QUASARS PROBING QUASARS. IV. JOINT CONSTRAINTS ON THE CIRCUMGALACTIC MEDIUM FROM ABSORPTION AND EMISSION

Bright emission in the Lyα line typically occurs in gas at T ∼ 10⁴ K; thus we consider a highly idealized model of a population of cool gas clouds in quasar halos. Several studies have previously explored similar models (Mo & Miralda-Escude 1996; Maller & Bullock 2004), and others have focused specifically on extended Lyα emission (Rees 1988; Haiman & Rees 2001; Haiman et al. 2000; Dijkstra & Loeb 2009). Nearly all of this work assumes the cool gas is in pressure equilibrium with a hot tenuous virialized plasma. Although there may be some evidence for pressure confinement from our absorption-line measurements (Prochaska & Hennawi 2009), our conclusions about Lyα emission do not depend on the existence of a hot phase.

We assume that the cool gas clouds observed in absorption are spherical with a single uniform hydrogen volume density n_H, and that they are spatially uniformly distributed throughout a spherical halo of radius R. Although extremely simple, this model is also relevant, to within order unity geometric corrections, to more complicated distributions which vary with radius, or the filamentary structures expected in the cold accretion picture. In what follows, we often employ averages over the projected area of the spherical halo on the sky which we denote with angle brackets, e.g., the average line-of-sight distance through the halo is $\langle s \rangle = 1/ A \int s dA = 4R/ 3$. Besides the volume density n_H and the size of the halo R, two other parameters completely specify the spatial distribution of the gas, namely the hydrogen column density of the individual clouds N_H, and the cloud covering factor f_C, which is similarly defined to be an average over the sphere. We focus on these four parameters (R, n_H, N_H, f_C) because they are either directly observable or are closely related to absorption-line observations.

All other quantities of interest can then be expressed in terms of these four parameters. For example, of particular interest for computing the Lyα surface brightness is the volume filling factor defined as the ratio of the volume occupied by the clouds to the total volume, f_V = n_cV_c, where n_c is the number density of clouds and V_c is their volume. The definition of the covering factor is

$$\begin{equation} f_{\rm C} \equiv \left\langle \int \frac{df_{\rm C}}{ds} ds\right\rangle , \end{equation} \tag{ 1 }$$

where (df_C/ds) = n_cσ_c, and σ_c is the cloud cross-sectional area. This gives $f_{\rm C} = n_{\rm c}\sigma _{\rm c}4R/ 3$ for our case of uniformly distributed clouds. Note that in general the covering factor can be larger than unity. The corresponding volume filling factor can be written as

$$\begin{equation} f_{\rm V}=\frac{3f_{\rm C}N_{\rm H}}{4Rn_{\rm H}}, \end{equation} \tag{ 2 }$$

and the total mass of cool gas is then

$$\begin{equation} M_{\rm c} = \frac{4}{3}\pi R^3 f_{\rm V} \frac{n_{\rm H} m_{\rm p}}{X}= \pi R^2 f_{\rm C}N_{\rm H} \frac{m_{\rm p}}{X} , \end{equation} \tag{ 3 }$$

where m_p is the mass of the proton and X = 0.76 is the hydrogen mass fraction.

As we will consider emission from objects at high redshift, for which the size of the emitting clouds could be unresolved by our instrument, the relevant observable is the surface brightness (erg s⁻¹ cm⁻² arcsec⁻²) averaged over the projected area of a sphere as measured from Earth. The specific intensity of Lyα from our spherical halo is given by integrating the equation of radiative transfer through the gas distribution

$$\begin{equation} I_{{\rm Ly}\alpha } = \int j_{{\rm Ly}\alpha } ds , \end{equation} \tag{ 4 }$$

where j_Lyα is the volume emissivity per steradian (erg s⁻¹ cm⁻³ster⁻¹). In this and the following sections we will ignore the attenuation of Lyα radiation caused by resonant scattering of gas within the halo itself or by the IGM, as well as extinction caused by dust. Scattering within the halo itself would only change the spatial distribution of the emission, and as we are averaging over apertures on the sky, this should produce small effects. We do however consider the resonant scattering of a point source of Lyα photons from the quasar CGM in Section 2.3. Scattering by the IGM is likely negligible for the redshifts we consider (Zheng et al. 2011). We estimate the impact of extinction by dust in the Appendix, and demonstrate that it should not result in a significant reduction of the Lyα emission.

We can then write the average surface brightness observed from Earth as

$$\begin{equation} {\rm SB}_{{\rm Ly}\alpha }\equiv \frac{1}{(1+z)^4}\langle I_{{\rm Ly}\alpha }\rangle = \frac{1}{(1+z)^4}\frac{1}{\pi R^2}\int j_{\rm {\rm Ly}\alpha }dV \end{equation} \tag{ 5 }$$

where the factor of (1 + z)⁻⁴ accounts for cosmological surface brightness dimming. The corresponding total Lyα luminosity is

$$\begin{eqnarray} L_{{\rm Ly}\alpha } &=& 4\pi ^2 (1+z)^4 R^2{\rm SB}_{{\rm Ly}\alpha }\nonumber\\ &=& 1.3{\; \times \; 10^{44}} \left(\frac{1+z}{3.0}\right)^{4}\!\! \left(\frac{R}{100\,{\rm kpc}}\right)^{2}\nonumber \\ &&\times \left(\frac{{\rm SB}_{{\rm Ly}\alpha }}{10^{-17}\,{\rm erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}}\right)\,{\rm erg\,s^{-1}}. \end{eqnarray} \tag{ 6 }$$

For a low density gas, it is possible to separate contributions to the total Lyα intensity into recombination and collisionally excited components (Chamberlain 1953). Observationally, extended Lyα emission could have contributions from both mechanisms. Collisional excitation of neutral atoms and the subsequent emission of Lyα can play an important role in the radiative cooling of T ∼ 10⁴ K gas, and we explore this cooling radiation from the CGM of quasars in detail in a companion paper (J. F. Hennawi & J. X. Prochaska 2013, in preparation). In what follows we consider recombination, which we will also refer to as Lyα fluorescence, from a source of ionizing photons. A central point source with an r⁻² dependence is assumed, but our estimates can easily be generalized to other spatial distributions. We consider the two limiting cases where the gas clouds are either optically thin $N_{\rm H\,\scriptsize{I}} \ll 10^{17.2}\,{\rm cm}^{-2}$ or optically thick $N_{\rm H\,\scriptsize{I}} \gg 10^{17.2}\,{\rm cm}^{-2}$ to Lyman continuum photons. At intermediate neutral columns ${\sim} N_{\rm H\,\scriptsize{I}} \sim 10^{17.2}\,{\rm cm}^{-2}$ our simple approximations break down and more detailed radiative transfer calculations are necessary. In each case, we consider the limit where light from the quasar has otherwise been unattenuated by resonant scattering or dust.

2.2.1. Optically Thin

The recombination emissivity from the cool clouds is

$$\begin{equation} j_{{\rm Ly}\alpha } = f_{\rm V}\frac{h\nu _{{\rm Ly}\alpha }}{4\pi }\eta _{\rm thin}n_{\rm e}n_{\rm p} \alpha _{\rm A}(T) , \end{equation} \tag{ 7 }$$

where n_e is the electron density, n_p is the proton density, η_thin = 0.42 is the fraction of recombinations which result in a Lyα photon in the optically thin limit (Osterbrock & Ferland 2006; Gould & Weinberg 1996), and the factor of f_V accounts for the fact that the emitting clouds fill only a fraction of the volume. The case A recombination coefficient α_A is a weak function of temperature T, and we evaluate it at T = 10, 000 K where α_A = 4.18 × 10⁻¹³ cm⁻³ s⁻¹ (Osterbrock & Ferland 2006).

The electron and proton densities are determined by photoionization equilibrium,

$$\begin{equation} n_{\rm H\,\scriptsize{I}}\Gamma = n_{\rm e}n_{\rm p}\alpha _{\rm A} , \end{equation} \tag{ 8 }$$

and the photoionization rate Γ is given by

$$\begin{equation} \Gamma = \frac{1}{4\pi r^2}\int _{\nu _{\rm LL}}^{\infty } \frac{L_{\nu }}{h\nu }\sigma _\nu d\nu. \end{equation} \tag{ 9 }$$

Here ν_LL is the frequency at the Lyman edge, and σ_ν is the hydrogen photoionization cross-section. We assume that the quasar's spectral energy distribution obeys the power-law form $L_{\nu } = L_{\nu _{\rm LL}} (\nu / \nu _{\rm LL})^{-\alpha _{\rm Q}}$, blueward of ν_LL and adopt a slope of α_Q = 1.57 consistent with the measurements of Telfer et al. (2002). The quasar ionizing luminosity is then parameterized by $L_{\nu _{\rm LL}}$, the luminosity at the Lyman edge. Throughout this paper we estimate $L_{\nu _{\rm LL}}$ by tying the Telfer et al. (2002) power-law spectrum to the composite quasar spectrum of Vanden Berk et al. (2001), which can then be compared to the SDSS optical photometry covering rest-frame UV wavelengths (see Appendix A of QPQ1 for a detailed description of this procedure).

For all cases of interest to us, radiation fields are sufficiently intense that optically thin gas will always be highly-ionized. Under this assumption, the neutral fraction $x_{\rm H\,\scriptsize{I}} \equiv ({n_{\rm H\,\scriptsize{I}}}/{n_{\rm H}}) \ll 1$, hence n_p ≈ n_H and $n_{\rm e} \approx (1 + Y/ 2X) n_{\rm H}$, where the helium and hydrogen mass fractions are Y and X, respectively, and we have assumed all helium is doubly ionized.

Putting this all together, the observed Lyα surface brightness is

$$\begin{eqnarray} {\rm SB}_{{\rm Ly}\alpha }&=&\frac{\eta _{\rm thin}h\nu _{{\rm Ly}\alpha }}{4\pi (1+z)^4}\alpha _{\rm A} \left(1 + \frac{Y}{2X} \right) n_{\rm H}f_{\rm C} N_{\rm H}\nonumber\\ &=& 7.7\times 10^{-19} \left(\frac{1+z}{3.0}\right)^{-4}\!\! \left(\frac{f_{\rm C}}{1.0}\right) \left(\frac{n_{\rm H}}{0.1\,{\rm cm^{-3}}}\right)\nonumber \\ &&\times \left(\frac{N_{\rm H}}{10^{20}\,{\rm cm^{-2}}}\right){\rm \,erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}. \end{eqnarray} \tag{ 10 }$$

Note that photoionization equilibrium implies that $j_{{\rm Ly}\alpha } \propto \Gamma n_{\rm H\,\scriptsize{I}}$, but since the neutral fraction is inversely proportional to photoionization rate $x_{\rm H\,\scriptsize{I}} \approx \alpha _{\rm A}n_{\rm H}(1 + Y/ 2X)/ \Gamma$, the net effect is that Lyα emission is independent of the luminosity of the ionizing source in the highly-ionized, optically-thin regime, provided that the ionizing source is bright enough to keep the gas highly ionized and optically thin. This fact can be exploited to obtain an expression for the neutral column density averaged over the area of the halo, $\langle N_{\rm H\,\scriptsize{I}}\rangle$, in terms of the observed surface brightness and ionizing luminosity:

$$\begin{eqnarray} \langle N_{\rm H\,\scriptsize{I}}\rangle &=& 2.2\times 10^{17}\left(\frac{1+z}{3.0}\right)^{-4} \!\!\left(\frac{L_{\nu _{\rm LL}}}{10^{30}\,{\rm erg\,s^{-1}\,Hz^{-1}}}\right)^{-1} \nonumber\\ &&\times \left(\frac{R}{100\,{\rm kpc}}\right)^{2}\!\! \left(\frac{{\rm SB}_{{\rm Ly}\alpha }}{10^{-17}\,{\rm erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}}\right)\,{\rm cm^{-2}}.\nonumber\\ \end{eqnarray} \tag{ 11 }$$

In order for the optically thin approximation to be valid, the individual clouds must have $N_{\rm H\,\scriptsize{I}} \ll 10^{17.2}\,{\rm cm^{-2}}$. In optically thin photoionization equilibrium, the neutral fraction scales as $x_{\rm H\,\scriptsize{I}} \propto r^2$ so that the largest neutral columns will be attained at R, the edge of the halo. It thus follows that $\langle N_{\rm H\,\scriptsize{I}}\rangle < {\rm max}(N_{\rm H\,\scriptsize{I}})$ and hence if $\langle N_{\rm H\,\scriptsize{I}}\rangle > 10^{17.2}\,{\rm cm}^{-2}$ then the clouds are definitely optically thick at some locations in the halo and the optically thin approximation breaks down. One can thus use Equation (11) to compute $\langle N_{\rm H\,\scriptsize{I}}\rangle$ and directly test whether the optically thin approximation is valid, given only the observed Lyα surface brightness and the size of the nebulae, provided that $L_{\nu _{\rm LL}}$ can be estimated, as in the case of Lyα nebulae observed around quasars. If $\langle N_{\rm H\,\scriptsize{I}}\rangle > 10^{17.2}\,{\rm cm}^{2}$ then the clouds cannot be optically thin. Note however that the converse is not true—both optically thin and optically thick clouds can result in $\langle N_{\rm H\,\scriptsize{I}}\rangle < 10^{17.2}\,{\rm cm}^{-2}$—a small value of $\langle N_{\rm H\,\scriptsize{I}}\rangle$ thus provides no definitive information about whether we are in the optically thin or thick regime.

