RADIATIVE TRANSFER IN A CLUMPY UNIVERSE. IV. NEW SYNTHESIS MODELS OF THE COSMIC UV/X-RAY BACKGROUND

In the case of LyC absorption by Poisson-distributed systems, the effective opacity between z_o and z is

where is the bivariate distribution of absorbers in redshift and column density along the line of sight, τ_c is the continuum optical depth at frequency ν′ = ν_o(1 + z′)/(1 + z_o) through an individual absorber,

where N_i and σ_i are the column densities and photoionization cross sections of ion i.

Besides photoelectric absorption, resonant absorption by the hydrogen and helium Lyman series will produce a sawtooth modulation of the radiation spectrum (Madau & Haardt 2009; Haiman et al. 1997). Continuum photons that are redshifted through the Lyα frequency, ν_α, are resonantly scattered until they redshift out of resonance: the only two Lyα line destruction mechanisms, two-photon decay and O iii Bowen fluorescence (Kallman & McCray 1980), can typically be neglected in the low metallicity, low density IGM. This is not true, however, for photons passing through a higher order Lyman-series resonance, which will be absorbed and degraded via a radiative cascade rather than escaping by redshifting across the line width. Since the line absorption cross section is a narrow, strongly peaked function, the effective line absorption optical depth for a photon observed at (z_o, ν_o < ν_n) that passed through a resonance at redshift z_n = (1 + z_o)(ν_n/ν_o) − 1, can be written as

where ν_n = ν_α × 4(1 − n⁻²)/3 is the frequency of the 1s → np Lyman-series transition (n ⩾ 3) and W_n is the rest equivalent width of the line expressed in wavelength units. This opacity is dominated by systems having line center optical depths of order unity, i.e., which lie at the transition between the linear and the flat part of the curve of growth.

Consider, for example, radiation observed at frequency below the Lyβ of hydrogen or helium, ν_o < ν_β. Photons emitted between z_o and z_β = (1 + z_o)(ν_β/ν_o) − 1 can reach the observer without undergoing resonant absorption. Photons emitted between z_β and z_γ = (1 + z_o)(ν_γ/ν_o) − 1 pass instead through the Lyβ resonance at z_β and are absorbed. Photons emitted between z_γ and z_δ = (1 + z_o)(ν_δ/ν_o) − 1 pass through both the Lyβ and Lyγ resonances before reaching the observer. The background intensity can then be written as (Madau & Haardt 2009)

where we have denoted with the symbol O(z_i, z_j) the “Olbers' integrals” on the right-hand side of Equation (2), calculated between redshifts z_i and z_j > z_i and with equal to the relevant continuum opacity,

In Equation (6), z_L = (1 + z_o)(ν_L/ν_o) − 1, ν_L is the frequency at the Lyman limit, and are the Lyman-series effective opacities at redshifts z_β, z_γ, z_δ, .... In the case of resonant absorption by H i, the LyC optical depth is zero in all O-integrals except the last, while in the case of He ii all terms must include photoelectric absorption by H i and He i (as well by He ii in the last term). Equation (6) is easily generalized to higher observed frequencies, ν_n < ν_o < ν_{n + 1}, to read

Note how, in the case of large resonant opacities, only sources between the observer and the “screen” redshift z_n = (1 + z_o)(ν_n/ν_o) − 1 corresponding to the frequency of the nearest Lyman-series line above ν_o will not be blocked from view: the background energy spectrum will show a series of discontinuities, peaking at frequencies just above each resonance, as the first integral in Equation (8) extends over the largest redshift path, and going to zero at resonance.

Each photon absorbed through a Lyman series resonance causes a radiative cascade that ultimately terminates either in a Lyα photon or in two-photon 2s → 1s continuum decay. In the former case the photon scatters until it is redshifted out of resonance, in the latter the photons escape to infinity without further interactions. Consider, for example, the absorption of a Lyβ photon. The excited 3p level is depopulated via 3p → 2s decay (Hα). In the low-density IGM, collisional l-mixing of the 2s–2p levels (Seaton 1959) is negligible, and the cascade can only terminate in two-photon 2s → 1s emission (Hirata 2006). Without l-mixing, the quantum selection rules forbid Lyβ photons from being converted into Lyα: by contrast, excitation of the 4p level by absorption of Lyγ can decay via the 3s or 3d levels to the 2p and ultimately produce Lyα. More generally, the fraction, f_n, of decays from an np state that generates Lyα photons can be determined from the selection rules and the decay probabilities. This fraction is found to increase as f_n = (0, 0.2609, 0.3078, 0.3259, …) for n = (3, 4, 5, 6, …) and to asymptote to 0.359 at large n (Pritchard & Furlanetto 2006). Note that this is valid in the approximation that the IGM is optically thick to higher-order Lyman-series transitions.

