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Chapter 1

Theoretical Framework: The Fundamentalsof the 21 cm Line

Steven R Furlanetto

We review some of the fundamental physics necessary for computing the highly-redshifted spin-flip background. We first discuss the radiative transfer of the 21 cmline and define the crucial quantities of interest. We then review the processes that setthe spin temperature of the transition, with a particular focus on Wouthuysen–Fieldcoupling, which is likely to be the most important process during and after theCosmic Dawn. Finally, we discuss processes that heat the intergalactic mediumduring the Cosmic Dawn, including the scattering of Lyα, cosmic microwavebackground (CMB), and X-ray photons.

1.1 Radiative Transfer of the 21 cm LineConsider a spectral line labeled 0 (the lower level) and 1 (the upper level). Theradiative transfer equation for the specific intensity Iν of photons at the relevantfrequency is

ϕ ν νπ

= − −νν

dIdℓ

hn A n B n B I

( )4

[ ( ) ], (1.1)1 10 0 01 1 10

where dℓ is a proper path length element, ϕ ν( ) is the line profile function, ni denotesthe number density of atoms at the different levels, and Aij and Bij are the Einsteincoefficients for the relevant transition (here, i and j are the initial and final states,respectively). For the 21 cm line, the line frequency is ν = 1420.4057 MHz21 . TheEinstein relations associate the radiative transition rates via =B g g B( / )10 0 1 01 and

ν=B A c h( /2 )10 102 3 , where g is the spin degeneracy factor of each state. For the 21 cm

transition, = × − −A 2.85 10 s1015 1 and =g g/ 31 0 .

The relative populations of hydrogen atoms in the two spin states determine thespin temperature, TS, through the relation

doi:10.1088/2514-3433/ab4a73ch1 1-1 ª IOP Publishing Ltd 2020

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎧⎨⎩⎫⎬⎭= − *n

n

g

gT

Texp , (1.2)1

0

1

0 S

where ≡ =*T E k/ 6810 B mK is equivalent to the transition energy E10. In almost allphysically plausible situations, ⋆T is much smaller than any other temperature,including TS, so all exponentials in temperature can be Taylor-expanded to leadingorder with high accuracy. Note, however, that TS implicitly assumes that the levelpopulations can be described by a single temperature—independent of each atom’svelocity. In detail, velocity-dependent effects must be considered in certain circum-stances (Hirata & Sigurdson 2007).

It is conventional to replace νI by the equivalent brightness temperature, νT ( )b ,required of a blackbody radiator (with spectrum νB ) such that =ν νI B T( )b . In thelow-frequency regime relevant to the 21 cm line, the Rayleigh–Jeans formula is anexcellent approximation to the Planck curve, so ν ν≈ νT I c k( ) /2b

2B

2.In this limit, the equation of radiative transfer along a line of sight through a

cloud of uniform excitation temperature TS becomes

ν ν′ = − + ′τ τ− −ν νT T e T e( ) (1 ) ( ) , (1.3)b S R

where ′ νT ( )b is the emergent brightness measured at the cloud and at redshift z, theoptical depth ∫τ α≡ν νds is the integral of the absorption coefficient (αν) along theray through the cloud, ′TR is the brightness of the background radiation field incidenton the cloud along the ray, and s is the proper distance. Because of the cosmologicalredshift, for the 21 cm transition an observer will measure an apparent brightness atEarth of ′ν ν= +T T z( ) ( )/(1 )b b 21 , where the observed frequency is ν ν= + z/(1 )21 .Henceforth we will work in terms of these observed quantities.

The absorption coefficient is related to the Einstein coefficients via

α ϕ ν νπ

= −hn B n B( )

4( ). (1.4)0 01 1 10

Because all astrophysical applications have ≫ *T TS , approximately three of fouratoms find themselves in the excited state ( ≈n n /30 1 ). As a result, the stimulatedemission correction represented by the first term is significant.

The fundamental observable quantity is the change in brightness temperatureinduced by the 21 cm line by a patch of the intergalactic medium (IGM), relative tothe incident radiation field. In most models, that incident field is simply the CMB,although if other sources create a low-frequency radio background at very highredshifts, or if there is a particular source behind the IGM patch along the line ofsight from the observer, a larger radio background may exist.