2.2.2. Optically Thick

If the clouds are optically thick to ionizing radiation, they will no longer emit Lyα proportional to the volume that they occupy because of self-shielding effects. Instead, a thin highly ionized skin will develop around each cloud, and nearly all recombinations and resulting Lyα photons will originate in this skin. The cloud will then behave like a special “mirror,” converting a fraction η_thick = 0.66 of the ionizing photons it intercepts into Lyα photons emitted at a uniform brightness (i.e., the specific intensity leaving the surface is isotropic, see Gould & Weinberg 1996). Simulations including both ionizing radiative transfer and Lyα resonant scattering (Cantalupo et al. 2005; Kollmeier et al. 2010) have shown, that for the case of a quasar illuminating an optically thick cloud, the true surface brightness is reduced from the “mirror” value by a factor of ≃ 2–4. This reduction occurs because illumination and radiative transfer effects redistribute photons over a wide solid angle. We introduce a geometric reduction factor f_gm to account for this reduction. For the case of a transversely illuminated spherical “mirror,” the resulting “half-moon” illumination pattern results in a $f_{\rm gm}=4/ (3\pi)=0.42$ (Adelberger et al. 2006). We set f_gm = 0.5, which is consistent with the range seen in radiative transfer simulations (Cantalupo et al. 2005; Kollmeier et al. 2010). At a distance R from a quasar, a mirror will emit a surface brightness

$$\begin{eqnarray} {\rm SB}_{{\rm Ly}\alpha }&=&\frac{f_{\rm gm}\eta _{\rm thick}h\nu _{{\rm Ly}\alpha }}{(1+z)^4}\frac{\Phi }{\pi }\nonumber\\ &=& 4.0\times 10^{-17}\left(\frac{1+z}{3.0}\right)^{-4}\!\! \left(\frac{f_{\rm gm}}{0.5}\right)\!\! \left(\frac{R}{100\,{\rm kpc}}\right)^{-2}\nonumber \\ &&\times \left(\frac{L_{\nu _{\rm LL}}}{10^{30}\,{\rm erg\,s^{-1}\,Hz^{-1}}}\right){\rm \,erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}} \end{eqnarray} \tag{ 12 }$$

where Φ (phot s⁻¹ cm⁻²) is the ionizing photon number flux

$$\begin{equation} \Phi = \int _{\nu _{\rm LL}}^{\infty } \frac{F_{\nu }}{h\nu } d\nu = \frac{1}{4\pi r^2}\int _{\nu _{\rm LL}}^{\infty } \frac{L_{\nu }}{h\nu } d\nu. \end{equation} \tag{ 13 }$$

The total observed flux from the mirror is simply proportional to the area of the mirror times this SB_Lyα.

Now consider the case of a population of cool optically thick clouds, hence also mirrors, but which are spatially unresolved by our instrument. In this regime we can write the Lyα volume emissivity as

$$\begin{equation} j_{{\rm Ly}\alpha } = \frac{\eta _{\rm thick}h\nu _{{\rm Ly}\alpha }}{4\pi }n_{\rm c}\sigma _{\rm c}\Phi. \end{equation} \tag{ 14 }$$

Note that there is no geometric reduction factor in this case. Conservation of photons requires that a fraction η_thick of the ionizing photons emerge as Lyα photons, and the emissivity must be isotropic. Again integrating the equation of radiative transfer and averaging over area (Equations (4) and (5)), we obtain

$$\begin{eqnarray} {\rm SB}_{{\rm Ly}\alpha }&=&\frac{\eta _{\rm thick}h\nu _{{\rm Ly}\alpha }}{4\pi (1+z)^4}f_{\rm C} \Phi (R/ \sqrt{3})\nonumber\\ &=& 6.0\times 10^{-17}\left(\frac{1+z}{3.0}\right)^{-4}\!\! \left(\frac{f_{\rm C}}{1.0}\right)\!\! \left(\frac{R}{100\,{\rm kpc}}\right)^{-2}\nonumber \\ && \times \left(\frac{L_{\nu _{\rm LL}}}{10^{30}\,{\rm erg\,s^{-1}\,Hz^{-1}}}\right){\rm \,erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}. \end{eqnarray} \tag{ 15 }$$

In the expression above it is understood that f_C is capped at f_C = 1, corresponding to the case where all ionizing photons are absorbed.

Thus in the optically thick limit, the emission depends only weakly on the amount of cool gas through the covering factor of the clouds, and scales linearly with the ionizing photon flux. This contrasts strongly with the highly ionized optically thin limit, where the surface brightness depends on the amount of cold gas through the combination f_Cn_HN_H, but not on the intensity of the ionizing source.

In this subsection we compute the surface brightness of extended Lyα emission, produced via resonant scattering of a central source of Lyα photons (i.e., the quasar) by the neutral gas in the CGM. By analogy with the optically thick fluorescence, the scattering surface brightness can be written as

$$\begin{equation} {\rm SB}_{{\rm Ly}\alpha }=\frac{h\nu _{{\rm Ly}\alpha }}{4\pi (1+z)^4}f_{\rm C}\Phi _{{\rm Ly}\alpha }(R/ \sqrt{3}) \end{equation} \tag{ 16 }$$

where f_C now represents the covering factor of gas which is optically thick in the Lyα transition (i.e., $N_{\rm H\,\scriptsize{I}} \gtrsim 10^{14}\,{\rm cm^{-2}}$). This will in general be much higher than the covering factor of clouds optically thick at the Lyman limit considered in Section 2.2.2. The quantity Φ_Lyα is the flux of ionizing photons emitted close enough to the Lyα resonance to be scattered by gas in motion around the quasar

$$\begin{equation} \Phi _{{\rm Ly}\alpha } \approx \frac{L_{\nu _{{\rm Ly}\alpha }}}{4\pi r^2h}\frac{\Delta \nu }{\nu _{{\rm Ly}\alpha }} = \frac{L_{\nu _{{\rm Ly}\alpha }}}{4\pi r^2h}\frac{W_{{\rm Ly}\alpha }}{\lambda _{{\rm Ly}\alpha }}, \end{equation} \tag{ 17 }$$

where we have introduced the rest-frame absorption EW W_Lyα, and implicitly assumed that it is a very weak function of radius, as we demonstrate next.

The volume averaged neutral column density through our gas distribution is $\langle N_{\rm H\,\scriptsize{I}}\rangle = f_{\rm C}N_{\rm H}\langle x_{\rm H\,\scriptsize{I}}\rangle$, where $\langle x_{\rm H\,\scriptsize{I}}\rangle$ is the volume averaged neutral fraction determined from photoionization equilibrium (Equation (8)). These relations imply an average Lyα optical depth

$$\begin{eqnarray} \langle \tau _{{\rm Ly}\alpha }\rangle &=& 253\left(\frac{N_{\rm H}}{10^{20}\,{\rm cm^{-2}}}\right)\!\left(\frac{n_{\rm H}}{0.1\,{\rm cm^{-3}}}\right)\!\left(\frac{b}{50\,{\rm km\, s}^{-1}}\right)^{-1}\nonumber \\ &&\times \left(\frac{f_{\rm C}}{1.0}\right)\!\!\left(\frac{R}{100\,{\rm kpc}}\right)^{2}\!\!\left(\frac{L_{\nu _{\rm LL}}}{10^{30}\,{\rm erg\,s^{-1}\,Hz^{-1}}}\right)^{-1} \end{eqnarray} \tag{ 18 }$$

where we have introduced the b-value describing turbulent and thermal motions of the absorbing gas. On the flat portion of the curve of growth 1 ≲ τ_Lyα ≲ 10⁴, the corresponding EW is (see, e.g., Draine 2011)

$$\begin{equation} W_{{\rm Ly}\alpha } \approx 0.99 \left(\frac{b}{50\,{\rm km\, s}^{-1}}\right)\left(\frac{\sqrt{\ln \left(\langle \tau _{{\rm Ly}\alpha }\rangle / \ln 2\right)}}{2.1}\right){\mathring{\rm{A}}}, \end{equation} \tag{ 19 }$$

so that EWs of W_Lyα ≃ 1 Å are expected. We can finally write

$$\begin{eqnarray} {\rm SB}_{{\rm Ly}\alpha } &=& 1.2\times 10^{-18}\left(\frac{1+z}{3.0}\right)^{-4}\!\! \left(\frac{f_{\rm C}}{1.0}\right)\!\!\left(\frac{R}{100\,{\rm kpc}}\right)^{-2}\nonumber \\ &&\times \left(\frac{W_{{\rm Ly}\alpha }}{1.0\,{\mathring{\rm{A}}}}\right) \left(\frac{L_{\nu _{{\rm Ly}\alpha }}}{10^{31}\,{\rm erg\,s^{-1}\,Hz^{-1}}}\right) \nonumber\\ && \times {\rm \,erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}} , \end{eqnarray} \tag{ 20 }$$

where it is again understood that f_C < 1 in Equation (20).

Because EW is such a weak function of optical depth on the saturated part of the curve of growth (Equation (19)), W_Lyα is approximately independent of the ionizing luminosity of the quasar as well as the details of the gas distribution. Thus the scattering surface brightness scales as SB_Lyα∝f_CΦ_Lyα, completely analogous to optically thick recombinations for which SB_Lyα∝f_CΦ (Equation (15)). However, scattering will typically be about ∼100 times fainter than optically thick recombinations if the covering factors are the same because $\eta _{\rm thick}\Phi / \Phi _{{\rm Ly}\alpha }\sim 100$, which results from the assumption of W_Lyα ≃ 1.0 Å, and from the relative amount of ionizing photons to Lyα photons emitted by a typical quasar. One expects this scattering emission to be important, therefore, only in regimes where the gas is optically thin at the Lyman limit. Indeed, the surface brightness of scattering (Equation (20)) and optically thin fluorescence (Equation (10)) can be comparable, but have very different dependencies on gas distribution and quasar luminosity. Throughout this work we estimate $L_{\nu _{{\rm Ly}\alpha }}$ by tying the composite quasar spectrum of (Vanden Berk et al. 2001) to the SDSS optical photometry (covering rest-frame UV wavelengths) and computing the peak of the Lyα emission line.