What is the Lyα diffuse flux produced by these Lyman-series cascades? Let be the background intensity measured just above the H i or He ii Lyn resonance at redshift z. The Lyn flux that is absorbed and converted into Lyα is , and the proper Lyα volume emissivity generated by this process can be written as

where δ(x) is the Dirac delta function. The additional flux observed at frequency ν_o ⩽ ν_α and redshift z_o from this process is then

(Madau & Haardt 2009), where 1 + z_α = (1 + z_o)(ν_α/ν_o) and the LyC optical depth is zero in the case of resonant absorption by H i. When summing up over all Lyman series lines, the term in square brackets must be replaced by . The same Lyn cascade also produces a two-photon continuum with emissivity given by

where the two-photon emission function f_ν is expressed in the photons per unit frequency interval and is symmetric about ν_α/2.

We note that the underlying assumption in Equations (9) and (11) is that every absorber is a source of unprocessed Lyα line and two-photon continuum radiation, i.e., that these photons escape into intergalactic space without appreciable local absorption. In the case of He ii Lyα emission, this requires negligible “in situ” destruction from dust, metals (O iii Bowen fluorence), and photoelectric absorption by H i, so that the Lyα photons diffuse into the wings and eventually escape from the production site into the IGM. This is a good approximation at the low metallicities that characterize intergalactic absorbers (Kallman & McCray 1980), even more so since the reprocessing of Lyman series photons occurs in a “skin” layer at the surface of an absorption system. In Section 4 we will show that this is a poor approximation in the case of the reprocessing of LyC radiation, a proper treatment of which requires a numerical solution of the radiative transfer equation within individual absorbers.

Inserting our into Equation (3), we can compute the (proper) LyC mean free path for 1 ryd photons as

This is plotted in the left panel of Figure 2 in the redshift range 3.5–4.5. At z = 3.5, the major contributors to the LyC opacity are, in order of decreasing magnitude, the high column-density Lyα forest (, 32%), the SLLSs (28%), the LLSs (20%), the low column-density Lyα forest (, 12%), and the DLAs (8%). A new method to directly measure the IGM LyC opacity along quasar sight lines has been recently presented by Prochaska et al. (2009). The approach analyzes the “stacked” spectrum of 1800 quasars drawn from the Sloan Digital Sky Survey (SDSS) to give an empirical determination of the mean free path λ⁹¹²_mfp. Our new opacity model agrees very well with the measurements of Prochaska et al. (2009) and produces a continuum opacity that is approximately half of that adopted in Paper II. The right panel of the same figure shows how our model also provides a good fit to the Lyα quasar transmission data over the entire redshift range 2 ≲ z ≲ 6.

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For a single population of absorbers described by Equation (12), the mean free path scales with frequency and redshift as λ_mfp(ν, z)∝(ν/ν₉₁₂)^{3(β − 1)}H⁻¹/(1 + z)^{γ + 1}. Given the multi-component distribution summarized in Table 1, we can readily compute the mean free path of ionizing radiation in the range 13.6 ⩽ hν < 48.4 eV under the assumption that He i continuum absorption at energies above 24.2 eV can be neglected (photons between 48.4 and 54.4 eV are reprocessed by He ii Lyman series resonance absorption, see Section 2.2). For ease of use in analytical calculations, we fit our numerical results for the mean free path as

where s ≡ ν/ν₉₁₂. Both the normalization A(s) and the exponent γ(s) are well fit by third-order polynomials of the form

Numerical values of the best-fit polynomial coefficients are given in Table 2: the fitting function is adjusted to be continuous in value at the redshifts where it changes slope. As discussed above, the fit is only valid in the range 1 ⩽ s ⩽ 3.56. Close to the hydrogen Lyman edge, and at early enough epochs, only “local” radiation sources—sources within a mean free path of a few tens of Mpc—contribute to the ionizing background intensity, and one can neglect cosmological effects such as source evolution and frequency shifts. In this “source-function” approximation, 4πJ₉₁₂(z) ≈ ₉₁₂(z) λ⁹¹²_mfp(z).

An iterative solution to the equations of cosmological radiative transfer that included a detailed numerical calculation of the ionization and temperature structure of individual absorbers at every timestep would be a very computing-intensive task. To properly treat the self-shielding of LyC radiation, in Paper II we modeled absorbers as semi-infinite slabs, developed a “steplike” approximation to the function , and used an analytical escape probability formalism to include the recombination emission from absorbers. Fardal et al. (1998) solved the local radiative transfer problem via an integral equation (the Milne solution for a gray atmosphere) for the number of photoionizations at any optical depth in a given slab. They also devised an approximation formula to η that closely followed the numerical results. Faucher-Giguère et al. (2009) have recently generalized the treatment of Fardal et al. (1998) and applied a similar fitting formula to the results of a code that self-consistently solves the photoionization equilibrium balance, including the influence of recombination radiation. Here, we follow a similar method: under the assumption of Jeans length thickness for the absorbers, we solve the ionization and thermal structure of a slab of finite width illuminated by an external isotropic radiation field J_ν, and derive analytical approximations for the ratios η and ζ as a function of . Details of our calculations are provided in the Appendix. We parameterize the external background flux as a power law, J_ν = J₉₁₂(ν/ν₉₁₂)^−α, and as in Faucher-Giguère et al. (2009) divide the intensity above 54.4 eV by a factor of 10 to mimic a cosmological UV filtered spectrum.

