Implications of Intergalactic Opacity

# Evolution of the Intergalactic Opacity: Implications for the Ionizing Background, Cosmic Star Formation, and Quasar Activity

Claude-André Faucher-Giguère1 , Adam Lidz1 , Lars Hernquist1 , Matias Zaldarriaga1 2
1affiliation: Department of Astronomy, Harvard University, Cambridge, MA, 02138, USA; cgiguere@cfa.harvard.edu.
2affiliation: Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, USA.
###### Abstract

We investigate the implications of the intergalactic opacity for the evolution of the cosmic ultraviolet luminosity density and its sources. Our main constraint follows from our measurement of the Lyman- forest opacity at redshifts from a sample of 86 high-resolution quasar spectra. In addition, we impose the requirements that intergalactic HI must be reionized by and HeII by , and consider estimates of the hardness of the ionizing background from HI to HeII column density ratio measurements. The derived hydrogen photoionization rate is remarkably flat over the Ly forest redshift range covered. Because the quasar luminosity function is strongly peaked near , the lack of redshift evolution indicates that star-forming galaxies likely dominate the photoionization rate at , and possibly at all redshifts probed. Combined with direct measurements of the galaxy UV luminosity function, this requires only a small (emissivity weighted) fraction of galactic hydrogen ionizing photons to escape their source for galaxies to solely account for the entire ionizing background. Under the assumption that the galactic UV emissivity traces the star formation rate (which is the case if the escape fraction and dust obscuration are constant with redshift), current state-of-the-art observational estimates of the star formation rate density, which peak similarly to quasars at , appear to underestimate the total photoionization rate at by a factor , are in tension with the most recent determinations of the UV luminosity function, and fail to reionize the Universe by if extrapolated to arbitrarily high redshift. A star formation history peaking earlier, as in the theoretical model of Hernquist & Springel, fits the Ly forest photoionization rate well, reionizes the Universe in time, and is in better agreement with the rate of gamma-ray bursts observed by Swift. Quasars suffice to doubly ionize helium by , provided that most of the HeII ionizing photons they produce escape into the intergalactic medium, and likely contribute a non-negligible and perhaps dominant fraction of the hydrogen ionizing background at their peak.

Cosmology: theory, diffuse radiation — methods: data analysis — galaxies: formation, evolution — quasars: general, absorption lines
slugcomment: Submitted to ApJ and revised following the referee’s report

## 1. Introduction

The opacity of the Lyman- (Ly) forest is set by a competition between hydrogen photoionizations and recombinations (Gunn & Peterson, 1965) and can thus serve as a probe of the photonization rate (e.g. Rauch et al., 1997). The hydrogen photoionization rate  is a particularly valuable quantity as it is an integral over all sources of ultraviolet (UV) radiation in the Universe,

 Γ(z)=4π∫∞νHIdνhνJν(z)σ(ν), (1)

where is the angle-averaged specific intensity of the background, is the photoionization cross section of hydrogen, and the integral is from the Lyman limit to infinity. As such, it bears a signature of cosmic stellar and quasistellar activity that is not subject to the completeness issues to which direct source counts (e.g., Steidel et al., 1999; Bouwens et al., 2004; Sawicki & Thompson, 2006; Bouwens et al., 2007; Yoshida et al., 2006; Hopkins et al., 2007) are prone. In addition, unlike the radiation backgrounds directly observed on Earth, the Ly forest is a local probe of the high-redshift UV background, as only sources at approximately the same redshift contribute to  at any point in the forest (Haardt & Madau, 1996).

The hydrogen photoionization rate is not only a powerful observational tracer of cosmic radiative history, but is also a key ingredient in cosmological simulations and theoretical models of galaxy formation. Photoionization heating can, for example, suppress the formation of dwarf galaxies (Efstathiou, 1992; Thoul & Weinberg, 1996; Weinberg et al., 1997a). While the photoionization rate of hydrogen by itself provides limited information on the shape of the background spectrum, it can be combined with the photoionization rates of other species (helium, for example) to provide a direct measure of the background hardness (e.g., Croft et al., 1997; Sokasian et al., 2003a; Bolton et al., 2006). The background spectrum sets the temperature of the gas and so it must be specified in any hydrodynamical simulation of structure formation (e.g., Hernquist et al., 1996; Davé et al., 1999; Springel & Hernquist, 2003b). Knowledge of the ionizing background spectrum is also necessary to infer the mass density of metals from observations of absorption by particular ions (“ionization corrections”) (e.g., Schaye et al., 2003; Aguirre et al., 2004, 2007).

Recent results make it especially worthwhile to reconsider the implications for the ionizing sources, usually assumed to be a combination of quasars and star-forming galaxies (e.g., Madau et al., 1999), of the Ly forest opacity in the light of new data. The comoving star formation rate (SFR) density as estimated from the observed luminosity of galaxies, for instance, is in tension with the independently derived evolution of the stellar mass density (e.g., Hopkins & Beacom, 2006). This suggests problems with observational estimates of the cosmic SFR, to which  is sensitive. The abundance of intergalactic metals as traced by CIV appears consistent with being constant from to perhaps (Songaila, 2001; Pettini et al., 2003; Ryan-Weber et al., 2006), suggestive of an early period of nucleosynthetic activity that may be at odds with estimates of the SFR density that decline above (Hopkins, 2004; Hopkins & Beacom, 2006). Stellar mass density measurements at (e.g., Labbé et al., 2006; Yan et al., 2006; Eyles et al., 2007; Stark et al., 2007a) also suggest significant early star formation. In addition, deep searches utilizing magnification by foreground galaxy clusters (Richard et al., 2006; Stark et al., 2007b) have yielded surprisingly large abundances of low-luminosity galaxies, in tension with blank field surveys and perhaps even requiring an upturn in their abundance with increasing redshift. At lower redshifts, Rauch et al. (2007) discovered a large population of faint Ly emitters at in an extremely deep 92-hour exposure with VLT/FORS2, most of which would not have been selected in existing Lyman-break surveys. This underscores the definite possibility that photometric-selection surveys may presently miss a substantial fraction of the cosmic luminosity density originating in low-luminosity objects.

Here, we use improved measurements of the intergalactic opacity at to obtain a measurement of the hydrogen photoionization rate . We present a detailed comparison with direct source counts of quasars and star-forming galaxies and ask whether these can account for the measured . Finally, we consider the implications for the estimates of the cosmic SFR density and the escape fraction of ionizing photons from galaxies, which relates the soft UV emission to the photons that actually contribute to photoionizations.

The observational constraints we use are described in §2. In §3, we derive the hydrogen photoionization rate implied by our recent measurement of the Ly forest opacity (Faucher-Giguère et al., 2008c) and explore potential systematic effects. In §4, we describe how to convert the photoionization rate to a UV luminosity density. We investigate the implications for the cosmic sources of UV radiation, quasars and star-forming galaxies in §5. We also consider the constraints imposed by the reionizations of hydrogen and helium in this section. We revisit the star formation history in §6 before discussing our results in §7 and summarizing our conclusions §8.

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 . Some of the results presented here were already reported in a concise companion Letter (Faucher-Giguère et al., 2008a). They are described here with the full details of the analysis and underlying assumptions, as well as supporting arguments.

