Ly\alpha Emitting Galaxies as a Probe of Reionization

# Lyα Emitting Galaxies as a Probe of Reionization

Mark Dijkstra   Institute of Theoretical Astrophysics, University of Oslo, Postboks 1029, 0858 Oslo, Norway, mark.dijkstra@astro.uio.no   MPI fuer Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany
• Abstract: The Epoch of Reionization (EoR) represents a milestone in the evolution of our Universe. Star-forming galaxies that existed during the EoR likely emitted a significant fraction () of their bolometric luminosity as Ly line emission. However, neutral intergalactic gas that existed during the EoR was opaque to Ly emission that escaped from galaxies during this epoch, which makes it difficult to observe. The neutral intergalactic medium (IGM) may thus reveal itself by suppressing the Ly flux from background galaxies. Interestingly, a ‘sudden’ reduction in the observed Ly flux has now been observed in galaxies at . This review contains a detailed summary of Ly radiative processes: I describe (i) the main Ly emission processes, including collisional-excitation & recombination (and derive the origin of the famous factor ‘0.68’), and (ii) basic radiative transfer concepts, including e.g. partially coherent scattering, frequency diffusion, resonant versus wing scattering, optically thick versus ’extremely’ optically thick (static/outflowing/collapsing) media, and multiphase media. Following this review, I derive expressions for the Gunn-Peterson optical depth of the IGM during (inhomogeneous) reionization and post-reionization. I then describe why current observations appear to require a very rapid evolution of volume-averaged neutral fraction of hydrogen in the context of realistic inhomogeneous reionization models, and discuss uncertainties in this interpretation. Finally, I describe how existing & futures surveys and instruments can help reduce these uncertainties, and allow us to fully exploit Ly emitting galaxies as a probe of the EoR.

Keywords: cosmology: dark ages, reionization, first stars — galaxies: intergalactic medium, high redshift — radiative transfer — scattering — ultraviolet: galaxies

## 1 Introduction

The Epoch of Reionization (EoR) represents a milestone in the evolution of our Universe. It represents the last major phase transformation of its gas from a cold (i.e. a few tens of K) and neutral to a fully ionized, hot (i.e. K) state. This transformation was likely associated with the formation of the first stars, black holes and galaxies in our Universe. Understanding the reionization process is therefore intimately linked to understanding the formation of the first structures in our Universe – which represents one of the most basic and fundamental questions in astrophysics.

Reionization is still not well constrained: anisotropies in the Cosmic Microwave Background (CMB) constrain the total optical depth to scattering by free electrons111Inhomogeneous reionization further affects the CMB anistropies on arcminute scales via the kinetic SZ (kSZ) effect. However, current measurement of the power-spectrum on these scales do not allow for stringent constraints (see Mesinger et al. 2012, for a discussion). to be (Komatsu et al. 2011; Hinshaw et al. 2013; Planck Collaboration et al. 2013). For a reionization history in which the Universe transitions from fully neutral to fully ionized at redshift over a redshift range , this translates to . Gunn-Peterson troughs that have been detected in the spectra of quasars (e.g. Becker et al. 2001; Fan et al. 2002; Mortlock et al. 2011; Venemans et al. 2013) suggest that the intergalactic medium (IGM) contained a significant neutral fraction (with a volume averaged fraction , see e.g. Wyithe & Loeb 2004; Mesinger & Haiman 2007; Bolton et al. 2011; Schroeder et al. 2013). Moreover, quasars likely inhabit highly biased, overdense regions of our Universe, which probably were reionized earlier than the Universe as a whole. It has been shown that existing quasar spectra at are consistent with a significant neutral fraction, even at (Mesinger 2010; McGreer et al. 2011).

The constraints obtained from quasars and the CMB thus suggest that reionization was a temporally extended process that ended at , but that likely started at (e.g. Pritchard et al. 2010; Mitra et al. 2012). These constraints are consistent with those obtained measurements of the temperature of the IGM at (Theuns et al. 2002; Hui & Haiman 2003; Raskutti et al. 2012), observations of the Ly damping wing in gamma-ray burst after-glow spectra (Totani et al. 2006; McQuinn et al. 2008), and Ly emitting galaxies (e.g. Haiman & Spaans 1999; Malhotra & Rhoads 2004; Kashikawa et al. 2006).