Figure 3 shows the resulting ratios and for a range of input spectra and for the representative intensity value at 1 ryd of 10⁻²²  erg cm⁻² s⁻¹ Hz⁻¹ sr⁻¹. The function η remains constant at low H i columns, as long as the optically thin approximation holds. As the neutral hydrogen column increases, the slab first becomes optically thick to He ii-ionizing radiation and η increases. Slabs with even larger columns become optically thick to H i LyC: they are characterized by a highly ionized surface layer and an almost fully neutral core. This is the reason for the rapid decrease of η after the peak and the consequent trend of ζ toward the neutral limit, ζ → n_He/n_H. As in Fardal et al. (1998) and Faucher-Giguère et al. (2009), we calculate the column from the equation

where , , and A and B are constants fitted to our numerical results. To make use of the above expression, one must further specify the ionization rates Γ_i to be used in the R_i terms together with a relation between electron density and . It is in this second step that our approach differs from that of Faucher-Giguère et al. (2009). These authors used the optically thin limit for Γ_i, which provides a poor approximation to the numerical results. Here, we first compute the ionization rates in the optically thin limit,

and derive η^thin using Equation (21). For a given (input value) of , we then calculate . We then write a first-order approximation to the He ii ionization rate at the face of the slab,

The analogous expression for H i is

Finally, we compute the R_i factors for H i and He ii in Equation (24) as

The recombination rates depend on the gas temperature. We found that our numerical results can be fit with the simple scaling

where J_{912, −22} ≡ J₉₁₂/10⁻²²  erg cm⁻² s⁻¹ Hz⁻¹ sr⁻¹ is adequate for our purposes. The weak dependence on J is related to the fact that, for , cooling is largely provided by collisionally excited line radiation rather than by recombinations. With this simplified treatment, we have been able to fit the numerically obtained values of η for a broad range of input spectra. The best-fit curves shown in the left panel of Figure 3 have been obtained taking A = 0.02 and B = 0.25 in Equation (24), and

for the electron density. Here, and . The above relation can be derived assuming Jeans length thickness for the absorbers and optically thin photoionization equilibrium (Schaye 2001; see also Faucher-Giguère et al. 2009). A simple approximation for ζ can also be derived, once η is obtained. We use Equation (20), , with for  cm⁻². At larger columns we apply a linear (in log space) extrapolation to the limiting vale ζ → n_He/n_H assumed to be reached at  cm⁻².

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According to Ueda et al. (2003), who combined various surveys from the HEAO-1, ASCA, and Chandra satellites, the hard 2–10 keV quasar luminosity function (HXLF) follows a luminosity-dependent density evolution with a cutoff redshift (above which the evolution stops) that increases with luminosity. At the present epoch, the intrinsic (i.e., before absorption) HXLF of all AGNs (including both Type 1's and Type 2's) is best represented by

in the luminosity range 10^41.5–10^46.5  erg s⁻¹, where ϕ_* = 2190 Gpc⁻³ and L_* = 10^43.94  erg s⁻¹. This changes with cosmic time (for redshift up to 3) as

where the evolution factor is

and

Here, e₁ = 4.23, e₂ = −1.5, z*_c = 1.9, and L_a = 10^44.6  erg s⁻¹ (Ueda et al. 2003). An extension of the HXLF up to z ∼ 5 by Silverman et al. (2008) shows a steeper decline in the number of z > 3 AGNs with an evolution rate similar to that found by studies of optically selected QSOs. The new fit requires a much stronger evolution above the cutoff redshift, e₂ = −3.27, than previously found by Ueda et al. (2003, e₂ = −1.5). In the following, we shall use Ueda et al.'s (2003) HXLF best-fit parameters together with the Silverman et al. (2008) value for e₂.

For the intrinsic spectrum before absorption, we assume the standard power law multiplied by an exponential,

with α = 0.9 (Nandra & Pounds 1994). The high-energy cutoff, E_c = 460 keV, is fixed by the shape of the XRB turnover above 30 keV. These seed photons are then reflected toward the observer by a semi-infinite cold disk close to the primary emitter. This reflection component, commonly detected in the X-ray spectra of nearby Seyfert galaxies (Nandra & Pounds 1994), is comparable to the direct flux around 30 keV, decreases rapidly toward lower energies, and flattens the overall spectral slope above 10 keV (Lightman & White 1988).