Consider photons incident on the patch from this background. If any redshift intoresonance with the 21 cm line, they can interact with the cloud—but only for a shorttime, as they will redshift out of resonance as the universe continues to expand.Thus, the Hubble expansion rate sets an effective path length through the cloud,simply equal to the distance the photon travels while it remains within the lineprofile. The total absorption can be calculated by integrating the IGM density across

The Cosmic 21-cm Revolution

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this interval, in an exactly analogous procedure to the calculation of the Gunn–Peterson Lyα optical depth (Field 1959; Gunn & Peterson 1965; Scheuer 1965). Theresult is

τπ ν

=+ ∥ ∥

hc Ak T

x nz dv dr

332 (1 ) ( / )

(1.5)10

310

B S 102

HI H

⎡⎣⎢

⎤⎦⎥δ≈ + + +

∥ ∥z

xT

H z zdv dr

0.0092 (1 ) (1 )( )/(1 )

/, (1.6)3/2 HI

S

where nH is the hydrogen number density, xHI is the neutral fraction, and ∥ ∥dv dr/ isthe velocity gradient along the line of sight (here scaled to the Hubble flow). In thesecond part, TS is in Kelvins, and we have scaled the density to the mean value bywriting δ= ¯ + +n n z(1 ) (1 )H H

0 3 , where n̄H0 is the mean comoving density today. Note

that this expression assumes a delta-function line profile, an assumption that breaksdown in regimes where the peculiar velocity gradient is large. A more carefulapproach is required in those cases, though note that such regions are rare in mostscenarios (Mao et al. 2012).

In most circumstances, the CMB provides the background radiation source, thetemperature for which is γT z( ). Then, ′ = γT T z( )R , so that we are observing thecontrast between high-redshift hydrogen clouds and the CMB. Because the opticaldepth is so small, we can then expand the exponentials in Equation (1.3), and

ν τ≈−

+γ

νTT T z

z( )

( )

1(1.7)b

S0

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥δ≈ + + − +γ

∥ ∥x z

T z

TH z z

dv dr9 (1 ) (1 ) 1

( ) ( )/(1 )/

mK. (1.8)HI1/2

S

Thus, <T 0b if < γT TS , yielding an absorption signal; otherwise, it appears inemission relative to the CMB. Both regimes are likely important for the high-zuniverse. Note thatTb saturates if ≫ γT TS , but the absorption can become arbitrarilylarge if ≪ γT TS . The observability of the 21 cm transition therefore hinges on thespin temperature; in the next section, we will describe the mechanisms that controlthat factor.

Of course, the other factors—the density, velocity, and ionization fields—are alsoimportant to understanding the 21 cm signal. The density field evolves throughcosmological structure formation, and that same evolution drives the velocity field—both of which we will describe briefly in Chapter 3. The ionization field depends,in most scenarios, on astrophysical sources, and it will be described in detail inChapter 2. For now, we will simply note that so long as stars drive reionization, the“two-phase” approximation is very accurate: the mean free path of ionizing photonsis so short that regions around ionizing sources are essentially fully ionized, whilethose outside of those H II regions are nearly fully neutral. Thus to a goodapproximation, we can take =x 0HI or 1.

The Cosmic 21-cm Revolution

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1.2 The Spin TemperatureThree competing processes determineTS: (i) absorption of CMB photons (as well asstimulated emission), (ii) collisions with other particles, and (iii) scattering of UVphotons. In the presence of the CMB alone, the spin states reach thermal equilibrium( = γT TS ) on a timescale of∼ = × +γ*

−T T A z/( ) 3 10 (1 )105 1 yr—much shorter than the

age of the universe at all redshifts after cosmological recombination, indicating thatCMB coupling establishes itself rapidly. Indeed, all the relevant processes adjust onvery short timescales (compared to the Hubble time), so equilibrium is an excellentapproximation.

However, the other two processes break this coupling. We let C10 and P10 be thede-excitation rates (per atom) from collisions and UV scattering, respectively. Wealso let C01 and P01 be the corresponding excitation rates. In equilibrium, the spintemperature is then determined by

+ + + = + +n C P A B I n C P B I( ) ( ), (1.9)1 10 10 10 10 CMB 0 01 01 01 CMB

where ICMB is the specific intensity of CMB photons at ν21. With the Rayleigh–Jeansapproximation, Equation (1.9) can be rewritten as

=+ +

+ +γ α

α

−− − −

TT x T x T

x x1, (1.10)S

11

c K1

c1

c

where xc and xα are coupling coefficients for collisions and UV scattering,respectively, and TK is the gas kinetic temperature. Here we have used the principleof detailed balance through the relation

⎛⎝⎜

⎞⎠⎟= ≈ −− ⋆⋆

CC

g

ge

TT

3 1 . (1.11)T T01

10

1

0

/

K

K

We have also defined the effective color temperature of the UV radiation fieldTc via

⎛⎝⎜

⎞⎠⎟≡ − ⋆P

PTT

3 1 . (1.12)01

10 c

In the limit in which →T Tc K (usually a good approximation), Equation (1.10) maybe written

⎛⎝⎜

⎞⎠⎟− = +

+ +−γ α

α

γT

Tx x

x x

T

T1

11 . (1.13)

S

c

c K

We must now calculate xc, xα, and Tc, which we shall do in the next subsections.

