Saas-Fee Lecture Notes: Physics of Lyman Alpha Radiative Transfer
PPhysics of Ly α Radiative Transfer
Mark Dijkstra (Institute of Theoretical Astrophysics, University of Oslo) (Dated: April 7, 2017)
Lecture notes for the 46 th Saas-Fee winterschool, 13-19 March 2016, Les Diablerets, Switzerland.See a r X i v : . [ a s t r o - ph . GA ] A p r Contents
1. Preface
2. Introduction
3. The Hydrogen Atom and Introduction to Ly α Emission Mechanisms α Emission Mechanisms 9
4. A Closer Look at Ly α Emission Mechanisms & Sources
5. Astrophysical Ly α Sources
6. Step 1 Towards Understanding Ly α Radiative Transfer: Ly α Scattering Cross-section α Cross-Section 266.4. Voigt Profile of Ly α Cross-Section 27
7. Step 2 Towards Understanding Ly α Radiative Transfer: The Radiative Transfer Equation α Cross Section 307.2. II: Volume Emission Term 307.3. III: Scattering Term 307.4. IV: ‘Destruction’ Term 367.5. Ly α Propagation through HI: Scattering as Double Diffusion Process 38
8. Basic Insights and Analytic Solutions α Transfer through Uniform, Static Gas Clouds 398.2. Ly α Transfer through Uniform, Expanding and Contracting Gas Clouds 428.3. Ly α Transfer through Dusty, Uniform & Multiphase Media 43
9. Monte-Carlo Ly α Radiative Transfer α Monte-Carlo Radiative Transfer 479.3. Extracting Observables from Ly α Monte-Carlo Simulations in 3D Simulations 499.4. Accelerating Ly α Monte-Carlo Simulations 51
10. Ly α Transfer in the Universe α Radiative Transfer during Reionization 60
11. Miscalleneous Topics I: Polarization α Scattering: The Polarizability of the Hydrogen Atom 6711.2. Ly α Propagation through HI: Resonant vs Wing Scattering 6911.3. Polarization in Monte-Carlo Radiative Transfer 70
12. Applications beyond Ly α : Wouthuysen-Field Coupling and 21-cm Cosmology/Astrophysics References
1. PREFACE
These lectures notes cover my 8 lectures on the ‘Physics of Ly α Radiative Transfer’, which I gave at the 46 th Saas-Fee winter school that was held in Les Diablerets, Switzerland on March 13-19 2016. These lectures aimedat offering basic insights into Ly α radiative processes including emission processes and Ly α radiative transfer, andhighlighting some of the exciting physics associated with these processes. The notes discuss some derivation ingreater detail than what was discussed during the lectures. Feel free to contact me with any questions/comment onthese notes. I plan to update & expand these notes in time.The Ly α transfer problem is an exciting problem to learn about and work on. Ly α transfer is deeply rooted inquantum 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, thecircum-galactic medium, the intergalactic medium, reionization, 21-cm cosmology and astrophysics. In these lecturesI will describe the basics of Ly α radiative processes and transfer. These lectures are aimed to be self-contained, andare (hopefully) suitable for anyone with an undergraduate degree in astronomy/physics.Throughout these notes, I denote symbols that represent vectors in bold print . I use CGS units, as is common inthe literature. Table I provides an overview of (some of the) symbols that appear throughout these notes.
2. INTRODUCTION
Half a century ago, Partridge & Peebles (1967) predicted that the Ly α line should be a good tracer of star forminggalaxies at large cosmological distances. This statement was based on the assumption that ionizing photons thatare emitted by young, newly formed stars are efficiently reprocessed into recombination lines, of which Ly α containsthe largest flux. In the past two decades the Ly α line has indeed proven to provide us with a way to both find andidentify galaxies out to the highest redshifts (currently as high as z = 8 .
7, see Zitrin et al., 2015). In addition, we donot only expect Ly α emission from (star forming) galaxies, but from structure formation in general (e.g. Furlanettoet al. 2005). Galaxies are surrounded by vast reservoirs of gas that are capable of both emitting and absorbingLy α radiation. Observed spatially extended Ly α nebulae (or ‘blobs’) indeed provide insight into the formation &evolution of galaxies, in ways that complement direct observations of galaxies.Many new instruments & telescopes are either about to be, or have just been, commissioned that are ideal fortargeting the redshifted Ly α line. The sheer number of observed Ly α emitting sources is expected to increase by morethan two orders of magnitude at all redshifts z ∼ −
7. For comparison, this boost is similar to that in the numberof known exoplanets as a result of the launch of the Kepler satellite. In addition, sensitive integral field unit spectro-graphs will allow us to ( i ) detect sources that are more than an order of magnitude fainter than what has been possibleso 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 α emission from the environment of galaxies is visible.In order to optimally benefit form this rapidly growing body of data, we must understand the radiative transfer ofLy α photons. Ly α transfer depends sensitively on the distribution and kinematics of neutral gas, which complicatesinterpretations of Ly α observations. On the other hand, the close interaction of the Ly α radiation field and gaseousflows in and around galaxies implies that the Ly α line contains information on the scattering medium, and may thuspresent an opportunity to learn more about atomic hydrogen in gaseous flows in and around galaxies . New instruments/telescopes that will revolutionize our ability to target Ly α emission: the Hobby-Eberly Telescope Dark EnergyExperiment (HETDEX, http://hetdex.org/ ) will increase the sample of Ly α emitting galaxies by orders of magnitude at z ∼ − ) will provide a similar boost out to z ∼
7. Integral Field UnitSpectrographs such as MUSE ( , see Bacon et al. 2010) and theKeck Cosmic Web Imager ( , and Martin et al. 2014 for observationscarried out with the
Palomar
Cosmic Web Imager) will allow us to map out spatially extended Ly α emission down to ∼
10 times lowersurface brightness levels, and take spatially resolved spectra. In the (near) future, telescopes such as the James Webb Space Telescope(JWST, ) and ground based facilities such as the Giant Magellan Telescope ( ) andESO’s E-ELT ( , (TMT). To underline this point: recent observations of Ly α sources (see e.g. CR7, Sobral et al. 2015) have triggered discussion on the formationof direct collapse black holes (e.g. Pallottini et al. 2015), Population III stars (e.g. Visbal et al. 2016), and on the structure of multiphasegases in and around galaxies (McCourt et al. 2017). These recent developments are not discussed in these lecture notes, and reflectthat only a handful of new observations of the Ly α have already started highlighting that exciting science can be done with the line. TABLE I Symbol DictionarySymbol Definition k B Boltzmann constant: k B = 1 . × − erg K − h P Planck constant: h P = 6 . × − erg s¯ h reduced Planck constant: ¯ h = h P π m p proton mass: m p = 1 . × − m e electron mass: m e = 9 . × − g q electron charge: q = 4 . × − esu c speed of light: c = 2 . × cm s − ∆ E ul Energy difference between upper level ’u’ and lower level ’l’ (in ergs) ν ul photon frequency associated with the transition u → l (in Hz) f ul the oscillator strength associated with the transition u → l (dimensionless) A ul Einstein A-coefficient of the transition u → l (in s − ) B ul Einstein B-coefficient of the transition u → l : B ul = h P ν ul c A ul (in erg cm − s − ) B lu Einstein B-coefficient of the transition l → u : B lu = g u g l B ul α A/B case A /B recombination coefficient (in cm s − ) α nl recombination coefficient into state ( n, l ) (in cm s − ) g u/l statistical weight of upper/lower level of a radiative transition (dimensionless) ν α photon frequency associated with the Ly α transition: ν α = 2 . × Hz ω α angular frequency associated with the Ly α transition: ω α = 2 πν α λ α wavelength associated with the Ly α transition: λ α = 1215 .
67 ˚A A α Einstein A-coefficient of the Ly α transition: A α = 6 . × s − T gas temperature (in K) v th velocity dispersion (times √ v th = q k B Tm p v turb turbulent velocity dispersion b Doppler broadening parameter : b = p v + v ∆ ν α Doppler induced photon frequency dispersion:∆ ν α = ν α bc (in Hz) x ’normalized’ photon frequency: x = ( ν − ν α ) / ∆ ν α (dimensionless) σ α ( x ) Ly α absorption cross-section at frequency x (in cm ), σ α ( x ) = σ α, φ ( x ) σ α, Ly α absorption cross-section at line center, σ α, = 5 . × − (cid:16) T K (cid:17) − / cm φ ( x ) Voigt profile (dimensionless) a v Voigt parameter: a v = A α / [4 π ∆ ν α ] = 4 . × − ( T / K) − / I ν specific intensity (in erg s − Hz − cm − sr − ) J ν angle averaged specific intensity (in erg s − Hz − cm − sr − )
3. THE HYDROGEN ATOM AND INTRODUCTION TO LY α EMISSION MECHANISMS3.1. Hydrogen in our Universe
It has been known from almost a century that hydrogen is the most abundant element in our Universe. In 1925Cecilia Payne demonstrated in her PhD dissertation that the Sun was composed primarily of hydrogen and helium.While this conclusion was controversial at the time, it is currently well established that hydrogen accounts for themajority of baryonic mass in our Universe: the fluctuations in the Microwave Background as measured by the Plancksatellite (Planck Collaboration et al., 2016) imply that baryons account for 4 .
6% of the Universal energy density,and that hydrogen accounts for 76% of the total baryonic mass. The remaining 24% is in the form of helium (seeFig 1). The leading constraints on these mass ratios come from Big Bang Nucleosynthesis, which predicts a hydrogenabundance of ∼
75% by mass for the inferred Universal baryon density (Ω b h = 0 . See https://en.wikipedia.org/wiki/Cecilia_Payne-Gaposchkin . FIG. 1 Relative contributions to Universal energy/mass density. Most ( ∼ ∼
25% is in the form of dark matter, which isa pressureless fluid which acts only gravitationally with ordinary matter. Only ∼
5% of the Universal energy content is in theform of ordinary matter like baryons, leptons etc. Of this component, ∼
75% of all baryonic matter is in Hydrogen, while theremaining 25% is almost entirely Helium. Observing lines associated with atomic hydrogen is therefore an obvious way to goabout studying the Universe. z > .
5, and from atomic hydrogen in the diffuse (neutral) intergalacticmedium represent the main science drivers for many low frequency radio arrays that are currently being developed,including the Murchinson Wide Field Array , the Low Frequency Array , The Hydrogen Epoch of Reionization Ar-ray (HERA) , the Precision Array for Probing the Epoch of Reionization (PAPER) , and the Square Kilometer Array .Similarly, the Ly α transition has also revolutionized observational cosmology: observations of the Ly α forest inquasar spectra has allowed us to measure the matter distribution throughout the Universe with unprecedented ac-curacy. The Ly α forest still provides an extremely useful probe of cosmology on scales that are not accessible withgalaxy surveys, and/or the Cosmic microwave background. The Ly α forest will be covered extensively in the lecturesby J.X. Prochaska. So far, the most important contributions to our understanding of the Universe from Ly α havecome from studies of Ly α absorption. However, with the commissioning of many new instruments and telescopes,there is tremendous potential for Ly α in emission. Because Ly α is a resonance line, and because typical astrophysicalenvironments are optically thick to Ly α , we need to understand the radiative transfer to be able to fully exploit theobservations of Ly α emitting sources. http://reionization.org/ http://eor.berkeley.edu/ FIG. 2 In the classical picture of the hydrogen atom, an electron orbits a central proton at v ∼ αc . The acceleration that theelectron experiences causes it to emit electromagnetic waves and loose energy. This causes the electron to spiral inwards intothe proton on a time-scale of ∼ − s. In the classical picture, hydrogen atoms are highly unstable, short-lived objects. The classical picture of the hydrogen atom is that of an electron orbiting a proton. In this picture, the electrostaticforce binds the electron and proton. The equation of motion for the electron is given by q r = m e v e r , (1)where q denotes the charge of the electron and proton, and the subscript ‘e’ (‘p’) indicates quantities related to theelectron (proton). The acceleration the electron undergoes thus equals a e = v e r = q r m e . When a charged particleaccelerates, it radiates away its energy in the form of electromagnetic waves. The total energy that is radiated awayby the electron per unit time is given by the Larmor formula, which is given by P = 23 q a e c . (2)The total energy of the electron is given by the sum of its kinetic and potential energy, and equals E e = m e v e − q r = − q r . The total time it takes for the electron to radiative away all of its energy is thus given by t = E e P = 3 c ra e = 3 r m e c q ≈ − s , (3)where we substituted the Bohr radius for r , i.e. r = a = 5 . × − cm. In the classical picture, hydrogen atomswould be highly unstable objects, which is clearly problematic and led to the development of quantum mechanics.In quantum mechanics, electron orbits are quantized : In Niels Bohr’s model of the atom, electrons can only reside indiscrete orbitals. While in such an orbital, the electron does not radiate. It is only when an electron transitions fromone orbital to another that it emits a photon. Quantitatively, the total angular momentum of the electron L ≡ m e v e r can only taken on discrete values L = n ¯ h , where n = 1 , , ... , and ¯ h denotes the reduced Planck constant (Table I).The total energy of the electron is then E e ( n ) = − q m e n ¯ h = − E n , (4)where E = 13 . n . The quantum mechanical picture of the hydrogen atomdiffers from the classical one in additional ways: the electron orbital of a given quantum state (an orbital characterized increasing orbital quantum number i n c r ea s i ng e n e r gy FIG. 3 The total energy of the quantum states of the hydrogen atom, and a simplified representation of the associated quantummechanical wavefunction describing the electron. The level denoted with ‘1s’ denotes the ground state and has a total energy E = − . n . The eccentricity/elongation of the wavefunction increases with quantum number l . The orientation ofnon-spherical wavefunction can be represented by a third quantum number m . by quantum number n ) does not correspond to the classical orbital described above. Instead, the electron is describedby a quantum mechanical wavefunction ψ ( r ), (the square of) which describes the probability of finding the electron atsome location r . The functional form of these wavefunctions are determined by the Schr¨odinger equation. We will notdiscuss the Schr¨odinger equation in these lectures, but will simply use that it implies that the quantum mechanicalwavefunction ψ ( r ) of the electron is characterized fully by two quantum numbers: the principal quantum number n ,and the orbital quantum number l . The orbital quantum number l can only take on the values l = 0 , , , ..., n − E and total angular momentum L .The diagram in Figure 3 shows the total energy of different quantum states in the hydrogen, and a sketch of theassociated wavefunctions. This Figure indicates that • The lowest energy state corresponds to the n = 1 state, with an energy of E = − . n = 1,the orbital quantum number l can only take on the value l = 0. This state with ( n, l ) = (1 ,
0) is referred toas 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 describesthe 1 s -state is spherically symmetric. The ‘size’ or extent of this wavefunction relates to the classical atomsize in that the expectation value of the radial position of the electron corresponds to the Bohr radius a , i.e. R dV r | ψ s ( r ) | = a . • The second lowest energy state, n = 2, has a total energy E = E /n = − . l = 0 and l = 1. The ‘2s’-state is again characterized by a spherically symmetricwavefunction, 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 electronbeing 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 /n = − . n (seee.g. https://en.wikipedia.org/wiki/Atomic_orbital for illustrations). Loosely speaking, the quantum
1 2 3 4increasing orbital quantum number i n c r ea s i ng e n e r gy cascades resulting in Ly α cascades not resulting in Ly α Ly α FIG. 4 Atoms in any state with n > | ∆ l | = 1. These transitions are indicated with colored lines connecting thedifferent quantum states. The green solid lines indicate radiative cascades that result in the emission of a Ly α photon, while red dotted lines indicate transitions that do not. We have omitted all direct radiative transitions np → s : this corresponds tothe ‘case-B’ approximation, which assumes that the recombining gas is optically thick to all Lyman-series photons, and thatthese photons would be re-absorbed immediately. The table in the lower right corner indicates Ly α production probabilitiesfrom various states: e.g. the probability that at atom in the 4 s state produces a Ly α photon is ∼ . Credit: from Figure 1of Dijkstra 2014, Lyman Alpha Emitting Galaxies as a Probe of Reionization, PASA, 31, 40D . number n denotes the extent/size of the wavefunction, l denotes its eccentricity/elongation. The orientation ofnon-spherical wavefunction can be represented by a third quantum number m . We discussed how in the classical picture of the hydrogen atom, the electron ends up inside the proton after ∼ − s. In quantum mechanics, the electron is only stable in the ground state (1s). The life-time of an atom in any excitedstate is very short, analogous to the instability of the atom in the classical picture. Transitions between differentquantum states have been historically grouped into series, and named after the discoverer of these series. The seriesinclude • The Lyman series . A series of radiative transitions in the hydrogen atom which arise when the electron goesfrom n ≥ n = 1. The first line in the spectrum of the Lyman series - named Lyman α (hereafter, Ly α ) -was discovered in 1906 by Theodore Lyman, who was studying the ultraviolet spectrum of electrically excitedhydrogen 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 ≥ n = 2. The series is named after JohannBalmer, who discovered an empirical formula for the wavelengths of the Balmer lines in 1885. The Balmer- α (hereafter H α ) transition is in the red, and is responsible for the reddish glow that can be seen in the famousOrion nebula. • Following the Balmer series, we have the
Paschen series ( n ≥ → n = 3), the Brackett series ( n ≥ → Pfund series ( n ≥ → δ is potentially an interesting probe ( ¨Ostlin & Hayes,2009-2016 private communication).Quantum mechanics does not allow radiative transitions between just any two quantum states: these radiativetransitions must obey the ‘selection rules’. The simplest version of the selection rules - which we will use in theselectures - is that only transitions of the form | ∆ l | = 1 are allowed. A simple interpretation of this is that photonscarry a (spin) angular momentum given by ¯ h , which is why the angular momentum of the electron orbital mustchange by ± ¯ h as well. Figure 4 indicates allowed transitions, either as green solid lines or as red dashed lines . Notethat the Lyman- β, γ, ... transitions (3 p → s , 4 p → s , ...) are not shown on purpose. As we will see later in thelectures, while these transitions are certainly allowed, in realistic astrophysical environments it is better to simply − R H−atom e (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) FIG. 5 Cooling radiation at the atomic level: an interaction between an electron and a hydrogen atom can leave the hydrogenatom in an excited state, which can produce a Ly α photon. The energy carried by the Ly α photon comes at the expense ofthe kinetic energy of the electron. Ly α emission by the hydrogen atom thus cools the gas. ignore them.Consider an electron in some arbitrary quantum state ( n, l ). The electron does not spend much time in thisstate, and radiatively decays down to a lower energy state ( n , l ). This lower energy state is again unstable [unless( n , l ) = (1 , n , l ). Ultimately, allpaths lead to the ground state, even those paths that go through the 2 s state. While the selection rules do notpermit transitions of the form 2 s → s , these transitions can occur, if the atom emits two photons (rather thanone). Because these two-photon transitions are forbidden, the life-time of the electron in the 2 s state is many ordersof magnitude larger than almost all other quantum states (it is ∼ p -state), and this quantum state is called ‘meta-stable’. The path from an arbitrary quantum state ( n, l ) to theground state via a sequence of radiative decays is called a ‘radiative cascade’.The green solid lines in Figure 4 show radiative cascades that result in the emission of a Ly α photon. The red dashedlines show the other radiative cascades. The table in the lower right corner shows the probability that a radiativecascade from quantum state ( n, l ) produces a Ly α photon. This probability is denoted with P ( n, l → ... → Ly α ). Forexample, the probability that an electron in the 2 s orbital gives rise to Ly α is zero. The probability that an electronin the 3 s orbital gives Ly α is 1. This is because the only allowed radiative cascade to the ground state from 3 s is3 s → p → s . This last transition corresponds to the Ly α transition. For n ≥ α Emission Mechanisms
A hydrogen atom emits Ly α once its electron is in the 2 p state and decays to the ground state. We mentionedqualitatively how radiative cascades from a higher energy state can give rise to Ly α production. Electrons can endup these higher energy quantum states (any state with n >
1) in two different ways:1.
Collisions.
The ‘collision’ between an electron and a hydrogen atom can leave the atom in an excited state, atthe expense of kinetic energy of the free electron. This process is illustrated in Figure 5. This process convertsthermal energy of the electrons, and therefore of the gas as whole, into radiation. This process is also referredto as Ly α production via ‘cooling’ radiation. We discuss this process in more detail in § § 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 α photon. As we discussed in § n, l ) produces a Ly α photon during the radiative cascadedown to the ground-state. If we sum over all these quantum states, and properly weigh by the probability thatthe freshly combined electron-proton pair ended up in state ( n, l ), then we can compute the probability that arecombination event gives us a Ly α photon. We discuss the details of this calculation in § upper panel of Figure 6 shows the total probability P (Ly α ) that a Ly α photon is emitted per recombinationevent as a function of gas temperature T . This plot contains two lines. The solid black line represents ‘Case-A’,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‘Case-B’, which refers to the case where we do not allow for ( i ) direct recombination into the ground state,0 FIG. 6 The top panel shows the probability P (Ly α ) that a recombination event leads to the production of a Ly α photon,as a function of gas temperature T . The upper dashed line ( lower solid line ) corresponds to ‘case B’ (‘case A’). The lowerpanel shows the recombination rate α ( T ) (in cm s − ) at which electrons and protons recombine. The solid line ( dashed line )represents case-B (case-A). The red open circles represent fitting formulae (Eq 5). Credit: from Figure 2 of Dijkstra 2014,Lyman Alpha Emitting Galaxies as a Probe of Reionization, PASA, 31, 40D . which produces an ionizing photon, and ( ii ) radiative transitions of the higher order Lyman series, i.e. Ly β ,Ly γ , Ly δ ,.... Case-B represents that most astrophysical gases efficiently re-absorb higher order Lyman seriesand ionizing photons, which effectively ‘cancels out’ these transitions (see § T = 10 K and case-B recombination, we have P (Ly α ) = 0 .
68. This value‘0 . is often encountered during discussions on Ly α emitting galaxies. It is worth keeping in mind that theprobability P (Ly α ) increases with decreasing gas temperature and can be as high as P (Ly α ) = 0 .
77 for T = 10 K (also see Cantalupo et al. 2008). The red open circles represent the following two fitting formulae P A (Ly α ) = 0 . − .
165 log T − . T ) − . (5) P B (Ly α ) = 0 . − .
106 log T − . T ) − . , where T ≡ T / K. The fitting formula for case-B is taken from Cantalupo et al. (2008).
4. A CLOSER LOOK AT LY α EMISSION MECHANISMS & SOURCES
The previous section provided a brief description of physical processes that give rise to Ly α emission. Here, wediscuss these in more detail, and also link them to astrophysical sources of Ly α . Collisions involve an electron and a hydrogen atom. The efficiency of this process depends on the relative velocity ofthe two particles. The Ly α production rate therefore includes the product of the number density of both species, andthe rate coefficient q s p ( P [ v e ]) which quantifies the velocity dependence of this process ( P [ v e ] denotes the velocitydistribution of electrons). If we assume that the velocity distribution of electrons is given by a Maxwellian distribution,1 FIG. 7
Left:
Velocity averaged collision strengths h Ω lu i are plotted as a function of temperature for the transitions 1 s → s ( dotted line ), 1 s → p ( dashed line ), and for their sum 1 s → black solid line ) as given by Scholz et al. (1990); Scholz &Walters (1991). Also shown are velocity averaged collision strengths for the 1 s → red solid line , obtained by summing overall transitions 3 s, p and 3 d ), and 1 s → blue solid line , obtained by summing over all transitions 4 s, p, d and 4 f ) as givenby Aggarwal et al. (1991). Evaluating the collision strengths becomes increasingly difficult towards higher n (see text). Right:
The blue-dotted line in the top panel shows the total cooling rate per H-nucleus that one obtaines by collisionally exciting Hatoms into all states n ≤
4. For comparison, the red dashed line shows the total cooling rate as a result of collisional excitationof the 2p state, which is followed by a downward transition through emission of a Ly α photon. All cooling rates increase rapidlyaround T ∼ K. The lower panel shows the ratio (in%) of these two cooling rates. This plot shows that ∼
60% of the totalgas cooling rate is in the form of Ly α photons at T ∼ K, and that this ratio decreases to ∼ −
50% towards higher gastemperatures. At the gas temperatures at which cooling via line excitation is important,
T < ∼ K (see text), ∼ − α photons. Also shown for comparison as the black solid line is the often used fitting formula ofBlack (1981), and modified following Cen (1992). then P [ v e ] is uniquely determined by temperature T , and the rate coefficient becomes a function of temperature, q s p ( T ). The total Ly α production rate through collisional excitation is therefore R Ly α coll = n e n H q s p cm − s − . (6)In general, the rate coefficient q lu is expressed in terms of a ‘velocity averaged collision strength’ h Ω lu i as q lu = h P (2 πm e ) / ( k B T ) / h Ω lu i g l exp (cid:16) − ∆ E lu k B T (cid:17) = 8 . × − T − / h Ω lu i g l exp (cid:16) − ∆ E lu k B T (cid:17) cm s − . (7)Calculating the collision strength is a very complex problem, because for the free-electron energies of interest, thefree electron spends a relatively long time near the target atom, which causes distortions in the bound electron’swavefunctions. Complex quantum mechanical interactions may occur, and especially for collisional excitation intohigher-n states, multiple scattering events become important (see Bely & van Regemorter, 1970, and referencestherein). The most reliable collision strengths in the literature are for the 1 s → nL , with n < L < d (Aggarwal1983, Scholz et al. 1990, Osterbrock & Ferland 2006). The left panel of Figure 7 shows the velocity averaged collisionstrength for several transitions. There exist some differences in the calculations between these different groups.Collisional excitation rates still appear uncertain at the 10-20% level. The subscripts ‘l’ and ‘u’ refer to the ‘lower’ and ‘upper’ energy states, respectively FIG. 8 This figure shows the temperature dependence of the cooling rate of primordial gas, under the assumption of collisionalionization equilibrium (i.e. the ionization state of the gas is set entirely by its temperature). Gas cooling due to collisionalexcitation of atomic hydrogen becomes important at log
T /K ∼
4, and reaches a maximum at log
T /K ∼ .
2, beyond whichcollisions can ionize away atomic hydrogen. At log
T /K > ∼ . As we mentioned above, radiation is produced in collisions at the expense of the gas’ thermal energy. The totalrate at which the gas looses thermal energy, i.e. cools, per unit volume is dE th dV dt = n e n H C ( T ) , (8)where C ( T ) = X u q s → u ∆ E s → u erg cm s − . (9)Here, the sum is over all excited states ’u’. The blue-dotted line in the top right panel of Figure 7 shows C ( T )including collisional excitation into all states n ≤
4. The cooling rate rises by orders of magnitude around T ∼ K, and reflects the strong temperature-dependence of the number density of electrons that are moving fast enough toexcite the hydrogen atom. For comparison, the red dashed line shows the contribution to C ( T ) from only collisionalexcitation into the 2p state, which is followed by a downward transition through emission of a Ly α photon. The lowerpanel shows the ratio (in%) of these two rates. This plot shows that ∼
60% of the total gas cooling rate is in the formof Ly α photons at T ∼ K, and that this ratio decreases to ∼ −
50% towards higher gas temperatures. The black solid line is an often-used analytic fitting formula by Black (1981) C ( T ) = 7 . × − exp (cid:16) − T (cid:17) (1 + T / ) erg cm s − . (10)It is good to keep in mind that over the past few decades, the hydrogen collision strengths have changed quitesubstantially, which can explain the difference between these curves.The cooling rate per unit volume depends on the product of C ( T ), n e , and n H , and therefore on the ionizationstate of the gas. If we also assume that the ionization state of the gas is determined entirely by its temperature (thegas is then said to be in ‘collisional ionization equilibrium’), then the total cooling rate per unit volume is a functionof temperature only (and overall gas density squared). Figure 8 shows that the cooling curve increases dramaticallyaround T ∼ K, which is due to the corresponding increase in C ( T ) (see Fig 7). The cooling curve reaches a3maximum at log T ∼ .