## 2. Observational Constraints

The main observational constraint we use is our measurement of the Ly forest effective optical depth between and (Faucher-Giguère et al., 2008c). This measurement, based on 86 quasar spectra obtained with the HIRES and ESI spectrographs on Keck and with MIKE on Magellan, is the most precise to date from high-resolution, high-signal-to-noise data in this redshift interval. We use our measurement corrected for continuum bias binned in intervals with the contribution of metals to absorption in the forest subtracted following the results of Schaye et al. (2003). For convenience, these data points are reproduced in Table 1. In Faucher-Giguère et al. (2008c), we provided estimates of the systematic uncertainty in arising from the continuum and metal corrections. In this paper, we are however most concerned with the derived photoionization rate. As the systematic uncertainties in converting to (involving the thermal history of the IGM) are larger than those from the continuum and metal corrections, we simply use the statistical error bars on and separately explore the systematic effects influencing our inference of in this work.

We further employ the constraints set by the reionizations of hydrogen and helium. Specifically, we use the fact that hydrogen is reionized by , as indicated by the absence of a complete Gunn & Peterson (1965) trough in the spectra of quasars at this redshift (see, e.g. Fan et al., 2006, for a review of the present observational constraints on reionization). We assume that helium is doubly ionized by , as supported by UV spectra of the HeII Ly forest of quasars. (Jakobsen et al., 1994; Reimers et al., 1997; Hogan et al., 1997; Heap et al., 2000; Smette et al., 2002; Reimers et al., 2005; Fechner et al., 2006).

Finally, we use estimates of the spectrum of the softness of the UV background (Bolton et al., 2006) from measurements of the HI to HeII column density ratio obtained by combining HI and HeII Ly forest spectra (Zheng et al., 2004; Shull et al., 2004).

## 3. THE PHOTOIONIZATION RATE FROM THE Lyα Forest

We begin by deriving the hydrogen photoionization rate implied our measurement of the Ly forest opacity. The method we employ is based on the mean level of absorption in the forest and is termed the “flux decrement” method (Rauch et al., 1997). For the reader who is not concerned with the details, we immediately present the results in Figure 1, with numerical values tabulated in Table 1. The figure compares our measurement to previous estimates based on the flux decrement method that overlap in redshift (Rauch et al., 1997; McDonald & Miralda-Escudé, 2001; Meiksin & White, 2004; Tytler et al., 2004; Bolton et al., 2005; Kirkman et al., 2005). We refer to the appendix of Faucher-Giguère et al. (2008b) for a detailed numerical summary of the previous measurements and to §7.6 for a discussion.

### 3.1. Lyα Effective Optical Depth

The Ly effective optical depth is defined as

 τeff(z)≡−ln[⟨F⟩(z)], (2)

where is the mean transmission of the Ly forest at redshift . Neglecting redshift-space distortions,

 F=exp(−τ), (3)

where

 τ=πe2fLyαmeνLyα1H(z)R(T)nHIIneΓ (4)

is the local Gunn & Peterson (1965) optical depth. Here, and are the charge and mass of the electron, and are the Ly oscillator strength and frequency, is the Hubble parameter, is the temperature-dependent hydrogen recombination rate, and are the proper number densities of HII and electrons, and is the hydrogen photoionization rate.

We consider a gas made purely of hydrogen and helium, with mass fractions and , respectively. To describe its ionization state, we define . Then

 nHII=ρcritΩbmpXx(1+δ)(1+z)3 (5)

and

 ne=ρcritΩbmp[Xx+0.25Y(yII+2yIII)](1+δ)(1+z)3, (6)

where is the local overdensity, and we have used the fact that a helium atom weighs approximately . To a good approximation,

 R(T)≈R0T−0.7, (7)

with (e.g., Hui & Gnedin, 1997), and the Hubble parameter is given by the Friedmann equation,

 H(z)=H0√Ωm(1+z)3+ΩΛ. (8)

Assuming that the intergalactic gas follows a power-law temperature-density relation of the form

 T=T0(1+z)β (9)

(Hui & Gnedin, 1997), equation (4) implies that

 τ=A(z)(1+δ)2−0.7β, (10)

with

 A(z) ≡πe2fLyαmeνLyα(ρcritΩbmp)21H(z) Xx[Xx+0.25Y(yII+2yIII)]R0T−0.70Γ(1+z)6. (11)

Note that equation (9) is often expressed in term of the parameter in the literature. An isothermal temperature-density relation corresponds to or, equivalently, .

Given a probability density function (PDF) for the gas density ,

 ⟨F⟩(z)=∫∞0dΔP(Δ;z)exp(−τ). (12)

Note that neglecting redshift-space distortions from thermal broadening and peculiar velocities is well-motivated here since is to first order independent of the smoothing of the field induced by these distortions.111The mean of is not strictly independent of redshift space distortions, since these really smooth the field, of which is a nonlinear function. This however amounts to a small effect. For any specified , a measurement of thus determines the evolution of , which itself fixes the ratio if the cosmological parameters and ionization state of the gas are known (eq. 3.1).

### 3.2. Gas Density PDF

Miralda-Escudé et al. (2000) derived an approximate analytical functional form for the volume-weighted gas density PDF,

 PV(Δ)=Aexp[−(Δ−2/3−C0)22(2δ0/3)2Δ−b]. (13)

This formula is motivated by the facts that the gas in the low-density voids, where gravity is negligible, should expand at nearly constant velocity, whereas the distribution in high-density collapsed structures should be determined by their power-law profile, .222Note that Miralda-Escudé et al. (2000) use the notation for . We have altered their notation in order to avoid confusion with the temperature-relation slope parameter defined in this work (eq. 9).

The parameters and are fixed by requiring the total volume and mass probabilities to be normalized to unity. Fits to a numerical simulation at 2, 3, and 4 yields the parameters given in Table 2, with ; at , , corresponding to isothermal halos, is assumed.

Rauch et al. (1997) compared the density distributions from simulations using very different numerical methods (Eulerian versus Lagrangian smoothed particle hydrodynamics) and cosmologies (CDM vs. Einstein-de Sitter). They found excellent agreement in spite of the different simulation parameters. Miralda-Escudé et al. (2000) attributed this similarity to the fact that the density distribution should depend mostly on the amplitude of the density fluctuations at the Jeans scale, both simulations having approximately the same.

Our use of the Miralda-Escudé et al. (2000) PDF in deriving is certainly an imperfect approximation. In general, the details of the gas density PDF will depend on the exact cosmological model (in particular through the power spectrum of density fluctuations), the thermal state of the gas, as well as on the numerical properties of the simulation. Bolton et al. (2005), in particular, studied the dependences of the inferred photoionization rate on these parameters, demonstrating that they have a significant effect and quantifying their importance. Possibly of equal importance, however, is the entire thermal history of the Universe. In fact, the gas spatial distribution (through Jeans-like smoothing) depends not only on the present temperature of the gas, but also on its past state as it takes times for the gas to respond to changes (e.g., Gnedin & Hui, 1998). In addition, fluctuations in the magnitude of the UV background and associated heating rates (e.g., Meiksin & White, 2004), the inhomogeneous reionization of both hydrogen and helium (e.g., Hui & Haiman, 2003; Lai et al., 2006; Furlanetto & Oh, 2007a, b), and secondary feedback effects from galaxy formation and evolution (e.g., Kollmeier et al., 2006) may also affect the detailed properties of the intergalactic gas. A full investigation of these effects is beyond the scope of the present paper.

