Physics of Ly\(\alpha \) Radiative Transfer
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Abstract
The Ly\(\alpha \) transfer problem is an exciting problem to learn about and work on. Ly\(\alpha \) transfer is deeply rooted in quantum physics, it requires knowledge of statistics, statistical physics/thermodynamics, computational astrophysics, and has applications in a wide range of astrophysical contexts including galaxies, the interstellar medium, the circumgalactic medium, the intergalactic medium, reionization, 21cm cosmology and astrophysics. In these lectures I will describe the basics of Ly\(\alpha \) radiative processes and transfer. These lectures are aimed to be selfcontained, and are (hopefully) suitable for anyone with an undergraduate degree in astronomy/physics. Throughout these notes, I denote symbols that represent vectors in bold print. Throughout I will use CGS units, as is common in the literature. Table 1.1 provides an overview of (some of the) symbols that appear throughout these notes.
1.1 Introduction
Half a century ago, [205] predicted that the Ly\(\alpha \) line should be a good tracer of star forming galaxies at large cosmological distances. This statement was based on the assumption that ionizing photons that are emitted by young, newly formed stars are efficiently reprocessed into recombination lines, of which Ly\(\alpha \) contains the largest flux. In the past two decades the Ly\(\alpha \) line has indeed proven to provide us with a way to both find and identify galaxies out to the highest redshifts (currently as high as \(z=8.7\), see [290]). In addition, we do not only expect Ly\(\alpha \) emission from (star forming) galaxies, but from structure formation in general (e.g. [94]). Galaxies are surrounded by vast reservoirs of gas that are capable of both emitting and absorbing Ly\(\alpha \) radiation. Observed spatially extended Ly\(\alpha \) nebulae (or ‘blobs’) may indeed provide insight into the formation and evolution of galaxies, in ways that complement direct observations of galaxies.
Many new instruments and telescopes^{1} are either about to be, or have just been, commissioned that are ideal for targeting the redshifted Ly\(\alpha \) line. The sheer number of observed Ly\(\alpha \) emitting sources is expected to increase by more than two orders of magnitude at all redshifts \(z\sim \) 2–7. For comparison, this boost is similar to that in the number of known exoplanets as a result of the launch of the Kepler satellite. In addition, sensitive integral field unit spectrographs will allow us to (i) detect sources that are more than an order of magnitude fainter than what has been possible so far, (ii) take spectra of faint sources, (iii) take spatially resolved spectra of the more extended sources, and (iv) detect phenomena at surface brightness levels at which diffuse Ly\(\alpha \) emission from the environment of galaxies is visible.
In order to interpret this growing body of data, we must understand the radiative transfer of Ly\(\alpha \) photons from its emission site to the telescope. Ly\(\alpha \) transfer depends sensitively on the gas distribution and kinematics. This complicates interpretation of Ly\(\alpha \) observations. On the other hand, the close interaction of the Ly\(\alpha \) radiation field and gaseous flows in and around galaxies implies that observations of Ly\(\alpha \) contains information on the medium through which the photons were scattering, and may thus present an opportunity to learn more about atomic hydrogen in gaseous flows in and around galaxies.
1.2 The Hydrogen Atom and Introduction to Ly\(\alpha \) Emission Mechanisms
1.2.1 Hydrogen in Our Universe
Because of its prevalence throughout the Universe, lines associated with atomic hydrogen have provided us with a powerful window on our Universe. The 21cm hyperfine transition was observed from our own Milky Way by Ewen and Purcell in [81], shortly after it was predicted to exist by Jan Oort in 1944. Observations of the 21cm line have allowed us to perform precise measurements of the distribution and kinematics of neutral gas in external galaxies, which provided evidence for dark matter on galactic scales (e.g. [33]). Detecting the redshifted 21cm emission from galaxies at \(z>0.5\), and from atomic hydrogen in the diffuse (neutral) intergalactic medium represent the main science drivers for many low frequency radio arrays that are currently being developed, including the Murchinson Wide Field Array,^{3} the Low Frequency Array,^{4} The Hydrogen Epoch of Reionization Array (HERA),^{5} the Precision Array for Probing the Epoch of Reionization (PAPER),^{6} and the Square Kilometer Array.^{7}
Similarly, the Ly\(\alpha \) transition has also revolutionized observational cosmology: observations of the Ly\(\alpha \) forest in quasar spectra has allowed us to measure the matter distribution throughout the Universe with unprecedented accuracy. The Ly\(\alpha \) forest still provides an extremely useful probe of cosmology on scales that are not accessible with galaxy surveys, and/or the Cosmic microwave background. The Ly\(\alpha \) forest will be covered extensively in the lectures by J. X. Prochaska. So far, the most important contributions to our understanding of the Universe from Ly\(\alpha \) have come from studies of Ly\(\alpha \) absorption. However, with the commissioning of many new instruments and telescopes, there is tremendous potential for Ly\(\alpha \) in emission. Because Ly\(\alpha \) is a resonance line, and because typical astrophysical environments are optically thick to Ly\(\alpha \), we need to understand the radiative transfer to be able to fully exploit the observations of Ly\(\alpha \) emitting sources.
1.2.2 The Hydrogen Atom: The Classical and Quantum Picture
Symbol dictionary
Symbol  Definition 

\(k_\mathrm{B}\)  Boltzmann constant: \(k_B=1.38\times 10^{16}\) erg K\(^{1}\) 
\(h_\mathrm{P}\)  Planck constant: \(h_P=6.67\times 10^{27}\) erg s 
\(\hbar \)  Reduced Planck constant: \(\hbar =\frac{h_\mathrm{P}}{2\pi }\) 
\(m_\mathrm{p}\)  Proton mass: \(m_p=1.66\times 10^{24}\) 
\(m_\mathrm{e}\)  Electron mass: \(m_\mathrm{e}=9.1\times 10^{28}\) g 
q  Electron charge: \(q=4.8\times 10^{10}\) esu 
c  Speed of light: \(c=2.9979\times 10^{10}\) cm s\(^{1}\) 
\(\varDelta E_{ul}\)  Energy difference between upper level ‘u’ and lower level ‘l’ (in ergs) 
\(\nu _{ul}\)  Photon frequency associated with the transition \(u \rightarrow l\) (in Hz) 
\(f_{ul}\)  The oscillator strength associated with the transition \(u \rightarrow l\) (dimensionless) 
\(A_{ul}\)  Einstein Acoefficient of the transition \(u \rightarrow l\) (in s\(^{1}\)) 
\(B_{ul}\)  Einstein Bcoefficient of the transition \(u \rightarrow l\): \(B_{ul}=\frac{2h_P\nu _{ul}^3}{c^2}A_{ul}\) (in erg cm\(^{2}\) s\(^{1}\)) 
\(B_{lu}\)  Einstein Bcoefficient of the transition \(l \rightarrow u\): \(B_{lu}=\frac{g_u}{g_l}B_{ul}\) 
\(\alpha _{A/B}\)  Case A /B recombination coefficient (in cm\(^{3}\) s\(^{1}\)) 
\(\alpha _{nl}\)  Recombination coefficient into state (n, l) (in cm\(^{3}\) s\(^{1}\)) 
\(g_{u/l}\)  Statistical weight of upper/lower level of a radiative transition (dimensionless) 
\(\nu _{\alpha }\)  Photon frequency associated with the Ly\(\alpha \) transition: \(\nu _{\alpha }=2.47\times 10^{15}\) Hz 
\(\omega _{\alpha }\)  Angular frequency associated with the Ly\(\alpha \) transition: \(\omega _{\alpha }=2\pi \nu _{\alpha }\) 
\(\lambda _{\alpha }\)  Wavelength associated with the Ly\(\alpha \) transition: \(\lambda _{\alpha }=1215.67\) Å 
\(A_{\alpha }\)  Einstein Acoefficient of the Ly\(\alpha \) transition: \(A_{\alpha }=6.25 \times 10^8\) s\(^{1}\) 
T  gas temperature (in K) 
\(v_\mathrm{th}\)  Velocity dispersion (times \(\sqrt{2}\)): \(v_\mathrm{th}=\sqrt{\frac{2k_BT}{m_p}}\) 
\(v_\mathrm{turb}\)  Turbulent velocity dispersion 
b  Doppler broadening parameter : \(b=\sqrt{v^2_\mathrm{th}+v^2_\mathrm{turb}}\) 
\(\varDelta \nu _{\alpha }\)  Doppler induced photon frequency dispersion:\(\varDelta \nu _{\alpha }=\nu _{\alpha }\frac{b}{c}\) (in Hz) 
x  ‘Normalized’ photon frequency: \(x=(\nu \nu _{\alpha })/\varDelta \nu _{\alpha }\) (dimensionless) 
\(\sigma _{\alpha }(x)\)  Ly\(\alpha \) Absorption crosssection at frequency x (in cm\(^2\)), \(\sigma _{\alpha }(x)=\sigma _{\alpha ,0}\phi (x)\) 
\(\sigma _{\alpha ,0}\)  Ly\(\alpha \) Absorption crosssection at line center, \(\sigma _{\alpha ,0}=5.9\times 10^{14}\Big (\frac{T}{10^4\mathrm{K}}\Big )^{1/2}\mathrm{cm}^{2}\) 
\(\phi (x)\)  Voigt profile (dimensionless) 
\(a_v\)  Voigt parameter: \(a_v=A_{\alpha }/[4\pi \varDelta \nu _{\alpha }]=4.7\times 10^{4}(T/10^4 \mathrm{K})^{1/2}\) 
\(I_\nu \)  Specific intensity (in erg s\(^{1}\) Hz\(^{1}\) cm\(^{2}\) sr\(^{1}\)) 
\(J_\nu \)  Angle averaged specific intensity (in erg s\(^{1}\) Hz\(^{1}\) cm\(^{2}\) sr\(^{1}\)) 
The lowest energy state corresponds to the \(n=1\) state, with an energy of \(E=13.6\) eV. For the state \(n=1\), the orbital quantum number l can only take on the value \(l=0\). This state with \((n,l)=(1,0)\) is referred to as the ‘1s’state. The ‘1’ refers to the value of n, while the ‘s’ is a historical way (the ‘spectroscopic notation’) of labelling the ‘\(l=0\)’state. This Figure also indicates (schematically) that the wavefunction that describes the 1sstate is spherically symmetric. The ‘size’ or extent of this wavefunction relates to the classical atom size in that the expectation value of the radial position of the electron corresponds to the Bohr radius \(a_0\), i.e. \(\int dV r \psi _{1s}(\mathbf{r})^2=a_0\).
The second lowest energy state, \(n=2\), has a total energy \(E=E_0/n^2=3.4\) eV. For this state there exist two quantum states with \(l=0\) and \(l=1\). The ‘2s’state is again characterized by a spherically symmetric wavefunction, but which is more extended. This larger physical extent reflects that in this higher energy state, the electron is more likely to be further away from the proton, completely in line with classical expectations. On the other hand, the wavefunction that describes the ‘2p’state (\(n=2\), \(l=1\)) is not spherically symmetric, and consists of two ‘lobes’. The elongation that is introduced by these lobes can be interpreted as the electron being on an eccentric orbit, which reflects the increase in the electron’s orbital angular momentum.
The third lowest energy state \(n=3\) has a total energy of \(E=E_0/n^2=1.5\) eV. The size/extent of the orbital/wavefunction increases further, and the complexity of the shape of the orbitals increases with n (see e.g. https://en.wikipedia.org/wiki/Atomic_orbital for illustrations). Loosely speaking, the quantum number n denotes the extent/size of the wavefunction, l denotes its eccentricity/elongation. The orientation of nonspherical wavefunction can be represented by a third quantum number m.
1.2.3 Radiative Transitions in the Hydrogen Atom: Lyman, Balmer, ..., Pfund, .... Series
The Lyman series. A series of radiative transitions in the hydrogen atom which arise when the electron goes from \(n \ge 2\) to \(n = 1\). The first line in the spectrum of the Lyman series—named Lyman \(\alpha \) (hereafter, Ly\(\alpha \))—was discovered in 1906 by Theodore Lyman, who was studying the ultraviolet spectrum of electrically excited hydrogen gas. The rest of the lines of the spectrum were discovered by Lyman in subsequent years.
The Balmer series. The series of radiative transitions from \(n \ge 3\) to \(n = 2\). The series is named after Johann Balmer, who discovered an empirical formula for the wavelengths of the Balmer lines in 1885. The Balmer\(\alpha \) (hereafter H\(\alpha \)) transition is in the red, and is responsible for the reddish glow that can be seen in the famous Orion nebula.
Following the Balmer series, we have the Paschen series (\(n \ge 4 \rightarrow n=3\)), the Brackett series (\(n \ge 5 \rightarrow 4\)), the Pfund series (\(n \ge 6 \rightarrow 5\)), .... Especially Pfund\(\delta \) is potentially an interesting probe ([199]2016 private communication).
Quantum mechanics does not allow radiative transitions between just any two quantum states: these radiative transitions must obey the ‘selection rules’. The simplest version of the selection rules—which we will use in these lectures—is that only transitions of the form \(\varDelta l=1\) are allowed. A simple interpretation of this is that photons carry a (spin) angular momentum given by \(\hbar \), which is why the angular momentum of the electron orbital must change by \(\pm \hbar \) as well. Figure 1.4 indicates allowed transitions, either as green solid lines or as red dashed lines. Note that the Lyman\(\beta ,\gamma ,...\) transitions (\(3p\rightarrow 1s\), \(4p \rightarrow 1s\), ...) are not shown on purpose. As we will see later in the lectures, while these transitions are certainly allowed, in realistic astrophysical environments it is better to simply ignore them.
