A DIRECT MEASUREMENT OF THE INTERGALACTIC MEDIUM OPACITY TO H i IONIZING PHOTONS

The observed high transmission of z ∼ 3 quasars at rest wavelengths λ_r blueward of H i Lyα reveals that the intergalactic medium (IGM) is highly ionized (Gunn & Peterson 1965). The presence of the Lyα forest demands an intense, extragalactic ultraviolet background (EUVB) radiation field. The quasars themselves provide a significant fraction of the required ionizing flux, buoyed by the emission from more numerous yet fainter star-forming galaxies. Several recent studies have argued that the latter population dominates the EUVB at z ≳ 3 (Faucher-Giguère et al. 2008a; Cowie et al. 2009; Dall'Aglio et al. 2009), where the quasar population likely declines (e.g., Fan et al. 2004). These assertions, however, hinge on the opacity of the IGM to ionizing radiation via H i Lyman limit absorption (κ_LL), which directly impacts estimates of the EUVB measured from the integrated quasar and stellar ionizing emissivity.

Traditionally, κ_LL has been estimated from the incidence of so-called Lyman limit systems (LLSs) via surveys of quasar spectroscopy (e.g., Lanzetta 1991; Storrie-Lombardi et al. 1994; Péroux et al. 2003). Observationally, one can rather easily identify systems with large optical depths τ₉₁₂ ≳ 2 and the majority of these surveys have probed to this limit. The integrated opacity of the IGM, however, includes and is likely dominated by gas with τ₉₁₂ < 1, the so-called partial LLSs and Lyα forest clouds. Systems with these H i column densities (N_H i ≈ 10¹⁴–10¹⁷ cm^-2) are especially difficult to survey because the strong lines of the Lyman series (e.g., Lyα, Lyβ) lie on the flat portion of the curve of growth, and they exhibit only weak absorption at the Lyman limit. Therefore, current estimates of κ_LL are based on an extrapolation/interpolation of the frequency of systems with N_H i < 10¹⁴ cm⁻² and N_H i > 10^17.5 cm⁻² (Madau et al. 1999; Schirber & Bullock 2003; Faucher-Giguere et al. 2009). Current constraints on κ_LL span over a magnitude of uncertainty, especially at z ≳ 4.

In this Letter, we introduce a new technique to estimate κ_LL that avoids the traditional line-counting statistics of the IGM. We analyze the average rest-frame spectra of 1800 z > 3.5 quasars drawn from the Sloan Digital Sky Survey Data Release 7 (SDSS-DR7; Abazajian et al. 2009). As an ensemble at a common redshift, these "stacked" spectra show the exponential drop in flux at λ < λ₉₁₂ = c/ν₉₁₂ = 911.76 Å from the integrated opacity of the IGM. We precisely evaluate κ_LL in a series of small redshift intervals covering z ≈ 3.6–4.3 to explore redshift evolution. We adopt a cosmology with H₀ = 72 h₇₂ km s⁻¹ Mpc⁻¹, Ω_m = 0.3, and Ω_Λ = 0.7, and report proper lengths unless specified.

The H i Lyman limit opacity of the IGM has traditionally been expressed as an effective optical depth τ_eff,LL estimated from the observationally constrained N_H i frequency distribution f(N_H i, z). An ionizing photon (ν ⩾ ν₉₁₂) emitted from a quasar with redshift z = z_q will redshift to 1 Ryd at z = z₉₁₂ ≡ (ν₉₁₂/ν)(1 + z_q) − 1. The effective optical depth that this photon experiences by the IGM Lyman limit opacity is then (compared to Meiksin & Madau 1993)