In what follows, we instead take a simple approach based on the Miralda-Escudé et al. (2000) PDF in deriving the hydrogen photoionization rate, focusing on its implications for the ionizing sources. To estimate the effect of this assumed PDF, we have repeated our analysis with the gas density PDF from the Q5 hydrodynamical simulation of Springel & Hernquist (2003b). This simulation of a CDM universe with has a box side length of 10 Mpc, with dark matter particles and as many baryonic ones. This PDF yields a at higher than the one inferred using the Miralda-Escudé et al. (2000) by , all other parameters fixed. While this is a non-negligible effect, it is comparable to the statistical error we quote (Table 1) and it is relatively small in comparison with the total plausible systematic error, the magnitude of which can be estimated from the scatter between the existing measurements (Figure 1). We caution that more work is clearly needed to refine the accuracy of the measured , but note that the conclusions that we develop here are broadly robust to an order unity uncertainty in its normalization that may reasonably be expected. On the other hand, our results are consistently derived and finely binned over a large redshift range, clearly constraining the constancy of barring any strongly redshift-dependent systematic effect.

### 3.3. Thermal History of the IGM

Although the thermal history of the IGM may only have a modest effect on the gas density PDF, the parameters and must be known in order to unambiguously solve for the evolution of the background hydrogen photoionization rate, . In this paper, we do not attempt to infer and directly from the data; rather, we use existing measurements, which we combine with our theoretical understanding of the thermal physics of the IGM in order to assess the robustness of our conclusions. We begin by reviewing this physics before examining the present observational constraints.

#### 3.3.1 Theory of the Thermal History of the IGM

A detailed formalism for the thermal history of the IGM can be found in Hui & Gnedin (1997), or more concisely in the appendix of Hui & Haiman (2003). As we will ultimately base our analysis on empirical measurements, we simply summarize the most important physical processes here.

For given initial conditions, the temperature of a given fluid element is uniquely determined by a competition between cooling and heating processes. In the low-density () IGM of the Universe, photoheating and adiabatic cooling typically drive the thermal evolution. Photoheating proceeds in two main regimes. During reionization events, the ionization fraction of a particular atomic species (with H and He being most important by a large factor) changes by a factor of order unity, inducing a large boost in the IGM temperature. In an already ionized plasma, photoheating occurs through the ionization of small residual fractions of HI, HeI, or HeII. In equilibrium, this process is balanced by recombinations. Gas elements further heat or cool simply because of adiabatic contraction or expansion; the overall expansion of the Universe, for example, induces a temperature fall-off .

Ultimately, the IGM temperature loses memory of reionization events, reaching a “thermal asymptote” determined by the balance between heating and cooling processes in the ionized plasma. The thermal asymptote depends only on the shape of the background spectrum. For a power-law ionizing background it is given by

 T0= 2.49×104 K(0.417+0.047yII+0.583yIII) (14) ×(2+α)−1/1.7(1+z4.9)0.53

(Hui & Gnedin, 1997).

What are relevant initial conditions? Hydrogen reionization can heat the IGM by a few times K (Miralda-Escudé & Rees, 1994; Abel & Haehnelt, 1999; Tittley & Meiksin, 2007), with negligible pre-reionization temperatures K expected from heating by X-rays (Oh, 2001; Venkatesan et al., 2001; Kuhlen et al., 2006; Furlanetto, 2006). While we know for certain that the Universe is reionized by from quasar spectra (e.g., Fan et al., 2006), the Thomson scattering optical depth measured by WMAP (Spergel et al., 2007) indicates that it is likely to have proceeded at . Similar entropy injection may occur at lower redshifts () during the reionization of HeII (e.g., Furlanetto & Oh, 2007b).

The parameters of the temperature-density relation (eq. 9) can in principle be solved for by calculating the evolution of the temperature of fluid elements as a function of density. Uncertainties in the redshifts of reionization events and their character, and on the evolution of the ionizing background spectrum however preclude a reliable ab initio calculation of the temperature-density relation. We therefore turn to empirical estimates.

#### 3.3.2 Thermal History Measurements

The thermal evolution of the IGM can be probed using the Ly forest itself. Indeed, the Jeans-like smoothing of the gas depends on its pressure, which in turn is set by its temperature. In addition, Ly forest spectral features are thermally broadened. Both of these effects increase the characteristic width of Ly absorption lines.

In general, Hubble broadening (arising from the differential Hubble flow across individual absorbers) and peculiar velocities also contribute to the line widths (Meiksin, 1994; Weinberg et al., 1997b; Hui & Rutledge, 1999; Theuns et al., 2000). On “velocity caustics,” where the peculiar velocities of the gas particles cancel the Hubble gradient, however, the line width should be uniquely determined by the gas temperature. This results in a cut-off in the distribution of line width versus column density that traces the temperature-density relation of the gas (Schaye et al., 1999, 2000; Ricotti et al., 2000; McDonald et al., 2001).

An alternative, but physically related, method to estimate the IGM temperature from the Ly forest is to consider its transmission small-scale power spectrum. In this case, the finite temperature of the gas suppresses the small-scale power (e.g., Theuns et al., 2000; Theuns & Zaroubi, 2000; Zaldarriaga et al., 2001; Theuns et al., 2002; Zaldarriaga, 2002). Zaldarriaga et al. (2001) used this dependence of the small-scale power to jointly estimate and from the power spectrum measurement from eight Keck/HIRES spectra of McDonald et al. (2000). The constraints on obtained after marginalizing over are K at , K at , and K at .333These values are given at in Hui & Haiman (2003).

We adopt the Zaldarriaga et al. (2001) values, consistent with those obtained by McDonald et al. (2001) from the same data set but with an independent line-fitting method, in our analysis. In the range , we linearly interpolate between their measurements; for , we linearly extrapolate based on the and values. In our fiducial model, we assume that the IGM has been cooling adiabatically, , for . For the slope of the temperature-density relation, we take the early reionization limit .

The errors in our inferred (Figure 1 and Table 1) include both the uncertainty in our measurement and on .

### 3.4. Ionization State of the IGM

We adopt the mass fractions and for hydrogen and helium, respectively (Burles et al., 2001). At the redshifts of interest, quasar spectra indicate that the Universe is fully ionized, , to better than one part in a thousand. While helium is likely also almost fully ionized at (, ), as indicated by observations of the HeII Ly forest, little is known about its ionization state at higher redshifts. Owing to its similar ionization potential, helium is expected to be singly ionized simultaneously to hydrogen. A number of observational lines of evidence (for a review, see Faucher-Giguère et al., 2008c) as well as observationally calibrated theoretical models (Sokasian et al., 2002; Wyithe & Loeb, 2003; Furlanetto & Oh, 2007b) suggest that HeII is reionized around . In our inference of , we assume that helium is fully ionized at all epochs probed; may be overestimated by up to 8% at the highest redshifts if our measurement reaches pre-HeII reionization times, owing to an overestimate of the number of free electrons (eq. 6).

### 3.5. Systematic Effects from the Thermal History

We conclude this section by exploring systematic effects that may arise if our assumptions regarding the thermal history of the IGM are in error with respect to our derived photoionization rate.

#### 3.5.1 Temperature at Mean Density

For fixed cosmology, ionization state of the IGM, and temperature-density relation slope , constrains the quantity 3.1). An error in the temperature of the IGM at mean density, , will therefore directly result in a corresponding error in the inferred . Although our error bars on include a contribution from the statistical uncertainty in the measured , it is conceivable that these measurements suffer from systematic effects. Figure 2 shows exactly how the inferred depends on the assumed at . We focus on this particular redshift as we will be interested in the robustness of a high at the highest redshifts probed when we consider the sources of the UV background (§5.3).