In this review I will discuss why Ly emitting galaxies provide a unique probe of the EoR, and place particular emphasis on describing the physics of the relevant Ly radiative processes. Throughout, ‘Ly emitting galaxies’ refer to all galaxies with ‘strong’ Ly emission (what ‘strong’ means is clarified later), and thus includes both LAEs (Ly emitters) and Ly emitting drop-out galaxies. We refer to LAEs as Ly emitting galaxies that have been selected on the basis of their Ly line. This selection can be done either in a spectroscopic or in a narrow-band (NB) survey. NB surveys apply a set of color-color criteria that define LAEs. This typically requires some excess flux in the narrow-band which translates to a minimum EW of the line. It is also common in the literature to use the term LAE to refer to all galaxies for which the Ly EWEW (irrespective of how these were selected).The outline of this review is as follows: In § 2 I give the general radiative transfer equation that is relevant for Ly. The following sections contain detailed descriptions of the components in this equation:

• In § 3 I summarize the main Ly emission processes, including collisional-excitation & recombination. For the latter, I derive the origin of the factor (which denotes the number of Ly photons emitted per recombination event), which is routinely associated with case-B recombination. I will also describe why and where departures from case-B may arise, which underlines why star-forming galaxies are thought to have very strong Ly emission lines, especially during the EoR (§ 3).

• In § 4 I describe the basic radiative transfer concepts that are relevant for understanding Ly transfer. These include for example, partially coherent scattering, frequency diffusion, resonant versus wing scattering, and optically thick versus ’extremely’ optically thick in static/ outflowing/ collapsing media.

After this review, I discuss our current understanding of Ly transfer at interstellar and intergalactic level in § 5. With this knowledge, I will then discuss the impact of a neutral intergalactic medium on the visibility of the Ly emission line from galaxies during the EoR (§ 6). I will then apply this to existing observations of Ly emitting galaxies, and discuss their current constraints on the EoR. This discussion will show that existing constraints are still weak, mostly because of the limited number of known Ly emitting galaxies at . However, we expect the number of known Ly emitting galaxies at to increase by up to two orders of magnitude. I will discuss how these observations (and other observations) are expected to provide strong constraints on the EoR within the next few years in § 8. Table 1 provides a summary of symbols used throughout this review.

The change in the intensity of radiation at frequency that is propagating into direction (where ) is given by (e.g. Rybicki & Lightman 1979)

 n⋅∇I(ν,n)=−α(ν)I(ν,n)+j(ν,n) (1) +∫dΩ′∫dn′I(ν′,n′)R(ν′,ν,n′,n),

where

• the attenuation coefficient , in which denotes the Ly absorption cross section, and denotes the dust absorption cross section per hydrogen nucleus. I give an expression for in § 4.1, and for in § 4.5.

• denotes the volume emissivity (energy emitted per unit time, per unit volume) of Ly photons, and can be decomposed into . Here, / denotes the contribution from recombination (see § 3.1)/collisional-excitation (see § 3.2).

• denotes the ‘redistribution function’, which measures the probability that a photon of frequency propagating into direction is scattered into direction and to frequency . In § 4.2 we discuss this redistribution function in more detail.

Eq 1 is an integro-differential equation, and has been studied for decades (e.g. Chandrasekhar 1945; Unno 1950; Harrington 1973; Neufeld 1990; Yang et al. 2011; Higgins & Meiksin 2012). Eq 1 simplifies if we ignore the directional dependence of the Ly radiation field (which is reasonable in gas that is optically thick to Ly photons)222Ignoring the directional dependence of allows us replace the term , and replace with the angle averaged intensity ., and the directional dependence of the redistribution function. Rybicki & dell’Antonio (1994) showed that the ‘Fokker-Planck’ approximation - a Taylor expansion in the angle averaged intensity in the integral - allows one to rewrite Eq 1 as a differential equation (also see Higgins & Meiksin 2012):

 dJ(ν)dτ=(ΔνD)22∂∂νϕ(ν)∂J(ν)∂ν, (2)

where we replaced the attenuation coefficient [] with the optical depth , in which denotes a physical infinitesimal displacement. Furthermore, we set and for simplicity. Eq 2 is a diffusion equation. Ly transfer through an optically thick medium is therefore a diffusion process: as photons propagate away from their source, they diffuse away from line centre. That is, the Ly transfer process can be viewed as diffusion process in real and frequency space.

## 3 Lyα Emission

Unlike UV-continuum radiation, the majority of Ly line emission typically does not originate in stellar atmospheres. Instead, Ly line emission is predominantly powered via two other mechanisms. In the first, ionizing radiation emitted by hot young O and B stars ionize their surrounding, dense interstellar gas, which recombines on a short timescale, (e.g. Hui & Gnedin 1997). A significant fraction of the resulting recombination radiation emerges as Ly line emission (see e.g. Johnson et al. 2009, Raiter et al. 2010, Pawlik et al. 2011 and § 3.1).

In the second, Ly photons are emitted by collisionally-excited HI. As we discuss briefly in § 3.2, the collisionally-excited Ly flux emitted by galaxies appears subdominant to the Ly recombination radiation, but may become more important towards higher redshifts.

