Photon number conservation and the large-scale 21 cm power spectrum in
  semi-numerical models of reionization
  Semi-numerical models of the reionization of neutral hydrogen (HI) based on
the excursion set (ES) approach are known to violate photon number conservation
at the few per cent level. In this work, we highlight a more severe, previously
unrecognized shortcoming of ES models: the large-scale 21 cm power spectrum
(equivalently, HI bias b_HI) is a relatively strong function of the spatial
resolution used to generate ES ionization maps. We trace this problem to the
fact that photon non-conservation in these models arises from a
resolution-dependent mixture of spatially resolved, photon non-conserving
bubbles, and partially ionized grid cells which are perfectly photon-conserving
by construction. We argue that this inevitably leads to a resolution-dependence
of b_HI, with the correct, converged value only emerging at very coarse
resolution. Quantitatively, we find that b_HI can be non-converged by as much
as ~20-25% in conservative ES implementations with grid sizes Delta x=5-10
cMpc/h, even when photon non-conservation is as small as ~3-4%. Thus, although
numerically efficient, ES ionization maps coarse enough to produce a converged
HI bias would wash out all topological features of the ionization field at
scales k>~0.05 h/cMpc. We therefore present a new, explicitly photon conserving
(PC) semi-numerical algorithm which distributes photons isotropically around
sources while also accounting for anisotropic overlaps between nearby bubbles.
Our PC algorithm predicts a resolution-independent value of b_HI consistent
with the result of low-resolution ES maps, thus serving as a useful compromise
between standard ES implementations and more expensive radiative transfer
simulations.