According to the AGN unification scheme, obscuring matter at a distance of several parsecs from the central powerhouse blocks our line of sight to the active nucleus. When our view is unobscured, we see a Type 1 AGN; when our view is occulted, photons of all energies from the far IR to several keV are absorbed, and in these bands we can only detect the nucleus in scattered light. Ueda et al. (2003) found the following expression for the observed (normalized) distribution of absorbing N_H columns:

where the parameter

accounts for the fact that the fraction of absorbed sources is smaller at higher luminosities. Here, ψ_max = (1 + )/(3 + ), = 1.7, ψ₄₄ = 0.47, β = 0.1, and ∫²⁴₂₀f(L, N_H) dlog N_H = 1. Sources absorbed by a column larger (smaller) than 10²²  cm⁻² are defined as X-ray Type 2 (Type 1) AGNs. It is assumed that “Compton-thick” AGNs with columns log N_H > 24 are not present in samples detected below 10 keV. Such a population is added by extrapolating the N_H function above log N_H > 24, keeping the same normalization up to log N_H = 25 as well as the same cosmological evolution of Compton-thin AGNs (Ueda et al. 2003).

We then follow Madau et al. (1993) and model the thick blocking material that covers most of the solid angle around the central X-ray source as a homogeneous spherical cloud of cold material and column N_H. The radiation transfer is computed with a Monte Carlo code constructed using the photon-escape weighing method of Pozdnyakov et al. (1983). We set the electron temperature equal to zero, use the full Klein–Nishina scattering cross section, adopt the bound-free opacity associated with standard cosmic-abundance material from Morrison & McCammon (1983), and ignore the iron Kα emission line in the spectra. Each Monte Carlo run uses 10⁶ photons and produces as output an absorbed spectrum, S_E(N_H). After being reprocessed by cold material along the line of sight, the emergent specific intensity forms a hump, whose position and width are determined by the competition of bound-free absorption at low energies, and Compton downscattering and exponential roll-off of the primary spectrum at high energies (Madau et al. 1993). A small spectral component, equal to 2.5% of the primary incident power and representing the flux scattered into the line of sight by electrons in the warm ionized medium, is added to the transmitted Type 2 flux. The absorbed spectra are then averaged over the N_H-distribution corresponding to a given luminosity, and normalized to the unabsorbed 2–10 keV flux,

to yield a flux normalized, luminosity-dependent, average AGN SED. The X-ray proper emissivity as a function of redshift is then obtained by simply integrating over the HXLF,

where we set L_min = 10^41.5  erg s⁻¹ and L_max = 10⁴⁸  erg s⁻¹.

The model described above is able to reproduce a number of X-ray observations, from the evolution of AGNs in the soft and hard X-ray bands, to the XRB. The quasar comoving emissivity at 2 keV and 10 keV is plotted in Figure 5, while a global fit to the XRB is shown in the left panel of Figure 6. The absolute XRB flux is still affected by rather large uncertainties: our model reproduces well the background intensity measured by HEAO-1 and BeppoSAX, but the HEAO-1 A2 data are lower by about 20% with respect to the determinations by, e.g., XMM and RXTE at energies below 10 keV. Figure 6 (right panel) depicts the broadband quasar comoving emissivity per logarithmic bandwidth, ν_ν/(1 + z)³, as a function of photon energy from the optical to hard X-rays. In terms of energy output, the composite spectrum for λ < 5000 Å is characterized by two broad bumps, one in the UV at 10 eV and another in the X-ray region at 30 keV (a third peak in the mid-infrared, see, e.g., Sazonov et al. 2004, can be neglected for the present purposes). While X-rays dominate the energy output at z = 0, the peak of the emitted power moves increasingly toward the UV at redshifts above 1.

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Stellar population synthesis codes do not typically include nebular line emission. Here, we provide a simple estimate of the Lyα emission from hydrogen recombinations in the interstellar medium of galaxies. In case B recombination, about 68% of all the absorbed LyC photons will be converted locally into Lyα (Osterbrock 1989). The Lyα proper volume emissivity can then be written as

where

and _ν is the proper volume emissivity from galaxies. We assume here that Lyα suffers the same dust extinction as LyC, a simple treatment that is unlikely to capture the complex radiative transfer physics of the Lyα line as it propagates through the dusty ISM (see, e.g., Caplan & Deharveng 1986; Neufeld 1991; Dijkstra 2009; Scarlata et al. 2009; Dayal et al. 2010). Inserting Equation (57) into (2) yields the additional flux observed at ν_o ⩽ ν_α from galaxy Lyα:

where 1 + z_α = (ν_α/ν_o)(1 + z_o). We have neglected collisionally excited Lyα emission, as this is only about 10%–20% of the recombination term (Dayal et al. 2010). A similar contribution is also expected in the emitted spectrum of dense absorbers like the SLLSs and DLAs, while collisional excitation is always negligible in lower column density systems.

