UV Background Spectrum

# A New Calculation of the Ionizing Background Spectrum and the Effects of HeII Reionization

Claude-André Faucher-Giguère11affiliation: Department of Astronomy, Harvard University, Cambridge, MA, 02138, USA; cgiguere@cfa.harvard.edu. , Adam Lidz11affiliation: Department of Astronomy, Harvard University, Cambridge, MA, 02138, USA; cgiguere@cfa.harvard.edu. , Matias Zaldarriaga11affiliation: Department of Astronomy, Harvard University, Cambridge, MA, 02138, USA; cgiguere@cfa.harvard.edu. 22affiliation: Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, USA. , Lars Hernquist11affiliation: Department of Astronomy, Harvard University, Cambridge, MA, 02138, USA; cgiguere@cfa.harvard.edu. ,
###### Abstract

The ionizing background determines the ionization balance and the thermodynamics of the cosmic gas. It is therefore a fundamental ingredient to theoretical and empirical studies of both the IGM and galaxy formation. We present here a new calculation of its spectrum that satisfies the empirical constraints we recently obtained by combining state-of-the-art luminosity functions and intergalactic opacity measurements.

In our preferred model, star-forming galaxies and quasars each contribute substantially to the HI ionizing field at , with galaxies rapidly overtaking quasars at higher redshifts as quasars become rarer. In addition to our fiducial model, we explore the physical dependences of the calculated background and clarify how recombination emission contributes to the ionization rates. We find that recombinations do not simply boost the ionization rates by the number of reemitted ionizing photons as many of these rapidly redshift below the ionization edges and have a distribution of energies. A simple analytic model that captures the main effects seen in our numerical radiative transfer calculations is given.

Finally, we discuss the effects of HeII reionization by quasars on both the spectrum of the ionizing background and on the thermal history of the IGM. In regions that have yet to be reionized, the spectrum is expected to be almost completely suppressed immediately above 54.4 eV while a background of higher-energy ( keV) photons permeates the entire universe owing to the frequency-dependence of the photoionization cross section. We provide an analytical model of the heat input during HeII reionization and its effects on the temperature-density relation.

Cosmology: theory, diffuse radiation — galaxies: formation, evolution, high-redshift — quasars: absorption lines
slugcomment: Submitted to ApJ

## 1. Introduction

The cosmic baryons give the ultraviolet (UV) background a particularly important standing among radiation backgrounds. In fact, the ionization potentials of both hydrogen and helium11113.6, 24.6, and 54.6 eV for HI, HeI, and HeII, respectively., which together account for 99% of the baryonic mass density (e.g., Burles et al., 2001), correspond to electromagnetic wavelengths in the UV regime. The UV background therefore governs the ionization state of intergalactic gas and furthermore plays a key role in its thermal evolution through photoheating. As such, it is an essential input to cosmological hydrodynamic simulations (e.g., Hernquist et al., 1996; Katz et al., 1996b; Davé et al., 1999; Springel & Hernquist, 2003) as well as to observational studies of the intergalactic medium (IGM).

The ionizing background can for example suppress the abundance of dwarf galaxies and the amount of cool gas in low-mass galaxies that do form both by modifying the cooling function through the ionization balance and by heating the gas before it collapses (Efstathiou, 1992; Quinn et al., 1996; Thoul & Weinberg, 1996; Weinberg et al., 1997). It is also crucially important for any simulation of the Ly forest, since the absence of a Gunn & Peterson (1965) trough in the spectra of quasars up to (e.g., Fan et al., 2002, 2006b; Becker et al., 2007) indicates that the IGM is highly ionized up to at least that redshift. Since the optical depth of the Ly forest is directly tied to the hydrogen photoionization rate (e.g., Rauch et al., 1997; McDonald & Miralda-Escudé, 2001; Meiksin & White, 2003; Tytler et al., 2004; Kirkman et al., 2005; Bolton et al., 2005; Faucher-Giguère et al., 2008a, b), it is important to know the latter accurately. In addition, the UV background determines the photoionization rate of helium, which is of particular relevance given the growing interest in studying HeII reionization, which may occur at redshifts for which a wealth of observational data are already available and upcoming (§7). The full spectrum of the UV background is perhaps most important in the study of metal ions, such as SIV and CIV, where relating the ionic abundances to elemental abundances or cosmic metal mass density requires ionizing corrections (e.g., Cowie et al., 1995; Songaila & Cowie, 1996; Songaila, 2001; Schaye et al., 2003; Boksenberg et al., 2003; Aguirre et al., 2004; Simcoe et al., 2004; Aguirre et al., 2007; Fechner & Richter, 2008). Finally, the spectrum of the UV background obviously depends on its sources and its study can therefore teach much about the sources responsible for keeping the IGM ionized, as well as reionizing it (e.g., Miralda-Escudé, 2003; Bolton & Haehnelt, 2007; Faucher-Giguère et al., 2008a, b).

Following early work (Miralda-Escude & Ostriker, 1990; Shapiro et al., 1994; Giroux & Shapiro, 1996), Haardt & Madau (1996) (see also Fardal et al., 1998) pioneered calculations of the UV background spectrum in their study of radiative transfer in a clumpy universe. Their model and some variants (e.g., Haardt & Madau, 2001) have since been extensively used in several hundreds of studies in the literature. Over a decade after their original calculation, the empirical constraints on the UV backgrounds and its sources have however improved dramatically. Larger and deeper surveys at all wavelengths have constrained the quasar luminosity function to both fainter magnitudes and higher redshifts (e.g., Boyle et al., 2000; Miyaji et al., 2000; Fan et al., 2001; Ueda et al., 2003; Fan et al., 2004; Croom et al., 2004; Richards et al., 2005; Barger et al., 2005; Hasinger et al., 2005; Richards et al., 2006; Brown et al., 2006; Hopkins et al., 2007; Fontanot et al., 2007; Siana et al., 2008). At the same time, our understanding of the population of high-redshift star-forming galaxies has tremendously expanded thanks to the application of the Lyman break selection technique to ever more ambitious surveys (e.g., Steidel et al., 1999; Sawicki & Thompson, 2006; Yoshida et al., 2006; Bouwens et al., 2007; Reddy et al., 2008). Detailed studies of the absorption properties of the IGM, particularly by HI and HeII, have also provided particularly valuable constraints on the UV background. These constraints are especially relevant for the UV background as the IGM is sensitive to the integral of the UV photons emitted by all sources, regardless of whether these are directly detected. Moreover, the IGM constraints probe the density of ionizing photons and thus circumvent the need to assume an escape fraction relating the luminosity of quasars and galaxies measured redward of the Lyman limit to their net output of ionizing radiation.

In a series of previous papers, we have measured the intergalactic Ly opacity (Faucher-Giguère et al., 2008d) and derived empirical constraints on the UV background and its sources incorporating also information on the reionization of HI and HeII, as well as column density ratios (Faucher-Giguère et al., 2008a, b). Specifically, we found that the HI photoionization rate is remarkably constant over the redshift interval . Since the quasar luminosity function peaks strongly around , star-forming galaxies most likely dominate the ionizing background beyond . The column density ratios however indicate that quasars likely do contribute a large fraction of the ionizing background at their peak. In this paper, we use these constraints as a basis for a new calculation of the full spectrum of the UV background. In addition to the improved empirical input, we reexamine many of the assumptions entering the original Haardt & Madau (1996) calculation. As we will show, we find that the original calculation likely overestimated the contribution of recombination emission to the ionizing background by a factor of a few. Of perhaps greatest interest, the original calculation completely neglected the effects of HeII reionization, so that simulators have usually resorted to artificial prescriptions to complement the Haardt & Madau (1996) spectrum. Here, we explicitly discuss the effects of HeII reionization on the UV background spectrum as well as on the thermal history of the IGM and provide a physical framework to implement them.