Consider an electron in some arbitrary quantum state (n, l). The electron does not spend much time in this state, and radiatively decays down to a lower energy state \((n',l')\). This lower energy state is again unstable [unless \((n',l')=(1,0)\)], and the electron again radiatively decays to an even lower energy state \((n'',l'')\). Ultimately, all paths lead to the ground state, even those paths that go through the 2s state. While the selection rules do not permit transitions of the form \(2s\rightarrow 1s\), these transitions can occur, if the atom emits two photons (rather than one). Because these twophoton transitions are forbidden, the lifetime of the electron in the 2s state is many orders of magnitude larger than almost all other quantum states (it is \({\sim }8\) orders of magnitude longer than that of the 2pstate), and this quantum state is called ‘metastable’. The path from an arbitrary quantum state (n, l) to the ground state via a sequence of radiative decays is called a ‘radiative cascade’.
The green solid lines in Fig. 1.4 show radiative cascades that result in the emission of a Ly\(\alpha \) photon. The red dashed lines show the other radiative cascades. The table in the lower right corner shows the probability that a radiative cascade from quantum state (n, l) produces a Ly\(\alpha \) photon. This probability is denoted with \(P(n,l\rightarrow ...\rightarrow \mathrm{Ly}\alpha )\). For example, the probability that an electron in the 2s orbital gives rise to Ly\(\alpha \) is zero. The probability that an electron in the 3s orbital gives Ly\(\alpha \) is 1. This is because the only allowed radiative cascade to the ground state from 3s is \(3s \rightarrow 2p \rightarrow 1s\). This last transition corresponds to the Ly\(\alpha \) transition. For \(n\ge 4\) the probabilities become nontrivial, as we have to compute the likelihood of different radiative cascades. We discuss this in more detail in the next section.
1.2.4 Ly\(\alpha \) Emission Mechanisms
 1.
Collisions. The ‘collision’ between an electron and a hydrogen atom can leave the atom in an excited state, at the expense of kinetic energy of the free electron. This process is illustrated in Fig. 1.5. This process converts thermal energy of the electrons, and therefore of the gas as whole, into radiation. This process is also referred to as Ly\(\alpha \) production via ‘cooling’ radiation. We discuss this process in more detail in Sect. 1.3.1, and in which astrophysical environments it may occur in Sect. 1.4.
 2.
Recombination. Recombination of a free proton and electron can leave the electron in any quantum state (n, l). Radiative cascades to the ground state can then produce a Ly\(\alpha \) photon. As we discussed in Sect. 1.2.3, we can compute the probability that each quantum state (n, l) produces a Ly\(\alpha \) photon during the radiative cascade down to the groundstate. If we sum over all these quantum states, and properly weigh by the probability that the freshly combined electronproton pair ended up in state (n, l), then we can compute the probability that a recombination event gives us a Ly\(\alpha \) photon. We discuss the details of this calculation in Sect. 1.3.2. Here, we simply discuss the main results.
The upper panel of Fig. 1.6 shows the total probability \(P(\mathrm{Ly}\alpha )\) that a Ly\(\alpha \) photon is emitted per recombination event as a function of gas temperature T. This plot contains two lines. The solid black line represents ‘CaseA’, which refers to the most general case where we allow the electron and proton to recombine into any state (n, l), and where we allow for all radiative transitions permitted by the selection rules. The dashed black line shows ‘CaseB’, which refers to the case where we do not allow for (i) direct recombination into the ground state, which produces an ionizing photon, and (ii) radiative transitions of the higher order Lyman series, i.e. Ly\(\beta \), Ly\(\gamma \), Ly\(\delta \),.... CaseB represents that most astrophysical gases efficiently reabsorb higher order Lyman series and ionizing photons, which effectively ‘cancels out’ these transitions (see Sect. 1.3.2 for more discussion on this). This Figure shows that for gas at \(T=10^4\) K and caseB recombination, we have \(P(\mathrm{Ly}\alpha )=0.68\). This value \(`0.68'\) is often encountered during discussions on Ly\(\alpha \) emitting galaxies. It is worth keeping in mind that the probability \(P(\mathrm{Ly}\alpha )\) increases with decreasing gas temperature and can be as high as \(P(\mathrm{Ly}\alpha )=0.77\) for \(T=10^3\) K (also see [42]). The red open circles represent the following two fitting formulaewhere \(T_4 \equiv T/10^4\) K. The fitting formula for caseB is taken from [42].$$\begin{aligned} P_\mathrm{A}(\mathrm{Ly}\alpha )=0.410.165 \log T_40.015(T_4)^{0.44}\nonumber \\ P_\mathrm{B}(\mathrm{Ly}\alpha )=0.6860.106 \log T_40.009(T_4)^{0.44}, \end{aligned}$$(1.5)
1.3 A Closer Look at Ly\(\alpha \) Emission Mechanisms and Sources
The previous section provided a brief description of physical processes that give rise to Ly\(\alpha \) emission. Here, we discuss these in more detail, and also link them to astrophysical sources of Ly\(\alpha \).
1.3.1 Collisions
1.3.2 Recombination

‘caseA’ recombination: recombination takes place in a medium that is optically thin at all photon frequencies. In this case, direct recombination to the ground state is allowed and \(n_\mathrm{min}=1\).

‘caseB’ recombination: recombination takes place in a medium that is opaque to all Lyman series^{11} photons (i.e. Ly\(\alpha \), Ly\(\beta \), Ly\(\gamma \), ...), and to ionizing photons that were emitted following direct recombination into the ground state. In the socalled ‘on the spot approximation’, direct recombination to the ground state produces an ionizing photon that is immediately absorbed by a nearby neutral H atom. Similarly, any Lyman series photon is immediately absorbed by a neighbouring H atom. This case is quantitatively described by setting \(n_\mathrm{min}=2\), and by setting the Einstein coefficient for all Lyman series transitions to zero, i.e. \(A_{np,1s}=0\).
The probability \(P(\mathrm{Ly}\alpha )\) that we obtain from Eq. 1.15 was plotted in Fig. 1.6 assuming caseA (solid line) and caseB (dashed line) recombination. The temperature dependence comes in entirely through the temperature dependence of the statespecific recombination coefficients \(\alpha _{nl}(T)\). As we mentioned earlier, for caseB recombination, we have \(P(\mathrm{Ly}\alpha )=0.68\) at \(T=10^4\) K. It is worth keeping in mind that our calculations technically only apply in a low density medium. For ‘high’ densities, collisions can ‘mix’ different llevels at a fixed n. In the limit of infinitely large densities, collisional mixing should cause different \(l\)levels to be populated following their statistical weigths [i.e \(n_{nl} \propto (2l1)\)]. Collisions can be important in realistic astrophysical conditions, as we discuss in more detail in Sect. 1.4.
1.4 Astrophysical Ly\(\alpha \) Sources
Now that we have specified different physical mechanisms that give rise to the production of a Ly\(\alpha \) photon, we discuss various astrophysical sites of Ly\(\alpha \) production.
1.4.1 Interstellar HII Regions
1.4.2 The Circumgalactic/Intergalactic Medium (CGM/IGM)
Not only nebulae are sources of Ly\(\alpha \) radiation. Most of our Universe is in fact a giant Ly\(\alpha \) source. Observations of spectra of distant quasars reveal a large collection of Ly\(\alpha \) absorption lines. This socalled ‘Ly\(\alpha \) forest’ is discussed in more detail in the lecture notes by X. Prochaska. Observations of the Ly\(\alpha \) forest imply that the intergalactic medium is highly ionized, and that the temperature of intergalactic gas is \(T\sim 10^4\) K. Observations of the Ly\(\alpha \) forest can be reproduced very well if we assume that gas is photoionized by the Universal “ionizing background” that permeates the entire Universe, and that is generated by adding the contribution from all ionizing sources.^{17} The residual neutral fraction of hydrogen atoms in the IGM is \(x_\mathrm{HI} \equiv \frac{n_\mathrm{HI}}{n_\mathrm{HI}+n_\mathrm{HII}}=\frac{n_e \alpha _\mathrm{B}(T)}{\varGamma _\mathrm{ion}}\), where \(\varGamma _\mathrm{ion}\) denotes the photoionization rate by the ionizing background (with units s\(^{1}\)).
The direct environment of galaxies, also known as the ‘circum galactic medium’ (CGM), represents a complex mixture of hot and cold gas, of metal poor gas that is being accreted from the intergalactic medium and metal enriched gas that is driven out of either the central, massive galaxy or from the surrounding lower mass satellite galaxies. Figure 1.12 shows a snapshot from a cosmological hydrodynamical simulation [3] which nicely illustrates this complexity. A disk galaxy (total baryonic mass \(\sim 2 \times 10^{10} M_{\odot }\)) sits in the center of the snapshot, taken at \(z=3\). The blue filaments show dense gas that is being accreted. This gas is capable of selfshielding. The red gas has been shock heated to the virial temperature (\(T_\mathrm{vir} \sim 10^6\) K) of the dark matter halo hosting this galaxy. The green clouds show metal rich gas that was driven out of smaller galaxies. This complex mixture of gas produces Ly\(\alpha \) via all channels described above: there exists fully ionized gas that is emitting recombination radiation with a surface brightness given by Eq. 1.28, the densest gas is capable of selfshielding and will emit both recombination and cooling radiation.
 1.
Ly\(\alpha \) emission extends further the UV continuum in nearby star forming galaxies. The left panel of Fig. 1.13 shows an example of a falsecolor image of galaxy # 1 from the Lyman Alpha Reference Sample [79, 107]. In this image, red indicates H\(\alpha \), green traces the farUV continuum, while blue traces the Ly\(\alpha \).
 2.
Stacking analyses have revealed the presence of spatially extended Ly\(\alpha \) emission around Lyman Break Galaxies [117, 253] and Ly\(\alpha \) emitters at surface brightness levels in the range SB\(\sim 10^{19}10^{18}\) erg s\(^{1}\) cm\(^{2}\) arcsec\(^{2}\) [170, 186].
 3.
Deep imaging with MUSE has now revealed emission at this level around individual star forming galaxies, which further confirms that this emission is present ubiquitously [274].
 4.
These previously mentioned faint halos are reminiscent of Ly\(\alpha \) ‘blobs’, which are spatially extended Ly\(\alpha \) sources not associated with radio galaxies (more on these next, [117, 143, 252]). A famous example of “blob # 1” is shown in the right panel of Fig. 1.13 (from [143]). This image shows a ‘pseudocolor’ image of a Ly\(\alpha \) blob. The red and blue really trace radiation in the red and blue filters, while the green traces the Ly\(\alpha \). The upper right panel shows how large the Andromeda galaxy would look on the sky if placed at \(z=3\), to put the size of the blob in perspective. The brightest Ly\(\alpha \) blobs have line luminosities of \(L_{\alpha } \sim 10^{44}\) erg s\(^{1}\), though recently two monstrous blobs have been discovered, that are much brighter than this: (i) the ‘Slug’ nebula with a Ly\(\alpha \) luminosity of \(L_{\alpha }\sim 10^{45}\) erg s\(^{1}\) [44], (ii) the ‘Jackpot’ nebula, which has a luminosity of \(L_{\alpha }\sim 2 \times 10^{44}\) erg s\(^{1}\), and contains a quadruplequasar system [125].
 5.
The most luminous Ly\(\alpha \) nebulae have traditionally been associated (typically) with Highredshift Radio Galaxies (HzRGs, e.g. [172, 219, 267]) with luminosities in excess of \(L\sim 10^{45}\) erg s\(^{1}\).
Finally, Ly\(\alpha \) cooling radiation gives rise to spatially extended Ly\(\alpha \) radiation [83, 109], and provides a possible explanation for Ly\(\alpha \) ‘blobs’ [69, 85, 100, 221]). In these models, the Ly\(\alpha \) cooling balances ‘gravitational heating’ in which gravitational binding energy is converted into thermal energy in the gas. Precisely how gravitational heating works is poorly understood. Haiman et al. [109] propose that the gas releases its binding energy in a series of ‘weak’ shocks as the gas navigates down the gravitational potential well. These weak shocks convert binding energy into thermal energy over a spatially extended region,^{19} which is then reradiated primarily as Ly\(\alpha \). We must therefore accurately know and compute all the heating rates in the ISM [43, 85, 221] to make a robust prediction for the Ly\(\alpha \) cooling rate. These heating rates include for example photoionization heating, which requires coupled radiationhydrodynamical simulations (as [221]), or shock heating by supernova ejecta (e.g. [242]).
The previous discussion illustrates that it is possible to produce spatially extended Ly\(\alpha \) emission from the CGM at levels consistent with observations, via all mechanisms described in this section.^{20} This is one of the main reasons why we have not solved the question of the origin of spatially extended Ly\(\alpha \) halos yet. In later lectures, we will discuss how Ly\(\alpha \) spectral line profiles (and polarization measurements) contain physical information on the scattering/emitting gas, which can help distinguish between different scenarios.
1.5 Step 1 Towards Understanding Ly\(\alpha \) Radiative Transfer: Ly\(\alpha \) Scattering Crosssection
The goal of this section is to present a classical derivation of the Ly\(\alpha \) absorption crosssection. This classical derivation gives us the proper functional form of the real crosssection, but that differs from the real expression by a factor of order unity, due to a quantum mechanical correction. Once we have evaluated the magnitude of the crosssection, it is apparent that most astrophysical sources of Ly\(\alpha \) emission are optically thick to this radiation, and that we must model the proper Ly\(\alpha \) radiative transfer.