2, which is because at higher T collisional ionization of hydrogen removes neutral hydrogen,which eliminates the collisional excitation cooling channel. For a cosmological mixture of H and He, collisionalexcitation of singly ionized Helium starts dominating at log T ∼ . The capture of an electron by a proton generally results in a hydrogen atom in an excited state ( n, l ). Once anatom is in a quantum state ( n, l ) it radiatively cascades to the ground state n = 1, l = 0 via intermediate states( n , l ). The probability that a radiative cascade from the state ( n, l ) results in a Ly α photon is given by P ( n, l → Ly α ) = X n ,l P ( n, l → n , l ) P ( n , l → Ly α ) . (11)This may not feel satisfactory, as we still need to compute P ( n , l → Ly α ), which is the same quantity but for n < n .In practice, we can compute P ( n , l → Ly α ) by starting at low values for n , and then work towards increasinglyhigh n . For example, the probability that a radiative cascade from the ( n, l ) = (3 ,
1) state (i.e. the 3p state) producesa Ly α photon is 0, because the selection rules only permit the transitions (3 , → (2 ,
0) and (3 , → (1 , n, l ) = (3 ,
2) state (i.e. the 3dstate) will certainly produce a Ly α photon, since the only permitted cascade is (3 , → (2 , Ly α → (1 , , → (2 , Ly α → (1 ,
0) , and P (3 , → Ly α ) = 1. For n >
3, multipleradiative cascades down to the ground state are generally possible, and P ( n, l → Ly α ) takes on values other than0 or 1 (see e.g. Spitzer & Greenstein, 1951, for numerical values). Figure 4 also contains a table that shows theprobability P ( n , l → Ly α ) for n ≤ n, l ) into multiple states ( n , l ), thenthis probability is given by the ‘branching ratio’ which represents the ratio of the decay rate into state ( n , l ) into allpermitted states, i.e. P ( n, l → n , l ) = A n,l,n ,l P n ,l A n,l,n ,l , (12)in which A n,l,n ,l denotes the Einstein A-coefficient for the nl → n l transition, where the quantum mechanicalselection rules only permit transitions for which | l − l | = 1. The probability that a radiative cascade from an arbi-trary quantum state ( n, l ) gives rise to a Ly α photon can be computed once we know the Einstein-coefficients A n,l,n ,l .The probability that an arbitrary recombination event results in a Ly α photon follows naturally, if we know theprobability that recombination leaves the atom in state ( n, l ). That is P (Ly α ) = ∞ X n min n − X l =0 α nl ( T ) α tot ( T ) P ( n, l → Ly α ) , (15) As we referred to in § Green solid lines depictradiative cascades that result in a Ly α photon, while red dotted lines depict radiative cascades that do not yield a Ly α photon. This coefficient is given by A n,l,n ,l = 64 π ν ul h P c max( l , l )2 l + 1 e a [ M ( n, l, n , l )] , (13)where fundamental quantities e , c , h P , and a are given in Table I, h P ν ul denotes the energy difference between the upper (n,l) andlower (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 ): M ( n, l, n , l ) = Z ∞ P n,l ( r ) r P n ,l ( r ) dr. (14)Analytic expressions for the matrix M ( n, l, n , l ) that contain hypergeometric functions were derived by Gordon (1929). For the Ly α transition M ( n, l, n , l ) = M (2 , , ,
0) = √ / n, l ) state, in which α tot denotes the totalrecombination coefficient α tot ( T ) = P ∞ n min P n − l =0 α nl ( T ). The temperature-dependent state specific recombinationcoefficients α nl ( T ) can be found in for example Burgess (1965) and Rubi˜no-Mart´ın et al. (2006). The value of n min depends on the physical conditions of the medium in which recombination takes place, and two cases bracket therange 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 n min = 1. • ‘case-B ’ recombination: recombination takes place in a medium that is opaque to all Lyman series photons(i.e. Ly α , Ly β , Ly γ , ...), and to ionizing photons that were emitted following direct recombination into theground state. In the so-called ‘on the spot approximation’, direct recombination to the ground state producesan ionizing photon that is immediately absorbed by a nearby neutral H atom. Similarly, any Lyman seriesphoton is immediately absorbed by a neighbouring H atom. This case is quantitatively described by setting n min = 2, and by setting the Einstein coefficient for all Lyman series transitions to zero, i.e. A np, s = 0.The probability P (Ly α ) that we obtain from Eq 15 was plotted in Figure 6 assuming case-A ( solid line ) and case-B( dashed line ) recombination. The temperature dependence comes in entirely through the temperature dependenceof the state-specific recombination coefficients α nl ( T ). As we mentioned earlier, for case-B recombination, we have P (Ly α ) = 0 .
68 at T = 10 K. It is worth keeping in mind that our calculations technically only apply in a low densitymedium. For ‘high’ densities, collisions can ‘mix’ different l -levels 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 ∝ (2 l − §
5. ASTROPHYSICAL LY α SOURCES
Now that we have specified different physical mechanisms that give rise to the production of a Ly α photon, wediscuss various astrophysical sites of Ly α production. Interstellar HII regions are the most prominent sources of Ly α emission in the Universe. Hot, (mostly) massive andyoung stars produce ionizing photons in their atmospheres which are efficiently absorbed in the interstellar medium,and thus create ionized HII regions. Recombining protons and electrons give rise to Ly α , H α , etc lines. These linesare called ‘nebular’ lines. One of the most famous nebulae is the Orion nebula, which is visible with the naked eye inthe constellation of Orion. The reddish glow is due to the H α line, which at λ = 6536 ˚A falls in the middle of thered part of the visual spectrum, that is produced as recombination emission. We showed previously that there is a P (Ly α ) = 0 .
68 probability that a Ly α photon is produced per case-B recombination event at T = 10 K. A similaranalysis can yield the probability that an H α photon is produced is: P (H α ) ∼ .
45. The total ratio of the Ly α to H α flux is therefore ∼
8. It is interesting to realize that the total Ly α luminosity that is produced in the Orion nebulais almost an order of magnitude larger than the flux contained in the H α line, which is prominently visible in Figure 9.Recombinations in HII regions in the ISM balance photoionization by ionizing photons produced by the hot stars.The total recombination rate in an equilibrium HII region therefore equals the total photoionization rate in thenebulae, i.e. the total rate at which ionizing photons are absorbed in the HII region. If a fraction f ionesc of ionizingphotons is not absorbed in the HII region (and hence escapes), then the total Ly α production rate in recombinationsis ˙ N recLy α = P (Ly α )(1 − f ionesc ) ˙ N ion ≈ . − f ionesc ) ˙ N ion , case − B , T = 10 K , (16) 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 case-C/D/E/... recombination todescribe recombination in a medium that is optically thick to Balmer/Paschen/Bracket/... series photons. The condition of equilibrium is generally satisfied in ordinary interstellar HII regions. In expanding HII regions, e.g. those that exist inthe intergalactic medium during cosmic reionization (which is discussed later), the total recombination rate is less than the total rateat which ionising photons are absorbed FIG. 9 The Orion nebula is one of the closest nearby nebulae, and visible with the naked eye. The reddish glow is due to H α that was produced as recombination radiation. These same recombination events produce a Ly α flux that is about 8 × larger.Recombination line emission is often referred to as ‘nebular’ emission. Credit: ESA/Hubble . where ˙ N ion ( ˙ N recLy α ) denotes the rate at which ionizing (Ly α recombination) photons are emitted. The production rateof ionizing photons, ˙ N ion , relates to the abundance of short-lived massive stars, and therefore closely tracks the starformation rate. For a Salpeter (1955) initial mass function (IMF) with mass limits M low = 0 . M (cid:12) and M high = 100 M (cid:12) we have (Kennicutt, 1998)˙ N ion = 9 . × × SFR(M (cid:12) / yr) s − ⇒ L α = 1 . × × SFR(M (cid:12) / yr) erg s − (Salpeter , Z = Z (cid:12) ) . (17)Equation 17 strictly applies to galaxies with continuous star formation over timescales of 10 years or longer. Thisequation is commonly adopted in the literature. A Kroupa IMF gives us a slightly higher Ly α luminosity for a givenSFR: L α = 1 . × × SFR(M (cid:12) / yr) erg s − (Kroupa , Z = Z (cid:12) ) . (18)Finally, for a fixed IMF the Ly α production rate increases towards lower metallicities: stellar evolution modelscombined with stellar atmosphere models show that the effective temperature of stars of fixed mass become hotterwith decreasing gas metallicity (Tumlinson & Shull 2000, Schaerer 2002). The increased effective temperature of starscauses a larger fraction of their bolometric luminosity to be emitted as ionizing radiation. We therefore expect galaxiesthat formed stars from metal poor (or even metal free) gas, to be strong sources of nebular emission. Schaerer (2003)provides the following fitting formula for ˙ N ion as a function of absolute gas metallicity Z gas log ˙ N ion = − . × (log Z gas + 9 . . + 53 .
81 + log SFR(M (cid:12) / yr) (Salpeter) . (19) Warning : note that this fitting formula is valid for a Salpeter IMF in the mass range M = 1 − M (cid:12) . If wesubstitute Z = Z (cid:12) = 0 .
02 we get log ˙ N ion = 53 .
39, which is a factor of ∼ . α luminosity. However, at Z < ∼ . Z (cid:12) departures from case-B increases the Ly α luminosity relative to case-B(e.g. Raiter et al., 2010). This increase of the Ly α 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. It is useful to recall that solar metallicity Z (cid:12) = 0 . n = 2 level, and which can transfer atoms from the 2 s into the 2 p states; ( ii ) harder ionizingspectra emitted by metal poor(er) stars. When higher energy photons (say E γ > ∼
50 eV) photoionize a hydrogen atom,then it releases an electron with a kinetic energy that is E = E γ − . ∼
25 hydrogen atoms in a fully neutralmedium. Raiter et al. (2010) provide a simple analytic formula which captures these effects:˙ N recLy α = f coll P (1 − f ionesc ) ˙ N ion (non − case B) , (20)where P ≡ h E γ, ion i / . h E γ, ion i denotes the mean energy of ionising photons . Furthermore, f coll ≡ an HI b + cn HI , in which a = 1 . × − , b = 1 . c = 1 . × − , and n HI denotes the number density of hydrogennuclei. Eq 20 resembles the ‘standard’ equation, but replaces the factor 0.68 with P f coll , which can exceed unity.Eq 20 implies that for a fixed IMF, the Ly α luminosity may be boosted by a factor of a few. Incredibly, for certainIMFs the Ly α line may contain 40% of the total bolometric luminosity of a galaxy. Not only nebulae are sources of Ly α radiation. Most of our Universe is in fact a giant Ly α source. Observations ofspectra of distant quasars reveal a large collection of Ly α absorption lines. This so-called ‘Ly α forest’ is discussed inmore detail in the lecture notes by X. Prochaska. Observations of the Ly α forest imply that the intergalactic mediumis highly ionized, and that the temperature of intergalactic gas is T ∼ K. Observations of the Ly α forest can bereproduced very well if we assume that gas is photoionized by the Universal ”ionizing background” that permeatesthe entire Universe, and that is generated by adding the contribution from all ionizing sources . The residual neutralfraction of hydrogen atoms in the IGM is x HI ≡ n HI n HI + n HII = n e α B ( T )Γ ion , where Γ ion denotes the photoionization rate bythe ionizing background (with units s − ).From § ∼ .
68 Ly α photons. First, we note thatthe recombination time of a proton and electron in the intergalactic medium is t rec ≡ α B ( T ) n e = 9 . × (cid:18) z (cid:19) − (cid:18) δ (cid:19) − T . yr , (22)where we used that fully ionized gas at mean density of the Universe, δ = 0, has n e = ¯ n = Ω m ρ crit , µm p (1 + z ) ∼ × − (1 + z ) cm − , in which ρ crit , = 1 . × − h ( H ≡ h km s − Mpc − ) denotes the critical density ofthe Universe today. We also approximated the case-B recombination coefficient as α B ( T ) = 2 . × − T − . cm s − , in which we have adopted the notation T ≡ ( T / K). We can compare this number to the Hubble time whichis t Hub ≡ H ( z ) ∼ z ] / − / Gyr (where the last approximation is valid strictly for z (cid:29) δ = 0) thus exceeds the Hubble time for most of the existence of the Universe. Only at That is, h E γ, ion i ≡ h P R ∞ . dνf ( ν ) R ∞ . dνf ( ν ) /ν , where f ( ν ) denotes the flux density. Another useful measure for the ‘strength’ of the Ly α line is the equivalent width (EW, which was discussed in much more detail in thelectures by J.X. Prochaska) of the line: EW ≡ Z dλ ( F ( λ ) − F ) /F , (21)which measures the total line flux compared to the continuum flux density just redward (as the blue side can be affected by intergalacticscattering, see § α line, F . For a Salpeter IMF in the range 0 . − M (cid:12) , Z = Z (cid:12) , the UV-continuum luminosity density, L UV ν , relates to SFR as L UV ν = 8 × × SFR( M (cid:12) / yr) erg s − Hz − . The corresponding equivalent width of the Ly α line would beEW ∼
70 ˚A (Dijkstra & Westra 2010). The equivalent width can reach a few thousand ˚A for Population III stars/galaxies forming starswith a top-heavy IMF (see Raiter et al. 2010). All sources within a radius equal to the mean free path of ionizing photons, λ ion . For more distant sources ( r > λ ion ), the ionizing fluxis reduced by an additional factor exp( − r/λ ion ). L FIG. 10 The adopted geometry for calculating the Ly α surface brightness of recombining gas in the IGM at redshift z . z > ∼ α recombination radiation (see e.g. Martinet al. 2014).Ly α emission from the CGM/IGM differs from interstellar (nebular) Ly α emission in two ways: ( i ) Ly α emissionfrom the CGM/IGM occurs over a spatially extended region. Spatially extended Ly α emission is better characterizedby its surface brightness (flux per unit area on the sky) than by its overall flux. We present a general formalism forcomputing the Ly α surface brightness below; and ( ii ) Ly α emission is powered by external sources. For example,recombination radiation from the CGM/IGM balances photoionization by either the ionizing background (generateby a large numbers of star forming galaxies and AGN), or a nearby source. This conversion of externally - i.e. not within the same galaxy (or even cloud) - generated ionizing radiation into Ly α is known as ‘ fluorescence ’ (e.g. Hogan& Weymann 1987, Gould & Weinberg 1996). Fluorescence generally corresponds to emission of radiation by somematerial following absorption by radiation at some other wavelength. Certain minerals emit radiation in the opticalwhen irradiated by UV radiation. Fluorescent materials cease to glow immediately when the irradiating source isremoved. In the case of Ly α , fluorescently produced Ly α is a product of recombinaton cascade, just as the case ofnebular Ly α emission.Here, we present the general formalism for computing the Ly α surface brightness level. The total flux ‘Flux’ fromsome redshift z that we receive per unit solid angle d Ω equals (see Fig 10 for a visual illustration of the adoptedgeometry) Flux d Ω = d A ( z )Flux dA = d ( z ) dA Luminosity4 πd ( z ) (23)where we used the definition of solid angle d Ω ≡ dA/d ( z ), in which d A ( z ) denotes the angular diameter distanceto redshift z . We also used that Flux = Luminosity / [4 πd ( z )], in which d L ( z ) denotes the luminosity distance toredshift z . We know that d L ( z ) = d A ( z )(1 + z ) . Furthermore, the total luminosity that we receive from dA dependson the length of the cylinder L , as Luminosity= (cid:15) Ly α × dA × L . Here, (cid:15) Ly α denotes the Ly α emission per unit volume.Plugging all of this into Eq (23) we get Flux d Ω = (cid:15) Ly α L π (1 + z ) ≡ S, (24)where we defined S to represent surface brightness. We compute the surface brightness for a number of scenarios next:1. Recombination in the diffuse, low density IGM.
First, we compute the surface brightness of Ly α from thediffuse, low density, IGM, denoted with S IGM . The expansion of the Universe causes photons that are emitted over8
HIgas in CIE at T
Fully ionised Partially ionised
FIG. 11 A schematic representation of the dominant processes that give rise to extended Ly α emission in the CGM/IGM. Left: the surface brightness from fluorescence from recombination in fully ionized gas scales as S fl , I ∝ n e n p λ J , where λ J denotes theJeans length (see Eq 28). This surface brightness can also be written as S fl , I ∝ n e N H where N H denotes the total columnof ionized gas (see Eq 27). Right: sufficiently dense clouds can self-shield and form a neutral core of gas, surrounded by anionized ‘skin’. The thickness of this skin is set by the mean free path of ionizing photons, λ mfp . The surface brightness ofrecombination emission that occurs inside this skin is S fl , II ∝ Γ ion , where Γ ion denotes the photoionization rate (see Eq 31).The neutral core can also produce Ly α radiation following collisional excitation. The surface brightness of this emission is S cool ∝ exp ( − E Ly α /k B T ). a line-of-sight length L to be spread out in frequency by an amount dν α ν α = dvc = H ( z ) Lc ⇒ L = cdν α ν α H ( z ) . (25)When we substitute this into Eq 24 we get S IGM = c(cid:15) Ly α π (1 + z ) H ( z ) dν α ν α ≈ − (cid:18) z (cid:19) . (cid:18) δ (cid:19) (cid:18) dν α /ν α . (cid:19) T − . erg s − cm − arcsec − for z (cid:29) , (26)where we obtained numerical values by substituting that for recombining gas we have (cid:15) Ly α = 0 . h p ν α n e n p α B ( T )(see § n e = n p ∝ (1 + z ) ). We adopted dν ∼ . ν , which represents the width of narrow-bandsurveys, which have been adopted in searches for distant Ly α emitters. For comparison, stacking analysesand deep exposure with e.g. MUSE go down to SB ∼ − erg s − cm − arcsec − (e.g. Rauch et al. 2008,Steidel et a. 2010, Matsuda et al. 2012, Momose et al. 2014, Wisotzksi et al. 2016, Xue et al. 2017).Directly observing recombination radiation from a representative patch of IGM is still well beyond our capa-bilities, but would be fantastic as it would allow us to map out the distribution of baryons throughout the Universe.2. Fluorescence from recombination in fully ionized dense gas (Fluorescence case I).
Eq 26 shows that thesurface brightness increases as (1 + δ ) , and the prospects for detection improve dramatically for overdensities of δ (cid:29)
1. The largest overdensities of intergalactic gas are found in close proximity to galaxies. We therefore expectrecombining gas in close proximity to galaxies to be potentially visible in Ly α . Note however that this denser gasin close proximity to galaxies is not comoving with the Hubble flow, and Eq 25 cannot be applied. Instead, we needto specify the line-of-sight size of the cloud L . Schaye (2001) has shown that the characteristic size of overdense,growing perturbations is the local Jeans length λ J ≡ c s √ Gρ ≈ z ] / − / (1 + δ ) − / T / kpc. If we adopt9that L = λ J , then we obtain S fl , I = (cid:15) Ly α λ J π (1 + z ) ≈ . × − (cid:18) z (cid:19) / (cid:18) δ (cid:19) / T − . erg s − cm − arcsec − . (28)For higher densities the enhanced recombination efficiency of the gas gives rise to an enhanced equilibrium neutralfraction, i.e. x HI ∝ n e . With an enhanced neutral fraction, the gas more rapidly ‘builds up’ a neutral column density N HI > ∼ − cm − , above which the cloud starts self-shielding against the ionizing background. Quantitatively, ifgas is photoionized at a rate Γ ion , then the total column density of neutral hydrogen through the cloud is N HI = λ J ( δ ) × n H α B ( T ) n e Γ ion ≈ × (cid:18) δ (cid:19) / (cid:18) Γ ion − s − (cid:19) − (cid:18) z (cid:19) / T − . cm − , (29)where we used that n HI = x HI n H = α B ( T ) n e Γ ion . The gas becomes self-shielding when N HI > ∼ σ − ∼ cm − , whichtranslates to δ > ∼ ion / − s − ) / ([1 + z ] / − T . (Schaye 2001, also see Rahmati et al. 2013 for a muchmore extended discussion on self-shielding gas). Once gas become denser than this, it can self-shield against anionizing background, and form a neutral core surrounded by an ionized ‘skin’ (see Fig 11).3. Fluorescence from recombination from the skin of dense clouds (Fluorescence case II).
For dense cloudsthat are capable of self-shielding, only the ionized ‘skin’ emits recombination radiation. The total surface brightnessrecombination radiation from this skin depends on its density and thickness, the latter depending directly on theamplitude of the ionizing background. This can be most clearly seen from Eq 24, and replacing L = λ mfp , where λ mfp denotes the mean free path of ionizing photons into the cloud. This mean free path is given by λ mfp = 1 n HI σ ion ⇒ λ mfp = 1 σ ion x HI n H = Γ ion σ ion α B ( T ) n e , (30)where we used that x HI = α B ( T ) n e / Γ ion . Substituting λ mfp for L into Eq 24 gives for the surface brightness of theskin: S fl , II = 0 . h p ν α Γ ion π (1 + z ) σ ion ≈ . × − (cid:18) Γ ion − s − (cid:19) (cid:18) z (cid:19) − erg s − cm − arcsec − , (31)where the T -dependence has cancelled out. A more precise calculation of the surface brightness of fluorescentLy α emission, which takes into account the spectral shape of the ionizing background as well as the frequencydependence of λ mfp , is presented by Cantalupo et al. (2005, note that this calculation introduces only a minorchange to the calculated surface brightness).4. ‘Cooling’ by dense, neutral gas. The neutral core of the cloud - the part of the cloud which is trulyself-shielded - produces Ly α radiation through collisional excitation. As we mentioned earlier, the rate at whichLy α is produced in collisions depends sensitively on temperature ( ∝ exp( − E Ly α /k b T ), see Eq 7). Note however,that this process is a cooling process, and thus must balance some heating mechanism. Once we know the heatingrate of the gas, we can almost immediately compute the Ly α production rate.The direct environment of galaxies, also known as the ‘circum galactic medium’ (CGM), represents a complexmixture of hot and cold gas, of metal poor gas that is being accreted from the intergalactic medium and metal enrichedgas that is driven out of either the central, massive galaxy or from the surrounding lower mass satellite galaxies.Figure 12 shows a snap-shot from a cosmological hydrodynamical simulation (Agertz et al., 2009) which nicelyillustrates this complexity. A disk galaxy (total baryonic mass ∼ × M (cid:12) ) sits in the center of the snap-shot, taken Another way to express S fl , I is by replacing Ln p = N H , where N H denotes the total column density of hydrogen ions (i.e. protons),which yields (see Hennawi et al. 2015) S fl , I = 0 . h p ν α n e α B ( T ) N H π (1 + z ) ≈ . × − (cid:18) z (cid:19) − (cid:16) n e − cm − (cid:17) (cid:18) N H cm − (cid:19) T − . erg s − cm − arcsec − . (27) FIG. 12 A snap-shot from a cosmological hydrodynamical simulation by Agertz et al. (2009) which illustrates the complexityof the circumgalactic gas distribution. A disk galaxy sits in the center of the snap-shot, taken at z = 3. The blue filaments show dense gas that is being accreted. The red gas has been shock heated to the virial temperature of the dark matter halohosting this galaxy ( T vir ∼ K). The green clouds show metal rich gas that was stripped from smaller galaxies. This complexmixture of circumgalactic gas produces Ly α radiation through recombination, cooling, and fluorescence. Credit: from Figure 1of Agertz et al. 2009, Disc formation and the origin of clumpy galaxies at high redshift, MNRAS, 397L, 64A . at z = 3. The blue filaments show dense gas that is being accreted. This gas is capable of self-shielding. The red gashas been shock heated to the virial temperature ( T vir ∼ K) of the dark matter halo hosting this galaxy. The greenclouds show metal rich gas that was driven out of smaller galaxies. This complex mixture of gas produces Ly α via allchannels described above: there exists fully ionized gas that is emitting recombination radiation with a surface bright-ness given by Eq 28, the densest gas is capable of self-shielding and will emit both recombination and cooling radiation.We currently have observations of Ly α emission from the circum-galactic medium at a range of redshifts, coveringa range of surface brightness levels:1. Ly α emission extends further the UV continuum in nearby star forming galaxies. The left panel of Figure 13shows an example of a false-color image of galaxy L yman A lpha R eference S ample ( ¨Ostlin et al.2014, Guaita et al. 2015). In this image, red indicates H α , green traces the far-UV continuum, while blue tracesthe Ly α .2. Stacking analyses have revealed the presence of spatially extended Ly α emission around Lyman Break Galaxies(Hayashino et al., 2004; Steidel et al., 2010) and Ly α emitters at surface brightness levels in the range SB ∼ − − − erg s − cm − arcsec − (Matsuda et al., 2012; Momose et al., 2014).3. Deep imaging with MUSE has now revealed emission at this level around individual star forming galaxies, whichfurther confirms that this emission is present ubiquitously (Wisotzki et al., 2016).4. These previously mentioned faint halos are reminiscent of Ly α ‘blobs’, which are spatially extended Ly α sources not associated with radio galaxies (more on these next, Francis et al. 1996, Steidel et al. 2000, Matsuda et al.2004, Dey et al. 2005, Matsuda et al. 2011, Prescott et al. 2012). A famous example of ”blob right panel of Figure 13 (from Matsuda et al. 2011). This image shows a ‘pseudo-color’ image of a Ly α blob. The red and blue really trace radiation in the red and blue filters, while the green traces the Ly α . The upper right panel shows how large the Andromeda galaxy would look on the sky if placed at z = 3, to put thesize of the blob in perspective. The brightest Ly α blobs have line luminosities of L α ∼ erg s − (e.g. Deyet al. 2005), though recently several monstrous blobs have been discovered, that are much brighter than this,including the ‘Slug’ nebula with a Ly α luminosity of L α ∼ erg s − (Cantalupo et al. 2014, also see Cai etal. 2017 for a similar monster), and the ‘Jackpot’ nebula, which has a luminosity of L α ∼ × erg s − (andcontains a quadruple-quasar system, Hennawi et al. 2015).5. The most luminous Ly α nebulae have traditionally been associated (typically) with High-redshift Radio Galaxies(HzRGs, e.g. McCarthy 1993, Reuland et al. 2003, Van Breugel et al. 2006) with luminosities in excess of L ∼ erg s − .1 FIG. 13
Left:
A false color image of ‘LARS1’ (galaxy α Reference Sample,
Credit: from Figure 1 of Hayes etal. 2013 c (cid:13) AAS. Reproduced with permission. ). Red indicates H α emission, while green traces far-UV continuum. The bluelight traces the Ly α which extends much further than other radiation. Right:
A pseudo color image of Ly α blob 1 (LAB1).Here, red and blue light traces emission from the V and B bands, respectively. The green light traces Ly α emission ( Credit:from Figure 2 of Matsuda et al. 2011, The Subaru Ly α blob survey: a sample of 100-kpc Ly α blobs at z= 3, MNRAS, 410L,13M ). For comparison, the upper right shows the Andromeda galaxy to get a sense for the scale of ‘giant’ Ly α blobs. The origin of extended Ly α emission is generally unclear. To reach surface brightness levels of ∼ − − − ergs − cm − arcsec − we need a density exceeding (see Eq 28) δ > ∼ (cid:0) . × − . × (cid:1) × (cid:18) z (cid:19) − / T − . (32)To keep this gas photoionized requires a large Γ ion :Γ ion > ∼ (5 − × − (cid:18) z (cid:19) . T − . s − , (33)which is ∼ − times larger than the values inferred from observations of the Ly α forest at this redshift. In casethe gas starts to self-shield, then Eq 31 shows that in order to reach S ∼ − − − erg s − cm − arcsec − weneed an almost identical boost in Γ ion ∼ − − − s − . This enhanced intensity of the ionizing radiation fieldis expected in close proximity to ionizing sources (e.g. Mas-Ribas & Dijkstra, 2016): the photoionization rate at adistance r from a source that emits ˙ N ion ionizing photons per second isΓ = ˙ N ion σ ion f ionesc πr = 5 × − (cid:18) f ionesc . (cid:19) ˙ N ion s − ! (cid:18) r
10 kpc (cid:19) − , (34)where f ionesc denotes the fraction of ionizing photons that escapes from the central source into the environment. Theproduction rate of ionizing photons can be linked to the star formation rate via Eq 19. Alternatively, ionizing radiationmay be powered by an accretion disk surrounding a black hole of mass M BH . Assuming Eddington accretion onto theblack hole, and adopting a template spectrum of a radio-quiet quasar, we have˙ N ion = 6 . × (cid:18) M BH M (cid:12) (cid:19) s − (35)which assumes a broken power-law spectrum of the form f ν ∝ ν − . for 1050 ˚A < λ < f ν ∝ ν − . for λ < α halos around each of the 17 brightest radio-quietquasars with MUSE (also see North et al. 2012 for earlier hints of the presence of extended Ly α halos around a highfraction of radio-quiet quasars, and see Haiman & Rees 2001 for an early theoretical prediction).Finally, Ly α cooling radiation gives rise to spatially extended Ly α radiation (Haiman et al. 2000, Fardal et al.2001), and provides a possible explanation for Ly α ‘blobs’ (Dijkstra & Loeb 2009, Goerdt et al. 2010, Faucher-Gigu`ereet al. 2010, Rosdahl & Blaizot 2012, Martin et al. 2015). In these models, the Ly α cooling balances ‘gravitationalheating’ in which gravitational binding energy is converted into thermal energy in the gas. Precisely how gravitationalheating works is poorly understood. Haiman et al. (2000) propose that the gas releases its binding energy in a seriesof ‘weak’ shocks as the gas navigates down the gravitational potential well. These weak shocks convert bindingenergy into thermal energy over a spatially extended region , which is then reradiated primarily as Ly α . We musttherefore accurately know and compute all the heating rates in the ISM (Faucher-Gigu`ere et al. 2010, Cantalupo etal. 2012, Rosdahl & Blaizot 2012) to make a robust prediction for the Ly α cooling rate. These heating rates includefor example photoionization heating, which requires coupled radiation-hydrodynamical simulations (as Rosdahl &Blaizot, 2012), or shock heating by supernova ejecta (e.g. Shull & McKee, 1979).The previous discussion illustrates that it is possible to produce spatially extended Ly α emission from the CGM atlevels consistent with observations, via all mechanisms described in this section . This is one of the main reasons whywe have not solved the question of the origin of spatially extended Ly α halos yet. In later lectures, we will discuss howLy α spectral line profiles (and polarization measurements) contain physical information on the scattering/emittinggas, which can help distinguish between different scenarios.