There are two important points that can be made about the surface brightness of extended Lyα emission, and its implications for the supply of cool gas. First, optically thin and optically thick fluorescence can result in comparable levels of emission, as can pure resonant scattering of a central source of Lyα photons. To complicate matters further, cooling radiation, that is collisional excitation of Lyα, can also lead to comparable emission. In general, the only way to determine which mechanism is at work is to use diagnostics from additional emission lines. For instance, the detection of strong extended Balmer line emission would be provide compelling evidence for recombinations, and effectively rule out scattering and cooling. But given that Balmer lines are redshifted into the near-IR for z > 2, detecting even Hα would be a daunting observational task because of the low expected surface brightness. Furthermore, even if one is certain that recombination is the physical mechanism, it is still typically impossible to ascertain the ionization state of the gas, and hence whether it is optically thin or thick.⁵

The second and related point is that measurements of the Lyα surface brightness alone cannot be used to determine the cool gas mass. For recombinations, the surface brightness scales as SB_Lyα∝f_Cn_HN_H if the gas is optically thin (Equation (10)), or as SB_Lyα∝Φf_C if it is optically thick (Equation (15)), whereas for scattering the scaling is SB_Lyα∝Φ_Lyαf_C (Equation (20)). The total cool gas mass scales with the different combination M_c∝f_CN_H (Equation (3)). Even if we have perfect knowledge of the mechanism (recombinations versus scattering) and the ionization state (optically thin or optically thick), the cool gas mass is not uniquely specified by the Lyα emission alone. Note that the situation is even worse for Lyα cooling radiation because of an exponential dependence on an unknown gas temperature (J. F. Hennawi & J. X. Prochaska 2013, in preparation). Thus even in the best-case scenario where Balmer line observations point to recombinations, one will not know whether the gas is optically thin or thick, and hence which dependence on physical parameters to use.

The following worked example illustrates the aforementioned issues. Consider a relatively bright z = 2 quasar with apparent magnitude of i = 17.0. Using the procedure in Appendix A of Hennawi et al. (2006a), we determine the specific luminosity to be, $\log _{10} L_{\nu _{\rm LL}}= 31.3$ at the Lyman limit, and $\log _{10} L_{\nu _{{\rm Ly}\alpha }}= 32.0$ in Lyα. Now consider three distinct cool gas distributions, one optically thin to ionizing photons with parameters (R, n_H, N_H, f_C) = (100 kpc, 0.15 cm⁻³, 10²¹ cm⁻², 1.0), another optically thick to ionizing photons with (R, n_H, N_H, f_C) = (100 kpc, 100 cm⁻³, 10²⁰ cm⁻², 0.01), and a third optically thin to ionizing photons but optically thick to Lyα photons with (R, n_H, N_H, f_C) = (100 kpc, 30 cm⁻³, 10¹⁷ cm⁻², 2.0). These very different cool gas distributions result in nearly identical average Lyα surface brightness SB_Lyα ≈ 1.1–1.2 × 10⁻¹⁷ erg s⁻¹ cm⁻² arcsec⁻² over an R = 100 kpc halo. For the first two models the emission is powered by photoionization, whereas for the third it is pure Lyα scattering. But note also that for the first distribution, Lyα scattering contributes at very nearly the same surface brightness level ∼1.1 × 10⁻¹⁷ erg s⁻¹ cm⁻² arcsec⁻² as optically thin photoionization, implying that disentangling the two mechanisms via say detection of Balmer lines would be challenging. These sizes and surface brightness levels are in line with the brightest Lyα blobs surveyed by Matsuda et al. (2004, 2011) (rescaled to z = 2) as well as the extended Lyα nebulae detected around z ∼ 2 HzRGs (e.g., Villar-Martín et al. 2007) and radio-loud quasars (e.g., Heckman et al. 1991b). For the first optically thin model the total cool gas mass is M_c = 3.3 × 10¹¹ M_☉, for the second optically thick model it is M_c = 2.5 × 10⁸ M_☉, and for the Lyα scattering case it is M_c = 6.6 × 10⁷ M_☉. Thus variations of ∼5000 in the total gas mass can all lead to comparable levels of extended Lyα emission, and even if one were sure that photoionization was powering the emission (from Balmer lines) and/or detected the source of ionizing photons (an unobscured quasar), there is no way to determine whether the gas is optically thin or thick. Plugging the observed surface brightness SB_Lyα, distance R, and ionizing luminosity $L_{\nu _{\rm LL}}$ into Equation (11) for the area averaged neutral column gives $\langle N_{\rm H\,\scriptsize{I}}\rangle = 1.3{\; \times \; 10^{16}}\,{\rm cm}^{-2}$, and thus we cannot distinguish an optically thin from an optically thick scenario.

In the previous subsection we saw that a variety of physical processes and a wide range of gas distributions can all lead to comparable Lyα emission. Hence, the great value of quasar pairs is that absorption-line observations of b/g quasars can provide crucial information about the physical properties of the gas in the CGM of the f/g quasar. Indeed, all the physical parameters (R, n_H, N_H, f_C) can be determined from absorption line observations, provided that the covering factor is high enough for the clouds to be occasionally intercepted by b/g sightlines. For example, we map out the covering factor of optically thick absorption as a function of impact parameter (see Figure 2) in Section 3.4 (see also Hennawi et al. 2006a; Hennawi & Prochaska 2007; Prochaska et al. 2013), and find a high covering factor of cool, optically thick gas around all bright quasars at z ∼ 2. In Prochaska & Hennawi (2009), we conducted detailed absorption line modeling of a single projected pair and determined that the column densities of the absorbing clouds are typically N_H ≃ 10²⁰ cm⁻² and volume densities n_H ≃ 0.1–1 cm⁻³, although it is unclear whether this single system is representative. Armed with this knowledge of the gas distribution, one can then interpret the physical implications of the presence or absence of extended line emission.

Note however that the properties of gas with a very low covering factor f_C ∼ 0.01 are much more challenging to measure, since thousands of background sources would be required to map it out. Thus a crucial caveat in comparing absorption line observations to emission line observations is that the gas responsible for the detected emission may arise from clouds with a much lower covering factor than the gas dominating the absorption. Nevertheless, one can still make quantitative statements about the expected emission from the component of gas which is detected in absorption, which will have the highest covering factor. This is the approach adopted in this work.

But even if quasar pairs are useful for measuring the properties of the quasar CGM, why use quasar pairs, as opposed to individual isolated quasars, to searching for diffuse emission lines? Consider the following four scenarios which are illustrated schematically in Figure 1. (1) The absorbers/emitters are large resolved clouds at a “random” location in the quasar CGM (Figure 1(a)). In this case, the background sightlines pinpoint the locations of gas and thereby determine where to look for emission. This approach is akin to that taken by Adelberger et al. (2006), who purported to detect quasar fluorescence from an optically thick absorber detected in a b/g sightline. For this case of large resolved absorbers, the relevant SB estimate is the mirror approximation (see Equation (12)). (2) Another possibility is that the clouds are small and hence unresolved by our instrument, but that the gas resides in large resolved filamentary complexes which cover a fraction of the quasar halo (Figure 1(b)). For this case, the b/g sightline intersects a filament by construction, and identifies the location where emission is expected via longslit observations of the quasar pair. Note that the average SBs computed in Section 2 would underestimate the true SB in this case, since they include a reduction by the covering factor f_C, due to regions devoid of gas. (3) It could also be that the clouds are small and hence unresolved, but some quasars have cool gas complexes and others do not. This bimodality in gas supply could result in the “hits” and “misses” which give rise to the observed covering factor (Figure 1(c)). In this case, the quasar pairs are helpful since the background sightline identifies which objects to search for emission around, and the average SBs that we computed in Section 2 are again underestimated because the covering factor includes objects with no gas. (4) Finally, the clouds could be small and hence unresolved, and distributed rather uniformly in the quasar halo (Figure 1(d)). In this case the background sightline does not provide us with useful information about where to search for emission. The average SB appropriately accounts for the dilution by the covering factor f_C, and this is the SB that we expect to detect at any spatial position along the slit. For this case, a search for diffuse emission around isolated individual quasars, i.e., not residing in pairs, would be just as effective.

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Modern spectroscopic surveys select against close pairs of quasars because of fiber collisions. For the SDSS, the finite size of optical fibers precludes discovery of pairs with separation <55″.⁷

Thus, to find pairs with sub-arcminute separations, additional follow-up spectroscopy is required both to spectroscopically confirm quasar pair candidates, and to obtain spectra of sufficient quality to search for absorption line systems.

In an ongoing survey, close quasar pair candidates are selected from a photometric quasar catalog (Bovy et al. 2011, 2012), and are confirmed via spectroscopy on 4 m class telescopes including the 3.5 m telescope at Apache Point Observatory (APO), the Mayall 4m telescope at Kitt Peak National Observatory (KPNO), the Multiple Mirror 6.5 m Telescope, and the Calar Alto Observatory (CAHA) 3.5 m telescope. Our continuing effort to discover quasar pairs is described in Hennawi (2004), Hennawi et al. (2006b), and Hennawi et al. (2010). To date about 350 pairs of quasars have been uncovered with impact parameter R_⊥ < 300 kpc and z_fg > 1.6.⁸ Projected pair sightlines were then observed with Keck and Gemini to obtain “science quality” high signal-to-noise ratio (S/N) moderate resolution spectra. A subset of higher resolution spectra at echellette and echelle resolution have also been obtained, but these are not used in this work. This observational program has several science goals: measure the small-scale quasar clustering of quasars (Hennawi et al. 2006b; Myers et al. 2008; Hennawi et al. 2010; Shen et al. 2010), to measure small scale transverse Lyα forest correlations (A. Rorai et al., in preparation), to characterize the transverse proximity effect (J. F. Hennawi et al. 2013a, in preparation), and to use the b/g sightline to characterize the CGM of the f/g quasar (Hennawi et al. 2006a; Hennawi & Prochaska 2007; Prochaska & Hennawi 2009; Prochaska et al. 2013), which is relevant to the goals of this paper.

High S/N, moderate resolution slit spectra of ∼100 quasars were obtained with Keck and Gemini in observing runs spanning from 2004 until 2008. All quasars observed have z > 1.6, which is the lower limit for detecting Lyα set by the atmospheric cutoff. About two-thirds of the targeted pairs consist of projected pairs of quasars (Δv > 2500 km s⁻¹) at different redshifts; the rest were physically associated binary quasars.

For the Keck observations, we used the Low Resolution Imaging Spectrograph (LRIS; Oke et al. 1995), either in longslit mode targeting only the two quasars in the pair, or in multi-slit mode with custom designed slitmasks, which allowed placement of slits on other known quasars or quasar candidates in the field. Typically the slitmask or slit was rotated such that both quasars in the close pair could be observed on the same slit. LRIS is a double spectrograph with two arms giving simultaneous coverage of the near-UV and red. We used the D460 dichroic with the 1200 lines mm⁻¹ grism blazed at 3400 Å on the blue side, resulting in wavelength coverage of ≈3300–4200 Å. The dispersion of this grism is 0.50 Å pixel⁻¹ and our 1″ slits give a resolution of FWHM ≃ 150 km s⁻¹. These data provide the coverage of Lyα at z_fg for all of our pairs. On the red side we typically used the R600/7500 or R600/10000 gratings with a tilt chosen to cover the Mg ii emission line at the f/g quasar redshift, useful for determining accurate systemic redshifts of the quasars. Occasionally the R1200/5000 grating was also used to give additional bluer wavelength coverage. The higher dispersion, better sensitivity, and extended coverage in the red provided high S/N spectra of the Mg ii emission line and also enabled a more sensitive search for metal-line absorption in the b/g quasar (see J. X. Prochaska et al. 2013, in preparation). Some of our older data also used the lower-resolution 300/5000 grating on the red-side covering the wavelength range 4700–10, 000 Å. About half of our LRIS observations were taken after the atmospheric dispersion corrector was installed, which reduced slit-losses (for point sources) in the UV.