1.2.1 Collisional Coupling

We will first consider collisional excitation and de-excitation of the hyperfine levels,which become important in dense gas. The coupling coefficient for collisions withspecies i is

The Cosmic 21-cm Revolution

1-4

κ≡ =γ γ

⋆ ⋆xCA

TT

nA

TT

, (1.14)ii

ii

c10

10

10

10

where κ i10 is the rate coefficient for spin de-excitation in collisions (with units of

cm3 s−1). The total xc is the sum over all relevant species i, including collisions with(1) neutral hydrogen atoms, (2) free electrons, and (3) protons.

These rate coefficients can be calculated by the quantum mechanical crosssections of the relevant processes (Zygelman 2005; Furlanetto & Furlanetto2007a, 2007b). We will not list them in detail but show the rates in Figure 1.1.Although the atomic cross section is small, in the unperturbed IGM collisionsbetween neutral hydrogen, atoms nearly always dominate these rates because theionized fraction is small. Free electrons can be important in partially ionized gas;collisions with protons are only important at the lowest temperatures.

Given the densities relevant to the IGM, collisional coupling is quite weak in anearly neutral, cold medium. Thus, the local density must be large in order for thisprocess to effectively fixTS. A convenient estimate of their importance is the criticaloverdensity, δcoll, at which =x 1c for H–H collisions:

⎡⎣⎢

⎤⎦⎥

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟δ κ

κ+ =

Ω +T h z1 0.99

(88 K)( )

0.023 701

, (1.15)b

coll10

10 K2

2

Figure 1.1. De-excitation rate coefficients for H–H collisions (dashed line), H–e− collisions (dotted line),and H–p collisions (solid line). Note that the net rates are also proportional to the densities of theindividual species, so H–H collisions still dominate in a weakly ionized medium. Reproduced fromFurlanetto & Furlanetto (2007b), by permission of Oxford University press on behalf of the RoyalAstronomical Society.

The Cosmic 21-cm Revolution

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where 88 K is the expected IGM temperature at + =z1 70.1 In the standard picture,at redshifts <z 70, ≪x 1c and → γT T ;S by ∼z 30 the IGM essentially becomesinvisible. However, κ10 is extremely sensitive toTK in this low-temperature regime. Ifthe universe is somehow heated above the fiducial value, the threshold density canremain modest: δ ≈ 1coll at z = 40 if =T 300K K.

1.2.2 The Wouthuysen–Field Effect

We must therefore appeal to a different mechanism to render the 21 cm transitionvisible during the era of the first galaxies. This is known as the Wouthuysen–Fieldmechanism (named after the Dutch physicist Siegfried Wouthuysen and Harvardastrophysicist George Field, who first explored it; Wouthuysen 1952; Field 1958).Figure 1.2 illustrates the effect. This shows the hyperfine sublevels of the S1 and P2states of H I and the permitted transitions between them. Suppose a hydrogen atomin the hyperfine singlet state absorbs a Lyα photon. The electric dipole selection rulesallow Δ =F 0, 1, except that = →F 0 0 is prohibited (here F is the total angularmomentum of the atom). Thus, the atom must jump to either of the central P2states. However, these same rules now allow electrons in either of these excited statesto decay to the S1 1/2 triplet level.2 Thus, atoms can change hyperfine states throughthe absorption and spontaneous re-emission of a Lyα photon (or indeed, anyLyman-series photon; see below).

The Wouthuysen–Field coupling rate depends ultimately on the total rate (peratom) at which Lyα photons scattered through the gas,

Figure 1.2. Level diagram illustrating the Wouthuysen–Field effect. We show the hyperfine splittings ofthe S1 and P2 levels. The solid lines label transitions that can mix the ground-state hyperfine levels, whilethe dashed lines label complementary allowed transitions that do not participate in mixing. Reproducedfrom Pritchard & Furlanetto (2006), by permission of Oxford University press on behalf of the RoyalAstronomical Society.

1Note that this is smaller than the CMB temperature at this time, because the IGM gas cools faster (due toadiabatic expansion) once Compton scattering becomes inefficient at ∼z 150.2Here we use the notation LF J , where L and J are the orbital and total angular momentum of the electron.