1.5.1 Interaction of a Free Electron with Radiation: Thomson Scattering
The power of reemitted radiation is not distributed isotropically across the sky. A useful way to see this is by considering what we see if we observe the oscillating electron along direction \(\mathbf{k}_\mathrm{out}\). The apparent acceleration that the electron undergoes is reduced to \(\hat{a}(t)\equiv a(t)\sin \varPsi \), where \(\varPsi \) denotes the angle between \(\mathbf{k}_\mathrm{out}\) and the oscillation direction, i.e \(\cos \varPsi \equiv \mathbf{k}_\mathrm{out}\cdot \mathbf{e}_\mathrm{E}\). Here, the vector \(\mathbf{e}_\mathrm{E}\) denotes a unit vector pointing in the direction of the Efield. The reduced apparent acceleration translates to a reduced power in this direction, i.e \(P_\mathrm{out}(\mathbf{k}_\mathrm{out})\propto \hat{a}^2(t) \propto \sin ^2 \varPsi \).
1.5.2 Interaction of a Bound Electron with Radiation: Lorentzian CrossSection
In the absence of the electromagnetic field, this yields a quadractic equation for \(\omega \), namely \(\omega ^2+i\varGamma \omega +\omega _0^2=0\). This equation has solutions of the form \(\omega =\frac{i\varGamma }{2}\pm \sqrt{\omega _0^2\varGamma ^2/4}\). We assume that \(\omega _0\gg \varGamma \) (which is the case for Ly\(\alpha \) as we see below), which can be interpreted as meaning that the electron makes multiple orbits around the nucleus before there is a ‘noticeable’ change in its position due to radiative energy losses. The solution for x(t) thus looks like \(x(t)\propto \exp (\varGamma t/2)\cos \omega _0t\). The solution indicates that the electron keeps orbiting the proton with the same natural frequency \(\omega _0\), but that it spirals inwards on a characteristic timescale \(\varGamma ^{1}\) (also see Sect. 1.2.2). We will evaluate \(\varGamma \) later.
 In the presence of an electromagnetic field, substituting \(x(t)=x\exp (i\omega t)\) yields the following solution for the amplitude x:where we used that \((\omega ^2\omega ^2_0)=(\omega \omega _0)(\omega +\omega _0)\approx 2\omega _0(\omega \omega _0)\). This last approximation assumes that \(\omega \approx \omega _0\). It is highly relevant for Ly\(\alpha \) scattering, where \(\omega \) and \(\omega _0\) are almost always very close together (meaning that \(\omega \omega _0/\omega _0 \ll 10^{2}\)).$$\begin{aligned} x=\frac{qE_0}{m_e}\frac{1}{\omega ^2\omega ^2_0+i\omega \varGamma }\overset{\omega \approx \omega _0}{\sim }\frac{qE_0}{2m_e\omega _0}\frac{1}{\omega \omega _0+i\varGamma /2}, \end{aligned}$$(1.45)
 1.
The function is sharply peaked on \(\omega _0\), at which \(\sigma _0\equiv \sigma _\mathrm{CL}(\omega _0)=\frac{3 \lambda _0^2}{8\pi }\sim 7\times 10^{11}\) cm\(^{2}\), where \(\lambda _0=2 \pi c /\omega _0\) corresponds to the wavelength of the electromagnetic wave with frequency \(\omega _0\). The crosssection falls off by \(\gtrsim 5\) orders of magnitude with \(\omega \omega _0/\omega _0 \sim 10^{4}\). Note that the crosssection is many, many orders of magnitude larger than the Thomson crosssection. This enhanced crosssection represents a ‘resonance’, an ‘unusually strong response of a system to an external trigger’.
 2.
The \(\sigma \propto \omega ^4\)dependence implies that the atom is slightly more efficient at scattering more energetic radiation. This correspond to the famous Rayleigh scattering regime, which refers to elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the radiation (see https://en.wikipedia.org/wiki/Rayleigh_scattering), and which explains why the sky is blue,^{23} and the setting/rising sun red.
1.5.3 Interaction of a Bound Electron with Radiation: Relation to Ly\(\alpha \) CrossSection
 1.
The parameter \(\varGamma \) reduces by a factor of \(f_{\alpha }=0.4162\), which is known as the ‘oscillator’ strength, i.e. \(\varGamma \rightarrow f_{\alpha }\varGamma \equiv A_{\alpha }\). Here, \(A_{\alpha }\) denotes the Einstein Acoefficient for the Ly\(\alpha \) transition.
 2.
In detail, a simple functional form (Lorentzian, Eq. 1.48, or see Eq. 1.49 below) for the Ly\(\alpha \) crosssection does not exist. The main reason for this is that if we want to evaluate the Ly\(\alpha \) crosssection far from resonance, we have to take into account the contributions to the crosssection from the higherorder Lymanseries transitions, and even photoionization. When these contributions are included, the expression for the crosssection involves squaring the sum of all these contributions (e.g. [16, 17, 188]). Accurate approximations to this expression are possible close to the resonance(s)—when \(\omega \omega _0/\omega _0 \ll 1\), which is the generally the case for practical purposes—and these approximations are in excellent agreement with our derived crosssection (see Eq. 14 in [188], and use that \(\omega \approx \omega _0\), and that \(\Lambda _{12}=\varGamma /2\pi \)).
1.5.4 Voigt Profile of Ly\(\alpha \) CrossSection
The solid line in the right panel of Fig. 1.17 shows the LAB frame cross section—this is also known as the Voigtprofile—as given by Eq. 1.55 as a function of the dimensionless frequency x. The red dashed line (green dotdashed line) represent the cross section where we approximated the Voigt function as \(\exp (x^2)\) (\(a_v/[\sqrt{\pi }x^2]\)). Clearly, these approximations work very well in the relevant regimes. Note that this Figure shows that a decent approximation to Voigt function at all frequencies is given simply by the sum of these two terms, i.e. \(\phi (x)\approx \exp (x^2)+a_v/(\sqrt{\pi }x^2)\) (provided that \(x \gtrsim \sqrt{a_v}\) [148]). This approximation fails in a very narrow frequency regime where the transition from core to wing occurs. A useful fitting function that is accurate at all x is given in [256]. Also shown for comparison as the blue dotted line is the symmetric single atom cross section (which was shown in the left panel as the solid line). Figure 1.17 shows that close to resonance, this single atom cross section provides a poor description of the real cross section. This is because Doppler motions ‘smear out’ the sharply peaked crosssection. Far in the wing however, the single atom crosssection provides an excellent fit to the velocity averaged Voigt profile.
One of the key results from this section is that the Ly\(\alpha \) crosssection, evaluated at line center and averaged over the velocity distribution of atoms, is tremendous at \(\sigma _{\alpha ,0}(T)\sim 5.9\times 10^{14}(T/10^4\mathrm{K})^{1/2}\) cm\(^{2}\), which is \({\sim }11\) orders of magnitude than the Thomson crosssection. That is, an electron bound to the proton is \({\sim }11\) orders of magnitude more efficient at scattering radiation than a free electron when the frequency of that radiation closely matches the natural frequency of the transition. This further emphasises that the electron ‘resonantly scatters’ the incoming radiation. To put these numbers in context, it is possible to measure the hydrogen column density, \(N_\mathrm{HI}\), in nearby galaxies. The observed intensity in the 21cm line translates to typical HI column densities of order \(N_\mathrm{HI} \sim 10^{19}10^{21}\) cm\(^{2}\) [33, 53, 140]), which translates to line center optical depths of Ly\(\alpha \) photons of order \(\tau _0 \sim 10^{7}10^8\). This estimate highlights the importance of understanding the transport of Ly\(\alpha \) photon out of galaxies. They generally are not expected to escape without interacting with hydrogen gas.
1.6 Step 2 Towards Understanding Ly\(\alpha \) Radiative Transfer: The Radiative Transfer Equation
1.6.1 I: Absorption Term: Ly\(\alpha \) Cross Section
The opacity \(\alpha ^\mathrm{HI}_{\nu }(s)=\big [n_l(s)\frac{g_l}{g_u}n_u(s)\big ]\sigma (\nu )\). The second term within the square brackets corrects the absorption term for stimulated emission. In most astrophysical conditions all neutral hydrogen atoms are in their electronic ground state, and we can safely ignore the stimulated emission term (see e.g. the Appendix of [23, 66]). That is, in practice we can simply state that \(\alpha _{\nu }(s)=n_l(s)\sigma _{\alpha }(\nu )=n_\mathrm{HI}(s)\sigma _{\alpha }(\nu )\). We have derived expressions for the Ly\(\alpha \) absorption crosssection in Sect. 1.5.
1.6.2 II: Volume Emission Term
1.6.3 III: Scattering Term
The ‘redistribution function’ \(R(\nu ,\nu ',\mathbf{n},\mathbf{n}')\) describes the probability that radiation that was originally propagating at frequency \(\nu '\) and in direction \(\mathbf{n}'\) is scattered into frequency \(\nu \) and direction \(\mathbf{n}\) (see e.g. Zanstra [130, 266, 284] for early discussions). In practise, this probability depends only on the angle between \(\mathbf{n}\) and \(\mathbf{n}'\), i.e. \(R(\nu ,\nu ',\mathbf{n},\mathbf{n}')=R(x_\mathrm{out},x_\mathrm{in},\mu )\), in which \(\mu \equiv \cos \theta =\mathbf{n}\cdot \mathbf{n}'\). We have also switched to standard dimensionless frequency coordinates (first introduced in Eq. 1.53), and denote with \(x_\mathrm{out}\) (\(x_\mathrm{in}\)) the dimensionless frequency of the photon after (before) scattering. Formally, \(R(x_\mathrm{out},x_\mathrm{in},\mu )dx_\mathrm{out}d\mu \) denotes the probability that a photon of frequency \(x_\mathrm{in}\) was scattered by an angle in the range \(\mu \pm d\mu /2\) into the frequency range \(x_\mathrm{out} \pm dx_\mathrm{out}/2\). Thus, \(R(x_\mathrm{out},x_\mathrm{in},\mu )\) is normalized such that \(\int _{1}^{1}d\mu \int _{\infty }^{\infty }dx_\mathrm{out}R(x_\mathrm{in},x_\mathrm{out},\mu )=1\).
In most astrophysical conditions, the energy of the Ly\(\alpha \) photon before and after scattering is identical in the frame of the absorbing atom. This is because the lifetime of the atom in its 2p state is only \(t =1/A_{\alpha } \sim 10^{9}\) s. In most astrophysical conditions, the hydrogen atom in this state is not ‘perturbed’ over this short timeinterval, and energy conservation forces the energy of the photon to be identical before and after scattering. Because of random thermal motions of the atom, energy conservation in the atom’s frame translates to a change in the energy of the incoming and outgoing photon that depends on the velocity of the atom and the scattering direction (see Fig. 1.18 for an illustration). This type of scattering is known as ‘partially coherent’ scattering.^{28}
Examples of \(R(x_\mathrm{out}\mu ,x_\mathrm{in})\) as a function of \(x_\mathrm{out}\) are plotted in Fig. 1.20. In the left panel we plot \(R(x_\mathrm{out}\mu ,x_\mathrm{in})\) for \(\mu =0.5\), and \(x_\mathrm{in}=3.0\) (red line), \(x_\mathrm{in}=0.0\) (black line), and \(x_\mathrm{in}=4.0\) (blue line). These frequencies were chosen to represent scattering in the wing (for \(x_\mathrm{in}=4.0\)), in the core (for \(x_\mathrm{in}=0.0\)) and in the transition region (\(x_\mathrm{in}=3.0\), see Fig. 1.17). In the right panel we used \(\mu =0.5\). Figure 1.20 shows clearly that (i) for the vast majority of photons \(x_\mathrm{in}x_\mathrm{out} \lesssim \) a few. That is, the frequency after scattering is closely related to the frequency before scattering (ii) \(R(x_\mathrm{out}\mu ,x_\mathrm{in})\) can depend quite strongly on \(\mu \). This is most clearly seen by comparing the curves for \(x_\mathrm{in}=3.0\) in the left and right panels.
There are three important properties of these redistribution functions that play an essential role in radiative transfer calculations. We discuss these next.
 1.Photons that scatter in the wing of the line are pushed back to the line core by an amount \(\frac{1}{x_\mathrm{in}}\) [197], i.e.Demonstrating this requires some calculation. The expectation value for \(\varDelta x\) per scattering event is given by$$\begin{aligned} \boxed {\langle \varDelta x  x_\mathrm{in} \rangle =\frac{1}{x_\mathrm{in}}}. \end{aligned}$$(1.73)where the factor of \(\frac{1}{2}\) reflects that \(\int _{1}^1 P(\mu )d\mu =2\) (see Eq. 1.41). For simplicity we will assume isotropic scattering, for which \(P(\mu )=1\). We previously presented an expression for \(R(x_\mathrm{out}\mu ,x_\mathrm{in})\) (see Eq. 1.71). Substituting this expression into the above equation yields$$\begin{aligned} \langle \varDelta x  x_\mathrm{in} \rangle&\equiv \int _{\infty }^{\infty }\varDelta x R(x_\mathrm{out}x_\mathrm{in})dx_\mathrm{out} \nonumber \\&=\frac{1}{2}\int _{\infty }^{\infty } dx_\mathrm{out} \int _{1}^{1}d\mu \varDelta x P(\mu )R(x_\mathrm{out}\mu ,x_\mathrm{in}), \end{aligned}$$(1.74)(1.75)(1.76)
This result is very important: it implies that as a Ly\(\alpha \) photon is far in the wing at \(x_\mathrm{in}\), resonant scattering exerts a ‘restoring force’ which pushes the photon back to line resonance. This restoring force generally overwhelms the energy losses resulting from atomic recoil: Eq. 1.65 indicates that recoil introduces a much smaller average \(\varDelta x \sim 2.6 \times 10^{4}(T/10^4\mathrm{K})^{1/2}\), i.e. if a photon finds itself on the red side of line center, then the restoring force pushes the photon back more to line center than recoil pulls it away from it.