6. STEP 1 TOWARDS UNDERSTANDING LY α RADIATIVE TRANSFER: LY α SCATTERING CROSS-SECTION
The goal of this section is to present a classical derivation of the Ly α absorption cross-section. This classicalderivation gives us the proper functional form of the real cross-section, but that differs from the real expression bya factor of order unity, due to a quantum mechanical correction. Once we have evaluated the magnitude of thecross-section, it is apparent that most astrophysical sources of Ly α emission are optically thick to this radiation, andthat we must model the proper Ly α radiative transfer.The outline of this section is as follows: we first describe the interaction of a free electron with an electromagneticwave (i.e. radiation) in the classical picture (see § § bound to a proton. The classical picture of this interaction allowsus to derive the Ly α absorption cross-section up to a numerical factor of order unity (see § α cross-section in § Figure 14 shows the classical view of the interaction of a free electron with an incoming electro-magnetic wave. Theelectromagnetic wave consists of an electric field (represented by the red arrows ) and a magnetic field (represented bythe blue arrows ). The amplitude of the electric field at time t varies as E ( t ) = E sin ωt , where ω denotes the angularfrequency of the wave. The electron is accelerated by the electric field by an amount | a e | ( t ) = q | E | ( t ) m e . The total powerradiated by this electron is given by the Larmor formula, i.e. P out ( t ) = q | a e ( t ) | c . The time average of this radiatedpower equals h P out i = q E c m e h sin ωt i = q E c m e . The total power per unit area transported by an electromagnetic wave(i.e. the flux) is F in = cE π . We define the cross-section as the ratio of the total radiated power to the total incident It is possible that a significant fraction of the gravitational binding energy is released very close to the galaxy (e.g. when gas free-fallsdown into the gravitational potential well, until it is shock heated when it ‘hits’ the galaxy: Birnboim & Dekel, 2003). It has beenargued that some compact Ly α emitting sources may be powered by cooling radiation (as in Birnboim & Dekel 2003, Dijkstra 2009,Dayal et al. 2010). After these lectures, Mas-Ribas et al. (2017) showed that extended Ly α emission can also be produced by faint satellite galaxies whichare too faint to be detected individually (also see Lake et al. 2015). electric field d i r e c t i o n m a g n e t i c fi e l d FIG. 14 Classical picture of the interaction of radiation with a free electron. The electric field of the incoming wave acceleratesthe free-electron. The accelerated electron radiates, and effectively scatters the incoming electromagnetic wave. The cross-section for this process is given by the Thomson cross-section. The angle Ψ denotes the angle between the direction of theoutgoing electro-magnetic wave, and the oscillation direction of the electron (which corresponds to the electric vector of theincoming electro-magnetic wave). flux, i.e. σ T = P out F in = q E m e c cE π = 8 π r e ≈ . × − cm , (36)where r e = q m e c = 2 . × − cm denotes the classical electron radius.The power of re-emitted radiation is not distributed isotropically across the sky. A useful way to see this is byconsidering what we see if we observe the oscillating electron along direction k out . The apparent acceleration thatthe electron undergoes is reduced to ˆ a ( t ) ≡ a ( t ) sin Ψ, where Ψ denotes the angle between k out and the oscillationdirection, i.e cos Ψ ≡ k out · e E . Here, the vector e E denotes a unit vector pointing in the direction of the E-field. Thereduced apparent acceleration translates to a reduced power in this direction, i.e P out ( k out ) ∝ ˆ a ( t ) ∝ sin Ψ.The outgoing radiation field therefore has a strong directional dependence with respect to e E . Note however, thatfor radiation coming in along some direction k in , the electric vector can point in an arbitrary direction within the planenormal to k in . For unpolarized incoming radiation, e E is distributed uniformly throughout this plane. To computethe angular dependence of the outgoing radiation field with respect to k in we need to integrate over e E . That is P out ( k out | k in ) ∝ Z P out ( k out | e E ) P ( e E | k in ) d e E . (37)We can solve this integral by switching to a coordinate system in which the x − axis lies along k in , and in which k out lies in the x − y plane. In this coordinate system e E must lie in the y − z plane. We introduce the angles φ and θ andwrite k in = (1 , , , k out = (cos θ, , sin θ ) , e E = (0 , cos φ, sin φ ) . (38) We have adopted the notation of probability theory. In this notation, the function p ( y | b ) denotes the conditional probability densityfunction (PDF) of y given b . The PDF for y is then given by p ( y ) = R p ( y | b ) 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 ( y | b ) p ( b ). y xz k out k in e E FIG. 15 The geometry adopted for calculating the phase-function associated with Thomson scattering.
We can then also get that cos Ψ ≡ k out · e E = sin φ sin θ . This coordinate system is shown in Figure 15, which showsthat θ denotes the angle between k in and k out . For this choice of coordinates Eq 37 becomes P out ( k out | k in ) ∝ Z π dφ sin Ψ (39)where we have used that P out ( k out | e E ) = P out sin Ψ, and that P ( e E | k in ) = Constant (i.e. e E is distributed uniformlyin the y − z plane). We have omitted all numerical constants, because we are interested purely in the angulardependence of P out . We will determine the precise constants that should preceed the integral below. If we substitutesin Ψ = 1 − cos Ψ = 1 − sin φ sin θ ≡ − A sin φ ( A ≡ sin θ ), we get P out ( k out | k in ) ∝ Z π dφ [1 − A sin φ ] ∝ Z π dφ [1 + A − A − A sin φ ] ∝ ... ∝ (1 + cos θ ) . (40)Figure 15 shows that our problem of interest is cylindrically symmetric around the x -axis, we therefore have that P out ( k out | k in ) only depends on θ , and we will write P out ( k out | k in ) = P out ( θ ) for simplicity.The angular dependence of the re-emitted radiation is quantified by the so-called Phase-function (or the angularredistribution function ), which is denoted with P ( θ ). P out ( θ ) F in ≡ σ T P ( θ )4 π , with Z − dµP ( µ ) = 2 (cid:18) i . e . Z d Ω P ( µ ) = 4 π (cid:19) , (41)where µ ≡ cos θ . Note that the phase-function relates to the differential cross-section simply as dσd Ω ≡ σP ( θ )4 π .There are two important examples of the phase-function we encounter for Ly α transfer. The first is the one wederived above, and describes ‘dipole’ or ‘Rayleigh’ scattering: P ( µ ) = 34 (1 + µ ) dipole . (42)The other is for isotropic scattering: P ( µ ) = 1 isotropic . (43)As we will see further on, the phase function associated with Ly α scattering is either described by pure dipolescattering, or by a superposition of dipole and isotropic scattering.5 We can understand the expression for the Ly α absorption cross-section via an analysis similar to the one describedabove. The main difference with the previous analysis is that the electron is not free, but instead orbits the protonat a natural (angular) frequency ω . We will treat the electron as a harmonic oscillator with natural frequency ω .In the classical picture, the electron radiates as it accelerates and spirals inward. To account for this we will assumethat the harmonic oscillator is damped. This damped harmonic oscillator with natural frequency ω is ‘forced’ bythe incoming radiation field that again has angular frequency ω . The equation of motion for this forced, dampedharmonic oscillator is ¨ x + Γ ˙ x + ω x = qm E ( x, t ) = qm E exp( iωt ) , (44)where x can represent one of the Cartesian coordinates that describe the location of the electron in its orbit. Theterm Γ ˙ x denotes the friction (or damping) term, which reflects that in the classical picture of a hydrogen atom, theelectron spirals inwards over a short timescale (see § F = qE ) that is exerted by the electric field. We have represented the electro-magnetic wave as E ( t ) = E exp( iωt )(which is a more general way of describing a wave than what we used when considering the free electron. It turns outthat this simplifies the analysis). We can find solutions to Eq 44 by substituting x ( t ) = x exp( iωt ). We discuss twosolutions here: • In the absence of the electromagnetic field, this yields a quadractic equation for ω , namely − ω + i Γ ω + ω = 0.This equation has solutions of the form ω = i Γ2 ± p ω − Γ /
4. We assume that ω (cid:29) Γ (which is the case forLy α as we see below), which can be interpreted as meaning that the electron makes multiple orbits around thenucleus before there is a ‘noticeable’ change in its position due to radiative energy losses. The solution for x ( t )thus looks like x ( t ) ∝ exp( − Γ t/
2) cos ω t . The solution indicates that the electron keeps orbiting the protonwith the same natural frequency ω , but that it spirals inwards on a characteristic timescale Γ − (also see § • In the presence of an electromagnetic field, substituting x ( t ) = x exp( iωt ) yields the following solution for theamplitude x : x = qE m e ω − ω + iω Γ ω ≈ ω ∼ qE m e ω ω − ω + i Γ / , (45)where we used that ( ω − ω ) = ( ω − ω )( ω + ω ) ≈ ω ( ω − ω ). This last approximation assumes that ω ≈ ω .It is highly relevant for Ly α scattering, where ω and ω are almost always very close together (meaning that | ω − ω | /ω (cid:28) − ).With the solution for x ( t ) in place, we can apply the Larmor formula and compute the time averaged power radiatedby the accelerated electron: h P out i = 2 q h| ¨ x |i c = 2 q ω c q E × m e ω ω − ω ) + Γ / . (46)Where the highlighted factor of 2 comes in from the time-average: h E i = E /
2. As before, the time average of thetotal incoming flux h F in i (in erg s − cm − ) of electromagnetic radiation equals h F in i = cE π . We therefore obtain anexpression for the cross-section as: σ ( ω ) = h P out ih F in i = 8 π h P out i cE = 16 πq ω m e ω c ω − ω ) + Γ / σ T ω ω ( ω − ω ) + Γ / , (47)where we substituted the expression for the Thomson cross section σ T = πq m e c (see Eq 36). We can justify this picture as follows: we define the x − y plane to be the plane in which the electron orbits the proton. The x − coordinateof the electron varies as x ( t ) = x cos ω t . The x-component of the electro-static force on the electron varies as F x = F e xr , in which F e = q r . That is, the equation of motion for the x-coordinate of the electron equals ¨ x = − kx , where k = q /r . FIG. 16 The Lorentzian profile σ CL ( ω ) for the Ly α absorption cross-section. This represents the absorption cross-section ofa single atom . When averaging over a collection of atoms with a Maxwellian velocity distribution, we obtain the Voigt profile(shown in Fig 17). The expression for σ ( ω ) can be recast in a more familiar form when we use the Larmor formula to constrain Γ. Theequation of motion shows that ‘friction/damping force’ on the electron is F = − m e Γ ˙ x . We can also write this forceas F = dp e dt = m e p e ddt p e m e = m e p e dE kin dt = − m e p e P out , where P out denotes the emitted power. Setting the two equal givesus a relation between Γ and P out : − Γ m e ˙ x = P out . Using that ˙ x = iω x , ¨ x = − ω x , and the Larmor formula givesus Γ = q ω m e c = 15 × s − . With this the ’classical’ expression for the Ly α cross section can be recast as σ CL ( ω ) = 3 λ π Γ ( ω/ω ) ( ω − ω ) + Γ / ⇒ σ CL ( ν ) = 3 λ π (Γ / π ) ( ν/ν ) ( ν − ν ) + Γ / π , (48)where the subscript ‘CL’ stresses that we obtained this expression with purely classical physics. If we ignore the( ω/ω ) term, then the functional form for σ ( ω ) is that of a Lorentzian Profile . In the second equation we re-expressedthe cross-section as a function of frequency ν = ω/ (2 π ).Figure 16 shows σ CL ( ω ) for a narrow range in ω . There are two things to note:1. The function is sharply peaked on ω , at which σ ≡ σ CL ( ω ) = λ π ∼ × − cm − , where λ = 2 πc/ω corresponds to the wavelength of the electromagnetic wave with frequency ω . The cross-section falls off by > ∼ | ω − ω | /ω ∼ − . Note that the cross-section is many, many orders of magnitudelarger than the Thomson cross-section. This enhanced cross-section represents a ‘resonance’, an ‘ unusuallystrong response of a system to an external trigger ’.2. The σ ∝ ω -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 otherelectromagnetic 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 , and the setting/risingsun red. α Cross-Section
The derivation from the previous section was based purely on classical physics. A full quantum-mechanical treatmentof the Ly α absorption cross-section is beyond the scope of these lectures, and - interestingly - still subject of ongoingresearch (e.g. Lee 2003, Bach & Lee 2014, 2015, Alipour et al. 2015). A clear recent discussion on this can be foundin Mortlock (2016). There are two main things that change: This is not always the case in the Netherlands or Norway. FIG. 17 In the left panel we compare the different atom frame Ly α cross sections. The black solid line shows the Lorentziancross-section. The red dashed line shows the cross-section given in Peebles (1993), and the blue dotted line shows the quantummechanical cross-section from Mortock & Hirata (in prep, also see Mortlock 2016). In the right panel the solid line shows thegas frame (velocity averaged absorption) cross section as given by Eq 55 as a function of the dimensionless frequency x (seetext). The red dashed line ( green dot-dashed line ) represent the cross section where we approximated the Voigt function asexp( − x ) ( a v / [ √ πx ]). Clearly, these approximations work very well in their appropriate regimes. Also shown for comparisonas the blue dotted line is the symmetric single atom Lorentzian cross section (which was shown in the left panel as the blacksolid line ). Close to resonance, this single atom cross section provides a poor description of the real cross section, but it doesvery well in the wings of the line (see text).
1. The parameter Γ reduces by a factor of f α = 0 . → f α Γ ≡ A α . Here, A α denotes the Einstein A-coefficient for the Ly α transition.2. In detail, a simple functional form (Lorentzian, Eq 48, or see Eq 49 below) for the Ly α cross-section does notexist. The main reason for this is that if we want to evaluate the Ly α cross-section far from resonance, we have totake into account the contributions to the cross-section from the higher-order Lyman-series transitions, and evenphotoionization. When these contributions are included, the expression for the cross-section involves squaringthe sum of all these contributions (e.g. Bach & Lee, 2014, 2015; Mortlock, 2016). Accurate approximations tothis expression are possible close to the resonance(s) - when | ω − ω | /ω (cid:28)
1, which is the generally the casefor practical purposes - and these approximations are in excellent agreement with our derived cross-section (seeEq 14 in Mortlock, 2016, and use that ω ≈ ω , and that Λ = Γ / π ).The left panel of Figure 17 compares the different Ly α cross sections. The black solid line shows the Lorentziancross-section. The red dashed line shows σ P ( ω ), which is the cross section that is given in Peebles (1993): σ ( ω ) = 3 λ α π A α ( ω/ω α ) ( ω − ω α ) + A α ( ω/ω α ) / . (49)If we ignore the ( ω/ω α ) term in the denominator, then this corresponds exactly to the cross-section that we derivedin our classical analysis (see Eq 48, with Γ → A α ). This cross-section is obtained from a quantum mechanicalcalculation, and under the assumption that the hydrogen atom has only two quantum levels (the 1s and 2p levels).The blue dotted line shows the cross-section obtained from the full quantum mechanical calculation (see Mortlock2016 for a discussion, which is based on Mortlock & Hirata in prep). This Figure shows that these cross-sections differonly in the far wings of the line profile. Close to resonance, the line profiles are practically indistinguishable. α Cross-Section
In the previous section we discussed the Ly α absorption cross section in the frame of a single atom. Because eachatom has its own velocity, a photon of a fixed frequency ν will appear Doppler boosted to a slightly different frequencyfor each atom in the gas. To compute the Ly α absorption cross section for a collection of moving atoms, we mustconvolve the single-atom cross section with the atom’s velocity distribution. That is,8 σ α ( ν, T ) = Z d v σ α ( ν | v ) f ( v ) , (50)where σ α ( ν | v ) f denotes the Ly α absorption cross-section at frequency ν for an atom moving at 3D velocity v (theprecise velocity v changes the frequeny in the frame of the atom). Suppose that the photon is propagating in adirection n . We decompose the atoms three-dimensional velocity vectors into directions parallel ( v || ) and orthogonal( v ⊥ ) to n . These components are independent and f ( v ) d v = f ( v || ) g ( v ⊥ ) dv || d v ⊥ . The absorption cross-sectiondoes not depend on v ⊥ because the frequency of the photon that the atoms ’sees’ does not depend on v ⊥ . We cantherefore write σ α ( ν, T ) = Z d v ⊥ g ( v ⊥ ) | {z } =1 Z ∞−∞ dv || σ α ( ν | v || ) f ( v || ) . (51)For a Maxwell-Boltzmann distribution f ( v || ) dv || = (cid:16) m p πk B T (cid:17) / exp (cid:16) − m p v || kT (cid:17) . If we insert this into Eq 51 andsubstitute Eq 48 for σ α ( ν | v || ) = σ ( ν ), where ν = ν (cid:16) − v || c (cid:17) we find σ α ( ν, T ) = 3( λ A α ) π (cid:16) m p πk B T (cid:17) / Z ∞−∞ dv || exp (cid:16) − m p v || kT (cid:17)h ν (cid:16) − v || c (cid:17) − ν α i + A α π , (52)where we dropped the term ( ν/ν ) , which is accurate for ν ≈ ν (which is practically always the case). We substitute y = r m p v || k B T ≡ v || /v th , and define the Voigt parameter a v ≡ A α π ∆ ν α = 4 . × − ( T / K) − / in which ∆ ν α = ν α v th /c .We can then recast this expression as σ α ( ν, T ) = 3( λ A α ) π (cid:16) m p πk B T (cid:17) / v th ∆ ν α Z ∞−∞ dy exp( − y ) h ν ∆ ν α (cid:16) − yv th c (cid:17) − ν α ∆ ν α i + a v . (53)We finally introduce the dimensionless frequency variable x ≡ ( ν − ν α ) / ∆ ν α , which we use to express ν = ν α (1+ xv th /c ).If we substitute this back into Eq 53 and drop the second order term with ( v th /c ) , then we get σ α ( ν, T ) = 3 λ A α π √ π ∆ ν α Z ∞−∞ dy exp ( − y )( x − y ) + a v ≡ λ a v √ π H ( a v , x ) ≡ σ α, ( T ) φ ( x ) = (54)= 5 . × − (cid:16) T K (cid:17) − / φ ( x ) cm and the Voigt function as (e.g. Chluba & Sunyaev, 2009) φ ( x ) ≡ H ( a v , x ) = a v π Z ∞−∞ e − y dy ( y − x ) + a v = ( ∼ e − x [exp( a v )erfc( a v )] ∼ e − x core; ∼ a v √ πx wing . (55)Note that throughout this review we will use both φ ( x ) and H ( a v , x ) to denote the shape of the Ly α line profile .The transition between core and wing occurs approximately when exp( − x ) = a v / ( √ πx ).The solid line in the right panel of Figure 17 shows the LAB frame cross section - this is also known as theVoigt-profile - as given by Eq 55 as a function of the dimensionless frequency x . The red dashed line ( green dot-dashedline ) represent the cross section where we approximated the Voigt function as exp( − x ) ( a v / [ √ πx ]). Clearly, theseapproximations work very well in the relevant regimes. Note that this Figure shows that a decent approximation to One has to be a bit careful because in the literature occasionally φ ( x ) = H ( a v , x ) / √ π , because in this convention the line profile isnormalized to 1, i.e. R φ ( x ) dx = 1. In our convention φ ( x = 0) = 1, while the normalization is R φ ( x ) dx = √ π . φ ( x ) ≈ exp( − x ) + a v / ( √ πx )(Krolik, 1989, provided that | x | > ∼√ a v ). This approximation fails in a very narrow frequency regime where thetransition from core to wing occurs. A useful fitting function that is accurate at all x is given in Tasitsiomi (2006).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 17 shows that close to resonance, this single atom cross section provides a poordescription of the real cross section. This is because Doppler motions ‘smear out’ the sharply peaked cross-section.Far in the wing however, the single atom cross-section provides an excellent fit to the velocity averaged Voigt profile.One of the key results from this section is that the Ly α cross-section, evaluated at line center and averaged overthe velocity distribution of atoms, is tremendous at σ α, ( T ) ∼ . × − ( T / K) − / cm − , which is ∼
11 ordersof magnitude than the Thomson cross-section. That is, an electron bound to the proton is ∼
11 orders of magnitudemore efficient at scattering radiation than a free electron when the frequency of that radiation closely matches thenatural frequency of the transition. This further emphasises that the electron ‘resonantly scatters’ the incomingradiation. To put these numbers in context, it is possible to measure the hydrogen column density, N HI , in nearbygalaxies. The observed intensity in the 21-cm line translates to typical HI column densities of order N HI ∼ − cm − (Bosma 1978, Kaberla et al. 2005, Chung et al. 2009), which translates to line center optical depths of Ly α photons of order τ ∼ − . This estimate highlights the importance of understanding the transport of Ly α photon out of galaxies. They generally are not expected to escape without interacting with hydrogen gas.
