Astrophysics from the 21-cm background
CChapter 2Astrophysics from the 21-cm background
Jordan Mirocha (McGill University)The goal of this chapter is to describe the astrophysics encoded by the 21-cm back-ground. We will begin in § z intergalactic medium. Then, in § § z Intergalactic Medium
In this section we provide a general introduction to the intergalactic medium (IGM) and howits properties are expected to evolve with time. We will start with a brief recap of the 21-cm brightness temperature (2.1.1), then turn our attention to its primary dependencies, theionization state and temperature of the IGM ( § § § α coupling. Readersfamiliar with the basic physics may skip ahead to § The differential brightness temperature of a patch of the IGM at redshift z and position x isgiven by δ T b ( z , x ) (cid:39) ( + δ )( − x i ) (cid:18) Ω b , h . (cid:19) (cid:18) . Ω m , h + z (cid:19) / (cid:18) − T R T S (cid:19) , (2.1) Refer back to Chapter 1 for a more detailed introduction. a r X i v : . [ a s t r o - ph . C O ] S e p CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND where δ is the baryonic overdensity relative to the cosmic mean, x i is the ionized fraction, T R is the radiation background temperature (generally the CMB, T R = T CMB ), and T − ≈ T − + x c T − + x α T − α + x c + x α . (2.2)is the spin temperature, which quantifies the level populations in the ground state of thehydrogen atom, and itself depends on the kinetic temperature, T K , and “colour temperature”of the Lyman- α radiation background, T α . Because the IGM is optically thick to Ly- α photons, the approximation T α ≈ T K is generally very accurate.The collisional coupling coefficients, x c , themselves depend on the gas density, ionizationstate, and kinetic temperature (see [160] for details). The radiative coupling coefficient, x α ,depends on the Ly- α intensity, J α , via x α = S α + z J α J α , (2.3)where J α , ≡ π T (cid:63) e f α A T γ , m e c . (2.4) J α is the angle-averaged intensity of Ly- α photons in units of s − cm − Hz − sr − , S α isa correction factor that accounts for variations in the background intensity near line-center[19, 47, 61], m e and e are the electron mass and charge, respectively, f α is the Ly- α oscillatorstrength, T γ , is the CMB temperature today, and A is the Einstein A coefficient for the 21-cm transition.A more detailed introduction to collisional and radiative coupling can be found in Chapter1. For the purposes of this chapter, the key takeaway from Equations 2.1-2.3 is simplythat the 21-cm background probes the ionization field, kinetic temperature field, and Ly- α background intensity. We quickly review the basics of non-equilibrium ionization chemistryin the next sub-section ( § α background in § As described in the previous section, the 21-cm brightness temperature of a patch of the IGMdepends on the ionization and thermal state of the gas, as well as the incident Ly- α intensity .The evolution of the ionization and temperature are coupled and so must be evolved self-consistently. The number density of hydrogen and helium ions in a static medium can be Note that Ly- α photons can transfer energy to the gas (see, e.g., [149]) though we omit this dependencefrom the current discussion (see Ch. 1.) .1. PROPERTIES OF THE HIGH- Z INTERGALACTIC MEDIUM dn H II dt = ( Γ H I + γ H I + β H I n e ) n H I − α H II n e n H II (2.5) dn He II dt = ( Γ He I + γ He I + β He I n e ) n He I + α He III n e n He III − ( β He II + α He II + ξ He II ) n e n He II − ( Γ He II + γ He II ) n He II (2.6) dn He III dt = ( Γ He II + γ He II + β He II n e ) n He II − α He III n e n He III .. (2.7)Each of these equations represents the balance between ionizations of species H I , He I ,and He II , and recombinations of H II , He II , and He III . Associating the index i withabsorbing species, i = H I , He I , He II , and the index i (cid:48) with ions, i (cid:48) = H II , He II , He III , wedefine Γ i as the photo-ionization rate coefficient, γ i as the rate coefficient for ionization byphoto-electrons [133, 50, ; see Ch. 1], α i (cid:48) ( ξ i (cid:48) ) as the case-B (dielectric) recombination ratecoefficients, β i as the collisional ionization rate coefficients, and n e = n H II + n He II + n He III as the number density of electrons.While the coefficients α , β , and ξ only depend on the gas temperature, the photo- andsecondary-ionization coefficients, Γ and γ , depend on input from astrophysical sources (see § ddt (cid:18) k B T K n tot µ (cid:19) = f heat ∑ i n i Λ i − ∑ i ζ i n e n i − ∑ i (cid:48) η i (cid:48) n e n i (cid:48) − ∑ i ψ i n e n i − ω He II n e n He II . (2.8)Here, Λ i is the photo-electric heating rate coefficient (due to electrons previously bound tospecies i ), ω He II is the dielectric recombination cooling coefficient, and ζ i , η i (cid:48) , and ψ i arethe collisional ionization, recombination, and collisional excitation cooling coefficients, re-spectively, where primed indices i (cid:48) indicate ions H II , He II , and He III , and unprimed indices i indicate neutrals H I , He I , and He II . The constants in Equation (2.8) are the total num-ber density of baryons, n tot = n H + n He + n e , the mean molecular weight, µ , Boltzmann’sconstant, k B , and the fraction of photo-electron energy deposited as heat, f heat (sometimesdenoted f abs ) [133, 50]. Formulae to compute the values of α i , β i , ξ i , ζ i , η i (cid:48) , ψ i , and ω He II ,are compiled in, e.g., [46, 64]. Terms involving helium become increasingly important in amedium irradiated by X-rays.These equations do not yet explicitly take into account the cosmic expansion, which di-lutes the density and adds an adiabatic cooling term to Eq. 2.8, however these generalizationsare straightforward to implement in practice. For the duration of this chapter we will operatewithin this simple chemical network, ignoring, e.g., molecular species like H and HD whosecooling channels are important in primordial gases. Though an interesting topic in their ownright, molecular processes reside in the “subgrid” component of most 21-cm models, giventhat they influence how, when, and where stars are able to form (see § CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND
In order to build intuition for the progression of ionization and heating in the IGM it isinstructive to consider the impact of a single point source of UV and X-ray photons on itssurroundings. Many early works focused on such 1-D radiative transfer problems [159, 142].In principle, this is the ideal way to simulation reionization – iterating over all sources in acosmological volume and for each one applying 1-D radiative transfer techniques over thesurrounding 4 π steradians. In practice, such approaches are computationally expensive, andwhile they provide detailed predictions [108, 105, 54], more approximate techniques arerequired to survey the parameter space and perform inference (see Chapter 4).In 1-D, the change in the intensity of a ray of photons, I ν , is a function of the path length, s , the emissivity of sources along the path, j ν , and the absorption coefficient, α ν , dI ν = j ν − α ν I ν . (2.9)If considering a point source, j ν =
0, we can integrate this radiative transfer equation (RTE)to obtain I ν ( s ) = I ν , exp (cid:20) − (cid:90) s α ν ( s (cid:48) ) ds (cid:48) (cid:21) , (2.10)i.e., the intensity of photons declines exponentially along the ray. It is customary to definethe optical depth, d τ ν ≡ α ν ds , (2.11)in which case we can write I ν ( s ) = I ν , e − τ ν . (2.12)In the reionization context, the optical depth of interest is that of the IGM, which is composedof (almost) entirely hydrogen and helium , in which case the optical depth is τ ν = ∑ i σ ν , i N i (2.13)where i = H I , He I , He II , and N i = (cid:82) s ds (cid:48) n i ( s (cid:48) ) is the column density of each species along theray.With a solution for I ν ( s ) in hand, one can determine the photoionization and heating ratesby integrating over all photon frequencies and weighting by the bound-free absorption crosssection for each species. For example, the photoionization rate coefficient for hydrogen canbe written as Γ H I ( s ) = (cid:90) ∞ ν H I σ H I I ν ( s ) d ν h ν (2.14)where ν H I is the frequency of the hydrogen ionization threshold, h ν = . Note that there will be small-scale absorption as well, though in most models this is unresolved, andparameters governing photon escape are used to quantify this additional opacity (see § § .1. PROPERTIES OF THE HIGH- Z INTERGALACTIC MEDIUM Γ i = A i (cid:90) ∞ ν i I ν e − τ ν (cid:16) − e − ∆ τ i , ν (cid:17) d ν h ν (2.15) γ i j = A j (cid:90) ∞ ν j (cid:18) ν − ν j ν i (cid:19) I ν e − τ ν (cid:16) − e − ∆ τ j , ν (cid:17) d ν h ν (2.16) Λ i = A i (cid:90) ∞ ν i ( ν − ν i ) I ν e − τ ν (cid:16) − e − ∆ τ i , ν (cid:17) d νν . (2.17)The normalization constant in each expression is defined as A i ≡ L bol / n i V sh ( r ) , where V sh isthe volume of a shell in this 1-D grid of concentric spherical shells, each having thickness ∆ r and volume V sh ( r ) = π [( r + ∆ r ) − r ] /
3, where r is the distance between the origin and theinner interface of each shell. We denote the ionization threshold energy for species i as h ν i . I ν represents the SED of radiation sources, and satisfies (cid:82) ν I ν d ν =
1, such that L bol I ν = L ν .Note that the total secondary ionization rate for a given species is the sum of ionizations dueto the secondary electrons from all species, i.e., γ i = f ion ∑ j γ i j n j / n i .These expressions preserve photon number by inferring the number of photo-ionizationsof species i in a shell from the radiation incident upon it and its optical depth [1], ∆ τ i , ν = n i σ i , ν ∆ r . (2.18)This quantity is not to be confused with the total optical depth between source and shell, τ ν = τ ν ( r ) , which sets the incident radiation field upon each shell, i.e., τ ν ( r ) = ∑ i (cid:90) r σ i , ν n i ( r (cid:48) ) dr (cid:48) = ∑ i σ i , ν N i ( r ) (2.19)where N i is the column density of species i at distance r from the source.In words, Equations 2.15-2.17 are propagating photons from a source at the origin, withbolometric luminosity L bol , and tracking the attenuation suffered between the source andsome volume element of interest at radius r , e − τ , and the attenuation within that volumeelement, ∆ τ , which results in ionization and heating. In each case, we integrate over the con-tribution from photons at all frequencies above the ionization threshold, additionally modi-fying the integrands for γ i j and Λ i with ( ν − ν i ) -like factors to account for the fact that boththe number of photo-electrons (proportional to ( ν − ν j ) / ν i ) and their energy (proportional to ν − ν i ) determine the extent of secondary ionization and photo-electric heating. Equations2.15-2.17 can be solved once a source luminosity, L bol , spectral shape, I ν , and density profileof the surrounding medium, n ( r ) , have been specified .Figure 2.1 shows an example 1-D radiative transfer model including sources of UV andX-ray photons. Because the mean free paths of UV photons are short, they generate sharp In practice, to avoid performing these integrals on each step of an ODE solver (for Eqs. 2.5-2.8), the resultscan be tabulated as a function of τ or column density, N i , where τ i , ν = σ i , ν N i [142, 98, 67]. CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND
Figure 2.1:
Ionization and temperature profile around stellar UV and X-ray sourcesin a one-dimensional radiative transfer model [67].