We review the basic equations of cosmological radiative transfer and the column density distribution of HI absorbers in §2. In §3, we study the ionization structure of individual absorbers and derive approximations to be used in the cosmological solution. §4 is devoted to the calculation of the contribution recombinations to the cosmological emissivity. Empirically calibrated calculations of the UV background spectrum, derived quantities, and their dependences on input parameters are presented in §5. In §6, we investigate how the calculated spectra (including recombination emission) and the corresponding ionization rates depend on input parameters. The effects of HeII reionization are investigated in §7. We finally compare our results with previous work in §8 and conclude in §9.

A series of appendices supplement the main text with technical details. In Appendix A, we describe our photoionization code. Appendix B contains technical aspects of our treatment of recombination emission, while Appendix C presents an analytic model of how this recombination emission boosts the photoionization rates. Appendix D analytically discusses spectral filtering in different regimes to aid in interpreting our results. Appendix E finally references atomic physics quantities used in our calculations.

Throughout, we assume a cosmology with , as inferred from the Wilkinson Microwave Anisotropy Probe (WMAP) five-year data in combination with baryon acoustic oscillations and supernovae (Komatsu et al., 2008). Unless otherwise stated, all error bars are . Table 1 defines many symbols used here.

In this work, we are first concerned with the specific intensity of the diffuse cosmological UV background averaged over both space and angle, which we denote by . The basic equations of cosmological radiative transfer were particularly well summarized by Haardt & Madau (1996), on which we base our treatment below. The specific intensity satisfies the radiative transfer equation,

 (∂∂t−νH∂∂ν)Jν=−3HJν−cανJν+c4πϵν, (1)

where is the Hubble parameter, is the speed of light, is the proper absorption coefficient, and is the proper emissivity. Integrating equation (1) and expressing the result in terms of redshift gives

 Jν0(z0)=14π∫∞z0dzdldz(1+z0)3(1+z)3ϵν(z)exp[−¯τ(ν0,z0,z)], (2)

where , the proper line element , and the “effective optical depth” 222This quantity is often denoted by . We use a different notation here to distinguish it from the effective optical depth owing to Ly absorption measured from quasar spectra (e.g., Faucher-Giguère et al., 2008d). quantifies the attenuation of photons of frequency at redshift that were emitted at redshift by the relation , where the average is over all lines of sights from to . For Poisson-distributed absorbers, each of column density ,

 ¯τ(ν0,z0,z)=∫zz0dz′∫∞0dNHI∂2N∂NHI∂z′(1−e−τν), (3)

where is the column density distribution versus redshift (Paresce et al., 1980).

Note that these expressions neglect the clustering of sources and sinks of radiation in both the Poisson distribution assumption and in assuming that the spatial average separates into in the integrand of equation 2. The Poisson distribution assumption should be very good since the mean free path of ionizing photons of hundreds of comoving Mpc at most redshifts of interest (see §7) far exceeds the correlation length comoving Mpc of the Ly forest absorbers (e.g., McDonald et al., 2000; Faucher-Giguère et al., 2008c). While absorbers likely do cluster around sources, this effect can be viewed as being incorporated into the definition of the escape fraction. The above formalism will however break down in certain regimes where sources are rare and in particular during HeII reionization. We discuss these cases in §7. As in equation 2, we henceforth drop the explicit averaging brackets around the emissivity .

The optical depth shortward of the Lyman limit will be dominated by the photoelectric opacity of hydrogen and helium,

 τν=NHIσHI(ν)+NHeIσHeI(ν)+NHeIIσHeII(ν), (4)

where the and are the column densities and photoionization cross sections of ion . Only the distribution of is reasonably well determined over a large redshift interval. We will therefore make use of relations between and the column densities of helium established in §3. In our calculations, we will prescribe as well as the specific emissivity of the ionizing sources, , based on our previous empirical studies (Faucher-Giguère et al., 2008d, a, b).

### 2.2. HI Column Density Distribution

Following previous work and consistent with empirical constraints, we parameterize the column density distribution with power laws in and :

 ∂2N∂z∂NHI={N0,lowN−βHI(1+z)γlowz≤zlowN0N−βHI(1+z)γz>zlow. (5)

The transition at accounts for the flattening of the redshift evolution observed at (e.g., Weymann et al., 1998; Kim et al., 2001, 2002) theoretically understood to arise from the drop in intensity of the ionizing background at low redshifts (Theuns et al., 1998; Davé et al., 1999; Bianchi et al., 2001; Scott et al., 2002). We fix .

For a steep column density distribution with , most of the contribution to at the Lyman limit arises from systems of optical depth near unity ( cm). We thus focus on the values of the power-law indices and that are most appropriate in this neighborhood. Stengler-Larrea et al. (1995) find that with and provides a good fit at least up to for systems with cm, whereas the column density power law is well-fitted by (Misawa et al., 2007). The constant in equation 5 is related to by , where cm. We use these values, with , in fiducial calculations but explore varying these parameters in §6.

Before proceeding, we note that column density distribution is largely unconstrained above , the highest redshift at which the Lyman limit system abundance has been measured. We therefore simply extrapolate from lower redshifts and caution that our calculation of the ionizing background may become inaccurate in this regime. In particular, we expect the extrapolation to become unreliable at , where the evolution of the effective optical depth measured from the spectra of quasars diverges rapidly from the power law fitting the data below this redshift (Fan et al., 2006b, though see Becker et al. (2007) for an opposing point of view), perhaps owing to HI reionization.

## 3. The Ionization Structure of Individual Absorbers

### 3.1. Overview

In this section, we study the photoionization equilibrium structure of individual cosmic absorbers composed of hydrogen and helium (with mass fraction 75% and 25%, respectively) as a function of the illuminating radiation background. This serves two purposes: close the set of equations of cosmological radiative transfer (§2) and allow us to more realistically calculate the contribution of recombination lines to the ionizing background spectrum (§4). To this end, we have developed a code that self-consistently solves the photoionization equilibrium balance, including the influence of recombination radiation. This code provides more accurate solutions than previous approximations with semi-infinite geometry and an escape probability formalism (Haardt & Madau, 1996) or gray cross sections (Fardal et al., 1998). To alleviate the text, the details of our photoionization calculations are provided in Appendix A. We assume our absorbers to be slabs of thickness equal to the Jeans scale of the gas, which is a function of the assumed temperature K Schaye (2001). This temperature is consistent with the line-fitting analysis of McDonald et al. (2001) and with the Ly forest power spectrum analysis of Zaldarriaga et al. (2001) for the IGM at mean density.

### 3.2. NHeII and NHeI from NHI

As only the column density distribution of HI is reasonably well constrained, the first application of our photoionization calculations is to obtain relations giving and in terms of . In Figure 1, we show the numerical results for external spectra with from bottom up. To ensure that these test spectra are representative of the UV background, we suppress the power laws by a factor of 10 above the HeII ionization edge.