 2.The r.m.s change in the photon’s frequency as it scatters corresponds to 1 Doppler width [197].This can be derived with a calculation that is very similar to the above calculation.$$\begin{aligned} \boxed {\sqrt{\langle \varDelta x^2x_\mathrm{in}\rangle }=1}. \end{aligned}$$(1.77)
 3.
\(R(x_\mathrm{out}x_\mathrm{in})=R(x_\mathrm{in}x_\mathrm{out})\). This can be verified by substituting \(u=y\varDelta x\) into Eq. 1.71. After some algebra one obtains an expression in which y replaces u, in which \(x_\mathrm{out}\) replaces \(x_\mathrm{in}\), and in which \(x_\mathrm{out}\) replaces \(x_\mathrm{in}\).
1.6.4 IV: ‘Destruction’ Term
 1.Dust. Dust grains can absorb Ly\(\alpha \) photons. The dust grain can scatter the Ly\(\alpha \) photon with a probability which is given by its albedo, \(A_\mathrm{d}\). Dust plays an important role in Ly\(\alpha \) radiative transfer, and we will return to this later (see Sect. 1.9.1). Absorption of a Ly\(\alpha \) photon by a dust grain increases its temperature, which causes the grain to reradiate at longer wavelengths, and thus to the destruction of the Ly\(\alpha \) photon. This process can be included in the radiative transfer equation by replacingwhere \(\sigma _\mathrm{dust}(x)\) denotes the total dust crosssection at frequency x, i.e. \(\sigma _\mathrm{dust}(x)=\sigma _\mathrm{dust,a}(x)+\sigma _\mathrm{dust,s}(x)\), where the subscript ‘a’ (‘s’) stands for ‘absorption’ (‘scattering’). Equation 1.78 indicates that the crosssection \(\sigma _\mathrm{dust}(x)\) is a crosssection per hydrogen atom. This definition implies that \(\sigma _\mathrm{dust}(x)\) is not just a property of the dust grain, as it must also depend on the number density of dust grains (if there were no dust grains, then we should not have to add any term). In addition to this, the dust absorption cross section \(\sigma _\mathrm{dust,s}(x)\) (and also the albedo \(A_\mathrm{d}\)) must depend on the dust properties. For example, [153] shows that \(\sigma _\mathrm{dust}=4 \times 10^{22}(Z_\mathrm{gas}/0.25Z_{\odot })\) cm\(^{2}\) for SMC type dust (dust with the same properties as found in the Small Magellanic Cloud), and \(\sigma _\mathrm{dust}=7 \times 10^{22}(Z_\mathrm{gas}/0.5Z_{\odot })\) cm\(^{2}\) for LMC (Large Magellanic Cloud) type dust. Here, \(Z_\mathrm{gas}\) denotes the metallicity of the gas. This parametrization of \(\sigma _\mathrm{dust}\) therefore assumes that the number density of dust grains scales linearly with the overall gas metallicity, which is a good approximation for \(Z\gtrsim 0.3 Z_{\odot }\), but for \(Z \lesssim 0.3 Z_{\odot }\) the scatter in dusttogas ratio increases (e.g. [78, 218, 235]). Figure 1.23 shows \(\sigma _\mathrm{dust}\) for dust properties inferred for the LMC (solid line), and SMC (dashed line). This Figure shows that the frequency dependence of the dust absorption cross section around the Ly\(\alpha \) resonance is weak, and in practise it can be safely ignored. Interestingly, as we will discuss in Sect. 1.7.3, dust can have a highly frequency dependent impact on the Ly\(\alpha \) radiation field, in spite of the weak frequency dependence of the dust absorption crosssection.$$\begin{aligned} n_\mathrm{HI}\sigma _{\alpha ,0}\phi (x) \rightarrow n_\mathrm{HI}[\sigma _{\alpha ,0}\phi (x)+\sigma _\mathrm{dust}(x)], \end{aligned}$$(1.78)
 2.Molecular Hydrogen. Molecular hydrogen has two transitions that lie close to the Ly\(\alpha \) resonance: (a) the \(v=12P(5)\) transition, which lies \(\varDelta v=99\) km s\(^{1}\) redward of the Ly\(\alpha \) resonance, and (b) the \(12R(6)\) transition which lies \(\varDelta v=15\) km s\(^{1}\) redward of the Ly\(\alpha \) resonance. Vibrationally excited \(H_2\) may therefore convert Ly\(\alpha \) photons into photons in the \(H_2\) Lyman bands ([189], and references therein), and thus effectively destroy Ly\(\alpha \). This process can be included in a way that is very similar to that of dust, and by includingwhere \(f_{\mathrm{H}_2}\equiv n_{\mathrm{H}_2}/n_\mathrm{HI}\) denotes the molecular hydrogen fraction. This destruction process is often overlooked, but it is important to realize that Ly\(\alpha \) can be destroyed efficiently by molecular hydrogen. Neufeld [189] provides expressions for fraction of Ly\(\alpha \) that is allowed to escape as a function of \(f_{\mathrm{H}_2}\) and HI column density \(N_\mathrm{HI}\).$$\begin{aligned} n_\mathrm{HI}\sigma _{\alpha ,0}\phi (x) \rightarrow n_\mathrm{HI}[\sigma _{\alpha ,0}\phi (x)+f_{\mathrm{H}_2}\sigma _{\mathrm{H}_2}(x)], \end{aligned}$$(1.79)
 3.Collisional Mixing of the 2 s and 2 p Levels. Ly\(\alpha \) absorption puts a hydrogen atom in its 2p state, which has a lifetime of \(t=A^{1}_{\alpha } \sim 10^{9}\) s. During this short time, there is a finite probability that the atom interacts with nearby electrons and/or protons. These interactions can induce transitions of the form \(2p \rightarrow 2s\). Once in the 2sstate, the atom decays back to the groundstate by emitting two photons. This process is known as collisional deexcitation of the 2p state. Collisional deexcitation from the 2p state becomes more probable at high gas densities, and is predominantly driven by free protons. The probability that this process destroys the Ly\(\alpha \) photon, \(p_\mathrm{dest}\), at any scattering event is given by \(p_\mathrm{dest}=\frac{n_pC_\mathrm{2p2s}}{n_pC_\mathrm{2p2s}+A_{\alpha }}\). Here, \(n_\mathrm{p}\) denotes the number density of free protons, and \(C_\mathrm{2p2s}=1.8 \times 10^{4}\) cm\(^{3}\) s\(^{1}\) (e.g. and references therein [61]) denotes the collisional rate coefficient. This process can be included by rescaling the scattering redistribution function (see Sect. 1.6.3) toThis ensures that for each scattering event, there is a finite probability (\(p_\mathrm{dest}\)) that a photon is destroyed.$$\begin{aligned} R(x_\mathrm{out}x_\mathrm{in}) \rightarrow R(x_\mathrm{out}x_\mathrm{in})\times (1p_\mathrm{dest}). \end{aligned}$$(1.80)
 4.
Other. There are other processes that can destroy Ly\(\alpha \) photons, but which are less important. These can trivially be included in MonteCarlo codes that describe Ly\(\alpha \) radiative transfer (see Sect. 1.8). These processes include: (i) Ly\(\alpha \) photons can photoionize hydrogen atoms not in the ground state. The photoionisation crosssection from the \(n=2\) level by Ly\(\alpha \) photons is \(\sigma ^{\mathrm{Ly}\alpha }_\mathrm{ion}=5.8 \times 10^{19}\) cm\(^{2}\) (e.g. [57], p 108). This requires a nonnegligible populations of atoms in the \(n=2\) state which can occur in very dense media (see e.g. [23]) (ii) Ly\(\alpha \) photons can detach the electron from the H\(^\) ion. The crosssection for this process is \(\sigma =5.9\times 10^{18}\) cm\(^{2}\) (e.g. [240]) for Ly\(\alpha \) photons, which is almost an order of magnitude larger than the photoionisation crosssection from the \(n=2\) level at the Ly\(\alpha \) frequency. So, unless the H\(^{}\) number density exceeds \(0.1[n_\mathrm{2p}+n_\mathrm{2s}]\), where \(n_\mathrm{2s/2p}\) denotes the number density of Hatoms in the 2s/2p state, this process is not important.
1.6.5 Ly\(\alpha \) Propagation Through HI: Scattering as Double Diffusion Process
 1.
First, we assume that the Ly\(\alpha \) radiation field is isotropic, i.e. scattering completely eliminates any directional dependence of \(I_{\nu }(\mathbf{n})\). Under this assumption we can replace the intensity \(I_{\nu }(\mathbf{n})\) with the angleaveraged intensity \(J_{\nu }\equiv \frac{1}{4\pi }\int d\varOmega I_{\nu }(\mathbf{n})\).
 2.
Second, we replace frequency \(\nu \) with the dimensionless frequency variable x, introduced in Sect. 1.5.4.
 3.
Third, we ignore destruction processes and rewrite \(\alpha ^\mathrm{HI}_{\nu }(s)=n_\mathrm{HI}(s)\sigma _{\alpha }(\nu )=n_\mathrm{HI}\sigma _{\alpha ,0}\phi (x)\).
 4.Fourth, we define \(d\tau =n_\mathrm{HI}(s)\sigma _{\alpha ,0}ds\) and obtainwhere \(S_x(\tau )\equiv j_x(\tau )/(n_\mathrm{HI}\sigma _{\alpha ,0})\) denotes the ‘source’ function. In the integral x denotes the frequency of the photon after scattering, and \(x'\) denotes the frequency of the photon before scattering. Equation 1.81 is an integrodifferential equation, which is notoriously difficult to solve.$$\begin{aligned} \frac{\partial J(x)}{\partial \tau }=\phi (x)J(x)+S_{x}(\tau )+\int dx' \phi (x')J(x')R(xx'), \end{aligned}$$(1.81)
1.7 Basic Insights and Analytic Solutions
1.7.1 Ly\(\alpha \) Transfer Through Uniform, Static Gas Clouds
We consider a source of Ly\(\alpha \) photons in the center of a static, homogeneous sphere, whose linecenter optical depth from the center to the edge equals \(\tau _0\), where \(\tau _0\) is extremely large, say \(\tau _0=10^7\). This linecentre optical depth corresponds to an HI column density of \(N_\mathrm{HI}=1.6 \times 10^{20}\) cm\(^{2}\) (see Eq. 1.54). We further assume that the central source emits all Ly\(\alpha \) photons at line center (i.e. \(x=0\)). As the photons resonantly scatter outwards, they diffuse outward in frequency space. Figure 1.24 illustrates that as the photons diffuse outwards in real space, the spectral energy distribution of Ly\(\alpha \) flux, J(x), broadens. If we were to measure the spectrum of Ly\(\alpha \) photons crossing some arbitrary radial shell, then we would find that J(x) is constant up to \(\sim \pm x_\mathrm{max}\) beyond which it drops off fast. For Ly\(\alpha \) photons in the core of the line profile, the mean free path is negligible compared to the size of the sphere: the mean free path at frequency x is \([\tau _0\phi (x)]^{1}\) in units of the radius of the sphere. Because each scattering event changes the frequency of the Ly\(\alpha \) photon, the mean free path of each photon changes with each scattering event. From the shape of the redistribution function (see Fig. 1.21) we expect that on rare occasions Ly\(\alpha \) photons will be scattered further from resonance into the wing of the line (i.e. \(x \gtrsim 3\)). In the wing, the mean free path of the photon increases by orders of magnitude.
1.7.2 Ly\(\alpha \) Transfer Through Uniform, Expanding and Contracting Gas Clouds
Our previous analysis focussed on static gas clouds. Once we allow the clouds to contract or expand, no analytic solutions are known (except when \(T=0\), see below). We can qualitatively describe what happens when the gas clouds are not static.
There exists one analytic solution to radiative transfer equation through an expanding medium: [162] derived analytic expressions for the radial dependence of the angleaveraged intensity \(J(\nu ,r)\) of Ly\(\alpha \) radiation as a function of distance r from a source embedded within a neutral intergalactic medium undergoing Hubble expansion, (i.e. \(v_\mathrm{out}(r)=H(z)r\), where H(z) is the Hubble parameter at redshift z). Note that this solution was obtained assuming completely coherent scattering (i.e. \(x_\mathrm{out}=x_\mathrm{in}\) which corresponds to the special case of \(T=0\)), and that the photons frequencies change during flight as a result of Hubble expansion. Formally it describes a somewhat different scattering process than what we discussed before (see for a more detailed discussion [63]). However, [66] have compared this analytic solution to that of a MonteCarlo code that uses the proper frequency redistribution functions and find good agreement between the MonteCarlo and analytic solutions.
