THE z = 5 QUASAR LUMINOSITY FUNCTION FROM SDSS STRIPE 82^*

Over the 10 year duration of the survey, the SDSS I/II repeatedly imaged a 25 × 100° stripe centered at zero declination in the Southern Galactic Cap (Abazajian et al. 2009). This repeat imaging increased in frequency during the latter part of operations as part of the SDSS Supernova Survey (Frieman et al. 2007). By the end of operations, over 100 repeat imaging scans were obtained along Stripe 82 (its designation according to the survey geometry; Stoughton et al. 2002). As with the main survey, the imaging was obtained with a drift-scan camera (Gunn et al. 1998) mounted on the 2.5 m Sloan telescope (Gunn et al. 2006), with nearly simultaneous 54.1 s exposures in five broad optical bands (ugriz; Fukugita et al. 1996). The quality restrictions applied to the main survey imaging (photometric and good-seeing conditions; Ivezić et al. 2004; Hogg et al. 2001) were relaxed for the Supernova Survey in order to increase the temporal coverage, resulting in a greater range of data quality.

Jiang et al. (2009) describe deep riz images of Stripe 82 constructed from the coaddition of 50–60 individual imaging scans at each position on the sky. Images with seeing poorer than 2'' or an r-band sky background brighter than 19.5 mag arcsec⁻² were rejected. The remaining images were weighted by their transparency, seeing, and sky background noise, and coadded using the SWARP software (Bertin et al. 2002). The final coadded images reach a depth roughly 2 mag fainter than single-epoch SDSS images, and were used to discover z ∼ 6 quasars as faint as z_AB = 22.2 (Jiang et al. 2009). These data are not the only coadded images available for Stripe 82: Annis et al. (2011) produced coadded images made using a similar process but with stricter image quality criteria resulting fewer imaging scans used at each sky position, while Huff et al. (2011) created image stacks optimized for weak lensing shear studies.

Our starting point is the riz coadded images from Jiang et al. (2009), which we now extend to include the u and g bands (L. Jiang et al., in preparation). The deep imaging in the bluer SDSS bands is being used to discover ultra-luminous Lyman break galaxies at z > 2.5 (F. Bian et al., in preparation). In this work, we extend the Jiang et al. (2009) search for faint, z ∼ 6 quasars to lower redshifts (z ∼ 5), as the deep g-band photometry combined with the redder bands allows reliable color selection of quasars in this redshift range. The depth of our coadds is similar to that of the Annis et al. (2011) coadds (see their Figure 7); however, as described in Section 4.2 we encountered much greater contamination when using the catalogs from the Annis et al. (2011) coadds for quasar selection.

We use Sextractor (Bertin & Arnouts 1996) to produce object catalogs from the deep ugriz images. Object detection was performed in the i band, and fluxes and other measurement parameters in the other bands were derived from apertures centered on the i-band object position using the dual-image mode of Sextractor. All fluxes and magnitudes from the coadded Stripe 82 imaging quoted in this work are taken from aperture photometry using a 18 diameter. The typical seeing in the coadded images¹⁸ is 14, 13, 11, 10, and 11 in u, g, r, i, and z, respectively.

The geometry of the Stripe 82 coadded imaging follows that of the parent SDSS imaging.¹⁹ We photometrically calibrated individual fields in the coadded catalogs by deriving zero points from the übercalibrated (Padmanabhan et al. 2008) SDSS DR8 (Aihara et al. 2011) photometry (derived from the best individual imaging scan at each location in Stripe 82), using stars with 15.5 < r < 20.5. We thus tie the aperture photometry from Sextractor on a field-by-field basis to point-spread function (PSF) photometry from SDSS. This process results in zero points that account for seeing and photometric variations between the fields, but not within the fields. However, the large number of images contributing to the coadds tends to average over effects such as varying sky backgrounds and PSF shapes. In general, we find the photometry is highly consistent between fields and agrees quite well with SDSS photometry for the brighter objects. Figure 1 shows that the depth across the Stripe is highly uniform, and Figure 2 shows the depths reached in the ugriz bands in the coadded imaging.

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Quasars at z ∼ 5 have similar fluxes in the i and z bands. We adopt the i band as our detection band as it provides greater a greater signal-to-noise ratio. However, at z ∼ 5 the Lyα emission line and the Lyα forest are within the i band. The fluxes in the band that defines our detection threshold are thus subject to the substantial variances in Lyα equivalent widths (EWs), the incidence of strong Lyα absorption systems, and variations in the mean IGM opacity. The z band is less affected by these issues (although it does contain C iv in emission); however, it is significantly noisier than the i band because of lower CCD sensitivity and a higher sky background. We will discuss these issues further in Section 5.

We further note that our survey is restricted to less than the full area on Stripe 82 for two reasons. At the time that we performed target selection, the coadds corresponding to the two uppermost camera columns had problems with the sky background in the g band, rendering them inadequate for our purposes. We thus imposed a cutoff of δ < 085 when selecting candidates. In addition, the western edge of Stripe 82 (near 20^h) approaches the Galactic plane. Attempting color selection of quasars in this region would result in overwhelming stellar contamination (see, e.g., Figure 1 of Vanden Berk et al. 2005). We thus restrict our survey to objects with R.A. >20^h32^m (this region also lies outside the UKIDSS imaging area described in the next section). The final area of the deep imaging we used for z ∼ 5 quasar selection is 235 deg², extending from 20^h32^m to 4^h, and from −125 to +085 (Figure 1).

The UKIDSS (Lawrence et al. 2007) consists of multiple infrared imaging surveys with the UKIRT Wide Field Camera (WFCAM; Casali et al. 2007), including a wide-area component (the Large Area Survey or LAS) that covers ∼4000 deg² to a depth of J_AB ≈ 20.6 (5σ). The LAS includes Stripe 82, providing shallow infrared imaging of our candidates and additional leverage in discriminating high-redshift quasars from stellar contaminants. We first apply a loose color cut to identify potential quasar candidates from the coadded optical imaging,

$$\begin{equation*} (r-i) > 1.0 \quad {\rm and}\; (i-z) < 0.625[(r-i)-1.0] + 0.2, \end{equation*}$$

then queried the DR8plus release in the WFCAM Science Archive (WSA; Hambly et al. 2008) for infrared counterparts.

Quasars at z ∼ 5 have (i − J)_AB ∼ 0 (see Section 3), thus at the faint limit of our survey most of our target objects do not have a counterpart in the UKIDSS catalogs. However, stars with similar optical colors to z ∼ 5 quasars (namely, M and L dwarf stars) are redder in the near-IR (0.5 < (i − J)_AB < 1.5, see Figure 5) and thus are relatively brighter at infrared wavelengths. We downloaded J-band image cutouts from the WSA for all of the candidates pre-selected from the optical imaging, then performed aperture photometry using the IRAF aper task at the i-band position from the SDSS coadded imaging, measured in 2'' diameter apertures. We validated our photometry by checking against the UKIDSS catalog measurements for brighter objects (utilizing the UKIDSS calibration as described in Hodgkin et al. 2009). The aperture photometry reaches J ≈ 21.3 at 3σ, sufficient to discriminate the typical stellar contaminants at our limit of i = 22.

At the time the initial candidate selection was performed, the publicly available UKIDSS data on Stripe 82 did not extend to R.A.  <  21^h36^m. We imaged some of the candidates lacking J coverage on 2011 October 14–15 with the MMT Smithsonian Widefield Infrared Camera (SWIRC; Brown et al. 2008). Each object was observed in the J band using a nine-point dither pattern with 30 s exposures at each position, for a total integration time of 4.5 minutes. The typical seeing was 06–09. Images were dark-corrected, flat-fielded, sky-subtracted, shifted, and combined using standard IRAF routines. Object photometry was measured in 2'' diameter apertures using IRAF routines, and calibrated using observations of UKIRT Faint Standards.