The Cosmic 21-cm Revolution

1-6

∫πσ ν ν ϕ ν=α ν αP d J4 ( ) ( ), (1.16)0

where σ σ ϕ ν≡ν α( )0 is the local Lyα absorption cross section, σ π≡ αe m c f( / )e02 ,

=αf 0.4162 is the oscillator strength of the Lyα transition, ϕ να( ) is the Lyαabsorption profile, and Jν is the angle-averaged specific intensity of the backgroundradiation field.3

Transitions to higher Lyman-n levels have similar effects (Hirata 2006;Pritchard & Furlanetto 2006). Suppose that a UV photon redshifts into theLyman-n resonance as it travels through the IGM. After absorption, it can eitherscatter (by the electron decaying directly to the ground state) or cascade through aseries of intermediate levels and produce a sequence of photons. The direct decayprobability for any level is ∼0.8, so a Lyman-n photon will typically scatter ≈Nscatt

− ∼→−P(1 ) 5nP S1

1 times before instead initiating a decay cascade. In contrast, Lyαphotons scatter hundreds of thousands of times before being destroyed, usually byredshifting all the way across the (very wide) Lyα profile. As a result, coupling fromthe direct scattering of Lyman-n photons is suppressed compared to Lyα by a largefactor.

However, Lyman-n photons can still be important because of their cascadeproducts, as shown in Figure 1.3. Following Lyβ absorption, the only permitteddecays are to the ground state (regenerating a Lyβ photon and starting the processagain) or to the S2 level. The Hα photon produced in the →P S3 2 transition (andindeed any photon produced in a decay to an excited state) escapes to infinity. Thus,the atom will eventually find itself in the S2 state, which decays to the ground state

Figure 1.3. Decay chains for Lyβ and Lyγ excitations. We show Lyman-n transitions by dashed curves, Lyα bythe dotted–dashed curve, cascades by solid curves, and the forbidden →S S2 1 transition by the dotted curve.Reproduced from Pritchard & Furlanetto (2006), by permission of Oxford University press on behalf of theRoyal Astronomical Society.

3 By convention, we use the specific intensity in units of photons cm−2 Hz−1 s−1 sr−1 here, which is conservedduring the expansion of the universe (whereas a definition in terms of energy instead of photon number issubject to redshifting).

The Cosmic 21-cm Revolution

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via a forbidden two-photon process with =→A 8.2S S2 1 s−1. These photons will alsoescape to infinity, so coupling from Lyβ photons can be completely ignored.4

But now consider excitation by Lyγ, also shown in Figure 1.3. This can cascade(through S3 or D3 ) to the P2 level, in which case the original Lyman-n photon is“recycled” into a Lyα photon, which then scatters many times through the IGM.Thus, the key quantity for determining the coupling induced by Lyman-n photons isthe fraction f n( )rec of cascades that terminate in Lyα photons. Our discussion in theprevious paragraph shows that =f n( 3)rec vanishes, but detailed quantum mechan-ical calculations show that the higher states all have ∼f 1/3rec (Hirata 2006;Pritchard & Furlanetto 2006).

Focusing again on the Lyα photons themselves, we must relate the total scatteringrate Pα to the indirect de-excitation rate P10 (Field 1958; Meiksin 2000). Let us firstlabel the S1 and P2 hyperfine levels a–f, in order of increasing energy, and let Aij andBij be the spontaneous emission and absorption coefficients for transitions betweenthese levels. We write the background intensity at the frequency corresponding tothe →i j transition as Jij. Then,

∝+

++

P B JA

A AB J

AA A

. (1.17)01 ad addb

da dbae ae

eb

ea eb

The first term contains the probability for an a → d transition (B Jad ad), together withthe probability for the subsequent decay to terminate in state b; the second term isthe same for transitions to and from state e (see Figure 1.2). Next, we need to relateeach Aij to the total spontaneous decay rate from the P2 level, = ×αA 6.25 10 Hz8 ,the total Lyα spontaneous emission rate. This can be accomplished using a sum rulestating that the sum of decay intensities (g Ai ij) for transitions from a given nFJ to allthe ′ ′n J levels (summed over ′F ) is proportional to +F2 1, which implies that therelative strengths of the permitted transitions are then (1, 1, 2, 2, 1, 5), where wehave ordered the lines by (initial, final) states (bc, ad, bd, ae, be, bf). With ourassumption that the background radiation field is constant across the individualhyperfine lines, we find = αP P(4/27)10 (Meiksin 2000).

The coupling coefficient xα is then

= ≡αα

γα

α

ν

⋆xPA

TT

SJJ

427

. (1.18)c10

The second part evaluates Jν “near” line center and sets ≡ ×ν−J 1.165 10c 10

+ z[(1 )/20] photons cm−2 sr−1 Hz−1 s−1. Sα is a correction factor that accountsfor (complicated) radiative transfer effects in the intensity near the line center(see below). The coupling threshold νJ c for =α αx S can also be written in terms of thenumber of Lyα photons per hydrogen atom in the universe, which we denote˜ = +ν

−J z0.0767 [(1 )/20]c 2. This threshold is relatively easy to achieve in practice.

4 In a medium with very high number density, atomic collisions can mix the two angular momentum states, butthat process is unimportant in the IGM.