Since these relations enter within three nested integrals (eqs. 1 and 2), it is necessary to develop analytical approximations that are fast to evaluate. It would be impractical to use self-consistent numerical photoionization calculations at each redshift and for each column density in the cosmological solution. Defining , when both HI and HeII are optically thin and in the limit of nearly complete ionization we have

 ηthin=ΓHIΓHeIIαAHeIIαAHIY4X. (6)

For fixed external background and increasing , an absorber first becomes optically thick in HeII, at which point increases rapidly with . The absorber then becomes optically thick in HI as well and, owing to the greater abundance of hydrogen, finally rapidly overtakes . This leads to the plateau, increase, and then decrease of with respect to seen in the numerical calculations. Similar behavior is found in three-dimensional radiative transfer simulations of the IGM (Maselli & Ferrara, 2005).

Fardal et al. (1998) give a fitting formula derived under the assumptions of negligible and :

 Y16XτHI1+AτHIIHI=τHeII+τHeII1+BτHeIIIHeII, (7)

where , and , and we have generalized their result to allow for arbitrary coefficients and . These fitting coefficients depends on, in particular, the relation between and and we will not be using the same model as these authors. Although our numerical calculations do not a priori assume a relation between and , we must assume one in order to make use of the analytic approximation in equation 7. The approximation curves in Figure 1 can be reproduced by taking cm cms. This relation is approximately derived under the assumptions of Jeans length thickness of the absorbers and optically thin photoionization equilibrium at K. Figure 1 shows that and give a good fit to our numerical results for a wide range of external illuminating spectra. The fitting formula has the exact optically thin limit; the asymptotic divergence from the numerical results as is unimportant as most of the HeII opacity arises in systems with . Although we have assumed specific (but varying) spectral shapes in determining the fitting parameters and , the Fardal et al. (1998) derivation of the functional form in equation 7 illustrates how the relation between and depends principally on the photoionization rates and . The relation should therefore hold well in general.

Obtaining a physically-motivated analytic approximation to is more difficult since is not readily known (for , we know that in the almost-completely ionized case, so that the ionized fractions do not appear explicitly in eqs 6 and 7). Because the ionization potential of HeI is relatively close to that of HI, their ionization states are similar and since helium is less abundant by a factor of 12 by number, HeI should contribute relatively little to the ionizing opacity. This intuition is supported by the right panel of Figure 1, which shows that versus is for illuminating spectra considered. After hardening by IGM filtering, both star-forming galaxies and quasars are expected to produce roughly flat spectra between the HI and HeI ionization edges (§5), so that a representative case is , yielding . We therefore approximate in our cosmological calculations and verify using toy cases of constant that this is justified in §6.

## 4. Recombination Emission

### 4.1. Cosmological Emissivity

The cosmic absorbers not only act as sinks but also as sources of ionizing radiation as a certain fraction of ionizations are followed by the reemission of other ionizing photons via recombinations (Fardal & Shull, 1993; Haardt & Madau, 1996; Fardal et al., 1998). This recombination emission must be taken into account because it may boost the photoionization rates and also since line reemission can imprint significant narrow features in the ionizing background spectrum. Our approach to include this recombination contribution is based on the self-consistent numerical calculations of recombination emission from individual absorbers using the code outlined in the previous section and detailed in Appendix A. This again differs from Haardt & Madau (1996), who used an analytical escape probability formalism and made the assumption of a constant source function within the absorbers (which breaks down in the very optically thick systems), and from the treatment of Fardal et al. (1998) and thus provides a check of these results.

For each recombination process of interest, we calculate the emergent specific intensity owing to the process given the external illuminating spectrum numerically using our photoionization code. The cosmological recombination emissivity for this process is then an average over the column density distribution:

 ϵrecν=4πdzdl∫∞0dNHI∂2N∂z∂NHIIrecν(NHI). (8)

Since in general depends on the spectrum of the ionizing background, which is not known a priori and evolves at each step in the redshift integration, it is again necessary to obtain an analytical approximation for this function that scales appropriately with the external background, as it is not practical to perform self-consistent numerical calculations in the cosmological solution. We develop these analytical approximations in Appendix B for each recombination process of interest.

### 4.2. Recombination Processes

For hydrogen, the only ionizing process is direct recombination to the ground state, which produces a 1 Ryd HI LyC photon. For helium, both recombinations to HeII and to HeI can in principle produce ionizing photons. In cosmological conditions, HeI plays a negligible role 3.2 and 6; Haardt & Madau, 1996) and we will ignore it in our reemission calculations. Three permitted HeII recombination channels lead to the reemission of ionizing photons: HeII LyC recombinations directly to the ground state, indirect recombinations leading to HeII Ly emission, and recombinations to the excited level resulting in Balmer continuum (BalC) photons. These respectively give photons of energy 4, 3, and 1 Ryd. Higher HeII Lyman-series photons could also produce ionizing photons, but we assume case B conditions in which these are ultimately degraded into lower-energy photons, the only ones of which that can ionize hydrogen being HeII Ly. We do not include forbidden two-photon recombination processes as these are energetically subdominant and do not result in distinctive emission features.

In calculating the contribution of reemission to the photoionization rates, it is important to model the finite width of the recombination lines. If HI LyC reemission is incorrectly modeled as a function, the reemission photons are immediately redshifted below the HI ionization edge and are lost as contributors to the ionizing background. For continuum recombinations, the line profile is well approximated by

 ϕrec(ν)=(ν/νrec)−1exp(−hν/kT)Γ(0, hνrec/kT)θ(ν−νrec)νrec, (9)

where is the temperature of the gas and is the Heaviside function which is 1 for and 0 otherwise (Appendix E). Electrons in gas at higher temperature tend to have large kinetic energy and so give rise to higher-energy recombination photons that take longer to redshift below the ionization edge. In Appendix E, we show that broadening owing to thermal and peculiar motion is negligible relative the width of the profile in equation 9. For Ly emission, either by HI or HeII, a function profile is however appropriate because of the narrow intrinsic line width (much smaller than the mean free path by which photons are redshifted before being reabsorbed) and its distance from the ionization edges. While resonant scattering radiative transfer effects can broaden Ly emission lines by km s (e.g., Neufeld, 1990; Zheng & Miralda-Escudé, 2002; Dijkstra et al., 2006; Verhamme et al., 2006), this width is negligible in comparison to the cosmological redshift broadening.

The analytical approximations for the recombination emission from individual absorbers are compared to the full numerical calculations in Figure 2 for the ionizing processes. In all cases, (where we define ) is maximum for HI LyC reemission, as expected since hydrogen recombinations are more frequent owing to its greater abundance and these recombination photons have the largest photoionization cross section, and is equal to about 10%. The helium recombination processes all contribute at the level or less. Note, however, that HeII LyC reemission will contribute more significantly to the HeII ionizing background and that processes which contribute negligibly to the photoionization rates can still imprint important narrow features in the background spectrum that can be important for metal line studies. The agreement between the numerical calculations and analytical approximations is generally good and the approximations scale well for different spectral indices. Discrepancies of a factor of a few exist over some column density intervals, particularly for the HeII BalC and HeII LyC processes. These processes are complex in their details that depend on the non-monotonic relative ionization of hydrogen and helium (Fig. 1) but their contributions are nevertheless reasonably well captured and fortunately subdominant to the photoionization rates. In contrast, the dominant contribution of reemission to the hydrogen ionizing background, HI LyC emission, involves only hydrogen and is accurately and robustly approximated.

Figure 3 compares the analytical approximation for HI Ly approximation to the full numerical solutions. In this case, both the optically thin and optically thick limits are accurately captured, resulting in an excellent approximation at all column densities that scales correctly with the external illuminating spectrum.