A total of 38 objects were observed with SWIRC. This includes 10 objects already spectroscopically confirmed as quasars but that lacked UKIDSS coverage or had low S/N in the J band. These objects were observed as a check on our J-band color selection for quasars selected solely by optical colors. The remaining SWIRC targets were taken from a sample of objects selected as high-redshift quasar targets based on optical colors but that lacked UKIDSS coverage. These objects are mainly at 309° < α_J2000 < 360°, the region of our survey with the highest stellar density. Nearly all of these sources were readily identified as stellar contaminants from bright detections in the SWIRC imaging, and would have greatly reduced the purity of our spectroscopic sample had they not been observed with SWIRC.

Figure 3 displays postage stamp images of a typical quasar within our survey. The i band clearly has the highest S/N. The non-detection in UKIDSS and the faint detection in the deeper SWIRC imaging are expected for a z ∼ 5 quasar; a contaminating star would have been easily identified from the infrared imaging.

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Fan (1999) outlines a procedure for generating simulated quasar spectra using a simple empirical model for their spectral properties at UV/optical wavelengths. The simulated spectra are integrated through survey bandpasses to generate simulated fluxes (colors) that can be used to define selection criteria, and to estimate their completeness. This requires that the spectral model reliably captures the diversity of quasar spectral features, and thus accurately reproduces observed quasar colors. The basic components of the quasar model are a broken power-law continuum, prominent UV/optical emission lines, pseudo-continuum from Fe complexes, and redshift-dependent Lyα forest absorption due to intervening neutral hydrogen. We have used an updated version of this model (described below) to reproduce the colors of ∼60, 000 quasars from the SDSS-III/BOSS in the range 2.2 < z < 3.5 (Ross et al. 2012). We apply the model at higher redshift under the assumption that the distribution of quasar spectral energy distributions (SEDs) does not evolve with redshift (Kuhn et al. 2001; Yip et al. 2004; Jiang et al. 2006). This approach can be compared to, e.g., cloning the spectra of lower redshift quasars to higher redshift (Chiu et al. 2005; Willott et al. 2005), which carries with it any selection function imprinted on the lower redshift sample; or using a small set of quasar templates (Mortlock et al. 2012), which may not capture objects with unusual features.

Each quasar is assigned a power law continuum with a break at 1100 Å. The blue slope is drawn from a normal distribution with μ(α) = −1.7 and σ(α) = 0.3 (Telfer et al. 2002)²⁰; the distribution for the red slope is μ(α) = −0.5 and σ(α) = 0.3. We add to this continuum emission lines with Gaussian profiles, where the Gaussian parameters (wavelength, EW, and FWHM) are drawn from normal distributions. These distributions are derived from fitting composite spectra of BOSS quasars in luminosity bins. These distributions recover trends in the mean and scatter of the line parameters as a function of continuum luminosity, e.g., the Baldwin effect (Baldwin 1977), and blueshifted lines (Gaskell 1982; Richards et al. 2011). Finally, we include Fe emission using the template of Vestergaard & Wilkes (2001), scaling the template in segments to match the Fe emission in the composite spectra. We do not include a contribution from quasars with unusually weak emission lines, which account for ∼6% of quasars at z > 4.2 (Diamond-Stanic et al. 2009), or from broad absorption line (BAL) quasars, which tend to have moderately redder colors (e.g., Weymann et al. 1991; Brotherton et al. 2001; Reichard et al. 2003).

The quasar model for the BOSS employs a prescription for the Lyα forest based on the work of Worseck & Prochaska (2011). This forest model is calibrated to observations at z < 4.6. We extend the model to higher redshifts by using the observed number densities of high column density systems from Songaila & Cowie (2010). For the lower column density systems ($\log N_{{\rm H}\,{\scriptsize {I}}} < 17.2$), we begin with the parameters of Worseck & Prochaska (2011), and follow their method of deriving the number densities by simulating a large number of sight lines and matching the mean free paths (mfps) of the simulated sightlines to the observations of Songaila & Cowie (2010). We also force continuity between the column density distribution function ($dn/dN_{{\rm H\,{\scriptsize {I}}}}dz$) at low and high column densities, and check the derived effective optical depth (τ_eff) against the measurements of Songaila (2004) and Fan et al. (2006a). There remains significant uncertainty in forest parameters at high redshift, but our simple color selection method is relatively insensitive to the model for the Lyα forest. To account for the degree of uncertainty in the forest absorption, we also include a "low" and "high" forest model, scaling the number densities of forest absorbers up and down by 10%, matching the scatter in measurements by Songaila & Cowie (2010). After comparison with observed quasar colors, we find that the model with a mean forest density 10% greater than the best-fit value from the mfp measurements provides the best match. Figure 4 shows an example simulated quasar spectrum from our models, and compares the Lyα forest observables (mfp and τ_eff) obtained from our simulated forest spectra to observations of high-redshift quasars.

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Based on the results from the simulated spectra, we concluded that the most efficient use of telescope time was to focus on a fairly narrow redshift range, 4.7 < z < 5.1, where the colors of stars and quasars are best separated and the selection efficiency is high. Figure 5 shows the gri, riz, and riJ colors of red stars and high-redshift quasars in our sample, as well as the selection criteria we adopted:

• 1.  
  i < 22.0,
• 2.  
  S/N(u) < 2.5,
• 3.  
  g − r > 1.8 OR S/N(g) < 3.0,
• 4.  
  r − i > 1.2,
• 5.  
  i − z < 0.625((r − i) − 1.0),
• 6.  
  i − z < 0.55,
• 7.  
  i − J < ((r − i) − 1.0) + 0.56,

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where all magnitudes are in the AB system, measured within fixed apertures (as described in Section 2), and have been corrected for Galactic extinction using the maps of Schlegel et al. (1998). S/N measurements are obtained from the aperture fluxes and uncertainties given by Sextractor.

We apply no morphological criteria in our candidate selection after finding that the number of potentially resolved objects (Sextractor CLASS_STAR <0.8) is small (<10% of the total selected), indicating that contamination from compact galaxies is negligible.

These criteria are similar but not identical to the cuts that define the z > 4.5 inclusion region in the SDSS quasar selection algorithm, as given in Richards et al. (2002). We extend the selection nearly 2 mag fainter than the SDSS main survey by utilizing the coadded imaging on Stripe 82. We define dropout criteria in the u and g bands through S/N criteria within fixed apertures rather than a magnitude cut. Although the Stripe 82 coadded imaging is highly uniform (e.g., Figure 1), using a S/N threshold more fully utilizes the u-band depth at a given location, as z ∼ 5 quasars are not expected to have any detectable flux in this band. We bracket the redshift range of the search to z ≳ 4.7 by requiring a red r − i color and to z ≲ 5.1 by requiring a blue i − z color. Finally, we take advantage of the available infrared data from UKIDSS, effectively imposing a veto on objects that are too red in i − J. This cut is made using the aperture flux ratios in the i and J bands, so that objects undetected in the UKIDSS J band—as expected for z ∼ 5 quasars—pass the final cut without imposing a S/N threshold.

The i-band detection catalog contains 1.5 M objects. We apply loose pre-selection cuts in the riz colors (see middle panel of Figure 5 and Section 2.2) to reduce this list to ∼10, 000 potential candidates for which UKIDSS images are downloaded and analyzed. The near-IR photometry is highly useful as stellar veto: of the ∼10, 000 pre-selected candidates from the optical imaging, only ∼650 are rejected by the i − J color cut we adopt (Section 3.2) using the UKIDSS catalog photometry, while ∼4500 are rejected based on the aperture fluxes. After applying the final color criteria listed above, we have 92 candidates to a limit of i_AB = 22.