$$\begin{eqnarray} \tau _{\rm eff,LL}(z_{912},z_q) &=& \int \nolimits _{z_{912}}^{z_q} \int \nolimits _0^\infty f(N_{\rm H\,\mathsc{i}},z^{\prime })\nonumber\\ &&\times \lbrace 1 - \exp [ - N_{\rm H\,\mathsc{i}}\sigma _{\rm ph}(z^{\prime }) ] \rbrace dN_{\rm H\,\mathsc{i}}dz^{\prime }, \end{eqnarray} \tag{ 1 }$$

where σ_ph is the photoionization cross section evaluated at the photon frequency. In practice, this approach is subject to large uncertainties because (1) f(N_H i, z) is poorly constrained for systems with τ₉₁₂ ≲ 1, (2) observational surveys rarely measure f(N_H i, z) at the same epoch, forcing interpolation and extrapolation, and (3) estimates of f(N_H i, z) for the LLS may always be subject to large systematic uncertainty (Prochaska et al. 2009, in preparation). This traditional approach may never yield a precise and robust estimate of τ_eff,LL.

Our new approach is to directly measure τ_eff,LL through analysis of averaged ensembles of quasar spectra. In Figure 1, we present the stacked spectrum of 150 mock quasar spectra at z ≈ 3.6. Each spectrum was given a unique emission redshift z_q and spectral energy distribution³ (SED; normalized at 1450 Å), and then was blanketed with Lyman series and Lyman limit absorption from an assumed f(N_H i, z) distribution (Dall'Aglio et al. 2008; Worseck & Prochaska 2009, in preparation). The spectra were degraded to the nominal spectral resolution of the SDSS spectrometer (FWHM = 150 km s⁻¹) and Gaussian noise was added to give a distribution⁴ of signal-to-noise ratio (S/N) values at λ_r = 1450 Å. The data were then averaged without weighting.

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Inspecting Figure 1, one identifies the effective opacity of the Lyβ forest at λ_r ≈ 1000 Å and corresponding decrements in the spectrum at Lyγ and Lyδ. One then observes a steep drop in the flux starting at λ_r ≈ 920 Å due to the opacity of higher order Lyman series lines of optically thick absorbers (e.g., damped Lyα systems). The continued decline at λ_r < λ₉₁₂, however, is dominated by the continuum opacity of H i. At all wavelengths, the scatter in the stacked spectrum is due to small-scale variance in IGM absorption, not the noise in the individual spectra.

Overplotted in Figure 1 is the flux model f = f₉₁₂exp [ − τ_eff,LL(z)] with τ_eff,LL evaluated from the input f(N_H i, z) distribution (Equation (1)) and f₉₁₂, the flux at λ = λ₉₁₂, estimated from the data. This is a good model of the stacked spectrum; even though the underlying average SED evolves as f_λ ∝ λ^2.4, the analysis is performed over too small a wavelength interval to note its evolution. The evaluation gives τ_eff,LL = 1 at (z_q − z₉₁₂) = 0.22 corresponding to a proper mean free path λ⁹¹²_mfp = 36.9 h⁻¹₇₂ Mpc at z = 3.6.

Now consider an alternate evaluation of τ_eff,LL which follows the standard definition of optical depth,

$$\begin{equation} \tau _{\rm eff,LL}(r,\nu) = \int \nolimits _0^r \! \kappa _{\rm LL}(r^{\prime },\nu) \, dr^{\prime }, \end{equation} \tag{ 2 }$$

where the integral is evaluated to an arbitrary proper distance from the quasar. In an expanding universe, an ionizing photon emitted by the quasar will be attenuated by the Lyman limit opacity κ_LL until it is redshifted to hν = 1 Ryd at z = z₉₁₂. If the photon is not absorbed by IGM line opacity from gas at z < z₉₁₂, it may be observed today at a wavelength λ_obs = (1 + z₉₁₂)c/ν₉₁₂. During the photon's travel from z_q to z₉₁₂, the opacity κ_LL evolves because of the decreasing frequency (redshift) and also from changes to the physical conditions of the universe (e.g., the expanding proper distance). We separate the frequency and radial dependencies in the opacity as follows,

$$\begin{equation} \kappa _{\rm LL}(r,\nu) \equiv {\tilde{\kappa }}_{912}(r) \left(\,\frac{\nu }{\nu _{\rm 912}} \, \right) ^{-3}, \end{equation} \tag{ 3 }$$

where the frequency dependence related to σ_ph is approximate. This treatment also ignores stimulated emission, i.e., it assumes τ_eff,LL is dominated by "clouds" with τ₉₁₂ ≲ 1.