## 5. Empirically Calibrated Spectra

### 5.1. Quasar and Stellar Emissivities

Having established efficient approximations for the radiative transfer within individual absorbers (§3 and 4), we proceed to include these in the solution of the cosmological radiative transfer problem (§2). Our prescriptions for the sources of ionizing radiation are based on the empirical constraints obtained in Faucher-Giguère et al. (2008d, a, b). Note, however, that these prescriptions can easily be modified to accommodate further constraints: our numerical code can compute the ionizing background for arbitrary input emissivities. We explore variations about these fiducial parameters in §6. Here, we consider two dominant known sources of ionizing radiation: quasars and star-forming galaxies.

For the quasar emissivity, we use the quasar luminosity function of Hopkins et al. (2007) based on a large set of observed quasar luminosity functions in the infrared, optical, soft and hard X-rays, as well as emission line measurements. Denoting by the emissivity at 4400 Å and assuming ,

 ϵQSO,comB=∫∞0dLBdϕdLBLBν|4400 \AA, (10)

where is the -band luminosity function in comoving units. The emissivity shortward of 4400 Å is calculated assuming that quasars have a spectral index at 2500-4400 Å, 0.8 at 1050-2500 Å (Madau et al., 1999), and shortward of 1050 Å. In order to match the total HI photoionization rate measured from the Ly forest and to account for uncertainties in converting from the emissivity at 4400 Å to the photoionization rate, we allow this emissivity to be normalized by a constant factor (see §5.2). In our fiducial model, (Telfer et al., 2002) but note that other studies have found both softer and harder spectra, with a significant variance about the mean (e.g., Zheng et al., 1997; Scott et al., 2004).

For the stellar emissivity, we assume that the emissivity is proportional to the star formation rate density,

 ϵ⋆,comν=K˙ρcom⋆, (11)

with the observationally-calibrated proportionality constant accounting for the efficiency of conversion of mass into ionizing photons. We use the theoretical star formation history of Hernquist & Springel (2003) developed from a combination of hydrodynamical simulations (Springel & Hernquist, 2003) and simple analytical arguments. In Faucher-Giguère et al. (2008b), we found that this model provides a better fit at high redshifts to the opacity of the Ly forest over , is easier to reconcile with hydrogen reionization completing by , and is in better agreement with the rate of long gamma-ray bursts observed by Swift than many of the existing measurements based on galaxy surveys, among which there is still a wide dispersion. We use equation 45 of Hernquist & Springel (2003) to scale their fiducial model to the cosmology assumed in this work. We assume, motivated by the theoretical starburst calculations of Kewley et al. (2001), that star-forming galaxies have a spectral index between 1 and 4 Ryd. This model is applicable for the stellar populations calculated with the PEGASE code using the Clegg & Middlemass (1987) atmosphere models for Wolf-Rayet stars. While different theoretical assumptions lead to significant variance in the 14 Ryd spectrum, this model provides the best observational match to the hard starburst spectra inferred by optical line diagnostics by Kewley et al. (2001). We assume that they effectively emit no harder photons, the theoretical calculations showing a break of several orders of magnitude at the HeII ionization edge. In the UV spectrum redward of 1 Ryd, we take , consistent with the LBGs observed by Shapley et al. (2003). Finally, we assume that the stellar emissivity has a discontinuity of a factor of 4 at the Lyman limit. While this factor is neither well constrained empirically or observationally, it only affects our predicted spectra (normalized to the measured ionizing background) at energies less than 1 Ryd, which we do not attempt to accurately model in this work.

The emissivities are converted to proper units before being inserted in the solution to the cosmological radiative transfer solution in equation 2 and the total emissivity is then .

### 5.2. Results

In Figure 4, we show the calculated cosmological UV background spectra at for the fiducial model above, with and without the recombination processes included. Since only quasars are assumed to contribute photons above the 4 Ryd HeII ionization edge, only them contribute to the HeII recombination lines and photoionization rate.

In Figure 5, we show the integrated photoionization rates of HI and HeII, as well as the fractional contribution of recombination lines with respect to the total background including both stars and quasars. The quasar contribution to the HI ionizing background increases toward as the peak of the quasar luminosity function is approached; the photoionization rate is dominated by stellar emission. The fractional recombination contribution to the HI photoionization rate ranges from 5% to 17% over the interval , significantly smaller than the % fraction of HI recombinations that are directly to the ground state. The relatively small contribution of recombinations to the ionizing background owes to a combination of the saturation of reemission in optically thick systems (Fig. 2), leakage of the reemitted photons at the ionizing edge, and the frequency dependence of the photoionization cross section (§6.2 and Appendix C).

The fractional contribution of HeII recombinations to the HeII photoionization rate is also relatively small for the same reason, but is more difficult to calculate accurately at redshifts in our model. In order to obtain an accurate value, in addition to the HeII LyC reemission line to be well resolved on the computational frequency grid, the mean free path of HeII ionizing photons must also be well resolved by the redshift grid. In our calculation, quasars produce a negligible and rapidly dropping HeII photoionization rate at while star-forming galaxies maintain a roughly constant HI photoionization rate. In these conditions, the ratio tends to infinity and the HeII mean free path to zero, making it exceedingly difficult to resolve it. Fortunately, the total HeII photoionization rate in this regime is so small that its fractional enhancement from recombinations is of little practical importance. Moreover, in this regime HeII reionization may well be still underway and the HeII ionizing background consequently modified, as elaborated on in §7. In Figure 5, we indicate the portion of poor convergence by a dashed curve segment; the turnover of around is likely an artifact and we in fact expect it to continue to increase slightly toward higher redshifts owing to the reduced leakage (§6.2).

The total HI photoionization rate matches the value s derived from the Ly forest at , subject to the constraint that quasars must contribute a large fraction near their peak (Faucher-Giguère et al., 2008a, b). This was done by normalizing the nominal quasar contribution (§5.1) by a factor of 0.36 and normalizing the stellar contribution so as to provide the rest of the ionizing photons. The renormalization of the quasar contribution can be justified by uncertainties in the mean free path of HI ionizing photons (a direct product of the prescribed HI column density distribution), in their escape fraction, and in the quasar spectral template (see discussion in Faucher-Giguère et al., 2008b). These uncertain factors enter in the conversion from the quasar luminosity to the photoionization rate. Since we wish to reproduce the more robustly constrained photoionization rate measured from the Ly forest, we adjust the normalization accordingly.

Although our calculations are normalized to match the hydrogen photoionization rate measured from the Ly forest, it is important to emphasize that this measurement and hence the normalization of the spectra calculated here are somewhat uncertain. The measurement was obtained using the flux decrement method (e.g., Rauch et al., 1997), in which we solve for the value of needed to produce the measured mean transmission of the Ly forest. Two important sources of systematic uncertainty are the assumed IGM temperature (since the flux decrement constrains only the combination ) and the gas density distribution (whose details depend on the cosmological parameters and thermal history). Another potential worry is that the measured Ly forest mean transmission may be increasingly biased high toward high redshifts (inducing a redshift-dependent error) as the continuum level is increasingly absorbed and difficult to estimate directly. We have however quantified and corrected for this effect in our measurement (Faucher-Giguère et al., 2008d) and so it should not affect our results. In the end, we expect the measured to be accurate within a factor , with the possible errors mostly systematic and weakly dependent on redshift. For a more exhaustive discussion of the uncertainties of the measured , see §3 of Faucher-Giguère et al. (2008b).