The SDSS I/II spectroscopic survey concluded with the DR7 release (Abazajian et al. 2009). Spectra were reduced with the standard SDSS pipeline (Stoughton et al. 2002). Quasar target selection is described in Richards et al. (2002); high-redshift quasars are targeted through two methods, both to a limit of i = 20.2. First, outliers from the stellar locus in griz space are identified as high-redshift quasar candidates, and second, various color cuts aim to extend the quasar yield within specific redshift ranges beyond the locus outlier selection. Schneider et al. (2010) present a catalog of over 100,000 spectroscopically confirmed quasars drawn from ≈9380 deg² of the DR7 spectroscopic footprint (hereafter DR7QSO). This catalog includes 191 quasars at 4.7 < z < 5.1 to a limit of i ≲ 20.2, of which 184 are selected by our color criteria based on their SDSS flux measurements.

We utilize the DR7 data in several ways. First, we matched our candidate list to the DR7QSO catalog, as well as to the full DR7 spectroscopic database. We identified eight of our candidates as confirmed 4.7 < z < 5.1 quasars in DR7QSO. There were three additional quasars in this redshift range that did not meet our color criteria. None of our candidates matched to non-quasars among the 1.6 million objects with spectra in DR7.

We further employ the DR7 quasars to determine the bright end of the luminosity function, by constructing a uniform sample of DR7QSO quasars in our redshift range drawn from the SDSS DR7 imaging footprint. We will describe this sample in Section 6.1.1.

The BOSS quasar survey is primarily designed to find quasars at 2.2 < z < 3.5 for the purpose of Lyα forest studies (McDonald & Eisenstein 2007; Ross et al. 2012). The Extreme Deconvolution algorithm (XDQSO) used as the primary quasar targeting method in BOSS is tuned to this redshift range and is highly efficient (Bovy et al. 2011). However, a large section of Stripe 82 was observed by BOSS in Fall 2010 as BOSS Chunk 11 (Ross et al. 2012); during these observations quasars were targeted through a combination of color selection (using XDQSO) and variability selection (Palanque-Delabrouille et al. 2011). The highest redshift quasar identified by the main BOSS targeting in Stripe 82, including variability, is at z = 4.46.

In addition, we are conducting ancillary science programs in BOSS that take advantage of unused fibers once the primary survey targets (galaxies and mid-z quasars) have been allocated fibers (Dawson et al. 2012). These programs extend the gri and riz color cuts described in Richards et al. (2002) to fainter quasars by utilizing areas with multiple epochs of imaging in SDSS I/II due to overlapping or repeated scans. On Stripe 82 these ancillary programs utilized photometry from the ∼20-epoch coadded images available in the DR7 CAS and described in Annis et al. (2011); these observations are included in BOSS DR9 (Ahn et al. 2012). We will analyze the quasars identified in overlap regions outside of Stripe 82 in future work.

The selection criteria for z > 4.6 quasars are similar to the riz inclusion region in Richards et al. (2002), but extended to a faint limit of i = 21.5 and using the catalogs from the Annis et al. (2011) coadded imaging for target selection. On Stripe 82, this program was allocated 283 targets, of which 221 received spectra, but only 22 were high-z quasars. Of those, 10 were previously known from SDSS DR7; thus the program yielded only 12 new quasars. All 12 meet our selection criteria and are included in this study. The ancillary program targets that are not quasars fall mainly into two categories: (1) objects with fluxes contaminated by nearby bright stars, and (2) spurious or moving objects (e.g., asteroids). We matched the failed targets to the coadded image catalogs and found that either they did not have matches in our coadded imaging, or the matches did not meet our selection criteria.

BOSS spectra are collected with a fiber-fed, multi-object spectrograph (Smee et al. 2012) and reduced with a pipeline described in Bolton et al. (2012). We visually examined all of the spectra for ancillary program targets on Stripe 82, as well as any classified by the pipeline as having z > 3.6. We also cross-checked our examinations against the DR9 Quasar Catalog as described in Pâris et al. (2012). As with SDSS, we cross-checked our candidate list against all spectra in BOSS, not just confirmed quasars, again finding no matches to non-quasars. Figure 6 displays the BOSS spectra of z > 4.7 quasars on Stripe 82.

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The remaining candidates on Stripe 82 (∼70) were mostly fainter than i = 21 and required spectroscopic observations on larger telescopes. We first observed four candidates at the Magellan Clay 6.5 m on 2011 June 11–13 using the Magellan Echellette spectrograph (MAGE; Marshall et al. 2008). MAGE provides full coverage from 3100 Å to 1 μm at a resolution of ∼5800 using the 07 slit. The typical seeing during this run was 06.

Magellan spectra were reduced using MASE (Bochanski et al. 2009), an IDL-based pipeline designed for MAGE data. Wavelength calibration was provided by ThAr lamps observed shortly after the targets and at a similar airmass, and the standard star Feige 110 was used for flux calibration. Of the four targets observed at Magellan, three are z ∼ 5 quasars (Figure 7; one object had been previously observed in BOSS), and one is a star.

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The bulk of our spectroscopic observations occurred at the MMT 6.5 m telescope using the Red Channel spectrograph. We used the 270 mm⁻¹ grating centered at 7500 Å,²¹ providing coverage from 5500 Å to 9700 Å. We used either the 1'' or 15 slit based on the seeing, providing resolutions of R ∼ 640 and R ∼ 430, respectively.

Observations were conducted on 2011 June 23–24, 2011 October 1–4, and 2012 May 27–28. During 2011 June nine objects were observed in poor seeing (15–3''); five were confirmed as quasars. Conditions during the 2011 October run were fair with ≲ 1'' seeing; however, most of the run was lost to thick clouds and high humidity, particularly during the early part of the night. As a result, we obtained few spectra for candidates at 20^h < R.A. < 22^h. In total, 52 targets were observed in October, of which 41 were confirmed quasars. In the 2012 May run, 17 additional candidates with 20^h < R.A. < 24^h were observed, resulting in 11 new high-redshift quasars. Conditions during this run were excellent, with 07–12 seeing throughout. In general we valued efficiency over quality. Therefore the exposure times were short, between 5 and 15 minutes with typically a single exposure per target, and many of the spectra have a low signal-to-noise ratio. However, the quasars among the observed targets are easily identified by their prominent emission lines, while the few contaminants can be ruled out as high-redshift quasars by the lack of either emission lines or a strong spectral break toward blue wavelengths.

Data were processed using standard longslit reduction techniques through a combination of Pyraf²² and python routines, including bias subtraction, pixel level corrections from flat fields generated from internal lamps, and sky subtraction using a polynomial background fit along the slit direction. Cosmic rays were identified and masked using the LACOS routines (van Dokkum 2001). Wavelength calibration was provided from an internal HeNeAr lamp, and then corrected on a per-image basis using night sky lines (primarily the OH line list given by Rousselot et al. 2000). Standard stars were observed between one and three times per night and used for flux calibration. However, the conditions were highly variable and the absolute flux calibration of the spectra is not reliable. Figure 8 shows the MMT spectra obtained for high-redshift quasars on Stripe 82.

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In this section, we note objects that have radio detections, uncertain identifications from the spectroscopy, and unusual spectral features.

J221941.90+001256.2 (z = 4.30), J224524.27+002414.2 (z = 5.16). These two objects have radio counterparts at 1.4 GHz from Very Large Array imaging of Stripe 82 (Hodge et al. 2011). They are the only sources with counterparts in that catalog, which is derived from imaging over 92 deg² to a depth of 52 μJy beam⁻¹. J221941.90+001256.2 has a peak flux density of F_{1.4, pk} = 0.92 ± 0.07 mJy beam⁻¹ and J224524.27+002414.2 has F_{1.4, pk} = 1.09 ± 0.06 mJy beam⁻¹. Both sources also have counterparts in the Faint Images of the Radio Sky at Twenty-Centimeters (FIRST) catalog (Becker et al. 1995), with peak flux densities of F_{1.4, pk} = 0.87 ± 0.10 mJy beam⁻¹ and F_{1.4, pk} = 0.92 ± 0.10 mJy beam⁻¹, respectively. Neither is included in our uniform sample as they lie outside the defined redshift range. J221941.90+001256.2 also shows strong BAL features.