The Gemini data were taken with the Gemini Multi-Object Spectrograph (GMOS; Hook et al. 2004) on the Gemini North facility. We used the B$1200\_G5301$ grating which has 1200 lines mm⁻¹ and is blazed at 5300 Å. The detector was binned in the spectral direction resulting in a pixel size of 0.47 Å, and the 1″ slit corresponds to a FWHM ≃ 125 km s⁻¹. The slit was rotated so that both quasars in a pair could be observed simultaneously. The wavelength center depended on the redshift of the quasar pair being observed. But most of the targets considered here are at z ∼ 2.2–2.5, so the grating was typically centered at 4500 Å, giving coverage from 3750 to 5225 Å. The Gemini CCD has two gaps in the spectral direction, corresponding to 9 Å at our resolution. The wavelength center was thus dithered by 15–50 Å between exposures to obtain full wavelength coverage in the gaps. The Gemini observations were conducted over three classical runs during UT 2004 April 21–23, UV 2004 November 16–18, and UT 2005 March 13–16 (Program IDs: GN-2004A-C-5, GN-2004B-C-4, GN-2005A-C-9, GN-2005A-DD-4).

Total exposure times ranged from 600 to 21800 s, for the Keck and Gemini observations, depending on the magnitudes of the targets. This was typically broken up into several individual exposures with exposure times of 600–1800 s depending on the total planned exposure time. The motivation for the shorter (total) integrations was to conduct a survey to build up a statistical sample of b/g absorption line spectra at small impact parameter from the f/g quasars. The goal of the longer integrations was to obtain high-quality spectra of a subset of quasar pairs and to search for low SB Lyα emission near the f/g quasar. The S/N in Lyα forest region of the b/g quasar varies considerably, but it is almost always S/N >5 per pixel for the data we consider here.

In QPQ5 and QPQ6 (J. X. Prochaska et al. 2013, in preparation) we analyze the statistical properties of the CGM near quasars using our full data set of “science quality” projected-pair spectra, which combines spectra from the SDSS (Abazajian et al. 2005) and BOSS (Ahn et al. 2012) spectroscopic surveys with higher quality data from 8 m class telescopes. Here we consider a subset of this full data set constituting 68 sightlines, and survey the spectra for optically thick absorption coincident with the f/g quasar redshift. We first describe the additional criteria applied to the full data set to arrive at the 68 sightline parent sample, and then summarize the procedure used in Prochaska et al. (2013) to identify optically thick absorbers coincident with the f/g quasars.

The present study concerns a sensitive search for extended Lyα emission, which may coincide with the location of an optically thick absorber detected in a neighboring background sightline. To this end, we restrict our parent sample to only the subset of quasars which have an implied SB_Lyα > 5 × 10⁻¹⁸ erg s⁻¹ cm⁻² arcsec⁻² given by the mirror approximation in Equation (12), where the radius R is set to be the impact parameter R_⊥ of the projected quasar pair. This surface brightness limit is chosen because it approximately matches the median 2σ surface brightness limit that our longslit spectroscopic observations achieve (see the next section for detailed discussion of surface brightness limits).

The only other criteria imposed are that we must have obtained a spectroscopic observation of the quasar pair with Gemini or Keck covering Lyα at the f/g quasar redshift, and we require that the velocity difference between the f/g and b/g quasar be >2000 km s⁻¹ to ensure that the two quasars are not physically associated. This minimum velocity difference is many times larger than both the expected line-of-sight velocity dispersion in a z ∼ 2 quasar environments ∼300 km s⁻¹, and the typical redshift errors ∼270–660 km s⁻¹ (see below). We further culled the sample by excluding pairs whose b/g quasar exhibits strong broad absorption lines (BALs), as evidenced by large C iv or Lyα EWs. Mild BALs were also excluded if the BAL absorption clearly coincided with the velocity window about Lyα at the f/g quasar redshift.

The remaining quasar pairs were visually inspected and searched for significant Lyα absorption within a velocity window of |Δv| = 1500 km s⁻¹ about the f/g quasar redshift. This velocity window is chosen to accommodate physically associated absorbers with extreme kinematics ≃ 1000 km s⁻¹, as well as to bracket uncertainties in the f/g quasar systemic redshifts. Quasar redshifts determined from the rest-frame ultraviolet emission lines (redshifted into the optical at z ∼ 2) can differ by up to 1000 km s⁻¹ from the systemic frame, because of outflowing/inflowing material in the broad line regions of quasars (Gaskell 1982; Tytler & Fan 1992; Vanden Berk et al. 2001; Richards et al. 2002b; Shen et al. 2007). Systemic redshifts are estimated by combining the line-centering procedure used in QPQ1 with the recipe in Shen et al. (2007) for combining measurements from different emission lines. The resulting typical 1σ redshift uncertainties using this technique are in the range σ_z ≃ 270–770 km s⁻¹ depending on which emission lines are used. This analysis considered all available spectra on the f/g quasar including the public SDSS and BOSS data sets. We refer the reader to QPQ6 for further details.

The Lyα transition saturates at $N_{\rm H\,\scriptsize{I}} \simeq 10^{14}\,{\rm cm}^{-2}$, and between column densities of $N_{\rm H\,\scriptsize{I}} \simeq 10^{14}{--}10^{19}\,{\rm cm}^{-2}$ the curve of growth is flat, and W_Lyα is a poor proxy for hydrogen column density. In principle, additional leverage may be provided by simultaneously fitting higher order Lyman transitions, or the detection of Lyman limit absorption, which requires $N_{\rm H\,\scriptsize{I}} > 10^{16.5}\,{\rm cm}^{-2}$. However, nearly all of our sightlines are at redshifts z < 2.6, for which the Lyman limit is below the atmospheric cutoff,⁹ and so constructing a complete and fully certifiable sample of optically thick absorbers with $N_{\rm H\,\scriptsize{I}} > 10^{17.2}\,{\rm cm}^{-2}$ is extremely challenging given our wavelength coverage. These challenges are exacerbated by noise in the spectra and the line blending which inevitably occurs at moderate resolution (FWHM ≃ 150 km s⁻¹).

We identified the strongest absorption line in the ±1500 km s⁻¹ velocity window about the f/g quasar redshift, and measured the Lyα EW W_Lyα. All systems with W_Lyα > 1 Å were flagged for Voigt profile fitting of the Lyα transition, to estimate the H i column density. For reference, a Lyα equivalent threshold of W_Lyα > 1.5 Å (W_Lyα > 2 Å) corresponds to column densities of roughly $N_{\rm H\,\scriptsize{I}}\gtrsim 10^{17}\,{\rm cm}^{-2}$ ($N_{\rm H\,\scriptsize{I}}\gtrsim 10^{19}\,{\rm cm}^{-2}$) if the absorption is dominated by a single component; but this need not be the case.

The H i analysis was complemented by a search for metal-lines at the f/g quasar redshift in the clean continuum region redward of the Lyα forest of the b/g quasar. The narrow metal lines provide a more accurate redshift for the absorption line system (than obtainable from Lyα) and, if present, they can help distinguish optically thick absorbers from blended Lyα forest lines. We focused on the strongest low-ion transitions commonly observed in DLAs (e.g., Prochaska et al. 2001): Si ii λ1260, 1304, 1526, O i λ1302, C ii λ1334, Al ii λ1670, Fe ii λ1608, 2382, 2600, Mg ii λ2796, 2803; and the strong high-ionization transitions commonly seen in LLSs: C iv λ1548, 1550 and Si iv λ1393, 1402. Any systems with secure metal-line absorption were also flagged for Voigt profile fitting. Again, this analysis occasionally benefited from the public data sets of SDSS and BOSS whose broad wavelength coverage complements the more limited coverage of our large-aperture spectroscopy.

Given the EW for Lyα absorption, our Voigt profile fits for the $N_{\rm H\,\scriptsize{I}}$, and the presence/absence of low-ion metal absorption, objects were classified into three categories: optically thick, ambiguous, or optically thin. Objects which show obvious damping wings or strong (EW >0.3 Å) low-ion metal absorption are classified as optically thick. For cases for which metal lines are weak, not covered by our spectral coverage, or significantly blended with the Lyα forest of the b/g quasar, an object is classified as optically thick only if it has W_Lyα ⩾1.7 Å. Table 1 lists relevant quantities for the final sample of 29 objects classified as optically thick, around which we search for extended Lyα emission.

Table 1. Properties of Projected Quasar Pairs Sample Surveyed for Diffuse Lyα Emission

