Preparatory Material for Mike Norman’s PiTP Lecture

This lecture concerns numerical methods for simulating cosmological reionization. At the heart of this are various methods for solving the radiative transfer equation for H/He ionizing radiation in 3D. I will begin with a brief overview of the observational constraints on cosmic reionization, summarizing the Annual Reviews article of Fan, Carilli and Keating (2006).

I will then review the physics of ionization fronts, including their classification, kinematics, and stability. A useful summary of the former is in Chapter 2 of Dan Whalen’s PhD thesis ; some publications on I-front instabilities are Whalen & Norman (2008a) andWhalen & Norman (2008b).

I then survey the various numerical methods that have been employed for simulating cosmological reionization. These can be broadly divided into methods that ignore the coupling between ionization and fluid dynamics (“post processing methods”), and those that include the coupling between ionization and fluid dynamics (“self-consistent methods”). Post processing methods are appropriate for evolving R-type ionization fronts through pre-computed density fields. These have been reviewed and compared in Iliev et al. (2006). Self-consistent methods solve the coupled of equations of radiative transfer, chemical ionization, and gas photoheating at various levels of approximation, and are appropriate for all types of I-fronts, especially D-type fronts that drive shock waves. These are reviewed and compared in Iliev et al. (2009).

Whether self-consistent or post-processing, the calculations require solution of the equation of radiative transfer in 3D. This is fundamentally a 6D + time first order hyperbolic equation for the specific intensity I. All methods in use today reduce the dimensionality of the problem through an astute choice of angle, frequency sampling techniques. Foremost among the former are ray tracing techniques Abel & Wandelt (2002); Monte Carlo methods Maselli et al. (2003); and Eddington factor methods Gnedin & Abel (2001), Paschos et al. (2007).

Finally, I will introduce a new method based on implicit flux-limited diffusion which promises to be more accurate in regimes where radiative transfer, ionization, and fluid dynamics are tightly coupled (i.e., in denser gas). The method is described in complete detail in Reynolds et al. (2009) and more concisely in Norman et al. (2009).

I will conclude with a brief discussion of the effects of the radiation spectrum on the properties of the I-front, and various methods to simulate the propagation of high energy photons ahead of the I-front.

== Coding Exercise

As an exercise, write a 1D code that advances an I-front in a static, uniform density medium by equating the number of ionizing photons absorbed in a cell with the number of ionizations in a neutral hydrogen gas (so-called photon conserving algorithm). The algorithm is described in Whalen & Norman (2006) for 1D spherical geometry, although you can do it in planar geometry. Verify that in the frame of the I-front, the flux of neutral atoms balances the flux of ionizing photons reaching the I-front (ignore relativistic effects).