### 3.1 Recombination Radiation: The Origin of the Factor ‘0.68’

The volume Ly emissivity following recombination is often given by

 jrec(n,Tν)=0.68nenpαB(T)ϕ(ν)Eα/(√πΔνD), (3)

where eV, / denotes the number density of free electrons/protons, and denotes the Voigt profile (normalised to , in which Hz quantifies thermal broadening of the line). Expressions for are given in § 4.1. The factor denotes the fraction of recombination events resulting in Ly and is derived next.

The capture of an electron by a proton generally results in a hydrogen atom in an excited state . Once an atom is in a quantum state it radiatively cascades to the ground state , via intermediate states . The probability that a radiative cascade from the state results in a Ly photon is given by

 P(n,l→Lyα)= (4) =∑n′,l′P(n,l→n′,l′)P(n′,l′→Lyα).

That is, the probability can be computed if one knows the probability that a radiative cascade from lower excitation states results in the emission of a Ly photon, and the probabilities that the atom cascades into these lower excitation states, . This latter probability is given by

 P(n,l→n′,l′)=An,l,n′,l′∑n′′,l′′An,l,n′′,l′′, (5)

in which denotes the Einstein A-coefficient for the transition333This coefficient is given by (6) where fundamental quantities , , , and are given in Table 1, denotes the energy difference between the upper (n,l) and lower (n’,l’) state. The matrix involves an overlap integral that involves the radial wavefunctions of the states and : (7) Analytic expressions for the matrix that contain hypergeometric functions were derived by Gordon (1929). For the Ly transition (Hoang-Binh 1990)..

The quantum mechanical selection rules only permit transitions for which , which restricts the total number of allowed radiative cascades. Figure 1 schematically depicts permitted radiative cascades in a four-level H atom. Green solid lines depict radiative cascades that result in a Ly photon, while red dotted lines depict radiative cascades that do not yield a Ly photon.

Figure 1 also contains a table that shows the probability for . For example, the probability that a a radiative cascade from the state (i.e. the 3p state) produces a Ly photon is , because the selection rules only permit the transitions and . The first transition leaves the H-atom in the 2s state, from which it can only transition to the ground state by emitting two photons (Breit & Teller 1940). On the other hand, a radiative cascade from the state (i.e. the 3d state) will certainly produce a Ly photon, since the only permitted cascade is . Similarly, the only permitted cascade from the 3s state is , and . For , multiple radiative cascades down to the ground state are generally possible, and takes on values other than or (see e.g. Spitzer & Greenstein 1951).

The probability that an arbitrary recombination event results in a Ly photon is given by

 P(Lyα)=∞∑nminn−1∑l=0αnl(T)αtot(T)P(n,l→Lyα) (8)

where the first term denotes the fraction of recombination events into the state, in which denotes the total recombination coefficient . The temperature-dependent state specific recombination coefficients can be found in for example (Burgess 1965) and Rubiño-Martín et al. (2006). The value of depends on the physical conditions of the medium in which recombination takes place, and two cases bracket the range of scenarios commonly encountered in astrophysical plasmas:

• ‘case-A’ 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 .

• ‘case-B’ recombination: recombination takes place in a medium that is opaque to all Lyman series444At gas densities that are relevant in most astrophysical plasmas, hydrogen atoms predominantly populate their electronic ground state (), and the opacity in the Balmer lines is generally negligible. In theory one can introduce case-C/D/E/… recombination to describe recombination in a medium that is optically thick to Balmer/Paschen/Bracket/… series photons. photons (i.e. Ly, Ly, Ly, …), and to ionizing photons that were emitted following direct recombination into the ground state. In the so-called ‘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 , and by setting the Einstein coefficient for all Lyman series transitions to zero, i.e. .

Figure 2 shows the total probability (Eq 8) that a Ly photon is emitted per recombination event as a function of gas temperature , assuming case-A recombination (solid line), and case-B recombination (dashed line). For gas at K and case-B recombination, we have . This value is often encountered during discussions on Ly emitting galaxies. It is worth keeping in mind that the probability increases with decreasing gas temperature and can be as high as for K (also see Cantalupo et al. 2008). The red open circles represent the following two fitting formulae

 PA(Lyα)=0.41−0.165logT4−0.015(T4)−0.44 (9) PB(Lyα)=0.686−0.106logT4−0.009(T4)−0.44,

where K. The fitting formula for case-B is taken from Cantalupo et al. (2008).