Finally, it is worth pointing out that in expanding/contracting media, the number of scattering events \(N_\mathrm{scat}\) and the total trapping time both decrease. The main reason for this is that in the presence of bulk motions in the gas, it becomes easier to scatter Ly\(\alpha \) photons into the wings of the line profiles where they can escape more easily. The reduction in trapping time has been quantified by [32], and in shell models (which will be discussed in Sect. 1.9.1 and Fig. 1.38) by Dijkstra and Loeb ([66], see their Fig. 1.6).
1.7.3 Ly\(\alpha \) Transfer Through Dusty, Uniform and Multiphase Media
Dust also destroys UVcontinuum photons, but because Ly\(\alpha \) photons scatter and diffuse spatially through the dusty medium, the impact of dust on Ly\(\alpha \) and UVcontinuum is generally different. This can affect the ‘strength’ (i.e. the equivalent width) of the Ly\(\alpha \) line compared to the underlying continuum emission. In a uniform mixture of HI gas and dust, Ly\(\alpha \) photons have to traverse a larger distance before escaping, which increases the probability to be destroyed by dust. In these cases we expect dust to reduce the EW of the Ly\(\alpha \) line. The ISM is not smooth and uniform however, which can drastically affect Ly\(\alpha \) radiative transfer. The interstellar medium is generally thought to consist of the ‘cold neutral medium’ (CNM), the ‘warm neutral/ionized medium’ (WNM/WIM), and the ‘hot ionized medium’ (HIM, see e.g. the classical paper by [173]). In reality, the cold gas is not in ‘clumps’ but rather in a complex network of filaments and sheets. Ly\(\alpha \) transfer calculations through realistic ISM models have only just begun, partly because modeling the multiphase nature of the ISM with simulations is a difficult task which requires extremely high spatial resolution. There is substantially more work on Ly\(\alpha \) transfer through ‘clumpy’ media that consists of cold clumps containing neutral hydrogen gas and dust, embedded within a (hot) ionized, dust free medium [77, 113, 136, 190], and which represent simplified descriptions of the multiphase ISM.
Laursen et al. [157] and Duval et al. [79] have recently shown however that—while clumpy media facilitate Ly\(\alpha \) escape—EW boosting only occurs under physically unrealistic conditions in which the clumps are very dusty, have a large covering factor, have very low velocity dispersion and outflow/inflow velocities, and in which the density contrast between clumps and interclump medium is maximized. While a multiphase (or clumpy) medium definitely facilitates the escape of Ly\(\alpha \) photons from dusty media, EW boosting therefore appears uncommon. We can understand this result as follows: the preferential destruction of UVcontinuum photons over Ly\(\alpha \) requires at least a significant fraction of Ly\(\alpha \) photons to avoid seeing the dust by scattering off the surface of the clumps. How deep the Ly\(\alpha \) photons actually penetrate, depends on their meanfree path, which depends on their frequency in the frame of the clumps. If clumps are moving fast, then it is easy for Ly\(\alpha \) photons to be Doppler boosted into the wing of the line profile (in the clump frame), and they would not scatter exclusively on the clump surfaces. Finally, we note that the conclusions of [79, 136] apply to the EWboost averaged over all photons emerging from the dusty medium. Gronke and Dijkstra [103] have investigated that for a given model, there can be directional variations in the predicted EW, with large EW boosts occurring in a small fraction of sightlines in directions where the UVcontinuum photon escape fraction was suppressed, thus partially restoring the possibility of EW boosting by a multiphase ISM.
1.8 MonteCarlo Ly\(\alpha \) Radiative Transfer
Analytic solutions to the radiative transfer equation (Eq. 1.58) only exist for a few idealised cases. A modern approach to solve this equation is via MonteCarlo methods, which refer to a ‘broad class of computational algorithms [...] which change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations’ (S. Ulam, see https://en.wikipedia.org/wiki/Monte_Carlo_method).^{35}
In Ly\(\alpha \) MonteCarlo radiative transfer, we represent the integrodifferential equation (Eq. 1.58) by a succession of random scattering events until Ly\(\alpha \) photons escape [5, 22, 24, 41, 63, 75, 85, 91, 146, 151, 153, 160, 162, 196, 210, 239, 256, 269, 279, 285, 286]. Details on how the MonteCarlo approach works can be found in many papers (see e.g. the papers mentioned above, and Chaps. 6–8 of [155], for an extensive description). I will first provide a brief description of drawing random variables, which is central to the MonteCarlo method. Then I will describe MonteCarlo Ly\(\alpha \) radiative transfer.
1.8.1 General Comments on MonteCarlo Methods
1.8.2 Ly\(\alpha \) MonteCarlo Radiative Transfer
 1.
We first randomly draw a position, \(\mathbf{r}\), from which the photon is emitted from the emissivity profile^{36} \(j_{\nu }(\mathbf{r})\) (see Eq. 1.58). We the assign a random frequency x, which is drawn from the Voigt function \(\phi (x)\), and a random propagation direction \(\mathbf{k}\).
 2.
We randomly draw the optical depth \(\tau \) the photon propagates into from the distribution \(P(\tau )=\exp (\tau )\).
 3.
We convert \(\tau \) into a physical distance s by (generally numerically) inverting the line integral \(\tau =\int _0^s d\lambda n_\mathrm{HI} (\mathbf{r}')\sigma _{\alpha }(x'[\mathbf{r}'])\), where \(\mathbf{r}' = \mathbf{r}+\lambda \mathbf{k}\) and \(x'=x\mathbf{v}(\mathbf{r}')\cdot \mathbf{k}/(v_\mathrm{th})\). Here, \(\mathbf{v}(\mathbf{r}')\) denotes the 3D bulk velocity vector of the gas at position \(\mathbf{r}'\). Note that \(x'\) is the dimensionless frequency of the photon in the ‘local’ frame of the gas at \(\mathbf{r}'\).
 4.
Once we have selected the scattering location, we need to draw the thermal velocity components of the atom that is scattering the photon (we only need the thermal velocity components, as we work in the local gas frame). As in Sect. 1.6.3, we decompose the thermal velocity of the atom into a direction parallel to that of the incoming photon, \(v_{}\) (or its dimensionless analogue u, see Eq. 1.66), and a 2Dvelocity vector perpendicular to \(\mathbf{k}\), namely \(\mathbf{v}_{\perp }\). We discussed in Sect. 1.6.3 what the functional form of the conditional probability P(ux) (see discussion below Eq. 1.68), and apply the rejection method to draw u from this functional form (see the Appendix [285] for a functional form of T(ux)). The 2 components of \(\mathbf{v}_{\perp }\) can be drawn from a MaxwellBoltzmann distribution (see e.g. [63]).
 5.
Once we have determined the velocity vector of the atom that is scattering the photon, we draw an outgoing direction of the photon after scattering, \(\mathbf{k}_\mathrm{out}\), from the phasefunction, \(P(\mu )\) (see Eq. 1.41 and Sect. 1.5.1). We will show below that this procedure of generating the atom’s velocity components and random new directions generates the proper frequency redistribution functions, as well as their angular dependence.
 6.
Unless the photon escapes, we replace the photon propagation direction and frequency and go back to (1). Once the photon escapes we record information we are interested in such as the location of last scattering, the frequency of the photon, the thermal velocity components of the atom that last scattered the photon, the number of scattering events the photon underwent, the total distance it travelled through the gas, etc.
It is important to test Ly\(\alpha \) MonteCarlo codes in as many ways as possible. Figure 1.32 shows a minimum set of tests MonteCarlo codes must be able to reproduce. These comparisons with analytic solutions test different aspects of the code.
The histograms in the top left panel show MonteCarlo realizations of the Ly\(\alpha \) spectra emerging from a uniform spherical gas cloud, in which Ly\(\alpha \) photons are injected in the center of the sphere, at the line center (i.e. \(x=0\)). The total line center optical depth, \(\tau _0\) from the center to the edge is \(\tau _0=10^5\) (blue), \(\tau _0=10^6\) (red) and \(\tau _0=10^7\) (green). Overplotted as the black dotted lines are the corresponding analytic solutions (see Eq. 1.92, but modified for a sphere, see [63]). The agreement is perfect at high optical depth (\(a_v\tau \gtrsim 10^3\)). At lower optical depth, \(a_v\tau _0 \lesssim 10^3\), the analytic solutions are not expected to be accurate any more (see Sect. 1.7.1). Because the analytic solutions were obtained under the assumption that scattering occured in the wing, the agreement between analytic and MonteCarlo techniques at high \(\tau _0\) only confirms that the MonteCarlo procedure accurately describes scattering in the wing of the line profile. This comparison does not test core scattering, which make up the vast majority of scattering events (see Sect. 1.7.1). The fact that Ly\(\alpha \) spectra emerging from optically (extremely) thick media is insensitive to core scattering implies that we need additional tests to test core scattering. However, it also implies we can skip these corescattering events, which account for the vast majority of all scattering events. That is, MonteCarlo simulations can be ‘accelerated’ by skipping core scattering events. We discuss how we can do this in more detail in Sect. 1.8.4.
In the upper right panel the colored histograms show MonteCarlo realization of the frequency redistribution functions, \(R(x_\mathrm{out},x_\mathrm{in})\) (see Eq. 1.72), for \(x_\mathrm{in}=0\) (blue), \(x_\mathrm{in}=2\) (red) and \(x_\mathrm{in}=5\) (green). The solid lines are the analytic solutions given by Eq. 1.72 (here for dipole scattering). This comparison tests individual Ly\(\alpha \) core scattering events, and thus complements the test we described above.
In the lower left panel the circles show the total number of scattering events that a Ly\(\alpha \) photon experiences before it escapes from a slab of optical thickness \(2 \tau _0\) in a MonteCarlo simulation. Overplotted as the red–solid line is the theoretical prediction that \(N_\mathrm{scat}=C\tau _0\), with \(C=1.1\) (see Eq. 1.89, [1, 114, 189]). The break down at low \(\tau _0\) corresponds to the range of \(\tau _0\) where analytic solutions are expected to fail. This test provides another way to test core scattering events, as \(N_\mathrm{scat}\) is set by the probability that a Ly\(\alpha \) photon is first scattered sufficiently far into the wing of the line such that it can escape in a single ‘excursion’. This test also shows how accurate the analytic prediction is, despite the fact that the derivation presented in Sect. 1.7.1 (following [1]) did not feel like it should be this accurate. This plot also underlines how computationally expensive Ly\(\alpha \) transfer can be if we simulate each scattering event.
1.8.3 Extracting Observables from Ly\(\alpha \) MonteCarlo Simulations in 3D Simulations
In MonteCarlo radiative transfer calculations applied to arbitrary 3D gas distributions, extracting observables requires (a bit) more work than recording the location of last scattering, the photon’s frequency etc. This is because formally we are interested only in a tiny subset of Ly\(\alpha \) photons that escape, and end up in the mirror of our telescope. We denote the direction from the location of last scattering towards the telescope with \(\mathbf{k}_\mathrm{t}\). The mirror of our telescope only subtends a solid angle \(d\varOmega _\mathrm{telescope}=\frac{dA_\mathrm{telescope}}{d^2_\mathrm{A}(z)}\), where \(dA_\mathrm{telescope}\) denotes the area of the mirror, and \(d_\mathrm{A}(z)\) denotes the angular diameter distance to redshift z. The probability that a Ly\(\alpha \) photon in our MonteCarlo simulation escapes from the scattering medium into this tiny solid angle is negligible.
An example of an image generated with the peeling algorithm is shown in Fig. 1.34. Here, a Ly\(\alpha \) source is at the origin of a cartesian coordinate system. Each of the 3 coordinate axes has 2 spheres of HI gas at identical distances from the origin. There are no hydrogen atoms outside the sphere, and the Ly\(\alpha \) scattering should only occur inside the 6 spheres. Resulting images (taken from Dijkstra and Kramer 2012) from 6 viewing directions are shown in the left panel of Fig. 1.34. The ‘darkness’ of a pixel represents its Ly\(\alpha \) surface brightness. The average of these 6 images is shown in the box. The right panel shows a closeup view of the image associated with the sphere on the \(+y\)axis. The side of the sphere facing the Ly\(\alpha \) source (on the bottom at \(y=0\)) is brightest. The red line in the inset shows an analytic calculation of the expected surface brightness, under the assumption that the sphere as a whole is optically thin (only this assumption allows for analytic solutions). This image shows that the Peeling algorithm gives rise to the sharp features in the surface brightness profiles that should exist. Note that the alternative method we briefly mentioned above, which averages over all viewing directions within some angle \(\varDelta \alpha \) from \(\mathbf{k}_\mathrm{t}\), would introduce some blurring to these images.
1.8.4 Accelerating Ly\(\alpha \) MonteCarlo Simulations
1.9 Ly\(\alpha \) Transfer in the Universe
Previous sections discussed the basics of the theory describing Ly\(\alpha \) transfer through optically thick media. The goal of this section is to discuss what we know about Ly\(\alpha \) transfer in the real Universe. We decompose this problem into several scales: (i) Ly\(\alpha \) photons have to escape from the interstellar medium (ISM) of galaxies into the circum galactic/intergalactic medium (CGM/IGM). We discuss this in Sect. 1.9.1. We then go to large scales, and describe the subsequent radiative transfer through the CGM/IGM at lower redshift (Sect. 1.9.2) and at higher redshift when reionization is still ongoing (Sect. 1.9.4).