The Cosmic 21-cm Revolution

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To complete the coupling calculation, we must determine Tc and the correctionfactor Sα. The former is the effective temperature of the UV radiation field, definedin Equation (1.12), and is determined by the shape of the photon spectrum at theLyα resonance. The effective temperature of the radiation field must matter, becausethe energy deficit between the different hyperfine splittings of the Lyα transition(labeled bc, ad, etc. above) implies that the mixing process is sensitive to the gradientof the radiation spectrum near the Lyα resonance. More precisely, the proceduredescribed after Equation (1.17) yields

⎛⎝⎜

⎞⎠⎟ν

ν= +

+≈ + νP

P

g

gn nn n

d nd

3 1ln

, (1.19)01

10

1

0

ad ae

bd be0

where ν=ν νn c J /22 2 is the photon occupation number. Thus, by comparison toEquation (1.12), we find

ν= − νh

k Td n

dln

. (1.20)B c

A simple argument shows that ≈T Tc K (Field 1959): so long as the medium isextremely optically thick, the enormous number of Lyα scatterings forces the Lyαprofile to be a blackbody of temperature TK near the line center. This condition iseasily fulfilled in the high-redshift IGM, where τ ≫α 1. In detail, atomic recoilsduring scattering tilt the spectrum to the red and are primarily responsible forestablishing this equilibrium (Field 1959).

The physics of the Wouthuysen–Field effect are actually much more complicatedthan naively expected because scattering itself modifies the shape of Jν near the Lyαresonance (Chen 2004). In essence, the spectrum must develop an absorption featurebecause of the increased scattering rate near the Lyα resonance. Photons lose energyat a fixed rate by redshifting, but each time they scatter, they also lose a smallamount of energy through recoil. Momentum conservation during each scatteringslightly decreases the frequency of the photon. The strongly enhanced scattering ratenear line center means that photons “flow” through that region more rapidly thanelsewhere (where only the cosmological redshift applies), so the amplitude of thespectrum must be smaller. Meanwhile, the scattering in such an optically thickmedium also causes photons to diffuse away from line center, broadening the featurewell beyond the nominal line width.

If the fractional frequency drift rate is denoted by A, continuity requiresAνn = constant. Because A increases near resonance, the number density must

fall. On average, the energy loss (or gain) per scattering is (Chen 2004)

⎛⎝⎜

⎞⎠⎟

νΔ = −EE

hm c

TT

1 , (1.21)recoil

p2

K

c

where the first factor comes from recoil off an isolated atom and the second factorcorrects for the distribution of initial photon energies; the energy loss vanishes when

=T Tc K, and when <T Tc K, the gas is heated by the scattering process.

The Cosmic 21-cm Revolution

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To compute Sα, we must calculate the photon spectrum near Lyα. We begin withthe radiative transfer equation in an expanding universe (written in comovingcoordinates), and again using units of photons cm−2 sr−1 Hz−1 s−1 for Jν:

∫σ

ϕ ν νν

ν ν ν ψ ν

∂∂

= − + ∂∂

+ ′ ′ +

να ν α

ν

ν′

cnJt

J HJ

d R J C t

1( )

( , ) ( ) ( ).(1.22)H 0

The first term on the right-hand side describes absorption, the second describesredshifting due to the Hubble flow, and the third accounts for re-emission followingabsorption. ν ν′R( , ) is the “redistribution function” that specifies the frequency of anemitted photon, which depends on the relative momenta of the absorbed andemitted photons as well as the absorbing atom. The last term accounts for theinjection of new photons (via, e.g., radiative cascades that result in Lyα photons): Cis the rate at which they are produced and ψ ν( ) is their frequency distribution.

The redistribution function R is the difficult aspect of the problem, but it can besimplified if the frequency change per scattering (typically of order the absorptionline width) is “small.” In that case, we can expand ν′J to second order in ν ν− ′( ) andrewrite Equation (1.22) as a diffusion problem in frequency. The steady-state versionof Equation (1.22) becomes, in this so-called Fokker–Planck approximation (Chen2004),

⎛⎝⎜

⎞⎠⎟A D ψ− + + =d

dxJ

dJdx

C x( ) 0, (1.23)

where ν ν ν≡ − Δαx ( )/ D, νΔ D is the Doppler width of the absorption profile,A is thefrequency drift rate, and D is the diffusivity. The Fokker–Planck approximation isvalid so long as (i) the frequency change per scattering ( ν∼Δ D) is smaller than thewidth of any spectral features, and either (iia) the photons are outside the line corewhere the Lyα line profile is slowly changing, or (iib) the atoms are in equilibriumwith ≈T Tc K.