Recombinations in HII regions in the ISM are balanced by photoionization in equilibrium HII regions. The total recombination rate in an equilibrium HII region therefore equals the total photoionization rate, or the total rate at which ionizing photons are absorbed in the HII region (in an expanding HII region, the total recombination rate is less than the total rate at which ionising photons are absorbed). If a fraction of ionizing photons is not absorbed in the HII region (and hence escapes), then the total Ly production rate in recombinations can be written as

 ˙NrecLyα=P(Lyα)(1−fesc)˙Nion≈ (10)

where () denotes the rate at which ionizing (Ly recombination) photons are emitted. The equation on the second line is commonly adopted in the literature. The ionizing emissivity of star-forming galaxies is expected to be boosted during the EoR: stellar evolution models combined with stellar atmosphere models show that the effective temperature of stars of fixed mass become hotter with decreasing gas metallicity (Tumlinson & Shull 2000; Schaerer 2002). The increased effective temperature of stars causes a larger fraction of their bolometric luminosity to be emitted as ionizing radiation. We therefore expect galaxies that formed stars from metal poor (or even metal free) gas during the EoR, to be strong sources of nebular emission. Schaerer (2003) provides the following fitting formula for as a function of absolute gas metallicity555It is useful to recall that solar metallicity . , ,which is valid for a Salpeter IMF in the mass range .

A useful measure for the ‘strength’ of the Ly line (other than just its flux) is given by the equivalent width

 EW≡∫dλ(F(λ)−F0)/F0, (11)

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 § 5.2) of the Ly line, . For ‘regular’ star-forming galaxies (Salpeter IMF, solar metallicity) the maximum physically allowed restframe EW is EW Å (see e.g. Schaerer 2003; Laursen et al. 2013, and references therein). Reducing the gas metallicity by as much as two orders of magnitude typically boosts the EW, but only by (Laursen et al. 2013). A useful way to gain intuition on EW is that EWFWHM(relative peak flux density). That is, for typical observed (restframe) FWHM of Ly lines of FWHM Å, EW= Å corresponds to having a relative flux density in the peak of the line that is times that in the continuum.

### 3.2 Collisionally-excited (a.k.a ‘Cooling’) Radiation

Ly photons can also be produced following collisional-excitation of the transition when a hydrogen atoms deflects the trajectory of an electron that is passing by. The Ly emissivity following collisional-excitation is given by

 jcoll(n,T,ν)=ϕ(ν)nenHIq1s2pEα/(√πΔνD), (12)

where denotes the number density of hydrogen atoms and

 q1s2p(T)=8.63×10−6T−1/2⟨Ω1s2p⟩exp(−EαkBT)cm3s−1 (13)

where denotes the velocity averaged collision strength, which depends weakly on temperature. The top panel of Figure 3 shows the temperature dependence of for the (dotted line), (dashed line), and for their sum (black solid line) as given by Scholz et al. (1990); Scholz & Walters (1991). Also shown are velocity averaged collision strengths for the (red solid line, obtained by summing over all transitions and ), and (blue solid line, obtained by summing over all transitions and ) as given by Aggarwal et al. (1991). The bottom panel shows the collision coupling parameter for the same transitions. This plot shows that collisional coupling to the level increases by orders of magnitude magnitude when K. The actual production rate of Ly photons can be even more sensitive to , as both sharply increases with and sharply decreases with within the same temperature range (under the assumption that collisional ionisation balances recombination, which is relevant in e.g self-shielded gas, see e.g. Fig 1 in Thoul & Weinberg 1995).

This process converts thermal energy of the gas into radiation, and therefore cools the gas. Ly cooling radiation has been predicted to give rise to spatially extended Ly radiation (Haiman et al. 2000; Fardal et al. 2001), and provides a possible explanation for Ly ‘blobs’ (Dijkstra & Loeb 2009; Goerdt et al. 2010; Faucher-Giguère et al. 2010; Rosdahl & Blaizot 2012). In these models, the Ly 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. (2000) 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, which is then reradiated primarily as Ly. It is possible that a significant fraction of the gravitational binding energy is released very close to the galaxy (e.g. when gas free-falls down into the gravitational potential well, until it is shock heated when it ‘hits’ the galaxy Birnboim & Dekel 2003). It has been argued that some compact Ly emitting sources may be powered by cooling radiation (as in Birnboim & Dekel 2003; Dijkstra 2009; Dayal et al. 2010). Recent hydrodynamical simulations of galaxies indicate that the fraction of Ly flux coming from galaxies in the form of cooling radiation increases with redshift, and may be as high as at (Dayal et al. 2010; Yajima et al. 2012). However, one should take these numbers with caution, because the predicted Ly cooling luminosity depends sensitively on the gas temperature of the ‘cold’ gas (i.e. around T K, as illustrated by the discussion above). It is very difficult to reliably predict the temperature of this gas, because the gas’ short cooling time drives the gas temperature to a value where its total cooling rate balances its heating rate. Because of this thermal equilibrium, we must accurately know and compute all the heating rates in the ISM (Faucher-Giguère et al. 2010; Cantalupo et al. 2012; Rosdahl & Blaizot 2012) to make a robust prediction for the Ly cooling rate. These heating rates include for example photoionization heating, which requires coupled radiation-hydrodynamical simulations (as Rosdahl & Blaizot 2012), or shock heating by supernova ejecta (e.g. Shull & McKee 1979).