The total optically thin photoionization rate of hydrogen, , is shown in Figure 8 as a function of redshift (left panel). For comparison, we have also plotted the individual contributions of the QSO population that dominates at low redshift and of the galaxy population that reionize the IGM at early times, together with the empirical measurements from the Lyα forest effective opacity by Bolton & Haehnelt (2007), Becker et al. (2007), and Faucher-Giguère et al. (2008a), and from the quasar proximity effect by Calverley et al. (2011). The fractional recombination contribution to increases from 9% at z = 0 to 18% at z = 4 to up to 37% at z ≳ 7: it does so because the mean free path of recombination photons decreases with look-back time and a smaller fraction of such photons gets redshifted below the ionization threshold before capture (Faucher-Giguère et al. 2009). While the total H i photoionization rate provides a good match to the data, we note that there are large systematic uncertainties in the measurements as these depend on the assumed IGM temperature and gas density distribution.

The right panel of the same figure depicts the optically thin photoheating rates per ion (see Equation (A7) with for the definition) of hydrogen, , and He ii, , for the quasars + galaxies and quasar-only models. The addition of a galaxy component boosts the H i rate as it increases the emissivity of hydrogen-ionizing photons at fixed H i opacity (the latter being determined by the observations). The opposite is true for He ii photoheating (as well as He ii photoionization), as galaxies do not contribute to the emissivity above 4 ryd: this increases the predicted He ii opacity (again at fixed H i opacity) and causes a large break in the background spectrum at 4 ryd and a smaller photoheating rate. While the Compton heating rate per electron is many orders of magnitude (about 7 dex at redshift 3) below the H i photoheating rate (note the different normalization of the heating rates plotted in Figure 8), it is a non-negligible source of heating for very underdense, highly ionized regions: the Compton heating rate for intergalactic gas at overdensity 0.1, temperature T = 10⁴ K, and redshifts z = (1, 2, 3) is (53, 16, 4)% of the total photoheating rate. Table 3 tabulates the optically thin photoionization and photoheating rates of hydrogen and helium predicted by our “quasars + galaxies” model for use, e.g., in cosmological hydrodynamics simulations of the Lyα forest.