7. STEP 2 TOWARDS UNDERSTANDING LY α RADIATIVE TRANSFER: THE RADIATIVE TRANSFER EQUATION
The specific intensity I ν ( r , n , t ) is defined as the rate at which energy crosses a unit area, per solid angle, perunit time, as carried by photons of energy h p ν in the direction n , i.e. I ν ( r , n , t ) = d E ν d Ω dtdA . The change in thespectral/specific intensity of radiation at a location r that is propagating in direction n at time t is (e.g. Meiksin,2009) n · ∇ I ν ( r , n , t ) + 1 c ∂I ( r , n , t ) ∂t = − α ν ( r ) I ( r , t ) + j ν ( r , n , t ) . (56)Here, α ν ( r ) I ( r , t ) in the first term on the RHS denotes the ‘attenuation coefficient’, which accounts for energy loss(gain) due to absorption (stimulated emission). In static media, we generally have that α ν ( r ) is isotropic , which iswhy we dropped its directional dependence. The emission coefficient j ν ( r , n , t ) describes the local specific luminosityper solid angle, per unit volume. The random orientation of atoms/molecules (again) generally causes emissioncoefficient to be isotropic.For Ly α radiation, photons are not permanently removed following absorption, but they are generally scattered.To account for this scattering, we must add a third term to the RHS on Eq 56: n · ∇ I ν ( r , n , t ) + 1 c ∂I ( r , n , t ) ∂t = − α ν ( r ) I ( r , t ) + j ν ( r , n , t ) + Z dν Z d n α ν ( r ) I ν ( r , n , t ) R ( ν, ν , n , n ) . (57)The third term accounts for energy redistribution as a result of scattering. In this term, the so-called ‘ redistributionfunction ’ R ( ν, ν , n , n ) describes the probability that radiation that was originally propagating at frequency ν andin direction n is scattered into frequency ν and direction n . We focus on equilibrium solutions and omit the timedependence in Eq 57. Furthermore, we introduce the coordinate s , which measures distance along the direction n .The change of the intensity of radiation that is propagating in direction n with distance s is then dI ν ( s, n ) ds = − [ α HI ν ( s ) | {z } I: absorption + α dest ν ( s ) | {z } IV: ‘destruction ] I ν ( s, n ) + j ν ( s ) | {z } II: emission + Z dν Z d ˆ n α ν ( s ) I ν ( s, ˆ n ) R ( ν, ν , n , ˆ n ) | {z } III: scattering . (58)In the following subsections we will discuss each term I-IV on the right hand side in more detail. For radiation at some fixed frequency ν close to the Ly α resonance, the opacity α ν ( r , n ) depends on n for non-static media. Thisdirectional dependence is taken into account when performing Monte-Carlo Ly α radiative transfer calculations (to be described in § α Cross Section
The opacity α HI ν ( s ) = (cid:2) n l ( s ) − g l g u n u ( s ) (cid:3) σ ( ν ). The second term within the square brackets corrects the absorptionterm for stimulated emission. In most astrophysical conditions all neutral hydrogen atoms are in their electronicground state, and we can safely ignore the stimulated emission term (see e.g. the Appendix of Dijkstra & Loeb 2008a,and Dijkstra et al. 2016). That is, in practice we can simply state that α ν ( s ) = n l ( s ) σ α ( ν ) = n HI ( s ) σ α ( ν ). We havederived expressions for the Ly α absorption cross-section in § The emission term is given by j ν ( s ) = φ ( ν ) hν α π / ∆ ν α (cid:16) n e n HI q s p + n e n HII α ( T ) f Ly α ( T ) (cid:17) , (59)where φ ( ν ) is the Voigt profile (see § π accounts for the fact that Ly α photons are emitted isotropicallyinto 4 π steradians, and the factor of √ π ∆ ν α arises when converting φ ( x ) to φ ( ν ). The first term within the bracketsdenotes the Ly α production rate as a result of collisional excitation of H atoms by (thermal) electrons, and is givenby Eq 7, in which the collision strength Ω s p ( T ) can be read off from Figure 7. The second term within the bracketsdenotes the Ly α production rate following recombination, in which both the recombination coefficient and the Ly α production probability depend on both temperature and the opacity of the medium to ionizing and Lyman seriesphotons (see Fig 6, and § The ’redistribution function’ R ( ν, ν , n , n ) describes the probability that radiation that was originally propagatingat frequency ν and in direction n is scattered into frequency ν and direction n (see e.g. Henyey 1940, Zanstra 1949,Unno 1952, Hummer 1962 for early discussions). In practise, this probability depends only on the angle between n and n , i.e. R ( ν, ν , n , n ) = R ( x out , x in , µ ), in which µ ≡ cos θ = n · n . We have also switched to standarddimensionless frequency coordinates (first introduced in Eq 53), and denote with x out ( x in ) the dimensionlessfrequency of the photon after (before) scattering. Formally, R ( x out , x in , µ ) dx out dµ denotes the probability that aphoton of frequency x in was scattered by an angle in the range µ ± dµ/ x out ± dx out / R ( x out , x in , µ ) is normalized such that R − dµ R ∞−∞ dx out R ( x in , x out , µ ) = 1.In the remainder of this section, we will compute the redistribution functions R ( x in , x out , µ ). Following Lee (1974)we will employ the notation of probability theory. In this notation, the function p ( y | b ) denotes the conditionalprobability density function (PDF) of y given b . The PDF for y is then given by p ( y ) = R p ( y | b ) 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 ( y | b ) p ( b ). We could just as wellhave written that p ( y, b ) = p ( b | y ) p ( y ), and by setting p ( y | b ) p ( b ) = p ( b | y ) p ( y ) we get ‘Bayes theorem’ which states that p ( y | b ) = p ( b | y ) p ( y ) p ( b ) . (60)We will use this theorem on several occasions below. We can write the conditional joint PDF for x out and µ given x in as R ( x out , µ | x in ) = R ( x out | µ, x in ) P ( µ | x xin ) . (61)An additional quantity that is of interest in many radiative transfer problems (see for example § x out given x in : R ( x out | x in ) = Z − dµR ( x out | µ, x in ) P ( µ | x xin ) . (62) Recall that φ ( x ) was normalized to R φ ( x ) dx = √ π (Eq 55). Substituting dx = dν/ ∆ ν α (see discussion below Eq 53) gives R φ ( ν ) dν = √ π ∆ ν α . In the lectures I illustrated this with an example in which y denotes my happiness, and in which b denotes the number of snowballs thatwere thrown in my face in the previous 30 minutes. Clearly, p ( y ) will be different when b = 0 or when b (cid:29) E out E in H E out E in Hatom frame gas frame
FIG. 18 In most astrophysical media, Ly α scattering is partially coherent : energy conservation implies that the energy of thephoton before and after scattering is the same in the frame of the atom (the ‘atom frame’ shown on the left ). In the gas framehowever (shown on the right ) the energy before and after scattering is different. This is because the random thermal motionof the atom induces Doppler shifts to the energy of the photon before and after scattering by an amount that depends on theatoms velocity, and on the scattering angle. Each scattering event induces small changes to the frequency of the Ly α photon.In this case, the atom is moving into the incoming Ly α photon, which causes it to appear at a higher frequency. Scatteringby 90 ◦ does not induce any additional Doppler boost in this example, and the outgoing photon has a higher frequency (whichis why we drew it blue, to indicate its newly acquired blue-shift). The picture on the left also illustrates the recoil effect: inthe atom frame, momentum conservation requires that the hydrogen atom moves after the scattering event. This newly gainedkinetic energy of the H-atom comes at the expense of the energy of the Ly α photon. This recoil effect is generally small. To complete our calculation we have to evaluate the PDFs R ( x out | µ, x in ) and P ( µ | x xin ). We will do this next.In most astrophysical conditions, the energy of the Ly α photon before and after scattering is identical in theframe of the absorbing atom. This is because the life-time of the atom in its 2 p state is only t = 1 /A α ∼ − s. Inmost astrophysical conditions, the hydrogen atom in this state is not ‘perturbed’ over this short time-interval, andenergy conservation forces the energy of the photon to be identical before and after scattering. Because of randomthermal motions of the atom, energy conservation in the atom’s frame translates to a change in the energy of theincoming and outgoing photon that depends on the velocity of the atom and the scattering direction (see Fig 18 foran illustration). This type of scattering is known as ‘partially coherent’ scattering .For notational clarity, we denote the propagation direction and dimensionless frequency of the photon before (after)scattering with k in and x in ( k out and x out ). We assume that the scattering event occurs off an atom with velocityvector v . Doppler boosting between the atom and gas frame corresponds to x atomin = x gasin − v · k in v th , gas → atom; x gasin = x atomin + v · k in v th , atom → gas . (63)Notice the signs in these equations. When v and k in point in the same direction, the atom is moving away from theincoming photon, which reduces the frequency of the photon in the atom’s frame. The expressions are the same forthe outgoing photon. Partially coherent scattering dictates that x atomin = x atomout , which allows us to write down therelation between x gasout and x gasin as x gasout = x gasin − v · k in v th + v · k out v th . (64) It is possible to repeat the analysis of this section under the assumption that ( i ) the energy of the photon before and after scattering isidentical, which is relevant when the gas has zero temperature. This corresponds to ‘completely coherent’ scattering ( ii ) the energy ofthe re-emitted photon is completely unrelated to the atom of the incoming photon. This can happen in very dense gas where collisionsperturb the atom while in the 2 p state. This case corresponds to ‘completely incoherent’ scattering. z xy k out k in H FIG. 19 Schematic depiction of the coordinate system that we used to describe the scattering event. The scattering plane isspanned by the wavevectors of the photon before ( k in = [1 , , k out = [ µ, p − µ , x − y plane. If we decompose the atom’s velocity vector ( the blue dashed vector )into components parallel ( v || , red dotted vector ) and orthogonal ( v ⊥ , red dotted vector ) to the propagation direction of theincoming photon, then we can derive convenient expressions for the conditional probabilities R ( x out | x in , µ ) (see Fig 20) and R ( x out | x in ) (see Fig 21). We always work in the gas frame and will drop the ‘gas’ superscript from now on. Eq 64 still misses the effect of atomic recoil , which is illustrated in the drawing on the left of Figure 18: in the atom frame, the momentum of thephoton prior to scattering is h p ν in /c and points to the right. The momentum of the photon after scattering equals h p ν out /c and points up. Momentum would therefore not be conserved in the atom’s frame, if it were not for the newlyacquired momentum of the hydrogen atom itself. This newly acquired momentum corresponds to newly acquiredkinetic energy, which in turn came at the expense of the energy of the Ly α photon. The energy of the Ly α photonis therefore (strictly) not conserved exactly, but reduced by a small amount in each scattering event. This effect isnot relevant when k in = k out (i.e. µ = 1), and maximally important when k in = − k out (i.e. µ = − x out = x in − v · k in v th + v · k out v th + g ( µ −
1) + O ( v /c ) | {z } recoil , (65)where g = hν α m p v th c = 2 . × − ( T / K) − / is the fractional amount of energy that is transferred per scatteringevent (Field, 1959). Throughout the remainder of this calculation we will ignore recoil. This may be counter-intuitive,as the change in x is of the order ∼ − for each scattering event , and - as we will see later - Ly α photons canscatter (cid:29) times. However, (fortunately) this process does not act cumulatively as we will discuss in more detailat the end of this section. Adams (1971) first showed that recoil can generally be safely ignored.For simplicity, but without loss of generality, we define a coordinate system such that k in = (1 , , k out =( µ, p − µ , v || ) and orthogonal ( v y and v z ) to k in , then v = ( v || , v y , v z ) and x out = x in − v || v th + v || µv th + v y p − µ v th ≡ x in − u + uµ + w p − µ , (66) We will not derive this here. The derivation is short. First show that the total momentum of the atom after scattering equals p Hout = h p ν in c √ − µ , where µ = k in · k out . This corresponds to a total kinetic energy E e = [ p Hout ] m p , which must come at the expenseof the Ly α photon. We therefore have ∆ E = h P ∆ ν = m p (cid:16) h p ν in c (cid:17) (1 − µ ), which corresponds to ∆ x = hν α m p v th c if we approximate that ν in ≈ ν α . u = v || /v th and w = v y /v th (see e.g. Ahn et al.,2000). Note that the value of v z is irrelevant in this equation, which is because it does not induce any Doppler booston either the incoming or outgoing photon.We were interested in calculating R ( x out | µ, x in ). We can do this by using Eq 66 and applying probability theory.We first write R ( x out | µ, x in ) = N Z ∞−∞ du Z ∞−∞ dwR ( x out | µ, x in , u, w ) P ( u | µ, x in ) P ( w | µ, x in ) , (67)where N is a normalization factor. Eq 66 states that when x out , x in , µ are fixed, then for a given u , solutions onlyexist when w ≡ w u = x out − x in + u − uµ √ − µ . In other words, R ( x out | µ, x in , u, w ) is only non-zero when w = w u . We cantherefore drop the integral over w and write R ( x out | µ, x in ) = N Z ∞−∞ duP ( u | µ, x in ) P ( w u | µ, x in ) . (68)The conditional absorption probabilities for both w and u cannot depend on the subsequent emission direction, asthe re-emission process is set by quantum mechanics of the wavefunction describing the electron. Therefore we have P ( u | µ, x in ) = P ( u | x in ) and P ( w u | µ, x in ) = P ( w u | x in ). Also note that w denotes the normalized velocity in a directionperpendicular to k in . The incoming photon’s frequency, x in - and therefore the absorption probability - cannot dependon w . Therefore, P ( w u | x in ) = P ( w u ) = exp( − w u ) / √ π , where we assumed a Maxwell-Boltzmann distribution of theatoms’ velocities.The expression for P ( u | x in ) is a bit more complicated. From Bayes Theorem (Eq 60) we know that P ( u | x in ) = P ( x in | u ) P ( u ) /P ( x in ), in which P ( x in | u ) denotes the absorption probability for a single atom that has a speed u , and P ( x in ) can be interpreted as a normalization factor. The scattering probability of Ly α photons off atoms with velocitycomponent u must scale with the (single-atom) cross-section, i.e. P ( x in | u ) ∝ σ α ( x in | u ) = λ α π A α [ ω α ( x in − u ) v th /c ] + A α / (see Eq 49). If we substitute this into Eq 68 and absorb all factors that can be pulled out of the integral into thenormalization constant N , then we get R ( x out | µ, x in ) = N Z ∞−∞ du exp( − u )( x in − u ) + a v exp h − (cid:16) ∆ x − u ( µ − p − µ (cid:17) i , (69)where we introduced ∆ x ≡ x out − x in . It is possible to get an analytic expression for N : R ( x out | µ, x in ) = a v π / p − µ φ ( x in ) Z ∞−∞ du exp( − u )( x in − u ) + a v exp h − (cid:16) ∆ x − u ( µ − p − µ (cid:17) i . (71)Examples of R ( x out | µ, x in ) as a function of x out are plotted in Figure 20. In the left panel we plot R ( x out | µ, x in ) for µ = − .
5, and x in = − . red line ), x in = 0 . black line ), and x in = 4 . blue line ). These frequencies were chosento represent scattering in the wing (for x in = 4 . x in = 0 .
0) and in the transition region ( x in = − . right panel we used µ = 0 .
5. Figure 20 shows clearly that ( i ) for the vast majority of photons | x in − x out | < ∼ a few. That is, the frequency after scattering is closely related to the frequency before scattering ( ii ) R ( x out | µ, x in ) can depend quite strongly on µ . This is most clearly seen by comparing the curves for x in = − . left and right panels . 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): N − ( µ ) = Z ∞−∞ dx out Z ∞−∞ du exp( − u )( x in − u ) + a v exp h − (cid:16) ∆ x − u ( µ − p − µ (cid:17) i = (70) Z ∞−∞ du exp( − u )( x in − u ) + a v Z ∞−∞ d ∆ x exp h − (cid:16) ∆ x − u ( µ − p − µ (cid:17) i = φ ( x in ) πa v p − µ √ π, where we used that d ∆ x = dx out . We rewrote the term in red using the definition of φ ( x ) (see Eq 55), and that the term in blue is aGaussian in ∆ x . FIG. 20 R ( x out | µ, x in ) is shown as a function of x out . In the left panel we plot curves with µ = − .
5, and x in = − . red line ), x in = 0 . black line ), and x in = 4 . blue line ). In the right panel we changed the sign of µ to µ = 0 .
5. This Figure showsthat for the vast majority of photons | x in − x out | < ∼ a few. Thus the photon frequencies before and after scattering are closelyrelated. Furthermore, R ( x out | µ, x in ) can depend quite strongly on µ . This is most clearly seen by comparing the curves for x in = − . left and right panels . In many radiative transfer problem we are mostly interested in the conditional PDF R ( x out | x in ) (see 62) for whichone needs to know the ‘conditional phase function’ P ( µ | x in ). We introduced the concept of the phase function in § i ) isotropic scattering, for which P ( µ ) = 1, and dipole scattering for which P ( µ ) = (1+ µ ). Aswe will discuss in more detail in § α scattering represents a superposition of dipole and isotropic scattering. Aswe will see, there is a weak dependence of the phase function P ( µ ) to x in , but only over a limited range of frequencies(see § P ( µ | x in ) = P ( µ ). The conditional PDF R ( x out | x in )for isotropic and dipole scattering are given by R A ( x out | x in ) = Z − dµR ( x out | µ, x in ) (72) R B ( x out | x in ) = 34 Z − dµR ( x out | µ, x in ) (cid:16) µ (cid:17) , where subscript ‘ A ’ (‘ B ’) refers to isotropic (dipole) scattering. Figure 21 shows type-A and type-B frequencyredistribution functions for x in = − . , . .
0. Because Ly α scattering is often a superposition of these two,the actual directionally averaged redistribution functions are intermediate between the two cases. As these two casesagree quite closely, they both provide a decent description of actual Ly α scattering.There are three important properties of these redistribution functions that play an essential role in radiative transfercalculations. We discuss these next.1. Photons that scatter in the wing of the line are pushed back to the line core by an amount − x in (Osterbrock,1962), i.e. h ∆ x | x in i = − x in . (73)Demonstrating this requires some calculation. The expectation value for ∆ x per scattering event is given by h ∆ x | x in i ≡ Z ∞−∞ ∆ x R ( x out | x in ) dx out = 12 Z ∞−∞ dx out Z − dµ ∆ xP ( µ ) R ( x out | µ, x in ) , (74)where the factor of reflects that R − P ( µ ) dµ = 2 (see Eq 41). For simplicity we will assume isotropic scattering,for which P ( µ ) = 1. We previously presented an expression for R ( x out | µ, x in ) (see Eq 71). Substituting this5 FIG. 21 The conditonal PDF R ( x out | x in ), obtained by marginalizing R ( x out | x in , µ ) (shown in Fig 20) over µ , is shown forisotropic scattering ( red solid lines , R A ( x out | x in )) and dipole scattering ( black dotted lines , R B ( x out | x in )) for T = 10 K. Forthis gas temperature, R ( x out | x in ) is very similar for isotropic and dipole scattering. expression into the above equation yields h ∆ x | x in i = a v π / φ ( x in ) Z − dµ p − µ Z ∞−∞ du exp( − u )( x in − u ) + a v Z ∞−∞ d ∆ x ∆ x exp h − (cid:16) ∆ x − u ( µ − p − µ (cid:17) i , (75)where we replaced the integral over x out with an integral over ∆ x . Note that the term in blue also represents a‘standard’ integral, which equals u ( µ − p π (1 − µ ). The factor containing µ ’s can be taken outside of theintegral over d ∆ x . The integral over µ simplifies to R − dµ ( µ −
1) = −
2, and we are left with a single integralover u . We will further assume that | x in | (cid:29)
1, in which case we get h ∆ x | x in i ∼ − a v πφ ( x in ) Z ∞−∞ du u exp( − u )( x in − u ) ∼ − a v πφ ( x in ) x Z ∞−∞ du u exp( − u ) (cid:16) |{z} odd =0 + 2 ux in | {z } even =0 (cid:17) == − a v πφ ( x in ) x Z ∞−∞ du u exp( − u ) = − a v πφ ( x in ) x √ π = − x in , (76)where the minus sign appeared after performing the integral over µ . In the last step we used that φ ( x in ) = a v / ( √ πx ) in the wing of the line profile. This photon is more likely absorbed by atoms that are movingtowards the photon, as it would appear closer to resonance for these atoms (i.e. P ( u | x in ) is larger for thosephotons with v · k in < α photon is far in the wing at x in , resonant scattering exertsa ‘restoring force’ which pushes the photon back to line resonance. This restoring force generally overwhelmsthe energy losses resulting from atomic recoil: Eq 65 indicates that recoil introduces a much smaller average∆ x ∼ − . × − ( T / K) − / , i.e. if a photon finds itself on the red side of line center, then the restoringforce 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 (Osterbrock 1962). p h ∆ x | x in i = 1 . (77) Namely that R ∞−∞ dx x exp( − a [ x − b ] ) = b p π/a with a = (1 − µ ) − and b = u ( µ − x in FIG. 22 Visual explanation of why resonant scattering tends to push Ly α photons far in the wing of the line profile back tothe line core. This represents a Ly α photon far in the red wing of the line profile (i.e. x in (cid:28) h ∆ x | x in i = − x − . This can be derived with a calculation that is very similar to the above calculation.3. R ( x out | x in ) = R ( x in | x out ). This can be verified by substituting u = y − ∆ x into Eq 71. After some algebra oneobtains an expression in which y replaces u , in which x out replaces x in , and in which x out replaces x in .These three properties of the redistribution functions offer key insights into the Ly α radiative transfer problem. Absorption of a Ly α photon by a hydrogen atom is generally followed by re-emission of the Ly α photon. Moreover,Ly α photons can be absorbed by something different than hydrogen atoms, which also leads to their destruction. Webriefly list the most important processes below:1. Dust.
Dust grains can absorb Ly α photons. The dust grain can scatter the Ly α photon with a probabilitywhich is given by its albedo, A d . Dust plays an important role in Ly α radiative transfer, and we will return tothis later (see § α photon by a dust grain increases its temperature, which causes thegrain to re-radiate at longer wavelengths, and thus to the destruction of the Ly α photon. This process can beincluded in the radiative transfer equation by replacing n HI σ α, φ ( x ) → n HI [ σ α, φ ( x ) + σ dust ( x )] , (78)where σ dust ( x ) denotes the total dust cross-section at frequency x , i.e. σ dust ( x ) = σ dust , a ( x ) + σ dust , s ( x ), wherethe subscript ‘a’ (‘s’) stands for ‘absorption’ (’scattering’). Eq 78 indicates that the cross-section σ dust ( x ) is across-section per hydrogen atom . This definition implies that σ dust ( x ) is not just a property of the dust grain, asit must also depend on the number density of dust grains (if there were no dust grains, then we should not haveto add any term). In addition to this, the dust absorption cross section σ dust , s ( x ) (and also the albedo A d ) mustdepend on the dust properties. For example, Laursen et al. (2009a) shows that σ dust = 4 × − ( Z gas / . Z (cid:12) )cm − for SMC type dust (dust with the same properties as found in the Small Magellanic Cloud), and σ dust =7 × − ( Z gas / . Z (cid:12) ) cm − for LMC (Large Magellanic Cloud) type dust. Here, Z gas denotes the metallicity ofthe gas. This parametrization of σ dust therefore assumes that the number density of dust grains scales linearlywith the overall gas metallicity, which is a good approximation for Z > ∼ . Z (cid:12) , but for Z < ∼ . Z (cid:12) the scatter indust-to-gas ratio increases (e.g. Draine et al. 2007, R´emy-Ruyer et al. 2014, Schneider et al. 2016). Figure 23shows σ dust for dust properties inferred for the LMC ( solid line ), and SMC ( dashed line ). This Figure showsthat the frequency dependence of the dust absorption cross section around the Ly α resonance is weak, and inpractise it can be safely ignored. Interestingly, as we will discuss in § α radiation field, in spite of the weak frequency dependence of the dust absorptioncross-section.7 FIG. 23 This Figure shows grain averaged absorption cross section of dust grains per hydrogen atom for SMC/LMC type dust( solid/dashed line , see text). The inset shows the cross section in a narrower frequency range centered on Ly α , where thefrequency dependence depends practically linearly on wavelength. However, this dependence is so weak that in practise it canbe safely ignored ( Credit: from Figure 1 of Laursen et al. 2009a c (cid:13) AAS. Reproduced with permission. ). We we discuss in § α resonance, dust can have ahighly frequency dependent impact in Ly α spectra. Molecular Hydrogen.
Molecular hydrogen has two transitions that lie close to the Ly α resonance: ( a ) the v = 1 − P (5) transition, which lies ∆ v = 99 km s − redward of the Ly α resonance, and ( b ) the 1 − R (6)transition which lies ∆ v = 15 km s − redward of the Ly α resonance. Vibrationally excited H may thereforeconvert Ly α photons into photons in the H Lyman bands (Neufeld, 1990, and references therein), and thuseffectively destroy Ly α . This process can be included in a way that is very similar to that of dust, and byincluding n HI σ α, φ ( x ) → n HI [ σ α, φ ( x ) + f H σ H ( x )] , (79)where f H ≡ n H /n HI denotes the molecular hydrogen fraction. This destruction process is often overlooked, butit is important to realize that Ly α can be destroyed efficiently by molecular hydrogen. Neufeld (1990) providesexpressions for fraction of Ly α that is allowed to escape as a function of f H and HI column density N HI .3. Collisional Mixing of the s and p Levels. Ly α absorption puts a hydrogen atom in its 2 p state, whichhas a life-time of t = A − α ∼ − s. During this short time, there is a finite probability that the atom interactswith nearby electrons and/or protons. These interactions can induce transitions of the form 2 p → s . Oncein the 2 s -state, the atom decays back to the ground-state by emitting two photons. This process is known ascollisional deexcitation of the 2 p state. Collisional de-excitation from the 2 p state becomes more probable athigh gas densities, and is predominantly driven by free protons. The probability that this process destroys theLy α photon, p dest , at any scattering event is given by p dest = n p C n p C + A α . Here, n p denotes the number densityof free protons, and C = 1 . × − cm s − (e.g. Dennison et al., 2005, and references therein) denotes thecollisional rate coefficient. This process can be included by rescaling the scattering redistribution function (see § R ( x out | x in ) → R ( x out | x in ) × (1 − p dest ) . (80)This ensures that for each scattering event, there is a finite probability ( p dest ) that a photon is destroyed.4. Other.
There are other processes that can destroy Ly α photons, but which are less important. These cantrivially be included in Monte-Carlo codes that describe Ly α radiative transfer (see § i ) Ly α photons can photoionize hydrogen atoms not in the ground state. The photoionisation cross-section from the n = 2 level by Ly α photons is σ Ly α ion = 5 . × − cm (e.g. Cox 2000, p 108). This requires anon-negligible populations of atoms in the n = 2 state which can occur in very dense media (see e.g. Dijkstraet al. 2016) ( ii ) Ly α photons can detach the electron from the H − ion. The cross-section for this process is8 σ = 5 . × − cm − (e.g. Shapiro & Kang, 1987) for Ly α photons, which is almost an order of magnitudelarger than the photoionisation cross-section from the n = 2 level at the Ly α frequency. So, unless the H − number density exceeds 0 . n + n ], where n / denotes the number density of H-atoms in the 2s/2p state,this process is not important. α Propagation through HI: Scattering as Double Diffusion Process
Scattering of Ly α photons is often compared to a diffusion process, in which the photons undergo a random walk inspace and frequency as they scatter off H atoms. Indeed, known analytic solutions to the radiative transfer equation arepossible only under certain idealized scenarios (which are discussed below), for which the radiative transfer equationtransforms into a diffusion equation. To demonstrate how this transformation works, and to gain some insight intothis diffusion process we rewrite the transfer equation (Eq 58) as a diffusion equation. We first simplify Eq 58 in anumber of steps:1. First, we assume that the Ly α radiation field is isotropic, i.e. scattering completely eliminates any directionaldependence of I ν ( n ). Under this assumption we can replace the intensity I ν ( n ) with the angle-averaged intensity J ν ≡ π R d Ω I ν ( n ).2. Second, we replace frequency ν with the dimensionless frequency variable x , introduced in § α HI ν ( s ) = n HI ( s ) σ α ( ν ) = n HI σ α, φ ( x ).4. Fourth, we define dτ = n HI ( s ) σ α, ds and obtain ∂J ( x ) ∂τ = − φ ( x ) J ( x ) + S x ( τ ) + Z dx φ ( x ) J ( x ) R ( x | x ) , (81)where S x ( τ ) ≡ j x ( τ ) / ( n HI σ α, ) denotes the ‘source’ function. In the integral x denotes the frequency of thephoton after scattering, and x denotes the frequency of the photon before scattering. Eq 81 is an integro-differential equation, which is notoriously difficult to solve.Fortunately, this integro-differential can be transformed into a diffusion equation by Taylor expanding J ( x ) φ ( x )around x as we demonstrate next. To keep our notation consistent with § x in ( x out ). To shorten the notation, we define f ( x ) ≡ J ( x ) φ ( x ). We would like to rewrite f ( x in ) as f ( x in ) = f ( x out ) + ( x in − x out ) ∂f∂x + 12 ( x in − x out ) ∂ f∂x + ... = f ( x out ) − ∆ x ∂f∂x + 12 ∆ x ∂ f∂x + ..., (82)where the derivatives are evaluated at x out . Dropping terms that contain ∆ x , ∆ x , ... , we can write the scatteringterm as Z dx in f ( x in ) R ( x out | x in ) ≈ R ∞−∞ dx in R ( x out | x in ) (cid:16) f ( x out ) − ∆ x ∂f∂x + ∆ x ∂ f∂x + ... (cid:17) == R ( x in | x out )= R ( x out | x in ) f ( x out ) Z ∞−∞ dx in R ( x in | x out ) | {z } =1 − ∂f∂x Z ∞−∞ dx in ∆ xR ( x in | x out ) | {z } h ∆ x | x in i = − x out , Eq 76 + ∂ f∂x Z ∞−∞ dx in ∆ x R ( x in | x out ) | {z } h ∆ x | x in i =1 , Eq 77 == f ( x out ) + 1 x out ∂f∂x + 12 ∂ f∂x (83)This term can be further simplified by replacing f ( x out ) with φ ( x out ) J ( x out ) and taking the derivatives. Note thatwe derived Eq 77 and Eq 76 assuming that | x | (cid:29)
1. This same assumption allows us to further simplify the equationas we see below. For brevity we only write φ for φ ( x out ) etc. The term then becomes Jφ + φx out ∂J∂x + Jx out ∂φ∂x + 12 ∂∂x (cid:18) φ ∂J∂x + J ∂φ∂x (cid:19) = Jφ − φx out ∂J∂x + Jφx + φ ∂ J∂x = (84) Jφ (cid:18) x (cid:19) + 12 ∂∂x φ ∂J∂x ≈ Jφ + 12 ∂∂x φ ∂J∂x , Ly α Radiative Transfer as a (Double) Diffusion Process
J(x) x τ =0.2R cl J(x) x τ =0.7R cl FIG. 24 Ly α scattering can be described as a double diffusion process, in which photons diffuse in frequency space as theyscatter. In this case photons are emitted in the center of the sphere, at line center. As photons scatter outwards, the spectrumof Ly α photons broadens. Analytic solutions to the Ly α transfer equation indicate that the spectrum remains remarkablyconstant over a range of frequencies (this range increases as we move outward from the center of the sphere). where we used that ∂φ∂x = − φx , ∂ φ∂ x = φx (see Eq 55), and that | x out | (cid:29) ∂J∂τ = 12 ∂∂x φ ( x ) ∂J∂x + S x ( τ ) . (85)This equation is known as the ‘Fokker-Planck’ transfer equation , and corresponds to a diffusion equation with adiffusion coefficient φ ( x ) (see https://en.wikipedia.org/wiki/Diffusion_equation ), where τ takes the role of thetime variable and x denotes the role of space variable. Eq 85 therefore states that as photons propagate outwards (asa result of scattering), they diffuse in frequency direction (but with a slight tendency for the photons to be pushedback to the line core). As photons diffuse further into the wings of the absorption line profile, their mean free pathincreases, which in turn increases their escape probability. Also note that
8. BASIC INSIGHTS AND ANALYTIC SOLUTIONS
In this section, we show that the concept of Ly α transfer as a diffusion process in real and frequency space canoffer intuitive insights into some basic aspects of the Ly α transfer process. These aspects include ( i ) how many timesa Ly α photon scatters, ( ii ) how long it takes to escape, and ( iii ) the emerging spectrum for a static, uniform mediumin § § § α Transfer through Uniform, Static Gas Clouds
We consider a source of Ly α photons in the center of a static, homogeneous sphere, whose line-center opticaldepth from the center to the edge equals τ , where τ is extremely large, say τ = 10 . This line-centre optical depth There exists various corrections to this Fokker Planck equation. Basko (1981) shows how adding recoil can be incorporated by replacing φ ( x ) ∂J∂x → φ ( x ) ∂J∂x + h P ∆ ν α Jk B T (see Rybicki 2006). Rybicki (2006) introduces further corrections based on requiring that the redistribu-tion satisfies ‘detailed balance’ (microscopic reversability of a process: in equilibrium, each elementary process should be equilibratedby its reverse process), which requires that R ( ν in , ν out ) ν exp (cid:16) − h P ν out k B T (cid:17) = R ( ν out , ν in ) ν exp (cid:16) − h P ν in k B T (cid:17) instead of our assumed R ( ν in , ν out ) = R ( ν out , ν in ). N HI = 1 . × cm − (see Eq 55). We further assume that the centralsource emits all Ly α photons at line center (i.e. x = 0). As the photons resonantly scatter outwards, they diffuseoutward in frequency space. Figure 24 illustrates that as the photons diffuse outwards in real space, the spectralenergy distribution of Ly α flux, J ( x ), broadens. If we were to measure the spectrum of Ly α photons crossing somearbitrary radial shell, then we would find that J ( x ) is constant up to ∼ ± x max beyond which it drops off fast. ForLy α 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 [ τ φ ( x )] − in units of the radius of the sphere . Because each scattering eventchanges the frequency of the Ly α photon, the mean free path of each photon changes with each scattering event.From the shape of the redistribution function (see Fig 21) we expect that on rare occasions Ly α photons will bescattered further from resonance into the wing of the line (i.e. | x | > ∼ h ∆ x | x in i = − x in ). We therefore expect photons that findthemselves in the wing of the line profile, at frequency x , to scatter N scat ∼ x times before returning to the core.During this ‘excursion’ back to the core, the photon will diffuse a distance D ∼ √ N scat × λ mfp ( x ) ≈ √ N scat / [ τ φ ( x )]away from the center of the sphere (recall that this is in units of the radius of the sphere). If we now set thisdisplacement equal to the size of the sphere, i.e. D = √ N scat / [ τ φ ( x )] = x/ [ τ φ ( x )] = 1, and solve for x using that φ ( x ) = a v / [ √ πx ], we find (Adams, 1972; Harrington, 1973; Neufeld, 1990) x p = ± [ a v τ / √ π ] / , (86)where x p denotes the frequency at which the emerging spectrum peaks. Photons that are scattered to frequencies | x | < | x p | will return to line center before they escape from the sphere (where they have negligible chance to escape).Photons that are scattered to frequencies | x | > | x p | can escape more easily, but there are fewer of these photonsbecause: ( i ) it is increasingly unlikely that a single scattering event displaces the photon to a larger | x | , and ( ii )photons that wish to reach | x | (cid:29) | x p | through frequency diffusion via a series of scattering events are likely to escapefrom the sphere before they reach this frequency. We can also express the location of the two spectral peaks at ± x p in terms of a velocity off-set and an HI column density as∆ v p = | x p | v th ≈ (cid:16) N HI cm − (cid:17) / (cid:16) T K (cid:17) / km s − . (87)That is, the full-width at half maximum of the Ly α line can exceed 2∆ v p ∼
300 km s − for a static medium in whichthe thermal velocity dispersion of the atoms is only ∼
10 km s − . Ly α scattering thus broadens spectral lines, whichimplies that we must exercise caution when interpreting observed Ly α spectra.We showed above that the spectrum of Ly α photons emerging from the center of an extremely opaque object tohave two peaks at x p ∼ ± [ a v τ / √ π ] / . More generally, x p = ± k [ a v τ / √ π ] / , where k is a constant of order unitywhich depends on geometry (i.e. k = 1 . k = 0 .
