On the accuracy of common moment-based radiative transfer methods for
  simulating reionization
  Cosmological simulations of reionization often treat radiative transfer by
solving for the monopole and dipoles of the intensity field and by making
ansatz for the quadrupole moments to close the system of equations. We
investigate the accuracy of the most common closure methods, i.e. Eddington
tensor choices. We argue that these algorithms are most likely to err after
reionization and study quasi-analytic test problems mimicking these situations:
large-scale post-reionization ionizing background fluctuations and radiative
transfer in a predominantly ionized medium with discrete absorbers. We show
that OTVET and M1 over-ionize self-shielding absorbers when fixing the
background photoionization rate, leading to 30-40% higher emissivity to balance
the increased recombination. This over-ionization results in a simulation run
with these algorithms having a factor of ~2 lower average metagalactic
photoionization rate relative to truth given an ionizing emissivity.
Furthermore, these algorithms are unlikely to reproduce ionizing background
fluctuations on scales below the photon mean path: OTVET tends to overpredict
the fluctuations there when the simulation box is smaller than twice the mean
free path and underpredict otherwise, while M1 drastically underpredicts these
fluctuations. As a result, these numerical methods are likely not sufficiently
accurate to interpret the Ly$\alpha$ forest opacity fluctuations observed after
reionzation. We show that a high number of angular directions need to be
followed to capture the post-reionization ionizing background fluctuations
accurately with ray-tracing codes. Lastly, we argue that the strong dependence
of the post-reionization ionizing background on the value of the reduced speed
of light found in many simulations signals that the ionizing photon mean free
path is several times larger in such simulations than the observationally
measured value.