Quasar Pair	B/G Quasar	F/G Quasar	zbga	zfgb	$\sigma _{z_{\rm fg}}$c	zabsd	δve	Radiof	θg	R⊥h	$\log _{10}L_{\nu _{\rm LL}}$i	${\rm SB_{\rm mir}}$j	SB1σk	texpl	Inst.m	WLyαn	$\log _{10}\!N_{\rm H\,\scriptsize{I}}$o
SDSS J0203−0002	J020341.34−000235.1	J020340.40−000201.8	1.984	1.735	272	1.7335	−219	<0.98	36.1	316	30.4	1.31	0.75	1200	LRIS	$\phantom{0}2.6\pm 0.2$	18.9 ± 0.3
SDSS J0239−0106	J023946.45−010644.2	J023946.43−010640.5	3.129	2.292	272	2.3024	988	<0.98	3.7	32	30.1	36.14	0.14	7200	LRIS	10.3 ± 0.3	20.3 ± 0.2
SDSS J0250−0047	J025039.82−004749.6	J025038.68−004739.2	2.447	1.851	662	1.8547	429	<0.93	20.0	175	31.0	14.97	0.21	2400	LRIS	$\phantom{0}5.9\pm 0.2$	19.7 ± 0.4
SDSS J0750+2724	J075008.27+272404.5	J075009.25+272405.2	1.802	1.771	272	1.7691	−248	<0.99	13.1	115	30.0	4.46	0.28	2400	LRIS	$\phantom{0}2.3\pm 0.2$	<18.9
SDSS J0752+4011	J075259.14+401118.2	J075259.81+401128.2	2.121	1.883	272	1.8830	−21	<0.98	12.6	110	29.9	2.87	0.28	1200	LRIS	$\phantom{0}3.8\pm 0.2$	18.8 ± 0.3
SDSS J0800+3542	J080048.74+354231.3	J080049.89+354249.6	2.066	1.981	272	1.9824	179	<0.97	23.1	201	30.6	3.82	0.31	3139	LRIS	$\phantom{0}2.0\pm 0.1$	18.7 ± 0.4
SDSS J0814+3250	J081419.58+325018.7	J081420.37+325016.1	2.213	2.173	792	2.1790	574	<0.92	10.3	88	30.3	9.06	0.32	12,600	GMOS	$\phantom{0}2.3\pm 0.2$	18.8 ± 0.3
SDSS J0837+3837	J083757.13+383722.4	J083757.91+383727.1	2.251	2.059	272	2.0612	199	8.61	10.3	89	30.5	14.60	0.39	4530	LRIS	$\phantom{0}2.5\pm 0.1$	18.6 ± 0.3
SDSS J0841+3921	J084159.26+392140.0	J084158.47+392121.0	2.214	2.041	272	2.0475	654	<1.02	21.1	183	30.4	2.59	0.23	1800	LRIS	$\phantom{0}3.9\pm 0.2$	19.2 ± 0.2
SDSS J0856+1158	J085656.05+115802.7	J085655.75+115802.0	2.483	1.767	272	1.7641	−299	10.89	4.5	40	30.6	153.06	0.15	4800	LRIS	16.0 ± 0.2	21.0 ± 0.2
SDSS J0924+3929	J092416.72+392914.6	J092417.65+392920.3	2.080	1.885	272	1.8868	170	<0.97	12.2	106	30.3	7.42	0.37	1800	LRIS	$\phantom{0}1.5\pm 0.1$	<18.8
SDSS J0938+5317	J093804.84+531743.1	J093804.22+531743.9	2.323	2.063	272	2.0675	412	<0.93	5.6	49	29.9	12.72	0.29	1800	LRIS	$\phantom{0}1.6\pm 0.1$	<18.8
SDSS J1006+4804	J100627.10+480429.9	J100627.47+480420.0	2.591	2.297	272	2.3015	367	<0.94	10.6	90	30.0	3.48	0.23	1800	LRIS	$\phantom{0}4.1\pm 0.2$	19.1 ± 0.2
SDSS J1025+5820	J102554.77+582017.0	J102553.47+582012.0	2.567	1.956	662	1.9534	−287	<0.88	11.4	99	30.0	4.25	0.37	1500	LRIS	$\phantom{0}2.4\pm 0.2$	19.1 ± 0.3
SDSS J1041+5630	J104129.27+563023.5	J104121.90+563001.3	2.264	2.054	272	2.0589	468	<1.03	65.0	562	31.0	1.08	0.38	900	LRIS	$\phantom{0}1.9\pm 0.1$	<18.8
SDSS J1045+4351	J104506.39+435115.3	J104508.88+435118.2	3.011	2.428	272	2.4366	726	<0.89	27.1	229	30.4	1.08	0.36	1460	LRIS	$\phantom{0}1.7\pm 0.1$	<18.8
SDSS J1145+0322	J114546.22+032251.9	J114546.54+032236.7	2.011	1.765	272	1.7644	−86	<1.02	15.9	139	29.9	2.40	0.62	800	LRIS	$\phantom{0}1.5\pm 0.2$	<18.9
SDSS J1204+0221b	J120416.69+022111.0	J120417.47+022104.7	2.528	2.436	44	2.4403	375	<0.94	13.3	112	30.2	2.72	0.18	7321	GMOS	$\phantom{0}4.9\pm 0.2$	19.6 ± 0.2
SDSS J1306+6158	J130603.55+615835.2	J130605.19+615823.7	2.175	2.109	272	2.1074	−136	—	16.3	141	30.2	2.82	0.24	3600	LRIS	10.7 ± 0.2	20.2 ± 0.2
SDSS J1358+2738	J135849.54+273756.9	J135849.71+273806.9	2.127	1.900	272	1.8930	−684	<0.92	10.2	89	29.9	4.65	0.31	2400	LRIS	$\phantom{0}2.1\pm 0.1$	<18.8
SDSS J1413+2715	J141337.96+271511.0	J141337.18+271517.1	1.965	1.689	272	1.6885	−94	<0.91	12.0	105	29.6	2.06	0.94	1860	LRIS	$\phantom{0}1.7\pm 0.1$	<18.8
SDSS J1420+1603	J142054.92+160342.9	J142054.42+160333.3	2.057	2.020	272	2.0224	273	<0.96	12.0	104	30.6	13.99	0.58	1050	LRIS	$\phantom{0}5.1\pm 0.2$	<19.5
SDSS J1427−0121	J142758.74−012136.2	J142758.89−012130.4	2.353	2.281	272	2.2821	100	30.63	6.2	53	30.6	42.30	0.14	21,800	GMOS	$\phantom{0}2.8\pm 0.1$	<18.8
SDSS J1431+5727	J143109.22+572726.4	J143109.67+572729.0	2.063	1.679	272	1.6835	499	<0.95	4.5	39	29.6	17.64	0.35	2400	LRIS	$\phantom{0}2.1\pm 0.2$	19.0 ± 0.3
SDSS J1442+0137	J144231.91+013734.8	J144232.92+013730.4	2.272	1.807	272	1.8068	8	<0.97	15.7	137	30.3	5.87	0.30	1800	LRIS	$\phantom{0}3.2\pm 0.2$	19.1 ± 0.3
SDSS J1444+3113	J144427.96+311314.0	J144429.40+311321.2	1.795	1.734	272	1.7370	294	<0.79	19.8	173	31.2	28.64	1.60	600	LRIS	$\phantom{0}1.5\pm 0.1$	<18.8
SDSS J1508+3635	J150812.80+363530.3	J150814.06+363529.4	2.105	1.848	272	1.8514	347	<1.00	15.2	133	30.7	14.45	0.34	1800	LRIS	$\phantom{0}2.0\pm 0.1$	<18.8
SDSS J1553+1921	J155325.89+192137.7	J155325.61+192141.0	2.098	2.016	662	2.0130	−326	<0.97	5.1	44	29.7	9.92	0.27	1605	LRIS	$\phantom{0}7.7\pm 0.2$	20.1 ± 0.2
SDSS J1605+5113	J160546.67+511322.9	J160547.61+511330.5	1.844	1.785	662	1.7817	−342	<0.98	11.6	102	30.4	12.24	1.08	600	LRIS	$\phantom{0}1.8\pm 0.2$	<18.8

Notes. ^aRedshifts of b/g quasar. ^bRedshifts of f/g quasar. ^cError on f/g quasar redshift (km s⁻¹). ^dAbsorber redshift. Determined from metal-line transitions when present; otherwise, it corresponds to the approximate centroid of Lyα. ^eVelocity difference between absorber and f/g quasar (km s⁻¹). ^fPeak 20 cm radio flux F_peak (mJy) of f/g quasars which had a match in the source catalog from the VLA/FIRST survey (Becker et al. 1995). For f/g quasars without FIRST matches we list the 5σ upper limit based on the sky-rms coverage maps. One source labeled—was not covered by FIRST. ^gAngular separation between f/g and b/g quasar (arcsec). ^hImpact parameter at f/g quasar redshift. ⁱSpecific luminosity at the Lyman limit (erg s⁻¹ Hz⁻¹). ^jPredicted mirror surface brightness (10⁻¹⁷ erg s⁻¹ cm⁻² arcsec⁻²) according to Equation (12). ^kObserved 1σ surface brightness limit (10⁻¹⁷ erg s⁻¹ cm⁻² arcsec⁻²). ^lExposure time (s). ^mRest-frame Lyα equivalent width (Å). ⁿNeutral hydrogen column density (cm⁻²). ^oFor SDSSJ 1204+0221 we use the f/g quasar redshift and error from near-IR spectroscopy published in Prochaska & Hennawi (2009).

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The false positives and completeness of our optically thick sample are sources of concern. Line blending, in particular, can significantly depress the continuum near the Lyα profile biasing the W_Lyα and the estimated column density high. Based on the analysis of our current sample, and comparisons of objects observed at moderate and echellette/echelle resolution (see, e.g., Hennawi et al. 2006a), we estimate that false positives, that is non optically-thick objects, in our sample is lower than 10%. This low contamination follows from our relatively conservative criteria for defining a system as optically thick. Specifically, there are three cases for which we are less confident in the optically thick designations (see Table 1; J1041+5630, J1045+4351, J1444+3113) but we nevertheless include them in the following analysis. Removing them would make no difference to our conclusions aside from reducing the sample size by 10%. Note that saturated Lyα forest lines with $N_{\rm H\,\scriptsize{I}} < 10^{17.2}\,{\rm cm}^{-2}$ are essentially indistinguishable from optically thick absorbers with low metallicity. As such, we expect the false positive rate to be smaller than our incompleteness, and we thus consider our estimate of the covering factor of optically thick gas (see Figure 2) to be conservative lower limits.

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Finally, given that there is suggestive evidence that radio-loud quasars have brighter Lyα emission nebulae and a higher detection frequency than radio-quiet quasars (Heckman et al. 1991b), and that HzRGs at z ∼ 2 typically exhibit bright (SB_Lyα ∼ 10⁻¹⁶ erg s⁻¹ cm⁻² arcsec⁻²) large-scale (∼100 kpc) Lyα nebulae (e.g., McCarthy et al. 1990; McCarthy 1993; van Ojik et al. 1996; Nesvadba et al. 2006; Binette et al. 2006; Reuland et al. 2007; Villar-Martín et al. 2007; Miley & De Breuck 2008), we consider the radio properties of our f/g quasars. The Faint Images of the Radio Sky at Twenty cm survey (FIRST; Becker et al. 1995) used the Very Large Array (VLA) to produce a map of the 20 cm (1.4 GHz) sky with a beam size of 54 and an rms sensitivity of about 0.15 mJy beam⁻¹. The survey covers the same 10,000 deg² sky region covered by the SDSS imaging, has a typical detection threshold of 1 mJy, and an astrometric accuracy of 005. Following (Ivezić et al. 2002), we match our f/g quasars to the FIRST catalog with a matching radius of 15. From our parent sample of 68 projected pairs, only 59 sources were covered by the FIRST imaging footprint, and of these, 4/59 have matches in the FIRST catalog (F_peak ≳ 1 mJy), which is consistent with roughly ∼10% of quasars being radio-loud (Ivezić et al. 2002). This is to be expected since our f/g quasars are typical SDSS quasars. Visual inspection of the FIRST images indicates that none show evidence of a complex radio morphology (i.e., core-jet, core-lobe, and double-lobe; Magliocchetti et al. 1998; McMahon et al. 2002). Among the 29 f/g quasars with coincident optically thick absorption, 28 sources were covered by the FIRST footprint, and only 3/28 have matches in the FIRST catalog, again consistent with a ∼10% radio loud fraction. The radio fluxes for the three sources detected in the FIRST survey are listed in in Table 1; for the sources which were not detected we list 5σ upper limits based on the FIRST sky-rms coverage maps. Given that only ∼10% of our f/g quasars have radio counterparts ≳ 1 mJy, both in our parent sample and in the sample with coincident absorbers, we conclude that quasars which we study here are predominantly radio-quiet.

The distribution of f/g quasar redshifts and transverse separations for all 68 projected quasar pair sightlines in our parent sample are shown in the scatter plot in Figure 2. The 29 black filled symbols indicate sightlines which have an optically thick absorber within an interval δv = [ − 1500, 1500] km s⁻¹ of the f/g quasar redshift (the median absolute offset of these systems is |δv| = 256 km s⁻¹), which are objects that we search around for extended Lyα emission. Open black circles indicate sightlines classified as ambiguous, and green sightlines which are optically thin. Note that the closest pairs are predominantly at redshift z ∼ 2, because this is where quasar selection, and hence quasar pair selection, is most efficient (see Richards et al. 2002a; Hennawi et al. 2006b; Bovy et al. 2011, 2012).

It is clear from Figure 2 that the covering factor of absorbers is very high for small separations: at least 28 out of 48 sightlines with R_⊥ < 200 kpc have signatures of optically thick gas coincident with the f/g quasar, implying a >58% covering fraction. This high covering factor was previously noted in Hennawi et al. (2006a) based on a much smaller pair sample (11 sightlines with R_perp < 200 kpc) which are a subset of our current parent sample (see also QPQ5). Note again that because of the significant incompleteness discussed in Section 3.3, this high covering factor should be interpreted as a conservative lower limit, albeit subject to Poisson uncertainty.

The distribution of points in Figure 2, or equivalently the dependence of covering factor on impact parameter, can be used to measure the quasar–absorber correlation function, which is equivalently the spatial profile of the absorbing clouds around the f/g quasar. In QPQ2, we analyzed the much smaller sample of pairs from QPQ1 and determined a correlation length of $r_0=9.2^{+1.5}_{-1.7}\, h^{-1}{\rm \ Mpc}$ (comoving) assuming a power law correlation function, ξ∝r^−γ, with γ = 1.6. The high covering factor of absorbers quantified by the correlation function, implies a comparably high incidence of proximate absorbers along the line-of-sight; however, this absorption is not observed in our f/g quasars, or in other isolated quasars. This anisotropic clustering of absorbers around quasars indicates that the transverse direction is less likely to be illuminated by ionizing photons than the line-of-sight, either because of anisotropic emission which could result if the f/g quasar is obscured from some vantage points, or because the quasar emits radiation intermittently, on a timescale comparable to the transverse light-crossing time. We will return to this important inference in the context of the search for Lyα fluorescence in Section 5.