92 for a sphere[Dijkstra et al. 2006]). This derivation required that photons escaped in a single excursion : that is, photonsmust have been scattered deep enough into the wing (which starts for | x | >
3, see Fig 17) to be able to undergo anon-negligible number of wing scattering events before returning to core. So formally our analysis is valid only when x (cid:29) α photons first start their excursion. Another way of phrasing this requirement is that x p (cid:29)
3, or- when expressed in terms of an optical depth τ - when a v τ = √ π ( x p /k ) > ∼ x p / . Indeed, analytic solutionsof the full spectrum emerging from static optically thick clouds appear in good agreement with full Monte-Carlocalculations (see §
9) when a v τ > ∼ α transfer that we can discuss: the first is the mean number of scatteringevents that each Ly α photon undergoes, N scat . This number of scattering events was calculated by Adams (1972), Apart from a small recoil effect that can be safely ignored (Adams 1971), photons are equally likely to scatter to the red and blue sidesof the resonance. Escape in a ‘single excursion’ can be contrasted with escape in a ‘single flight’: gases with lower N HI can become optically thin to Ly α photons when they first scattered into the wing of the line profile. For example, gas with N HI = 10 cm − has a line center opticaldepth τ = 5 . × ( T/ K) − / (Eq 55). However, Figure 17 shows that the cross-section is > ∼ | x | > ∼
3. A photon that first scattered into the wing would be free to escape from this gas without further scattering. Key quantities in Ly α transfer. Line center optical depth Peak frequencies of spectrum Number of scattering events Trapping time
FIG. 25 Ly α radiative transfer through a static, uniform sphere has several interesting features. We denote the line-centeroptical depth from the center to the edge of the sphere with τ . Photons emitted in the center of the sphere diffuse infrequency space as they scatter outward, and emerge with a characteristic double peaked spectrum, for which the peaks occurat x p = ± [ a v τ / √ π ] / . It takes an average of N scat ∼ τ scattering events for a Ly α photon to escape, which ‘traps’ thephoton inside the cloud for a time t trap = | x p | t cross (where t cross = R/c , | x p | ≈ N HI / cm − ) / ( T / K) − / ). after observing that his numerical results implied that the Ly α spectrum of photons was flat within some frequencyrange centered on x = 0 (as in Fig 24), where the range increases with distance from the center of the sphere. Adams(1972) noted that for a flat spectrum, the scattering rate at frequency x is ∝ φ ( x ). Since partially coherent scatteringdoes not change the frequency much (recall that p h ∆ x | x in i = 1, see Eq 77), we expect that if we pick a randomphoton after a scattering event, then the probability that this photon lies in the range x ± dx/ φ ( x ) dx (thismeans that we are more likely to pick a photon close to the core, which simply reflects that these photons scattermore). Adams (1972) then noted that photons at x typically scatter x times before returning to the core, and arguesthat the probability that a photon first scattered into the frequency range x ± dx/ φ ( x ) dx/x . The probabilitythat a photon scatters to some frequency which allows for escape in a single excursion then equals P esc = Z − x p −∞ φ ( x ) dxx + Z ∞ x p φ ( x ) dxx = 2 Z ∞ x p φ ( x ) dxx ≈ wing a v √ πx = Eq 86 √ π τ k . (88)We thus expect photons to escape after N scat = 1 /P esc scattering events, which equals N scat = Cτ , (89)where C ≡ k √ π ≈ . C ≈ . α photons typically scatter ∼ τ times before escaping from extremely opaque media. This differs from standard random walks where a photonwould scatter ∝ τ times before escaping. The reduced number of scattering events is due to frequency diffusion,which forces photons into the wings of the line, where they can escape more easily.We can also estimate the time it takes for a photon to escape. We now know that photons escape in a singleexcursion with peak frequency x p = ± [ a v τ / √ π ] / . During this excursion the photon scattered x times in thewing of the line profile. We also know it took N scat = 0 . τ scattering events on average before the excursionstarted. Generally, N scat = 0 . τ (cid:29) x , and the vast majority of scattering events occurred in the line core, wherethe mean free path was very short. The total distance that the photon travelled while scattering in the core is D core = N scat λ mfp ∼ τ × τ − ∼
1, where we used that the mean free path at line center is τ − (in units ofthe radius of the sphere). The total distance that was travelled in the wing - i.e. during the excursion - equals D wing ∼ x × [ τ φ ( x p )] − = x √ π/ ( a v τ ). If we now substitute the expression for x p , and add the distance travelledin the core and in the wing we get D = D core + D wing ≈ (cid:18) a v τ √ π (cid:19) / = 1 + | x p | ≈ | x p | , (90)2 -100 -50 0 50 100x = ( ν - ν )/ ∆ν D J( τ , x ) τ = 10 τ = 10 τ = 10 FIG. 26 Ly α spectra emerging from a uniform spherical, static gas cloud surrounding a central Ly α source which emits photonsat line centre x = 0. The total line-center optical depth, τ increases from τ = 10 ( narrow histogram) to τ = 10 ( broadhistogram ). The solid lines represent analytic solutions ( Credit: from Figure A2 of Orsi et al. 2012, ‘Can galactic outflowsexplain the properties of Ly α emitters?’, MNRAS, 425, 87O ). where we used that | x p | (cid:29)
1, i.e. that D wing (cid:29) D core . If we express this in terms of travel time, then the vastmajority of scattering events take up negligible time. The total time it takes for Ly α photons to escape - or the totaltime for which they are ‘trapped ’ - thus equals t trap = | x p | t cross , | x p | ≈ N HI / cm − ) / ( T / K) − / . (91)This is an interesting, and often overlooked result: for τ ∼ we now know that Ly α photons scatter 10 milliontimes on average before escaping. Yet, they are only trapped for ∼
15 light crossing times, which is not long.The last thing I will only mention is that the diffusion equation (Eq 85) can be solved analytically for the angleaveraged intensity J ( x, τ ) for static, uniform gaseous spheres, slabs, and cubes. These solutions were presentedby Harrington (1973) & Neufeld (1990), who showed (among other things) that the angle-averaged Ly α spectrumemerging from a semi-infinite ‘slab’ has the following analytic solution: J ( x ) = √ √ πaτ x hq π | x | aτ i ! , (92)when the photons were emitted in the center of the slab. Here, τ denotes the line center optical depth from the centerto the edge of the slab (which extends infinitely in other directions). Similar solutions have been derived for spheres(Dijkstra et al., 2006) and cubes (Tasitsiomi, 2006b). The solid lines shown in Figure 26 shows analytic solutions for J ( x ) emerging from spheres for τ = 10 , and τ = 10 . Histograms show the spectra emerging from Monte-Carlosimulations of the Ly α radiative transfer process (see § x max = ± . a v τ ) / ∼ x p [ x max = ± . a v τ ) / ], where x p is the characteristic frequency estimated above (seeEq 86). α 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 analyticsolutions are known (except when T = 0, see below). We can qualitatively describe what happens when the gasclouds are not static.Consider an expanding sphere: the predicted spectral line shape must also depends on the outflow velocityprofile v out ( r ). Qualitatively, photons are less likely to escape on the blue side (higher energy) than photons onthe red side of the line resonance because they appear closer to resonance in the frame of the outflowing gas.3 FIG. 27 This Figure illustrates the impact of bulk motion of optically thick gas to the emerging Ly α spectrum of Ly α : The green lines show the spectrum emerging from a static sphere (as in Fig 26). In the left/right panel the HI column density fromthe centre to the edge of the sphere is N HI = 2 × /2 × cm − . The red/blue lines show the spectra emerging froman expanding/a contracting cloud. Expansion/contraction gives rise to an overall redshift/blueshift of the Ly α spectral line( Credit: from Figure 7 of Laursen et al. 2009b c (cid:13) AAS. Reproduced with permission ). Moreover, as the Ly α photons are diffusing outward through an expanding medium, they loose energy because thedo ’work’ on the outflowing gas (Zheng & Miralda-Escud´e, 2002). Both these effects combined enhance the redpeak, and suppress the blue peak, as illustrated in Figure 27 (taken from Laursen et al. 2009b). In detail, howmuch the red peak is enhanced, and the blue peak is suppressed (and shifted in frequency directions) depends onthe outflow velocity and the HI column density of gas . Not unexpectedly, the same arguments outlined abovecan be applied to a collapsing sphere: here we expect the blue peak to be enhanced and the red peak to besuppressed (e.g. Zheng & Miralda-Escud´e 2002, Dijkstra et al. 2006). It is therefore thought that the Ly α line shapecarries information on the gas kinematics through which it is scattering. As we discuss in § α spectral line profile has been used to infer properties of the medium through which they are scattering.There exists one analytic solution to radiative transfer equation through an expanding medium: Loeb & Rybicki(1999) derived analytic expressions for the radial dependence of the angle-averaged intensity J ( ν, r ) of Ly α radiationas a function of distance r from a source embedded within a neutral intergalactic medium undergoing Hubbleexpansion, (i.e. v out ( r ) = H ( z ) r , where H ( z ) is the Hubble parameter at redshift z ). Note that this solution wasobtained assuming completely coherent scattering (i.e. x out = x 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 asomewhat different scattering process than what we discussed before (see Dijkstra et al., 2006, for a more detaileddiscussion). However, Dijkstra & Loeb (2008a) have compared this analytic solution to that of a Monte-Carlo codethat uses the proper frequency redistribution functions and find good agreement between the Monte-Carlo andanalytic solutions.Finally, it is worth pointing out that in expanding/contracting media, the number of scattering events N scat andthe 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 α photons into the wings of the line profiles where they can escape more easily. Thereduction in trapping time has been quantified by Bonilha et al. (1979), and in shell models (which will be discussedin § α Transfer through Dusty, Uniform & Multiphase Media
The interstellar medium (ISM) contains dust. A key difference between a dusty and dust-free medium is that inthe presence of dust, Ly α photons can be destroyed during the scattering process when the albedo (also known as thereflection coefficient) of the dust grains (see § A d <
1. Dust therefore causes the ‘escape fraction’ ( f α esc ), which 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 α spectrum: see http://bit.ly/man-alpha . FIG. 28 This figure illustrates that dust can suppress the Ly α spectral line profile in a highly frequency dependent way, eventhough the dust absorption cross-section barely varies over the same frequency range ( Credit: from Figure 9 of Laursen et al.2009b c (cid:13) AAS. Reproduced with permission ). The main reason for this is that dust limits the distance Ly α photons can travelbefore they are destroyed, which limits how much they can diffuse in frequency space. Dust therefore has the largest impact atfrequencies furthest from line center. denotes the fraction Ly α photons that escape from the dusty medium, to fall below unity, i.e. f α esc <
1. For a mediumwith a uniform distribution of gas and dust, the probability that a Ly α photon is destroyed by a dust grain increaseswith the distance travelled by the Ly α photon. We learned before that as photons diffuse spatially, they also diffusein frequency. This implies that dust affects the emerging Ly α spectral line profile in a highly frequency-dependent way,even though the absorption cross-section for this process is practically a constant over the same range of frequencies .Specifically, if we compare Ly α spectra emerging from two scattering media which differ only in their dust content,then we find that the impact of the dust is largest at large frequency/velocity off-sets from line center (see Fig 28).Dust also destroys UV-continuum photons, but because Ly α photons scatter and diffuse spatially through thedusty medium, the impact of dust on Ly α and UV-continuum is generally different. This can affect the ‘strength’(i.e. the equivalent width) of the Ly α line compared to the underlying continuum emission. In a uniform mixture ofHI gas and dust, Ly α photons have to traverse a larger distance before escaping, which increases the probability tobe destroyed by dust. In these cases we expect dust to reduce the EW of the Ly α line. The ISM is not smooth anduniform however, which can drastically affect Ly α radiative transfer. The interstellar medium is generally thought toconsist of the ‘cold neutral medium’ (CNM), the ‘warm neutral/ionized medium’ (WNM/WIM), and the ‘hot ionizedmedium’ (HIM, see e.g. the classical paper by McKee & Ostriker 1977). In reality, the cold gas is not in ‘clumps’ butrather in a complex network of filaments and sheets. Ly α transfer calculations through realistic ISM models haveonly just begun, partly because modeling the multiphase nature of the ISM with simulations is a difficult task whichrequires extremely high spatial resolution. There is substantially more work on Ly α transfer through ‘clumpy’ mediathat consists of cold clumps containing neutral hydrogen gas and dust, embedded within a (hot) ionized, dust freemedium (Neufeld 1991, Hansen & Oh 2006, Laursen et al. 2013, Gronke et al. 2016), and which represent simplifieddescriptions of the multi-phase ISM.Clumpy media facilitate Ly α escape from dusty interstellar media, and can help explain the detection of Ly α emission from dusty galaxies such as submm galaxies (e.g. Chapman et al. 2005), and (U)LIRGs (e.g Nilsson& Møller 2009, Martin et al. 2015). In a clumpy medium dust can even increase the EW of the Ly α line, i.e.preferentially destroy (non-ionizing) UV continuum photons over Ly α photons: Ly α photons can propagate freelythrough the ‘interclump’ medium. Once a Ly α photon enters a neutral clump, its mean free path can be substantiallysmaller than the clump size. Because scattering is partially coherent, and the frequency of the photon in the clumpframe changes only by an r.m.s amount of p h ∆ x | x in i = 1 (see Eq 77), the mean free path remains roughly thesame. Since the photon penetrated the clump by a distance corresponding to τ ∼
1, the photon is able to escapeafter each scattering event with a significant probability. A Ly α photon that penetrates a clump is therefore likelyto escape after ∼ sourceclouds intercloud medium FIG. 29 Schematic illustration of how a multiphase medium may favour the escape of Ly α line photons over UV-continuumphotons: Solid / dashed lines show trajectories of Ly α /UV-continuum photons through clumpy medium. If dust is confined tothe cold clumps, then Ly α may more easily escape than the UV-continuum ( Credit: from Figure 1 of Neufeld 1991 c (cid:13) AAS.Reproduced with permission. ). larger number of scattering events. The Ly α photons effectively scatter off the clump surface (this is illustrated inFig 29), thus avoiding exposure to dust grains. In contrast, UV continuum photons will penetrate the dusty clumpsunobscured by hydrogen and are exposed to the full dust opacity. Clumpy, dusty media may therefore preferentiallylet Ly α photons escape over UV-continuum.Laursen et al. (2013) and Duval et al. (2014) have recently shown however that - while clumpy media facilitateLy α 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 thedensity contrast between clumps and interclump medium is maximized. While a multiphase (or clumpy) mediumdefinitely facilitates the escape of Ly α photons from dusty media, EW boosting therefore appears uncommon. Wecan understand this result as follows: the preferential destruction of UV-continuum photons over Ly α requires atleast a significant fraction of Ly α photons to avoid seeing the dust by scattering off the surface of the clumps. Howdeep the Ly α photons actually penetrate, depends on their mean-free path, which depends on their frequency in theframe of the clumps. If clumps are moving fast, then it is easy for Ly α photons to be Doppler boosted into the wingof the line profile (in the clump frame), and they would not scatter exclusively on the clump surfaces. Finally, wenote that the conclusions of Laursen et al. (2013) and 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, therecan be directional variations in the predicted EW, with large EW boosts occurring in a small fraction of sightlines indirections where the UV-continuum photon escape fraction was suppressed, thus partially restoring the possibility ofEW boosting by a multiphase ISM.
9. MONTE-CARLO LY α RADIATIVE TRANSFER
Analytic solutions to the radiative transfer equation (Eq 58) only exist for a few idealised cases. A modernapproach to solve this equation is via
Monte-Carlo methods, which refer to a ‘broad class of computational algorithms[...] which change processes described by certain differential equations into an equivalent form interpretable as asuccession of random operations’ (S. Ulam, see https://en.wikipedia.org/wiki/Monte_Carlo_method ) . The term ‘Monte-Carlo’ was coined by Ulam & Metropolis as a code-name for their classified work on nuclear weapons (radiationshielding, and the distance that neutrons would likely travel through various materials). ‘Monte-Carlo’ was the name of the casinowhere Ulam’s uncle had a (also classified) gambling addiction. I was told most of this story over lunch by M. Baes. For questions, pleasecontact him. FIG. 30 The rejection method provides a simple way to randomly draw variables x from arbitrary probability distributions P ( x ) (see text). In Ly α Monte-Carlo radiative transfer, we represent the integro-differential equation (Eq 58) by a succession ofrandom scattering events until Ly α photons escape (Lee & Ahn 1998, Loeb & Rybicki 1999, Ahn et al. 2001, Zheng& Miralda-Escud´e 2002, Cantalupo et al. 2005, Dijkstra et al. 2006, Verhamme et al. 2006, Tasitsiomi 2006, Semelinet al. 2007, Laursen et al. 2009a, Pierleoni et al. 2009, Faucher-Gigu`ere et al. 2010, Kollmeier et al. 2010, Zheng etal. 2010, Barnes et al. 2011, Forero-Romero et al. 2011, Orsi et al. 2012, Yajima et al. 2012, Behrens & Niemeyer2013, Gronke & Dijkstra 2014, Lake et al. 2015). Details on how the Monte-Carlo approach works can be found inmany papers (see e.g. the papers mentioned above, and Chapters 6-8 of Laursen, 2010, for an extensive description).I will first provide a brief description of drawing random variables, which is central to the Monte-Carlo method. ThenI will describe Monte-Carlo Ly α radiative transfer. Central in Monte-Carlo methods is generating random numbers from probability distributions. If we denote aprobability distribution of variable x with P ( x ), the P ( x ) dx denotes the probability that x lies in the range x ± dx/ R ∞−∞ dxP ( x ) = 1. The cumulative probability is C ( x ) ≡ P ( < x ) = R x −∞ P ( x ) dx , where clearly C ( x → −∞ ) = 0 and C ( x → ∞ ) = 1. We can draw a random x from the distribution P ( x ) by randomly generating anumber ( R ) between 0 and 1. We then transform R into x by inverting its cumulative probability distribution C ( x ),i.e. R ≡ C ( x ) → R ≡ Z x −∞ P ( x ) dx . (93)This implies that we need to ( i ) be able to integrate P ( x ), and ( ii ) be able to invert the integral equation. Generally,there is no analytic way of inverting Eq 93. One way of randomly drawing x from P ( x ) is provided by the rejectionmethod . This method consists of picking another T ( x ) which lies above P ( x ) everywhere, and which we can integrateand invert analytically. We are completely free to pick T ( x ) however we want. Because T ( x ) lies above P ( x )everywhere, R T ( x ) dx = A , where A >
1. We first generate x from T ( x ) by generating a random number R between 0 and 1 and inverting R ≡ A Z x −∞ T ( x ) dx . (94)Once we have x we then generate another random number R ∈ between 0 and 1, and accept x as our random pickwhen R ≤ P ( x ) T ( x ) . (95)7 FIG. 31
Red histograms show the distributions of 10 ( left panel ) and 10 ( right panel ) randomly drawn values for x from itsPDF P ( x ) by using the rejection method. In this case we randomly truncated a Gaussian PDF with σ = 1 by suppressing thePDF by an arbitrary function f ( x ) only for x > f ( x ) < black solid line . If we repeat this procedure a large number of times, and make a histogram for x , we can see that this traces P ( x )perfectly (this is illustrated with an example in Fig 31). Eq 95 shows that we ideally want T ( x ) to lie close to P ( x )in order not to have to reject the majority of trials (the better the choice for T ( x ), the smaller the rejected fraction). α Monte-Carlo Radiative Transfer
Here, we briefly outline the basic procedure that describes the Monte-Carlo method applied to Ly α photons. Foreach photon in the Monte-Carlo simulation:1. We first randomly draw a position, r , from which the photon is emitted from the emissivity profile j ν ( r ) (seeEq 58). We the assign a random frequency x , which is drawn from the Voigt function φ ( x ), and a randompropagation direction k .2. We randomly draw the optical depth τ the photon propagates into from the distribution P ( τ ) = exp( − τ ).3. We convert τ into a physical distance s by (generally numerically) inverting the line integral τ = R s dλ n HI ( r ) σ α ( x [ r ]), where r = r + λ k and x = x − v ( r ) · k / ( v th ). Here, v ( r ) denotes the 3D bulk velocity vector of the gas at position r . Note that x is the dimensionless frequency of the photon in the ‘local’frame of the gas at r .4. Once we have selected the scattering location, we need to draw the thermal velocity components of the atom thatis scattering the photon (we only need the thermal velocity components, as we work in the local gas frame). Asin § v || (or its dimensionless analogue u , see Eq 66), and a 2D-velocity vector perpendicular to k , namely v ⊥ . Wediscussed in § P ( u | x ) (see discussion below Eq 68),and apply the rejection method to draw u from this functional form (see the Appendix Zheng & Miralda-Escud´e2002 for a functional form of T ( u | x )). The 2 components of v ⊥ can be drawn from a Maxwell-Boltzmanndistribution (see e.g. Dijkstra et al. 2006).5. Once we have determined the velocity vector of the atom that is scattering the photon, we draw an outgoingdirection of the photon after scattering, k out , from the phase-function, P ( µ ) (see Eq 41 and § For arbitrary gas distributions, the emissivity profile is a 3D-field. We can still apply the rejection method. One way to do this is todiscretize the 3D field j ν ( x, y, z ) → j ν ( i, j, k ), where we have N x , N y , and N z of cells into these three directions. We can map this3D-array onto a long 1D array j ν ( m ), where m = 0 , , ..., N x × N y × N z , and apply the rejection method to this array. FIG. 32 This Figure shows an example set of tests that Monte-Carlo codes must be able pass (
Credit: from Figure 1 ofDijkstra et al. 2006 c (cid:13) AAS. Reproduced with permission. ). In the top left panel show Monte-Carlo calculations of the Ly α spectra emerging from a uniform spherical gas cloud, in which Ly α photons are injected in the center of the sphere, at the linecenter (i.e. x = 0). The total line center optical depth, τ from the center to the edge is τ = 10 ( blue ), τ = 10 ( red ) and τ = 10 ( green ). Overplotted as the black dotted lines are the analytic solutions. The agreement is perfect at high optical depth( a v τ > ∼ ). Upper right panel:
The colored histograms show the frequency redistribution functions, R ( x out , x in ), for x in = 0( blue ), x in = 2 ( red ) and x in = 5 ( green ), as generated by the Monte-Carlo simulation. The solid lines are the analytic solutionsgiven by Eq 73. Lower left panel:
The total number of scattering events that a Ly α photon experiences before it escapes from aslab of optical thickness 2 τ according to a Monte-Carlo simulation ( circles ). Overplotted as the red–solid line is the theoreticalprediction by Harrington (1973). Lower right panel: . The spectrum emerging from an infinitely large object that undergoesHubble expansion. The histogram is the output from our code, while the green–solid line is the (slightly modified) solutionobtained by Rybicki & Loeb (1999) using their Monte Carlo algorithm (see Dijkstra et al. 2006 for more details).