Finally, we must also note that for precise work with hydrodynamical simulations, the simulated Ly forest mean transmission should always be compared with the measured value and the photoionization rates renormalized if necessary. In fact, even if the correct ionizing background (with the correct normalization) is prescribed, the simulated mean transmission may be slightly off if, for example, the temperature of the IGM is incorrect reproduced. This is particularly likely to occur if the effects of HI and HeII reionization (see §7) are not explicitly modeled.

## 6. Dependences on Input Parameters

Even after fixing the stellar and quasar emissivities for our fiducial model, the spectrum calculations depend on a number of parameters. It is useful to investigate how the calculated spectrum and its integrals depend on these as their values are only known to limited precision. This also provides a physical understanding of the shape of the calculated spectra. We begin by considering the dependences of the overall ionizing background spectrum in §6.1 and focus on the recombination contribution in §6.2.

### 6.1. Overall Spectrum

In Figure 6, we show how the spectrum changes when the constant ratio , the HI column density distribution power-law index , and the stellar and quasar spectral indices and are individually varied. In each case, all other parameters are fixed to the fiducial model of the previous section. Even for a constant ratio , a factor more than one hundred times that expected in our fiducial calculation (§3.2), HeI absorbs only a very small fraction of the spectrum shortward of its ionization edge. It is therefore a good approximation to neglect it in our cosmological calculations. The HI column density distribution power-law index determines the spectral hardening just above the ionization edges following (Appendix D) as well as the depth of the absorption edges. Note that the depth of the HeII absorption edge is more sensitive to ; this arises because the column density distribution is normalized to the abundance of HI Lyman limit systems (§2.2) so that it is fixed in these calculations while the abundance of the HeII Lyman limit systems varies. The stellar and quasar spectral indices simply determine the spectral slopes of the background prior to hardening. For fixed emissivity at the Lyman limit, the stellar spectral index has only a modest effect on the amplitude of the spectrum because it is truncated at 4 Ryd. The quasar spectral index, assumed to extend to infinity, has a more drastic overall impact toward high energies: as and spectral hardening becomes negligible, different spectral indices result in a ratio of . At 10 keV, for example, this ratio is 735 for and ; redshifted from to , this falls in the bandpass of x-ray observatories such as Chandra and XMM-Newton. As most (%) of the soft x-ray background has already been resolved into AGNs (e.g., Hasinger et al., 2005), the x-ray background is a powerful probe of the high-energy quasar spectral energy distribution, although a proper analysis requires the inclusion of obscured quasars, which we do not explicitly consider in this work (e.g., Gilli et al., 2007).

In Figure 7, we explore how the hydrogen photoionization rate is affected by the redshift evolution of the column density distribution. In the left panel, we vary the redshift at which the redshift evolution of the column density distribution flattens (§2.2) from to . Interestingly, this has a minimal impact on the redshift evolution of the photoionization rate even if it does significantly change the mean free path of HI ionizing photons at these redshifts. This is easily understood as a consequence of the fact that the universe effectively becomes transparent at a “breakthrough” redshift (Madau et al., 1999), below which the mean free path becomes so large the local ionizing background is not limited by the latter but by the cosmological horizon. As shown in the right panel, the high-redshift declines more rapidly with more a rapid increase in the abundance of absorbers with redshift, or large , translating into a more rapidly diminishing mean free path. At present, although more than a decade old, the best constraints on the abundance of the Lyman limit systems (Storrie-Lombardi et al., 1994, 1996; Stengler-Larrea et al., 1995) are relatively loose and mostly nonexistent beyond . As future measurements refine these and push toward higher redshifts, it is possible that these will give more credence to one of the alternative values of plotted here.

### 6.2. Recombination Contribution

The contribution of recombinations to the photoionization rates, , is a subtle question as it depends on several factors. It not only depends on the number of reemitted ionizing photons integrated over the distribution of absorbers (§4) but also crucially on the energy at which these photons are reemitted as well as on their redshifted energy at the point of evaluation of the photoionization rate.

The LyC recombination line processes, most important for the boosting the ionization rates, reemit ionizing photons just above the ionization edges of HI or HeII. Since the ionizing background at a given point is sourced along its past light cone, its photons have generally redshifted slightly from their emission energy. As a result, many recombination photons with initial energy just above their corresponding ionization edges quickly redshift below these edges and are lost as contributors to the ionization rates. The fraction of ionizing recombination photons lost in this way depends on two factors: 1) the recombination line profile which determines how far above the ionization edge an ionizing photon is reemitted and 2) the mean free path of ionizing photons which determines how long the photons have to redshift before they are absorbed. In the limit of a mean free path of zero length, the recombination photons cannot redshift before they are reabsorbed and no photons are lost; as the mean free path increases, more photons leak out of the ionizing range. Similarly, a narrow line profile concentrates the recombination photons just above the ionization edges, leading to a high probability of leakage, while a wider one allows them to remain longer in the ionizing range. Since the mean free path of ionizing photons is determined by the column density distribution and the recombination line width is determined by the temperature of the absorbers, these are important parameters for the recombination contribution. Finally, as recombination photons are reemitted at energies above the ionization edges and subsequently redshift, the frequency dependence of the photoionization cross section changes the weight they receive in the photoionization rates.

All these effects are self-consistently treated when solving the radiative transfer equation 1. Figure 8 shows how much the HI photoionization rate is increased by recombination emission as a function of the normalization of the column density distribution, its power-law slope (see eq. 5), and the temperature of the absorbers. We also show the case of the column density distribution assumed by Haardt & Madau (1996), in which optically thin and optically thick absorbers have different redshift evolutions, leading to a redshift-dependent effective slope of the distribution (steeper at high redshifts).

Because the mean free path decreases with increasing abundance of Lyman limit systems (the normalization ), fewer recombination photons leak out of the ionizing range and so the ratio of the photoionization rate with and without recombinations, , increases. The ratio also increases with the steepness of the column density distribution since more recombinations occur in optically thin systems, from which practically all the recombination photons escape into into the IGM, as opposed to in optically thick systems that trap a large fraction. Higher gas temperatures result in wider recombination lines so that fewer photons are lost owing to redshifting as well as a higher fraction of recombinations directly to the ground state (; Appendix C). A competing effect is that the recombination photons of high-temperature absorbers tend to have higher energies and receive less weight in the photoionization rate. The net effect is however relatively weak on for the relevant temperature K. Note that for any given set of parameters the ratio tends to increase toward higher redshifts since the mean free path is lower at the higher cosmological densities. This behavior is however not seen below the breakthrough redshift , where the photoionization rate is limited by the cosmological horizon rather than by the mean free path.

In Appendix C, we develop a quantitative analytic model that captures and clarifies these effects and agrees well with the full numerical calculations presented here.

## 7. Reionization Events

The previous calculations have implicitly assumed that the universe is reionized in both HI and HeII. This assumption is most evident in the case of HI, for which we have used a column density distribution measured from the Ly forest. The assumption creeps in for HeII reionization during which there are large HeII patches the inside which the HeII photoionization rate is very low in comparison to within ionized bubbles. In each region, the mapping between and depends on the local spectrum and will in general be very inhomogeneous. The IGM opacity to HeII ionizing photons will therefore be poorly approximated by using a globally-averaged spectrum to map from to . In this work, we do not attempt to model HI reionization (and the likely simultaneous reionization of HeI), which occurs at the limit of the present observational reach at (e.g., Fan et al., 2006a; Dunkley et al., 2008).