From top to bottom, this includesthe hydrogen neutral fraction, neutral helium fraction, singly-ionized helium fraction, andkinetic temperature, while the left and right columns indicate different time snapshots aftersources are first turned on. Different lines adopt different source models, from a “stars only”model (solid red), to hybrid models with stars and X-ray sources with different power-lawspectra (dashed and dotted curves). .1. PROPERTIES OF THE HIGH- Z INTERGALACTIC MEDIUM . The mostlyneutral “bulk IGM” outside of bubbles is affected predominantly by X-rays, which havemean free paths long enough to escape the environments in which they are generated (thoughsee § While the procedure outlined in the previous section is relevant to small-scale ionization andheating, i.e., that which is driven a single (or perhaps a few) source(s) close to a volumeelement of interest, it is also instructive to consider the ionization and heating caused bya population of sources separated by great distances. In this limit, rather than consideringthe luminosity of a single source at the origin of a 1-D grid, we treat the volume-averagedemissivity of sources, ε ν , in a large “chunk” of the Universe, and solve for the evolution ofthe mean intensity in this volume, J ν .The transfer equation now takes its cosmological form, i.e., (cid:18) ∂∂ t − ν H ( z ) ∂∂ ν (cid:19) J ν ( z ) + H ( z ) J ν ( z ) = c π ε ν ( z )( + z ) − c α ν J ν ( z ) (2.20)where J ν is the mean intensity in units of erg s − cm − Hz − sr − , ν is the observed fre-quency of a photon at redshift z , related to the emission frequency, ν (cid:48) , of a photon emitted atredshift z (cid:48) as ν (cid:48) = ν (cid:18) + z (cid:48) + z (cid:19) , (2.21) α ν = n σ ν is the absorption coefficient, not to be confused with recombination rate coeffi-cient, α H II , and ε ν is the co-moving emissivity of sources.The optical depth, d τ = α ν ds , experienced by a photon at redshift z and emitted at z (cid:48) isan integral along a cosmological line element, summed over all absorbing species , i.e., τ ν ( z , z (cid:48) ) = ∑ j (cid:90) z (cid:48) z n j ( z (cid:48)(cid:48) ) σ j , ν (cid:48)(cid:48) dldz (cid:48)(cid:48) dz (cid:48)(cid:48) (2.22) In practice one then solves two sets of equations like Eqs. 2.5-2.8 – one for each phase of the IGM. In thefully-ionized phase, the ionized fraction represents a volume-filling fraction, while in the “bulk IGM” phase, itretains its usual meaning. In general, one must iteratively solve for τ ν and J ν . However, in many models the bulk of cosmic re-heating precedes reionization, in which case τ ν can be tabulated assuming a fully neutral IGM. This approachprovides a considerable speed-up computationally and remains accurate even when reionization and reheatingpartially overlap [92]. CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND
The solution to Equation 2.20 isˆ J ν ( z ) = c π ( + z ) (cid:90) z f z ε (cid:48) ν ( z (cid:48) ) H ( z (cid:48) ) e − τ ν dz (cid:48) . (2.23)where z f is the “first light redshift” when astrophysical sources first turn on, H is the Hubbleparameter, and the other variables take on their usual meaning .With the background intensity in hand, one can compute the rate coefficients for ion-ization and heating. These coefficients are equivalent to those for the 1-D problem (Eqs.2.15-2.17), though the intensity of radiation at some distance R from the source has beenreplaced by the mean background intensity, Γ i ( z ) = π n i ( z ) (cid:90) ν max ν min ˆ J ν σ ν , i d ν (2.24) γ i j ( z ) = π ∑ j n j (cid:90) ν max ν min ˆ J ν σ ν , j ( h ν − h ν j ) d ν h ν (2.25) ε X ( z ) = π ∑ j n j (cid:90) ν max ν min ˆ J ν σ ν , j ( h ν − h ν j ) d ν (2.26)Then, the ionization state and temperature of the gas can be updated accordingly via Equa-tions 2.5-2.8.Figure 2.2 shows predictions for the evolution of the mean ionized fraction and kinetictemperature of the IGM [80, 112] using the two-phase IGM picture described above. Whilecurrent observations are consistent with reionization occurring relatively rapidly at z (cid:46) α Coupling
On scales large and small, the 21-cm background will only probe the kinetic temperature ofthe gas if the Ly- α background intensity is strong enough to couple the spin temperature tothe kinetic temperature. Determining the Ly- α background intensity, J α , requires a specialsolution to the cosmological radiative transfer equation (Eq. 2.20). Two effects separate thisproblem from the generic transfer problem outlined in § . < h ν / eV < . α background issourced both by photons redshifting into the line resonance as well as those produced incascades downward from higher n transitions [115].As a result, it is customary to solve the RTE in each Ly- n frequency interval separately.Within each interval, bounded by Ly- n line on its red edge and Ly- n + This equation can be solved efficiently on a logarithmic grid in x ≡ + z [58, 92], in which case photonsredshift seamlessly between frequency bins over time. .1. PROPERTIES OF THE HIGH- Z INTERGALACTIC MEDIUM
Predictions for the evolution of the mean properties of the IGM.