1.9.1 Interstellar Radiative Transfer
^{38}Understanding interstellar Ly\(\alpha \) radiative transfer requires us to understand gaseous flows in a multiphase ISM, which lies at the heart of understanding star and galaxy formation. Modelling the neutral component of interstellar medium is an extremely challenging task, as it requires resolving the multiphase structure of interstellar medium, and how it is affected by feedback from starformation (via supernova explosions, radiation pressure, cosmic ray pressure, etc). Instead of taking an ‘abinitio’ approach to understanding Ly\(\alpha \) transfer, it is illuminating to use a ‘topdown’ approach in which we try to constrain the broad impact of the ISM on the Ly\(\alpha \) radiation field from observations (for this also see the lecture notes by M. Ouchi and M. Hayes for more extended discussions of the observations).
We first focus on observational constraints on the escape fraction of Ly\(\alpha \) photons, \(f^{\alpha }_\mathrm{esc}\). To estimate \(f^{\alpha }_\mathrm{esc}\) we would need to compare the observed Ly\(\alpha \) luminosity to the intrinsic Ly\(\alpha \) luminosity. The intrinsic Ly\(\alpha \) luminosity corresponds to the Ly\(\alpha \) luminosity that is actually produced. The best way to estimate the intrinsic Ly\(\alpha \) luminosity is from some other nonresonant nebular emission line such as H\(\alpha \). The observed H\(\alpha \) luminosity can be converted into an intrinsic H\(\alpha \) luminosity once nebular reddening is known (from joint measurements of e.g. the H\(\alpha \) and H\(\beta \) lines see lecture notes by M. Hayes). Once the intrinsic H\(\alpha \) luminosity is known, then we can compute the intrinsic Ly\(\alpha \) luminosity assuming caseB (or caseA) recombination. This procedure indicates that \(f^{\alpha }_\mathrm{esc}\sim 12\%\) at \(z\sim 0.3\) [60] and \(z\sim 5\%\) at \(z\sim 2\) [120].
The dependence of \(f^{\alpha }_\mathrm{esc}\) on dust content of galaxies is an intuitive result, as it is practically the only component of the ISM that is capable of destroying Ly\(\alpha \). However, there is more to Ly\(\alpha \) escape. This is probably best illustrated by nearby starburst galaxy 1Zwicky18 (shown in Fig. 1.36). This is a metal poor, extremely blue galaxy, and it has even been argued to host Population III stars (i.e. stars that formed our of primordial gas). We would expect this galaxy to have a high \(f^{\alpha }_\mathrm{esc}\). However, the spectrum of 1Zwicky18 (also shown in Fig. 1.36) shows strong Ly\(\alpha \) absorption. In contrast, the more enriched (\(Z\sim 0.10.3 Z_{\odot }\)) nearby galaxy ESO 350 does show strong Ly\(\alpha \) emission (see Fig. 1.1 of [149]). The main difference between the two galaxies is that ESO 350 shows evidence for the presence of outflowing gas. Kunth et al. [149] observed that for a sample of 8 nearby starburst galaxies, 4 galaxies that showed evidence for outflows showed Ly\(\alpha \) in emission, while no Ly\(\alpha \) emission was detected for the 4 galaxies that showed no evidence for outflows (irrespective of the gas metallicity of the galaxies). These observations indicate that gas kinematics is a key parameter that regulates Ly\(\alpha \) escape [13, 149, 220, 275]. This result is easy to understand qualitatively: in the absence of outflows, the Ly\(\alpha \) sources are embedded within a static optically thick scattering medium. The traversed distance of Ly\(\alpha \) photons is enhanced compared to that of (nonionizing) UV continuum photons (see Sect. 1.7.1), which makes them more ‘vulnerable’ to destruction by dust. In contrast, in the presence of outflows Ly\(\alpha \) photons can be efficiently scattered into the wing of the line profile, where they can escape easily.
As modelling the outflowing component in interstellar medium is an extremely challenging task (as we mentioned in the beginning of this section), simplified representations, such as the popular ‘shell model’, have been invoked. In the shell model the outflow is represented by a spherical shell with a thickness that is 0.1\(\times \) its inner/outer radius. Figure 1.38 summarizes the different ingredients of the shell model. The two parameters that characterize the Ly\(\alpha \) sources are (i) its equivalent width (EW) which measures the ‘strength’ of the source compared to the underlying continuum, (ii) its full width at half maximum (FWHM) which denotes the width of the spectral line prior to scattering. This width may reflect motions in the Ly\(\alpha \) emitting gas. The main properties that characterise the shell are its (i) HIcolumn density, \(N_\mathrm{HI}\), (ii) outflow velocity, \(v_\mathrm{sh}\), (iii) ‘bparameter’ \(b^2\equiv v^2_\mathrm{turb}+v^2_\mathrm{th}\). Here, \(v_\mathrm{th}=\sqrt{2k_\mathrm{B}T/m_\mathrm{p}}\) (which we encountered before), and \(v_\mathrm{turb}\) denotes its turbulent velocity dispersion; (iv) its dust content (e.g. [7, 269, 270]).
In spite of its success, there are two issues with the shellmodels: (i) gas in the shells has a single outflow velocity and a small superimposed velocity dispersion, while observations of lowionization absorption lines indicate that outflows typically cover a much wider range of velocities (e.g. [126, 150]); and (ii) observations of lowionization absorption lines also suggest that outflows—while ubiquitous—do not completely surround UVcontinuum emitting regions of galaxies. Observations by [139] show that the maximum lowionization covering fraction is \(100\%\) in only 2 out of 8 of their \(z>2\) galaxies (also see [124], who find evidence for a low covering factor of optically thick, neutral gas in a small fraction of lower redshift Lyman Break Analogues). There is thus some observational evidence that there exist sight lines that contain no detectable lowionization (i.e. cold) gas, which may reflect the complex structure associated with outflows which cannot be captured with spherical shells. Two caveats are that (a) the inferred covering factors are measured as a function of velocity (and can depend on spectral resolution, see e.g. but [139] discuss why this is likely not an issue in their analysis [214]). Gas at different velocities can cover different parts of the source, and the outflowing gas may still fully cover the UV emitting source. This velocitydependent covering is nevertheless not captured by the shellmodel; (b) the lowionization metal absorption lines only probe enriched cold (outflowing) gas. Especially in younger galaxies it may be possible that there is additional cold (outflowing) gas that is not probed by metal absorption lines.
Shibuya et al. [241] have shown that Ly\(\alpha \) line emission is stronger in galaxies in which the covering factor of lowionization material is smaller (see their Fig. 1.10, also see [262]). Similarly, [138] found the average absorption line strength in lowionization species to decrease with redshift, which again coincides with an overall increase in Ly\(\alpha \) flux from these galaxies [251]. Besides dust, the covering factor of HI gas therefore plays an additional important role in the escape of Ly\(\alpha \) photons. These cavities may correspond to regions that have been cleared of gas and dust by feedback processes (see [191, 192] who describe a simple ‘blowout’ model).
In short, dusty outflows appear to have an important impact on the interstellar Ly\(\alpha \) radiative process, and give rise to redshifted Ly\(\alpha \) lines. Low HIcolumn density holes further facilitate the escape of Ly\(\alpha \) photons from the ISM, and can alter the emerging spectrum such that Ly\(\alpha \) photons can emerge closer to the galaxies’ systemic velocities ([25, 105, 106, 116], also see [288]).
1.9.2 Transfer in the Ionised IGM/CGM
HI gas that exists outside of the galaxy can further scatter Ly\(\alpha \) that escaped from the ISM. If this scattering occurs sufficiently close to the galaxy, then this radiation can be detected as a low surface brighteness glow (e.g. [286, 287]). As we showed previously in Fig. 1.13, observations indicate that spatially extended Ly\(\alpha \) halos appear to exist generally around starforming galaxies (see e.g. [96, 107, 117, 118, 119, 123, 170, 186, 191, 199, 217, 274]). Understanding what fraction of the Ly\(\alpha \) flux in these halos consists of scattered Ly\(\alpha \) radiation that escaped from the ISM, and what fraction was produced insitu (as recombination, cooling, and/or fluorescence, see Sect. 1.4) is still an open question,^{42} which we can address with integral field spectographs such as MUSE and the Keck Cosmic Web Imager. Polarization measurements (see Sect. 1.10) should also be able to distinguish between in situproduction and scattering [23, 67, 122], although these differences can be subtle [263].
1.9.3 Intermezzo: Reionization
Reionization refers to the transformation of the intergalactic medium from fully neutral to fully ionized. For reviews on the Epoch of Reionization (EoR) we refer the reader to e.g. [18, 95, 187], and the recent book by [184]. The EoR is characterized by the existence of patches of diffuse neutral intergalactic gas, which provide an enormous source of opacity to Ly\(\alpha \) photons: the GunnPeterson optical depth is \(\tau _\mathrm{GP}\sim 10^6\) (see Eq. 1.99) in a fully neutral medium. It is therefore natural to expect that detecting Ly\(\alpha \) emitting galaxies from the EoR is hopeless. Fortunately, this is not the case, as we discuss in Sect. 1.9.4.
Ideally, one would like to simulate reionization by performing full radiative transfer calculations of ionising photons on cosmological hydrodynamical simulations. A number of groups have developed codes that can perform these calculations in 3D (e.g. [55, 89, 98, 99, 132, 206, 245, 248, 260]). These calculations are computationally challenging as one likes to simultaneously capture the large scale distribution of HII bubbles, while resolving the photon sinks (such as Lyman Limit systems) and the lowest mass halos (\(M\sim 10^8\) M\(_{\odot }\)) which can contribute to the ionising photon budget (see e.g. [261]). Modeling reionization contains many poorly known parameters related to galaxy formation, the ionising emissivity of starforming galaxies, their spectra etc. Alternative, faster ‘seminumeric’ algorithms have been developed which allow for a more efficient exploration of the full parameter space (e.g. [72, 164, 180, 246]). These seminumeric algorithms utilize excursionset theory to determine if a cell inside a simulation is ionized or not [93]. Detailed comparisons between full radiation transfer simulations and seminumeric simulations show both methods produce very similar ionization fields [283].
The picture of reionization that has emerged from analytical consideration and largescale simulations is one in which the early stages of reionization are characterized by the presence of HII bubbles centered on overdense regions of the Universe, completely separated from each other by a neutral IGM [93, 132, 176]. The ionized bubbles grew in time, driven by a steadily growing number of starforming galaxies residing inside them. The final stages of reionization are characterized by the presence of large bubbles, whose individual sizes exceeded tens of cMpc (e.g. [164, 283]). Ultimately these bubbles overlapped (percolated), which completed the reionization process. The predicted redshift evolution of the ionization state of the IGM in a realistic reionization model is shown in Fig. 1.42. This Figure illustrates the inhomogeneous, temporally extended nature of the reionization process.
1.9.4 Intergalactic Ly\(\alpha \) Radiative Transfer during Reionization
Current models indicate that if the observed reduction in Ly\(\alpha \) flux from galaxies at \(z>6\) is indeed due to reionization—which is plausible (see [184] for recent constraints on the reionization history from a suite of observations)—then this requires a volume filling factor of diffuse neutral gas which exceeds \(\langle x_\mathrm{HI} \rangle \gtrsim 40\%\), which implies that reionization is still ongoing at \(z \sim \) 6–7 (see [75] for a review). This constraint is still uncertain due to the limited number of Ly\(\alpha \) galaxies at \(z\sim 6\) and \(z\sim 7\), but this situation is expected to change, especially with large surveys for highz Ly\(\alpha \) emitters to be conducted with Hyper SuprimeCam. These surveys will enable us to measure the variation of IGM opacity on the sky at fixed redshift, and constrain the reionization morphology (see [136, 137, 182]).
1.10 Miscalleneous Topics I: Polarization
In Sect. 1.5.1 we discussed how the intensity of scattered radiation \(I \propto \sin ^2\varPsi \), where \(\cos \varPsi \equiv \mathbf{k}_\mathrm{out} \cdot \mathbf{e}_\mathrm{E}\) in which \(\mathbf{e}_\mathrm{E}\) denotes the normalized direction of the electric vector (see Fig. 1.15). Similarly, we can say that the amplitude of the electricfield scales as \(E \propto \sin \varPsi \) (note at \(I \propto E^2\)), i.e. we project the electric vector onto the plane perpendicular to \(\mathbf{k}_\mathrm{out}\) (see the left panel of Fig. 1.46). This same argument can be applied to demonstrate that a free electron can transform unpolarized into a polarized radiation if there is a ‘quadrupole anisotropy’ in the incoming intensity: the right panel of Fig. 1.46 shows a free electron with incident radiation from the left and from the top. If the incident radiation is unpolarized, then the electric field vector points in arbitrary direction in the plane perpendicular to the propagation direction. Consider scattering by 90\(^{\circ }\). If we apply the projection argument, then for radiation incident from the left we only ‘see’ the component of the Efield that points upward (shown in blue). Similarly, for radiation coming in from the top we only see the Efield that lies horizontally. The polarization of the scattered radiation vanishes if the blue and red components are identical, which—for unpolarized radiation—requires that the total intensity of radiation coming in from the top must be identical to the that coming in from the left. For this reason, electron scattering can polarize the Cosmic Microwave Background if the intensity varies on angular scales of \(90^{\circ }\). If fluctuation exist on these scales, then the CMB is said to have a nonzero quadrupole moment. Similarly, if there were a point source irradiating the electron from the top, then we would also expect only the red Evector to be transmitted.