The reionization of HeII was however likely delayed until the rise of the quasar luminosity function, at redshifts that are immediately accessible to observations and some understanding of its effects can be obtained by studying the ratio . In fact, while stellar spectra are theoretically expected to have a strong break at the HeII ionization edge and therefore have little impact on the HeII ionization state, quasars have power-law far-UV spectra that extend well into the x-rays (§5.1). Theoretical calculations based on the quasar luminosity function in fact indicate that quasars can reionize HeII by (e.g., Sokasian et al., 2002; Wyithe & Loeb, 2003; Furlanetto & Oh, 2008; Faucher-Giguère et al., 2008b). A number of lines of evidence, based HI and HeII Ly forests as well as on the evolution of metal line ratios, also suggest that the IGM is undergoing changes that could be associated with HeII reionization at these redshifts (for a review of these lines of evidence, see Faucher-Giguère et al., 2008d). While alternative candidate sources of HeII reionization exist – such as possible HeII ionizing emission from galaxies (e.g., Furlanetto & Oh, 2008), high-redshift x-rays (e.g., Oh, 2001b; Ricotti & Ostriker, 2004), or thermal emission from shock heated gas (Miniati et al., 2004) – quasars are the best established and most likely. Large fluctuations observed in the HeII ionizing background toward , which can be explained by the small number density of bright objects, also lend support to the quasar hypothesis (Zheng et al., 2004; Shull et al., 2004; Bolton et al., 2006) and we will therefore concentrate on this scenario.

### 7.1. The Ionizing Background During HeII Reionization

Recently, McQuinn et al. (2008) performed detailed radiative transfer simulations of HeII reionization in large boxes up to 430 comoving Mpc on a side (for previous simpler treatments, analytic and numerical, see Sokasian et al., 2002; Bolton et al., 2004; Gleser et al., 2005; Paschos et al., 2007; Furlanetto & Oh, 2008; Bolton et al., 2008). These simulations used realistic models for the quasar sources based on the luminosity function of Hopkins et al. (2007) and with physically and empirically motivated prescriptions for the triggering of quasars in massive halos as well as of quasar light curves (see, e.g., Hopkins et al., 2005a, b, 2006, 2008). A striking result of this work is the remarkable complexity of HeII reionization, in particular of the HeII ionizing radiation field, likely rendering the detailed resulting structure beyond analytic tractability. Nevertheless, some intuition on the spectrum and magnitude of the ionizing background during HeII reionization can be gained by considering idealized cases. We consider two such cases: 1) a single quasar at the center of an isolated ionized bubble and 2) a point in a large HeII patch that has yet to be reionized.

Key insight into the ionizing background is gained by considering relevant physical scales. The left panel of Figure 9 compares the mean free paths (calculated as in Appendix D) of 1 Ryd (HI ionizing) and 4 Ryd (HeII ionizing) photons to the mean separation between the sources of the ionizing background, while the right panel shows scales relevant to understanding the possible effects of HeII reionization by quasars at . The HeII ionizing mean free paths are calculated by converting the HI column densities to HeII assuming a constant s and varying . Since emission from star-forming galaxies provides most of the hydrogen photoionization rate at 5) and the mean separation between galaxies is much smaller than the HI ionizing mean free path at all redshifts considered,333At redshifts , the estimated mean free path relies on an extrapolation of the measured column density distribution and so the conclusion should accordingly be treated with caution. In particular, the conclusion is likely to break down if HI reionization ends at (e.g., Fan et al., 2002, 2006b). it is a good approximation to treat the stellar emissivity as a uniform volume average as in equation 2. It is also similarly the case for HI ionizing quasar emissivity at redshifts , where quasars are relatively abundant and the HI ionizing mean free path large, though with larger fluctuations expected from the smaller number of quasars within each mean free path (for more detailed studies of UV background fluctuations, see Zuo, 1992b, a; Fardal & Shull, 1993; Croft et al., 1999, 2002; Gnedin & Hamilton, 2002; Meiksin & White, 2003, 2004; Croft, 2004). Thus, it is a reasonable approximation at all redshifts to calculate the ionizing background between 1 and 4 Ryd using a volume average emissivity.

The situation is however quite different beyond 4 Ryd, where continuum opacity owing to HeII dominates. In fact, the mean free path of HeII ionizing photons at these energies, which depends on the local HeII photoionization rate, is generally smaller than the mean free path of 1 Ryd HI ionizing photons since even quasars produce relatively few photons above 4 Ryd. For example, near the peak of the quasar luminosity function at , (Bolton et al., 2006). The relative rarity of quasars and shortness of the HeII ionizing mean free path combine to create a situation in which often a single bright quasar contributes to the local HeII ionizing flux. This is the case even after HeII reionization has completed and results in substantial fluctuations in the ionizing background above 4 Ryd. These fluctuations could be important for metal absorption line studies and will be addressed in future work (for recent studies of the HeII ionizing background fluctuations, see Bolton et al., 2006; Furlanetto, 2008, 2009). Prior to the complete reionization of HeII, ionized bubble walls will further limit the exposure of a given point to the HeII ionizing fields of distant quasars. The radii of HeIII bubbles depend, at least until they percolate, on the ionizing luminosity of the central quasars and the duration for which these have been shining. This gives rise to a wide range of scales, depending on the specific quasar model, but by the middle of HeII reionization (determined by an ionized fraction ) bubble radius comoving Mpc (corresponding to a few to 20-25 proper Mpc at ) are representative (e.g., Furlanetto & Oh, 2008; McQuinn et al., 2008). A volume average uniform emissivity is then clearly inappropriate.

#### 7.1.1 Quasar Within an Isolated Ionized Bubble

Consider first a point within an isolated HeIII bubble occupied by a single quasar at the center, . Locally neglecting cosmological effects, the specific intensity of a radial ray is given by

 Iν=Iν(r=0)e−τν(r). (12)

In addition to the intensity being exponentially suppressed, the spectral shape is altered by the frequency dependence of the optical depth:

 (13)

where the last equality holds approximately just above the HeII ionization edge and we have neglected the fractionally small HI continuum opacity. It follows that the magnitude of the specific intensity is set by the optical depth at the HeII ionization edge to the source, with the spectral shape entirely determined by the frequency dependence of the photoionization cross section at a given optical depth, in addition to the intrinsic spectrum of the source.

In the left panel of Figure 10, we show a quasar spectrum is hardened as a function of , the optical depth at the HeII ionization edge from the source. The curves are pictorially labeled assuming a HeII ionizing mean free path comparable to the HeIII bubble size, so that is near the edge of an isolated bubble centered on the quasar. We also show a post HeII reionization case in which the mean free path is sufficiently large to contain several quasar sources, with the HI column density distribution and softness parameter measured at (Bolton et al., 2006). This limit is representative of the hardening in the calculations of §5. Note that as , the spectral shape can be arbitrarily hardened just above the HeII ionization edge. As and , the spectrum returns to the unfiltered case.