Left:
Predic-tions for the mean neutral fraction of the IGM as a function of redshift compared to severalobservational constraints [80]. Magenta curve includes all galaxies brighter than UV mag-nitude M UV < −
12, while green curve includes only brighter galaxies with M UV < − Right:
Predictions for the mean kinetic temperature of the IGM [112] for different assump-tions about how efficiently galaxies produce X-rays (parameterized via f X and the fractionof X-ray energy absorbed in the IGM, f abs ; see § PAPER [111, 113]. Note that the
PAPER limit has since been revised [20, 68].0
CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND can ionize hydrogen or helium . As a result, any photon starting its journey just redward ofLy- β will travel freely until it redshifts into the Ly- α resonance, while photons originatingat bluer wavelengths will encounter Ly- n resonances (with n > α photons.We can thus write the mean Ly- α background intensity as (cid:98) J α ( z ) = c π ( + z ) n max ∑ n = f n rec (cid:90) z ( n ) max z ε (cid:48) ν ( z (cid:48) ) H ( z (cid:48) ) dz (cid:48) (2.27)where f n rec is the “recycling fraction,” that is, the fraction of photons that redshift into a Ly- n resonance that ultimately cascade through the Ly- α resonance [115]. The upper bound ofthe definite integral, 1 + z ( n ) max = ( + z ) (cid:2) − ( n + ) − (cid:3) − n − , (2.28)is set by the horizon of Ly- n photons – a photon redshifting through the Ly- n resonance at z could only have been emitted at z (cid:48) < z ( n ) max , since emission at slightly higher redshift wouldmean the photon redshifted through the Ly ( n + ) resonance. The sum over Ly- n levels in Eq.2.27 is generally truncated at n max =
23 [4] since the horizon for such photons is smaller thanthe typical ionized bubble sourced by an individual galaxy. As a result, any Ly- α photonsgenerated by such high- n cascades are “wasted” as far as the spin temperature is concerned,as they will most likely have redshifted out of resonance before reaching any neutral gas.Ly- α emission produced by recombinations in galactic HII regions is generally neglectedfor the same reason. Though there are some assumptions built into the n max estimate, thetotal Ly- α photon budget is relatively insensitive to the exact value of n max [4, 115].Note that in general the mean free path of photons between Lyman series resonances isvery long, which makes tracking them in numerical simulations very expensive. For exam-ple, a photon emitted just redward of Ly- β and observed at the Ly- α frequency at redshift z has traveled a distance d β → α (cid:39) h − (cid:18) Ω m , . + z (cid:19) − / cMpc , (2.29)where we have assumed the high- z approximation Ω Λ (cid:28) Ω m . This exceeds a Hubble lengthat high- z , meaning most of the Wouthuysen-Field coupling at very early times must comefrom photons originating just blueward of their nearest Ly- n resonance. Despite their longmean free paths, fluctuations in the Ly- α background inevitably arise [4, 3, 62]. However,this background is expected to become uniform (and strong) relatively quickly, meaning ingeneral the 21-cm background is only sensitive to J α at the earliest epochs (see § In the previous section we outlined the basic equations governing the ionization and tem-perature evolution of the IGM without actually specificying the sources of ionization and There is in principle a small opacity contribution from H , though we neglect this in what follows as the H fraction in the IGM is expected to be small. .2. SOURCES OF THE UV AND X-RAY BACKGROUND . Instead, we used a placeholder emissivity, ε ν , to encode the integrated emissionsof sources at frequency ν within some region R . We will now write this emissivity as anintegral over the differential luminosity function (LF) of sources, dn / dL ν , i.e., ε ν ( z , R ) = (cid:90) ∞ dL ν dndL ν . (2.30)where ν refers to the rest frequency of emission at redshift z .It is common to rewrite the emissivity as an integral over the DM halo mass function(HMF), dn / dm , multiplied by a conversion factor between halo mass and galaxy light, dm / dL ν , i.e., ε ν ( z , R ) = (cid:90) ∞ m min dm dndm dmdL ν , (2.31)where m min is the minimum mass of DM halos capable of hosting galaxies. Because dn / dm is reasonably well-determined from large N-body simulations of structure formation [114,132, 144], much of the modeling focus is on the mass-to-light ratio, dm / dL ν , which encodesthe efficiency with which galaxies form in halos and the relative luminosities of differentkinds of sources within galaxies (e.g., stars, compact objects, diffuse gas) that emit at differ-ent frequencies .The main strength of the 21-cm background as a probe of high- z galaxies is now apparent:though 21-cm measurements cannot constrain the properties of individual galaxies, they canconstrain the properties of all galaxies, in aggregate, even those too faint to be detecteddirectly . As a result, it is common to forego detailed modeling of the mass-to-light ratio andinstead relate the emissivity to the fraction of mass in the Universe in collapsed halos, ε ν ( z , R ) = ρ b f coll ( z , R ) ζ ν , (2.32)where ρ b is the baryon mass density, the collapsed fraction is an integral over the HMF, f coll = ρ − m (cid:90) ∞ m min dmm dndm (2.33)and ζ ν is an efficiency factor that quantifies the number of photons emitted at frequency ν per baryon of collapsed mass in the Universe. It is generally modeled as ζ ν = f ∗ N ν f esc , ν , (2.34)where f ∗ is the star formation efficiency (SFE), in this case defined to be the fraction ofbaryons that form stars, N ν is the number of photons emitted per stellar baryon at somefrequency ν , and f esc , ν is the fraction of those photons that escape into the IGM. One coulddefine additional ζ factors to represent, e.g., emission from black holes or exotic particles,in which case f ∗ and N ν would be replaced by some black hole or exotic particle production Note that some (at least roughly) model-independent constaints on the properties of the IGM should beattainable with future 21-cm measurements [22, 21, 95]. Most models consider regions R that are sufficiently large that one can assume a well-populated HMF,though at very early times this approximation may break down, rendering stochasticity due to poor HMFsampling an important effect. CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND efficiencies. In practice, most often three ζ factors are defined: ζ = ζ ion , ζ X , and ζ α , i.e., oneefficiency factor for each radiation background that influences the 21-cm signal. A minimalmodel for the 21-cm background thus contains four parameters: m min , ζ , ζ X , and ζ α . Notethat ζ X and ζ α are often replaced by the parameters f X and f α , where the latter are definedsuch that f X = f α = ζ X and ζ α .Because the factors constituting ζ are degenerate with each other, at least as far as 21-cmmeasurements are concerned, they generally are not treated separately as free parameters.However, it is still useful to consider each individually in order to determine a fiducial valueof ζ and explore deviations from that fiducial model. In addition, inclusion of ancillarymeasurements may eventually allow ζ to be decomposed into its constituent parts [94, 110,57]. For the remainder of this section, we focus on plausible values of f ∗ , N ν and f esc , ν , andthe extent to which these quantities are currently understood. Though a first-principles understanding of star formation remains elusive, the bulk propertiesof the star-forming galaxy population appear to obey simple scaling relationships. In thissection, we outline the basic strategies used to infer the relationships between star formationand dark matter halos, and how such relationships can be used to inform 21-cm models.The simplest description of the galaxy population follows from the assumption that eachdark matter halo hosts a single galaxy. With a model for the abundance of DM halos, i.e.,the halo mass function (HMF), many of which are readily available [114, 132], one can then“abundance match” halos with measured galaxy abundances [10, 43], i.e., n ( > L h ) = (cid:90) ∞ L dndL (cid:48) dL (cid:48) = n ( > m h )= (cid:90) ∞ m h dndm (cid:48) h dm (cid:48) h . (2.35)This procedure reveals the mapping between mass and light, dL / dm h , upon repeated inte-gration over a grid of L h values, solving for the M h value needed for abundances to match.Results of this simple procedure show that galaxy luminosity is a function of both halomass and cosmic time [146, 100, 6, 139, 78, 138, 79]. While the HMF can be readily used topredict the abundances of halos out to arbitrary redshifts, one shortcoming of this approachis that any evolution in dL / dM must be modeled via extrapolation. As a result, predictionsfor deeper and/or higher redshift galaxy surveys may not be physically motivated.To avoid the possibility of unphysical extrapolations of dL / dm h , it is becoming morecommon to parameterize the galaxy – halo connection from the outset, effectively resultingin forward models for galaxy formation that link galaxy star formation rate (SFR), ˙ m ∗ , tohalo mass, m h , or mass accretion rate (MAR), ˙ m h , e.g.,˙ m ∗ ( z , m h ) = ˜ f ∗ ( z , m h ) m h ( z , m h ) . (2.36)or ˙ m ∗ ( z , m h ) = f ∗ ( z , m h ) ˙ m h ( z , m h ) . (2.37) .2. SOURCES OF THE UV AND X-RAY BACKGROUND Relationship between halos and star formation recovered via semi-empiricalmodeling [139].
Left:
Star formation efficiency as a function of halo mass for a variety ofdifferent approaches, including constant metallicity “Z-const”, an evolving metallicity model“Z-evo”, a model with SMC dust instead of the standard relation from [88] (see § Right:
Modeled luminosity function at z = f ∗ and f ∗ , to explicitly indicatewhether tied to m h or ˙ m h , is left as a flexible function to be calibrated empirically . Inthe MAR-based model, one of course requires a model for the halo MAR as well as theHMF, though such models are readily available from the results of numerical simulations[81, 145], or modeled approximately from the HMF itself [51]. Both approaches are used inthe literature, and while inferred SFRs are largely in agreement, there is some difference inthe interpretation of the models, which we revisit below in § z measurements mostlyprobe the rest UV spectrum of galaxies, so it is customary to link the SFR with the rest1600 ˚ A luminosity of galaxies, L ( z , m h ) = l ˙ m ∗ ( z , m h ) (2.38)where l is of order 10 erg s − Hz − ( M (cid:12) / yr ) − according to commonly-used stellarpopulation synthesis models, assuming constant star formation [71, 31, 23]. The precisevalue depends on stellar metallicity, binarity, and initial mass function (IMF), and variesfrom model to model. We will revisit the details of stellar spectra in § § Note that ˜ f ∗ in Equation 2.36 necessarily has units of time − , whereas f ∗ in Eq. 2.37 is dimensionless. CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND and below ∼ M (cid:12) . Such models generally agree that star formation is inefficient, withpeak values f ∗ (cid:46) .