1.10.1 Quantum Effects on Ly\(\alpha \) Scattering: The Polarizability of the Hydrogen Atom
In order to accurately describe how H atoms scatter Ly\(\alpha \) radiation, we must consider the finestructure splitting of the 2p level. The spin of the electron causes the 2p state quantum state to split into the \(2p_{1/2}\) and \(2p_{3/2}\) levels, which are separated by \({\sim }10\) Ghz (see Fig. 1.49). The notation that is used here is \(nL_{J}\), in which \(\mathbf{J}=\mathbf{L}+\mathbf{s}\) denotes the total (orbital +spin) angular momentum of the electron. The \(1s_{1/2}\rightarrow 2p_{1/2}\) and \(1s_{1/2}\rightarrow 2p_{3/2}\) is often referred as the Kline and Hline, respectively.
It turns out that a quantum mechanical calculation yields that \(E_1=\frac{1}{2}\) for the H transition, while \(E_1=0\) for the K transition (e.g. [6, 37, 112, 159]). When a Ly\(\alpha \) scattering event goes through the Ktransition, the hydrogen atom behaves like an isotropic scatterer. This is because the wavefunction of the \(2p_{1/2}\) state is spherically symmetric (see [273]), and the atom ‘forgot’ which direction the photon came from or which direction the electric field was pointing to. For the \(2p_{3/2}\) state, the wavefunction is not spherically symmetric and contains the characteristic ‘double lobes’ shown in Fig. 1.3. The hydrogen in the \(2p_{3/2}\) state thus has some memory of the direction of the incoming Ly\(\alpha \) photon and its electric vector, and behaves partially as a classical dipole scatterer, and partly as an isotropic scatterer.
 1.
\(E_1=0\) at \(\lambda =\lambda _K\) and that \(E_1=\frac{1}{2}\) at \(\lambda =\lambda _H\), which agrees with earlier studies (e.g. [37, 112]), and which reflects what we discussed above.
 2.
\(E_1\) is negative for most wavelengths in the range \(\lambda _H< \lambda < \lambda _K\). The classical analogue to this would be that when an atom absorbs a photon at this frequency, that then the electron oscillates along the propagation direction of the incoming wave, which is strange because the electron would be oscillating in a direction orthogonal to the direction of the electric vector of the electromagnetic wave. However, scattering at these frequencies is very unlikely (see Sect. 1.10.2).
 3.
\(E_1=1\) when a photon scatters in the wings of the line, which is arguably the most bizarre aspect of this plot. Stenflo [254] points out that, when a Ly\(\alpha \) photon scatters in the wing of the line profile, it goes simultaneously through the \(2p_{1/2}\) and \(2p_{3/2}\) states, and as a result, the bound electron is permitted to behave as if it were free.
1.10.2 Ly\(\alpha \) Propagation Through HI: Resonant Versus Wing Scattering
1.10.3 Polarization in MonteCarlo Radiative Transfer
Incorporating polarization in a MonteCarlo is complicated if you want to do it correctly. A simple procedure was presented by [12, 162], which is accurate for wingscattering only. In practise this is often sufficient, as Ly\(\alpha \) wing photons are the ones that are most likely to escape from a scattering medium, and are thus most likely to be observed.
Consider a Ly\(\alpha \) photon in the MonteCarlo simulation that was propagating in a direction \(\mathbf{k}_\mathrm{in}\), and with an electric vector \(\mathbf{e}_\mathrm{E}\). It was then scattered towards the observer. The polarization of this photon when it is observed can be obtained as follows: we know that when a photon reaches us, its propagation direction, \(\mathbf{k}_\mathrm{out}\), is perpendicular to the plane of the sky. We therefore know that the polarization vector \(\mathbf{e}'_\mathrm{E}\) must lie in the plane of the sky. The linear polarization measures the difference in intensity when measured in the two orthogonal directions in the plane of the sky (denoted previously with \(I_{}\) and \(I_{\perp }\)). We now define \(\mathbf{r}\) to be the vector that connects the location of last scattering to the Ly\(\alpha \) source, projected onto the sky. Both \(\mathbf{r}\) and \(\mathbf{e}'_\mathrm{E}\) therefore lie in the plane of the sky, and we let \(\chi \) denote the angle between them (i.e. \(\cos \chi \equiv \mathbf{r}\cdot \mathbf{e}'_\mathrm{E}/\mathbf{r}\)). The photon then contributes \(\cos ^2 \chi \) to \(I_l\) and \(\sin ^2 \chi \) to \(I_r\). This geometry is depicted in Fig. 1.52. This procedure was tested successfully by [162] against analytic solutions obtained by Schuster [238].
The previous approach assigns electric vectors to each Ly\(\alpha \) photon in the MonteCarlo simulation, and therefore implicitly assumes that each individual Ly\(\alpha \) photon is \(100\%\) linearly polarized. It would be more realistic if we could assign a fractional polarization to each photon, which would be more representative of the radiation field. Recall that the phasefunctions depend on the polarization of the radiation field. An alternative way of incorporating polarization which allows fractional polarization to be assigned to individual Ly\(\alpha \) photons is given by the densitymatrix formalism described in [8]). In this formalism all polarization information is encoded in 2 parameters (the 2 parameters reflect the degrees of freedom for a massless spin0 ‘particle’) within the ‘density matrix’. We will not discuss this formalism in this lecture. Both methods should converge for scattering in optically thick gas, but they have not been compared systematically yet (but see [49] for recent work in this direction).
1.11 Applications Beyond Ly\(\alpha \): WouthuysenField Coupling and 21cm Cosmology/Astrophysics
1.11.1 The 21cm Transition and its Spin Temperature
The 21cm transition is a highly forbidden transition with a natural lifetime of \(t\equiv A_{21}^{1}\sim (2.87\times 10^{15}\mathrm{s}^{1})^{1}\sim 1.1 \times 10^7\) yr (one way to interpret this long lifetime is to connect it to the low probability of the electron and proton overlapping). The 21cm line has been observed routinely in nearby galaxies, and has allowed us to map out the distribution and kinematics of HI gas in galaxies. Observations of the kinematics of HI gas have given us galaxy rotation curves, which further confirmed the need for dark matter on galaxy scales. Because of its intrinsic faintness, it is difficult to detect HI gas in emission beyond \(z\gtrsim 0.5\) until the Square Kilometer Array (SKA) becomes operational.
1.11.2 The 21cm Brightness Temperature
1.11.3 The Spin Temperature and the WouthuysenField Effect
1.11.4 The Global 21cm Signal
Adiabatic expansion of the Universe causes \(T_\mathrm{CMB} \propto (1+z)\) at all redshifts. At \(z \gtrsim 100\), the gas temperature remains coupled to the CMB temperature because of the interaction between CMB photons and the small fraction of free electrons that exist because recombination is never ‘complete’ (i.e. there is a residual fraction of protons that never capture an electron to form a hydrogen atom). When \(T_\mathrm{CMB}=T_\mathrm{gas}\), we must have \(T_\mathrm{s}=T_\mathrm{CMB}\), and therefore that \(\delta T_\mathrm{b}(\nu )=0\) mK, which corresponds to the highz limit in the right panel.
At \(z \lesssim 100\), electron scattering can no longer couple the CMB and gas temperatures, and the (nonrelativistic) gas adiabatically cools faster than the CMB as \(T_\mathrm{gas} \propto (1+z)^2\). Because \(T_\mathrm{gas} < T_\mathrm{CMB}\), we must have that \(T_\mathrm{s} < T_\mathrm{CMB}\) and we see the 21cm line in absorption. When \(T_\mathrm{gas}\) first decouples from \(T_\mathrm{CMB}\) the gas densities are high enough for collisions to keep \(T_\mathrm{s}\) locked to \(T_\mathrm{gas}\). However, at \(z\sim 70\) (\(\nu \sim 20\) MHz) collisions can no longer couple \(T_\mathrm{s}\) to \(T_\mathrm{gas}\), and \(T_\mathrm{s}\) crawls back to \(T_\mathrm{CMB}\), which reduces \(\delta T_\mathrm{b}(\nu )\) (at \(\nu \sim \) 20–50 MHz, i.e. \(z\sim \)70–30).
The first stars, galaxies, and accreting black holes emitted UV photons in the energy range \(E=10.213.6\) eV. These photons can travel freely through the neutral IGM, until they redshift into one of the Lyman series resonances, at which point a radiative cascade can produce Ly\(\alpha \). The formation of the first stars thus generates a Ly\(\alpha \) background, which initiates the WFcoupling, which pushes \(T_\mathrm{s}\) back down to \(T_\mathrm{gas}\). The onset of Ly\(\alpha \) scattering—and thus the WF coupling—causes \(\delta T_\mathrm{b}(\nu )\) to drop sharply at \(\nu \gtrsim 50\) Mhz (\(z \lesssim 30\)).
At some point the radiation of stars, galaxies, and black holes starts heating the gas. Especially Xrays produced by accreting black holes can easily penetrate deep into the cold, neutral IGM and contain a lot of energy which can be converted into heat after they are absorbed. The left panel thus has \(T_\mathrm{gas}\) increase at \(z\sim 20\), which corresponds to onset of Xray heating. In the right panel this onset occurs a bit earlier. This difference reflects that the redshift of all the features (minima and maxima) in \(\delta T_\mathrm{b}(\nu )\) are model dependent, and not wellknown (more on this below). With the onset of Xray heating (combined with increasingly efficient WF coupling to the buildup of the Ly\(\alpha \) background) drives \(\delta T_\mathrm{b}(\nu )\) up until it becomes positive when \(T_\mathrm{gas} > T_\mathrm{CMB}\).
Finally, \(\delta T_\mathrm{b}(\nu )\) reaches yet another maximum, which reflects that neutral, Xray heated gas is reionized away by the ionizing UVphotons emitted by star forming galaxies and quasars. When reionization is complete, there is no diffuse intergalactic neutral hydrogen left, and \(\delta T_\mathrm{b}(\nu ) \rightarrow 0\).
In detail, the onset and redshift evolution of Ly\(\alpha \) coupling, Xray heating, and reionization depend on the redshift evolution of the number densities of galaxies, and their spectral characteristics. All these are uncertain, and it is not possible to make robust predictions for the precise shape of the global 21cm signature. Instead, one of the main challenges for observational cosmology is to measure the global 21cm signal, and from this constrain the abundances and characteristic of first generations of galaxies in our Universe. Detecting the global 21cm is challenging, but especially the deep absorption trough that is expected to exist just prior to the onset of Xray heating at \(\nu \sim 70\) MHz is something that may be detectable because of its characteristic spectral shape. It is interesting that the presence of this absorption feature relies on the presence of a Ly\(\alpha \) background, which must be strong enough to enable WFcoupling.
Footnotes
 1.
Examples of new instruments/telescopes that will revolutionize our ability to target the Ly\(\alpha \) emission line: the HobbyEberly Telescope Dark Energy Experiment (HETDEX, http://hetdex.org/ ) will increase the sample of Ly\(\alpha \) emitting galaxies by orders of magnitude at \(z \sim \) 2–4; Subaru’s Hyper SuprimeCam (http://www.naoj.org/Projects/HSC/) is expected to provide a similar boost out to \(z\sim 7\). Integral Field Unit Spectrographs such as MUSE (https://www.eso.org/sci/facilities/develop/instruments/muse.html, also see the Keck Cosmic Web Imager http://www.srl.caltech.edu/sal/keckcosmicwebimager.html) will allow us to map out spatially extended Ly\(\alpha \) emission down to \({\sim }10\) times lower surface brightness levels, and take spatially resolved spectra. In the (near) future, telescopes such as the James Webb Space Telescope (JWST, http://www.jwst.nasa.gov/ ) and ground based facilities such as the Giant Magellan Telescope (http://www.gmto.org/) and ESO’s EELT (http://www.eso.org/public/usa/telesinstr/eelt/, http://www.tmt.org/ (TMT).
 2.
 3.
 4.
 5.
 6.
 7.
 8.
The subscripts ‘l’ and ‘u’ refer to the ‘lower’ and ‘upper’ energy states, respectively.
 9.
 10.This coefficient is given by where fundamental quantities e, c, \(h_\mathrm{P}\), and \(a_0\) are given in Table 1.1, \(h_\mathrm{P}\nu _\mathrm{ul}\) denotes the energy difference between the upper (n,l) and lower (n’,l’) state. The matrix \(M(n,l,n',l')\) involves an overlap integral that involves the radial wavefunctions of the states (n, l) and \((n',l')\):
Analytic expressions for the matrix \(M(n,l,n',l')\) that contain hypergeometric functions were derived by [101]. For the Ly\(\alpha \) transition \(M(n,l,n',l')=M(2,1,1,0)=\sqrt{6}(128/243)\) [128].
 11.
At gas densities that are relevant in most astrophysical plasmas, hydrogen atoms predominantly populate their electronic ground state (\(n=1\)), and the opacity in the Balmer lines is generally negligible. In theory one can introduce caseC/D/E/... recombination to describe recombination in a medium that is optically thick to Balmer/Paschen/Bracket/... series photons.
 12.
So next time you look up and see Orion’s belt, try and remember that if it were not for the Ozone layer, you would be blasted with Ly\(\alpha \) flux strong enough to give you a firstdegree burn.
 13.
The condition of equilibrium is generally satisfied in ordinary interstellar HII regions. In expanding HII regions, e.g. those that exist in the intergalactic medium during cosmic reionization (which is discussed later), the total recombination rate is less than the total rate at which ionising photons are absorbed.
 14.
It is useful to recall that solar metallicity \(Z_{\odot }=0.02\).
 15.