6. Unless the photon escapes, we replace the photon propagation direction & frequency and go back to 1). Once thephoton escapes we record information we are interested in such as the location of last scattering, the frequencyof the photon, the thermal velocity components of the atom that last scattered the photon, the number ofscattering events the photon underwent, the total distance it travelled through the gas, etc.It is important to test Ly α Monte-Carlo codes in as many ways as possible. Figure 32 shows a minimum set of testsMonte-Carlo codes must be able to reproduce. These comparisons with analytic solutions test different aspects ofthe code.The histograms in the top left panel show Monte-Carlo realizations of the Ly α spectra emerging from a uniform9spherical gas cloud, in which Ly α photons are injected in the center of the sphere, at the line center (i.e. x = 0).The total line center optical depth, τ from the center to the edge is τ = 10 ( blue ), τ = 10 ( red ) and τ = 10 ( green ). Overplotted as the black dotted lines are the corresponding analytic solutions (see Eq 92, but modified fora sphere, see Dijkstra et al. 2006a). The agreement is perfect at high optical depth ( a v τ > ∼ ). At lower opticaldepth, a v τ < ∼ , the analytic solutions are not expected to be accurate any more (see § τ only confirms that the Monte-Carlo procedure accurately describes scatteringin the wing of the line profile. This comparison does not test core scattering, which make up the vast majority ofscattering events (see § α spectra emerging from optically (extremely) thick media is insensitiveto core scattering implies that we need additional tests to test core scattering. However, it also implies we canskip these core-scattering events, which account for the vast majority of all scattering events. That is, Monte-Carlosimulations can be ’accelerated’ by skipping core scattering events. We discuss how we can do this in more detail in § upper right panel the colored histograms show Monte-Carlo realization of the frequency redistributionfunctions, R ( x out , x in ) (see Eq 73), for x in = 0 ( blue ), x in = 2 ( red ) and x in = 5 ( green ). The solid lines are theanalytic solutions given by Eq 73 (here for dipole scattering). This comparison tests individual Ly α core scatteringevents, 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 α photon experiences beforeit escapes from a slab of optical thickness 2 τ in a Monte-Carlo simulation. Overplotted as the red–solid line is thetheoretical prediction that N scat = Cτ , with C = 1 . τ corresponds to the range of τ where analytic solutions are expected to fail. This test providesanother way to test core scattering events, as N scat is set by the probability that a Ly α photon is first scatteredsufficiently far into the wing of the line such that it can escape in a single ‘excursion’. This test also shows howaccurate the analytic prediction is, despite the fact that the derivation presented in § α transfer can beif we simulate each scattering event.The lower right panel shows Ly α spectrum emerging from a Ly α point source surrounded by an infinitely largesphere that undergoes Hubble expansion. The black histogram shows a Monte-Carlo realization. The green line represents a ‘pseudo-analytic’ solution of Loeb & Rybicki (1999): as we mentioned earlier in § J ( r, x ) where r denotes the distance fromthe galaxy. Unfortunately, their analytic solution does not apply to emerging spectrum because as r → ∞ , the Ly α photons have redshifted far enough into the wing that the IGM is optically thin to the Ly α photons, in which case theLy α radiative transfer problem cannot be reduced to a diffusion problem anymore. Though not shown here, Dijkstra& Loeb (2008) compared the analytic integrated U ( r ) ∝ R J ( r, x ) to that extracted from Monte-Carlo simulations,and found excellent agreement. The green line shown here represents the spectrum obtained from a simplifiedMonte-Carlo simulation (see Loeb & Rybicki 1999 for details), and agrees well with the full Monte-Carlo code. Thistest provides us with a way to test the code when bulk motions in the gas are present. α Monte-Carlo Simulations in 3D Simulations
In Monte-Carlo radiative transfer calculations applied to arbitrary 3D gas distributions, extracting observablesrequires (a bit) more work than recording the location of last scattering, the photon’s frequency etc. This is becauseformally we are interested only in a tiny subset of Ly α 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 k t . The mirror of ourtelescope only subtends a solid angle d Ω telescope = dA telescope d ( z ) , where dA telescope denotes the area of the mirror, and d A ( z ) denotes the angular diameter distance to redshift z . The probability that a Ly α photon in our Monte-Carlosimulation escapes from the scattering medium into this tiny solid angle is negligible. The frequency ˆ x here shows another dimensionless frequency variable that was used by Loeb & Rybicki (1999), and relates to ‘standard’dimensionless frequency x as ˆ x ∼ − . × x for T = 10 K (see Dijkstra et al. 2006). Peeling Algorithm
FIG. 33 A schematic representation of the peeling algorithm . This algorithm allows us to efficiently and accurately extractLy α observables (spectra, surface brightness profiles, etc) from arbitrary 3D gas distributions. This algorithm overcomes theproblem that in Monte-Carlo simulations, the probability that a Ly α photon escapes - and into the direction of the telescopesuch that it lands on the mirror - is practically zero due to the infinitesimally small angular scale of the telescope mirror whenviewed from the Ly α source. The peeling algorithm treats each scattering event as a point source with a (scattering inducedanisotropic) luminosity L α /N phot , where L α is the total luminosity of the Ly α source and N phot denotes the number of photonsused in the Monte-Carlo run to simulate the source. One way to get around this problem is by relaxing the restriction that photons must escape into the solid angle d Ω telescope centered on the direction k t , by increasing d Ω telescope to a larger solid angle. This approximation corre-sponds to averaging over all viewing directions within some angle ∆ α from the real viewing direction. If observableproperties of Ly α do not depend strongly on viewing direction, then this approximation is accurate. However, thismethod implies we are not using information from the vast majority of photons ( ∼ ∆ α / π ) that we used in theMonte-Carlo simulation. To circumvent this problem in a more efficient (and accurate) way is provided by the so-called ‘ peeling algorithm ’. This algorithm treats each scattering event in the simulation domain as a point sourcewith luminosity L α /N phot , where L α is the total Ly α luminosity of the source and N phot denotes the total number ofLy α photons used in the Monte-Carlo simulation to represent this source (see e.g. Yusef-Zadeh et al. 1984, Zheng &Miralda-Escud´e 2002, Tasitsiomi 2006). The total flux S we expect to get from each point source is S = L α πd ( z ) N phot × P ( µ t )2 e − τ t ( x t ) , (96)where d L ( z ) denotes the luminosity distance to redshift z . Furthermore, µ t ≡ k in · k t , in which k in (like before)denotes the propagation direction of the photon prior to scattering. The presence of the scattering phase-function P ( µ ) reflects that the ‘point source’ is not emitting isotropically, but that the emission in the direction µ t is enhancedby a factor of P ( µ t ) / k t (the factor of ‘2’ reflects that R dµ P ( µ ) = 2, see Eq 42). The factor e − τ t ( x t ) denotes the escape fraction in direction k t , where x t is the frequency of the photon it would have had if it trulyhad scattered in direction k t . This frequency x t can be obtained from Eq 66. Eq 96 should be applied for eachscattering event: for each scattering event there is a tiny/infinitesimal probablity - ∼ d Ω telescope π e − τ t ( x t ) - that theLy α scatters into the telescope-mirror. We formally have to reduce the weight of the photon by this probability aftereach scattering event (we are ‘peeling’ off the weight of this photon), though in practice this can be ignored becauseof the tiny probability that a photon scattered into the telescope mirror.An example of an image generated with the peeling algorithm is shown in Figure 34. Here, a Ly α source is at theorigin of a cartesian coordinate system. Each of the 3 coordinate axes has 2 spheres of HI gas at identical distancesfrom the origin. There are no hydrogen atoms outside the sphere, and the Ly α scattering should only occur insidethe 6 spheres. Resulting images (taken from Dijkstra & Kramer 2012) from 6 viewing directions are shown in the leftpanel of Figure 34. The ‘darkness’ of a pixel represents its Ly α surface brightness. The average of these 6 images isshown in the box. The right panel shows a close-up view of the image associated with the sphere on the + y -axis. The1 FIG. 34 Examples of Ly α images generated with the peeling algorithm (see text for a more detailed description, Credit: fromFigure A3 of Dijkstra & Kramer. 2012, ‘Line transfer through clumpy, large-scale outflows: Ly α absorption and haloes aroundstar-forming galaxies’, MNRAS, 424, 1672D ). side of the sphere facing the Ly α source (on the bottom at y = 0) is brightest. The red line in the inset shows ananalytic calculation of the expected surface brightness, under the assumption that the sphere as a whole is opticallythin (only this assumption allows for analytic solutions). This image shows that the Peeling algorithm gives rise tothe sharp features in the surface brightness profiles that should exist. Note that the alternative method we brieflymentioned above, which averages over all viewing directions within some angle ∆ α from k t , would introduce someblurring to these images. α Monte-Carlo Simulations
As we mentioned above, we do not care about the vast majority of core scattering events for Ly α transfer throughextremely optically thick media ( a v τ > ∼ ). We can accelerate Monte-Carlo simulations by ‘forcing’ photons intothe wing of the line profile. Ordinarily, the physical mechanism that puts a Ly α photon from the core into thewing of the line profile is an encounter with a fast moving atom. We can force the scattering atom to have a largevelocity when generating its velocity components. A simple way to do this is by forcing the velocity vector of theatom perpendicular to k in , v ⊥ , to be large (Ahn et al. 2002, Dijkstra et al. 2006). We know from § v ⊥ follows a 2D Maxwell-Boltzmann distribution g ( v ⊥ ) d v ⊥ ∝ v ⊥ exp( − v ⊥ ), where v ⊥ ≡ | v ⊥ | is the magnitude of v ⊥ .In dimensionless units u ⊥ ≡ v ⊥ /v th , and we have g ( u ⊥ ) du ⊥ = 2 πu ⊥ exp( − u ⊥ ) /π (see the discussion under Eq 68).We can force u ⊥ to be large by drawing it from a truncated Maxwell-Boltzmann distribution, which states that p ( u ⊥ ) du ⊥ = ( | u ⊥ | < x crit N u ⊥ exp( − u ⊥ ) | u ⊥ | > x crit . , (97)where N ensures that the truncated distribution function for u ⊥ is normalized. Furthermore, x crit is a parameterthat determines how far into the wing we force the Ly α photons. This parameter therefore sets how much Ly α transfer is accelerated. Clearly, one has to be careful when choosing x crit : forcing photons too far into the wing maycause them to escape at frequencies where frequency diffusion would otherwise never take them. Various authorshave experimented with choosing x crit based on the local HI-column density of a cell in a simulation, etc (Tasitsiomi2006a,b, Laursen et al. 2009a, Smith et al. 2015).
10. LY α TRANSFER IN THE UNIVERSE
Previous sections discussed the basics of the theory describing Ly α transfer through optically thick media. Thegoal of this section is to discuss what we know about Ly α transfer in the real Universe. We decompose this problem2 redshift − − − f e sc ( L y α ) De 08Co 10Ha 10Ca 10Gr 07Ou 08Da 07Sh 09Iy 06Ou 10Hi 10
FIG. 35 Observational constraints on the redshift-dependence of the volume averaged ‘effective’ escape fraction, f effesc , whichcontains constraints on the true escape fraction f α esc ( Credit: from Figure 1 of Hayes et al. 2011 c (cid:13) AAS. Reproduced withpermission ). into several scales: ( i ) Ly α photons have to escape from the interstellar medium (ISM) of galaxies into the circumgalactic/intergalactic medium (CGM/IGM). We discuss this in § § § Understanding interstellar Ly α 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 interstellarmedium is an extremely challenging task, as it requires resolving the multiphase structure of interstellar medium,and how it is affected by feedback from star-formation (via supernova explosions, radiation pressure, cosmic raypressure, etc). Instead of taking an ‘ab-initio’ approach to understanding Ly α transfer, it is illuminating to use a‘top-down’ approach in which we try to constrain the broad impact of the ISM on the Ly α radiation field from obser-vations (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 α photons, f α esc . To estimate f α esc wewould need to compare the observed Ly α luminosity to the intrinsic Ly α luminosity. The intrinsic Ly α luminositycorresponds to the Ly α luminosity that is actually produced. The best way to estimate the intrinsic Ly α luminosityis from some other non-resonant nebular emission line such as H α . The observed H α luminosity can be convertedinto an intrinsic H α luminosity once nebular reddening is known (from joint measurements of e.g. the H α and H β lines see lecture notes by M. Hayes). Once the intrinsic H α luminosity is known, then we can compute the intrinsicLy α luminosity assuming case-B (or case-A) recombination. This procedure indicates that f α esc ∼ −
2% at z ∼ . z ∼
5% at z ∼ α luminosity from the inferred star formation rate of galaxies (and apply Eq 17 or Eq 18). These starformation rates can be inferred from the dust corrected (non-ionizing) UV-continuum flux density and/or from the IRflux density (e.g. Kennicutt, 1998). These analyses have revealed that f α esc is anti-correlated with the dust-content ofgalaxies (Atek et al. 2009, Kornei et al. 2010, Hayes et al. 2011). This correlation may explain why f α esc increases withredshift from f α esc ∼ −
3% at z ∼ f α esc ∼ −
50% at z ∼ f α esc : Ly α photons that escape from galaxies can scatter frequentlyin the IGM (or circum-galactic medium) before reaching earth in a low surface brightness glow that cannot be This discussion represents an extended version of the discussion presented in the review by Dijkstra (2014). There is also little observational evidence for EW-boosting by a multiphase medium (e.g. Finkelstein et al., 2008; Scarlata et al., 2009). Credit: HST/NASA/ESA.
I Zwicky 18
Atek et al. 2009
FIG. 36 1Zwicky18. A nearby, metal poor, blue young star forming galaxy (
Image Credit: HST/NASA/ESA, and A. Aloisi ).While this galaxy is expected to be dust-poor, no Ly α is detected in emission ( Credit: Atek et al, A&A, 502, page 791-801,2009, reproduced with permission c (cid:13) ESO ). It is thought that this is because there are no (or little) outflows present in thisgalaxy, which could have facilitated the escape of Ly α . More enriched nearby star forming galaxies that do show Ly α inemission, show evidence for outflows (e.g. Kunth et al. 1998, Atek et al. 2008). detected yet (see § f α esc on dust content of galaxies is an intuitive result, as it is practically the only componentof the ISM that is capable of destroying Ly α . However, there is more to Ly α escape. This is probably bestillustrated by nearby starburst galaxy 1Zwicky18 (shown in Fig 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). Wewould expect this galaxy to have a high f α esc . However, the spectrum of 1Zwicky18 (also shown in Fig 36) showsstrong Ly α absorption. In contrast, the more enriched ( Z ∼ . − . Z (cid:12) ) nearby galaxy ESO 350 does showstrong Ly α emission (see Fig 1 of Kunth et al. 1998). The main difference between the two galaxies is thatESO 350 shows evidence for the presence of outflowing gas. Kunth et al. (1998) observed that for a sample of8 nearby starburst galaxies, 4 galaxies that showed evidence for outflows showed Ly α in emission, while no Ly α emission was detected for the 4 galaxies that showed no evidence for outflows (irrespective of the gas metallicityof the galaxies). These observations indicate that gas kinematics is a key parameter that regulates Ly α escape(Kunth et al. 1998, Atek et al. 2008, Wofford et al. 2013, Rivera-Thorsen et al. 2015). This result is easy tounderstand qualitatively: in the absence of outflows, the Ly α sources are embedded within a static optically thickscattering medium. The traversed distance of Ly α photons is enhanced compared to that of (non-ionizing) UVcontinuum photons (see § α photons can be efficiently scattered into the wing of the line profile, where they can escape easily.The role of outflows is apparent at all redshifts. Simultaneous observations of Ly α and other non-resonantnebular emission lines indicate that Ly α lines typically are redshifted with respect to these other lines by ∆ v Ly α .This redshift is more prominent for drop-out (Lyman break) galaxies, in which the average ∆ v Ly α ∼
460 kms − in LBGs (Steidel et al. 2010, Kulas et al. 2012), which is larger than the shift observed in LAEs, wherethe average ∆ v ∼
200 km s − (McLinden et al. 2011, Chonis et al. 2013, Hashimoto et al. 2013, Erb et al.2014, McLinden et al. 2014, Song et al 2014, Trainor et al. 2015, Prescott et al. 2015) . These observationsindicate that outflows generally affect Ly α radiation while it is escaping from galaxies. This is not surprising: The different ∆ v in LBGs and LAEs likely relates to the different physical properties of both samples of galaxies. Shibuya et al. (2014)argue that LAEs may contain smaller N HI which facilitates Ly α escape, and results in a smaller shift (also see Song et al., 2014). Ly α em. line IS abs. line 〈 Δ v Ly α 〉 ~ 450 km/s 〈 Δ v IS 〉 ~ -160 km/s FIG. 37 The Figure shows (some) observational evidence for the ubiquitous existence of cold gas in outflows in star-forminggalaxies, and that this cold gas affects the Ly α transport: the right panel shows the vast majority of low-ionization interstellar(IS) absorption lines are blueshifted relative to the systemic velocity of the galaxy, which is indicative of outflows (as illustratedin the left panel . Moreover, the right panel illustrates that the Ly α emission line is typically redshifted by an amount thatis ∼ − in the same galaxies . These observations are consistent with ascenario in which Ly α photons scatter back to the observer from the far-side of the nebular region (indicated schematically inthe left panel ). Credit: figure as a whole corresponds to Figure 12 of Dijkstra 2014, Lyman Alpha Emitting Galaxies as a Probeof Reionization, PASA, 31, 40D . outflows are detected ubiquitously in absorption in other low-ionization transitions (e.g. Steidel et al., 2010).Moreover, the Ly α photons appear to interact with the outflow, as the Ly α line is redshifted by an amountthat is correlated with the outflow velocity inferred from low-ionization absorption lines (e.g. Steidel et al. 2010,Shibuya et al. 2014). The presence of winds and their impact on Ly α photons is illustrated schematically in Figure 37.As modelling the outflowing component in interstellar medium is an extremely challenging task (as we mentionedin 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 × its inner/outerradius. Figure 38 summarizes the different ingredients of the shell model. The two parameters that characterizethe Ly α sources are ( i ) its equivalent width (EW) which measures the ‘strength’ of the source compared to theunderlying continuum, ( ii ) its full width at half maximum (FWHM) which denotes the width of the spectral lineprior to scattering. This width may reflect motions in the Ly α emitting gas. The main properties that characterisethe shell are its ( i ) HI-column density, N HI , ( ii ) outflow velocity, v sh , ( iii ) ‘b-parameter’ b ≡ v + v . Here, v th = p k B T /m p (which we encountered before), and v turb denotes its turbulent velocity dispersion; ( iv ) its dustcontent (e.g. Ahn et al., 2003; Verhamme et al., 2006, 2008).The shell-model can reproduce observed Ly α spectral line shapes remarkably well (e.g. Verhamme et al. 2008,Schaerer & Verhamme 2008, Dessauges-Zavadsky et al. 2010, Vanzella et al 2010, Hashimoto et al.2015, Yang et al.2016), though not always (see e.g. Barnes & Haehnelt 2010, Kulas et al. 2012, Chonis et al. 2013, Forero-Romeroet al. in prep.). One example of a good shell model fit to an observed spectrum is shown in Figure 39. Here, thegalaxy is a z ∼ . triangle.py ,Foreman-Mackey et al. 2013) shows constraints on the 6 shell model parameters, and their correlations. ‘Typical’HI column densities in shell models are N HI = 10 − cm − and v sh ∼ a few tens to a few hundreds km s − .For a limited range of column densities, the Ly α spectrum peaks at ∼ v sh . This peak consists of photons that5 The Shell Model
Parameters source.shell v shell Parameters shell. , Ly α source FIG. 38 The ‘shell model’ is a simplified representation of the Ly α transfer process on interstellar scales. The shell modelcontains six parameters, and generally reproduces observed Ly α spectral line profiles remarkably well (see Fig 39). scatter ’back’ to the observer on the far side of the Ly α source, and are then Doppler boosted to twice the outflowvelocity , where they are sufficiently far in the wing of the absorption cross section to escape from the medium (thecross section at ∆ v = 200 km s − is only σ α ∼ a few times 10 − cm , see Eq 55 and Fig 17).In spite of its success, there are two issues with the shell-models: ( i ) gas in the shells has a single outflow velocityand a small superimposed velocity dispersion, while observations of low-ionization absorption lines indicate thatoutflows typically cover a much wider range of velocities (e.g. Kulas et al. 2012, Henry et al. 2015); and ( ii )observations of low-ionization absorption lines also suggest that outflows - while ubiquitous - do not completelysurround UV-continuum emitting regions of galaxies. Observations by Jones et al. (2013) show that the maximumlow-ionization covering fraction is 100% in only 2 out of 8 of their z > a ) the inferred covering factors are measured as a function ofvelocity (and can depend on spectral resolution, see e.g. Prochaska, 2006, but Jones et al. 2013 discuss why thisis likely not an issue in their analysis). Gas at different velocities can cover different parts of the source, and theoutflowing gas may still fully cover the UV emitting source. This velocity-dependent covering is nevertheless notcaptured by the shell-model; ( b ) the low-ionization 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 bymetal absorption lines.Shibuya et al. (2014) have shown that Ly α line emission is stronger in galaxies in which the covering factor oflow-ionization material is smaller (see their Fig 10, also see Trainor et al. 2015). Similarly, Jones et al. (2012) foundthe average absorption line strength in low-ionization species to decrease with redshift, which again coincides withan overall increase in Ly α flux from these galaxies (Stark et al., 2010). Besides dust, the covering factor of HI gastherefore plays an additional important role in the escape of Ly α photons. These cavities may correspond to regionsthat have been cleared of gas and dust by feedback processes (see Nestor et al., 2011, 2013, who describe a simple‘blow-out’ model). This argument implicitly assumes that the scattering is partially coherent (see § ν/ν ∼− v out / c when they enter the shell, and an identical Doppler boost ∆ ν/ν ∼ − v out / c when they exit the shell in opposite direction (as isthe case for ‘back scattered’ radiation). In the case of partially coherent scattering, the frequency of the photon changes only little inthe frame of the gas (because q h ∆ x | x in i = 1), and the total Doppler boost equals the sum of the two Doppler boosts imparted uponentry and exit from the shell. v exp (km s − ) = . +9 . − . . . . . . l og N H I / c m − log N HI / cm − = . +0 . − . . . . . l og T / K log T/ K = . +0 . − . σ i ( k m s − ) σ i (km s − ) = . +16 . − . . . . . . τ d τ d = . +0 . − .
80 120 160 200 v exp (km s − ) E W i () . . . . . log N HI / cm − . . . . log T/ K
200 225 250 275 σ i (km s − ) . . . . . τ d
12 18 24 30 EW i ( ) EW i ( ) = . +2 . − . v ( km/s) DataBest fit
FIG. 39 This figure shows an example of an observed Ly α spectral line shape of a z ∼ . ImageCredit: Max Gronke ). In short, dusty outflows appear to have an important impact on the interstellar Ly α radiative process, and giverise to redshifted Ly α lines. Low HI-column density holes further facilitate the escape of Ly α photons from the ISM,and can alter the emerging spectrum such that Ly α photons can emerge closer to the galaxies’ systemic velocities(Behrens et al. 2014, Verhamme et al. 2015, Gronke & Dijkstra 2016, Dijkstra et al. 2016b, also see Zheng & Wallace2014). HI gas that exists outside of the galaxy can further scatter Ly α that escaped from the ISM. If this scattering occurssufficiently close to the galaxy, then this radiation can be detected as a low surface brighteness glow (e.g. Zheng et al.,2010, 2011). As we showed previously in Figure 13, observations indicate that spatially extended Ly α halos appearto exist generally around star-forming galaxies (see e.g. Fynbo et al. 1999, Hayashino et al. 2004, Hayes et al. 2005,2007, Rauch et al. 2008, ¨Ostlin et al. 2009, Steidel et al. 2011, Matsuda et al. 2012, Hayes et al. 2013, Momoseet al. 2014, Guaita et al. 2015, Wisotzki et al., 2016, Xue et al. 2017). Understanding what fraction of the Ly α flux in these halos consists of scattered Ly α radiation that escaped from the ISM, and what fraction was produced7 FIG. 40 The average fraction of photons that are transmitted though the IGM to the observer as a function of (restframe)wavelength. Overdense gas in close proximity to the galaxy - this gas can be referred to as ‘circum-galactic gas’ - enhances theopacity in the forest at velocities close to systemic (i.e v sys = 0). Inflowing circum-galactic gas gives rise to a large IGM opacityeven at a range of velocities redward of the Ly α resonance. Each line represents a different redshift. At wavelengths well onthe blue side of the line, we recover the mean transmission measured from the Ly α forest. Overdense gas at close proximityto the galaxy increases the IGM opacity close to the Ly α resonance (and causes a dip in the transmission curve, Credit: fromFigure 2 of Laursen et al. 2011 c (cid:13) AAS. Reproduced with permission ). in-situ (as recombination, cooling, and/or fluorescence, see §
5) is still an open question , which we can address withintegral field spectographs such as MUSE and the Keck Cosmic Web Imager. Polarization measurements (see § ν > ν α and/or x > α resonance due to the Hubble expansion of the Universe. Oncethese photons are at resonance there is a finite probability to be scattered. These photons are clearly not destroyed,but they are removed from the intensity of the radiation pointed at us. For an observer on Earth, these photons areeffectively destroyed. Quantitatively, we can take ‘sufficiently far’ to mean that r IGM > ∼ . r vir , where r vir denotes thevirial radius of halos hosting dark matter halos (Laursen et al., 2011), and r IGM denotes the radius beyond whichscattered radiation is effectively removed from obsevations. Clearly, r IGM depends on the adopted surface brightnesssensitivity and the total Ly α luminosity of the source, and especially with sensitive MUSE observations, it will beimportant to verify what ‘ r IGM ’ is. At r > r
IGM ‘intergalactic’ radiative transfer consists of simply suppressing theemerging Ly α flux at frequency ν em by a factor of T IGM ( ν em ) ≡ e − τ IGM ( ν em ) , where τ IGM ( ν ) equals τ IGM ( ν em ) = Z ∞ r IGM ds n HI ( s ) σ α ( ν [ s, ν em ]) . (98)As photons propagate a proper differential distance ds , the cosmological expansion of the Universe redshifts thephotons by an amount dν = − dsH ( z ) ν/c . Photons that were initially emitted at ν em > ν α will thus redshift intothe line resonance. Because σ α ( ν ) is peaked sharply around ν α (see Fig 17), we can approximate this integral bytaking n HI ( s ) and c/ν ≈ λ α outside of the integral. We make an additional approximation and assume that n HI ( s ) Modeling the distribution of cold, neutral gas in the CGM is also challenging as there is increasing observational support that theCGM of galaxies contains cold, dense clumps (of unknown origin) on scales that are not resolved with current cosmological simulations(Cantalupo et al., 2014; Hennawi et al., 2015). n HI ( z ), where ¯ n HI ( z ) = Ω b h (1 − Y He )(1 + z ) /m p denotes the mean number density of hydrogenatoms in the Universe at redshift z . If we evaluate this expression at ν em > ν α , i.e. at frequencies blueward of theLy α resonance, then we obtain the famous Gunn-Peterson optical depth (Gunn & Peterson, 1965): τ IGM ( ν em > ν α ) ≡ τ GP = ¯ n HI ( z ) λ α H ( z ) Z ∞ dν σ α ( ν ) = ¯ n HI ( z ) λ α H ( z ) f α πe m e c ≈ . × (cid:16) z (cid:17) / , (99)where we used that R dν σ ( ν ) = f α πe m e c (e.g. Rybicki & Lightman, 1979, p 102). The redshift dependence of τ IGM reflects that n HI ( z ) ∝ (1+ z ) and that at z (cid:29) H ( z ) ∝ (1+ z ) / . Eq 99 indicates that if the IGM were 100% neutral,it would be extremely opaque to photons emitted blue-ward of the Ly α resonance. Observations of quasar absorptionline spectra indicate that the IGM transmits an average fraction F ∼
85% [ F ∼ α photons at z = 2[ z = 4] (see lecture notes by X. Prochaska) which imply ‘effective’ optical depths of τ eff ≡ − ln[ F ] ∼ .