The Gemini/GMOS observations were conducted in longslit mode, whereas the LRIS observations include a mix of longslit and multislit exposures. All data were reduced using the LowRedux pipeline,¹⁰ which is a publicly available collection of custom codes written in the Interactive Data Language (IDL) for reducing slit spectroscopy. Individual exposures are processed using standard techniques, namely they are overscan and bias subtracted and flat fielded. Flat fielding is performed in two steps, using both a pixel flat to correct for pixel-to-pixel sensitivity variations, as well as spectroscopic illumination flats (taken with the sky in twilight) to flatten the larger scale illumination pattern arising from either instrument optics or imperfections in the slits. Cosmic rays and bad pixels are identified and masked in multiple steps. First, a sharpness detection algorithm is run on each image to identify and mask features smaller than the spectral/spatial PSF. Then further downstream in the reduction procedure, outliers from our models of the sky and the object are masked (see below). Wavelength solutions are determined from low order polynomial fits to arc lamp spectra, and then a wavelength map is obtained by tracing the spatial trajectory of arc lines across each slit. This wavelength map allows us to model the sky counts as a function of wavelength and slit position without needing to rectify the original data (e.g., Kelson 2003).

Our method for spectral PSF subtraction employs a novel custom algorithm. We treat sky and PSF subtraction as a coupled problem to obtain a multi-component model of the image counts. This model is composed of a sum of two-dimensional (2D) “basis functions,” which consists of a sky background and a PSF model for each object on the slit. This is typically just the two quasars (f/g and b/g), but in some cases it includes additional objects which landed on the slit in the region of interest. By construction, the sky-background has a flat spatial profile because our slits are flattened by the slit illumination function. For the object models, we first identify objects in an initial sky-subtracted image, and trace their trajectory across the detector. We then extract a 1D spectrum, normalize these sky-subtracted images by the total extracted flux, and fit a B-spline profile to the normalized spatial light profile of each object relative to the position of its trace. This non-parametric object profile has the flexibility to vary as a function of wavelength, and corrections to the initial trace are simultaneously determined and applied. Given this set of 2D basis functions, i.e., the flat sky and the object model profiles, we then minimize chi-squared for the best set of spectral B-spline coefficients (i.e., break points are spaced in wavelength) which are the spectral amplitudes of each basis component of the 2D model. In other words

$$\begin{equation} \chi ^2 = \sum _{i}^{N_{\rm pix}} \frac{\left({\rm DATA}_i - {\rm MODEL}_i\right)^2}{\sigma _i^2} \end{equation} \tag{ 21 }$$

where the sum is taken over all N_pix pixels in the image, “DATA” is the image, “MODEL” is a linear combination of 2D basis functions multiplied by B-spline spectral amplitudes, and σ is a model of the noise in the spectrum, i.e., σ² = SKY + OBJECTS + READNOISE. The result of this procedure are then full 2D models of the sky-background (SKY), all object spectra (OBJECTS), and the noise (σ²). We then use this model SKY to update the sky-subtraction, the individual object profiles are re-fit and the basis functions updated, and chi-square fitting is repeated. We iterate this procedure of object profile fitting and subsequent chi-squared modeling four times until we arrive at our final models for the sky background, the 2D spectrum of each object, and the noise.

Each exposure of a given target is modeled according to the above procedure which then allows us to subtract the sky and the object profiles from each individual image. These images are registered to a common frame by applying integer pixel shifts (to avoid correlating errors), and are then combined to form final 2D stacked sky-subtracted and sky-and-PSF-subtracted images. The individual 2D frames are optimally weighted by the ${\rm (S/ N)}^2$ of their extracted 1D spectra (using the b/g quasar), and bad pixels identified via sharpness filtering, as outliers in the chi-square fitting, or by sigma-clipping of the image stack, are masked. The final result of our data analysis are three images: (1) an optimally weighted average of (DATA − SKY), henceforth the “stacked sky-subtracted image,” (2) an optimally weighted average of (DATA − SKY − OBJECTS), henceforth the “stacked sky-and-PSF-subtracted image,” and (3) the noise model for these images σ². The final noise map is propagated from the individual noise model images taking into account weighting and pixel masking entirely self-consistently.

Extended Lyα emission will be manifest as residual flux in our 2D sky-and-PSF-subtracted images which is inconsistent with being noise. We thus search for Lyα emission by defining a χ image in analogy with Equation (21) above, but using the stacked images and corresponding propagated noise instead of the individual frames. This procedure allows us to visually assess the statistical significance of any putative emission feature. Figure 5 presents these images for the 29 objects in our optically thick sample (see Table 1). The middle panels show $\chi _{\rm sky+PSF} = ({\rm DATA} - {\rm SKY} - {\rm OBJECTS})/ \sigma$ for the stacked sky-and-PSF-subtracted images. Recall that an analogous χ² has been minimized by our modeling of each image. Thus if our model is an accurate description of the data, the distribution of pixel values in the χ_{sky + PSF} should be a Gaussian with unit variance. The lower panel of each image shows $\chi _{\rm sky}= ({\rm DATA} - {\rm SKY})/ \sigma$; the numerator is the stacked sky-subtracted (but not PSF-subtracted) image. The upper panels show smoothed maps which are helpful for identifying extended emission. Specifically, the smoothed images are given by

$$\begin{equation} \chi _{\rm smth} = \frac{{\rm CONVOL}[{\rm DATA} - {\rm SKY} - {\rm OBJECTS}]}{{\rm CONVOL}[\sigma ]}, \end{equation} \tag{ 22 }$$

where the CONVOL operation denotes smoothing of the stacked images with a symmetric Gaussian kernel (same spatial and spectral widths) with FWHM_smth = 235 km s⁻¹ (dispersion σ_smth = 100 km s⁻¹). For LRIS this FWHM_smth corresponds to 5.2–6.1 pixel, or 1.4–1.5 times the spectral resolution element, and 1.4–16 spatially.¹¹ For GMOS a FWHM_smth = 235 km s⁻¹ corresponds to 6.6–7.0 pixel or 1.4–1.5 times the spectral resolution, or 1.9–20 spatially.

Note that smoothing correlates pixels in the smoothed image, and furthermore the ratio of smoothed image to smoothed noise will no longer obey Gaussian statistics. Nevertheless, χ_smth proves to be a useful tool for identifying extended emission.

The following describes our procedure to flux calibrate our spectra and determine surface brightness limits. Standard star spectra were not typically taken immediately before/after our quasar pair observations. Instead, we construct a model sensitivity function of the Keck LRIS-B spectrograph by first fitting an aggregate of standard star spectra taken at different slit positions, which span the range of wavelength coverage available with a multi-slit setup. A similar procedure was employed for Gemini GMOS using standard star spectra taken at a variety of central wavelengths. We apply this archived sensitivity function to the b/g quasar spectrum, and then integrate the flux-calibrated 1D spectrum against the SDSS u-band filter curve. The sensitivity function is then rescaled to yield the correct SDSS u-band photometry. A comparison of our 1D spectra flux calibrated in this way to the SDSS spectra typically show agreement to within ∼10%–20% in the region of wavelength overlap. Given that SDSS spectra have spectrophotometric errors of ∼10% (Abazajian et al. 2005), that u-band photometric errors (used for our renormalization) are about 7% for a typical source with u ≃ 21, and that quasar variability over ≃ 5 year timescales (between when the SDSS photometry and our spectra were taken) could result in a ∼10% fluctuation (e.g., Schmidt et al. 2010), we consider ∼20% agreement between our spectra and the SDSS spectra to be reasonable.

This procedure of rescaling the sensitivity functions to match photometry is effective for point source flux-calibration, however a subtlety arises in relation to calibrating extended emission. The point source counts will be reduced because of improper object centering due to guiding errors or inaccuracies in slit/mask alignment, and because a fraction of the spatial PSF lies outside the slit area. These point-source slit-losses do not, however, reduce the amount of spatially resolved emission, which is of interest here. Thus, renormalizing to point-source photometry will tend to over-estimate our surface brightness detection limit and underestimate our sensitivity. Hence, our procedure is to apply the rescaled sensitivity functions (based on point source photometry) to our 2D images, but reduce them by a geometric slit-loss factor so that we properly treat extended emission.

To compute the slit-losses we use the measured spatial FWHM to determine the fraction of light going through our 10 slits, but we do not model centering errors. A useful check on our flux calibration procedure is to examine any residual variation in our renormalized sensitivity functions after the effects of the atmospheric extinction (i.e., due to airmass) and slit-losses are removed. These variations in the spectroscopic zero-point could be due either to transparency variations or to other systematic flux calibration errors. We find a 35% relative variation (1σ) about the mean zero-point, which is at a level that can be plausibly explained by transparency variations in clear but non-photometric conditions. If these variations in zero-point are indeed due to transparency fluctuations, then our procedure of renormalizing to the photometry effectively takes them out. But because we have no quantitative information about the transparency during our observations, we conservatively assume a relative error of ∼35% on our surface brightness limits.

Application of the wavelength dependent sensitivity function converts our stacked sky-and-PSF-subtracted images from units of electrons into erg s⁻¹ cm⁻²  Å⁻¹. We then tile these calibrated images with a uniform grid of aperture centers within ±3000 km s⁻¹ of the f/g quasar's redshifted Lyα line. Regions within ±70 of the f/g quasar trace on either side are excluded to avoid potential contamination from Lyα emission. Regions within ±3000 km s⁻¹ of the b/g quasar's Lyα and also within ±70 of the b/g quasar trace are also excluded, to similarly avoid Lyα contamination (this is relevant if the redshift difference of the pair δv < 3000 km s⁻¹). The spacing between these aperture centers is 10 spatially and Δv = 400 km s⁻¹ spectrally. We then compute SB limits in windows of 700 km s⁻¹ × 10 about each aperture center, which corresponds to an aperture of 700 km s⁻¹ × 1.0 arcsec² on the sky because we always used a 10 slit. The 700 km s⁻¹ spectral width is motivated by the observed kinematics of the metal-line absorbers associated with the optically thick gas we detect in absorption (Prochaska & Hennawi 2009; QPQ7, J. X. Prochaska et al. 2013, in preparation). If this gas emits Lyα, we expect the emission kinematics to have a comparable velocity width (in the absence of significant resonant broadening). Because our noise has been propagated self-consistently, we use the noise model for our stacked sky-and-PSF-subtracted image to determine the 1σ surface brightness limit for the ith aperture SB_{1σ, i}, and we take the mean value of $1/ N \sum _i^N {\rm SB}_{1\sigma ,i}$ of all apertures to be the surface brightness limit for the quasar pair SB_1σ, which is quoted in Table 1.

We can assess the accuracy of our noise model by examining the distribution of $\chi _{i} = {\rm SB}_i/ {\rm SB}_{1\sigma ,i}$, where SB_i is the extracted flux of the ith aperture in each 700 km s⁻¹ × 1.0 arcsec² window, and SB_{1σ, i} is the surface brightness limit based on our noise model at this location. In the absence of detectable extended Lyα emission the SB_i should be pure noise, and if our noise model is accurate, the χ_i should thus obey a Gaussian distribution with unit variance. The distribution of χ_i for one of our quasar pairs SDSSJ 0938+5317 is compared to a Gaussian in Figure 3. It is clear that our noise model provides an accurate description of the real noise in the stacked sky-and-PSF subtracted image. Systematic errors in the data reduction and sky and object modeling procedure are not accounted for in the purely statistical error SB_{1σ, i}, and thus we would expect to typically underestimate the true noise level. We can quantify these errors in our noise estimates by computing the standard deviation of χ_i after clipping >5σ outliers (resulting from, e.g., a small number of residual cosmic ray hits or bad pixels that remain unmasked by our data reduction procedure). For the distribution in Figure 3 we obtain σ_chi = 0.97, which indicates that for this case we actually slightly overestimated the noise. We similarly compute σ_chi for each quasar pair in our sample. The median value over all quasar pairs is 1.08, thus on average we tend to underestimate the noise by 8%. Because this is smaller than our estimated flux calibration error, we therefore quote statistical SB limits based only on our noise model as SB_1σ in Table 1.