Table 3. The Cosmic Background Photoionization and Photoheating Rates

z
	(s−1)	(eV s−1)	(s−1)	(eV s−1)	(s−1)	(eV s−1)
0.00	0.228E-13	0.889E-13	0.124E-13	0.112E-12	0.555E-15	0.114E-13
0.05	0.284E-13	0.111E-12	0.157E-13	0.140E-12	0.676E-15	0.138E-13
0.10	0.354E-13	0.139E-12	0.196E-13	0.174E-12	0.823E-15	0.168E-13
0.16	0.440E-13	0.173E-12	0.246E-13	0.216E-12	0.100E-14	0.203E-13
0.21	0.546E-13	0.215E-12	0.307E-13	0.267E-12	0.122E-14	0.245E-13
0.27	0.674E-13	0.266E-12	0.383E-13	0.331E-12	0.148E-14	0.296E-13
0.33	0.831E-13	0.329E-12	0.475E-13	0.408E-12	0.180E-14	0.357E-13
0.40	0.102E-12	0.405E-12	0.587E-13	0.502E-12	0.218E-14	0.429E-13
0.47	0.125E-12	0.496E-12	0.722E-13	0.615E-12	0.263E-14	0.514E-13
0.54	0.152E-12	0.605E-12	0.884E-13	0.751E-12	0.317E-14	0.615E-13
0.62	0.185E-12	0.734E-12	0.108E-12	0.911E-12	0.380E-14	0.732E-13
0.69	0.223E-12	0.885E-12	0.130E-12	0.110E-11	0.454E-14	0.867E-13
0.78	0.267E-12	0.106E-11	0.157E-12	0.132E-11	0.538E-14	0.102E-12
0.87	0.318E-12	0.126E-11	0.187E-12	0.157E-11	0.633E-14	0.119E-12
0.96	0.376E-12	0.149E-11	0.222E-12	0.186E-11	0.738E-14	0.139E-12
1.05	0.440E-12	0.175E-11	0.261E-12	0.217E-11	0.852E-14	0.159E-12
1.15	0.510E-12	0.203E-11	0.302E-12	0.251E-11	0.970E-14	0.181E-12
1.26	0.585E-12	0.232E-11	0.346E-12	0.287E-11	0.109E-13	0.202E-12
1.37	0.660E-12	0.262E-11	0.391E-12	0.323E-11	0.119E-13	0.221E-12
1.49	0.732E-12	0.290E-11	0.434E-12	0.357E-11	0.127E-13	0.237E-12
1.61	0.799E-12	0.317E-11	0.474E-12	0.387E-11	0.132E-13	0.247E-12
1.74	0.859E-12	0.341E-11	0.509E-12	0.413E-11	0.134E-13	0.253E-12
1.87	0.909E-12	0.360E-11	0.538E-12	0.432E-11	0.133E-13	0.252E-12
2.01	0.944E-12	0.374E-11	0.557E-12	0.444E-11	0.128E-13	0.244E-12
2.16	0.963E-12	0.381E-11	0.567E-12	0.446E-11	0.119E-13	0.229E-12
2.32	0.965E-12	0.382E-11	0.566E-12	0.438E-11	0.106E-13	0.207E-12
2.48	0.950E-12	0.375E-11	0.555E-12	0.422E-11	0.904E-14	0.178E-12
2.65	0.919E-12	0.363E-11	0.535E-12	0.398E-11	0.722E-14	0.145E-12
2.83	0.875E-12	0.346E-11	0.508E-12	0.368E-11	0.530E-14	0.111E-12
3.02	0.822E-12	0.325E-11	0.476E-12	0.336E-11	0.351E-14	0.775E-13
3.21	0.765E-12	0.302E-11	0.441E-12	0.304E-11	0.208E-14	0.497E-13
3.42	0.705E-12	0.279E-11	0.406E-12	0.274E-11	0.114E-14	0.296E-13
3.64	0.647E-12	0.257E-11	0.372E-12	0.249E-11	0.591E-15	0.168E-13
3.87	0.594E-12	0.236E-11	0.341E-12	0.227E-11	0.302E-15	0.925E-14
4.11	0.546E-12	0.218E-11	0.314E-12	0.209E-11	0.152E-15	0.501E-14
4.36	0.504E-12	0.202E-11	0.291E-12	0.194E-11	0.760E-16	0.267E-14
4.62	0.469E-12	0.189E-11	0.271E-12	0.181E-11	0.375E-16	0.141E-14
4.89	0.441E-12	0.178E-11	0.253E-12	0.170E-11	0.182E-16	0.727E-15
5.18	0.412E-12	0.167E-11	0.237E-12	0.160E-11	0.857E-17	0.365E-15
5.49	0.360E-12	0.148E-11	0.214E-12	0.146E-11	0.323E-17	0.156E-15
5.81	0.293E-12	0.123E-11	0.184E-12	0.130E-11	0.117E-17	0.624E-16
6.14	0.230E-12	0.989E-12	0.154E-12	0.112E-11	0.442E-18	0.269E-16
6.49	0.175E-12	0.771E-12	0.125E-12	0.952E-12	0.173E-18	0.128E-16
6.86	0.129E-12	0.583E-12	0.992E-13	0.783E-12	0.701E-19	0.674E-17
7.25	0.928E-13	0.430E-12	0.761E-13	0.625E-12	0.292E-19	0.388E-17
7.65	0.655E-13	0.310E-12	0.568E-13	0.483E-12	0.125E-19	0.240E-17
8.07	0.456E-13	0.219E-12	0.414E-13	0.363E-12	0.567E-20	0.155E-17
8.52	0.312E-13	0.153E-12	0.296E-13	0.266E-12	0.274E-20	0.103E-17
8.99	0.212E-13	0.105E-12	0.207E-13	0.191E-12	0.144E-20	0.698E-18
9.48	0.143E-13	0.713E-13	0.144E-13	0.134E-12	0.819E-21	0.476E-18
9.99	0.959E-14	0.481E-13	0.982E-14	0.927E-13	0.499E-21	0.326E-18
10.50	0.640E-14	0.323E-13	0.667E-14	0.636E-13	0.325E-21	0.224E-18
11.10	0.427E-14	0.217E-13	0.453E-14	0.435E-13	0.212E-21	0.153E-18
11.70	0.292E-14	0.151E-13	0.324E-14	0.314E-13	0.143E-21	0.106E-18
12.30	0.173E-14	0.915E-14	0.202E-14	0.198E-13	0.984E-22	0.752E-19
13.00	0.102E-14	0.546E-14	0.123E-14	0.122E-13	0.681E-22	0.531E-19
13.70	0.592E-15	0.323E-14	0.746E-15	0.749E-14	0.473E-22	0.373E-19
14.40	0.341E-15	0.189E-14	0.446E-15	0.455E-14	0.330E-22	0.257E-19
15.10	0.194E-15	0.110E-14	0.262E-15	0.270E-14	0.192E-22	0.154E-19

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Figure 9 shows the spectrum of the radiation background as a function of redshift for a “quasar-only” model, together with the old results from Paper II. The new spectra are characterized by a lower UV flux (by as much as a factor of three at 1 ryd and z = 3), smaller spectral breaks at 1 and 4 ryd because of the reduced H i and He ii LyC absorption, a sawtooth modulation by the Lyman series of H i and He ii that becomes more and more pronounced with increasing redshift, and a flatter soft X-ray spectrum.

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The addition of radiation from galaxies has little effect on the ionizing background at redshifts below 3, as shown in Figure 10, a consequence of our adopted redshift-dependent escape fraction. At higher redshifts the impact is more dramatic: a large boost at 1 ryd is associated with a much sharper He ii sawtooth and He ii absorption edge. As noted above regarding the photoheating rates, this arises because galaxy spectra are truncated at 4 ryd, and the large increase in the H-ionizing emissivity from the early galaxy population is not accompanied by a similar increase at the He ii edge. The net effect is a larger He ii opacity at fixed H i opacity. At z ≳ 5, the He i opacity of the IGM also starts building up (it is negligible at lower redshifts), and a small He i absorption edge can be discerned in the spectrum of the background at 24.6 eV. At z ≳ 3, the sawtooth modulation produced by resonant absorption in the Lyman series of intergalactic He ii (see Figure 11) is clearly a sensitive probe of the nature of the sources that keep the IGM ionized, and may be a crucial ingredient in the modeling of the abundances of metal absorption systems (Madau & Haardt 2009). The analogous sawtooth modulation produced by the H i Lyman series becomes significant above redshift 6 (see Figure 12) and may affect the photodissociation of molecular hydrogen during cosmological reionization (Haiman et al. 1997).