Note that the slight upward bump444The statistical significance of this feature is greater in the evolution than in the derived evolution of , as the statistical errors in the latter include contribution from the error in which is correlated between bins. For an analysis of this feature in , see Faucher-Giguère et al. (2008c). near in the derived (Fig. 1) may be an artifact of our smooth interpolation of the temperature between the sparse measurements at 2, 3, and 4. This procedure directly translates a downward bump in into an upward bump in . An alternative possibility is that the evolution of is featureless and that instead undergoes a rapid increase, for example owing to the reionization of HeII (theoretical work however suggests that the time scale for heating the IGM during HeII reionization is too long to explain such a narrow feature; Bolton et al., 2008; McQuinn et al., 2008). For a discussion of these degenerate effects, see Faucher-Giguère et al. (2008c). The detection of a bump in is more robust in that it is not subject to this degeneracy.

#### 3.5.2 Slope of the Temperature-Density Relation

From line-fitting analyses, Ricotti et al. (2000) and Schaye et al. (2000) have found some (albeit marginal) evidence that the temperature-density relation becomes isothermal near , perhaps owing to reheating from the reionization of HeII near this redshift. Comparing an improved measurement of the Ly forest flux PDF at by Kim et al. (2007) to a large set of hydrodynamical simulations with different cosmological parameters and thermal histories, Bolton et al. (2007) find evidence for an equation that is close to isothermal, or even inverted (). From a data set extending to , Becker et al. (2007) also find suggestive evidence that an inverted temperature-density relation with fits the observed Ly flux PDF best, if the underlying gas density PDF follows that of Miralda-Escudé et al. (2000). This suggests that the voids in the IGM may be significantly hotter and the thermal state of the low-density IGM more complex than previously assumed. These claims have not been corroborated by the results of McDonald et al. (2001) and Zaldarriaga et al. (2001), although both these analyses are consistent with isothermality at .

While the present observational evidence is scarce and difficult to interpret, some theoretical modeling of inhomogeneous HeII reionization also suggests a complex, possibly inverted temperature-density relation (Bolton et al., 2004; Gleser et al., 2005; Tittley & Meiksin, 2007; Furlanetto & Oh, 2007a). Detailed radiative transfer simulations of HeII reionization however find that the slope of the temperature-density relation is only moderately affected and that in actuality the correct slope is somewhere between the early hydrogen reionization limit () and the isothermal case () (McQuinn et al., 2008). These simulations also produce a large scatter in the temperature-density relation, suggesting that a power-law approximation is rather rough during HeII reionization.

How does the uncertainty in the slope of the temperature-density relation of the low-density IGM affect our inference of ? In Figure 3, we repeat our calculation of as determined from our effective optical depth measurement varying . We consider, in addition to the early hydrogen reionization limit, isothermal and inverted () temperature-density relations. The latter is consistent with the best-fit parameters found by Bolton et al. (2007) and Becker et al. (2007), but likely too extreme to be physically realistic based on the calculations of McQuinn et al. (2008).

As the temperature-density relation becomes flatter or inverted, the inferred at decreases, indicating that the Ly forest becomes more sensitive to underdense regions with . This is increasingly the case at higher redshifts. Near , the inferred is nearly independent of the slope of the temperature-density relation (indicating that the measurement is most sensitive to ), while at the early-reionization limit gives a that is about twice that of the inverted case.

Note that the systematic uncertainties in the temperature of the IGM at the mean density and on the slope of the temperature-density relation , especially if they conspire to affect the inferred in the same direction, could go some way in resulting in a declining photoionization rate toward high redshifts and therefore bringing it in better agreement with observationally derived star formation rates (see §5.3). However, the requirement that the Universe be reionized by also argues forcefully against a steep decline of star formation beyond 5.5). The latter argument, depending only on the magnitude of at through the normalization of the ionizing emissivity, is largely insensitive to the uncertainties in the thermal history at the redshifts probed by our Ly forest opacity measurement.

## 4. From Photoionization to Emissivity

Although is clearly related to the cosmic sources of ionizing photons, the sources are most closely related to their specific emissivity, . In this section, we compute the UV emissivity implied by our inferred photoionization rate, before proceeding to investigate its implications for quasars and star-forming galaxies in the next. Our formalism is based on Haardt & Madau (1996), Madau et al. (1999), and Schirber & Bullock (2003).

This specific intensity of the UV background obeys the cosmological radiative transfer equation,

 (∂∂t−νH∂∂ν)Jν=−3HJν−cανJν+c4πϵν, (15)

where is the Hubble parameter, is the speed of light, is the proper absorption coefficient, and is the proper emissivity. Integrating equation (15) 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)], (16)

where , the proper line element , and quantifies the attenuation of photons of frequency at redshift that were emitted at redshift . For Poisson-distributed absorbers, each of column density ,

 ¯τ(ν0,z0,z)=∫zz0dz′∫∞0dNHI∂2N∂NHI∂z′(1−e−τν), (17)

where is the column density distribution versus redshift and (Paresce et al., 1980). We neglect the continuum opacity owing to helium and heavier elements, which is a good approximation near the hydrogen Lyman edge. For the redshifts of interest here, the mean free path of UV ionizing photons is so short that the sources contributing to at a given point are largely local. In this local-source approximation (e.g., Schirber & Bullock, 2003), the solution to equation (16) reduces to

 Jν(z)≈14πΔl(ν, z)ϵν(z). (18)

We may thus trivially solve for the emissivity,

 ϵν(z)≈4πJν(z)Δl(ν, z). (19)

### 4.2. Mean Free Path

As the mean free path of ionizing photons is the key quantity relating to its sources, we pause to derive an improved value in the light of new data and estimate its uncertainty. We assume that the column density distribution of absorbers is well-approximated by power laws of the form

 ∂2N∂NHI∂z=N0N−βHI(1+z)γ, (20)

in which case the general solution to equation (17) is

 ¯τ(ν0, z0, z)=Γ(2−β)(β−1)(γ−3β+4) ×N0(νHIν0)3(β−1)σβ−1HI(1+z0)3(β−1) ×[(1+z)γ−3β+4−(1+z0)γ−3β+4]. (21)

Here, is the gamma-function and is not related to the photoionization rate. This reduces to the expression given by Madau et al. (1999) in the special case . The mean free path is obtained by differentiating, setting , and using the line element to convert to proper length:

 Δl(ν0, z)≈(β−1)cΓ(2−β)N0σβ−1HI(ν0νHI)3(β−1) ×1(1+z)γ+1H(z). (22)

For steep column density distributions with , as suggested by observations, most of the contribution to at the Lyman limit arises for systems of optical depth near unity ( cm). We thus focus on the values of the power-law indexes and that are most appropriate in this neighborhood. Stengler-Larrea et al. (1995) find that with and for systems with cm, whereas the column density distribution is well-fitted by (Misawa et al., 2007). The constant in equation (20) is related to by , where cm is the minimum column density accounted for in .

Figure 4 shows the corresponding mean free path at the Lyman limit, proper Mpc, and its propagated uncertainty. The smaller errors take into account only the uncertainty in the redshift evolution of the mean free path (). Also shown is the commonly used result of Madau et al. (1999), proper Mpc, which is seen to underestimate our value by a factor , albeit within the uncertainty. The redshift evolution differs by a power of 0.5 as we have adopted the power-law index from Stengler-Larrea et al. (1995) for the redshift evolution of Lyman limit systems, consistent with the results of Storrie-Lombardi et al. (1994).

