A SURVEY OF LUMINOUS HIGH-REDSHIFT QUASARS WITH SDSS AND WISE. II. THE BRIGHT END OF THE QUASAR LUMINOSITY FUNCTION AT z ∼ 5

Our SDSS+WISE selection technique and spectroscopic follow-up observations were discussed in detail in Paper I. Here we briefly review the basic steps. At z ∼ 5, most quasars are undetectable in the u band and g band because of the presence of strong Lyman limit systems, which are optically thick to the UV continuum radiation from quasars (Fan et al. 1999). The Lyα absorption systems also begin to dominate in the r band and Lyα emission moves to the i band. Therefore, the r − i/i − z color–color diagram was often used to select z ∼ 5 quasar candidates in previous studies (Fan et al. 1999; Richards et al. 2002; M13). However, with increasing redshift, the i − z color also becomes increasingly red and most z > 5.1 quasars enter the M star locus in the r − i/i − z color–color diagram, which makes it difficult to find z > 5.1 quasars with only the optical colors. Therefore, we added near-infrared colors from WISE photometry data in our selection. We used typical u, g drop-out methods but more relaxed r − i/i − z cuts to select candidates from the SDSS DR10 database. Then we cross-matched our candidates with the ALLWISE database using a 2'' match radius and used z-W1/W1-W2 cuts to remove more star contaminations by the following criteria. The exact selection criteria are given in Paper I.

$$\begin{eqnarray}&&z-W1\gt 2.5\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&W1-W2\gt 0.5\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&W1\lt 17.0,{\sigma }_{W2}\lt 0.2\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&z-W1\gt 2.8\,{\rm{or}}\,W1-W2\gt 0.7,\mathrm{if}\,i-z\gt 0.4.\end{eqnarray} \tag{ 4 }$$

We constructed our main luminous quasar candidate sample by limiting the SDSS z-band magnitudes to brighter than 19.5, and selected a total of 420 luminous z ∼ 5 quasar candidates. We removed 78 known quasars, one known dwarf and 231 candidates with suspicious detections, such as multiple peaked objects or being affected by bright star artifacts. We visually inspected images of each candidate and removed those 231 candidates. We selected 110 candidates with high image quality as our main candidate sample. Our spectroscopic follow-up campaign started in 2013 October. We observed 99 candidates from our main sample with the Lijiang 2.4 m telescope (LJT) and Xinglong 2.16 m telescope in China, the Kitt Peak 2.3 m Bok telescope and 6.5 m MMT telescope in the U.S., as well as the 2.3 m ANU telescope in Australia. 64 (64.6%) candidates have been identified as high-redshift quasars in the redshift range 4.4 ≲ z ≲ 5.5. As discussed in Paper I, due to the serious contamination from M-type stars, there is a gap in the previously published quasar redshift distribution at 5.2 < z < 5.7 with only 33 published quasars in this redshift range. Among our 64 newly identified quasars from main candidates sample, 9 quasars are at 5.2 < z < 5.7, which represents an increase of 27% in the number of known quasars in this redshift range. The details of spectroscopic observation and data reduction are also given in Paper I.

The redshifts of newly identified quasars are measured from Lyα, N v, O i/Si ii, C ii, Si iv, and C iv emission lines (any available) by an eye-recognition assistant for quasar spectra software (ASERA; Yuan et al. 2013). The typical redshift error is about 0.05 for Lyα-based redshift measurement and will be less for that based on more emission lines. We calculate M₁₄₅₀ in the AB system by fitting a power-law continuum ${f}_{\nu }\sim {\nu }^{{\alpha }_{\nu }}$ to the spectrum for each quasar. We assume an average quasar UV continuum slope of α_ν = −0.5 (Vanden Berk et al. 2001; see details in Paper I). Our 64 new quasars from the luminous quasar candidate sample are within the absolute magnitude range −29 < M₁₄₅₀ < −26.4. We calculate M₁₄₅₀ for previous known quasars using the same method. The known quasars are from the SDSS DR7 and DR12 quasar catalogs (Schneider et al. 2010; Pãris et al. 2016; McGreer et al. 2013; Schneider et al. 1991). When we removed the known quasars from our quasar candidate sample, we missed two known quasars. These two quasars from McGreer et al. (2009) and Schneider et al. (1991) were also spectroscopically observed by us, and thus we use our new spectra to do the M₁₄₅₀ calculation.