J223907.56+003022.6 (z = 5.09), J232741.35−002803.9 (z = 4.75), J021043.16−001818.4 (z = 5.05). These three objects also have FIRST counterparts, with peak flux densities of F_{1.4, pk} = 1.35 ± 0.10 mJy beam⁻¹, F_{1.4, pk} = 1.24 ± 0.12 mJy beam⁻¹, and F_{1.4, pk} = 6.82 ± 0.10 mJy beam⁻¹, respectively. All three are included in the uniform sample.

J211158.01+005302.6 (z = 4.98), J234730.56+002306.3 (z = 4.71). We identified these objects as quasars based on their discovery spectra; however, the spectra are noisy and the identifications were uncertain. We later confirmed them as quasars with MMT observations on 2012 August 25.

J211225.39−000141.3 (z = 4.67), J030315.05−000347.6 (z = 4.72). These spectra also have low S/N. Both objects appear to have a Lyman break feature (more evident in the two-dimensional spectra) and Lyα and N v emission features, with apparent N v absorption troughs. We include these objects as confirmed quasars. J030315.05−000347.6 may have BAL features.

J222018.48−010146.8 (z = 5.62), J000552.33−000655.6 (z = 5.86). These two quasars were targeted as part of an SDSS+UKIDSS high-z BOSS quasar ancillary program. J000552.33−000655.6 was first reported in Fan et al. (2004).

We use the simulations described in Section 3.1 to estimate the completeness of our selection criteria. To derive a selection function, we construct a grid of simulated quasars distributed evenly in (M₁₄₅₀, z) space with 200 quasars bin⁻¹ of ΔM = 0.1, Δz = 0.05. Each object is sampled from models for quasar emission properties as outlined in Section 3.1. The simulated quasars are then passed through the color cuts after adding photometric errors, and the fraction of objects within each bin that pass the cuts provides an estimate of the completeness for a given luminosity and redshift, under the assumption that our quasar model accurately represents the intrinsic distributions of quasar properties.

We generate two simulation grids. The first adopts photometric uncertainties typical of the SDSS main survey and uses asinh magnitudes; we use this simulation to derive the completeness of the SDSS quasar survey at z ∼ 5. The second uses photometric uncertainties from the coadded imaging and is used for the Stripe 82 sample. The uncertainties are determined by fitting a relation to the uncertainties of stellar objects in the Stripe 82 catalog as a function of band flux.

6.1.1. DR7 Completeness

The SDSS quasar selection algorithm (Richards et al. 2002) was not finalized until the survey was already in progress, thus to construct a statistical sample from DR7 we follow the method given in Richards et al. (2006) to identify DR7 quasars from regions with uniform target selection. This limits the final area to 6222 deg². We further cut the DR7 sample to only objects selected by the riz color inclusion regions (excluding objects selected only as griz stellar locus outliers); this results in the loss of only a few objects, but greatly simplifies the calculation of the selection function. After restricting to the uniform targeting area and applying the color cuts, DR7QSO provides 146 quasars with 4.7 < z < 5.1.²⁵

We use the simulated quasar photometry combined with the riz color cuts of Richards et al. (2002) to derive the selection function for z ∼ 5 quasars in the main SDSS sample (Figure 11). We add a 5% correction for photometric incompleteness (i.e., objects lost due to crowding or proximity to bright stars or galaxies; see Richards et al. 2006) and spectroscopic incompleteness of 5% (objects selected by the targeting algorithm but without a spectrum in DR7). We determined the latter incompleteness by querying the DR7 Target database for objects with the QSO_HIZ flag, finding that 5% lack spectra. The missing spectra are mainly due to fiber collisions that result when tiling spectroscopic targets, due to the restriction that fibers on a single plate must be separated by >55''.

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6.1.2. Stripe 82 Completeness

Although the Stripe 82 selection function is obtained from the same quasar model as for DR7, it is evident from Figure 11 that the Stripe 82 selection probes a different population. First, the Stripe 82 data are much deeper and thus the completeness remains high to lower luminosities. Also, the color selection criteria are somewhat different. Figure 12 compares the selection efficiencies for DR7 and Stripe 82 as a function of redshift. It is noteworthy that the selection function value is higher for fainter objects at some redshifts. This is because the selection function calculation averages over a range of colors in each (M, z) bin, with the colors depending on the continuum slopes, line strengths, and other features sampled from the model. At z ∼ 5, our color selection is quite sensitive to the Lyα EW, as shown in Figure 13. Because the EW increases at lower luminosities and Lyα is fully within the i band in our redshift range, the colors of fainter objects are redder in r − i and bluer in i − z than brighter objects, increasing their likelihood of meeting our selection criteria (see also Figure 5). This effect is not captured by models that do not account for the Baldwin effect.

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Our selection function models generally predict near-zero completeness for z ≳ 5.2 quasars; however, it is clear from Figure 10 that our color criteria do select a non-negligible number of quasars at these redshifts (particularly for the DR7 data). We found that models with increases in both the mean and scatter of the Lyα EW distribution provide a better fit to the observed colors (and the z > 5.2 redshift distribution) than our fiducial model. However, these models had little effect on our primary analysis, as we restrict to the range 4.7 < z < 5.1, where our fiducial model already predicts high completeness. We thus chose to maintain consistency with the BOSS analysis presented in Ross et al. (2012) and adopt the same quasar spectral model.

We estimate the photometric completeness to be 95% (see, e.g., Figure 7 of Annis et al. 2011). The spectroscopic coverage of the Stripe 82 candidates is high: ∼90% of candidates with i_AB < 21.5 have spectra. Near the faint limit, the completeness is a bit lower: 69% of candidates with 21.5 < i_AB < 21.7 and 63% of candidates with 21.7 < i_AB < 22 have spectra. Figure 14 shows our spectroscopic completeness as a function of observed magnitude, as well as the correction we use to account for the missing spectra in our luminosity function calculation. In total, 19 candidates do not have spectroscopic identifications; by applying this correction we assume that they are a random subsample of the full candidate set. This assumption is fair given that the choice of which targets to observe was largely constrained by sky location and weather conditions, not properties of the candidates themselves (such as color).

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The Stripe 82 survey includes 52 quasars at 4.7 < z < 5.1. The Stripe 82 sample is not only highly complete but also highly pure: out of 73 candidates with spectroscopic identifications, 71 are high-redshift quasars (z > 4), and 52 (71%) are quasars in the targeted redshift range. For completeness, we provide in Table 3 a list of the candidates that were either not observed spectroscopically (19 objects), or were found not to be high-redshift quasars (2 objects, both have no emission features and appear to be stellar continua).