That is, \( \langle E_{\gamma ,\mathrm{ion}} \rangle \equiv h_\mathrm{P}\frac{\int _{13.6\mathrm{eV}}^{\infty }d\nu f(\nu )}{\int _{13.6\mathrm{eV}}^{\infty }d\nu f(\nu )/\nu }\), where \(f(\nu )\) denotes the flux density.
 16.Another useful measure for the ‘strength’ of the Ly\(\alpha \) line is the equivalent width (EW, which was discussed in much more detail in the lectures by J.X. Prochaska) of the line:
which measures the total line flux compared to the continuum flux density just redward (as the blue side can be affected by intergalactic scattering, see Sect. 1.9.2) of the Ly\(\alpha \) line, \(F_0\). For a Salpeter IMF in the range \(0.1100M_{\odot }\), \(Z=Z_{\odot }\), the UVcontinuum luminosity density, \(L^\mathrm{UV}_{\nu }\), relates to SFR as \(L^\mathrm{UV}_{\nu }=8\times 10^{27}\times \mathrm{SFR}(M_{\odot }/\mathrm{yr})\) erg s\(^{1}\) Hz\(^{1}\). The corresponding equivalent width of the Ly\(\alpha \) line would be EW\({\sim }70\) Å [70]. The equivalent width can reach a few thousand Å for Population III stars/galaxies forming stars with a topheavy IMF (see [216]).
 17.
All sources within a radius equal to the mean free path of ionizing photons, \(\lambda _\mathrm{ion}\). For more distant sources (\(r>\lambda _\mathrm{ion}\)), the ionizing flux is reduced by an additional factor \(\exp (r/\lambda _\mathrm{ion})\).
 18.Another way to express \(S_\mathrm{fl,I}\) is by replacing \(Ln_{p}=N_\mathrm{H}\), where \(N_\mathrm{H}\) denotes the total column density of hydrogen ions (i.e. protons), which yields (see [125]).
 19.
It is possible that a significant fraction of the gravitational binding energy is released very close to the galaxy (e.g. when gas freefalls down into the gravitational potential well, until it is shock heated when it ‘hits’ the galaxy: [27]). It has been argued that some compact Ly\(\alpha \) emitting sources may be powered by cooling radiation (as in [27, 58, 68]).
 20.
 21.
We have adopted the notation of probability theory. In this notation, the function p(yb) denotes the conditional probability density function (PDF) of y given b. The PDF for y is then given by \(p(y)=\int p(yb)p(b)db\), where p(b) denotes the PDF for b. Furthermore, the joint PDF of y and b is given by \(p(y,b)=p(yb)p(b)\).
 22.
We can justify this picture as follows: we define the \(xy\) plane to be the plane in which the electron orbits the proton. The \(x\)coordinate of the electron varies as \(x(t)=x_0 \cos \omega _0 t\). The xcomponent of the electrostatic force on the electron varies as \(F_{x}=F_e\frac{x}{r}\), in which \(F_e=\frac{q^2}{r^2}\). That is, the equation of motion for the xcoordinate of the electron equals \(\ddot{x}=kx\), where \(k=q^2/r^3\).
 23.
This is not always the case in the Netherlands or Norway.
 24.
One has to be a bit careful because in the literature occasionally \(\phi (x)=H(a_v,x)/\sqrt{\pi }\), because in this convention the line profile is normalized to 1, i.e. \(\int \phi (x)dx=1\). In our convention \(\phi (x=0)=1\), while the normalization is \(\int \phi (x)dx=\sqrt{\pi }\).
 25.
For radiation at some fixed frequency \(\nu \) close to the Ly\(\alpha \) resonance, the opacity \(\alpha _{\nu }(\mathbf{r},\mathbf{n})\) depends on \(\mathbf{n}\) for nonstatic media. This directional dependence is taken into account when performing MonteCarlo Ly\(\alpha \) radiative transfer calculations (to be described in Sect. 1.8).
 26.
 27.
In the lectures I illustrated this with an example in which y denotes my happiness, and in which b denotes the number of snowballs that were thrown in my face in the previous 30 minutes. Clearly, p(y) will be different when \(b=0\) or when \(b \gg 1\).
 28.
It is possible to repeat the analysis of this section under the assumption that (i) the energy of the photon before and after scattering is identical, which is relevant when the gas has zero temperature. This corresponds to ‘completely coherent’ scattering (ii) the energy of the reemitted photon is completely unrelated to the atom of the incoming photon. This can happen in very dense gas where collisions perturb the atom while in the 2p state. This case corresponds to ‘completely incoherent’ scattering.
 29.
We will not derive this here. The derivation is short. First show that the total momentum of the atom after scattering equals \(p^\mathrm{H}_\mathrm{out}=\frac{h_\mathrm{p}\nu _\mathrm{in}}{c}\sqrt{22\mu }\), where \(\mu =\mathbf{k}_\mathrm{in}\cdot \mathbf{k}_\mathrm{out}\). This corresponds to a total kinetic energy \(E_e=\frac{[p^\mathrm{H}_\mathrm{out}]^2}{2m_\mathrm{p}}\), which must come at the expense of the Ly\(\alpha \) photon. We therefore have \(\varDelta E=h_\mathrm{P}\varDelta \nu =\frac{1}{m_\mathrm{p}}\left( \frac{h_\mathrm{p}\nu _\mathrm{in}}{c}\right) ^2(1\mu )\), which corresponds to \(\varDelta x=\frac{h\nu _{\alpha }}{m_pv_\mathrm{th} c}\) if we approximate that \(\nu _\mathrm{in}\approx \nu _{\alpha }\).
 30.This can be seen as follows (note that the second line contains colors to clarify how we got from the L.H.S to the R.H.S):
where we used that \(d\varDelta x=dx_\mathrm{out}\). We rewrote the term in Open image in new window using the definition of \(\phi (x)\) (see Eq. 1.55), and the term in Open image in new window is a Gaussian in \(\varDelta x\).
 31.
Namely that \(\int _{\infty }^{\infty }dx x \exp (a[xb]^2)=b\sqrt{\pi /a}\) with \(a=(1\mu ^2)^{1}\) and \(b=u(\mu 1)\).
 32.
Apart from a small recoil effect that can be safely ignored [1], photons are equally likely to scatter to the red and blue sides of the resonance.
 33.
Escape in a ‘single excursion’ can be contrasted with escape in a ‘single flight’: gases with lower \(N_\mathrm{HI}\) can become optically thin to Ly\(\alpha \) photons when they first scattered into the wing of the line profile. For example, gas with \(N_\mathrm{HI}=10^{17}\) cm\(^{2}\) has a line center optical depth \(\tau _0=5.9\times 10^3(T/10^4\mathrm{K})^{1/2}\) (Eq. 1.54). However, Fig. 1.17 shows that the crosssection is \(\gtrsim 4\) orders of magnitude smaller when \(x\gtrsim 3\). A photon that first scattered into the wing would be free to escape from this gas without further scattering.
 34.
Max Gronke has developed an online tool which allows users to vary column density, outflow/inflow velocity of the scattering medium, and directly see the impact on the emerging Ly\(\alpha \) spectrum: see http://bit.ly/manalpha.
 35.
The term ‘MonteCarlo’ was coined by Ulam and Metropolis as a codename for their classified work on nuclear weapons (radiation shielding, and the distance that neutrons would likely travel through various materials). ‘MonteCarlo’ was the name of the casino where Ulam’s uncle had a (also classified) gambling addiction. I was told most of this story over lunch by M. Baes. For questions, please contact him.
 36.
For arbitrary gas distributions, the emissivity profile is a 3Dfield. We can still apply the rejection method. One way to do this is to discretize the 3D field \(j_{\nu }(x,y,z) \rightarrow j_{\nu }(i,j,k)\), where we have \(N_x\), \(N_y\), and \(N_z\) of cells into these three directions. We can map this 3Darray onto a long 1D array \(j_{\nu }(m)\), where \(m=0,1,...,N_{x}\times N_y \times N_z\), and apply the rejection method to this array.
 37.
 38.
This discussion represents an extended version of the discussion presented in the review by [75].
 39.
 40.
 41.
This argument implicitly assumes that the scattering is partially coherent (see Sect. 1.6.3): photons experience a Doppler boost \(\varDelta \nu /\nu \sim v_\mathrm{out/c}\) when they enter the shell, and an identical Doppler boost \(\varDelta \nu /\nu \sim v_\mathrm{out/c}\) when they exit the shell in opposite direction (as is the case for ‘back scattered’ radiation). In the case of partially coherent scattering, the frequency of the photon changes only little in the frame of the gas (because \(\sqrt{\langle \varDelta x_\mathrm{in}^2x_\mathrm{in}\rangle }=1\)), and the total Doppler boost equals the sum of the two Doppler boosts imparted upon entry and exit from the shell.
 42.
 43.
Early studies defined the IGM to be all gas at \(r>11.5\) virial radii, which would correspond to the ‘circumgalactic’ medium by more recent terminology. Regardless of what we call this gas, scattering of Ly\(\alpha \) photons would remove photons from a spectrum of a galaxy, and redistribute these photons over faint, spatially extended Ly\(\alpha \) halos.
 44.
It is worth noting that these models predict that the IGM can reduce the observed Ly\(\alpha \) line by as much as \({\sim }30\%\) between \(z=5.7\) and \(z=6.5\) [156]. Observations of Ly\(\alpha \) halos around star forming galaxies provide hints that scattering in this CGM may be more prevalent at \(z=6.5\) than at \(z=5.7\), although the statistical significance of this claim is weak [186].
 45.
To make matters more confusing: selfshielding absorbers inside the ionized bubbles with sufficiently large HI column densities can be optically thick in the Ly\(\alpha \) damping wing, and can give rise to damping wing absorption as well. This damping wing absorption is included in \(\tau _\mathrm{HII}(z,\varDelta v)\).
 46.
Just as the Ly\(\alpha \) forest at lower redshifts—where hydrogen reionization was complete—contains neutral hydrogen gas with different densities, ionization states and column densities.
 47.
If a photon enters the first neutral patch on the blue side of the line resonance, then the total opacity of the IGM depends on whether the photon redshifted into resonance inside or outside of a neutral patch. If the photon redshifted into resonance inside patch ‘i’, then \(\tau _\mathrm{D}(z_\mathrm{g},\varDelta v)=\tau _\mathrm{GP}(z)x_\mathrm{HI,i}\). If on the other hand the photon redshifted into resonance in an ionized bubble, then we must compute the optical depth in the ionized patch, \(\tau _\mathrm{HII}(z,\varDelta v=0)\), plus the opacity due to subsequent neutral patches. Given that the ionized IGM at \(z=6.5\) was opaque enough to completely suppress Ly\(\alpha \) flux on the blueside of the line, the same likely occurs inside ionized HII bubbles during reionization because of (i) the higher intergalactic gas density, and (ii) the shorter mean free path of ionizing photons and therefore likely reduced ionizing background that permeates ionised HII bubbles at higher redshifts.
 48.
Scattering through an extremely opaque static medium gives rise to a spectrally broadened doublepeaked Ly\(\alpha \) spectrum (see Fig. 1.26). Of course, photons in the red peak start with a redshift as well, which boosts their visibility especially for large \(N_\mathrm{HI}\) (see Fig. 1.2 in also see [70, 111]).
 49.
Note that the transition from core to wing scattering occurs at \(x \sim 3\), see Fig. 1.17.
 50.
A spinning proton can be seen as a rotating charged sphere, which produces a magnetic field.
 51.
Eq. 1.118 follows from solving the radiative transfer equation, \(\frac{dI}{d\tau }=I+\frac{A_{10}}{B_{10}}\frac{1}{(3n_0/n_11)}\), in the (appropriate) limit that the neutral IGM is optically thin in the HI 21cm line, and that it therefore only slightly modifies the intensity I of the background CMB. It is common in radio astronomy to express intensity fluctuations as temperature fluctuations by recasting intensity as a temperature in the RayleighJeans limit: \(I_{\nu }(\mathbf{x})\equiv \frac{2k_\mathrm{B}T(\mathbf{x})}{\lambda }\).
 52.
Note that this equation uses that for each Ly\(\alpha \) scattering event, the probability that it induces a scattering event is \(P_\mathrm{flip}=\frac{4}{27}\) (see e.g. [65, 177]). This probability reduces by many orders of magnitude for wing scattering as \(P_\mathrm{flip} \propto x^{2}\) (see [65, 127]). In practise wing scattering contributes little to the overall scattering rate, but it is good to keep this in mind.
 53.
See [95] for an explanation of how to best pronounce Wouthuysen. Hint: it helps if you hold your breath 12 s before trying.
Notes
Acknowledgements
I thank Ivy Wong for providing a figure for a section which (unfortunately) was cut as a whole, Daniel Mortlock and Chris Hirata for providing me with tabulated values of their calculations which I used to create Fig. 1.17, Jonathan Pritchard for permission to use one of his slides for these notes, Andrei Mesinger for providing Fig. 1.42, and other colleagues for their permission to reuse Figures from their papers. I thank the astronomy department at UCSB for their kind hospitality when I was working on preparing these lecture notes. I thank the organizers of the school: Anne Verhamme, Pierre North, Hakim Atek, Sebastiano Cantalupo, Myriam Burgener Frick, to Matt Hayes, X. Prochaska, and Masami Ouchi for their inspiring lectures. I thank Pierre North for carefully reading these notes, and for finding and correcting countless typos. Finally, special thanks to the students for their excellent attendance, and for their interest and enthusiastic participation.
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