In principle, one can adopt any radial dependence for κ_LL. Expressing Equation (3) in redshift space, we have

$$\begin{equation} \kappa _{\rm LL}(z) = {\tilde{\kappa }}_{912}(z) \left(\,\frac{1+z}{1+z_{912}} \, \right) ^{-3}. \end{equation} \tag{ 4 }$$

With this functional form for the Lyman limit opacity, it is straightforward to integrate Equation (2) by adopting a Friedman–Walker cosmology where

$$\begin{equation} \frac{dr}{dz} \equiv \frac{c}{H(z) (1+z)} = \frac{c/H_0}{(1+z) \sqrt{\Omega _{\rm m} (1+z)^3 + \Omega _\Lambda }}. \end{equation} \tag{ 5 }$$

At z > 3, the universe is matter dominated and we can express dr/dz ≈ c/(H₀Ω^1/2_m)(1 + z)^−5/2. Altogether, we find

$$\begin{eqnarray} \tau _{\rm eff,LL}(z_{912},z_q) & =& \frac{c}{H_0 \Omega _{\rm m}^{1/2}} (1+z_{912})^3 \nonumber \\ && \times \int \nolimits _{z_{912}}^{z_q} {\tilde{\kappa }}_{912}(z^{\prime }) \, (1+z^{\prime })^{-11/2} \, dz^{\prime }. \end{eqnarray} \tag{ 6 }$$

In practice, we find that the analysis of a single stacked spectrum does not constrain the redshift evolution in κ_LL. Therefore, we have simply parameterized κ_LL by its value at z = z_q, i.e., ${\tilde{\kappa }}_{912}(z^{\prime }) = \kappa _{z_q}$. The thin solid curve in Figure 1 shows the resulting flux model for τ_eff,LL for a best-fit value $\kappa _{z_q}= 0.028 \; h_{72} \, {\rm Mpc}^{-1}$. The corresponding mean free path $\lambda _{\rm mfp}^{912}= 1/\kappa _{z_q}= 35.2 \, {h_{72}^{-1}\, {\rm Mpc}}$ is in excellent agreement with the traditional evaluation. We stress that the analysis was performed without any consideration of the quasar SEDs nor any consideration of evolution in the Lyman series line opacity. Although these contribute to the observed flux in the stacked spectrum, the exponential drop due to τ_eff,LL dominates over this and any other astrophysical aspect.

Our new approach provides a tight constraint on the effective mean free path near the quasar emission redshift (at z ≈ z_q). Because the stacked spectrum covers only several tenths in redshift below the Lyman limit, it imposes a very weak constraint on the redshift evolution of κ_LL. Instead, one must evaluate the stacked spectrum of quasars at a range of emission redshifts.

We apply the methodology to 1800 quasar spectra drawn from the SDSS-DR7. We began with the vetted quasar list from our survey of LLSs (Prochaska et al. 2009, in preparation) which avoids all purported quasars in the SDSS-DR7 that have erroneous redshift estimates or are not bona-fide quasars. We have also ignored all quasars with strong associated absorption in the C iv, N v, and/or O vi doublets (e.g., broad absorption line systems). We have not, however, removed quasars with evident Lyman limit absorption at z ≈ z_q. It is our goal to estimate the entire Lyman limit opacity that quasars experience, except for the influence of gas on parsec scales. Therefore, our estimates of κ_LL include opacity from the quasar's local galactic environment, i.e., its proximity region.