Table 3. Stripe 82 High-redshift Quasar Targets

R.A. (J2000)	Decl. (J2000)	gAB	rAB	iAB	zAB	JAB	Notes
20:37:53.64	+00:05:49.4		22.81 ± 0.05	20.84 ± 0.01	20.33 ± 0.03	...
20:49:47.51	+00:08:24.1	25.82 ± 0.59	23.38 ± 0.09	21.91 ± 0.03	21.64 ± 0.11	...	Star
20:59:54.11	−00:54:27.7	25.90 ± 0.76	23.11 ± 0.06	21.54 ± 0.02	21.24 ± 0.05	21.22 ± 0.15a
21:00:41.31	−00:52:03.4	25.58 ± 0.58	23.59 ± 0.10	21.79 ± 0.03	21.37 ± 0.06	20.94 ± 0.10a
21:05:17.96	−00:49:20.2		23.23 ± 0.07	21.49 ± 0.02	21.18 ± 0.06	20.86 ± 0.31
21:47:30.25	+00:12:37.5	25.32 ± 0.38	23.65 ± 0.11	21.92 ± 0.03	21.50 ± 0.09	20.82 ± 0.38
21:52:56.25	−01:06:13.8	25.01 ± 0.46	23.00 ± 0.07	21.56 ± 0.03	21.32 ± 0.07	20.66 ± 0.08a
00:03:32.25	+00:37:49.9	26.30 ± 0.72	23.18 ± 0.05	21.61 ± 0.02	21.37 ± 0.05	21.30 ± 0.32
00:05:31.71	−00:34:43.4		23.85 ± 0.10	21.91 ± 0.03	21.75 ± 0.07	21.77 ± 0.64
00:18:22.10	+00:25:23.1	26.20 ± 0.64	23.61 ± 0.08	21.79 ± 0.02	21.68 ± 0.08	...
00:23:53.63	−00:43:56.9	26.00 ± 0.52	23.64 ± 0.08	21.51 ± 0.02	21.00 ± 0.04	20.75 ± 0.07a
00:30:30.36	−00:04:17.7	25.78 ± 0.42	23.41 ± 0.06	21.78 ± 0.02	21.67 ± 0.06	21.69 ± 0.76
00:41:37.17	−00:30:39.4		22.94 ± 0.04	20.92 ± 0.01	20.45 ± 0.02	20.00 ± 0.15
01:26:55.05	−00:00:12.7		24.37 ± 0.16	21.87 ± 0.02	21.78 ± 0.10	21.09 ± 0.40	Star
01:40:14.61	−00:12:59.9	25.42 ± 0.30	23.30 ± 0.05	22.00 ± 0.02	21.81 ± 0.07	21.47 ± 0.60
02:38:09.21	−01:09:27.5	28.23 ± 4.72	23.44 ± 0.06	21.94 ± 0.03	21.83 ± 0.08	...
02:52:29.91	−00:48:13.1		23.74 ± 0.09	21.97 ± 0.03	21.77 ± 0.08	21.14 ± 0.41
03:09:37.66	+00:46:16.8	26.77 ± 1.16	23.32 ± 0.06	21.91 ± 0.03	21.74 ± 0.07	21.38 ± 0.37
03:40:50.33	+00:48:12.2	25.99 ± 0.61	23.23 ± 0.07	21.88 ± 0.03	21.73 ± 0.07	21.06 ± 0.23
03:54:23.45	+00:32:31.6	24.00 ± 0.44	22.92 ± 0.15	21.28 ± 0.04	21.05 ± 0.08	20.33 ± 0.18
03:56:55.52	−00:34:47.9	24.82 ± 0.65	22.48 ± 0.08	21.21 ± 0.03	21.09 ± 0.07	22.01 ± 1.16

Notes. Targets selected by our color criteria that either lack spectroscopic observations or are confirmed non-quasars. Magnitudes are given in the same format as Table 2. ^aJ magnitude from SWIRC.

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We first calculate the luminosity function from the combined SDSS main and Stripe 82 z ∼ 5 quasar samples by dividing the sample into discrete bins of luminosity and redshift. Guided by our completeness calculations, we restrict the sample to the interval 4.7 < z < 5.1, where the selection efficiency is relatively high. We refer to this as the uniform sample. We use a single redshift bin, ignoring any evolution of the QLF parameters over the width of the bin. In particular, the redshift evolution derived by Fan et al. (2001b) predicts a decline by a factor of ∼1.5 in space density from z = 4.7 to z = 5.1. We do not account for this evolution in the binned QLF, but it will be incorporated below in a maximum likelihood fit to each quasar. We calculate the binned luminosity function using the 1/V_a method (Schmidt 1968; Avni & Bahcall 1980), including the correction of Page & Carrera (2000). The calculation is performed separately on the DR7 and Stripe 82 data, accounting for the differences in sky area, depth, and selection criteria between the two samples.

Table 4 provides the binned QLF for both the SDSS main (DR7) and Stripe 82 samples. We include the number counts in each magnitude bin, as well as the corrected number counts after accounting for all sources of incompleteness. The binned QLF data is also displayed in Figure 15. The SDSS data show a steep drop in the number density at the bright end; from the combined data it is evident that the QLF becomes shallower toward lower luminosities. Fitting a single power law to the binned data from both surveys at the bright end (M₁₄₅₀ < −27.0) results in a steep slope of β = −3.7.

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Table 4. Binned QLF

M1450	N	Ncor	log Φa	$\sigma _\Phi$b
DR7
−28.05	3	4.8	−9.45	0.21
−27.55	5	7.7	−9.24	0.26
−27.05	30	42.0	−8.51	0.58
−26.55	57	85.1	−8.20	0.92
−26.05	51	81.5	−7.90	1.89
Stripe 82
−27.00	2	2.2	−8.40	2.81
−26.45	5	8.1	−7.84	6.97
−25.90	5	7.0	−7.90	5.92
−25.35	10	16.7	−7.53	10.23
−24.80	15	24.3	−7.36	11.51
−24.25	14	26.8	−7.14	19.90

Notes. ^aΦ is in units of Mpc⁻³ mag⁻¹. ^b$\sigma _\Phi$ is in units of 10⁻⁹ Mpc⁻³ mag⁻¹.

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Shen & Kelly (2012) have also calculated the binned QLF of SDSS quasars at z = 4.75, repeating the DR3 analysis of Richards et al. (2006) for the larger DR7 sample. Our methodology differs from that of Shen & Kelly (2012) in several respects. Most notably, we have recalculated the selection function and k-corrections using our new quasar model. In addition, we restrict the quasar sample to color-selected objects (excluding those identified only as stellar locus outliers). The Richards et al. (2006) selection function has a value of ∼1 at z ≳ 5, much higher than the values we obtain (see Figure 12), and inconsistent with our previous finding (Section 6.1.1) that few quasars were selected by locus outlier criteria alone. However, although our selection function disagrees with that of Richards et al. (2006) by as much as a factor of ∼2 at z ∼ 5, the highest redshift bin (4.5 < z < 5.0) in both Richards et al. (2006) and Shen & Kelly (2012) is dominated by objects near the low-redshift edge of the bin. Finally, we use a slightly different method for calculating the spatial area of the survey; however, we obtain a similar result (6222 deg² versus 6248 deg²). Figure 15 compares our binned QLF to Richards et al. (2006) and Shen & Kelly (2012); the agreement is generally good, though differences of ∼20%–30% may still be attributed to the different approaches used.

We now derive parametric fits to the observed data using maximum likelihood estimation (MLE). The maximum likelihood estimate for a luminosity function Φ(M, z) corresponds to the minimum of the log likelihood function

$$\begin{equation*} S = -2 \sum \limits _i^N \ln [\Phi (M_i,z_i)] + 2\int \int \Phi (M,z)p(M,z)\frac{dV}{dz} dM dz , \end{equation*}$$

where the first sum is over the observed quasars, and the second is over the full luminosity and redshift range of the sample and provides the normalization (Marshall et al. 1983; see Fan et al. 2001a and Kelly et al. 2008 for alternative derivations of the likelihood function). The term p(M, z) is the probability that a quasar with absolute magnitude M (for our purposes, M₁₄₅₀) and redshift z is included in the survey; i.e., the selection function as derived in Section 6.1, including all sources of incompleteness. Confidence intervals are derived from the likelihood function using a χ² distribution in ΔS = S − S_min (Lampton et al. 1976).

Over a wide redshift range, the QLF is found to be well fit by a double power law (Boyle et al. 1988),

$$\begin{equation*} \Phi (M,z) = \frac{\Phi ^*(z)}{10^{0.4(\alpha +1)(M-M^*)} + 10^{0.4(\beta +1)(M-M^*)}} , \end{equation*}$$

where M* is the characteristic luminosity at which the function changes slope from steep at the bright end (β) to shallow at the faint end (α). The four QLF parameters may evolve with redshift, possibly in an interdependent manner. Given the limited redshift range of our survey, we will only account for the steep decline in number density at high redshift using the fit of Fan et al. (2001b): Φ*(z) = Φ*(z = 6) × 10^{k(z − 6)}, with k = −0.47.²⁶

Even with a sample of nearly 200 quasars spanning ΔM ≈ 4, there are substantial degeneracies in fitting the data with this parameterization. In particular, there are strong covariances between the placement of the break luminosity and the indices of the power-law slopes. We thus perform several fits while fixing one or more parameters.

