Lyman-alpha radiative transfer during the EoR

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Adaptive time integration

The integration time step for updating the photoionization and photoheating rates, tupdate, is adaptative. We use a typical tupdate smaller than the dynamical time step tdyn. Furthermore, recombination, collisional ionization and cooling are treated with an integration time step tcool which is much smaller than tupdate.
The ionizing continuum radiative transfer is the repetition of two steps. i) The emission and propagation of Nph photon packets during the time interval treg and then ii) the update of physical quantities.
• i) All the sources emit Nph photon packets along random directions. Each RT cell records the number of absorbed photons and the absorbed energy during this period.
• ii) The physical quantities of the gas are updated with the integration time step treg. This update is applied even to the RT cells that absorbed no photon during treg. This is necessary in order to consider the effect of collisional ionization, recombination, adiabatic expansion, etc.
However, this regular update is not appropriate where the flux of photons is too high, especially close to the sources. In this case, the number of photons absorbed during treg is so high that it can exceed the total number of atoms available in the RT cell. To avoid this excess, we update the physical quantities and optical depth of the RT cell whenever the number of absorbed photons reaches a pre-set limit, for example 30% of the total number of neutral atoms in the RT cell. We update this RT cell with tupdate(< treg) which is equal to the time elapsed since the last update.
The hierarchical scale of the different time steps are specified in Fig. 2.3

X-ray radiative transfer

LICORICE can deal with the ray tracing of soft X-ray continuum (E ≤ 2 k eV) as well as UV continuum. The main effect of X-rays on the gas is heating rather than ionization. We included only soft X-rays because X-ray heating is dominated by soft X-rays, as harder X-rays have a mean free path comparable to the Hubble scale (Pritchard & Furlanetto, 2007). Here are some differences of the X-ray radiative transfer compared to the UV continuum.

Lyman line transfer

The last part of LICORICE, which deals with the Ly-α line radiative transfer, shares is main features with the second part, the Monte Carlo approach and the adaptive grid. The methods for resonant line scattering were presented in Semelin et al. (2007).
The implementation is similar to those of other existing codes. (e.g. Zheng & Miralda- Escud´e (2002); Cantalupo et al. (2005); Dijkstra et al. (2006); Verhamme et al. (2006); Tasitsiomi (2006)). Necessary improvements have been implemented in our code.

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The optical depth

The neutral hydrogen atoms absorb Lyman series photons as well as the ionizing photons, but absorbed Lyman series photons are re-emitted immediately by the spontaneous emission, also considered as scattering. The current version of LICORICE has only the Ly-α line; upper Lyman series lines will be included in a future work. Similar to the ionizing photon case, a Ly-α photon traveling through the intergalactic medium has a probability P(τ) = 1−e− of being scattered after traveling through an optical depth τ from its emission point. The optical depth of Ly-α scattering can be expressed as; τ = Z l 0 Z +∞ −∞ ds duk nH0 p(uk) σ ν 1 − vmacro k + uk c .

Tree SPH algorithm for dynamics

The first part of LICORICE is the dynamical part. The gravity and fluid dynamics are followed using a TreeSPH method. More details on different aspects of the implementation are described in Semelin & Combes (2002) and Semelin & Combes (2005).

Table of contents :

1 Introduction 
1.1 The epoch of reionization
1.1.1 A brief thermal history of the Universe
1.2 Observational constraints on reionization
1.2.1 Gunn-Peterson Troughs : Quasar absorption spectra
1.2.2 CMB polarization and temperature anisotropy
1.2.3 Other probes of reionization
1.3 21 cm line and reionization
1.3.1 The HI 21-cm probe
1.3.2 Brief review of numerical simulations of the EoR
1.4 Observing the 21 cm line
1.4.1 Tomography
1.4.2 Global signal
1.4.3 Power spectrum
1.4.4 Foreground Contamination
2 Numerical code – LICORICE 
2.1 Tree SPH algorithm for dynamics
2.1.1 Tree algorithm for gravitation
2.1.2 The SPH algorithm for hydrodynamics
2.1.3 Additional physics
2.2 Monte Carlo on an adaptive grid – continuum
2.2.1 Adaptive grid
2.2.2 Ionizing radiation field
2.2.3 Absorption probability of photon packets
2.2.4 Updating physical quantities
2.2.5 Adaptive time integration
2.2.6 X-ray radiative transfer
2.3 Lyman line transfer
2.3.1 The optical depth
2.3.2 Hubble Expansion
2.3.3 Scattering off atoms
2.3.4 Propagation
2.3.5 Acceleration scheme
2.3.6 Further improvements for future
2.4 Radiative Transfer Comparison Test
2.4.1 Static density field cases
2.4.2 Radiative-hydrodynamics cases
2.5 Performance
3 The simulated 21 cm signal I 
3.1 Physics of the 21 cm signal
3.1.1 Basic equations
3.1.2 The spin temperature
3.2 Lyman-alpha radiative transfer during the EoR
3.3 The simulated 21 cm signal during the EoR
4 The simulated 21 cm signal II 
4.1 Source Model
4.1.1 Star Formation Rate
4.1.2 Luminosity and SED of the stellar sources
4.1.3 X-ray source model
4.2 Simulations
4.2.1 Initial condition
4.2.2 Global history of reionization
4.3 Helium reionization
4.4 QSO index
4.5 Luminosity of the QSO
4.6 The 21-cm signal
5 Galaxy formation with LICORICE 
5.1 Initial conditions
5.2 Cooling rate and collisional equilibrium
5.3 Snapshot of the simulation
5.4 Discussion and prospects
6 Conclusion 
A Cross-sections and rate coefficients


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