1, and f ∗ (cid:46) .
01 in the less massive (but numerous) population of galaxiesresiding in halos M h (cid:46) M (cid:12) . As a result, if using the standard ζ modeling approach (seeEq. 2.34), reasonable fiducial f ∗ values are f ∗ ∼ . − .
1. Of course, evolution of the HMFimplies that representative values of f ∗ will also evolve with time, though discerning sucheffects will only be possible in 21-cm analyses focused on a broad frequency range. f ∗ Current high- z measurements support a relatively simple picture of star formation in earlygalaxies, in which galaxies maintain a rough equilibrium between inflow and outflow throughstellar feedback [8, 28, 30], the efficiency of which is a strong function of halo mass andperhaps time. Though there are quantitative differences amongst studies in the literature,which could arise due to different assumptions about dust, stellar populations, and/or dif-ferent definitions of f ∗ , there does appear to be a consensus that star formation is maxi-mally efficient (and feedback correspondingly inefficient) in M h (cid:39) . − M (cid:12) halos[79, 138, 100, 78, 139, 5]. The decline in the SFE below the peak is widely thought to bea signature of stellar feedback, while the decline in massive systems is likely due to shockheating and/or AGN feedback, which reduces the availability of cold gas [34, 25, 136].This general trend can be explained – but perhaps not understood – from relatively simplearguments. The basic idea is that star formation is fueled by the inflow of gas from the IGM,but that the overall rate of star formation in galaxies is self-regulated by feedback fromstellar winds and supernovae explosions, both of which expell gas that could otherwise formstars [29, 51]. As a result, galaxies forming in shallow gravitational potential wells are at adisadvantage simply because the escape velocity is lower, making it easier for supernovaeand winds to drive material out of the galaxy. However, the escape velocity depends on boththe mass and size of an object – if low-mass halos are sufficiently compact, they may be ableto retain enough gas to continue forming stars.To build some intuition for possible outcomes, it is common to model star formation asa balance between inflow and outflow [8, 28, 30], i.e.,˙ m ∗ = ˙ m b − ˙ m w (2.39)where m b is the accretion rate of baryons onto a halo and ˙ m w is the mass-loss rate throughwinds (and/or supernovae). If we relate mass-loss to star formation via “mass loading factor” η , ˙ m w ≡ η ˙ m ∗ , then we can write f ∗ = ˙ m ∗ ˙ m b = + η . (2.40)One can show that, for energy-conserving winds, η ∝ m / h ( + z ) − , while for momentum-conserving winds, η ∝ m / h ( + z ) − / [29, 51]. Though simple, these models provide somephysically-motivated guidance for extrapolating models to higher redshifts and/or fainter ob-jects than are probed by current surveys. Current measurements can still accommodate eitherscenario, largely due to (i) the small time baseline overwhich measurements are available and(ii) uncertainties in correcting for dust reddening (see § .2. SOURCES OF THE UV AND X-RAY BACKGROUND z galaxies. For example, one need not require that star formation operatein an equilibrium with inflow and outflow, in which case Eq. 2.36 may be a more sensiblechoice than 2.37. Many efforts are now underway to simulate galaxy formation using abinitio cosmological simulations [153, 130, 63, 108, 54], rather than using analytic or semi-analytic models. However, doing so self-consistently in statistically representative volumesis exceedingly computationally challenging, as a result, semi-analytic and semi-empiricalprescriptions for star formation in reionization modeling remain the norm in 21-cm mod-eling codes [94, 110, 101]. Though such approaches lack the spatial resolution to modelindividual galaxies or even groups of galaxies, including some information about the galaxypopulation permits joint modeling of 21-cm observables as well as high- z galaxy luminosityfunctions, stellar mass functions, and so on, and thus open up the possibility of tighteningconstraints on the properties of galaxies using a multi-wavelength approach. The very first generations of stars to form in the Universe did so under very different condi-tions than stars today, so it is not clear that the star formation models outlined in the previoussection apply. The first stars, by definition, formed from chemically-pristine material, sinceno previous generations of stars had existed to enrich the medium with heavy elements. Thishas long been recognized as a reason that the first stars are likely different than stars to-day [2, 13, 107, 157]. Without the energetically low-lying electronic transitions commonin heavy elements, hydrogen-only gas clouds cannot cool efficiently, as collisions energeticenough to excite atoms from n = n = α photons) imply temperatures of ∼ K, corresponding to virial masses oforder ∼ M (cid:12) . Such halos are increasingly rare at z (cid:38) M (cid:12) (cid:46) M (cid:12) . Hydrogenmolecules, H , can form using free electrons as a catalyst ,H + e − → H − + h ν (2.41)H − + H → H + e − , (2.42)These reactions are limited by the availability of free electrons and the survivability of H − ions. Even in the absence of astrophysical backgrounds, the formation of H is limited bythe CMB, which at the high redshifts of interest can dissociate the H − ion. [141] found thatthe molecular hydrogen fraction in high- z halos scales with the virial temperature as f H ≈ . × − (cid:18) T vir K (cid:19) . . (2.43) Setting T min ∼ K (or M min ∼ M (cid:12) ) is thus a way to roughly exclude the effects of PopIII-hosting“minihalos” in a 21-cm model. Dust is the primary catalyst of H formation in the local Universe, but of course is does not exist in thefirst collapsing clouds. Exotic models in which an X-ray background emerges before the formation of the first stars may affectearly star formation by boosting the electron fraction. CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND
Once the first stars form, the situation grows considerably more complicated. As will be de-tailed in the following section ( § ∼ . . − dissociation by the CMB as the most important mechanism capable of regulating starformation in chemically pristine halos .A substantial literature has emerged in the last ∼
20 years aimed at understanding the crit-ical LW background intensity, J LW , required to prevent star formation in high- z minihaloes.For example, [152] find M crit = . × (cid:18) + z (cid:19) − / ( + . ( π J LW ) . ) M (cid:12) (2.44)where J LW is the LW background intensity in units of 10 − erg s − cm − Hz − sr − . Inprinciple, M crit varies across the Universe from region to region as a function of the local LWintensity, but it is common to use the mean LW background intensity for simplicity. Notefinally that sufficiently dense clouds can self-shield themselves against LW radiation, whichis an important (and still uncertain) aspect of modeling LW feedback [55].While the LW background is responsible for setting the minimum halo mass requiredto host star formation, the maximum mass of Pop III halos, i.e., the mass at which halostransition from Pop III to Pop II star formation, depends on the interplay of many complexprocesses. For example, Pop III supernovae will inject metals into the ISM of their hostgalaxies, which can trigger the transition to Pop II star formation provided that at least somemetals are retained and efficiently mix into proto-stellar clouds. The timescales involvedare highly uncertain and may vary from halo to halo. Some halos may even be externallyenriched [135]. As a result, whereas UVLFs at high- z provide some insight into the Pop IISFE, the Pop III SFE, which encodes the complex feedback processes at play, is completelyunconstrained.Figure 2.4 shows some example predictions for the Pop III SFRD in a semi-analyticmodel of Pop III star formation. Clearly, the level of Pop III star formation is subject tomany unknowns, which results in a vast array of predictions spanning spanning ∼ ρ ∗ , III ∼ − M (cid:12) yr − cMpc − . Even a crude constraint on the Pop III SFRD would rule outentire classes of models and thus provide a vital constraint on star formation processes in theearliest halos.In 21-cm models it is common to neglect a detailed treatment of individual Pop III star-forming halos, and instead parameterize the impact of Pop II and Pop III halos separately.One way to do this is to assume all atomic cooling halos form Pop II stars (with ζ LW , II , ζ X , II ,etc.), and all minihalos form Pop III, with their own efficiency factors ζ X , II etc., and m min determined self-consistently from the emergent LW background intensity [36, 97]. We willrevisit the predictions of these models in § If the PopIII IMF is very bottom-heavy, the resulting IR background could continue to regulate star forma-tion via H − photo-detachment [155]. .2. SOURCES OF THE UV AND X-RAY BACKGROUND Predictions for the Pop III SFRD at high- z [84]. Upper panel shows resultsassuming a low-mass Salpeter-like Pop III IMF with different assumptions about the Pop IIand Pop III star formation prescriptions. Lower panels study the effects of Pop III IMF be-tween energy-regulated (left) and momentum-regulated (right) stellar feedback from Pop IIstars. Note that the Pop II SFRD inferred from current UVLF measurements is of order ∼ − M (cid:12) yr − cMpc − at z ∼ CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND
With some handle on the efficiency of star formation, we now turn our attention to the ef-ficiency with which stars generate UV photons, particularly in the Lyman continuum andLyman Werner bands . In principle the 21-cm background is also sensitive to the spectrumof even harder He-ionizing photons, since photo-electrons generated from helium ionizationcan heat and ionize the gas, while HeII recombinations can result in H-ionizing photons.The 21-cm signal could in principle also constrain the rest-frame infrared spectrum of starsin the early Universe, since IR photons can feedback on star-formation at very early timesthrough H − photo-detachment [155]. However, in this section, we focus only on the soft UVspectrum (10 . (cid:46) E (cid:46) . • The stellar initial mass function (IMF), ξ ( m ) , i.e., the number of relative number ofstars formed in different mass bins. Commonly-adopted IMFs include Salpeter [123],Chabrier [18], Kroupa [69], and Scalo [127] which are all generally power-laws withindices ∼ − . M ∗ < . M (cid:12) ). • Models for stellar evolution, i.e., how stars of different masses traverse the Hertzprung-Russell diagram over time. • Models for stellar atmospheres, i.e., predictions for the spectrum of individual stars asa function of their mass, age, and composition.With all these ingredients, one can synthesize the spectrum of a stellar population formedsome time t after a “burst,” L ν ( t ) = (cid:90) t dt (cid:48) (cid:90) ∞ m min dm ξ ( m ) l ν ( m , t (cid:48) ) (2.45)where l ν ( m , t ) is the specific luminosity of a star of mass m and age t , and we have assumedthat ξ is normalized to the mass of the star cluster, (cid:82) dm ξ ( m ) = M ∗ . Equation 2.45 can begeneralized to determine the spectrum of a galaxy with an arbitrary star formation history(SFH) composed of such bursts. Widely used stellar synthesis codes include STARBURST
BPASS [31], Flexible Stellar Population Synthesis (
FSPS ) [23], Stochastically Lightingup Galaxies (
SLUG ) [26], and the Bruzual & Charlot models [16].Generally, 21-cm models do not operate at level of SPS models because the 21-cm back-ground is insensitive to the detailed spectra and SFHs of individual galaxies. Instead, be-cause 21-cm measurements probe the relatively narrow intervals 10 . < h ν / eV < . h ν > . We use this definition here loosely. Technically, the LW band is ∼ . − . H . The Ly- α background is sourced by photonsin a slightly broader interval, ∼ . − . α photons. .2. SOURCES OF THE UV AND X-RAY BACKGROUND N ion and N α , which integrateover age and the details of the stellar SED, i.e., N ion = m − ∗ (cid:90) ∞ dt (cid:48) (cid:90) ∞ ν LL d ν h ν L ν ( t (cid:48) ) (2.46) N α = m − ∗ (cid:90) ∞ dt (cid:48) (cid:90) ν LL ν α d ν h ν L ν ( t (cid:48) ) (2.47)where ν LL is the frequency of the Lyman limit (13.6 eV) and ν α is the Ly- α frequency.UV emission is dominated by massive, short-lived stars, hence the integration from t = t = ∞ .Assuming a Scalo IMF, stellar metallicity of Z = Z (cid:12) /
20, using
STARBURST
99 SPSmodel, [4] report N ion = N α = n resonance. The general expectation is for N ion and N α increase for more top-heavy IMF and lower metallicity, meaning these values arelikely to increase for Pop III stars [14, 148, 128]. Similarly, binary evolution can effectivelyincrease the lifetimes of massive stars, leading to a net gain in UV photon production [137].Note that if simultaneously modeling the UVLF one generally assumes a constant starformation rate, in which case the UV luminosity of stellar populations asymptotes after a fewhundred Myr. As a result, it is common to use the results of SPS models (with continuousstar formation) at t =
100 Myr when converting UV luminosity to SFR, as in Eq. 2.38,though in reality the detailed star formation history is (generally) unknown. In this particularcase, we do not quantify the UV luminosity in photons per stellar baryon – instead, the exactvalues of f ∗ , l , N ion , and N α should all be determined self-consistently from the modelcalibration. In other words, one cannot simply change N ion and f ∗ independently, since theinferred value of f ∗ depends on the assumed stellar population model, and thus implicitly on N ion (see also § Even if the intrinsic stellar spectrum of galaxies were known perfectly, our ability to draw in-ferences about star formation are hampered by the presence of dust, which dims and reddensthe “true” spectrum of galaxies. The opacity of dust is an inverse function of wavelength,meaning its impact is greatest at short wavelengths [154]. Unfortunately, most observationsof high- z galaxies (so far) target the rest UV spectrum of galaxies , and thus must be con-siderably “dust corrected” before star formation rates and/or efficiencies are estimated.However, correcting for the effets of dust attenuation is not completely hopeless. If weassume that UV-heated dust grains radiate as blackbodies, we would expect to see an “in-frared excess” (IRX) in galaxies with redder-than-expected UV continuua, as UV reddeningis suggestive of dust attenuation. If we assume for simplicity a power-law UV continuum, f λ = f λ , λ β , we would expect an excessIRX ≡ F FIR F = (cid:82) ∞ f λ , ( − e − τ λ ) d λ f , e − τ λ (cid:18) F FIR F bol (cid:19) (2.48) This is simply due to the limited availability and sensitivity of near-infrared observations, which will soonbe greatly enhanced by the James Webb Space Telescope. CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND where τ λ is the wavelength-dependent dust opacity, equivalently written via 10 − . A = e − τ , where A is the extinction at 1600 ˚ A in magnitudes, f , is the intrinsic intensityat 1600 ˚ A , and F FIR / F bol is a correction factor that accounts for the fraction of the bolometricdust luminosity emitted in the far-infrared band of observation .An empirical constraint on the so-called IRX- β relation was first presented by [88], whofound A = . − . β obs , where β obs is the logarithmic slope of the observed rest UVspectrum. The intrinsic UV slope of young stellar populations is generally − (cid:46) β (cid:46) − − (cid:46) β obs (cid:46) − β relation, the cause of its scatter, andthe possibility that it evolves with time [102, 122]. Early efforts with ALMA [17, 11] arebeginning to test these ideas with rest-IR observations of z ∼ β indicative of reduced dust obscuration in higherredshift galaxies. However, such inferences are currently dependent on assumptions for thedust temperature – if dust at high- z is warmer than dust at low- z , the data may be consistentwith no evolution in A at fixed M UV or stellar mass. These uncertainties in correctinghigh- z rest-UV measurements for dust reddening may affect the normalization of f ∗ at thefactor of ∼ few level, and could also bias the shape of the inferred f ∗ depending on preciselyhow A scales with M UV (or m ∗ ). While photons with wavelengths longer than 912 ˚ A are most likely to be absorbed by dustgrains, as described in the previous section, photons with wavelengths shortward of 912 ˚ A will be absorbed by hydrogen and helium atoms. As a result, reionization models mustalso account for local attenuation, since the ionization state of intergalactic gas is of courseonly influenced only by the ionizing photons that are able to escape galaxies. The fraction ofphotons that escape relative to the total number produced is quantified by the escape fraction, f esc , and is the final component of the ionizing efficiency, ζ , introduced previously (see Eq.2.34).Current constraints on high- z galaxies and reionization suggest that f esc must be ∼ −
20% [121]. The result is model-dependent, however, as it relies on assumptions about theUV photon production efficiency in galaxies and extrapolations to source populations beyondcurrent detection limits. For example, if f esc depends inversely on halo mass, reionizationcan be driven by galaxies that have yet to be detected directly [41] .Numerical simulations now lend credence to the idea that escape fractions of ∼ −
20% are possible, with perhaps even larger f esc in low-mass halos [66, 156]. The basictrend is sensible: as the depth of halo potentials declines, supernovae explosions can moreeasily excavate clear channels through which photons escape. However, there is far from aconsensus on this issue. For example, the FIRE simulations do not see evidence that f esc depends on halo mass [73], and on average f esc (cid:46) Note that the above expression assumes that all heating is done by photons redward of the Lyman limit andneglects heating by line photons. This scenario is appealing because it can explain the very gradual evolution in the post-reionization ioniz-ing background, and rarity of galaxies leaking LyC radiation at 3 (cid:46) z (cid:46) .2. SOURCES OF THE UV AND X-RAY BACKGROUND f esc is set by very small-scalestructure in the ISM, rather than the depth of the host halo potential.21-cm measurements in principle open a new window into constraining f esc . If, forexample, the UV/SFR conversion factor is well known and dust can be dealt with (see § z galaxy LFs can isolate f ∗ and f esc [110,57]. Note, however, that this is still model-dependent, as f ∗ must be extrapolated to somelimiting UV magnitude or halo mass in order to obtain the total photon production rate perunit volume. Though stars themselves emit few photons at energies above the HeII-ionizing edge ( ∼ . § L X = . × (cid:18) ˙ M ∗ M (cid:12) yr − (cid:19) erg s − (2.49)where L X is defined here as the luminosity in the 0.5-8 keV band. This relation providesan initial guess for many 21-cm models, which add an extra factor f X to parameterize ourignorance of how this relation evolves with cosmic time. For example, [49] write L X = × f X (cid:18) ˙ M ∗ M (cid:12) yr − (cid:19) erg s − , (2.50)which is simply Equation 2.49 re-normalized to a broader energy range, 0 . < h ν / keV < × , assuming a power-law spectrum with spectral index α X = − .