Table 5 and Figure 16 show the results of several model fits with various choices for the fixed parameters. Uncertainties derived by varying a single parameter and calculating the likelihood for the best-fit solution for the other parameters are included. First, we fix the bright-end slope to β = −4.0. This value was chosen as it is approximately the slope derived from a single power-law fit to the brightest magnitude bins (Section 6.2). We choose to fix the bright-end slope as we find that our fits prefer high values for the break luminosity; interestingly, this implies that the bright-end slope is poorly constrained by our data (due to small numbers and limited luminosity range). This fit has a steep faint end slope and high break luminosity (α = −2.0 and $M_{1450}^*=-27.2$). We adopt these values as our best fit. Although the likelihoods can be improved by allowing even steeper values for β, this tends to drive the break luminosity near the limit of our data, where both parameters have considerable freedom while fitting.

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Table 5. MLE Fit Parameters

log Φ*(z = 6)a	$M_{1450}^*$	α	β
$-8.94^{+0.20}_{-0.24}$	$-27.21^{+0.27}_{-0.33}$	$-2.03^{+0.15}_{-0.14}$	−4.00
$-8.10^{+0.30}_{-0.25}$	$-25.88^{+0.60}_{-0.49}$	−1.50	$-3.12^{+0.28}_{-0.41}$
$-8.40^{+0.03}_{-0.03}$	−26.39	−1.80	−3.26

Notes. Parameters without uncertainty ranges are fixed during the maximum likelihood fitting. ^alog Φ*(z) = log Φ*(z = 6) + k(z − 6), with k = −0.47.

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We further explore the parameter degeneracies by calculating the joint likelihood ranges for each of the power-law slopes and $M_{1450}^*$. Figure 17 shows probability contours for α and $M_{1450}^*$, while allowing the other two parameters to vary. Figure 18 shows similar contours for β and $M_{1450}^*$. Comparison of these two figures shows that β is indeed more poorly constrained than α.

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Surveys of faint z ≲ 3 quasars typically find α = −1.5 (Hunt et al. 2004; Richards et al. 2005; Siana et al. 2008; Croom et al. 2009), while observations at high redshift favor a steeper value of α = −1.7 (Glikman et al. 2010; Ikeda et al. 2011; Masters et al. 2012), albeit with large uncertainties due to the difficulties of assembling large samples of faint quasars at high redshift. We find that a steeper value for the faint end slope is favored by our data, although α = −1.5 lies within our 3σ contour.

Figure 18 shows that fits to our data similarly prefer steep values of β. In fact, if we allow all of the parameters to be free, the maximum likelihood fitting tends to arbitrarily steep values for β. We consider fits with β < −4 to be effectively unconstrained, as this places the break luminosity at $M_{1450}^* \lesssim -27.4$, where we have only a handful of quasars. We thus impose a ceiling on the likelihood based on a fit with β = −4.0, and calculate probability contours relative to this fit. Nonetheless, it is clear that β is steep: β < −3.1 at 95% confidence. Further observations of bright quasars at z ∼ 5 are needed to better constrain the bright-end slope at this redshift.

The strong constraints on the steepness of the bright-end slope show that a flattening of the bright-end slope at high redshift, as found by, e.g., Richards et al. (2006), is not in agreement with our data. This is likely due to the fact that Richards et al. (2006) fit a single power law slope to the SDSS data under the assumption that the break luminosity was well below their flux limit; we will discuss this further in the following section.

We provide some context for these fits by comparing to results at lower and higher redshift. Ross et al. (2012) recently presented a QLF measurement at 2.2 < z < 3.5 from the BOSS DR9 quasar sample. Over this redshift range, the QLF is well-fit by a luminosity evolution and density evolution (LEDE) model, where the power law slopes have fixed values and the normalization and break luminosity evolve in a log-linear fashion. Specifically,

$$\begin{eqnarray} \log [\Phi ^{*}(z)] & = & \log [\Phi ^{*}(z=2.2)] + c_{1}(z-2.2), \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} M_{i,2}^{*}(z) & = &M_{i,2}^{*}(z=2.2) + c_{2}(z-2.2). \end{eqnarray} \tag{ 2 }$$

In Equation (2), M_{i, 2} ≡ M_i(z = 2) = M₁₄₅₀ − 1.486 is the absolute i-band magnitude at z = 2 (Richards et al. 2006), corresponding to rest-frame ∼2600 Å and assuming a spectral index of α_ν = −0.5 (f_ν∝ν^α). The evolution in M* and Φ* given by this model is shown in Figure 19, extrapolated to higher redshift to compare with our data. While evolution in the power law slopes is not well constrained by the data, our slopes are reasonably consistent with the values obtained from the BOSS data, with some indication that the faint end slope steepens toward higher redshift (Figures 17 and 18). However, the values for M* and Φ* do not agree with the simple extrapolation of the LEDE model from lower redshift. This can be clearly seen in Figure 20, which shows the LEDE prediction at z = 4.9 significantly overestimates our measurements. We will discuss a modified form to the LEDE model that provides a better fit to the high-redshift evolution in Section 6.6.

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At higher redshift, we compare to the z ∼ 6 QLF measurement reported by Willott et al. (2010). The last row of Table 5 shows the results of a fit where all the parameters except Φ* have been fixed to the best-fit values from Willott et al. (2010) for a fixed faint end slope of α = −1.8 (we use this value as it provides a better fit to our data than their α = −1.5 fit). Figures 17 and 18 show that the α, β, and $M_{1450}^*$ from the Willott et al. (2010) QLF lie near the ∼2σ contours from our constraints at z = 5; however, the uncertainties on the Willott et al. (2010) values are not available and would likely eliminate any tension between the fitted values at z = 5 and z = 6 (see, e.g., their Figure 6 for the uncertainties on β and $M_{1450}^*$ for a fit with α = −1.5). What is more clear is that the normalization is quite different: it is a factor of ∼1.6 higher than predicted by evolving the Willott et al. (2010) model to z = 4.9 using k = −0.47, as they adopted for their fit (they obtained log Φ*(z = 6) = −8.6, in contrast to the value of log Φ*(z = 6) = −8.4 we obtain when fitting the shape of their QLF to our z = 5 data). Figure 20 illustrates the discrepancy in the normalization from the evolved Willott et al. (2010) model; we will discuss this point further in Section 6.6.

The empirical QLF model of Hopkins et al. (2007b, hereafter HRH07) combines observations in the optical, X-ray, and mid-infrared bands to construct a bolometric QLF from z = 0 to z = 5. At high redshift the most constraining data in HRH07 comes from optical surveys, which have shown a flattening of the bright-end slope at high redshift (Schmidt et al. 1995; Fan et al. 2001b; Richards et al. 2006). In the HRH07 model, the break luminosity increases with redshift until z ∼ 2 and then turns over, such that it is a factor of ∼10 lower luminosity at z = 5 than at z = 2. At z = 5, the HRH07 model predicts a quite low break luminosity ($M_{1450}^* \approx -22.6$) and a shallow bright-end slope (β ≈ −2.5). Figure 15 shows that the HRH07 model agrees with our data at −27 < M₁₄₅₀ < −26, which is not surprising since at z ∼ 5 their fit is mainly to the Richards et al. (2006) data. The agreement at lower luminosities arises because the bright-end slope from HRH07 roughly agrees with our faint end slope. At the bright end, we attribute the disagreement between our data and the HRH07 QLF to a steeper slope and a much brighter $M_{1450}^*$ than predicted by their model. This demonstrates the substantial degeneracies in the QLF parameters, as the HRH07 values for Φ* and $M_{1450}^*$ at z = 5 would not even appear on Figure 19, even though the model itself provides a good fit to our data at M₁₄₅₀ ≳ −27. The key difference between our work and HRH07 (and by extension; Richards et al. 2006) is the increased survey area at the bright end, which is needed to extend above the break luminosity at this redshift and make the break in the luminosity function more evident.