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Figure 13 compares the broadband spectrum of the total extragalactic background light (EBL) from quasars and galaxies, predicted by CUBA at z = 0, with current EBL observations from the mid-IR to the γ-rays.

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It is interesting at this stage to use the quasar and galaxy ionizing emissivities of Sections 6, 7, and 8 and track the evolution of the volume filling factors of ionized hydrogen and doubly ionized helium regions in the universe as a function of cosmic time. As shown in Paper III, the volume filling factor of H ii regions, , is equal at any given instant t to the integral over cosmic time of the number ionizing photons emitted per hydrogen atom by all radiation sources present at earlier epochs,

minus the number of radiative recombinations per ionized hydrogen atom,

Here

with 〈f_esc〉 = 1 in the case of quasars, 〈n_H〉 = 1.9 × 10⁻⁷(1 + z)³ cm⁻³ is the mean hydrogen density of the expanding IGM, and 〈t_rec〉 is the volume-averaged hydrogen recombination timescale,

where α_B is the recombination coefficient to the excited states of hydrogen, χ = 1.08 accounts for the presence of photoelectrons from singly ionized helium, and is the clumping factor of ionized hydrogen. Differentiation yields the H i “reionization equation” of Paper III,

and its equivalent for expanding He iii regions,

where now includes only photons above 4 ryd (which are mostly absorbed by He ii), and the recombination timescale of doubly ionized helium, 〈t_{rec, He}〉, is about six times shorter than the hydrogen recombination timescale if H ii and He iii have similar clumping factors. We will not attempt here to model the reionization of He i, as this occurs nearly simultaneously to and cannot be readily decoupled from that of H i. The reionization equation (1) describes the transition from a neutral universe to a fully ionized one in a statistical way, independently, for a given emissivity, of the emission histories of individual radiation sources; (2) assumes that the mean free path of ionizing photons is much smaller than the horizon, i.e., that they are absorbed before being redshifted below the ionization edge; and (3) includes in the source term only those photons above the Lyman limit that escape into the IGM (〈f_esc〉 = 1 in the case of quasars). Photons that are absorbed in loco by dense interstellar gas do not enter in the source term, nor does the interstellar absorbing material contribute to the recombination rate. The volume-weighted clumping factor reflects only the nonuniformity of the ionized low-density IGM, the repository of most of the baryons in the universe, and its use in the recombination timescale is justified when the size of the ionized regions is large compared to the scale of the clumping.

When (the “pre-overlap” stage), individual ionization fronts propagate from star-forming early galaxies into the low-density IGM. The neutral phase shrinks as grows and H ii regions start to overlap. The radiation field remains highly inhomogeneous until the reionization process is completed at the “overlap epoch,” , when all the low-density IGM becomes highly ionized. Pockets of neutral gas remain in collapsed systems during the entire “post-overlap” stage (Gnedin 2000) and may manifest themselves as the SLLSs or DLA systems in quasar absorption spectra. We have integrated equation (64) assuming a gas temperature of 2 × 10⁴ K and a clumping factor for the IGM of

This is equal to the expression for C₁₀₀ (the clumping factor of gas below a threshold overdensity of 100) found at z ⩾ 6 in a suite of cosmological hydrodynamical simulations by Pawlik et al. (2009). These authors found that photoionization heating by a uniform UV background greatly reduces clumping as it smoothes out small-scale density fluctuations, and that the clumping factor at z = 6 is insensitive to the redshift at which the UV background is actually turned on (as long as reheating occurs at ≳ 9). We use an overdensity of 100 to differentiate between dense gas belonging to virialized halos and the diffuse intergalactic gas, and assume that the collapsed mass fraction is small. We also extrapolate Equation (66) down to z ≳ 2, and assume the same clumping factor for H ii and He iii.

The results of this “minimal reionization model” are shown in Figure 14. Cosmological H ii regions driven by star-forming galaxies overlap at redshift 6.7, and the hydrogen in the universe is half-ionized (by volume) at redshift 10. He iii regions driven by quasars overlap much later, at redshift 2.8, and their filling factor is only 4% at redshift 5. These overlap epochs are consistent with the SDSS spectra of z ∼ 6 quasars (Fan et al. 2006b), with numerical simulations of H i reionization (Gnedin & Fan 2006), and with observations of the He ii Lyα forest at z ≲ 3 (see, e.g., Worseck et al. 2011; Shull et al. 2010; Fechner et al. 2006; Heap et al. 2000, and references therein). A simple probe of the reionization history is the integrated optical depth to electron scattering τ_es, which depends on the path length through ionized gas along the line of sight to the CMB as

(Wyithe & Loeb 2003), where σ_T is the Thomson cross section. The seven-year WMAP results imply τ_es = 0.088 ± 0.015 (Jarosik et al. 2011). Our minimal reionization model assumes and yields an electron scattering opacity to the epoch of reionization of τ_es = 0.084, in good agreement with the observations.