It may be possible to observationally constrain the contribution of cooling radiation to the Ly luminosity of a source, through measurements of the Ly equivalent width: the larger the contribution from cooling radiation, the larger the EW. Ly emission powered by regular star-formation can have EW Å(see discussion above). Naturally, observations of Ly emitting galaxies whose EW significantly exceeds EW (as in e.g. Kashikawa et al. 2012), may provide hints that we are detecting a significant contribution from cooling. However, the same signature can be attributed population III stars (e.g. Raiter et al. 2010), and/or galaxies forming stars with a top-heavy initial mass function (IMF, e.g. Malhotra & Rhoads 2002), or stochastic sampling of the IMF (Forero-Romero & Dijkstra 2013). In theory one can distinguish cooling radiation from these other processes via the Balmer lines, because Ly cooling radiation is accompanied by an H luminosity that is times weaker, which is much weaker than expected for case-B recombination (where the H flux is times weaker, e.g. Dijkstra & Loeb 2009). Measuring the flux in the H line at these levels requires an IR spectrograph with a sensitivity comparable to that of JWST666For example, for a Ly source with an intrinsic luminosity of erg/s at , the corresponding H flux is erg/s/cm, which is too faint to be detected with existing IR spectrographs (here denotes the escape fraction of H photons). However, these flux levels can be reached at in sec with NIRSPEC on JWST, provided that the flux is in an unresolved point source (see http://www.stsci.edu/jwst/science/sensitivity/)..

Equation 10 was derived assuming case-B recombination. However, at significant departures from case-B are expected. These departures increase the Ly luminosity relative to case-B (e.g. Raiter et al. 2010). This increase of the Ly luminosity towards lower metallicities is due to two effects: (i) the increased temperature of the HII region as a result of a suppressed radiative cooling efficiency of metal-poor gas. The enhanced temperature in turn increases the importance of collisional processes. For example, collisional-excitation increases the population of H-atoms in the state, which can be photoionized by lower energy photons. Moreover, collisional processes can mix the populations of atoms in their and states; (ii) harder ionizing spectra emitted by metal poor(er) stars. Higher energy photons can in principle ionize more than 1 H-atom, which can boost the overall Ly production per ionizing photon. Raiter et al. (2010) provide a simple analytic formula which capture all these effects:

 ˙NrecLyα=fcollP(1−fionesc)˙Nion, (14)

where , in which denotes the mean energy of ionising photons777That is, , where denotes the flux density.. Furthermore, , in which , , , and denotes the number of density of hydrogen nuclei. Eq 14 resembles the ‘standard’ equation, but replaces the factor 0.68 with , which can exceed unity. Eq 14 implies that for a fixed IMF, the Ly luminosity may be boosted by a factor of a few. Incredibly, for certain IMFs the Ly line may contain of the total bolometric luminosity of a galaxy, which corresponds to a rest frame EW Å.

We point out that the collisional processes discussed here are distinct from the collisional-excitation process discussed above (in § 3.2), as they do not directly produce Ly photons. Instead, they boost the number of Ly photons that we can produce per ionising photon.

## 4 Lyα Radiative Transfer Basics

Ly radiative transfer consists of absorption followed by (practically) instant reemission, and hence closely resembles pure scattering. Here, we review the basic radiative transfer that is required to understand why & how Ly emitting galaxies probe the EoR.

It is common to express the frequency of a photon in terms of the dimensionless variable . Here, Hz denotes the frequency corresponding the Ly resonance, and . Here, denotes the temperature of the gas that is scattering the Ly radiation, and denotes the thermal speed.

### 4.1 The Cross Section

The frequency dependence of the Ly absorption cross-section, , is described well by a Voigt function. That is

 σα(x)=σ0×avπ∫+∞−∞dyexp(−y2)(x−y)2+a2v≡σ0×ϕ(x). (15) σ0=fα√πΔνDπe2mec=5.88×10−14(T/104K)−1/2cm2

where denotes the Ly oscillator strength, and denotes the Voigt parameter, and denotes the cross section at line center. We introduced the Voigt function888We adopt the normalization , which translates to . , which is plotted as the black solid line in the upper panel of Figure 4. This Figure also shows that the Voigt function is approximated accurately as

 ϕ(x)≈⎧⎨⎩ e−x2`core'% , i.e.|x|xcrit, (16)

where the transition from the Gaussian core (red dashed line) to the Lorentzian wing (blue dotted line)999The Ly absorption line profile of an individual HI atom is given by a Lorentzian function, . This Lorentzian profile is also plotted. The figure clearly shows that far from line center, we effectively recover the single atom or Lorentzian line profile. occurs at at the gas temperature of K that we adopted. An even more accurate fit - which works well even in the regime where we transition from core to wing - is given in Tasitsiomi (2006).