For the QLF determination, we define our sample of z ∼ 5 luminous quasars as follows.

• 1.  
  Quasars in the redshift range 4.7 ≤ z < 5.4. Our selection criteria yield low completeness at redshifts lower than 4.7 or higher than 5.4. The former is caused by the drop of $W1-W2$ color (See Figure 3 in Paper I), and the latter is caused by our $r-i/i-z$ limit. Therefore, we restrict our sample to the range of 4.7 ≤ z < 5.4 (see details in Section 3.2).
• 2.  
  Quasars in the luminosity range of M₁₄₅₀ ≤ −26.8. Our selection criteria yield a low completeness in the region with z > 5 and −26.8 < M₁₄₅₀ < −26.4. The mean completeness in this region is ∼4%. That is caused by our SDSS magnitude limit of z < 19.5. Therefore, we limit our sample to M₁₄₅₀ ≤ −26.8.
• 3.  
  Our selection covers the whole SDSS footprint without masked regions, which is a 14,555 square degree field.

Based on the criteria above, there are 45 newly identified quasars in the sample of Paper I and another 54 previously known quasars that satisfy our selection criteria. This is the final complete z ∼ 5 luminous quasar sample that we will use to determine the z ∼ 5 QLF. Figure 1 shows the redshift and M₁₄₅₀ distributions of both our newly identified luminous quasars and known quasars. Three of our new quasars are more luminous than any previously known quasars at z > 5. It is obvious that our discovery significantly expands the z ∼ 5 luminous quasar sample. Table 1 lists all 99 quasars in our sample used for the QLF determination.

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The magnitude limit of our main sample is SDSS z = 19.5, which is much brighter than the flux limit (5σ) of the SDSS survey. Therefore, within our magnitude limit, the SDSS detections can be considered as complete. Our survey adds ALLWISE W1 and W2 photometric data into the selection, and thus we need to consider the detection incompleteness caused by the shallower ALLWISE detection limit. We correct this by using the ALLWISE detection completeness from the Explanatory Supplement to the AllWISE Data Release Products¹⁰ , which is a function of frame coverage and flux in W1 and W2 bands respectively. Figure 2 represents the empirical models of 2D detection completeness in W1 and W2 bands. As shown, our sample limited with W1 < 17 and σ_W2 < 0.2 will be effected slightly by the detection incompleteness at the faint end.

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The ALLWISE coverage depends on the sky position. To take the position-dependence into account on completeness correction, we mapped the ALLWISE spatial surveying depth within the SDSS footprint. We first randomly generated ∼1,220,000 positions in the whole SDSS footprint and derived the ALLWISE coverage map in the SDSS footprint by matching(1') positions to the nearest ALLWISE sources. A detection with coverage ≤5 could be contaminated by random pixel variations such as cosmic rays because they are at or below the threshold for ALLWISE statistically viable outlier detection and rejection. So we removed all positions with frame coverage ≤5. There are only 269 (0.02%) positions with coverage ≤5. Figure 3 shows the distributions of W1/W2 frame coverages in the whole SDSS footprint. The average coverage is 36 in both W1 and W2 bands. The 10% and 90% tile coverage in W1/W2 band are 23/22 and 56/56. We use our ALLWISE coverage map to correct the detection incompleteness (see details in Section 3.2).