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Having quantified the formal errors in our sky-and-PSF-subtracted images, we now determine the significance level for a convincing detection. While we quote 1σ SB limits over 700 km s⁻¹ × 1.0 arcsec² apertures, our sensitivity to a given surface brightness is of course set by the size of the source and its spectral characteristics (i.e., kinematics). One can in principle always reach lower SB levels by averaging over larger spatial regions, and it is easier to detect narrow lines over the background. To address these issues we construct synthetic fluorescent sources and add them to the sky-and-PSF-subtracted image for a representative pair SDSSJ 0938+5317. This pair has z_bg = 2.32, z_fg = 2.07, an angular separation of θ = 56 corresponding to R_⊥ = 49 kpc, and was observed with LRIS, as are the bulk of our pairs. The total exposure time was 1800 s, allowing us to reach a surface brightness limit of SB_1σ = 0.29 × 10⁻¹⁷ erg s⁻¹ cm⁻² arcsec⁻² which is close to the median value of our sample SB_1σ = 0.31 × 10⁻¹⁷ erg s⁻¹ cm⁻² arcsec⁻² (Table 1). Figure 3 testifies to the accuracy of our noise model for this object, and illustrates that the surface brightness noise fluctuations obey Gaussian statistics. Our spectral resolution for these observations is FWHM =153 km s⁻¹ and the seeing measured from the quasar profiles is 079 corresponding to a spatial resolution of 7.0 kpc.

We construct synthetic sources with a uniform spatial surface brightness distribution, where the size is specified by a single dimension along the slit d_fake. Spectrally, we assume a Gaussian velocity profile specified by FWHM_fake. The amplitude of the synthetic sources is set such that their total surface brightness (i.e., integrated out spectrally) is equal to SB = [1, 2, 3, 4, 5, 10] × SB_1σ, which corresponds to SB_Lyα = [0.29, 0.57, 0.86, 1.1, 1.4, 2.9] × 10⁻¹⁷  erg s⁻¹ cm⁻² arcsec⁻² for SDSSJ 0938+5317.

We consider synthetic sources with three different spatial and kinematic distributions which represent our expectations for the sources that might occur near a bright quasar. The first is a large d_fake = 75 kpc source with kinematics FWHM_fake = 500 km s⁻¹, which is both spatially and spectrally resolved. This source represents the kind of emission expected if the quasar CGM is made up of a large population of small spatially unresolved clouds which nevertheless have sufficient covering factor, f_C, to result in detectable Lyα emission (see Figures 1(b)–(d)); we denote this model as “CGM.” In this scenario, the kinematics of the emission correspond to the motions traced by the metal-line absorbers as observed near quasars with b/g sightlines (Prochaska & Hennawi 2009; QPQ7, J. X. Prochaska et al. 2013, in preparation). The second synthetic source is meant to represent a spatially resolved Lyα source (see Figure 1(a)), which is the counterpart to the frequently observed optically thick absorption in b/g quasar sightlines (Figure 2). In this “LLS” scenario, the size is taken to be d_fake = 20 kpc, corresponding to about three resolution elements. The kinematics are assumed to be comparable to the narrow turbulent and/or thermally broadened line widths ∼20 km s⁻¹ characteristic of metal absorption in individual LLS absorption complexes, (i.e., spectrally unresolved Prochter et al. 2010). For the third synthetic source, denoted as “CORE,” we assume a size d_fake of two LRIS pixels corresponding to 4.8 kpc, and cold kinematics ∼20 km s⁻¹, such that it is both spatially and spectrally unresolved. This source is meant to mimic the properties of the candidate fluorescent emitters recently detected by Cantalupo et al. (2012) at ∼ Mpc distances from bright quasars. These fluorescent candidates tend to be compact, are spatially unresolved in ground-based seeing, and are believed to be the dense cores of proto-galaxies. For spatially unresolved sources, the total Lyα luminosity is the relevant quantity rather than the surface brightness, and the corresponding luminosity of our CORE sources are L_Lyα = [0.92, 1.8, 2.8, 3.7, 4.6, 9.2] × 10⁴¹ erg s⁻¹. Note that the median luminosity for the most compelling (i.e., those with W_Lyα > 240 Å) Cantalupo et al. (2012) fluorescent candidates is L_Lyα = 9.1  × 10⁴¹ erg s⁻¹, which is at the level of the 10σ model source we consider. Our model images are then convolved with the spectral and spatial resolution of our data, converted into counts using our sensitivity function, and Poisson realizations of the average counts are added to the real sky-subtracted and sky-and-PSF-subtracted images.

Figure 4 illustrates the detection significance of the synthetic source model profiles which have been added to the real data for SDSSJ 0938+5317. The lower panel shows the χ_sky with synthetic sources added. In this panel only, the synthetic sources are noiseless, and their brightness and relative significance can be directly compared to the real (noisy) 2D spectra of the two quasars shown in the central inset. Quantitatively, a uniform screen of surface brightness at SB_1σ = 0.29 × 10⁻¹⁷ erg s⁻¹ cm⁻² arcsec⁻² emitted at the f/g quasar Lyα with a top-hat velocity distribution matched to our 700 km s⁻¹ spectral extraction aperture would result in 0.44 e⁻ electrons pixel⁻¹, whereas the typical value of the 1σ noise in our stacked images (far from the quasars) is σ ≃ 4 e⁻ electrons pixel⁻¹.

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Inspection of Figure 4 suggests that we should be able to marginally detect “CGM”-like sources down to a level of ≳ 2 × SB_1σ, “LLS”-like sources down to ≳ 3 × SB_1σ, and the spatially and spectrally unresolved “CORE”-like sources to ≳ 10 × SB_1σ. We emphasize that these significance limits are rough guidelines, as the noise level varies with wavelength due to fluctuations in the sky level. For example, the reason why the 4σ sources in Figure 4 look scarcely more significant than the 3σ ones, is that the 4σ sources happen to land on a region where the sky-background is ∼10% higher. Furthermore, in some cases systematics in the data reduction limit our ability to make detections down to low significance levels. These systematics manifest as large-scale correlations in the χ maps, and are particularly conspicuous in the smoothed χ_smth images. In the absence of emission, the χ_{sky + object} image should be a random Poisson noise field because the pixel noise is completely uncorrelated, and thus the characteristic size of structures in the χ_smth should be comparable to the size of the smoothing kernel ≃ 5–6 pixel. Some of our χ maps in Figure 5 show correlated structure and spurious features due to systematics, the worst case being SDSSJ 0814+3250, where a large-scale gradient in the flux level is seen across the spatial direction. This large-scale gradient is likely due to a poorly determined slit-illumination function.

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In this subsection we compare our observational constraints on extended Lyα emission to predictions for recombination radiation from optically thick gas, as laid out in Section 2.2.2. We consider emission from spatially resolved “mirrors” and from an unresolved distribution of clouds in turn.

5.1.1. Fluorescent Emission in the Mirror Approximation

5.1.2. Fluorescent Emission from Unresolved Clouds

5.1.3. Why is Optically Thick Fluorescence Absent?

In the previous section, we argued that the absence of significant Lyα emission from optically thick gas in the f/g quasar CGM indicates that this material is not illuminated by the quasar. However, our longslit observations frequently reveal extended Lyα fuzz at a level of SB ∼ 0.5–5  ×  10⁻¹⁷ erg s⁻¹ cm⁻² arcsec⁻² at distances R_⊥ ≲ 50 kpc. To quantify the frequency of this extended emission, both authors independently visually inspected the maps in Figure 5 to search for compelling extended Lyα emission. Each object in our sample (see Table 1) was classified as having definite fuzz, ambiguous, or having no fuzz. Classifications were merged according to the following rules: (1) an object is classified as definite or no fuzz only if both authors agreed on that classification; (2) all other objects were considered ambiguous. Out of our sample of 29 systems, this yields 10 objects with definite fuzz, eight ambiguous objects, and 11 for which no fuzz is detected. Given that this fuzz tends to be asymmetrically distributed, and that our single-slit orientation covers only a fraction of the area around the quasar, the true detection rate of such fuzz is expected to be significantly higher. Our detection rate is hence about a factor of two lower than the frequency of ∼50%–70% deduced from the more recent studies of this small-scale Lyα fuzz (Christensen et al. 2006; Courbin et al. 2008; Hennawi et al. 2009; North et al. 2012), although limited statistics, sample inhomogeneities, differing methodologies, and our single position angle preclude a quantitative comparison.

We now interpret this small-scale Lyα fuzz in terms of the emission mechanisms discussed in Section 2. Consider our typical f/g quasar at z ≃ 2, which has $\log _{10} L_{\nu _{\rm LL}}\simeq 30.3$. If we assume R_⊥ = 35 kpc as characteristic of the impact parameters at which we detect the fuzz, the average surface brightness for optically thick fluorescence given by Equation (15) is ${\rm SB}_{{\rm Ly}\alpha } = 1.0{\; \times \; 10^{-15}}(f_{\rm C}/ 1.0)\,{\rm erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}$. It is thus quite clear that small-scale fuzz cannot be arising from the same population of clouds giving rise to the high covering factor f_C ≳ 0.50 of optically thick absorbers at R_⊥ < 50 kpc in Figure 2, since the implied surface brightness would be two orders of magnitude higher. To be consistent with optically thick emission, the covering factor of clouds responsible for the small-scale fuzz needs to instead be much lower f_C ≃ 0.005–0.05, and this low covering factor matches the values deduced in previous work on quasar nebulae (e.g., Heckman et al. 1991b).

Considering together both the detected small-scale fuzz emission, which implies a small covering factor f_C ≃ 0.005–0.05, and the lack of large-scale nebulosity from gas with a high f_C ≳ 0.50 covering factor established by b/g sightlines, the picture that emerges is that clouds with a range of densities and covering factors exist in the CGM around quasars. In the absence of ionizing photons, a low density n_H ∼ 0.01–1 cm⁻³ population dominates the covering factor, and it is this gas that we typically observe in absorption with b/g sightlines. In contrast, the denser clouds (n_H ∼ 10–100 cm⁻³) cover only a tiny fraction of the line-of-sight ∼0.01. When illuminated by the f/g quasar, the lower density (high covering factor) clouds are photoionized by the intense quasar radiation, whereas the dense clouds (low covering factor) can self-shield and survive. It is then these dense clouds which are responsible for the small scale Lyα fuzz that we detect in our sample, and which have been previously detected around quasars (e.g., Bunker et al. 2003; Weidinger et al. 2004; Christensen et al. 2006; Francis & McDonnell 2006; Courbin et al. 2008; Hennawi et al. 2009; North et al. 2012). This scenario could also apply to the Lyα blobs and HzRGs. Indeed, it is intriguing that detailed modeling of the spectra of EELRs around quasars and FR II radio galaxies at low redshift z ≲ 0.5, similarly show evidence for two phase media composed of an abundant low density medium (n_H ∼ 1 cm⁻³) and much rarer high density clouds (n_H ∼ 400 cm⁻³ Stockton et al. 2002, 2006; Fu & Stockton 2006, 2007, 2009). It thus seems plausible that the small-scale fuzz that we detect in Lyα are high-redshift analogs of the EELRs detected around low redshift z < 0.5 type-I (e.g., Stockton et al. 2006; Husemann et al. 2013) and type-II (Greene et al. 2011) quasars, in [O iii] and Balmer lines.