It is important to point out that the Voigt function itself (as given by Eq 15) only represents an approximation to the real frequency dependence of the absorption cross section. The Voigt profile is derived through a convolution of a Gaussian profile (describing the thermal velocity-distribution of HI atoms) with the Lorentzian profile (see above). A common modification of the Lorentzian is given in e.g. Peebles (1993, 1969), where the absorption cross section for a single atom includes an additional -dependence, as is appropriate for Rayleigh scattering101010This additional term arises naturally in a classical calculation in which radiation of frequency scatters off an electron that orbits the proton at a natural frequency .. In this approximation we have (also see Schroeder et al. 2013)

 σα,Ray(x)=σα(x)(ννα)4=σα(x)(1+xvth/c)4, (17)

which gives rise to slightly asymmetric line profiles (as shown by the grey solid line in Fig 4). However, even this still represents an approximation (as pointed out in Peebles 1969). A complete quantum-mechanical derivation of the frequency-dependence of the Ly absorption cross section has been presented only recently by Lee (2013), which can be captured by the following correction to the Voigt profile:

 σα,Lee(x)=σα(x)(1−1.792xvth/c). (18)

This cross section is shown as the green dotted line. In contrast to the Rayleigh-approximation given above, the red wing is strengthened relative to the pure Lorentzian proÞle, which Lee (2013) credits to positive interference of scattering from all other levels. The lower panel shows the fractional difference of the three cross sections , , and .

Although the Voigt function does not capture the full frequency dependence of Ly absorption cross section far in the wings on the line, it clearly provides an accurate description of the cross section near the core. The fast reduction in the cross section outside of the core (here at km s) enables Ly photon to escape more easily from galaxies, which - with HI column densities in excess of cm - are extremely opaque to Ly photons. This process is discussed in more detail in the next sections.

### 4.2 Frequency Redistribution R(ν′,ν): Resonance vs Wing Scattering

Absorption of an atom is followed by re-emission on a time scale s. At interstellar and intergalactic densities, atoms are very likely not ‘perturbed’ in such short times scales111111The rate at which atoms in the state interact with protons is given by s. For electrons the corresponding rate is reduced by a factor of . The interaction with other neutral atoms is negligible., and the atom re-emits a photon with an energy that equals that of the absorbed photon when measured in the atom’s frame. Because of the atom’s thermal motion however, in the lab frame the photon’s energy will be Doppler boosted. The photon’s frequency before and after scattering are therefore not identical but correlated, and the scattering is referred to as ‘partially coherent’ (completely coherent scattering would refer to the case where the photons frequency before and after scattering are identical).

We can derive a probability distribution function (PDF) for the photons frequency after scattering, , given its frequency before scattering, . Expressions for these ‘redistribution functions’121212It is worth pointing out that these redistribution functions are averaged over the direction in which the outgoing photon is emitted, i.e. (19) where denotes the ‘phase function’, and describes the probability that lies in the range . We stress that the redistribution functions depend strongly on outgoing direction. Expressions for can be found in Dijkstra & Kramer (2012). can be found in e.g. Lee (1974, also see Unno 1952, Hummer 1962). Redistribution functions that describe partially coherent scattering have been referred to as ‘type-II’ redistribution functions (where type-I would refer to completely incoherent scattering).

Figure 5 shows examples of type-II redistribution functions, , as a function of for (this Figure was kindly provided by Max Gronke). This Figure shows that (i) varies rapidly with , and (ii) for the probability of being scattered back to becomes vanishingly small. Before we discuss why this latter property of the redistribution function has important implications for the scattering process, we first explain that its origin is related to ‘resonant’ vs ‘wing’ scattering.

Figure 6 shows the PDF of the frequency of a photon, in the atoms frame (), for two incoming frequency (black solid line, here labelled as ) and (red dashed line). The black solid line peaks at . That is, the photon with is most likely scattered by an atom to which the photon appears exactly at line centre. That is, the scattering atom must have velocity component parallel to the incoming photon that is times . This requires the atom to be on the Maxwellian tail of the velocity distribution. Despite the smaller number of atoms that can meet this requirement, there are still enough to dominate the scattering process. However, when the same process would require atoms that lie even further on the Maxwellian tail. These atoms are too rare to contribute to scattering. Instead, the photon at is scattered by the more numerous atoms with speeds close to . In the frame of these atoms, the photon will appear centered on (as shown by the red dashed line).

As shown above, a fraction of photons at scatter off atoms to which they appear very close to line centre. Thus, a fraction of these photons scatter ‘resonantly’. In contrast, this fraction is vanishingly small for the photons at (or more generally, for all photons with ). These photons do not scatter in the wing of the line. This is more than just semantics, the phase-function and polarisation properties of scattered radiation depends sensitively on whether the photon scattered resonantly or not (Stenflo 1980; Rybicki & Loeb 1999; Dijkstra & Loeb 2008).