The sample was further limited to the following criteria: (1) z_q ⩾ 3.5 to insure significant coverage of the Lyman limit in the SDSS spectra and minimize the likelihood that LLSs bias the quasar target probability (but see below); (2) S/N ⩾4 at λ_r = 1450 Å; (3) z_q < 4.35 to insure that a stack of 150 quasars covers a redshift interval Δz < 0.4. Starting at z = 3.5, we constructed a series of bins of 150 quasars each to produce a stacked spectrum. Each quasar spectrum was normalized by the observed flux at λ_r = 1450 Å (in a 20 Å window) and shifted to its rest frame (nearest pixel). The full ensemble was then averaged (without weighting), ignoring bad pixels. A sample of three of the stacked spectra is given in Figure 2.

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As noted above, our analysis of τ_eff,LL includes contributions from the quasar's proximity region. In general, this corresponds to ≈10 Mpc or Δλ_r ≈ 9 Å (e.g., Dall'Aglio et al. 2008). From the stacked spectrum, Figure 2 shows that the flux at the "edge" of the proximity region (i.e., at λ_r ≈ 900 Å) is ≈0.75 times the flux at 912 Å, giving τ_eff,LL ≈ 0.3. In the absence of the IGM beyond the proximity region, the flux would begin to recover shortward of 900 Å as ν⁻³. It is evident, therefore, that opacity from the "true" IGM dominates our analysis. Furthermore, we find that our analysis yields models that extrapolate well to 912 Å, suggesting that the opacity of the proximity region follows the behavior of the general IGM. This is consistent with our finding that there are no strong differences in the incidence of LLSs, near quasars (Prochaska et al. 2009, in preparation).

Using the model for τ_eff,LL (Equation (6)), we have fitted the data to evaluate the opacity $\kappa _{z_q}$ at z ≈ z_q in a series of redshift intervals, each containing 150 quasars. We minimized χ² over rest-wavelength intervals starting at 905 Å (to minimize bias from strong Lyman series absorption in the proximity zone of the quasar) and extending down in wavelength corresponding to the larger of 3900 Å and z_q − z₉₁₂ = 0.4. The scatter in the stacked spectrum was estimated locally in 21 pixel bins centered at each data point and presumed to be Gaussian. The flux at λ₉₁₂ was estimated from the data but allowed to vary by 10% when minimizing χ². Because the observed scatter in the stacked spectrum is systematic (related to stochasticity in the IGM) and is not included in our model, one cannot estimate $\sigma (\kappa _{z_q})$ the uncertainty in $\kappa _{z_q}$ from standard χ² techniques. Instead, we performed a bootstrap analysis of 100 realizations of each stacked spectrum and estimated $\sigma (\kappa _{z_q})$ from the resulting distribution of $\kappa _{z_q}$ values.

Figure 3 presents the evaluations of $\kappa _{z_q}$ in terms of the mean free path ($\lambda _{\rm mfp}^{912}=1/\kappa _{z_q}$). The mean free path exhibits a peak (minimum in opacity) of nearly $50 \, {h_{72}^{-1}\, \rm Mpc}$ at z = 3.6 and declines with increasing redshift. We have parameterized the redshift evolution in λ⁹¹²_mfp with a simple linear regression: λ⁹¹²_mfp = λ₀ − b_λ(z_q − 3.6). Restricting the analysis to z ⩾ 3.59, a χ² minimization of this model to the binned evaluations of λ⁹¹²_mfp gives λ₀ = (48.4 ± 2.1) h⁻¹₇₂ Mpc and b_λ = (38.0 ± 5.3) h⁻¹₇₂ Mpc.