Solving for the spectrum including scattering thus reduces to specifyingA and D.The drift involves the Hubble flow, which sets A τ= − α

−H

1, where τα is the Gunn–Peterson optical depth for the Lyα line (Gunn & Peterson 1965; Scheuer 1965):

⎛⎝⎜

⎞⎠⎟τ

χν

= ≈ × +α

α

α

n z c

H zx

z( )

( )3 10

17

. (1.24)HI 5HI

3/2

Because it is uniform, the Hubble flow does not introduce any diffusion. Theremaining terms come from R and incorporate all the physical processes relevant toenergy exchange in scattering. The drift from recoil causes (Hirata 2006)

D ϕ= α x( )/2, (1.25)scatt

A η ϕ= − − α−x x( ) ( ), (1.26)scatt 0

1

The Cosmic 21-cm Revolution

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where ν ν≡ Δαx / D0 and η ν ν≡ Δαh m c( )/( )2p

2D . The latter is the recoil parameter

measuring the average loss per scattering in units of the Doppler width. The smallenergy defect between the hyperfine levels provides another source of slow energyexchange (Hirata 2006) and can be incorporated into the scattering in nearly thesame way as recoil.

We can now solve Equation (1.23) once we choose the boundary conditions,which essentially correspond to the input photon spectrum (ignoring scattering) andthe source function. Because the frequency range of interest is so narrow, two casessuffice: a flat input spectrum (which approximately describes photons that redshiftthrough the Lyα resonance, regardless of the initial source spectrum) and a stepfunction, where photons are “injected” at line center (through cascades or recombi-nations) and redshift away. In either case, the first integral over x in Equation (1.23)is trivial. At high temperatures, where spin flips are unimportant to the overallenergy exchange, we can write

⎧⎨⎩⎫⎬⎭ϕ η ϕ τ τ+ − + + =α α

− −dJdx

x x J K2 [ ( ) ] 2 / . (1.27)01 1

The integration constant K equals ∞J , the flux far from resonance, both for photonsthat redshift into the line and for injected photons at <x 0 (i.e., redward of linecenter); it is zero for injected photons at >x 0.

The formal analytic solution, when ≠K 0, is most compactly written in terms ofδ ≡ −∞ ∞J J J( )/J (Chen 2004):5

⎡⎣⎢

⎤⎦⎥∫ ∫δ η η

τ ϕ= − − + − ′

′α α

∞−

−x dy x x y

dxx

( ) 2 exp 2{ ( ) }2

( ). (1.28)J

x y

x

00

1

(An analogous form also exists for photons injected at line center.) The full problem,including the intrinsic Voigt profile of the Lyα line, must be solved numerically, butincluding only the Lorentzian wings from natural broadening allows a simplersolution (Furlanetto & Pritchard 2006). Fortunately, this assumption is quiteaccurate in the most interesting regime of <T 1000K K.

The crucial aspect of Equation (1.28) is that (as expected from the qualitativeargument) an absorption feature appears near the line center, with its depth roughlyproportional to η, our recoil parameter. The feature is more significant when TK issmall (because in that case the average effect of recoil is large). Figure 1.4 showssome example spectra (both for a continuous background and for photons injectedat line center).

Usually, the most important consequence is the suppression of the radiationspectrum at line center compared to the assumed initial condition. This decreases thetotal scattering rate of Lyα photons (and hence the Wouthuysen–Field coupling),with the suppression factor (defined in Equation (2.3)) as (Chen 2004)

5Here we assume the gas has a sufficiently high temperature that the different hyperfine subtransitions can betreated as one (Hirata 2006).

The Cosmic 21-cm Revolution

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∫ ϕ δ= ≈ − ⩽α α−∞

∞S dx x J x( ) ( ) [1 (0)] 1, (1.29)J

where the second equality follows from the narrowness of the line profile. Again, theLorentzian wing approximation turns out to be an excellent one; when ≫ ⋆T TK , thesuppression is (Furlanetto & Pritchard 2006)

⎜ ⎟ ⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎤⎦⎥

τ∼ −αα

−S

Texp 0.803

1 K 10. (1.30)K

2/3

6

1/3

Note that this form applies to both photons injected at line center as well as thosethat redshift in from infinity. As we can see in Figure 1.4, the suppression is mostsignificant in cool gas.

1.3 Heating of the Intergalactic MediumWe have seen that both collisions and the Wouthuysen–Field effect couple the spintemperature to the kinetic temperature of the gas. The 21 cm brightness temperaturetherefore depends on processes that heat the neutral IGM. (Note that photo-ionization heating is likely the most important mechanism in setting the IGMtemperature, because that process typically heats the gas to ∼T 104 K. However, bydefinition, that process only occurs when ionization is significant—and, in standardreionization scenarios, where ≈x 0HI so that the 21 cm signal vanishes.) We willreview several such mechanisms in this section.