Madau et al. (1999) motivate their steeper choice of from measurements of the redshift evolution of optically thin systems of lower column density (Press & Rybicki, 1993; Bechtold, 1994; Kim et al., 1997). However, if, for example, Lyman limit systems are associated with the outskirts of galaxies (e.g., Tytler, 1982; Bechtold et al., 1984; Katz et al., 1996b; Lanzetta, 1988; Sargent et al., 1989; Steidel, 1990; Misawa et al., 2004; Kohler & Gnedin, 2007), then their evolution may not follow that of absorbers in the diffuse IGM. We therefore prefer to use the column density distribution measurements directly probing Lyman limit systems. Although it is true that lower column density systems contribute non-negligibly to the Lyman continuum opacity, we note that instead choosing would lead to a steeper decrease of the mean free path with increasing redshift. This in turn would result in a larger requirement on the ionizing emissivity in order to produce the observed photoionization rate, exacerbating the tension we find in §5.3 and §6.3 with some estimates of the star formation rate density at high redshifts. In this sense, our choice is conservative and our conclusions would only be strengthened if we had significantly underestimated the redshift evolution of the absorption systems determining the mean free path.

We finally note that even for identical choices of (), our estimate of the mean free path would exceed that of Madau et al. (1999) by about 30% owing to their assumption of an Einstein-de Sitter cosmology, in contrast to the WMAP cosmology we adopt.

### 4.3. Emissivity

We finally turn to the calculation of the emissivity at the Lyman limit implied by our measurement of . Assuming that the ionizing background has a power-law spectrum blueward of the Lyman limit,

 JνHI=Γh(αHI+3)4πσHI, (23)

from which

 ϵνHI(z)≈Γ(z)h(αHI+3)σHIΔl(νHI, z), (24)

or, equivalently,

 Γ(z)≈σHIΔl(νHI, z)ϵνHI(z)h(αHI+3). (25)

We are also interested in the rest-frame UV emissivity probed by optical quasar and galaxy surveys. Following the galaxy survey literature, we define the UV emissivity to correspond to the wavelength 1500 Å, and denote the corresponding frequency by . If is the spectral index between the Lyman limit and 1500 Å, then

 ϵUV=1fesc(1500 \AA912 \AA)αUVϵνHI. (26)

The escape fraction accounts for the fact that, owing to Lyman continuum opacity associated with the host halo, only a certain fraction of the photons blueward of the Lyman limit escape from their source. The correct value depends on the nature of the sources, to which we now turn.

## 5. The Cosmic Sources of Uv Photons

The two known dominant sources of UV photons in the Universe are quasars and star-forming galaxies. We investigate the constraints that can be put on their populations using our measurement of the IGM Ly opacity.

Our starting assumption is that the luminosity function of quasars and their spectral energy distribution (SED) are better constrained than those of galaxies. In addition, we assume that all the photons capable of ionizing hydrogen emitted by quasars escape their host unimpeded, i.e. that quasars have an escape fraction of unity. This is often assumed to be the case on the theoretical basis that quasars are sufficiently powerful to fully ionize their surrounding gas (e.g., Wood & Loeb, 2000). Observational evidence that this is the case is provided by the proximity effect, which is clearly detected at roughly the level predicted if the SED of quasars does not suffer a break at the Lyman limit from intrinsic absorption (e.g., Scott et al., 2000; Goncalves et al., 2007; Dall’Aglio et al., 2008). Moreover, quasar spectra generally do not show DLAs associated with the quasar hosts. In contrast, DLAs of high column density are seen in most spectra of long-duration gamma-ray bursts occurring in star-forming galaxies and their Lyman-continuum opacity is understood to be responsible for the small escape fraction of inactive galaxies (Chen et al., 2007; Gnedin et al., 2007).

### 5.1. Quasar Emissivity

Hopkins et al. (2007) have used a large set of observed quasar luminosity functions in the infrared, optical, soft and hard X-rays, as well as emission line measurements, combined with recent estimates of the quasar column density distribution from hard X-ray and IR observations and measurements of their spectral shape from the radio through hard X-rays to estimate their bolometric luminosity function. Denoting by the emissivity at 4400 Å and assuming ,

 ϵcomB=∫∞0dLBdϕdLBLBν|4400 \AA, (27)

where is the -band luminosity function derived by Hopkins et al. (2007) from their full spectral and obscuration modeling in comoving units. To convert to an emissivity at the Lyman limit, we assume that quasars have a spectral index at 2500-4400 Å, 0.8 at 1050-2500 Å (Madau et al., 1999), and 1.6 from 1050 Å to 228 Å (Telfer et al., 2002). The corresponding photoionization rate is then obtained using equation (24).

Although equation 27 appears to assume isotropic quasar emissivity, it is more generally valid. For instance, if quasars emit with constant intensity in some directions and are completely obscured toward the others, then the overestimate of the total photon output of a quasar based on its observed magnitude and the assumption of isotropy will be exactly canceled by the fraction of quasars missed in the luminosity function. To generalize further, note that in a context in which quasars shine with different intensities in different directions, the observed quasar luminosity function can be interpreted as quantifying the number of (randomly oriented) lines of sights of given intensity from quasars. For the purpose of calculating the cosmological emissivity, one can imagine distributing these sight lines among isotropically emitting sources and recover equation 27.

Some biases could however be introduced in the calculation if the angular distribution of the quasar intensity has a more complex smoothly varying profile or depends on frequency (see §5.3).

We take a background spectral index shortward of the Lyman limit , appropriate if the spectrum is hardened by IGM filtering. This hardening arises from the frequency dependence of the mean free path of ionizing photons (eq. 4.2) and the relation between the UV background spectrum and the emissivity producing it (eq. 19): . Taking the approximate value for the column density distribution of absorbers, the spectrum is hardened by .555Note that this differs from the maximum hardening that would result from a homogeneous distribution of neutral hydrogen because of the photoionization cross-section dependence. The difference arises from the distribution of absorbers into discrete clumps and depends on the slope of the column density distribution. See, for example, the argument in Zuo & Phinney (1993).

### 5.2. Stellar Emissivity

Because only massive short-lived stars produce UV photons in significant amounts, the stellar emissivity is intimately connected to the cosmic SFR. To calculate the stellar contribution to the emissivity of ionizing photons, we consider both theoretical (Springel & Hernquist, 2003b; Hernquist & Springel, 2003) and observational estimates (Hopkins, 2004; Hopkins & Beacom, 2006, and references therein) of the cosmic star formation history, which we describe in §5.2.1.

The exact calculation of the emergent emissivity of ionizing photons from galaxies involves detailed assumptions about the initial mass function (IMF) of stars, stellar astrophysics, and the escape fraction of ionizing radiation, all of which are subject to substantial uncertainty. Under the assumption that these are constant across cosmic time, however, the emissivity is simply proportional to the SFR density,

 ϵν=K˙ρ⋆(1+z)3, (28)

where the factor of arises from the conversion from comoving SFR density to proper emissivity in our notation, and the constant depends on the uncertain astrophysics. The usefulness of this description arises from the fact that significant insight can be gained from the redshift evolution of the emissivity only. If the proper mean free path evolves as (appropriate if the line-of-sight number density of Lyman limit systems evolves as ; §4.2), then we can use the local-source approximation in equation (18) to relate the evolution of the comoving SFR density to that of the hydrogen photoionization rate:

 Γ=K′˙ρ⋆(1+z)−1, (29)

for some other constant .

