFP values seem to be true "islands."
• Figure 4: The sizes of ionized bubbles are much more sensitive to the MFP during the later stages of reionization. Each panel shows the size distributions $\bar{Q}^{-1} V dn/d\ln R$ of fully ionized bubbles at different redshifts and volume-filling fractions $\bar{Q}$ at various choices for the MFP (colors). All cases use $\xi = 0$. Also shown are the standard photon-counting results using the method outlined in FZH04 (black dotted lines). As $\lambda~\rightarrow~\infty$, the result of our model approaches that of FZH04, as expected. Slight deviations occur because the solution of FZH04 approximates the barrier as linear, whereas our solution does not. As the global ionized fraction increases towards the end of reionization, the differences in the size distributions for different MFP lengths become much more pronounced, indicating that the MFP has an increasing effect on the topology as reionization progresses.
• Figure 5: Fraction of fully ionized regions as a function of scale. Results are shown for a variety of MFP values (colors). All cases are at $z=6$ and $\left< x_{\rm HI} \right> = 0.1$, and a constant ionizing efficiency ($\xi = 0$). Results are shown using our model (solid lines), $\texttt{21cmFASTv1}$ with the same ionization criterion (dotted lines), as well as a Monte Carlo version of our model, which accounts for a finite number of subregions within each larger region (dashed lines; see section \ref{['s:one-point-PDF-description']} for more details on the use of subregions and appendix \ref{['a:21cmFAST-differences']} for details on the Monte Carlo version of the model). While this fraction should be monotonic, our model (both the analytic and Monte Carlo version) produces a turnover at small radii. This is because of the reduced amount of scatter in the mapping from density to ionized fraction on the smoothing scale (see appendix \ref{['a:21cmFAST-differences']} for more detail). In the analytic model there is no scatter, and the turnover is most severe. The Monte Carlo model includes some sources of scatter, resulting in a less severe turnover at smaller radii. In all cases, good agreement is shown at large scales ($R \gtrsim 15$ cMpc), where the number of subregions grows large and the scatter in the ionized fractions of the regions as a function of their density grows small, which are the scales relevant to this work.