15 [ τ eff ∼ . τ eff (cid:28) τ GP which is (of course) because the Universewas highly ionized at these redshifts. A common approach to model the impact of the IGM is to reduce the Ly α flux on the blue side of the Ly α resonance by this observed (average) amount, while transmitting all flux on the red side.The values of F and τ eff mentioned above are averaged over a range of frequencies. In detail, density fluctuationsin the IGM give rise to enhanced absorption in overdense regions which is observed as the Ly α forest. It is importantto stress that galaxies populate overdense regions of the Universe in which: ( i ) the gas density was likely higher thanaverage (see e.g. Fig 2 of Barkana, 2004), ( ii ) peculiar motions of gas attracted by the gravitational potential ofdark matter halos change the relation between ds and dν , ( iii ) the local ionising background was likely elevated. Wethus clearly expect the impact of the IGM at frequencies close to the Ly α emission line to differ from the meantransmission in the Ly α forest: Figure 40 shows the transmitted fraction of Ly α photons averaged over a large numberof sight lines to galaxies in a cosmological hydrodynamical simulation (Laursen et al. 2011). This Figure showsthat infall of over dense gas (and/or retarded Hubble flows) around dark matter halos hosting Ly α emitting galaxiescan give rise to an increased opacity of the IGM around the Ly α resonance, and even extending somewhat intothe red side of the Ly α line (Santos 2004, Dijkstra et al. 2007, Iliev et al. 2008, Laursen et al. 2011, Dayal et al. 2011).Because these models predict that the IGM can strongly affect frequencies close to the Ly α resonance, the overallimpact of the IGM depends strongly on the Ly α spectral line shape as it emerges from the galaxy (also see Haiman, 2002;Santos, 2004). This is illustrated by the lower three panels in Figure 41. For Gaussian and/or generally symmetricemission lines centered on the galaxies’ systemic velocities, the IGM can transmit as little as T IGM = 10 − That is, not only do we care about how much Ly α escapes from the dusty ISM, we must care as much about howthe emerging photons escape in terms of the line profile . The fraction T IGM (also indicated in Fig 41) denotes the‘IGM transmission’ and denotes the total fraction of the Ly α radiation emitted by a galaxy that is transmitted bythe IGM. The IGM transmission T IGM is given by the integral over the frequency-dependent transmission, e − τ IGM ( ν ) .This frequency-dependence can be expressed as a function of the dimensionless frequency variable x , or as a functionof the velocity offset ∆ v ≡ − xv th as: T IGM = Z ∞−∞ d ∆ v J α (∆ v ) exp[ − τ IGM ( z g , ∆ v )] (100)where J α (∆ v ) denotes the line profile of Ly α photons as they escape from the galaxy at z g . The IGM opacitydiscussed above originates in mildly over dense ( δ = 1 −
20, see Dijkstra et al. 2007), highly ionized gas. Anothersource of opacity is provided by Lyman-limit systems (LLSs) and Dampled Ly α absorbers (DLAs). The precise Early studies defined the IGM to be all gas at r > − . α photons would remove photons from a spectrum of a galaxy,and redistribute these photons over faint, spatially extended Ly α halos. It is worth noting that these models predict that the IGM can reduce the observed Ly α line by as much as ∼
30% between z = 5 . z = 6 . α halos around star forming galaxies provide hints that scattering in this CGM maybe more prevalent at z = 6 . z = 5 .
7, although the statistical significance of this claim is weak (Momose et al., 2014). transmitted fraction wavelengthblue red100% blue red blue red blue red `gaussian’ `outflow’ `inflow’ v sys FIG. 41 Schematic representation of the impact of a residual amount of intergalactic HI on the fraction of photons that isdirectly transmitted to the observer, T IGM for a set of different line profiles. The top panel shows that the Ly α forest (shownin the inset ) suppresses flux on the blue side of the Ly α line. The lower left figure shows that for lines centered on v sys = 0(here symmetric around v sys = 0 for simplicity), the IGM cuts off a significant fraction of the blue half of the line, and somefraction of the red half of the line. For lines that are redshifted [blueshifted] w.r.t v sys , larger [smaller] fraction of emitted Ly α photons falls outside of the range of velocities affected by the IGM. The line profiles set at the interstellar level thus plays a keyrole in the subsequent intergalactic radiative transfer process. Credit: from ‘Understanding the Epoch of Cosmic Reionization:Challenges and Progress’, Vol 423, Fig 3 of Chapter ’Constraining Reionization with Ly α Emitting Galaxies’ by Mark Dijkstra,2016, page 145-161, With permission of Springer . impact of these systems on Ly α radiation has only been studied recently (Bolton & Haehnelt 2013, Mesinger et al.2015, Choudhury et al. 2015, Kakiichi et al. 2016), and depends most strongly on how they cluster around Ly α emitting galaxies. Reionization refers to the transformation of the intergalactic medium from fully neutral to fully ionized. Forreviews on the Epoch of Reionization (EoR) we refer the reader to e.g. Barkana & Loeb (2001), Furlanetto etal. (2006), Morales & Wyithe (2010), and the recent book by Mesinger (2016). The EoR is characterized by theexistence of patches of diffuse neutral intergalactic gas, which provide an enormous source of opacity to Ly α photons:the Gunn-Peterson optical depth is τ GP ∼ (see Eq 99) in a fully neutral medium. It is therefore natural to expectthat detecting Ly α emitting galaxies from the EoR is hopeless. Fortunately, this is not the case, as we discuss in § FIG. 42 The predicted redshift evolution of the ionization state of the IGM in a realistic reionization model (
Credit: Figurekindly provided by Andrei Mesinger, published previously as Fig 14 in Dijkstra 2014, Lyman Alpha Emitting Galaxies as a Probeof Reionization, PASA, 31, 40D ). The white/black represents fully neutral/ionized intergalactic gas. This Figure demonstratesthe inhomogeneous nature of the reionization process which took place over an extended range of redshifts: at z >
16 the firstionized regions formed around the most massive galaxies in the Universe (at that time). During the final stages of reionization- here at z ∼ photons on cosmological hydrodynamical simulations. A number of groups have developed codes that can performthese calculations in 3D (e.g. Gnedin 2000, Sokasian et al. 2001, Ciardi et al. 2003, Mellema et al. 2006, Trac& Cen 2007, Pawlik & Schaye 2008, Finlator et al. 2009, So et al. 2014, Gnedin 2016). These calculations arecomputationally 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 ∼ M (cid:12) ) whichcan contribute to the ionising photon budget (see e.g. Trac & Gnedin, 2011). Modeling reionization containsmany poorly known parameters related to galaxy formation, the ionising emissivity of star-forming galaxies, theirspectra etc. Alternative, faster ‘semi-numeric’ algorithms have been developed which allow for a more efficientexploration of the full parameter space (e.g. Mesinger & Furlanetto 2007, Mesinger et al. 2011, Majumdar et al.2014, Sobacchi & Mesinger 2014). These semi-numeric algorithms utilize excursion-set theory to determine if a cellinside a simulation is ionized or not (Furlanetto et al., 2004). Detailed comparisons between full radiation transfersimulations and semi-numeric simulations show both methods produce very similar ionization fields (Zahn et al., 2011).The picture of reionization that has emerged from analytical consideration and large-scale simulations is one in whichthe early stages of reionization are characterized by the presence of HII bubbles centered on overdense regions of theUniverse, completely separated from each other by a neutral IGM (Furlanetto et al., 2004; Iliev et al., 2006; McQuinnet al., 2007). The ionized bubbles grew in time, driven by a steadily growing number of star-forming galaxies residinginside them. The final stages of reionization are characterized by the presence of large bubbles, whose individualsizes exceeded tens of cMpc (e.g. Zahn et al. 2011, Majumdar et al. 2014). Ultimately these bubbles overlapped(percolated), which completed the reionization process. The predicted redshift evolution of the ionization state of theIGM in a realistic reionization model is shown in Figure 42. This Figure illustrates the inhomogeneous, temporallyextended nature of the reionization process. α Radiative Transfer during Reionization
There are indications that we are seeing Ly α emission from galaxies in the reionization epoch: there is increasingobservational support for the claim that Ly α emission experiences extra opacity at z > left panel showsthe drop in the ‘ Ly α fraction ’ in the drop-out (Lyman Break) galaxy population. The ‘Ly α fraction’ denotes thefraction of continuum selected galaxies which have a ‘strong’ Ly α emission line (Stark et al. 2010). The ‘strength’ ofthe Ly α emission line is quantified by some arbitrary equivalent width threshold. The Ly α fraction rises out to z ∼ z > right panel shows the sudden evolution in the Ly α luminosity function of Ly α selected galaxies (LAEs). The Ly α luminosity function does not evolve much between z ∼ z ∼ z > ∼ z > ∼ . α luminosity function of LAEs andLy α fractions have been shown to be quantitatively consistent (see Dijkstra & Wyithe 2012, Gronke et al. 2015).During reionization we denote the opacity of the IGM in the ionized bubbles at velocity off-set ∆ v and redshift z with τ HII ( z, ∆ v ). This allows us to more explicitly distinguish this source of intergalactic opacity from the ‘damping wing’1 z~3 Ouchi et al. 2008z~3.7 Ouchi et al. 2008z~5.7 Ouchi et al. 2008 z~6.5 Ouchi et al. 2010z~7.0 Ota et al. 2010Compilation from Ono et al. 2012 FIG. 43 There are two independent observational indications that the Ly α flux from galaxies at z > left panel shows the drop in the ‘ Ly α fraction ’ in the drop-out (Lyman Break)galaxy population, the right panel shows the sudden evolution in the Ly α luminosity function of Ly α selected galaxies (LAEs).This evolution has been shown to be quantitatively consistent (see Dijkstra & Wyithe 2012, Gronke et al. 2015). Credit:from ‘Understanding the Epoch of Cosmic Reionization: Challenges and Progress’, Vol 423, Fig 1 of Chapter ’ConstrainingReionization with Ly α Emitting Galaxies’ by Mark Dijkstra, 2016, page 145-161, With permission of Springer . opacity, τ D ( z, ∆ v ), which is the opacity due to the diffuse neutral IGM and which is only relevant during reionization.In other words, τ HII refers to the Ly α opacity in neutral gas that survived in the ionized bubbles. This neutral gas canreside in dense self-shielding clouds , or as residual neutral hydrogen that survived in the ionized bubbles . Eq 100therefore changes to T IGM = Z ∞−∞ d ∆ v J α (∆ v ) exp[ − τ IGM ( z g , ∆ v )] , τ IGM ( z g , ∆ v ) = τ D ( z g , ∆ v ) | {z } reionization only + τ HII ( z g , ∆ v ) . (101)Decomposing τ IGM ( z g , ∆ v ) into τ D ( z g , ∆ v ) and τ HII ( z g , ∆ v ) is helpful as they depend differently on ∆ v : the left panel of Figure 44 shows the IGM transmission, e − τ D ( z g , ∆ v ) , as a function of velocity off-set (∆ v ) for the diffuse neutralIGM for x HI = 0 .
8. This Figure shows clearly that the neutral IGM affects a range of frequencies that extendsmuch further to the red-side of the Ly α resonance than the ionized IGM (compare with Fig 40). This large opacity τ D ( z g , ∆ v ) far redward of the Ly α resonance is due to the damping wing of the absorption cross-section (and not dueto gas motions at these large velocities), which is why we refer to it as the ‘damping wing optical depth’. The rightpanel shows that there is a contribution to the damping wing optical depth from neutral, self-shielding clouds (with N HI > ∼ cm − , also see the high-redshift curves in Fig 40) which can theoretically mimick the impact of a diffuseneutral IGM (Bolton & Haehnelt 2013), though this requires a large number density of these clouds (see Bolton &Haehnelt 2013, Mesinger et al. 2015, Choudhury et al. 2015, Kakiichi et al. 2016).Star-forming galaxies that are luminous enough to be detected with existing telescopes likely populated dark matterhalos with masses in excess of M > ∼ M (cid:12) (see e.g. Sobacchi & Mesinger 2015). These halos preferentially reside To make matters more confusing: self-shielding absorbers inside the ionized bubbles with sufficiently large HI column densities canbe optically thick in the Ly α damping wing, and can give rise to damping wing absorption as well. This damping wing absorption isincluded in τ HII ( z, ∆ v ). Just as the Ly α forest at lower redshifts - where hydrogen reionization was complete - contains neutral hydrogen gas with differentdensities, ionization states and column densities. Diffuse neutral patches v sys v sys Self-shielding systems
FIG. 44 Neutral gas in the intergalactic medium can give rise to a large ‘damping’ wing opacity τ D ( z g , ∆ v ) that extends farto the red side of the Ly α resonance, i.e. ∆ v (cid:29)
1. The left panel shows the IGM transmission, e − τ D ( z g , ∆ v ) , as a function ofvelocity off-set (∆ v ) for the diffuse neutral IGM for x HI = 0 .
8. The right panel shows the IGM transmission for a (large) numberof self-shielding clouds (see text). To obtain the total
IGM transmission, one should multiply the transmission curves shownhere and in Fig 40.
Credit: from ‘Understanding the Epoch of Cosmic Reionization: Challenges and Progress’, Vol 423, Fig 4of Chapter ’Constraining Reionization with Ly α Emitting Galaxies’ by Mark Dijkstra, 2016, page 145-161, With permission ofSpringer . Figures adapted from
Figures 2 and 4 of Mesinger et al. 2015, ‘Can the intergalactic medium cause a rapid drop inLy α emission at z > ?’, MNRAS, 446, 566 . in over dense regions of the Universe, which were reionized earliest. It is therefore likely that (Ly α emitting) galaxiespreferentially resided inside these large HII bubbles. This has an immediate implication for the visibility of the Ly α line. Ly α photons emitted by galaxies located inside these HII regions can propagate (to the extent that is permittedby the ionized IGM) - and therefore redshift away from line resonance - through the ionized IGM before encounteringthe neutral IGM. Because of the strong frequency-dependence of the Ly α absorption cross section, these photons areless likely to be scattered out of the line of sight inside the neutral IGM. A non-negligible fraction of Ly α photons maybe transmitted directly to the observer, which is illustrated schematically in Figure (45). Inhomogeneous reionizationthus enhances the prospect for detecting Ly α emission from galaxies inside HII bubbles (see Dijkstra, 2014, for a review,and an extensive list of references). It also implies that the impact of diffuse neutral intergalactic gas on the visibilityof Ly α flux from galaxy is more subtle than expected in models in which reionization proceeds homogeneously, andthat the observed reduction in Ly α flux from galaxies at z > α photons emitted by galaxiesthat lie outside of large HII bubbles, scatter repeatedly in the IGM. These photons diffuse outward, and are visibleonly as faint extended Ly α halos (Loeb & Rybicki 1999, Kobayashi et al. 2006, Jeeson-Daniel et al. 2012).We can quantify the impact of neutral intergalactic gas on the Ly α flux from galaxies following our analysis in § τ D . We first consider the simplest casein which a Ly α photon encounters one fully neutral patch which spans the line-of-sight coordinate from s b (‘b’ standsfor beginning) to s e (‘e’ stands for end): τ D ( ν ) = Z s e s b ds n HI ( s ) σ α ( ν [ s ]) = n HI ( s ) λ α H ( z ) Z ν e ( ν ) ν b ( ν ) dν σ α ( ν ) . (102)where we followed the analysis of § n HI ( s ) is constantacross this neutral patch. We eliminate n HI ( s ) by using the expression for the Gunn-Peterson optical depth in Eq 99,and recast Eq 102 as τ D ( ν ) = τ GP R ν e ( ν ) ν b ( ν ) dν σ α ( ν ) R ∞ dν σ α ( ν ) = τ GP R x e ( ν ) x b ( ν ) dx φ ( x ) R ∞ dx φ ( x ) . (103)The denominator can be viewed as a normalisation constant, and we can rewrite Eq 103 as τ D ( ν ) = τ GP √ π Z x e ( ν ) x b ( ν ) dx φ ( x ) , (104)where the factor of √ π enters because of our adopted normalisation for the Voigt profile φ ( x ).3 FIG. 45 This Figure schematically shows why inhomogeneous reionization boosts the visibility of Ly α emitting galaxies. Duringthe mid and late stages of reionization star-forming - and hence Ly α emitting - galaxies typically reside in large HII bubbles.Ly α photons emitted inside these HII bubbles can propagate - and redshift away from line resonance - through the ionized IGMbefore encountering the neutral IGM. The resulting reduced opacity of the neutral IGM (Eq 108) to Ly α photons enhances theprospect for detecting Ly α emission from those galaxies inside HII bubbles. Credit: from Figure 15 of Dijkstra 2014, LymanAlpha Emitting Galaxies as a Probe of Reionization, PASA, 31, 40D . During the EoR a Ly α photon emitted by a galaxy will generally propagate through regions that are alternatingbetween (partially) neutral and highly ionized. The more general case should therefore contain the sum of the opticaldepth in separate neutral patches: τ D ( ν ) = 1 √ π X i τ GP , i x HI , i Z x e,i ( ν ) x b,i ( ν ) dx φ ( x ) , (105)where we have placed τ GP within the sum, because τ GP depends on redshift as τ GP ∝ (1 + z i ) / , and therefore differsslightly for each patch ‘i’ at redshift z i , which has a neutral hydrogen fraction x HI , i .More specifically, the total optical depth of the neutral IGM to Ly α photons emitted by a galaxy at redshift z g with some velocity off-set ∆ v is given by Eq 105 with x b,i = − v th , i [∆ v + H ( z g ) R b,i / (1 + z g )], in which R b,i denotesthe comoving distance to the beginning of patch ‘i’ ( x e,i is defined similarly). Eq 105 must generally be evaluatednumerically. However, one can find intuitive approximations: for example, if we assume that ( i ) x HI , i = 1 for all ‘i’(i.e. all patches are fully neutral), ( ii ) z i ∼ z g , and ( iii ) that Ly α photons have redshifted away from resonance bythe time they encounter this first neutral patch , then τ D ( z g , ∆ v ) = τ GP ( z g ) √ π X i (cid:16) a v , i √ πx e,i − a v , i √ πx b,i (cid:17) ≡ τ GP ( z g ) √ π ¯ x D (cid:16) a v √ πx e − a v √ πx b, (cid:17) , (106)where x e,i = x e,i (∆ v ) and x b,i = x b,i (∆ v ). It is useful to explicitly highlight the sign-convention here: photons thatemerge redward of the Ly α resonance have ∆ v >
0, which corresponds to a negative x . Cosmological expansionredshifts photons further, which decreases x further. The a v / [ √ πx b,i ] is therefore less negative, and τ D is thus 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 whetherthe photon redshifted into resonance inside or outside of a neutral patch. If the photon redshifted into resonance inside patch ‘i’, then τ D ( z g , ∆ v ) = τ GP ( z ) x HI , i . If on the other hand the photon redshifted into resonance in an ionized bubble, then we must compute theoptical depth in the ionized patch, τ HII ( z, ∆ v = 0), plus the opacity due to subsequent neutral patches. Given that the ionized IGMat z = 6 . α flux on the blue-side of the line, the same likely occurs inside ionized HIIbubbles during reionization because of ( i ) the higher intergalactic gas density, and ( ii ) the shorter mean free path of ionizing photonsand therefore likely reduced ionizing background that permeates ionised HII bubbles at higher redshifts. x D , which is related to the volume fillingfactor of neutral hydrogen h x HI i in a non-trivial way (see Mesinger & Furlanetto 2008).Following Mesinger & Furlanetto (2008), we can ignore the term a v / [ √ πx e ] and write τ D ( z g , ∆ v ) ≈ τ GP ( z g ) π ¯ x D a v | x b, | = τ GP ( z g ) π ¯ x D A α c πν α v b, , (107)where x e denotes the frequency that photon has redshifted to when it exits from the last neutral patch, while x b, denotes the photon’s frequency when it encounters the first neutral patch. Because typically | x e | (cid:29) | x b, | we can dropthe term that includes x e . We further substituted the definition of the Voigt parameter a v = A α / (4 π ∆ ν α ), to rewrite x b, as a velocity off-set from line resonance when a photon first enters a neutral patch, ∆ v b, = ∆ v + H ( z g ) R b,i / (1+ z g ).Substituting numbers gives (Miralda-Escud´e 1998, Dijkstra & Wyithe 2010) τ D ( z g , ∆ v ) ≈ . x D (cid:16) ∆ v b,
600 km s − (cid:17) − (cid:16) z g (cid:17) / . (108)This equation shows that the opacity of the IGM drops dramatically once photons enter the first patch of neutralIGM with a redshift. This redshift can arise partly at the interstellar level, and partly at the intergalactic level:scattering off outflowing material at the interstellar level can efficiently redshift Ly α photons by a few hundredkm/s (see § α photons can undergo a larger cosmological subsequent redshift inside larger HIIbubbles, Ly α emitting galaxies inside larger HII bubbles may be more easily detected. Eq 108 shows that setting τ D = 1 for ¯ x D = x HI = 1 . v = 1380 km s − . This cosmological redshift reduces the damping wing opticaldepth of the neutral IGM to τ D < R > ∼ ∆ v/H ( z ) ∼ independent of z (because at a fixed R the corresponding cosmological redshift ∆ v ∝ H ( z ) ∝ (1 + z ) / , Miralda-Escud´e 1998). Thepresence of large HII bubbles during inhomogeneous reionization may have drastic implications for the prospects ofdetecting Ly α emission from the epoch.Current models indicate that if the observed reduction in Ly α flux from galaxies at z > h x HI i > ∼ z ∼ − α galaxies at z ∼ z ∼
7, but this situation is expected to change, especiallywith large surveys for high-z Ly α emitters to be conducted with Hyper Suprime-Cam. These surveys will enable usto measure the variation of IGM opacity on the sky at fixed redshift, and constrain the reionization morphology (seeJensen et al. 2013, 2014, Sobacchi & Mesinger 2015).