#### 5.2.1 Star Formation Rate Density

Employing hydrodynamic simulations of structure formation in a CDM cosmology, Springel & Hernquist (2003b) calculated the comoving SFR density , taking into account radiative heating and cooling of the gas, star formation, supernova feedback, and galactic winds based on the hybrid star-formation multiphase model detailed in Springel & Hernquist (2003a). They obtained a SFR density gradually rising by a factor of ten from and peaking at . Using simple analytic reasoning, Hernquist & Springel (2003) identified the processes that drive the evolution of the cosmic SFR in CDM universes, and derived an analytic model matching their simulation results to better than %.

They found that the cosmic SFR is described by two regimes. At early times, densities are sufficiently high and cooling times sufficiently short that abundant quantities of star-forming gas are present in all dark matter halos that can cool by atomic processes. In this gravitationally-dominated regime, generically rises exponentially as decreases, independent of the details of the physical model for star formation, but dependent on the normalization and shape of the cosmological power spectrum. At low redshifts, densities decline as the Universe expands to the point that cooling is inhibited, limiting the amount of star-forming gas available. In this regime, the SFR scales in proportion to the cooling rate within halos, .

These two regimes lead to a redshift of peak star formation which in general depends on the efficiency of star formation. For a star formation efficiency normalized to the empirical Kennicutt law (Kennicutt, 1989, 1998b) of disk galaxies at , the comoving SFR density in their simulations is well described by the analytic form

 ˙ρ⋆(z)=˙ρ⋆(0)χ21+α(χ−1)3exp(βχ7/4), (30)

where

 χ≡(H(z)H0)2/3 (31)

and , , and M yr Mpc are fitting parameters.

This fit was derived for their CDM cosmology with , , , and . For consistency, in this paper we use a modification of this fit adapted to the cosmology that we assume. Specifically, we use equation (45) of Hernquist & Springel (2003) to evaluate their analytic model for the cosmology. The resulting star formation history differs from their fiducial CDM model through the parameter governing the amplitudes of density fluctuations. Specifically, star formation is slightly delayed for with respect to the higher value .

Extending prior work by Hopkins (2004), Hopkins & Beacom (2006) examined the observational constraints on the cosmic star formation history based on a database of measurements in all relevant wavebands. To the luminosity density measurements, dust corrections (where necessary), SFR calibrations, and IMF assumptions are applied to derive the star formation history. These authors chose to use the parametric form of Cole et al. (2001) for their fits to the star formation history:

 ˙ρ⋆(z)=(a+bz)h1+(z/c)d, (32)

where . For a “modified Salpeter A” IMF (Baldry & Glazebrook, 2003), they find the best-fit parameters . Other IMFs, such as the one advocated by Baldry & Glazebrook (2003), affect this fit only in its overall normalization, provided they are constant with cosmic time. The IMF normalization will be irrelevant for the following arguments and we therefore only consider the modified Salpeter A case.

Although we will focus on the widely used Hopkins & Beacom (2006) fit for convenience in our study, we note that it is representative of many other current state-of-the-art observational estimates of the SFR density in that it peaks at and subsequently declines (e.g., Reddy et al., 2008). In addition to the smooth fit based on the Cole et al. (2001) parametric form, Hopkins & Beacom (2006) also present a piecewise linear fit of the same data in order to avoid artifacts resulting from the choice of a particular functional form. We have duplicated our analysis with their piecewise linear fit as well and reached practically identical conclusions. To simplify our presentation, we only show our calculations for the smooth fit.

### 5.3. Comparison with the Lyα Forest Photoionization Rate

To compare the photoionization rate measured from the Ly forest to what can be accounted for by the observed quasars and estimated star formation history, we proceed as follows. We begin by fixing the quasar contribution, with the working assumption that it is better constrained than that of stars. For each star formation history we consider, we then solve for the constant in equation (29) that minimizes the between our measured and the sum of the quasar and stellar contributions. We repeat the exercise for different normalizations of the quasar contribution to investigate how our conclusions would be affected if the escape fraction of quasars were less than unity, or if there were an error in the normalization of the mean free path or in the assumed spectral indexes.

The results are shown in Figure 5. The superpositions of the best-fit nearly flat contribution to from stars in the Hernquist & Springel (2003) model and of the quasar contribution peaked near estimated from the Hopkins et al. (2007) luminosity function match the observed total photoionization rate from the Ly forest reasonably well (panel A). However, if the stellar ionizing emissivity instead traces the fit of Hopkins & Beacom (2006) to the SFR density, then the stellar contribution to peaks similarly to that of quasars near . As a result, any superposition of the two contributions also peaks near . As the measured total is nearly flat between and , any such superposition falls short of accounting for it at by a factor while simultaneously overestimating its value by a factor at (panel B), in the best-fit case. Note that if we had assumed an abundance of Lyman-limit systems increasing with redshift as (as in Madau et al., 1999, and appropriate for lower column density systems) instead of , then the discrepancy at would be even larger. Unless the escape fraction of star-forming galaxies or the IMF evolve appropriately with redshift, our measurement of the total photoionization rate from the Ly forest is therefore in conflict with the best-fit SFR density of Hopkins & Beacom (2006). We will discuss potential issues with this SFR estimate in §6.

Note that in Figure 5 we have only shown quasar normalizations from 0.1 to 0.35, as for higher normalizations quasars alone overproduce the measured total photoionization rate, which is of course unphysical. The fact that a normalization strictly less than unity is required may indicate that the photoionization rate derived from the forest is too low either because of an overestimate of the mean free path of ionizing photons or inexact spectral assumptions. A smaller mean free path would yield, for fixed emissivity, a lower photoionization rate (eq. 25). The ionizing emissivity, in turn, is obtained from modeling the quasar spectrum between the -band at  Å and the Lyman edge at 912 Å. The photoionization rate moreover depends on the spectral shape shortward of 912 Å and therefore some level of uncertainty in the calculated value definitely arises from the assumed spectrum.

It could also indicate that the escape fraction of Lyman continuum photons from quasars is in fact less than unity. It is interesting to consider why this may be the case. Although (unlike GRBs occurring in star-forming galaxies) quasar hosts typically do not show DLAs, Lyman limit systems are observed to cluster around quasars, apparently preferentially in the transverse direction (Hennawi et al., 2006; Hennawi & Prochaska, 2007; Prochaska & Hennawi, 2008). It is possible that such systems absorb a significant fraction of the ionizing radiation emitted by the quasars they surround before these penetrate into the diffuse IGM, resulting into an “effective” escape fraction less than unity. A similar effect may also be in action around star-forming galaxies, although Lyman limit systems may cluster more weakly around the latter owing to their lower typical masses. As our argument here is based on the shape of the evolution of the photoionization rate, it is largely robust to the normalization uncertainties.

In any case, the apparent overproduction of the ionizing background by quasars alone highlights the uncertainties in converting from a luminosity function to a photoionization rate. In particular, models that are based on integrating the luminosity functions of the ionizing sources (e.g., Haardt & Madau, 1996) risk being significantly in error in normalization (see §7.5). Calibrating these models to the photoionization rate measured from the Ly forest using the flux decrement method, while itself subject to some systematic effects (§3), should yield more robust results.

We now proceed to consider further constraints on the cosmic ultraviolet luminosity density imposed by the reionization of hydrogen and helium.

### 5.4. Clumping Factor

As recombinations result from two-particle collisions, their average rate is proportional to the clumping factor of the gas. The clumping factor is therefore a key quantity in calculating the ionization history of the Universe. We pause to discuss its numerical value.