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ALLWISE coverage also affects the photometric errors of detected sources. The photometric error in W1/W2 will be a function of magnitude and coverage. We used all point sources in our ALLWISE sources sample discussed above to model the empirical magnitude–coverage–magnitude error relations for the ALLWISE W1 and W2 bands. The ALLWISE sensitivity improves approximately as the square root of the depth of coverage, σ ∝ 1/$\sqrt{{N}_{\mathrm{cov}}}$, N_cov is the number of frame coverage. Considering this, we first eliminated the effect of coverage on magnitude errors and then fit the relations between W1/W2 magnitude and coverage-corrected magnitude error. Based on the WISE all-sky magnitude-error relation (Wright et al. 2010), the final ALLWISE magnitude–coverage–error relation we obtained is

$$\begin{eqnarray}&&\sigma (m,{N}_{\mathrm{cov}})=a+[2.5/\mathrm{ln}(10)]n/{10}^{-0.4m}/\sqrt{{N}_{\mathrm{cov}}}\,.\end{eqnarray} \tag{ 5 }$$

Where m is the magnitude in W1/W2. Constant a is basic photometric error, equal to 0.01 in WLSE all-sky photometry (Wright et al. 2010). We found that a should be 0.022 for W1 and 0.019 for W2 in ALLWISE photometry. The best-fitted parameter n is 6.43e–8 for W1, 2.71e–7 for W2. Figure 4 shows our empirical model compared with observed data.

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To estimate the completeness of our selection criteria, we generate a sample of simulated quasars following the procedure in Fan (1999). M13 updated the spectral model of Fan (1999) and applied it to higher redshift, assuming that the quasar spectral energy distributions do not evolve with redshift (Kuhn et al. 2001; Yip et al. 2004; Jiang et al. 2006). We extend this model toward redder wavelengths to cover the ALLWISE W1, W2 bands for quasars at z = 4–6 (I. D. McGreer et al. 2016, in preparation). The quasar spectrum from M13 is modeled as a power-law continuum with a break at 1100 Å. For redder wavelength coverage, we added three new breaks at 5700, 10850, and 22300 Å. The slope (α_ν) from 5700 to 10850 Å follows a Gaussian distribution of μ(α) = −0.48 and σ(α) = 0.3; the middle range has a slope with the distribution of μ(α) = −1.74 and σ(α) = 0.3; and at the red end, the slope distribution has μ(α) = −1.17 and σ(α) = 0.3 (Glikman et al. 2006). The parameters of emission lines are derived from the composite quasar spectra (Glikman et al. 2006). Although the composite spectrum from Glikman et al. (2006) is constructed from fainter lower redshift quasars, it does not have obvious difference with composite spectra built from luminous quasars at high redshift (e.g., Selsing et al. 2016). It is the only one we know that can cover both W1 and W2 bands in the redshift range of our simulation (4 < z < 6). The IGM absorption model is the same as M13, which extends the Ly_α forest model based on the work of Worseck & Prochaska (2011) to higher redshift by using the observed number densities of high column density systems (Songaila & Cowie 2010). Compared to M13, we have made minor modifications for Fe emission. We use the template from Vestergaard & Wilkes (2001) for wavelengths shorter than 2200 Å. For 2200–3500 Å, we use the template from Tsuzuki et al. (2006), which separates the Fe ii emission from the Mg ii λ2798 line. A template from Boroson & Green (1992) covering 3500–7500 Å is also added.

Based on this model, we generate a sample of simulated quasars and then calculate the selection function of our color–color selection. We construct a grid of quasars in the redshift range of 4 < z < 6 and the luminosity range of −29.5 < M₁₄₅₀ < −25.5. A total of 314,000 simulated quasars has been generated and evenly distributed in the (M₁₄₅₀, z) space. There are ∼200 quasars in each (M₁₄₅₀, z) bin with ΔM = 0.1 and Δz = 0.05. We assign optical photometric errors, which are from the SDSS main survey, and photometric uncertainties of the W1 and W2 bands using the empirical magnitude–coverage–error relations discussed above. We added the ALLWISE detection completeness into the selection probability calculation.

We calculate the ALLWISE detection probability by randomly choosing a unique sky position from our 1,220,000 positions for each simulated quasar, and thus obtained an ALLWISE detection probability of each simulated quasar based on its frame's coverage and W1 and W2 magnitude. For each (M₁₄₅₀, z) bin (ΔM = 0.1 and Δz = 0.05) discussed above, we obtain a mean detection probability. Then we calculate the fraction of simulated quasars selected by our selection criteria in each (M₁₄₅₀, z) bin as the selection probability, shown in Figure 5.