11. MISCALLENEOUS TOPICS I: POLARIZATION Ly α radiative transfer involves scattering. Scattered radiation can be polarized. The polarization of electromagneticradiation measures whether there is a preferred orientation of its electric and magnetic components. Consider theexample that we discussed in § k in ).In § I ∝ sin Ψ, where cos Ψ ≡ k out · e E in which e E denotes the normalized direction of the electric vector (see Fig 15). Similarly, we can say that the amplitude of theelectric-field scales as E ∝ sin Ψ (note at I ∝ | E | ), i.e. we project the electric vector onto the plane perpendicularto k out (see the left panel of Figure 46). This same argument can be applied to demonstrate that a free electron cantransform unpolarized into a polarized radiation if there is a ‘quadrupole anisotropy’ in the incoming intensity: the right panel of Figure 46 shows a free electron with incident radiation from the left and from the top. If the incident Scattering through an extremely opaque static medium gives rise to a spectrally broadened double-peaked Ly α spectrum (see Fig 26).Of course, photons in the red peak start with a redshift as well, which boosts their visibility especially for large N HI (see Fig 2 inDijkstra & Wyithe, 2010, also see Haiman 2002). FIG. 46 This Figure illustrates how scattering by a free electron can polarize radiation. Both panels illustrate how scatteringby a free electron transmits only the electric vector of the incident radiation, projected onto the outgoing radiation direction(also see Fig 15). radiation is unpolarized, then the electric field vector points in arbitrary direction in the plane perpendicular to thepropagation direction. Consider scattering by 90 ◦ . If we apply the projection argument, then for radiation incidentfrom the left we only ‘see’ the component of the E-field that points upward (shown in blue ). Similarly, for radiationcoming in from the top we only see the E-field that lies horizontally. The polarization of the scattered radiationvanishes if the blue and red components are identical, which - for unpolarized radiation - requires that the totalintensity 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 ◦ . Iffluctuation exist on these scales, then the CMB is said to have a non-zero quadrupole moment. Similarly, if there werea point source irradiating the electron from the top, then we would also expect only the red E-vector to be transmitted.The first detections of polarization in spatially extended Ly α sources have been reported (Hayes et al. 2011b,Humphrey et al. 2013, Beck et al. 2016). The left panel in Figure 47 shows Ly α polarization vectors overlaidon a Ly α surface brightness map of ‘LAB1’ (Ly α Blob 1, see Hayes et al. 2011b). The lines here denote thelinear polarization (see a more detailed discussion of this quantity below), which denotes the preferred orientationof the electric vector of the observed Ly α radiation. The longer the lines, the more polarized the radiation.Figure 47 shows how the polarization vectors appear to form concentric circles around spots of high Ly α surfacebrightness. This is consistent with a picture in which most Ly α was emitted in the spots with high surfacebrightness, and then scattered towards the observer at larger distance from these sites. This naturally gives rise tothe observed polarization pattern (also see the right panel of Fig 47 for an artistic illustration of this, from Bower 2011).The previous discussion can be condensed into a compact equation if we decompose the intensity of the radiationinto a component parallel and perpendicular to the scattering plane, which is spanned by the propagation directions k in and k out (see Fig 48). We write I ≡ I || + I ⊥ , with I || ≡ | e || | I , in which e || denotes the component of e E in thescattering plane. Similarly, we have I ⊥ ≡ | e ⊥ | I . We define the scattering matrix , R , as I I ! = S S S S ! ≡ R I || I ⊥ ! , (109)where the total outgoing intensity is given by I = I + I . The scattering matrix quantifies the angular redistributionof both components of a scattered electromagnetic wave. For comparison, the phase-function quantified the angularredistribution of the intensity total I only. For the case of a free electron, the scattering matrix R Ray is given by R dip = 32 cos θ
00 1 ! . (110)This expression indicates that for scattering by an angle θ , e = cos θ e || while e = e ⊥ (see Fig 48). The totaloutgoing intensity I ≡ I + I = (cos θI || + I ⊥ ), which for unpolarized incoming radiation ( I || = I ⊥ = 0 . I )6 FIG. 47
Left Panel : Lines indicate the magnitude and orientation of the linear polarization of Ly α radiation in Ly α blob1, overlaid on the observed Ly α surface brightness (in gray scale , ( Reprinted by permission from Macmillan Publishers Ltd:Hayes et al, 2011b, Nature 476, 304H, copyright ). Ly α polarization vectors form concentric circles around the most luminousLy α ‘spots’ in the map. This is consistent with a picture in which Ly α emission is produced in the locations of high surfacebrightness, and where lower surface brightness regions correspond to Ly α that was scattered back into the line-of-sight at largerdistance from the Ly α source. This process is illustrated visually in the right panel ( Reprinted by permission from MacmillanPublishers Ltd: Bower, 2011, Nature 476, 288B, copyright ). reduces to I = (1 + cos θ ) I . The ratio I /I corresponds to the phase-function for Rayleigh scattering encounteredin § linear polarization of the radiation is defined as P ≡ I ⊥ − I || I || + I ⊥ , (111)and for the scattered intensity of unpolarized radiation we find P = 1 − cos θ θ . (112)Note how this reflects our previous discussion: unpolarized radiation that is scattered by 90 ◦ becomes 100% linearlypolarized (also see the right panel of Fig 46). For comparison, the scattering matrix for an ‘isotropic’ scatterer is R iso = 12 ! . (113)That is, the outgoing intensity I = I = 0 . α scattering.An electron that is bound to a hydrogen atom is confined to orbits defined by quantum physics of the atom (seeFig 3 and § p state (in § p state splits up into several substates) the hydrogen atom is in, the electron may have some memory of thedirection and polarization of the incoming photon. In other words, the wavefunction of the electron in the 2 p statemay be aligned along the polarization vector of the incoming photon. It turns out that scattering of Ly α photons byhydrogen atoms can be described as some linear combination of dipole and isotropic scattering, which is described as7 θ e’ e e’ z y xe e’e k in k out FIG. 48 I || ( I ) denotes the intensity of the radiation parallel to the scattering plane before (after) scattering (spanned by k in and k out , indicated in green ). Furthermore, I ⊥ ( I ) denotes the intensity perpendicular to this plane. The intensities relateto the polarization vectors e ⊥ and e || as I || ≡ ( e || · e || ) I etc. Classically, an incoming photon accelerates an electron in thedirection of the polarization vector e . The oscillating electron radiates as a classical dipole, and the angular redistribution ofthe outgoing intensity I = I + I scales as I ∝ θ , where cos θ = k in · k out . For scattering by an angle θ , e = cos θ e || while e = cos θ . For a classical dipole, the intensity I is therefore reduced by a factor of cos θ relative to I (see text). ‘scattering by anisotropic particles’ (Chandrasekhar, 1960). The scattering matrix for anisotropic particles is givenby (Chandrasekhar, 1960, Eq 250-258) R = 32 E cos θ
00 1 ! + 12 E ! , (114)where E + E = 1. Precisely what E and E are is deteremined by the quantum numbers that describe the electronin the 2 p state. The number E gives the relative importance of dipole scattering, and is sometimes referred to as the‘ polarizability ’, as it effectively measures how efficiently a scatterer can polarize incoming radiation. Both the angularredistribution - or the phase function - and the polarization of scattered Ly α radiation can therefore be characterizedentirely by this single number E , which is discussed in more detail next. α Scattering: The Polarizability of the Hydrogen Atom
In order to accurately describe how H atoms scatter Ly α radiation, we must consider the fine-structure splittingof the 2 p level. The spin of the electron causes the 2 p state quantum state to split into the 2 p / and 2 p / levels,which are separated by ∼
10 Ghz (see Fig. 49). The notation that is used here is nL J , in which J = L + s denotesthe total (orbital +spin) angular momentum of the electron. The 1 s / → p / and 1 s / → p / is often referredas the K-line and H-line, respectively.It turns out that a quantum mechanical calculation yields that E = for the H transition, while E = 0for the K transition (e.g. Hamilton 1947, Brandt & Chamberlain 1959, Lee et al. 1994, Ahn et al. 2002).When a Ly α scattering event goes through the K-transition, the hydrogen atom behaves like an isotropic scat-terer. This is because the wavefunction of the 2 p / state is spherically symmetric (see White 1934), and theatom ‘forgot’ which direction the photon came from or which direction the electric field was pointing to. Forthe 2 p / state, the wavefunction is not spherically symmetric and contains the characteristic ‘double lobes’shown in Figure 3. The hydrogen in the 2 p / state thus has some memory of the direction of the incoming Ly α photon and its electric vector, and behaves partially as a classical dipole scatterer, and partly as an isotropic scatterer.However, in reality the situation is more complex and Stenflo (1980) showed that E depends strongly on frequencyas E = (cid:0) ω K ω H (cid:1) ( b H + d H ) + 2 ω K ω H ( b H b K + d H d K ) b K + d K + 2 (cid:0) ω K ω H (cid:1) ( b H + d H ) , (115)8 H 1S
F=0F=1F=0F=1F=1F=2HFSFS K K H
FIG. 49 Schematic diagram of the energy levels of a hydrogen atom. The notation for each level is nL J , where n is the principlequantum number, and L denotes the orbital angular momentum number, and J the total angular momentum (‘total’ angularmomentum means orbital+spin angular momentum, i.e. J = L + S ). The fine structure splitting of the 2 p level shifts the 2 p / and 2 p / -level by ∆ E/h p ∼
11 Ghz (e.g. Brasken & Kyrola, 1998). A transition of the form 1 s / → p / (1 s / → p / ) isdenoted by a K (H) transition. The spin of the nucleus induces further ’hyperfine’ splitting of the line. This is illustrated in the right panel , where each fine structure level breaks up into two lines which differ only in their quantum number F which measuresthe total + nuclear spin angular momentum, i.e. F = J + I . Hyperfine splitting ultimately breaks up the Ly α transition intosix individual transitions. Fine and hyperfine structure splitting plays an important role in polarizing scattered Ly α radiation. where b H,K = ω − ω H,K and d H,K = ω H,K Γ H,K . Here, ω H = 2 πν H ( ω K = 2 πν K ) denotes the resonant angularfrequency of the H (K) transition, and Γ H,K = A α . The frequency dependence of E is shown in Figure 50, wherewe have plotted E as a function of wavelength λ . The black solid line shows E that is given by Eq 115. This plotshows that1. E = 0 at λ = λ K and that E = at λ = λ H , which agrees with earlier studies (e.g. Hamilton 1947, Brandt& Chamberlain 1959), and which reflects what we discussed above.2. E is negative for most wavelengths in the range λ H < λ < λ K . The classical analogue to this would be that whenan atom absorbs a photon at this frequency, that then the electron oscillates along the propagation directionof the incoming wave, which is strange because the electron would be oscillating in a direction orthogonal tothe direction of the electric vector of the electromagnetic wave. However, scattering at these frequencies is veryunlikely (see § E = 1 when a photon scatters in the wings of the line, which is arguably the most bizarre aspect of this plot.Stenflo (1980) points out that, when a Ly α photon scatters in the wing of the line profile, it goes simultaneouslythrough the 2 p / and 2 p / states, and as a result, the bound electron is permitted to behave as if it were free.The impact of ‘quantum interference’ on the scattering phase function is more than just interesting from a ‘funda-mental’ viewpoint, because it makes a physical distinction between ‘core’ and ‘wing’ scattering. This distinction arisesbecause E affects the scattering phase function: the phase function for wing scattering P wing ∝ (1 + µ ), while forcore scattering the phase function can behave like a (sometimes strange, i.e. when E <
0) superposition of isotropicand wing scattering (see Eq 117 for an example of how to compute this phase function from E ).For reference: there is hyperfine splitting in the fine structure lines that is induced by the spin of the proton whichcan couple to the electron spin, which induces further splitting of the line. This is illustrated in Figure 49, where eachfine structure level breaks up into two lines which differ only in their quantum number F which measures the total +nuclear spin angular momentum, i.e. F = J + I . The dependence of E on frequency when this hyperfine splitting isaccounted for has been calculated by Hirata (2006, his Appendix B. Note however, that E is not computed explicitly.Instead, the angular redistribution functions are given and it is possible to extract E from these). The formula for E is quite lengthy, and the reader is referred to Hirata (2006) for the full expression. We have overplotted as the reddotted line wavelength dependence of E when hyperfine splitting is accounted for. The general frequency dependence9 FIG. 50 The frequency dependence of the polarizability E of a hydrogen atom is shown. The black solid line shows E as afunction of λ when fine structure splitting is accounted for (Eq 115, taken from Stenflo 1980). This plot shows that E = 0 at λ = λ K and that E = at λ = λ H . Interestingly, E is negative for most wavelengths in the range λ H < λ < λ K . However,scattering at these frequencies is very unlikely (see § E = 1 when a photon scatters in the wings ofthe line. That is, for photons that are scattered in the wing of the line profile, the electron in the hydrogen atom behaves like aclassical dipole (i.e. as if it were a free electron!). The red dotted line shows the wavelength dependence of E when hyperfinesplitting is accounted for. Hyperfine splitting introduces an overall slightly higher level of polarization. The box in the lowerleft corner shows a close-up view of E near the resonances (see text). of E is not affected. However, there is an overall higher level of polarization. The box in the lower left corner ofFigure 50 shows a close-up view of E near the resonances. Around the H-resonance, hyperfine splitting introducesnew resonances (indicated by the blue dotted lines ) which affect E somewhat. At wavelengths longward of the Hresonance, the E is boosted slightly, which causes scattering through the K resonance to not be perfectly isotropic(another interpretation is that the hyperfine splitting breaks the perfect spherical symmetry of the 2 p / state). Anoverall boost in the polarizability as a result of hyperfine splitting has also been found by other authors (e.g. Brasken& Kyrola, 1998). α Propagation through HI: Resonant vs Wing Scattering
We highlighted the distinction between ’core’ versus ’wing’ scattering previously in §
11. As we already mentioned,thd polarizability can be negative E < γ H , K ≡ Γ H , K / [4 π ] ∼ Hz (cid:28) ν H − ν K = 1 . × Hz, and the absorption cross–section in the atom’srest-frame scales as, σ ( ν ) ∝ [( ν − ν H , K ) + γ , K ] − (see Eq 48). A Ly α photon is therefore much more likely to beabsorbed by an atom for which the photon appears exactly at resonance, than by an atom for which the photon has afrequency corresponding to a negative E . Quantitatively, the Maxwellian probability P that a photon of frequency x is scattered in the frequency range x at ± dx at / P ( x at | x ) dx at = aπH ( a v , x ) e − ( x at − x ) x + a v dx at , (116)where we used that P ( x at | x ) = P ( u at | x ) = P ( x | u at ) P ( u at ) /P ( x ), where u at = x − x at denotes the atom velocity inunits of v th (also see the discussion above Eq 69). The probability P ( x at | x ) is shown in Figure 51 for x = 3 . solidline ) and x = − . dashed line ). Figure 51 shows that for x = 3 .
3, photons are scattered either when they are exactlyat resonance or when they appear ∼ x at = 0 .
0. For frequencies x < ∼ α photon Note that the transition from core to wing scattering occurs at x ∼
3, see Fig 17. FIG. 51 The probability P ( x at , x ) (Eq. 116) that a photon of frequency x is scattered by an atom such that it appears at afrequency x at in the frame of the atom ( Credit: from Figure A2 of Dijkstra & Loeb. 2008, ‘The polarization of scattered Ly α radiation around high-redshift galaxies’, MNRAS, 386, 492D ). The solid and dashed lines correspond to x = 3 . x = − . x = 3 .
3, photons are either scattered by atoms to which they appear exactly at resonance (see inset), or towhich they appear ∼ x = −
5, resonant scattering is less important by orders of magnitude. Incombination with Fig 50, this figure shows that if a photon is resonantly scattered then E is either 0 or . appears at exactly at resonance in the frame of the atom. However, this is not the case any more for frequencies | x | > ∼
3. Instead, the majority of photons is scattered while it is in the wing of the absorption profile. This is illustratedby the dashed line which shows the case x = − .
0, for which resonant scattering is less likely by orders of magnitude.This discussion illustrates that the transition from ‘core’ to ‘wing’ scattering is continuous (though it occurs over anarrow range of frequencies). Photons can ‘resonantly’ scatter while they are in the wing with a finite probability,and vice versa. While in practise, this is not an important effect, it is worth keeping in mind.
Incorporating polarization in a Monte-Carlo is complicated if you want to do it correctly. A simple procedure waspresented by Angel (1969) and Rybicki & Loeb (1999), which is accurate for wing-scattering only. In practise this isoften sufficient, as Ly α wing photons are the ones that are most likely to escape from a scattering medium, and arethus most likely to be observed.Rybicki & Loeb (1999) assigned 100% linear polarization to each Ly α photon used in the Monte-Carlo simulationby attaching an (normalized) electric vector e E to the photon perpendicular to its propagation direction (seediscussion above Eq 37). For each scattering event, the outgoing direction ( k out ) is drawn from the phase-function P ∝ sin Ψ (see the discussion above Eq 37), where cos Ψ = k out · e (see Angel 1969, Rybicki & Loeb 1999, Dijkstra &Loeb 2008b for technical details). This approach thus accounts for the polarization-dependence of the phase functiondescribed previously in § α photon in the Monte-Carlo simulation that was propagating in a direction k in , and with an electricvector e E . It was then scattered towards the observer. The polarization of this photon when it is observed can beobtained as follows: we know that when a photon reaches us, its propagation direction, k out , is perpendicular to theplane of the sky. We therefore know that the polarization vector e E must lie in the plane of the sky. The linearpolarization measures the difference in intensity when measured in the two orthogonal directions in the plane ofthe sky (denoted previously with I || and I ⊥ ). We now define r to be the vector that connects the location of lastscattering to the Ly α source, projected onto the sky. Both r and e E therefore lie in the plane of the sky, and we let χ denote the angle between them (i.e. cos χ ≡ r · e E / | r | ). The photon then contributes cos χ to I l and sin χ to I r .This geometry is depicted in Figure 52. This procedure was tested successfully by Rybicki & Loeb (1999) againstanalytic solutions obtained by Schuster (1879).For core scattering the situation is more complicated. We know that scattering through the H -transition corresponds1 FIG. 52 This figure illustrates visually what the ‘polarization’ angle χ is, which can be used in Monte-Carlo calculations tocompute the linear polarization of scattered Ly α as a function of projected distance r ≡ | r | from the galaxy/Ly α source. to isotropic scattering, while scattering through the K -transition corresponds to scattering off an anisotropic particlewith E = 0 .
5. From their statistical weights we can infer that scattering through the K -transition is twice is likely asscattering through the H-transition ( g H = 2, g K = 4). The weighted average implies that core scattering correspondsto scattering by an anisotropic particle with E = 1 /
3. Eq 114 shows that the scattering matrix and phase functiontake on the following form: R = cos θ +
13 1313 56 ! ⇒ P ( µ ) = 1112 + 312 µ , (117)for unpolarized incident radiation. Formally, the frequency redistribution function for core scattering is thereforeneither given by R A ( x out | x in ) nor R B ( x out | x in ) (Eq 73), but by some intermediate form. Given the similarity of R A ( x out | x in ) and R B ( x out | x in ) (see Fig 21) this difference does not matter in practice. Note that this phase functioncan be implemented naturally in a Monte-Carlo simulation by treating 1 / / α photon in the Monte-Carlo simulation, and thereforeimplicitly assumes that each individual Ly α photon is 100% linearly polarized. It would be more realistic if we couldassign a fractional polarization to each photon, which would be more representative of the radiation field. Recall thatthe phase-functions depend on the polarization of the radiation field. An alternative way of incorporating polarizationwhich allows fractional polarization to be assigned to individual Ly α photons is given by the density-matrix formalism described in Ahn & Lee (2015). In this formalism all polarization information is encoded in 2 parameters (the 2parameters reflect the degrees of freedom for a mass-less spin-0 ‘particle’) within the ‘density matrix’. We will notdiscuss this formalism in this lecture. Both methods should converge for scattering in optically thick gas, but theyhave not been compared systematically yet (but see Chang et al. 2017 for recent work in this direction).
12. APPLICATIONS BEYOND LY α : WOUTHUYSEN-FIELD COUPLING AND 21-CM COSMOLOGY/ASTROPHYSICS12.1. The 21-cm Transition and its Spin Temperature Detecting the redshifted 21-cm line from neutral hydrogen gas in the young Universe is one of the main challengesof observational cosmology for the next decades, and serves as the science driver for many low frequency arraysthat were listed in § s ) of atomichydrogen. The energy difference arises due to coupling of the proton to the electron spin: the proton spin S p givesit a magnetic moment µ p = g p e ¯ h m p c S p , where the proton’s ‘g-factor’ is g p ∼ .
59. This magnetic dipole generates a A spinning proton can be seen as a rotating charged sphere, which produces a magnetic field.
1S 1 S S λ =21cm r FIG. 53 This figure illustrates the classical picture of the origin of the energy difference between the two hyperfine levels of theground state of atomic hydrogen. The proton spin generates a magnetic dipole moment which in turn generates a magneticfield. This magnetic field introduces an energy dependence to the orientation of the electron spin. In quantum mechanicshowever, the two states shown here have equal energy, unless the electron finds itself inside the proton (i.e unless r = 0). magnetic field which interacts with the magnetic moment of the electron ( µ e ) due to its spin. Classically, the energydifference between the two opposite electron spin states equals ∆ E = 2 | µ e || B p | , where B p denotes the magnetic fieldgenerated by the spinning proton. This is illustrated schematically in Figure 53. The 21-cm transition correspondsto the electron’s spin ‘flipping’ in the magnetic field generated by the proton. The 21-cm transition is therefore oftenreferred to as the ‘spin-flip’ transition. What is interesting about this classical picture is that it fails to convey thatquantum mechanically, the energy difference between the two hyperfine levels of the 1 s level is actually zero, unless r =0. In quantum mechanics, there is a finite probability that the electron finds itself inside the proton. Formally,this is what causes the different hyperfine levels of the ground state of atomic hydrogen to have different energies(special thanks to D. Spiegel for pointing this out to me).The 21-cm transition is a highly forbidden transition with a natural life-time of t ≡ A − ∼ (2 . × − s − ) − ∼ . × yr (one way to interpret this long lifetime is to connect it to the low probability of the electron and protonoverlapping). The 21-cm line has been observed routinely in nearby galaxies, and has allowed us to map out thedistribution & kinematics of HI gas in galaxies. Observations of the kinematics of HI gas have given us galaxyrotation 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 > ∼ . It is theoretically possible to detect the 21-cm line from neutral gas during the reionization epoch and even the‘Dark Ages’, which refers to the epoch prior to the formation of the first stars, black holes, etc., and when the onlysource of radiation was the Cosmic Microwave Background which had redshifted out of the visual band and into theinfrared. The visibility of HI gas in its 21-cm line can be expressed in terms of a differential brightness temperature , δT b ( ν ), with respect to the background CMB, and equals (e.g. Furlanetto et al. 2006, Morales & Wyithe 2010, andreferences therein): δT b ( ν ) ≈ x HI (1 + δ )(1 + z ) / (cid:18) − T CMB ( z ) T S (cid:19) (cid:20) H ( z )(1 + z ) dv || /dr (cid:21) mK , (118)where δ + 1 ≡ ρ/ ¯ ρ denotes the overdensity of the gas cloud, z the redshift of the cloud, T CMB ( z ) = 2 . z ) Kdenotes the temperature of the CMB, the factor in square brackets contains the line-of-sight velocity gradient dv || /dr .Finally, T S denotes the spin temperature (also known as the excitation temperature) of the 21-cm transition, which Eq 118 follows from solving the radiative transfer equation, dIdτ = − I + A B n /n − , in the (appropriate) limit that the neutralIGM is optically thin in the HI 21-cm line, and that it therefore only slightly modifies the intensity I of the background CMB. It iscommon in radio astronomy to express intensity fluctuations as temperature fluctuations by recasting intensity as a temperature in theRayleigh-Jeans limit: I ν ( x ) ≡ k B T ( x ) λ . The Wouthuysen-Field Effect S S λ =21cm Ly α absorption FIG. 54 Ly α photons that have a higher energy are (slightly) more likely to be absorbed from the 1 S / level because ofthe (again slightly) larger energy separation between this level and any n = 2 level. The precise spectrum of Ly α photonsaround the line resonance therefore affects n (number density of hydrogen atoms in the 1 S / level) and n (number densityof hydrogen atoms in the 1 S / level). Repeated scattering of Ly α photons in turn modifies the spectrum in such as a waythat Ly α scattering drives the spin temperature to the gas temperature. This is known as the Wouthuysen-Field effect. quantifies the number densities of hydrogen atoms in each of the hyperfine transitions, i.e. n n ≡ g g exp (cid:18) − hν k B T s (cid:19) = 3 exp (cid:18) − hν k B T s (cid:19) , (119)where n ( n ) denotes the number densities of hydrogen atoms in the ground (excited) level of the 21-cmtransition, and where we used that g = 3 and g = 1. The numerical prefactor of 9 mK in Eq 118 was de-rived assuming that the gas was undergoing Hubble expansion. For slower expansion rates, we increase the numberof hydrogen atoms within a fixed velocity (and therefore frequency) range, which enhances the brightness temperature.Eq 118 states that when T CMB ( z ) < T S , we have δT b ( ν ) >
0, and when T CMB ( z ) > T S we have δT b ( ν ) <
0. Thismeans that we see HI in absorption [emission] when T S < T CMB ( z ) [ T S > T CMB ( z )]. The spin temperature thus playsa key role in setting the 21-cm signal, and we briefly discuss what physical processes set T s below in § The spin temperature - i.e. how the two hyperfine levels are populated - is set by ( i ) collisions, which drive T S → T gas , ( ii ) absorption by CMB photons, which drives T S → T CMB (and thus that δT b →
0, see Eq 118), and ( iii )Ly α scattering. This is illustrated in Figure 54: absorption of Ly α photons can occur from any of the two hyperfinelevels. The subsequent radiative cascade back to the ground state can leave the atom in either hyperfine level. Ly α scattering thus mixes the two hyperfine levels, which drive T S → T α , where T α is known as the Ly α color temperature,which will be discussed below. Quantitatively, it has been shown that (e.g. Madau et al. 1997, Tozzi et al. 2000,Furlanetto et al. 2006) 1 T S = T − + x c T − + x α T − α x c + x α , (120)4
10 100 1000redshift redshift 61020
FIG. 55
Left panel:
Globally, averaged redshift evolution of T gas ( red solid line ) and the T CMB ( blue solid line ). Right panel:The ‘Global’ 21-signal, represents the sky-averaged 21-cm brightness temperature δT b ( ν ). See the main text for a descriptionof each of these curves. The global 21-cm signal constrains when the first stars, black holes, and galaxies formed and dependson their spectra. Credit: adapted from slides created by J. Pritchard (based on Pritchard & Loeb 2010, 2012) . where x c = C T ∗ A T gas denotes the collisional coupling coefficient, in which k B T ∗ denotes the energy difference betweenthe hyperfine levels, C denotes the collisional deexcitation rate coefficient. Furthermore, x α = P α A T α , in which P α denotes the Ly α scattering rate . The Ly α color temperature provides a measure of the shape of the spectrum nearthe Ly α resonance (Meiksin 2006, Dijkstra & Loeb 2008c): k B T α h P = R J ( ν ) σ α ( ν ) dν R ∂σ α ∂ν J ( ν ) dν = − R J ( ν ) σ α ( ν ) dν R ∂J∂ν σ α ( ν ) dν , (121)where σ α ( ν ) denotes the Ly α absorption cross-section (see Eq 55), and J ( ν ) denotes the angle-averaged intensity(see § α enters is illustrated inFig 54, which shows that higher frequency Ly α photons are (slightly) more likely to excite hydrogen atoms from theground (singlet) state of the 21-cm transition. The relative number of Ly α photons slightly redward and blueward ofthe resonance therefore affects the 21-cm spin temperature (Wouthuysen, 1952). Eq 121 indicates that if ∂J ( ν ) ∂ν < T α >
0. Interestingly, it has been demonstrated that repeated scattering of Ly α photons changes the Ly α spectrum around the resonance such that it drives T α → T gas (Field 1959). The resulting coupling between T s and T gas as a result of Ly α scattering is known as the ‘Wouthuysen-Field (WF) coupling. We generally expect T s tobe some weighted average of the gas and CMB temperature, where the precise weight depends on various quantitiessuch as gas temperature, density and the WF-coupling strength.There are two key ingredients in WF-coupling: The Ly α color temperature T α and the Ly α scattering rate. Theo-retically, Ly α scattering rates are boosted in close proximity to star forming galaxies because of the locally enhancedLy α background (Chuzhoy & Zheng 2007). However, this local boost in the WF-coupling strength can be furtheraffected by the assumed spectrum of photons emerging from the galaxy (galaxies themselves are optically thick toLy α and higher Lyman series radiation), which has not been explored yet. Eq 120 implies that the spin temperature T s is a weighted average of the gas and CMB temperature. The left panel of Figure 55 shows the universally (or globally) averaged temperatures of the gas ( red solid line ) and the CMB ( blue Note that this equation uses that for each Ly α scattering event, the probability that it induces a scattering event is P flip = (see e.g.Meiksin 2006, Dijkstra & Loeb 2008c). This probability reduces by many orders of magnitude for wing scattering as P flip ∝ x − (seeHirata 2006, Dijkstra & Loeb 2008c). In practise wing scattering contributes little to the overall scattering rate, but it is good to keepthis in mind. See Furlanetto et al. 2006 for an explanation of how to best pronounce ‘Wouthuysen’.
Hint: it helps if you hold your breath 12 secondsbefore trying. solid line ), and the corresponding ‘global’ 21-cm signature in the Right panel . We discuss these in more detail below: • Adiabatic expansion of the Universe causes T CMB ∝ (1 + z ) at all redshifts. At z > ∼ T CMB = T gas , we must have T s = T CMB ,and therefore that δT b ( ν ) = 0 mK, which corresponds to the high- z limit in the right panel . • At z < ∼ T gas ∝ (1 + z ) . Because T gas < T CMB , we must have that T s < T CMB and we see the 21-cm line in absorption. When T gas first decouples from T CMB the gas densities arehigh enough for collisions to keep T s locked to T gas . However, at z ∼
70 ( ν ∼
20 MHz) collisions can no longercouple T s to T gas , and T s crawls back to T CMB , which reduces δT b ( ν ) (at ν ∼ −
50 MHz, i.e. z ∼ − • The first stars, galaxies, and accreting black holes emitted UV photons in the energy range E = 10 . − . α . The formation of the first stars thus generatesa Ly α background, which initiates the WF-coupling, which pushes T s back down to T gas . The onset of Ly α scattering - and thus the WF coupling - causes δT b ( ν ) to drop sharply at ν > ∼
50 Mhz ( z < ∼ • At some point the radiation of stars, galaxies, and black holes starts heating the gas. Especially X-rays producedby accreting black holes can easily penetrate deep into the cold, neutral IGM and contain a lot of energy whichcan be converted into heat after they are absorbed. The left panel thus has T gas increase at z ∼
20, whichcorresponds to onset of X-ray heating. In the right panel this onset occurs a bit earlier. This difference reflectsthat the redshift of all the features (minima and maxima) in δT b ( ν ) are model dependent, and not well-known(more on this below). With the onset of X-ray heating (combined with increasingly efficient WF coupling tothe build-up of the Ly α background) drives δT b ( ν ) up until it becomes positive when T gas > T CMB . • Finally, δT b ( ν ) reaches yet another maximum, which reflects that neutral, X-ray heated gas is reionized awayby the ionizing UV-photons emitted by star forming galaxies and quasars. When reionization is complete, thereis no diffuse intergalactic neutral hydrogen left, and δT b ( ν ) → α coupling, X-ray heating, and reionization depend on the redshiftevolution of the number densities of galaxies, and their spectral characteristics. All these are uncertain, and it is notpossible to make robust predictions for the precise shape of the global 21-cm signature. Instead, one of the mainchallenges for observational cosmology is to measure the global 21-cm signal, and from this constrain the abundancesand characteristic of first generations of galaxies in our Universe. Detecting the global 21-cm is challenging, butespecially the deep absorption trough that is expected to exist just prior to the onset of X-ray heating at ν ∼ α background, which must be strong enough toenable WF-coupling. Acknowledgments
I thank Ivy Wong for providing a figure for a section which (unfortunately) was cut as a whole,Daniel Mortlock & Chris Hirata for providing me with tabulated values of their calculations which I used to createFig 17, Jonathan Pritchard for permission to use one of his slides for these notes, Andrei Mesinger for providing Fig 42,and other colleagues for their permission to re-use Figures from their papers. I thank the astronomy department atUCSB for their kind hospitality when I was working on preparing these lecture notes. I thank the organizers of theschool: 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 & enthusiastic participation.67
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