While the explanation of small-scale fuzz above is imminently plausible, and has indeed been the standard interpretation for nebulosities around both HzRGs (see, e.g., McCarthy et al. 1990; McCarthy 1993) and quasars (e.g., Heckman et al. 1991b), as emphasized in Section 2.4, the interpretation of Lyα emission alone is degenerate, and optically thin fluorescence and Lyα scattering can also result in comparable emission levels. First consider optically thin fluorescence from a population of clouds with (R, n_H, N_H, f_C) = (35 kpc, 1.0 cm⁻³, 10^20.1 cm⁻², 1.0). This results in a SB_Lyα = 10⁻¹⁷ erg s⁻¹ cm⁻² arcsec⁻² according to Equation (10), and it is rather intriguing that this column density N_H and volume density n_H are comparable to values deduced from our detailed absorption line modeling of SDSSJ 1204+0221 QPQ3, and the covering factor consistent with the large values we deduced in Section 3.4. In other words, if such clouds are not illuminated by the quasar, they could account for the optically thick absorption that we observe in b/g sightlines at R_⊥ < 200 kpc. But if illuminated at a distance R = 35 kpc, such clouds would be so highly ionized that they would appear optically thin in absorption (e.g., along the line-of-sight). Hence, the small scale fuzz could be arising from the illuminated (hence optically thin) analogs of the shadowed (hence optically thick) clouds that we detect in absorption in background sightlines.

But if this interpretation is correct, why do we not see fuzz at larger distances R_⊥ ≃ 100 kpc? Recall that in the optically thin regime, fluorescence does not depend on distance provided that the clouds remain optically thin, and at R_⊥ ≃ 100 kpc this would marginally be the case ($\log N_{\rm H\,\scriptsize{I}} \simeq 17.2$). Thus under this interpretation, there should be extended emission at R_⊥ ≃ 100 kpc which is just as bright as the small-scale fuzz. Such emission is definitely not observed in our data set. As the optically thin surface brightness scales with the combination f_CN_Hn_H, one solution would be to make this product substantially lower at R ∼ 100 kpc compared to R ∼ 35 kpc, so that the optically thin emission would then be too faint to be detected.

Now consider the SB levels from Lyα scattering. A model with (R, n_H, N_H, f_C) = (35 kpc, 0.1 cm⁻³, 10^20.0 cm⁻², 1.0), i.e., nearly identical to the optically thin case considered above, but with an order of magnitude lower volume density n_H, results in scattering SB_Lyα = 0.9 × 10⁻¹⁷ erg s⁻¹ cm⁻² arcsec⁻² (see Equation (20)) which is also bright enough to be the mechanism responsible for the small-scale Lyα fuzz. Furthermore, the lower volume density n_H of this configuration lowers the optically thin fluorescence discussed above to SB_Lyα = 8 × 10⁻¹⁹ erg s⁻¹ cm⁻² arcsec⁻², almost an order of magnitude fainter than our typical 2σ detection limit SB_Lyα ≃ 6 × 10⁻¹⁸ erg s⁻¹ cm⁻² arcsec⁻².

Thus both optically thick and optically thin fluorescence, as well as Lyα scattering can all plausibly explain the SB levels of the small-scale fuzz observed at distances R_⊥ < 50 kpc from quasars. Our simple estimates are crude, and more detailed numerical treatments which properly treat ionizing and resonant radiative transfer are required to improve their accuracy as well as make more detailed predictions about the SB profiles and the kinematics of the emission from these various processes. However, we believe that the key point, that the interpretation of the Lyα emission alone is not unique, is robust. We have argued that additional information about the gas distribution from absorption lines measurements can ameliorate this issue. For example, if the covering factor, column densities, and volume densities of the gas at R < 50 kpc can be statistically characterized, as we have undertaken in the QPQ series, then we can make better predictions for the optically thin and scattering surface brightness discussed above. Moreover, observations of additional emission line diagnostics, such as a non-resonant Balmer line (i.e., Hα), could distinguish between fluorescence and Lyα resonant scattering. The possibility will always remain that a very low covering factor component, which cannot be easily mapped with absorption lines, dominates the emission, as is the case for the low covering factor optically thick fluorescence discussed at the beginning of this section. Nevertheless, a comparison of absorption line properties to emission line measurements yields important constraints on the physical properties of the gas dominating the covering factor.

To conclude our discussion of the small-scale fuzz, we note that one can in principle also set a lower limit on the opening angle of the quasar emission based on the fact that fuzz is frequently detected extending to a distance R_⊥ ≃ 50 kpc. Unfortunately, given the unknown distance r to the gas, this is not particularly constraining. If we conservatively assume that the fuzz-emitting gas has to lie within the quasar halo at r < 200 kpc, then simple geometric considerations imply an opening angle $\Omega / 4\pi > 0.004$ (f_obscured < 0.996) corresponding to emission into a cone with half-angle >7°. If we instead assume the unknown line-of-sight distance R_∥ is comparable to R_⊥, then we obtain $\Omega / 4\pi > 0.04$ (f_obscured < 0.96) corresponding to a cone with half-angle >23°.

5.3.1. Previous Work on Lyα Fluorescence

5.3.2. The Lyα Emitter in SDSSJ 0856+1158

5.3.3. Clustering of LAEs around Quasars

The expected number of LAEs with rest-frame EW >W_Lyα within a volume V around a quasar can be written

$$\begin{equation} N_{\rm LAE}({>}W_{{\rm Ly}\alpha })= {\bar{n}}_{\rm LAE} P({>}W_{{\rm Ly}\alpha })\int _{V} dV [1 + \xi _{\rm QG}(r)], \end{equation} \tag{ 23 }$$

where ${\bar{n}}_{\rm LAE}$ is the average number density above the canonical rest-frame EW threshold W_Lyα > 20 Å for selecting LAEs, P(> W_Lyα) is the cumulative distribution of LAE EWs, and the integration over the correlation function ξ_QG quantifies the increase due to quasar–LAE clustering.

Although there is strong evidence that the powerful HzRGs trace large proto-cluster-like overdensities at z > 2 (e.g., Pentericci et al. 2000; De Breuck et al. 2004; Venemans et al. 2005; Kuiper et al. 2012), the clustering of galaxies around more typical radio-loud quasars like SDSSJ 0856+1158 and/or the radio-quiet quasars which dominate our sample, appears to be much more modest (e.g., Fynbo et al. 2001; Adelberger & Steidel 2005; Trainor & Steidel 2012). We thus assume that the clustering of LAEs around quasars at z ∼ 2 is the same as that of the continuum selected Lyman break galaxy (LBG) population at z ∼ 2.7. Trainor & Steidel (2012; see also Adelberger & Steidel 2005) measured the cross-correlation between LBGs and quasars at z ∼ 2.7, and assuming a power law form $\xi =(r/ r_{\rm 0,QG})^{-\gamma }$ with γ = 1.5, they measured a quasar–galaxy cross-correlation¹⁴ length of r_{0, QG} = 7.3 h⁻¹ Mpc (comoving). We integrate this correlation function over a cylindrical volume defined by an annular region [R_{⊥, min}, R_{⊥, max}] and a line-of-sight extent $\pm \Delta v/ [H(z)]$. The minimum radius is set to R_{⊥, min} = 50 kpc (proper), motivated by the fact that it could be difficult to distinguish LAE emitters from the ubiquitous small-scale Lyα fuzz seen around quasars. The maximum radius is set to R_{⊥, max} = 130 kpc (proper; 15″ at z ∼ 2), which is approximately the impact parameter of the LAE in SDSSJ 0856+1158 and the bright fluorescence candidate discovered near the quasar by Cantalupo et al. (2012). Along the line-of-sight, we take Δv = 1500 km s⁻¹, which encompasses both gravitational motions of galaxies in the quasar halo, quasar redshift errors, and also agrees with the interval used in the clustering analysis of Trainor & Steidel (2012). Furthermore, the 4 nm narrow-band filter used by Cantalupo et al. (2012) also translates to a line-of-sight velocity interval of 2900 km s⁻¹ or about ≃ 2 × Δv. Based on these assumptions, we find that over this volume, clustering alone enhances the number density of LAEs around quasars by a factor of 19 above the cosmic average.

For the background number density of LAEs ${\bar{n}}_{\rm LAE}$, we first need to decide which Lyα luminosity threshold to consider. The LAE in SDSSJ 0856+1158 has L_Lyα = 2.1 × 10⁴¹ erg s⁻¹, but owing to the long-exposure (4800 s) and relatively low redshift z = 1.76, this represents the most sensitive observation in our sample. As we wish to consider our sample in aggregate, we scale this value up by a factor of three, which represents the ratio of the luminosity limit in SDSSJ 0856+1158 to the median value of our sample. This value of L_Lyα = 6.4 × 10⁴¹ erg s⁻¹ is comparable to the luminosity of the fluorescent candidate detected closest to the quasar L_Lyα = 9.1 × 10⁴¹ erg s⁻¹ from Cantalupo et al. (2012), and also matches the median luminosity of their best fluorescence candidates. We use the Ciardullo et al. (2012) Schechter luminosity function fits to the z = 2.1 LAE survey data of Guaita et al. (2011), to determine that the comoving number density of LAEs with L_Lyα > 6.4 × 10⁴¹ erg s⁻¹ is ${\bar{n}}_{\rm LAE} = 5.1{\; \times \; 10^{-3}}\,h^3\,{\rm Mpc^{-3}}$. Ciardullo et al. (2012) also estimated the rest-frame EW distribution, and found that z = 2.1 LAEs selected to have rest-frame W_Lyα > 20 Å, follow an exponential distribution with rest-frame scale length W₀ = 50 Å. Hence the cumulative probability distribution of rest-frame EW is $P({>} W_{{\rm Ly}\alpha }) = \exp {[-(W_{{\rm Ly}\alpha } - 20\; {\mathring{\rm{A}}})/ W_0]}$. Given our conservative lower limit for the rest-frame W_Lyα > 50 Å of the LAE in SDSSJ 0856+1158 we obtain a probability P(> 50 Å) = 0.55, whereas our formal limit of W_Lyα > 173 Å gives P(> 173 Å) = 0.047. Putting all the pieces together (see Equation (23)), we find that the expected number of LAEs at impact parameter 50 kpc < R_⊥ < 130 kpc per quasar is N_LAE = 0.34 for our conservative lower limit W_Lyα > 50 Å and N_LAE = 0.029 for our formal lower limit W_Lyα > 173 Å. Note that this calculation is conservative and would tend to over-predict the number of LAEs near quasars for two reasons. First, we used the quasar–LBG correlation function, but LBGs are known to cluster slightly stronger than LAEs. Second, quasar clustering varies rather strongly with redshift (Shen et al. 2009) and so the Trainor & Steidel (2012) quasar–galaxy clustering measurements at z ∼ 2.7 probably overestimate the clustering around z ∼ 2 quasars.

But given that we expect N_LAE sources in the annular region between R_{⊥, min} and R_{⊥, max} around a given quasar, what is the probability that one such source should land on our slit given our observed sample? As we used our median luminosity limit above (i.e., the limit in SDSSJ 0856+1158 scaled up by three) to compute the number density of LAEs ${\bar{n}}_{\rm LAE}$, we assume that we could have detected a L_Lyα > 6.4 × 10⁴¹ erg s⁻¹ in 15/29 sources in our sample. Each longslit observation covers 3% of the annular region around a given quasar, hence the total number of LAEs expected in our sample is 15 × 0.03 × N_LAE, which equals 0.15 LAEs for our conservative EW limit, and 0.013 for our formal EW limit. Although our calculation is crude, and does not fully take into account the sensitivity variations in our sample, it seems rather unlikely that the clustering of LAEs around the quasar can trivially explain our discovery of a relatively high-EW LAE at R_⊥ = 131 kpc from the f/g quasar in SDSSJ 0856+1158.

5.3.4. Is SDSSJ 0856+1158 Powered by Fluorescence?