5, where α X is definedby L E ∝ E α X . Coupled with estimates for the star formation rate density at high- z , the L X -SFR relation suggests that X-ray binaries could be considerable sources of heating in thehigh- z IGM [49, 45, 92, 38, 75].The normalization of these empirical L X -SFR relations are not entirely unexpected, atleast at the order-of-magnitude level. For example, if one considers a galaxy forming stars ata constant rate, a fraction f • (cid:39) − of stars will be massive enough ( M ∗ > M (cid:12) ) to form ablack hole assuming a Chabrier IMF. Of those, a fraction f bin will have binary companions,2 CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND with a fraction f surv surviving the explosion of the first star for a time τ . If accretion ontothese black holes occurs in an optically thin, geometrically-thin disk with radiative efficiency ε • = . f . − = .
84 of the bolometric luminosity will originate in the 0.5-8 keVband. Finally, assuming these BHs are “active” for a fraction f act of the time, we can write[91, 97] L X ∼ × erg s − (cid:18) ˙ M ∗ M (cid:12) s − (cid:19) (cid:16) ε • . (cid:17) (cid:18) f • − (cid:19) (cid:18) f bin . (cid:19) (cid:18) f surv . (cid:19) (cid:18) τ
20 Myr (cid:19) (cid:18) f act . (cid:19) (cid:18) f . − . (cid:19) . (2.51)While several of these factors are uncertain, particularly f surv and f act , this expression pro-vides useful guidance in setting expectations for high redshift. For example, it has long beenpredicted that the first generations of stars were more massive on average than stars todayowing to inefficient cooling in their birth clouds. This would boost f • , and thus L X / ˙ M ∗ , solong as most stars are not in the pair-instability supernova (PISN) mass range, in which noremnants are expected.There are of course additional arguments not present in Eq. 2.51. For example, theMCD spectrum is only a good representation of HMXB spectra in the “high soft” state. Atother times, in the so-called “low hard” state, HMXB spectra are well fit by a power-law.The relative amount of time spent in each of these states is unknown. Figure 2.5 comparestypical HMXB spectra with the spectrum expected from hot ISM gas (see § L X -SFR relation may invoke the metallicity as adriver of changes in the relation with time and/or galaxy (stellar) mass. As the metallicitydeclines, one might expect the stellar IMF to change (as outlined above), however, the windsof massive stars responsible for transferring material to BHs will also grow weaker as theopacity of their atmospheres declines. As a result, increases in L X /SFR likely saturate belowsome critical metallicity. Observations of nearby, metal-poor dwarf galaxies support thispicture, with L X /SFR reaching a maximum of ∼
10x the canonical relation quoted in Eq.2.49 [89].
Though super-massive black holes (SMBHs) are exceedingly rare and thus unlikely to con-tribute substantially to the ionizing photon budget for reionization [60] (though see [76]),fainter – but more numerous – intermediate mass black holes (IMBHs) with 10 (cid:46) M • / M (cid:12) (cid:46) could have a measureable impact on the IGM thermal history [159, 120, 140]. Growingblack holes, if similar to their low- z counterparts, could also generate a strong enough radiobackground to amplify 21-cm signals [32], possibly providing an explanation for the ana-malous depth of the EDGES global 21-cm measurement [12]. Such scenarios cannot yet beruled out via independent measurements. For example, the unresolved fraction of the cosmicX-ray background still permits a substantial amount of accretion at z (cid:38) ∼
10% of the radio excess reported by ARCADE-2 [44, 134] must originate at z (cid:38) z (cid:38)
6, the signatures of BH growth in the 21-cm background areworth exploring in more detail. .2. SOURCES OF THE UV AND X-RAY BACKGROUND -1 E [keV] R [ M p c ] d E X / d l og ( E ) [ e V ] Figure 2.5:
Models for the X-ray spectra of star-forming galaxies.
Left:
Template CygnusX-1 HMXB spectrum (solid) compared with power-law (dashed), with mean free path shownon left scale and relative luminosity on right [38].
Right:
Typical HMXB spectrum (blue)compared to soft X-ray spectrum characteristic of bremmstrahlung emission from hot ISMgas [109].
While compact remnants of massive stars are likely the leading producer of X-rays in high- z star-forming galaxies, the supernovae events in which these objects are formed may not befar behind. Supernovae inject a tremendous amount of energy into the surrounding medium,which then cools either via inverse Compton emission (in supernova remnants; [106]) oreventually via bremsstrahling radiation (in the hot interstellar medium; ISM). Because thesesources are related to the deaths of massive stars their luminosity is expected to scale withSFR, as in the case of HMXBs. Indeed, [90] find that diffuse X-ray emission in nearbysources follows the following relation in the 0.5-2 keV band: L X = . × (cid:18) ˙ M ∗ M (cid:12) yr − (cid:19) erg s − (2.52)This luminosity is that from all unresolved emission, and as a result, is not expected to traceemission from the hot ISM alone. Emission from supernova remnants will also contributeto this luminosity, as will fainter, unresolved HMXBs and LMXBs. [90] estimate that ∼ −
40% of this emission may be due to unresolved point sources.Though the soft X-ray luminosity from hot gas appears to be subdominant to the HMXBcomponent in nearby galaxies, at least in total power, there are of course uncertainties in howthese relations evolve. Furthermore, the bremmstrahlung emission characteristic of hot ISMgas has a much steeper ∼ ν − . spectrum than inverse Compton ( ∼ ν − ) or XRBs ( ∼ ν − or ν − . ), and thus may heat more efficiently (owing to σ ∝ ν − cross section) provided softX-rays can escape galaxies.4 CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND
Though the mean free paths of X-rays are longer than those of UV photons, they still maynot all escape from galaxies into the IGM. For example, hydrodynamical simulations suggesttypical hydrogen column densities of N H I ∼ cm − in low-mass halos [27], which issubstantial enough to eliminate emission below ∼ . ζ X , the specificX-ray luminosity is modeled as L X , ν = L X , (cid:18) h ν (cid:19) α X exp [ − σ ν N H I ] (2.53)and the normalization, L X , , spectral index α X , and typical column density, N H I , are left asfree parameters. It is common to approximate this intrinsic attenuation with a piecewisemodel for L X , i.e., L X , ν = (cid:26) h ν < E min L X , (cid:0) h ν (cid:1) α X h ν ≥ E min (2.54)Note that N H I (or E min ) can be degenerate with the intrinsic spectrum, e.g., the SED ofHMXBs in the high-soft state exhibits a turn-over at energies h ν < High energy cosmic rays (CRs) produced in supernovae explosions offer another potentialsource of ionization and heating in the bulk IGM [103, 126, 70]), though most likely theeffects are only discernible in the thermal history. Simple models suggest that CRs can raisethe IGM temperature by ∼ −
200 K by z ∼
10 depending on the details of the CR spectrum[70]. CRs are thus a potentially important, though relatively unexplored, source of heatingin the high- z IGM.
So far we have assembled a simple physical picture of the IGM at high redshift ( § § CMFAST [86] including a 2-D slice ofthe δ T b field, the global 21-cm signal, and power spectrum on two spatial scales [87]. Timeproceeds from right to left from ∼
20 Myr after the Big Bang until the end of reionization ∼ .3. PREDICTIONS FOR THE 21-CM BACKGROUND Dark Ages Ly α coupling X-‐ray hea6ng Reioniza6on Figure 2.6:
Predictions from the 21
CMFAST
Evolution of Structure (EoS) model suite[87].