Figure 1.4. Background radiation field near the Lyα resonance at z = 10; ν ν ν≡ − Δαx ( )/ D is the normalizeddeviation from line center, in units of the Doppler width. The upper and lower sets are for continuous photonsand photons injected at line center, respectively. (The former are normalized to ∞J ; the latter have arbitrarynormalization.) The solid and dashed curves take =T 10K and 1000 K, respectively. Reproduced fromFurlanetto & Pritchard (2006), by permission of Oxford University press on behalf of the Royal AstronomicalSociety.

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1.3.1 The Lyα Background

The photons that trigger Lyα coupling exchange energy with the IGM through therecoil in each scattering event. The typical energy exchange per scattering is small(see Equation (1.21)), but the scattering rate is extremely large. If the net heating rateper atom followed the naive expectation, ν∼ ×α αP h m c( ) /2

p2, the kinetic temperature

would surpass Tγ soon after Wouthuysen–Field coupling becomes efficient.However, the details of radiative transfer radically change these expectations

(Chen 2004). In a static medium, the energy exchange must vanish in equilibriumeven though scattering continues at nearly the same rate. Scattering induces anasymmetric absorption feature near να (Figure 1.4), whose shape depends on thecombined effects of atomic recoils and the scattering diffusivity. In equilibrium, thelatter exactly counterbalances the former.

If we removed scattering, the absorption feature would redshift away as theuniverse expands. Thus, the energy exchange rate from scattering must simply bethat required to maintain the feature in place. For photons redshifting intoresonance, the absorption trough has the total energy

∫π ν νΔ = −α ν∞u c J J h d(4 / ) ( ) , (1.31)

where ∞J is the input spectrum, and we note that the νh factor converts from ourdefinition of specific intensity (which counts photons) to energy. The radiationbackground loses ε = Δα αH u per unit time through redshifting; this energy goes intoheating the gas. Relative to adiabatic cooling by the Hubble expansion, thefractional heating amplitude is

∫ε π ν ν δ= Δα α ∞

−∞

∞

k T n H zh

k TJ

cndx x

23 ( )

83

( ) (1.32)JB K H B K

D

H

⎛⎝⎜

⎞⎠⎟≈

+α

αT

xS z

0.80 101

. (1.33)K4/3

Here we have evaluated the integral for the continuum photons that redshift into theLyα resonance; the “injected” photons actually cool the gas slightly. The net energyexchange when Wouthuysen–Field coupling becomes important (at ∼α αx S ) istherefore just a fraction of a degree, and in practice, gas heating through Lyαscattering is generally unimportant (Chen 2004; Furlanetto & Pritchard 2006).

Fundamentally, Lyα heating is inefficient because scattering diffusivity cancelsthe effects of recoil. From Figure 1.4, we see that the background spectrum is weakeron the blue side of the line than on the red. The scattering process tends to move thephoton toward line center, with the extra energy deposited in or extracted from thegas. Because more scattering occurs on the red side, this tends to transfer energyfrom the gas back to the photons, mostly canceling the energy obtained throughrecoil.

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1.3.2 The Cosmic Microwave Background

The previous section shows that, when considered as a two-level process that acts inisolation, Lyα scattering has only a slight effect on the gas temperature. However, inreality, this Lyα scattering always occurs in conjunction with the scattering of CMBphotons within the 21 cm transition. The combination leads to an enhanced heatingrate (Venumadhav et al. 2018).

In essence, the process works as follows. The CMB photons scatter through thehyperfine levels of H I to heat those atoms above their expected temperature(determined in this simple case by adiabatic cooling). Meanwhile, Lyα photonsscatter through the gas as well. As they do so, they mix the hyperfine levels of the H I

ground state, as depicted in Figure 1.2—this is the Wouthuysen–Field effect. CMBscattering continues to heat the hyperfine level populations during the Lyαscattering, which then sweeps up this extra energy and ultimately deposits it asthermal energy through the net recoil effect.

We can estimate the energy available to this heating mechanism by consideringthe CMB energy reservoir (Venumadhav et al. 2018). The CMB energy density atthe 21 cm transition is π π ν= ≈ν ν γu c B c k T(4 / ) 8 ( / )21

2 3B . Over a redshift interval

Δ =z 1, the total energy that redshifts through the line is ν π νΔ ≈νu c8 ( / )213

+γk T z/(1 )B2. However, only a fraction τ10 actually interacts with the line. If all of

this energy is used for heating, the temperature change per H atom would be

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟τΔ ≈ Δ ≈ +

αν ν

−

−T

un

xz

T(3/2)5

120

10 KK. (1.34)CMB Ly 10

HHI

1/2

S

A more detailed calculation of the heating rate shows that it is somewhat slower,but it does amplify the effect of the Lyα heating alone by a factor of several(Venumadhav et al. 2018). In standard models of the early radiation backgrounds,the correction is still relatively modest, but it is not negligible. For example, in thefiducial model considered by Venumadhav et al. (2018), the Lyα heating on itsown modifies TK by ∼1%–5%, but with the CMB scattering included the effect is∼9%–15%. Additionally, the CMB scattering can be enhanced in some exoticphysics models that decrease the spin temperature substantially.