The rarity of quasars implies that around an individual object the specific intensity obeys equation 12, in which a single source is attenuated with distance, rather than a solution involving a volume average emissivity as in equation 2. Why, though, does the ordinary optical depth enter in equation 12 instead of the effective optical depth as in equation 2? Any given light ray is always attenuated according to the intervening ordinary optical depth . However, the optical depth between two points separated by a fixed distance (at fixed frequency and redshifts) fluctuates depending on their particular spatial positions because of the stochastic nature of the intervening absorbers. The effective optical depth captures the average attenuation through . It is an appropriate quantity for the ionizing background between 1 and 4 Ryd, where the local intensity is an average over the light received from sources in all directions within one mean free path. The radiation above 4 Ryd at a given point in the vicinity of a quasar prior to and during HeII reionization will often be dominated by the local quasar and therefore be uniquely attenuated as in equation 12.

#### 7.1.2 Point in a Large HeII Patch

A point within a HeII patch that has not yet been reionized444In reality, the HeII ionization fronts are quite smooth and extended since they are driven by a hard spectrum (e.g., McQuinn et al., 2008). Except at the very beginning, few points have been truly untouched by HeII reionization, but the discussion holds wherever the ionized fraction has not exceeded, say, . will see a similarly hardened spectrum, but with a much stronger suppression at the HeII ionization edge owing to the large intervening optical depth. The optical depth at the HeII ionization edge, as a function of redshift and path length , in a medium in which all the helium is assumed to be in the form of HeII is given by

 τneutralνHeII=σHeII(νHeII)nHeII(z)L=318(1+z4.5)4(L10 comoving Mpc). (14)

In HeII patch, the intensity of the background at the ionization edge is therefore expected be almost entirely suppressed. As and , however, the optical depth drops quickly and the intensity of the background recovers. The corresponding increase of the mean free path with energy leads to the presence of a spatially smooth high-energy radiation background permeating most of the cosmic volume, as seen for example in the numerical simulations of McQuinn et al. (2008). The right panel of Figure 10 shows how the fiducial spectrum of the ionizing background at , as calculated in §5, is altered shortward of the HeII ionization edge as a function of the intervening optical depth. Note, in particular, the tremendous HeII edge suppression even in the moderate case of .

### 7.2. Recombination Lines during HeII Reionization

The photoionization rate and ionization state of hydrogen are unaffected by the presence of HeII before and during HeII reionization apart from a small contribution by photons above 4 Ryd. Consequently, only the HeII recombination processes are significantly modified. Of these, the most important is HeII Ly which imprints a distinctive line feature at 3 Ryd (Fig. 4); HeII LyC and BalC only slightly smooth the spectrum at the HeII and HI ionization edges, respectively, and contribute only marginally to the photoionization rates (Figs. 2 and 5).

Equations B7 and B8 compactly capture the behavior to HeII Ly reemission. As explained in the previous section, before HeII reionization begins the background spectrum is almost completely suppressed above 4 Ryd by the large optical depth at these energies. Since HeII Ly reemission scales with the HeII ionizing spectrum (with saturation in the optically thick limit), it will be absent before the start of HeII reionization. Similarly, no HeII Ly should arise within HeII patches during HeII reionization. However, HeII Ly will be reemitted within ionized bubbles illuminated by the local quasars. As HeII reionization proceeds, the distance between neighboring bubbles should quickly become smaller than the mean free path for the LyC absorption of 3 Ryd photons by HI (at 1 Ryd, Fig. 9 shows the mean free path to be about 200 comoving Mpc at ; at 3 Ryd, eq. D3 predicts the mean free path to be longer by a factor of ) that governs the attenuation of HeII Ly radiation. In the regime in which this mean free path contains several ionized bubbles, the volume fraction of reionized HeII can be viewed as the fraction of the IGM reemitting in HeII Ly and the HeII Ly reemission line of the background spectrum can be expected to be about this fraction times the fully reionized value.

### 7.3. Heat Input During HeII Reionization

The photons that ionize HeII atoms in general carry more energy than the required. The residual energy is converted into kinetic energy of the resulting free electron and HeIII nucleus, with the frequent Coulomb collisions leading to rapid thermalization. This process of photoheating is at work at all times and for all species present. Its effect is however much more important during reionization, when atoms are being ionized at a much greater rate. The effects of HeII reionization on the thermal state of the gas in cosmological simulations has so far generally be modeled by artificially boosting the photoheating rate calculated from a prescribed spatially homogeneous background instantaneously(e.g., Bryan & Machacek, 2000; Theuns et al., 2002; Jena et al., 2005), or ignoring it altogether. This approach, of limited physical basis, is a serious limitation of these simulations given the growing body of evidence that HeII reionization occurs at observable redshifts and is certain to manifest itself to some extent.

While cosmological radiative transfer simulations are beginning to self-consistently treat gas thermodynamics during HeII reionization (e.g., Paschos et al., 2007; McQuinn et al., 2008), it is likely that the vast majority of simulations performed in the near to moderate future will not explicitly incorporate radiative transfer, either due to the computational cost or to the unavailability of an appropriate code. It therefore remains important to develop ways of approximately treating the effects of HeII reionization in those simulations. We examine this problem in this section. Specifically, we consider the questions: How much does HeII reionization heat the IGM? Over what timescale? And how can we approximately model its effects in standard cosmological N-body and hydrodynamical codes such as GADGET (Springel et al., 2001; Springel, 2005), Hydra (Pearce & Couchman, 1997), or Enzo (O’Shea et al., 2004)?

The simple analytic models that follow are motivated by and owe much to the physical picture of HeII reionization suggested by the radiative transfer calculations of McQuinn et al. (2008). We refer to that work for many original insights.

#### 7.3.1 Heat Input Calculation

In order to gain physical intuition, we begin with a simplified model. Suppose that all the photons up to frequency emitted by a population of sources with intrinsic spectral index are absorbed by HeII atoms. Then the mean energy injected into the IGM per ionization is given by

 ⟨Ei⟩=∫νmaxνHeIIdν/(hν)(hν−hνHeII)ν−αUV∫νmaxνHeIIdν/(hν)ν−αUV=hνHeII[αUVαUV−1(1−xαUV−1)(1−xαUV)−1]≈hνHeIIαUV−1(1−αUVxαUV−1), (15)

where and the last equality holds approximation for and . This equation neglects redshifting of the photons before absorption, which is a reasonable approximation if HeII reionization lasts at (e.g., Furlanetto & Oh, 2008; Faucher-Giguère et al., 2008b; McQuinn et al., 2008). Here, the effects of spectral filtering (e.g., Abel & Haehnelt, 1999; Bolton et al., 2004; Tittley & Meiksin, 2007; Bolton et al., 2008) are incorporated in the prescribed frequency cutoff .

If all the helium is initially in the form of HeII and hydrogen is fully ionized, the temperature increase is obtained by distributing the injected energy over all particles. After thermal equilibrium has been reached,

 ΔTHeII=23knHentot⟨Ei⟩=15550 K[αUVαUV−1(1−xαUV−1)(1−xαUV)−1]≈31100 K(0.5αUV−1)(1−αUVxαUV−1). (16)

Here, is the total number density of particles including free electrons. Note that the total number of particles is slightly less before HeII reionization owing to the smaller number of free electrons. The fractional change of is however negligible. Although the use of a sharp frequency cutoff is a simplification of the radiative transfer, McQuinn et al. (2008) show that a simple argument like this one gives a good estimate of the heat input determined from detailed radiative transfer simulations.