Top : 2-D slice of the brightness temperature field, with red colors indicating a coolIGM, blue colors indicative of a heated IGM, and black representing a null signal (eitherdue to ionization or T S = T CMB ). Middle:
Global 21-cm signal, with dashed line indicating δ T b = Bottom:
Evolution of the dimensionless 21-cm power spectrum, ∆ = k P ( k ) / π ,on two different scales, k = . − (dotted) and k = . − (solid).6 CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND
The Dark Ages:
As the Universe expands after cosmological recombination, Compton scat-tering between free electrons and photons keeps the radiation and matter temperaturein equilibrium. The density is high enough that collisional coupling remains effec-tive, and so T S = T K = T CMB . Eventually, Compton scattering becomes inefficient asthe CMB cools and the density continues to fall, which allows the gas to cool fasterthan the CMB. Collisional coupling remains effective for a short time longer and so T K initially tracks T S . This results in the first decoupling of T S from T CMB at z ∼ z ∼
80 ( ν ∼
15 MHz), which comes to an endas collisional coupling becomes inefficient, leaving T S to reflect T CMB once again.
Ly- α coupling: When the first stars form they flood the IGM with UV photons for thefirst time. While Lyman continuum photons are trapped near sources, photons withenergies 10 . < h ν / eV < . α resonance orcascade via higher Ly- n levels, giving rise to a large-scale Ly- α background capableof triggeiring Wouthuysen-Field coupling as they scatter through the medium (see alsoCh. 1 and § T S is driven back toward T K , which (in most models)still reflects the cold temperatures of an adiabatically-cooling IGM. X-ray Heating:
The first generations of stars beget the first generations of X-ray sources,whether they be the explosions of the first stars themselves or remnant neutron starsor black holes that subsequently accrete. Though the details change depending on theidentity of the first X-ray sources (see § T S > T CMB . Once T S (cid:29) T CMB , the 21-cm signal “saturates,” and subsequently is sensitive only to the density and ionizationfields. However, it is possible that heating is never “complete” in this sense beforereionization, meaning neutral pockets of IGM gas may remain at temperatures at orbelow T CMB until they are finally engulfed by ionized bubbles.
Reionization:
As the global star formation rate density climbs, the growth of ionized re-gions around groups and clusters of galaxies will continue, eventually culminating inthe completion of cosmic reionization. This rise in ionization corresponds to a declinein the amount of neutral hydrogen in the Universe capable of producing generating21-cm signals. As a result, the amplitude of the 21-cm signal, both in its mean andfluctuations, falls as reionization progresses. After reionization, neutral hydrogen onlyremains in systems over-dense enough to self-shield from the UV background.The particular model shown in Figure 2.6 [87] assumes that very faint galaxies domi-nate the UV and X-ray emissivity, which results in relatively early features in the 21-cmbackground, e.g., both the power spectrum and global 21-cm signal peak in amplitude at z ∼
18. Reionization and reheating occur later in scenarios in which more massive halosdominate the emissivity, and may even occur simultaneously, resulting in strong 21-cm sig-nals at z (cid:46)
12 [87, 94, 110].For the remainder of this section we focus on changes in the 21-cm signal wrought by pa-rameters of interest. We limit our discussion to the global 21-cm signal and power spectrum,though there are of course many other statistics one could use to constrain model parameters .3. PREDICTIONS FOR THE 21-CM BACKGROUND
Analytic models of bubble growth during the EoR [48].
Left:
Bubble sizedistributions at bubble filling fractions of Q = . , . , . , .
5, and 0.74, from left toright.
Middle:
Correlation function of ψ = x H I ( + δ ) (solid) at x H I = .
81 (top) and x H I = .
52 (bottom) as well as its constituent components, including the ionization auto-correlationfunction (dashed), density auto-correlation function (dotted), and cross-correlation functionbetween ionization and density (dot-dashed).
Right:
Dimensionless power spectrum of ψ fordifferent values of ζ , including ζ =
12 (thin) and ζ =
40 (thick), at several neutral fractions, x H I = .
96 (dotted), 0.8 (short-dashed), 0.5 (long-dashed), and 0.26 (solid).(see Chapter 4). We note that there is no consensus parameterization for models of galaxyformation or the 21-cm background, nor do all models incorporate the same physical pro-cesses or employ the same numerical techniques. As a result, in this section we make noeffort to closely compare or homogenize results from the literature, but instead draw exam-ples from many works in order to illustrate different aspects of the 21-cm background as aprobe of galaxy formation.
Generally written as ζ or ζ ion , the ionizing efficiency quantifes the number of Lyman contin-uum (LyC) photons that are produced in galaxies and escape into the IGM, i.e., ζ = f ∗ N ion f esc (see § (cid:46) z (cid:46) ν (cid:38)
130 MHz), during which the bulk of reionization likely takes place.Figure 2.7 shows predictions for the growth of ionized bubbles in the excursion set for-malism [48]. In time, bubbles grow larger, eventually reaching typical sizes of ∼ tens ofMpc during reionization. The two-point correlation function of the ionization field (middle)grows with time as well, peaking near the midpoint of reionization [72]. This rise and fall isreflected in the 21-cm power spectrum as well (right), here modeled in the “saturated limit” T S (cid:29) T CMB , in which case only fluctuations in ψ = x H I ( + δ ) need be considered. Largervalues of ζ (thicker lines in right panel of Fig. 2.7) result in stronger fluctuations on largescales and a suppression in the power on small scales.Figure 2.8 shows results from four different numerical simulations (RT post-processedon N-body simulation) [83], each differing in their treatment of ζ . The key difference is how ζ depends on halo mass – here, models span the range of ζ ∝ m − / h (S2) to ζ ∝ m / h (S3),8 CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND
Figure 2.8:
Ionization field in (
94 cMpc ) box for different ionizing efficiency assump-tions [83] as a function of redshift (top to bottom). Neutral gas is black, while ionizedregions are white. Each column shows results from a different simulation, with S1-S3 using ζ = constant, ζ ∝ m − / h , and ζ ∝ m / h , respectively. S4 is the same as S1 except only haloswith m h > M (cid:12) are included.including the case of ζ = constant (S1). As the ionizing emissivity becomes more heavilyweighted toward more massive, more rare halos (in S3 and S4), ionized structures grow largerand more spherical, while the smaller bubbles nearly vanish. This is a result of an increasein the typical bias of sources as ζ increases with m h – because more massive halos are moreclustered, ionizing photons from such halos combine to make larger ionized regions, whereasless clustered low-mass halos carve out smaller, more isolated ionized bubbles.From Fig. 2.8 it is clear that the behavior of ζ not only sets the timeline for reioniza-tion but also its topology. However, ζ is degenerate with m min , since the ionizing emissivitycan be enhanced both by increasing ζ directly or by increasing the number of star-forminghalos by decrasing m min (recall that the total number of ionizing photons emitted in a re-gion is N γ = ζ f coll ). Despite this degeneracy, power spectrum measurements expected tobe able to place meaningful constraints on both parameters [56]. The power spectrum on k ∼ . h − scales reliably peaks near the midpoint of reionization [72], meaning some(relatively) model-independent constraints are expected as well. .3. PREDICTIONS FOR THE 21-CM BACKGROUND Effects of Ly- α and X-ray efficiencies on the global 21-cm signal [117, 109]. Left:
Predictions for the global 21-cm signal showing sensitivity to the normalization of the L X -SFR relation, f X (top), and the production efficiency of Ly- α photons, f α (bottom) [117]. Right:
Predictions for evolution in the 21-cm power spectrum at k = . − for modelswith different X-ray spectra [109]. Blue curves indicate soft power-law spectra with indicesof α =
3, while red curves are indicative of hard spectra sources with α = .
8. Linestylesdenote different minimum virial temperatures, T min , and lower energy cutoffs for the X-raybackground, E . The progression of cosmic reheating is analogous in some respects to reionization thoughdriven by sources of much harder photons (see § § ζ X ), spectralcutoff ( E min ), and power-law slope of X-ray emission ( α X ). The combination of these pa-rameters can capture a variety of physical models and mimic the shape of more sophisticatedtheoretical models (e.g., the multi-color disk spectrum; [99]).Holding the SED fixed, variations in ζ X affect the thermal history much like ζ affects theionization history: increasing ζ X causes efficient heating to occur earlier in the Universe’shistory, resulting in lower frequency (and shallower) absorption troughs in the global 21-cmsignal, while the peak amplitude of the power spectrum also shifts to earlier times (at fixedwavenumber). Effects of f X on the global signal can be seen in the left panel of Figure 2.9(see also [91, 92, 38]).Allowing the SED of X-ray sources can change the story dramatically because the meanfree path of X-rays is a strong function of photon energy ( l mfp ∝ ν − due to the bound-freeabsorption cross section scaling σ ∝ ν − ; [150]). This strong energy dependence means thatphotons with rest energies h ν (cid:38) z (cid:38) CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND
Photons with h ν (cid:46) ∼ . α X = ∼ − α X = . L X ∝ h ν − α X .The hardness of X-ray emission is also controlled by the “cutoff energy,” E min , belowwhich X-ray emission does not escape efficiently from galaxies (see § E min could indicate an intrinsic turn-over in the X-rayspectra of galaxies, e.g., that expected from a multi-color disk spectrum [99]. Joint con-straints on ζ X , E min , and α X are thus required to help identify the sources of X-ray emissionin the early Universe and the extent to which their spectra are attenuated by their host galax-ies. Finally, it is important to note that interpreting ζ X is potentially more challenging thaninterpreting ζ given the additional parameters needed to describe the X-ray SED. One mustbe mindful of the fact that ζ X quantifies the X-ray production efficiency in some bandpass ,generally 0.5-8 or 0.5-2 keV. As a result, changing α X or E min may be accompanied by anormalization shift so as to preserve the meaning of ζ X . This degeneracy can be mitigatedto some extent by re-defining ζ X in the ( E min − α Efficiency
The production efficiency of Ly- α photons affects when the 21-cm background first “turnson” due to Wouthuysen-Field coupling (see Ch. 1 and § f α has on the global 21-cm signal [117]. For very largevalues, f α =
100 (blue), the dark ages come to an end at z ∼
30 ( ν ∼
45 MHz), triggering amuch deeper absorption trough than the fiducial model (with f α = f X =
1; black lines). Theintuition here is simple: at fixed f X , increasing f α drives T S → T K at earlier times, meaningthere has been less time for sources to heat the gas.Despite the very long mean-free paths of photons that source the Ly- α background, thereare still fluctuations in the background intensity J α [4, 3, 62]. As a result, there will befluctuations in the spin temperature, as different regions transition from T S ≈ T CMB to T S ≈ T K at different rates. The onset of Ly- α coupling is visible in the right panel of Figure 2.9, asthe power (at k=0.2 Mpc − ) departs from its gradual descent at z ∼
25 ( T min = K) and z ∼
33 ( T min = K).Because the Ly- α background is sourced by photons in a relatively narrow frequencyinterval, 10 . (cid:46) h ν / eV (cid:46) .