Many early calculations of the evolution of the global ionized fraction during reionization (e.g., Madau et al., 1999) assumed clumping factors derived from averaging over the entire gas content of high-resolution simulations (e.g., Gnedin & Ostriker, 1997; Springel & Hernquist, 2003b), which were as large as at the redshifts relevant for hydrogen reionization. Such clumping factors are however unrealistically high, as they include gas in collapsed halos which are associated with the galaxies that produce the ionizing photons. As the escape fraction already accounts for the effects of clumpy gas within these sources, the latter should not be included in calculating the clumping factor used to calculate the recombinations in the diffuse IGM.

Let us identify the size of collapsed structures with their virial radius, with the latter approximated by the radius within which the mean density is 200 times the mean cosmic density (e.g., Barkana & Loeb, 2001). Then if halos are further approximated as isothermal spheres, , their dimensionless overdensity at the virial radius is given by . A crude estimate of the clumping factor excluding gas within the virial radii of halos is then obtained by calculating using the Miralda-Escudé et al. (2000) PDF truncated at . Figure 6 illustrates the result of this calculation at , 4, and 6 for varying . As expected, the clumping factor decreases with increasing redshift for fixed .

For the redshifts relevant for hydrogen reionization, for . If HeII reionization proceeds between at , then slightly higher clumping factors are more appropriate. Of course, these estimates are rough and we will therefore present our calculations for a range of reasonable clumping factors.

### 5.5. Constraints from the Reionization of Hydrogen and Helium

The facts that hydrogen is reionized by and helium is fully ionized by constrain the production of ionizing photons at earlier times. We investigate these constraints here.

The rate of change of the mass-weighted ionized fraction of species can be written as

 dxidt=1ni∫∞νidνhνϵν−CRi(T)nexi, (33)

where is the recombination coefficient from the ionized to the ground state. Although many studies of reionization use the case-B recombination coefficient on the basis that recombinations directly to the ground state produce an ionizing photon and therefore have no net effect on the ionized fraction, most of the recombinations in a clumpy universe occur near dense regions (such as Lyman-limit systems), so that the reemitted photon is likely to be absorbed before it escapes into the diffuse IGM (Miralda-Escudé, 2003; Furlanetto & Oh, 2007b). We therefore use case-A recombination coefficients.

#### 5.5.1 Hydrogen Reionization

We first consider the case of hydrogen reionization. We assume that helium is singly ionized to HeII simultaneously () with hydrogen, with both fully neutral at very high redshifts unaffected by reionization (). We assume that stars are solely responsible for reionizing hydrogen and that the emissivity evolves with redshift following the SFR density estimates (eq. 28). Beyond , the highest redshift at which Zaldarriaga et al. (2001) measured the IGM temperature, we take the temperature to be constant at 22,000 K. We normalize the value of the specific emissivity at the Lyman limit to be that inferred from measured from the Ly forest at using equation (24), comoving erg s Hz cm. The corresponding escape fraction will be estimated by comparing with the 1500 Å luminosity density obtained by integrating over galaxies in §5.6. For this conversion, we have assumed a spectral index shortward of the Lyman limit for a high-redshift ionizing background dominated by star-forming galaxies. The theoretical starburst models of Kewley et al. (2001) produce intrinsic spectra with approximately between and , covering the wavelength interval that contributes most significantly to . This spectrum is then hardened, , by IGM filtering.

The redshift evolution of the specific intensity is assumed to follow the SFR density. Under the assumption that it does not evolve with redshift, this approach is free of assumptions in the escape fraction of galaxies. We investigate a wide range of clumping factors by successively setting and 10 (c.f. §5.4).

The results are shown in Figure 7 for the theoretical model of Hernquist & Springel (2003) and for the observational SFR history of Hopkins & Beacom (2006). Of particular note here is that the SFR density fit of Hopkins & Beacom (2006) (under the assumption we have made that it traces the ionizing emissivity) fails to reionize hydrogen by by a large factor even in the most optimistic case of no recombinations. This further supports the conclusion that the Hopkins & Beacom (2006) SFR density estimates decline too steeply at high redshifts , which we had reached using only the measurement of the Ly opacity at .

If the emissivity more closely follows the SFR density predicted by Hernquist & Springel (2003), then the Universe is reionized by provided that . This is lower than expected at according to our simple estimate in §5.4. At higher redshifts more representative of the middle of reionization, however, the clumping factor is likely smaller. In addition, if low-density gas is reionized first, then an effective clumping factor may result throughout most of reionization, because all the dense gas would remain locked up in self-shielded systems and should not be counted in the term (Miralda-Escudé et al., 2000). Even the Hernquist & Springel (2003) model may be in tension with estimates of the redshift of reionization from the optical depth to Thomson scattering from the WMAP 5-year data, which yield a sudden reionization redshift (Dunkley et al., 2008). This tension could be eased if baryonic mass were more efficiently converted into ionizing photons at very high redshifts, for example as a result of a top-heavy IMF. Alternatively, the gas consumption timescale could be shorter than the one fiducially assumed ( Gyr), which would push the peak of star formation to a higher redshift.

The conclusions of the present section are in general agreement with the similar analysis of Miralda-Escudé (2003), who found that the ionizing comoving emissivity cannot decline from up to by more than a factor 1.5 if reionization is to be complete at . Our results also support the picture of a “photon-starved” reionization similarly advocated by Bolton & Haehnelt (2007) based the Ly forest photoionization rate.

#### 5.5.2 Helium Reionization

We also consider the reionization of HeII to HeIII, which we assume to proceed through the action of quasars only, requiring hard photons of energy eV. In this case, we can rewrite the emissivity integral as

 ∫∞νHeIIdνhνϵν=fQSO,HeIIesc∫∞0dLB˙Ni,HeIIdΦdLB, (34)

where

 ˙Ni,HeII=1.95×1055 s−1(LB1012 L⊙) (35)

is the number of HeII ionizing photons emitted per unit time by a quasar of -band luminosity and is again the -band quasar luminosity function from Hopkins et al. (2007). We again investigate the wide range of clumping factors and . We examine escape fractions of HeII ionizing photons from quasars and in addition to unity. The results are shown in Figure 8. We find that quasars are able to full ionize helium by , as required by observations of the HeII forest, for escape fractions of unity. Escape fractions may however not be sufficient to fully reionize helium by this redshift unless the gas is unrealistically smoothly distributed (; but see comment in §5.5.1 regarding the possibility of an effective clumping factor during reionization).666If the emergent quasar emission is beamed, then the true fraction of ionizing photons escaping into the IGM may be smaller. Since quasars beamed away from us would not be accounted for in the B-band luminosity function, this is however the relevant effective escape fraction to use in equation (34).