Finally, we examine the z = 5 QLF from Fontanot et al. (2007). This work combines bright quasars from SDSS DR3 with faint quasars from the Great Observatories Origins Deep Survey (Dickinson et al. 2003) over the redshift range 3.5 < z < 5.2. Figure 20 compares their pure density evolution model (12 from their Table 3) to our data. We find that the Fontanot et al. (2007) model overestimates our QLF measurements at all luminosities. Ikeda et al. (2012) similarly found some tension between their constraint derived from observations of z ∼ 5 quasar candidates drawn from COSMOS (Scoville et al. 2007), and suggested this may be due to the completeness correction applied by Fontanot et al. (2007). We show the Ikeda et al. (2012) upper limit in Figure 20.

In conclusion, we find evidence for a steepening of the faint end slope, and no evidence in favor of an evolution in the bright-end slope, although obtaining strong constraints on the evolutionary forms of these parameters is difficult with existing data. On the other hand, we do see evidence for strong evolution in the break luminosity, as it brightens from $M_{1450}^* \approx -25.4$ at z = 2.5 to $M_{1450}^* \approx -27.2$ at z = 5 (Figure 19). This evolution has consequences for surveys where the faint limit is near the break luminosity, as single power law fits to such data would naturally find a flatter slope. The problem is evident in Figure 15, which shows that a single power law can describe the SDSS DR3 data from Richards et al. (2006), the full range of which is near the break luminosity. The possibility that high redshift fits to the bright end of the QLF may be biased by a higher break luminosity was put forward by Assef et al. (2011) and Shen & Kelly (2012). Based on our fits at z = 5 and the greater dynamic range of our survey, we find this scenario to be plausible; i.e., the flattening of the bright-end slope at high redshift reported previously (Schmidt et al. 1995; Fan et al. 2001b; Richards et al. 2006; Hopkins et al. 2007b) may be attributed instead to rapid evolution in the break luminosity.

Figure 21 compares our data to various theoretical models for the QLF at z = 5. Shen (2009) provides a theoretical prediction for the evolution of the QLF in a cosmological framework by relating the growth of SMBHs to the hierarchical assembly of their host dark matter halos. In this model, quasar activity is triggered by major mergers of halos. Their fiducial model reproduces the observed QLF at 0.5 < z < 4.5, but underpredicts the observed QLF at higher redshifts. Indeed, their fiducial model lies below our data at z = 5 (Figure 21). Shen (2009) also has a variant of this fiducial model that includes a faster redshift evolution in the normalization of the scaling relation between peak quasar luminosity and host halo mass, a redshift evolution in the scatter of this relation and an increase in the upper host halo mass above which quasar triggering is cut off exponentially in order to better fit the QLF at higher redshifts (see Section 4.4 of Shen 2009 for details). This alternative model provides a good fit to our data at the bright end, but overpredicts the number counts at the faint end.

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Conroy & White (2013) introduce a model in which quasars are tied to galaxies in a straightforward manner through the M_BH–M_gal relation, so that the evolution of the QLF is determined by the evolution of this relation. Their model includes two free parameters that are allowed to vary with redshift: the quasar duty cycle and the normalization of the M_BH–M_gal relation. Conroy & White (2013) constrain these parameters from existing QLF measurements at 0.5 < z < 4.75. The high redshift constraints mainly come from the SDSS (Richards et al. 2006) and have limited dynamic range, leading to significant uncertainties in the z ∼ 5 prediction (represented by the gray shaded region in Figure 21, see also Figure 3 in Conroy & White 2013). Overall, the Conroy & White (2013) model provides a good fit to the data.

Degraf et al. (2010) present a QLF prediction based on hydrodynamic simulations that include radiative cooling, star formation, black holes, and feedback processes. This work has since been updated to a larger simulation volume, with 100 h⁻¹ Mpc on a side and 2 × 1792³ particles (MassiveBlackII; C. DeGraf et al., in preparation). At each time step, the QLF is calculated from the active black holes, and the final QLF prediction is derived by time-averaging the individual measurements. This allows the bright end of the QLF to be estimated by catching the brief episodes of peak luminosity among the rare, massive black hole population. Figure 21 shows the prediction of this model in our redshift range, including an estimate for cosmic variance derived by comparing two simulation volumes (represented by the extent of the shaded region). This model generally agrees with our data near and just below the break luminosity. It appears to be somewhat steeper than the data at faint luminosities, though fainter measurements of the QLF from deeper optical data or at other wavelengths (e.g., X-rays) are needed to better constrain the model.

We remind the reader that our QLF only accounts for unobscured, Type I quasars. Our measurements are lower limits on the true density of actively accreting black holes, assuming some fraction are in an obscured (or even mildly extincted) growth phase. For example, comparison to X-ray surveys indicates that only ∼25% of quasars are unobscured at z = 4 (Masters et al. 2012). Some inconsistency with theoretical models that do not distinguish between unobscured and obscured quasars is thus expected. Furthermore, a luminosity dependence for the obscured fraction (e.g., Ueda et al. 2003) could further bias comparisons of the data with theoretical models.

A rapid decline in the comoving number density of quasars at high redshift was observed three decades ago (Osmer 1982). Following Fan et al. (2001a), we quantify this evolution in terms of the spatial density of quasars above a minimum luminosity within a redshift window. The density is derived from the 1/V_a method, where for each quasar the volume within which it would have been observed within a survey is

$$\begin{equation*} V_a = \int _{\Delta z} p(M_{1450},z) \frac{dV}{dz} dz , \end{equation*}$$

where p(M, z) is again the selection function for the survey. From this equation, the total spatial density and its uncertainty are estimated by

$$\begin{equation*} \rho = \sum _i\frac{1}{V_a^i} ,\quad \sigma (\rho)= \left[\sum _i\left(\frac{1}{V_a^i}\right)^2\right]^{1/2} , \end{equation*}$$

where the sum is over all quasars more luminous than M. This density is related to the luminosity function in that

$$\begin{equation} \rho ({<}M,z) = \int _{-\infty }^{M}\Phi (M,z) dM , \end{equation} \tag{ 3 }$$

where ρ(< M, z) is the space density of quasars more luminous than M. We choose to perform a sum over our data rather than an integration over the QLF as the latter requires extrapolation and model fitting. We calculate this quantity at z ∼ 4, z ∼ 5, and z ∼ 6, combining data from SDSS, our work, and multiple surveys at z ∼ 6. We choose a limit of M₁₄₅₀ = −26 as it corresponds to the lowest luminosity quasars in the SDSS sample at z ∼ 4.

At z = 4.25 we use the uniform quasar sample from the SDSS DR7 described in Section 6.1.1. This sample includes 311 quasars with 4.1 < z < 4.4, representing a factor of four increase in number over the SDSS DR3 results given in Richards et al. (2006). We adopt the selection function given in Table 1 of that work rather than recalculate it from our simulations, as the SDSS quasar target selection has a complicated dependence on the extent of the stellar locus in color space (Richards et al. 2002), which is captured by the Richards et al. (2006) selection function. At z = 4.9 we use our combined sample from the SDSS DR7 and from Stripe 82 and the selection functions presented in Section 6.1. Finally, at z = 6 we use the sample compiled by Willott et al. (2010), consisting of quasars from the SDSS main (Fan et al. 2001a, 2003, 2004, 2006b), SDSS deep (Jiang et al. 2008, 2009), and CFHTQS (Willott et al. 2010). We derive selection functions for the three z ∼ 6 quasar surveys using our quasar simulations, extended to higher redshift. The derived selection functions from our models agree well with those shown in Figure 4 of Willott et al. (2010).

