Accurate simulations of reionization using the reduced speed of light approximation

The reduced speed of light approximation has been employed to speed up radiative transfer simulations of reionization by a factor of $\gtrsim 5-10$. However, it has been shown to cause significant errors in the HI-ionizing background near reionization's end in simulations of representative cosmological volumes. This can bias inferences on the galaxy ionizing emissivity required to match observables, such as the Ly$\alpha$ forest. In this work, we show that using a reduced speed of light is, to a good approximation, equivalent to re-scaling the global ionizing emissivity in a redshift-dependent way. We derive this re-scaling and show that it can be used to ``correct'' the emissivity in reduced speed of light simulations. This approach of re-scaling the emissivity after the simulation has been run is useful in contexts where the emissivity is a free parameter. We test our method by running full speed of light simulations using these re-scaled emissivities and comparing them with their reduced speed of light counterparts. We find that for reduced speeds of light $\tilde{c} \geq 0.2$, the 21 cm power spectrum at $0.1 \leq k /[h{\rm Mpc}^{-1}] \leq 0.2$ and key Ly$\alpha$ forest observables agree to within $20\%$ throughout reionization, and often better than $10\%$. Position-dependent time-delay effects cause inaccuracies in reionization's morphology on large scales that produce errors up to a factor of $2$ for $\tilde{c} \leq 0.1$. Our method enables a factor of $5$ speedup of radiative transfer simulations of reionization in situations where the emissivity can be treated as a free parameter.