There are two useful & important expectation values of the redistribution functions for photons at (e.g. Osterbrock 1962; Furlanetto & Pritchard 2006):

 E(√Δx2|xin)=1 (20) E(Δx|xin)=−1xin |xin|≫1,

where , and the expectation values are calculated as .

The first equality states that the r.m.s. frequency change of the photon before and after scattering equals 1 Doppler width. This is an important result: a photon that is absorbed far in the wing of the line, will remain far in the wing after scattering, which facilitates the escape of photons (see below). The second equality states that for photons that are absorbed in the wing of the line, there is a slight tendency to be scattered back to the core, e.g. a photon that was at , will have an outgoing frequency around . The second equality also implies that photons at typically scatter times before they return to the core (in a static medium). These photons can travel a distance from where they were emitted. This should be compared with the distance that can be travelled by photons at . The path of photons in real space as they scatter in the wing of the line (i.e. at ) back towards the core is referred to as an ‘excursion’. The optical depth of a static uniform medium beyond which photons preferably escape in ‘excursions’ marks the transition from ‘optical thick’ to ‘extremely optical thick’. Finally, the second equality also allows us to estimate the spectrum of Ly photons that escape from an extremely opaque, static medium as is discussed in § 4.3 below.

### 4.3 Lyα Scattering in Static Media

Consider of source of Ly photons in the center of a static, homogeneous sphere, whose line-center optical depth from the center to the edge equals , where is extremely large (say ). We further assume that the central source emits all Ly photons at line center. The photons initially resonantly scatter in the core of the line profile, with a mean free path that is negligible small compared to the size of the sphere. Because the photons change their frequency during each scattering event, the photons ’diffuse’ in frequency space as well. We expect on rare occasions the Ly photons to be scattered into the wing of the line. The mean free path of a wing photon at frequency equals in units of line-center optical depth. Photons that are in the wing of the line scatter times before returning to the core, but will have diffused a distance from the center of the sphere. If we set this displacement equal to the size of the sphere, i.e. , and solve for using that , we find (Adams 1972; Harrington 1973; Neufeld 1990). Photons that are scattered to frequencies131313Apart from a small recoil effect that can safely ignored (Adams 1971), photons are equally likely to scatter to the red and blue sides of the resonance. will return to line center before they escape from the sphere (where they have negligible chance to escape). Photons that are scattered to frequencies can escape more easily, but there are fewer of these photons because: (i) it is increasingly unlikely that a single scattering event displaces the photon to a larger , and (ii) photons that wish to reach through frequency diffusion via a series of scattering events are likely to escape from the sphere before they reach this frequency.

We therefore expect the spectrum of Ly photons emerging from the center of an extremely opaque object to have two peaks at , where is a constant of order unity which depends on geometry (i.e. for a slab [Harrington 1973, Neufeld 1990], and for a sphere [Dijkstra et al. 2006]). This derivation required that photons escape in a single excursion. That is, photons must have been scattered to a frequency (see § 4.2). If for simplicity we assume that then escape in a single excursion - and hence the transition to extremely opaque occurs - when or when . Indeed, analytic solutions of the full spectrum emerging from static optically thick clouds appear in good agreement with full Monte-Carlo calculations (see § 4.6) when (e.g. Neufeld 1990; Dijkstra et al. 2006).

These points are illustrated in Figure 7 where we show spectra of Ly photons emerging from static, uniform spheres of gas surrounding a central Ly source (taken from Orsi et al. 2012, the assumed gas temperature is K). This Figure contains three spectra corresponding to different . Solid lines/histograms represent spectra obtained from analytic calculations/ Monte-Carlo simulations. This Figure shows the spectra contain two peaks, located at given above. The Monte-Carlo simulations and the analytic calculations agree well. For K, we have and , and we expect photons to escape in a single excursion, which is captured by the analytic calculations.

We can also express the location of the two spectral peaks in terms of a velocity off-set and an HI column density as

 (21)

That is, the full-width at half maximum of the Ly line can exceed km s for a static medium.

### 4.4 Lyα Scattering in an Expanding/Contracting Medium

For an outflowing medium, the predicted spectral line shape also depends on the outflow velocity, . Qualitatively, photons are less likely to escape on the blue side (higher energy) than photons on the red side of the line resonance because they appear closer to resonance in the frame of the outflowing gas. Moreover, as the Ly photons are diffusing outward through an expanding medium, they loose energy because the do ’work’ on the outflowing gas (Zheng & Miralda-Escudé 2002). Both these effects combined enhance the red peak, and suppress the blue peak, as illustrated in Figure 8 (taken from Laursen et al. 2009b). In detail, how much the red peak is enhanced, and the blue peak is suppressed (and shifted in frequency directions) depends on the outflow velocity and the HI column density of gas.