1.3.3 The X-Ray Background

Because they have relatively long mean free paths, X-rays from galaxies and quasarsare likely to be the most important heating agent for the low-density IGM (Madauet al. 1997). In particular, photons with > +E x z1.5 [(1 )/10]HI

1/3 1/2 keV have meanfree paths exceeding the Hubble length (Oh 2001). Lower-energy X-rays will beabsorbed in the IGM, depositing much of their energy as heat, as will a fraction ofhigher-energy X-rays.

X-rays heat the IGM gas by first photoionizing a hydrogen or helium atom. Theresulting “primary” electron retains most of the photon energy (aside from thatrequired to ionize it) as kinetic energy, which it must then distribute to the generalIGM through three main channels: (1) collisional ionizations, which produce more

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secondary electrons that themselves scatter through the IGM, (2) collisionalexcitations of He I (which produce photons capable of ionizing H I) and H I (whichproduces a Lyα background), and (3) Coulomb collisions with free electrons (whichdistributes the kinetic energy). The relative cross sections of these processesdetermines what fraction of the X-ray energy goes to heating f( )heat , ionizationf( )ion , and excitation ( fexcite); clearly, it depends on both the ionized fraction xi andthe input photon energy. Through these scatterings, the primary photoelectrons,with ∼T 106 K, rapidly cool to energies just below the Lyα threshold, <10 eV,and thus equilibrate with the other IGM electrons. After that, the electrons andneutrals equilibrate through elastic scattering on a timescale ∼ +t z5[10/(1 )] Myreq

3 .Because ≪ −t H z( )eq

1, the assumption of a single-temperature fluid is an excellentone.

The details of this process have been examined numerically (Shull & vanSteenberg 1985; Valdés & Ferrara 2008; Furlanetto & Johnson Stoever 2010), andFigure 1.5 shows some example results.6 Note that the deposition fractions aresmooth functions at high electron energies but, at low energies—where the atomicenergy levels become relevant—can be quite complex. A number of approximate fitshave been presented for the high-energy regime (Ricotti et al. 2002; Volonteri &Gnedin 2009), but they are not accurate over the full energy range. A crude butuseful approximation to the high-energy limit often suffices (Chen & Kamionkowski2004):

∼ +∼ ∼ −

f x

f f x

(1 2 )/3

(1 )/3,(1.35)

i

i

heat

ion excite

where xi is the ionized fraction. In highly ionized gas, collisions with free electronsdominate and →f 1;heat in the opposite limit, the energy is split roughly equallybetween these three processes. However, the complexity of the behavior at lowelectron energies—together with the increasing optical thickness of the IGM in thatregime, and the fact that most sources are brighter in this soft X-ray regime—suggests that a more careful treatment is needed for accurate work. Furlanetto &Johnson Stoever (2010) recommended interpolating the exact results.

1.3.4 Other Potential Heating Mechanisms

We close this section by noting that other heating mechanisms have been consideredin the literature. One possibility is the heating that accompanies structureformation. When regions collapse gravitationally, they are heated by adiabaticcompression (which we will discuss in Chapter 3), which is a minor effect. But, if theresulting gas flows converge at velocities above the (very small) sound speed, theycan also trigger shocks, which convert a large fraction of that kinetic energy intoheat. Analytic models and simulations suggest, however, that structure formation is

6Note that these results are relative to the initial X-ray energy; some others in the literature instead use presentresults relative to the primary electron’s energy.

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still sufficiently gentle during the Cosmic Dawn that these shocks will have littleeffect on the 21 cm signal (Furlanetto & Loeb 2004; Kuhlen et al. 2006; McQuinn &O’Leary 2012).

Finally, exotic mechanisms like dark matter annihilation or decay, primordialblack hole emission, and other speculative processes can also affect the thermalevolution of the IGM during the Dark Ages. We will discuss such possibilitiesfurther in Chapter 3.

Figure 1.5. Energy deposition from fast electrons. We show the fraction of the initial X-ray energy deposited inionization (upper left), heating (upper right), and collisional excitation (lower left), as a function of electronenergy and for several different ionized fractions xi. The lower right shows the fraction of the collisionalexcitation energy deposited in the H I Lyα transition, αfLy . Reproduced from Furlanetto & Johnson Stoever(2010), by permission of Oxford University press on behalf of the Royal Astronomical Society.

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