SCORCH. III. Analytical Models of Reionization with Varying Clumping
  Factors
  In the Simulations and Constructions of the Reionization of Cosmic Hydrogen
(SCORCH) project, we compare analytical models of the hydrogen ionization
fraction with radiation-hydrodynamic simulations. We derive analytical models
of the mass-weighted hydrogen ionization fraction from the local ionization
balance equations as a more accurate alternative to the widely adopted model
based on the volume filling factor. In particular, our model has a
recombination term quadratic in the ionization fraction, which is consistent
with the two-body interaction nature of recombination. Then, we use the
radiation-hydrodynamic simulations to study the clumping factors needed to
solve the analytical equations, and provide accurate fitting functions. We find
that the ionized hydrogen clumping factors from our radiative transfer
simulations are significantly different than those from other simulations that
use a uniform photoionization background. In addition to redshift dependence,
we also see the dependence of ionized hydrogen clumping factor on ionization
fraction, and we incorporate this into our fits. We calculate the reionization
histories using our analytical models and clumping factors and compare with
widely adopted models, and all of our models achieve $<7\%$ difference from
simulation results while the other models have $>20\%$ deviations. The Thomson
optical depths from reionization calculated from our analytical models result
in $<5\%$ deviation from simulations, while the previous analytical models have
$>20\%$ difference in and could result in biased conclusions of the IGM
reionization.