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Formally, our analysis also indicated a rise in the opacity at z < 3.6, i.e., the lower limits in Figure 3. This runs contrary to all expectation and current understanding of the IGM. Initially, we suspected that this measurement indicated a systematic error in the SDSS spectra at the bluest wavelengths (e.g., Bernardi et al. 2003). To test this hypothesis, we examined the (u − g) colors of the quasars in the first (z ≈ 3.5) and third (z ≈ 3.6) quasar bins. Figure 4 histograms the two (u − g) distributions. The colors of the z = 3.5 quasars are systematically redder than those at z = 3.6; this is the exact opposite of what one predicts if the IGM were monotonically increasing in opacity with redshift. We conclude that the quasars at z = 3.5 drawn from the SDSS are redder than the cohort at z = 3.6 because of an elevated incidence of the Lyman limit opacity, confirmed by analysis on the incidence of LLS (Prochaska et al. 2009, in preparation).

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We then explored whether this elevated opacity is related to observational bias in the SDSS quasar sample. We simulated the SDSS experiment by constructing mock quasar spectra at z ≈ 3.5 and z ≈ 3.6 with intrinsic SEDs having mean (u − g) color of 0.57 mag and standard deviation of 0.19 mag. These spectra were blanketed with IGM absorption assuming a monotonically increasing opacity with redshift. After restricting the quasar sample according to the SDSS color-selection criteria (Richards et al. 2002), we found the z ≈ 3.5 cohort has systematically redder colors than the higher redshift sample. Similarly, we find a correspondingly higher opacity inferred from the stacked spectrum. Because of the targeting criteria, the cohort of z ≈ 3–3.6 quasars in the SDSS spectroscopic database are systematically biased against having (u − g) < 1.5 which biases against sight lines without strong Lyman limit absorption. Our analysis indicates that the bias extends to z_q = 3.6, beyond which very few quasars are predicted to ever have such blue color (see Worseck & Prochaska 2009, in preparation, for further details). The results for z > 3.6 are presented in Table 1.

Table 1. Summary Table

z	〈zq〉	λanalysis (Å)	$\kappa _{z_q}$ (Mpc−1)	$\sigma (\kappa _{z_q})$ (Mpc−1)
[3.59,3.63]	3.61	[846, 905]	0.0209	0.0023
[3.63,3.67]	3.65	[839, 905]	0.0218	0.0021
[3.67,3.71]	3.68	[834, 905]	0.0224	0.0026
[3.71,3.76]	3.73	[835, 905]	0.0212	0.0020
[3.76,3.81]	3.78	[836, 905]	0.0257	0.0020
[3.81,3.86]	3.83	[837, 905]	0.0254	0.0024
[3.86,3.92]	3.89	[837, 905]	0.0243	0.0024
[3.92,4.02]	3.96	[839, 905]	0.0292	0.0026
[4.02,4.13]	4.08	[840, 905]	0.0341	0.0033
[4.13,4.34]	4.23	[842, 905]	0.0403	0.0036

Notes. Because of the color-criteria bias discussed in this Letter, we caution that the values for the first few bins may systematically underestimate $\kappa _{z_q}$ by 10%–30%. All wavelengths are in the quasar rest frame and all distances are proper. The assumed cosmology has Ω_m = 0.7, Ω_Λ = 0.7, and H₀ = 72 km s⁻¹ Mpc⁻¹.

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Our new technique provides the most precise measurements on the IGM opacity to H i ionizing radiation at any redshift. Previous estimates of κ_LL were limited by large uncertainties in the N_H i frequency distribution, especially the incidence of systems with N_H i ≈ 10¹⁷ cm⁻² (Madau et al. 1999; Schirber & Bullock 2003; Miralda-Escudé 2003; Meiksin & White 2004; Faucher-Giguère et al. 2008a). Our results indicate that systems dominating κ_LL at z > 3.6 must evolve more slowly than the N_H i ⩽ 10¹⁴ cm⁻² Lyα forest. In turn, we infer a flattening in the H i frequency distribution between N_H i ≈ 10¹⁵ and 10¹⁷ cm^-2 (Petitjean et al. 1993; O'Meara et al. 2007). We provide a full discussion on the implications for f(N_H i, z) in Prochaska et al. (2009, in preparation).