6, the timing of Wouthuysen-Field coupling and the amplitudeof fluctuations are relatively insensitive to the SED of sources. Similarly, because hydrogengas is transparent to these photons (except at the Ly- n resonances) these photons have anescape fraction f esc , LW (cid:38) .
5, at least in the far field limit [129], as their only impediment isH , which is quickly dissociated by stellar Lyman-continuum emission. .3. PREDICTIONS FOR THE 21-CM BACKGROUND
50 100 150 200 ν (MHz) −200−150−100−50050 δ T b ( m K ) c X =const . Z =0 . Z =0 . Z =0 . Z =0 . Z =0 .
50 100 150 200 ν (MHz) c X ∝ ( Z/ . − .
80 30 20 15 12 10 8 7 6 z − σ µ +2 σ τ e
80 30 20 15 12 10 8 7 6 z T S ( z =8 . Figure 2.10:
Effects of stellar metallicity on the global 21-cm signal [94].
Left:
Metallicityeffects assuming no link between stellar metallicity, Z ∗ , and X-ray luminosity. Right : Metal-licity effects assuming empirical relation between f X and Z ∗ [15]. Insets show predictionsfor CMB optical depth, τ e (left) and mean IGM spin temperature at z = . As shown in the previous sections, it is common to allow ζ , ζ X , and ζ α to vary independentlyas free parameters. However, if all features of the 21-cm background are driven by stars andtheir remnants, and the properties of such objects do not vary with time, then these efficiencyfactors will be highly correlated. For example, the number of Lyman continuum photonsproduced per unit star formation is inversely proportional to stellar metallicity, Z , as is theyield in the Lyman Werner band, so it may be more appropriate to use Z as the free parameterinstead of N ion and N LW . It is more difficult to connect the X-ray luminosity per baryon, N X , to Z as it depends on poorly understood details of the late stages of stellar evolution andcompact binaries [7]. However, observationally the L X -SFR relation (see § ζ X to Z [94].Figure 2.10 shows these effects on the global 21-cm signal [94]. In the left panel, no linkbetween L X /SFR and Z is assumed, while in the right panel the empirical relation with f X ∝ Z − . is adopted. In each case, though particularly in the left panel, the effect of metallicityis very small. This is because these models force a match to high- z UVLF measurements[10, 43], which means any change in Z ∗ also affects the 1600 ˚ A luminosity to which UVLFmeasurements are sensitive. As a result, changes in metallicity make galaxies more or lessbright in the UV, but to preserve UVLFs, the efficiency of star formation must compensate(see § f X depends on Z (right panel), the global 21-cm signal becomesmore sensitive to changes in Z because the change in X-ray luminosity can overcome thedecline in SFE as Z decreases [94].In reality, the metallicity is a function of galaxy mass and time, so the simple constant Z ∗ models above are of course simplistic. Note also that the models in Figure 2.10 only2 CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND include atomic cooling halos. As a result, observed signals peaking at lower frequencies(like the EDGES 78 MHz signal [12]) likely require minihalos and/or non-standard sourceprescriptions [97, 93].
The minimum halo mass (or equivalent virial temperature) for star formation sets the totalnumber of halos emitting UV and X-ray photons as a function of redshift and thus influencesall points in 21-cm background, unlike the ζ factors, which largely impact a single feature.Fiducial models often adopt the mass corresponding to a virial temperature of 10 K, sincegas in halos of this mass will be able to cool atomically, i.e., there is not an obvious barrierto star formation in halos of this mass. Reducing m min , as is justified if star formation inminihalos is efficient, results in a larger halo population, while increasing m min of coursereduces the halo population. Moreover, for models in which low-mass halos are the dominantsources of emission, the typical star-forming halo is less biased than that drawn from a modelin which high-mass halos dominate. As a result, changing m min in principle affects both thetiming of events in the 21-cm background as well as the amplitude of fluctuations.As shown in Figure 2.11 [85], m min indeed affects all features of the 21-cm background,both in the global signal and fluctuations (see also, e.g., [40, 96]). With no other changes tothe model the effects are largely systematic, i.e., the timing of features in the global signaland power spectrum are shifted without a dramatic change in their amplitude. Notice alsothat changing m min can serve to mimic the effects of including warm dark matter (e.g., reddotted vs. magenta dash-dotted curve), which suppresses the formation of small structuresthat would otherwise (presumably) host galaxies.Not depicted in Figure 2.11 is the possibility that m min evolves with time. Initially, only amild redshift-dependence is expected just from linking m min to a constant virial temperatureof ∼
500 K [141], which is required for molecular cooling and thus star formation to initiallybegin (see § m min will react to the LWbackground [59, 74, 152], and likely rise to the atomic cooling threshold, T min ∼ K, at z (cid:38)
10 [146, 84]. During reionization, this threshold may grow even higher, as ionizationinhibits halos from accreting fresh gas from which to form stars [53, 104, 158].Figure 2.12 shows 21-cm power spectra for various models of feedback in the first star-forming halos [39]. Both the strength of feedback and type of feedback (LW and/or baryon-velocity streaming in this particular example) change the power spectrum by a factor of ∼ − .3. PREDICTIONS FOR THE 21-CM BACKGROUND Effects of the minimum mass on the global 21-cm signal and 21-cm powerspectrum on k = . − scales [85]. Solid black, red dotted, and magenta dash-dottedcurves hold constant f X = m min by-hand (first two) and via a 2 keV warmdark matter particle (magenta). Left and right panels differ only in which sensitivity curvesare included for comparison.Figure 2.12: Effects of LW feedback on the 21-cm power spectrum [39].
Left:
Powerspectra with different feedback models, including no feedback (red), weak (blue), strong(green), and saturated (black). Dashed curves exclude the baryon-DM velocity offset effect[147]. Right: All models here include the baryon-velocity offset effect – linestyles indi-cate power spectra at three different redshifts. Note that because changes to the strength offeedback shift the timing of events, models are compared at fixed increments relative to the“heating redshift,” z , which in these models occurs between z ∼
15 and z ∼
18 [39]. Dashedlines are power spectra at z = z , while z = z + z = z + CHAPTER 2. ASTROPHYSICS FROM THE 21-CM BACKGROUND ν min (MHz) − − − − − δ T b ( ν m i n )( m K ) ˙ M ∗ , III [ M ⊙ yr − ] − − − − − − A (MHz) . . . . . . . W ( M H z m K − ) − . − . − . − . − . − . − . − . − . (cid:0) δT ′ b (cid:1) lo . . . . . . . . . . (cid:0) δ T ′ b (cid:1) h i ν min (MHz) − − − − − δ T b ( ν m i n )( m K ) f X, III − − A (MHz) . . . . . . . W ( M H z m K − ) − . − . − . − . − . − . − . − . − . (cid:0) δT ′ b (cid:1) lo . . . . . . . . . . (cid:0) δ T ′ b (cid:1) h i Figure 2.13:
Potential Pop III signatures in the global 21-cm signal [97].
Comparisonbetween Pop II-only and Pop III models performed in three diagnostic spaces, includingthe absorption trough position (left), the prominence of its wings, W , and its asymmetry, A (middle), and the mean slopes at frequencies above and below the extremum (right).Black contours enclose sets of PopII models generated by Monte Carlo sampling a viablerange of parameter space constrained by current observations, while the green polygons areslices through a 3-D Pop III model grid, first assuming assuming a fixed Pop III SFE andvarying the X-ray production efficiency (top row), and then for different SFE models hav-ing marginalized over all f X , III (bottom row). Measurements falling in regions of overlapbetween the green and black contours would have no clear evidence of PopIII, while mea-surements falling only within the green contours would be suggestive of PopIII. .4. SUMMARY
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