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The outcome of our minimal reionization model is rather sensitive to the assumed escape fraction of hydrogen-ionizing radiation at early epochs; this exceeds 50% at z ≳ 9 and reaches unity at z = 11.6. Had we assumed a maximum f_esc of 50% instead, the same model would yield at z = 6.2 and τ_es = 0.06. We also remark that, in the pre-overlap era, the background spectra shown in Figures 9–12 have only a formal meaning, as they describe a space-averaged radiation field that is in reality highly inhomogeneous. Recent spectra taken by the Cosmic Origins Spectrograph on the HST exhibit patchy He ii Gunn–Peterson absorption, with a mean He ii/H i abundance ratio that is 47 ± 42 at 2.4 < z < 2.73 and 209 ± 281 at z > 2.73 (Shull et al. 2010). In the redshift interval 2.4 < z < 2.73, our background spectrum yields a He ii/H i abundance ratio (in the optically thin limit) around 50–70, in good agreement with the observations. The predicted mean He ii/H i ratio increases rapidly toward high redshift to (280, 494, 887, 1615) at z = (3.42, 3.64, 3.87, 4.1) as galaxies start dominating the ionizing emissivity and the spectrum of the UVB steepens. The evolution of the He ii abundance and the fluctuating spectrum of the cosmic UVB in the pre-overlap era will be the subject of a subsequent paper.

The background spectra computed in the previous section are sensitive to a number of poorly determined input parameters. In this section, we briefly discuss just a few of the uncertainties inherent in our synthesis modeling of the UVB. The left panel of Figure 15 shows the adopted comoving quasar emissivity at 1 ryd (Equation (37)), together with the determinations by Meiksin (2005), Cowie et al. (2009), Bongiorno et al. (2007), Willott et al. (2010), and Siana et al. (2008). The poorly known faint-end slope of the quasar luminosity function at high redshift, incompleteness corrections, as well as the uncertain spectral energy distribution (SED) in the UV, all contribute to the large apparent discrepancies between different measurements.

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The escape fraction of ionizing radiation that leaks into the IGM and its unknown redshift evolution is the major source of uncertainty in the determination of the galaxy contribution to the UVB. To better gauge the impact of this parameter on our synthesis model, we show in the right panel of Figure 15 the spectrum of the comoving galaxy emissivity at four different epochs. The relatively large leakage of LyC photons assumed at early times is a crucial ingredient of our “minimal reionization model,” which yields an optical depth to Thomson scattering in agreement with WMAP results. A smaller escape fraction at high redshifts would lead to too-late reionization, while a significantly larger escape fraction at lower redshifts would produce a hydrogen photoionization rate that appears to be too high compared to the observations (see Figure 8).

We have also checked that, for a given IMF, uncertainties in the stellar population synthesis technique are relatively small. Figure 16 shows the emission rate of hydrogen-ionizing photons for an SSP, calculated as a function of age with the GALAXEV models of Bruzual & Charlot (2003), the Starburst99 models of Leitherer et al. (1999), and the FSPS models of Conroy et al. (2009). While the three packages use different stellar evolution tracks and spectral libraries, the total number of ionizing photons emitted agrees to within 10%. The figure also illustrates the significant effect of stellar metallicity: an SSP of metallicity 1/50 of solar emits 60% more hydrogen-ionizing photons over its lifetime than a solar metallicity SSP (Salpeter IMF, GALAXEV package).

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Finally, we address the effect of a change in the effective opacity of the IGM. In our parameterization, the shape of the distribution over the column density range of the LLSs is adjusted at every redshift for continuity with the SLLSs. As detailed in Section 3, this procedure yields the slopes β = 0.47, 0.61, 0.72, 0.82 at redshifts z = 2, 3, 4, 5, respectively. To gauge how an uncertainty in translates into an uncertainty in the UVB, we have run CUBA with the fixed values of β = 0.2 and β = 1 in the column density interval . The resulting UVB at z = 3 is shown in the right panel of Figure 16. Compared to our fiducial model, the flat β = 0.2 distribution generates a hydrogen photoionization rate () that is 29% lower and a He ii photoionization rate () that is 22% higher. This is because a larger opacity at 1 ryd from the LSSs results in a harder background spectrum, which in turn produces a smaller He ii opacity at 4 ryd. Conversely, in the steep β = 1 case, increases by 8% and decreases by 10%. Note, however, that the former model would significantly underestimate the 1 ryd photon mean free path compared to the measurements of Prochaska et al. (2009).