An interesting consistency check on the contribution of quasars to the ionizing background is provided by measurements of the HI to HeII column density ratio of absorbers, . Let us assume that the total hydrogen photoionization rate is a sum of a stellar contribution and a quasar contribution, with a fraction coming from stars:

 Γ=Γ⋆+ΓQSO; Γ⋆=f⋆Γ & ΓQSO=(1−f⋆)Γ. (36)

Further assume that only quasars contribute significantly to the HeII ionizing budget, so that . Again defining the helium terms in analogy to hydrogen, equation (24) implies that

 ΓQSOΓHeII= ⎛⎝ϵQSOνHIϵQSOνHeII⎞⎠(σHIσHeII)⎛⎝αQSOHeII+3αQSOHI+3⎞⎠ ×(Δl(νHI)Δl(νHeII)), (37)

where the superscripts indicate that only the quasar component is considered. Defining the ratio of the mean free paths and for an unhardened emissivity , this reduces to

 ΓQSOΓHeII=4αQSOunhard+1r, (38)

where we have used and , and assumed that the UV background is uniformly hardened by IGM filtering, so that . Combining the above expressions, we can solve for the fraction of the total contributed by quasars in term of the background softness parameter (where the HeII photoionization rate is defined in analogy to the hydrogen rate , but integrating only above the HeII photoelectric edge at 54.4 eV) and the ratio of the mean free paths:

 1−f⋆=4αQSOunhardrS, (39)

As both and depend on , measurements of the latter constrain . Using hydrodynamical simulations, Bolton et al. (2006) infer the softness parameter from obtained by fitting spectra of the HI and HeII Ly forests of the quasar HE 23474342 (Zheng et al., 2004; Shull et al., 2004). They find a softness parameter increasing from about at to about at , with large uncertainties. In Figure 9, we show the contribution of quasars to the hydrogen photoionization rate allowed by their measurement as a function of the ratio of the mean free paths at , and 2.8.

What is a reasonable value for the ratio of the mean free paths? Opacity owing to hydrogen only constrains the mean free path at the HeII photoelectric edge to be less than 8 times the value at the hydrogen Lyman edge (eq. 4.2). Helium opacity is however the limiting factor for the the mean free path of HeII ionizing photons. A rough estimate of the HeII mean free path can be obtained by assuming a constant ratio of column densities and making a Jacobian transformation to relate the column density of HeII to that of HI:

 ∂2N(NHeII)∂NHeII∂z=1r∂2N(NHI=NHeII/r)∂NHI∂z. (40)

Equation (4.2) may then be applied to express the HeII mean free path in terms of its column density distribution by simply making the replacement HIHeII.777This approach neglects the contribution of hydrogen opacity to the HeII mean free path. The latter is however negligible comparable to the HeII opacity for the measured column density ratios . Taking the ratio to the mean free path at the Lyman edge owing to hydrogen opacity, we find

 r=(4η)1−β. (41)

For the median column density ratios at (Zheng et al., 2004; Bolton et al., 2006) and approximating , we find . These ratios are plotted as vertical dashed lines in Figure 9 and suggest that quasars contribute a large (perhaps most) of the hydrogen ionizing background at , near the peak of the quasar luminosity function, with their contribution fractionally decreasing with increasing redshift.

We need to point out some uncertainties in this simple calculation. There are three main sources. First, in assuming a constant equal to the observed median, we have neglected the large fluctuations that are seen. A rough estimate of the uncertainty in the ratio of the mean free paths induced by the large fluctuations is obtained by considering the extreme ratios and , outside of which range the measured values are probably not reliable owing to difficulties introduced by background subtraction, the low signal-to-noise ratio of the spectra, line saturation, and blending of higher order Lyman series lines and metals (Reimers et al., 2004; Shull et al., 2004). These cases are indicated by the vertical dotted lines in Figure 9. Consistently, we have also used the softness parameter inferred by Bolton et al. (2006) for a spatially uniform UV background, which will be in error if the column density fluctuations arise from a fluctuating radiation field.

Second, the vast majority of the absorbers for which the column density ratio has been measured are optically thin to both HI and HeII ionizing photons, whereas the systems that are optically thick to HeII ionizing photons are likely to contribute significantly to limiting their mean free path. As the expected column density ratio varies non-monotonically with increasing column density ratio near the optically thick transition for reasonable background spectra (e.g., Haardt & Madau, 1996), it is unclear in which direction our optically thin calculation for the mean free path ratio may be biased.

Finally, we have neglected a possible anti-correlation between and (Reimers et al., 2004; Shull et al., 2004). This anti-correlation would tend to make our calculation of the HeII ionizing mean free path an overestimate, so that would be underestimated. In light of these uncertainties, the exact numerical values in Figure 9 should be interpreted with caution. Nevertheless, the extreme cases indicated by the dotted lines and the steep increase of the quasar contribution for convincingly argue that at their peak, quasars contribute a non-negligible fraction (%) of the hydrogen ionizing background, and possibly even dominate it.

A caveat to the above arguments is that it is not inconceivable that galaxies contribute a non-negligible fraction of HeII ionizing photons (Furlanetto & Oh, 2007b). For example, the composite LBG spectrum of Shapley et al. (2003) shows a significant HeII 1640 recombination line but no evidence for AGN contamination. In some scenarios, this Balmer- line may be accompanied by HeII ionizing photons. Associating the production of HeII ionizing photons with star formation, Furlanetto & Oh (2007b) show that a large escape fraction % for such photons is however required for LBGs to contribute comparably to quasars to the intergalactic HeII ionizing flux at . Although we cannot rule this out, the speculative nature of stars as a source of HeII ionizing photons and the large escape fraction that it would require motivate us to at present take the conservative view that this scenario is unlikely. Moreover, a significant contribution to UV emission at the HeII photoelectric edge from hot gas in galaxies and galaxy groups (Miniati et al., 2004) would decrease the expected fluctuations in the HI to HeII column density ratio, which appears difficult to reconcile with the large increase in the HeII opacity fluctuations toward (Bolton et al., 2006).

### 5.6. Direct Comparison with Galaxy UV Luminosity Functions and the Escape Fraction of Ionizing Photons

The arguments of this section have thus far fundamentally relied on the shape of the redshift evolution of the quasar emissivity and of the total photoionization rate. This is the reason we used analytic fits (which can be straightforwardly extrapolated to higher redshifts) to the star formation history, a quantity which at high redshifts is derived from the observed UV luminosity functions. As the UV luminosity functions themselves are the relevant quantities that involve the least unnecessary assumptions, it is of interest to compare them directly with the UV emissivity implied by our Ly forest measurement by equation (26).

To estimate the contribution of star-forming galaxies to the UV background, we consider recent determinations of the galaxy UV luminosity function from Lyman break (LBG) surveys at by Sawicki & Thompson (2006) (Keck Deep Fields), Bouwens et al. (2007) (Hubble Ultra Deep Field [HUDF] and other deep Hubble Space Telescope [HST] deep fields), Steidel et al. (1999) and Reddy et al. (2008) (Keck LBG), and Yoshida et al. (2006) (Subaru Deep Field). These were selected to be the most up-to-date measurements in the fields covered. The measured luminosity functions are summarized in Table 3, where in each case the parameters of the Schechter (1976) function are defined by

 ϕ(L)dL=ϕ⋆(LL⋆)αexp(−LL⋆)d(LL⋆) (42)

and the break magnitude is related to the characteristic luminosity by the usual AB-magnitude conversion:

 L⋆=4π(10 pc/cm)210−0.4(M⋆+48.60) erg s−1 (43)

(Oke & Gunn, 1983). Although the exact effective wavelength depends on the selection of the particular sample of interest, we will assume in our calculations that the UV luminosity functions are valid at 1500 Å and label the corresponding frequency by .

For a UV luminosity function given in comoving units the corresponding comoving UV emissivity is simply obtained by taking the first moment of the luminosity function,

 ϵcomUV=∫∞LmindLLϕ(L)=L⋆ϕ⋆Γ(α+2,Lmin/L⋆), (44)

where only sources above the minimum luminosity are accounted for and is the upper incomplete gamma-function.

Figure 10 shows the completeness fraction as a function of the faint-end slope <