What is the relevant value of ? A reasonable guess is the value such that the mean free path of photons of this frequency equals the “thickness” of HeII reionization. Photons of higher frequency (and therefore longer mean free path) will typically not be absorbed before HeII reionization is complete. The right panel of Figure 9 shows where the mean free path intersects the thickness of HeII reionization, assuming that the bulk of the latter takes place between and , for different values of . For this purpose, we calculate the mean free path assuming homogeneously distributed 50% ionized HeII at . Under these conditions (accounting for some uncertainty on the thickness of HeII reionization) we expect . Figure 11 shows the corresponding heat input owing to HeII reionization for different value of the spectral index . For spectral indexes 5.1), the heat input depends only weakly on our rough estimate of .

Having obtained simple estimates for the total heat input during HeII reionization, we proceed to make the derivation more rigorous, which also allows us to trace the time evolution of the heat injection. Specifically, we replace the sharp frequency cutoff by a calculation taking into account the fraction of photons emitted at each frequency at any given redshift that is absorbed during HeII reionization:

 ⟨Ei⟩(z)=∫∞zdt∫∞ν′HeIIdν′/(hν′)(hν′−hν′HeII)ϵQSO,comν′(z′)[1−e−τ(ν′, z, z′(t))]∫∞zdt∫∞ν′HeIIdν′/(hν′)ϵQSO,comν′(z′)[1−e−τ(ν′, z, z′(t))], (17)

where and

 τ(ν′, z, z′)=∫z′zdz′′dldz′′nHe(z′′)[1−yIII(z′′; αUV)]σHeII(ν′′=ν(1+z′′)(1+z′)) (18)

is the optical depth encountered by a photon of frequency emitted at redshift before reaching redshift . Here, is the proper number density of helium atoms and a fraction given by the reionization state is assumed to be homogeneously distributed in the form of HeII, taken to be the dominant source of opacity. Equation 17 is similar to equation 15, but with the high-frequency cutoff replaced by the smoothly varying fraction of photons absorbed for each frequency. In addition, the mean energy injected per ionization is calculated as a function of redshift, which allows us to trace the heat input over time. While the homogeneous IGM approximation is obviously a simplification, it is a reasonable assumption for this heuristic calculation. In fact, the potential error introduced by neglecting the inhomogeneities is most important for . However, the strong frequency dependence of the HeII photoionization cross section implies that the range of photon energy for which is narrow. Morever, for a quasar spectral index , the cruder estimate of Fig. 11 indicates that the heat input is only weakly sensitive to the exact maximum energy of the absorbed photons. These effects combine to make the uniform IGM approximation relatively robust for this particular calculation. Ultimately, though, the calculation is motivated by the fact that it reproduces the results of the full radiative transfer calculations of McQuinn et al. (2008) well.

Since at a given redshift , only a fraction of the HeII has been reionized, the temperature increase contributed by HeII reionization at that redshift, neglecting cooling, is given by

 ΔTHeII(z)=23knHentotyIII(z)⟨Ei⟩(z). (19)

The temperature of a cosmic gas parcel is in general determined by all the processes by which it gains heat and cools as it evolves, including adiabatic heating and cooling, shock heating, photoheating, Compton cooling off microwave background photons, and recombination cooling (e.g., Hui & Gnedin, 1997). Instructive intuition can however be gained from idealized solutions.

In the limit of early HI reionization (with the reionization of HeI assumed to proceed simultaneously), the temperature at mean density reaches a “thermal asymptote” determined by the competition between adiabatic cooling and photoheating and whose value depends on the HeII ionization state. For a power-law background spectrum just above the ionization edges, a good approximation to the thermal asymptote is given by

 Tasymp0(z)=2.49×104 K(0.464+0.536yIII)(2+αbg)−1/1.7(1+z4.9)0.53 (20)

(Hui & Haiman, 2003). To first order and ignoring inhomogeneities, the effect of HeII reionization is to inject additional heat to each gas parcel. As the universe expands, the extra heat is diluted by adiabatic cooling, , so that an estimate of the overall temperature evolution in the early HI reionization limit accounting for HeII reionization heat input is given by

 T(z)≈Tassymp0(z)+∫z∞dz′dΔTHeII(z′)dz′(1+z1+z′)2. (21)

Figure 12 shows thermal histories calculated using this equation assuming a background spectral index and the Hopkins et al. (2007) quasar luminosity function in the band for different spectral indices of the HeII ionizing sources . Harder spectral indices are seen to result in greater heat injections, which simply owes to the larger fraction of ionizations caused by high-energy photons. Moreover, the magnitude of the total heat input as a function of spectral index is consistent with the simpler estimates using a sharp frequency cutoff shown in Figure 11. At fixed band luminosity, harder spectral indices result in higher ionizing photon output rates and thus earlier HeII reionization.

The HeIII fraction in the above equations is obtained by counting the number of HeII ionizing photons emitted by quasars as in Faucher-Giguère et al. (2008b) and we have assumed a gas clumping factor .

#### 7.3.2 Scatter in the Temperature-Density Relation

The thermal history calculations of the previous section implicitly assumed that the universe is homogeneous at a mean density and that HeII reionization happens simultaneously throughout. In reality, the IGM is characterized by density fluctuations and the quasars that putatively drive HeII reionization turn on at different times at different locations owing to cosmic variance. These inhomogeneities imply that the IGM temperature is not fully described by a single redshift dependent number but in reality exhibits a temperature-density relation with some scatter about the mean at each redshift.

In the absence of HeII reionization, Hui & Gnedin (1997) showed that the temperature-density relation at is well approximated by a power law . In the limit of early HI reionization, as a result of the competition between photoheating and adiabatic cooling. We are interested in how this result is modified by HeII reionization. Equation 21 can be generalized to

 T(z)≈Tassymp0(z)Δβ+κ∫z∞dz′dΔTHeII(z′)dz′(1+z1+z′)2, (22)

where is set to the value that would be obtained without HeII reionization and is a stochastic factor that accounts for the fact that different regions are heated at different times by HeII reionization photoheating. Our task is then reduced to determining the distribution function of to estimate the scatter in the temperature-density relation.

One of the results highlighted by the radiative transfer simulations of McQuinn et al. (2008) is that much of the heating during HeII reionization by quasars results from ionizations by the diffuse background of high-energy photons with large mean free paths that penetrate into HeII patches before these are actually reionized by softer photons (for a different picture, see Bolton et al., 2008). In this picture, the longer a given region is exposed to the high-energy background before it is reionized, the more heat it receives; regions that are reionized last tend to be hotter. As an ansatz, again motivated by the work of McQuinn et al. (2008), we may thus posit that , where is an effective exposure time to the high-energy background. Note, though, that this will not be correct at the very beginning of HeII reionization before the background has had time to diffuse. We denote by the redshift at which a given gas parcel is reionized in HeII and set

 texp,eff(zHeII)≡∫zHeII∞dtyIII(z)(1−yIII(z)). (23)

The effective exposure time is thus the age of the universe at reionization of the gas parcel, weighted by the time-dependent ionized fraction, and saturating as the latter reaches order unity. The motivation for the weighting is that the heat injection is not only proportional to the raw exposure time, but also to the intensity of the high-energy background. The ionized fraction counts the number of ionizing photons emitted and is therefore a tracer of this high-energy background. The saturation factor approximates the fact the rate of heat input also scales with and is thus suppressed toward the end of reionization.

The PDF of reionization redshifts is also straightforwardly approximated from the ionized fraction evolution since the probability of reionization during a redshift interval scales as the rate at which ionizations occur at that time:

 P(zHeII; z)={yIII(zHeII)−1dyIIIdz(zHeII)zHeII≥z0zHeII

Energy conservation requires , so we set