Our analysis also gives the first direct description of the evolution in λ⁹¹²_mfp, albeit over a small redshift interval. Our results are well parameterized by a linear decrease in λ⁹¹²_mfp but can also be described by a (1 + z)^−γ power law with γ = 3.5–5.5. These values are consistent with the observed evolution in the incidence of LLSs (Prochaska et al. 2009, in preparation).

Our results refine recent inferences (e.g., Faucher-Giguère et al. 2008a; Dall'Aglio et al. 2009) that galaxies contribute significantly to the EUVB at z > 3. We assume the photoionization rate at z = 4 inferred from the effective Lyα opacity of the IGM (log Γ_IGM = −12.3; Faucher-Giguère et al. 2008b). Comparing this value against the photoionization rate inferred from quasars Γ_q using an emissivity ^q₉₁₂ = 2 × 10²⁴ erg s^-1 Hz⁻¹ Mpc⁻³ (comoving; Hopkins et al. 2007; Bongiorno et al. 2007) and adopting our estimate of λ⁹¹²_mfp at z = 4, we find Γ_q = 0.5 Γ_IGM. This suggests a modest but non-negligible contribution from galaxies to the EUVB at this redshift. The systematic uncertainties in ^q₉₁₂ and Γ_IGM are sufficiently large that one could recover Γ_IGM=Γ_q, but Γ_q would overpredict Γ_IGM at z = 3.5 given the observed rise in λ⁹¹²_mfp and ^q₉₁₂ with decreasing redshift.

The results also revise at least some previous estimates of the EUVB. Haardt & Madau (1996) and their subsequent analyses (CUBA), for example, have adopted an approximately three times shorter mean free path at z ∼ 4 than our analysis reveals. This implies that (1) the normalization of their EUVB spectrum is several times too low and (2) the EUVB spectrum is softer at energies of ≈1 Ryd. The latter point may help to reconcile apparent contradictions in the metal-line analysis of the IGM (Aguirre et al. 2008) without resorting to a large input from galaxies. Other analyses, however, have used estimates for λ⁹¹²_mfp that are in much better agreement with our results (e.g., Faucher-Giguere et al. 2009). An understanding of the full implications of our new constraints on λ⁹¹²_mfp awaits new calculations of the EUVB.

We identified a previously unreported systematic bias in the SDSS quasar spectroscopic sample against sight lines at z < 3.6 that are "clear" of optically thick absorbers. This bias has implications for a range of IGM analysis at z ∼ 3 including (1) the paucity of sight lines for studying He ii reionization (Worseck & Prochaska 2009, in preparation), (2) an overestimate of the incidence of LLSs (Prochaska et al. 2009, in preparation) and damped Lyα systems (Prochaska & Wolfe 2009), and (3) analysis of the Lyα forest. We caution that all existing studies of the IGM at z ∼ 3 using the SDSS database should be reviewed in light of this systematic bias.

In future work, we analyze these same spectra to constrain the SEDs of z > 3.5 quasars, infer the N_H i frequency distribution and Doppler parameters of gas with N_H i ≈ 10¹⁶ cm⁻², and isolate the opacity of the IGM far from the quasar's proximity region. The technique introduced here is easily extended to higher and lower redshifts by obtaining modest S/N, low-resolution spectroscopy of several hundred quasars. Future ground- and space-based programs will precisely estimate κ_LL from z_q ≈ 0.5 to 5.

We acknowledge valuable conversations with P. Madau, J. Hennawi, G. Richards, and S. Burles. J.X.P. and J.M.O are supported by NASA grant HST-GO-10878.05-A. J.X.P. and G.W. acknowledge support from NSF CAREER grant (AST-0548180) and NSF grant AST-0908910.