Photon (Non)Conservation in the Reduced Speed of Light Approximation and How to (Almost) Fix It

TL;DR

This work analyzes how the Reduced Speed of Light (RSL) approximation in radiative transfer can violate photon conservation, especially in optically thin regimes, and develops a method to count the missing photons for certain numerical schemes. It then proposes a background-photon framework to restore these missing photons on-the-fly, including a 4-equation background model and a simpler 1-equation diffusion-like scheme, with the 4-equation approach performing better in semi-realistic tests. Across idealized and semi-realistic reionization simulations, counting missing photons yields sub-percent conservation, but placing them correctly is challenging, particularly in voids during overlap, and full restoration only partially mitigates discrepancies with full-speed-of-light runs. The results suggest photon non-conservation as a plausible contributor to differences between major reionization simulations and highlight the need for larger, on-the-fly, full-speed-of-light treatments or improved background-based corrections, requiring community-wide effort.

Abstract

The "Reduced Speed of Light" (RSL) approximation is commonly used to speed up radiative transfer calculations in cosmological simulations. However, it has been shown previously that the RSL approximation leads to photon non-conservation in some regimes. I show that these missing photons can be counted exactly for some numerical schemes. Adding them back into a simulation, however, is a much harder task. I show one example of such a scheme, which achieves sub-percent accuracy on simple tests. Unfortunately, the scheme performs much worse on semi-realistic simulations of cosmic reionization, leading to a faster overlap and significant errors in the point-to-point comparison of the RSL radiation field with the reference simulation that maintains the full speed of light for the radiative transfer.

Figures (7)

• Figure 1: The number of ionizing photons per baryon (the main panels) and its ratio to the input value $N_{\rm TRUE} = t/(1\,{\rm Gyr})$ (the basement panels) as a function of time for several numerical schemes. Dotted lines show the true input value, while solid lines show the values counted in the simulation: the sum of all ionizations $\langle x_{\rm HII}\rangle$ and the number of ionizing photons in the radiation field, $N_{\rm HI} = \int N_\nu d\nu$. Ideally, the two are the same. The left panel shows the original schemes from equation (\ref{['eq:num']}), and the large deviation of the ratio from unity at $t>1\,{\rm Gyr}$ is the manifestation of photon non-conservation from Equations (\ref{['eq:rslthin']}) and (\ref{['eq:gammabar']}). The right panel also accounts for the missing photon contribution from Equation (\ref{['eq:lambda']}). Note that the range of the y-axis in the basement in the right panel is 50 times smaller than in the left panel.
• Figure 2: The same as Figure \ref{['fig:nobg']}, but now with the missing photons included in the background radiation field (left) and several tests with the cosmological expansion included (right).
• Figure 3: Point-wise comparison between the full speed of light solution (left panel) and the $\hat{c}=0.1c$ solution (middle panel) for the explicit M1-closure scheme. The rightmost panel shows the relative difference between the two. The comparison is at $t=500$ Myr, when the errors are approximately the largest.
• Figure 4: Point-to-point error distributions as functions of the "true", full speed of light radiation field for several representative tests, as listed in the legend. Solid lines show the median errors while dark and light shading spans 25%-75% and 5%-95% percentile ranges, respectively.
• Figure 5: Stellar mass density (left) and hydrogen ionization fraction (right) for two semi-realistic CROC-like simulations of cosmic reionization, as labeled in the legend. The black points with error bars are the data from Fan2006 and are shown merely for illustration - the semi-realistic simulations in this figure are not meant to be practical models of reionization.