Figure 22 shows the evolution of the space density of luminous quasars (M₁₄₅₀ < −26) from z = 4 to z = 6. Our measurement at z = 5 shows a decline in the space density by a factor of 1.8 from z = 4.25. Fan et al. (2001b) fit an exponential decline to the space density at high redshifts, finding that ρ(M₁₄₅₀ < −25.5, z) ∼ 10^kz with k = −0.47 at z > 3.6, about a factor of three per unit redshift. Brusa et al. (2009) reported a similar value (k = −0.43) using X-ray-selected quasars at z ∼ 3.0–4.5 (log L_X > 44.2). Our results from z = 4.25 to z = 5 are in good agreement with these results; we measure a decline of 2.4 ± 0.3 per unit redshift (k = −0.38).

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On the other hand, the evolution to z = 6 is more pronounced: a factor of 5.1 ± 1.5 per unit redshift (k = −0.7), or roughly twice the rate measured at lower redshift (Figure 22). The departure of the z = 6 measurement from the prediction based on the slope fit to the 4 ≲ z ≲ 5 data is significant at the 4.5σ level. If we integrate deeper for the two high-redshift bins, to a limit of M₁₄₅₀ < −24.5, the decline is even stronger. We obtain ρ(M₁₄₅₀ < −24.5, z = 4.9) = 4.2 ± 0.7 × 10⁻⁸ Mpc⁻³ and ρ(M₁₄₅₀ < −24.5, z = 6.0) = 0.5 ± 0.1 × 10⁻⁸ Mpc⁻³, corresponding to a factor of 6.5 ± 1.7 decline per unit redshift.

We check the consistency of our faint-end measurements by comparing to recent work at X-ray wavelengths with the Chandra-COSMOS survey (Civano et al. 2011), which is sensitive to moderate-luminosity (log L_{(2 − 10 keV)} > 44.15 erg s⁻¹) AGNs at z ∼ 5. In order to compare the two samples, we must integrate our double power law model to M₁₄₅₀ = −24.0 (adopting the α_ox relation from Young et al. 2010). Civano et al. (2011) identified three X-ray-selected Type I quasars in the redshift bin 4.2 < z < 5.4, which translates to an observed space density of Φ(4.2 < z < 5.4) = (2.5 ± 1.5) × 10⁻⁷ Mpc⁻³. Integrating our best-fit model for the QLF over that redshift bin, we obtain Φ(4.2 < z < 5.4) = 0.8 × 10⁻⁷ Mpc⁻³. Thus our QLF agrees with the Civano et al. (2011) results to within the ∼1σ uncertainties, especially when considering that to make the comparison we must extrapolate our QLF in both luminosity and redshift. The X-ray data do not strongly constrain the evolution to z ∼ 6 due to the small area of the Chandra-COSMOS survey (from zero z ∼ 6 objects in Civano et al. 2011; the decline from z = 5 to z = 6 is >2 per unit redshift).

We now relate the decline in the high-redshift quasar density to the evolutionary model for the QLF derived from the BOSS DR9 (Ross et al. 2012). In this LEDE model, log Φ* declines by 0.6 dex per unit redshift, while M* brightens by 0.68 mag per unit redshift (c₁ = −0.6 in Equation (1) and c₂ = −0.68 in Equation (2), respectively). Integrating this model (Equation (3)) shows that it significantly overpredicts the high-redshift number densities we have derived (Figure 22). This result suggests that the steeper decline in the high-redshift number counts begins at z ≲ 4, and is also consistent with Figure 19, which shows that the log Φ* predicted by the LEDE model is higher than the data at z ≳ 5, and is not well constrained at z ∼ 4. We thus modify the LEDE model from BOSS by steepening the slope of the log Φ*(z) evolution to c₁ = −0.7, and softening the slope of the $M_{1450}^*$ evolution to c₂ = −0.55. These modifications are within the 1σ uncertainties of the values derived from fitting BOSS data; Figure 22 shows that they bring the QLF model in agreement with the integral constraints from the z ∼ 4–5 data, but not the z = 6 point. At z = 4.9, this modified evolutionary model predicts $M_{1450}^*=-26.8$ and log Φ* = −8.0. This provides a good match to the values we obtained when fixing the slopes to the BOSS values during the MLE fit, $M_{1450}^*=-26.6$ and log Φ*(z = 4.9) = −7.9 (Table 5).

Finally, we consider further modifying the model for the evolution of M*. It is unlikely that M* continues to rise to very high luminosities; indeed, Figure 19 shows that the high-redshift fits for M* are somewhat below the LEDE prediction. We thus impose a maximum break luminosity of $M_{1450}^*=-27$, which in the modified LEDE model is reached at z ∼ 5.4. This choice causes a turnover in the high-redshift number densities that matches the data at z ∼ 6 (Figure 22). This model is somewhat arbitrary, but qualitatively, a model in which Φ* has a more rapid downward evolution at high redshift and M* brightens until z ∼ 4–5 and then levels off (or even turns over) provides a reasonable description of the high-redshift data. A quasar with $M_{1450}^*=-27$ radiating at the Eddington luminosity corresponds to a ∼2 × 10⁹M_☉ black hole; it seems reasonable to suppose that while the break luminosity evolves strongly at high redshift, it would not greatly exceed this value.

The number of ionizing photons required to maintain hydrogen ionization as a function of redshift can be found by balancing the ionizing photon emissivity with the density of hydrogen and the rate of recombinations. The recombination rate depends on the clumping factor, $C={\langle }n_H^2{\rangle } /{\langle }n_H{\rangle }^2$. Madau et al. (1999) present an equation for the ionizing photon density $\dot{N}_{\rm ion}$ that adopts a high value for the clumping factor (C = 30). More recent work suggests that the clumping factor is not so large; Meiksin (2005) argues that C ≈ 5 and McQuinn et al. (2011) use cosmological simulations that include the effects of LLSs and find C ≈ 2–3 at z = 6. Similarly, both Shull et al. (2012a) and Finlator et al. (2012) prefer a lower clumping factor of C ≈ 3 at z ∼ 6 in their reionization models.

We calculate the ionizing photon output for each quasar by assuming a broken power-law SED with an index of α_ν = −1.7 at ultraviolet wavelengths (Telfer et al. 2002), a break at 1100 Å, and an index of α_ν = −0.5 above the break.²⁷ Rescaling the Madau et al. (1999) equation and updating to our cosmology, we estimate the number of photons required to maintain full ionization at z = 5 to be $\dot{N}_{\rm ion} = 3.4\times 10^{50} (C/5) \, {\rm Mpc}^{-3}\, {\rm s}^{-1}$. Integrating the best-fit QLF model (with β = −4.0) to M₁₄₅₀ = −20 gives $\dot{N}_Q \sim 9.5 \times 10^{49}\, {\rm Mpc}^{-3}\, {\rm s}^{-1}$, or ≈28% of the number required (for the flatter faint-end slopes, the percentage lowers to ∼20%). For C = 2, quasars provide ∼70% of the photons required for hydrogen ionization, suggesting they may have played some role in maintaining ionization at z ∼ 5. However, the steeper decline of luminous quasars from z ∼ 5 to z ∼ 6 further reduces the likelihood that quasars were an important source of ionizing photons during the reionization epoch. Indeed, even when assuming a steep faint-end slope (α = −1.8), Willott et al. (2010) find that z ∼ 6 quasars generate <10% of the required ionizing photon background. This is also in good agreement with constraints from deep X-ray surveys (Barger et al. 2003; Fontanot et al. 2007) that limit the contribution from moderate luminosity AGNs at z > 6.