There exists one analytic solution to radiative transfer equation through an expanding medium: Loeb & Rybicki (1999) derived analytic expressions for the angle-averaged intensity of Ly radiation as a function of distance from a source embedded within a neutral intergalactic medium undergoing Hubble expansion141414This calculation assumed that which corresponds to the special case of ..

Not unexpectedly, the same arguments outlined above can be applied to an inflowing medium: here we expect the blue peak to be enhanced and the red peak to be suppressed (e.g. Dijkstra et al. 2006; Barnes & Haehnelt 2010). It is therefore thought that the Ly line shape carries information on the gas kinematics through which it is scattering. As we discuss in § 5.1, the shape and shift of the Ly spectral line profile has been used to infer properties of the medium through which they are scattering.

### 4.5 Lyα Transfer through a Dusty (Multiphase) Medium

Ly photons can be absorbed by dust grains. A dust grain can re-emit the Ly photon (and thus ‘scatter’ it), or re-emit the absorbed energy of the Ly photon as infrared radiation. The probability that the Ly photon is scattered, and thus survives its encounter with the dust grain, is given by the ‘albedo’ , where denotes the total cross section for dust absorption, and denotes the cross section for scattering. Both the albedo and absorption cross section depend on the dust properties. For example, Laursen et al. (2009a) shows that cm for SMC type dust (dust encountered in the Small Magellanic Cloud), and cm for LMC (Large Magellanic Cloud) type dust. Here, denotes the metallicity of the gas. Laursen et al. (2009a) further show that the frequency dependence of the dust absorption cross section around the Ly resonance can be safely ignored (see Figure 9).

A key difference between a dusty and dust-free medium is that in the presence of dust, Ly photons can be destroyed during the scattering process when . Dust therefore causes the ‘escape fraction’ (), which denotes the fraction Ly photons that escape from the dusty medium, to fall below unity, i.e. . Thus, while scattering of Ly photons off hydrogen atoms simply redistributes the photons in frequency space, dust reduces their overall number. Dust can similarly destroy continuum photons, but because Ly photons scatter and diffuse spatially through the dusty medium, the impact of dust on Ly and UV-continuum is generally different. In a uniform mixture of HI gas and dust, Ly 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 line.

The presence of dust does not necessarily reduce the EW of the Ly line. Dust can increase the EW of the Ly line in a ‘clumpy’ medium that consists of cold clumps containing neutral hydrogen gas and dust, embedded within a (hot) ionized, dust free medium (Neufeld 1991; Hansen & Oh 2006). In such a medium Ly photons can propagate freely through the interclump medium, and scatter only off the surface of the neutral clumps, thus avoiding exposure to dust grains. In contrast, UV continuum photons will penetrate the dusty clumps unobscured by hydrogen and are exposed to the full dust opacity. This is illustrated visually in Figure 10. Laursen et al. (2013) and Duval et al. (2014) have recently shown however that 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 photons from dusty media, EW boosting therefore appears uncommon. We note the conclusions of Laursen et al. (2013); Duval et al. (2014) apply to the EW-boost averaged over all photons emerging from the dusty medium. Gronke & Dijkstra (2014) 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 UV-continuum photon escape fraction was suppressed.

### 4.6 Monte Carlo Radiative Transfer

Analytic solutions to the radiative transfer equation (Eq 1) only exist for a few idealised cases. A modern approach to solve this equation is via Monte-Carlo151515It is worth pointing out that there exist numerous studies which explore alternatives to the Monte-Carlo approach, and focus on numerically solving approximations to the Ly radiative transfer equation (e.g. Roy et al. 2010; Yang et al. 2011; Higgins & Meiksin 2012; Yang et al. 2013)., in which scattering of individual photons is simulated until they escape (e.g Loeb & Rybicki 1999; Ahn et al. 2001; Zheng & Miralda-Escudé 2002; Cantalupo et al. 2005; Verhamme et al. 2006; Tasitsiomi 2006; Dijkstra et al. 2006; Semelin et al. 2007; Pierleoni et al. 2009; Kollmeier et al. 2010; Faucher-Giguère et al. 2010; Barnes et al. 2011; Zheng et al. 2010; Forero-Romero et al. 2011; Yajima et al. 2012; Orsi et al. 2012; Behrens & Niemeyer 2013)161616The codes presented by Pierleoni et al. (2009); Yajima et al. (2012) allow for simultaneous calculation of Ly and ionising photons.. Details on how the Monte-Carlo approach works can be found in many papers (see e.g. the papers mentioned above, and Chapters 6-8 of Laursen 2010, for an extensive description). To briefly summarise, for each photon in the Monte-Carlo simulations we first randomly draw a position, , at which the photon was emitted from the emissivity profile, a frequency from the Voigt function , and a random propagation direction