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As shown in Figure 5, after relaxing the traditional r − i/i − z color cut and adding the $W1-W2$ color, our color selection criteria show a high completeness at 4.8 < z < 5.2 and extend the selection region to z ∼ 5.4. Within the central bright region (4.8 < z < 5.2 and M₁₄₅₀ < −26.8), the mean completeness reaches 78%. Extended to the range of 4.7 < z < 5.4, the mean completeness is ∼60%. At a redshift lower than 4.5 or higher than 5.4, the completeness is below than 5%. At z < 4.7, the $W1-W2$ color becomes bluer; our $W1-W2$ > 0.5 cut will miss some quasars at z < 4.7 (see Figure 3 in Paper I ). However, the exact completeness is more sensitive to the assumption we made about the rest-frame optical continuum of high-redshift quasars at 4.5 < z < 4.7 due to the fast change of $W1-W2$ color here. The uncertainty of the simulation in this redshift range is higher than that at z > 4.7. Therefore, we restrict our quasar sample for the QLF calculation to z ≥ 4.7. At 5.2 < z < 5.4, although the completeness becomes lower, our selection has explored a higher redshift range with higher completeness than previous works at z ∼ 5. Thus, we limit our sample with z < 5.4.

To see how the ALLWISE coverage affects our selection function, we also calculate the selection function by assuming a fixed number of coverage, 10% tile ALLWISE coverage (N_cov = 23 for W1, N_cov = 22 for W2), shown in Figure 6. The difference in the selection function between using 10% tile coverage and position-dependent coverage is less than 5% at M₁₄₅₀ < −27 and increases to ∼10% at M₁₄₅₀ < −26.8, to 15%–20% at a fainter range. We also compare the QLF result base on 10% tile coverage and position-dependent coverage. The change of the parameters of best-fitted QLFs is ∼0.05, much smaller than error bars. We use the position-dependent coverage selection function to calculate the parametric QLF. When we calculate the selection probability of each quasar in our sample for a binned QLF measurement, we use the real ALLWISE coverage to calculate the ALLWISE detection completeness of each quasar. The agreement between binned QLF and best-fit parametric QLF (See Sections 4.1 and 4.2) shows that our ALLWISE coverage model and the selection function using mean detection incompleteness are reasonable.

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We spectroscopically observed 99 out of 110 candidates. The spectroscopic completeness reaches 100% at z_SDSS ≤ 19; at the fainter end, the completeness is lower but it still has a high value around 80%. The histogram of our observed and unobserved candidates is shown in Figure 7. The completeness is a function of z-band magnitude. We use this function to correct the incompleteness from spectral coverage, assuming the probability of an unobserved candidate to be a quasar is the same as in the observed sample. As shown, the quasar fraction in our unknown candidate sample becomes lower at the faint end. That is caused by the fact that there are more known quasars at z_SDSS > 19, and these known quasars are not plotted in this figure.

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To compute the binned QLF, we divide our sample into several bins. Due to the narrow redshift interval of our sample, we only use one redshift bin and do not include any evolution with redshift. We then divide our sample into 5 mag bins with ΔM₁₄₅₀ = 0.5 mag over the magnitude range −26.8 < M₁₄₅₀ < −29.3 (see Figure 1). We calculate the binned luminosity function by using the Page & Carrera (2000) modification of the 1/V_a method (Schmidt 1968; Avni & Bahcall 1980) for flux limit correction. The final selection function is applied after all incompleteness corrections have been applied for each quasar. The result for the binned QLF and number counts are listed in Table 2. In the table, the number counts and corrected number counts derived by applying all incompleteness corrections are denoted as N and N_cor respectively. The result is also displayed in Figure 8 as red squares together with the binned QLF data from the SDSS main (black) and Stripe 82 (blue) samples in M13 for comparison. Data from M13 have been corrected to z = 5.05 by using the quasar redshift evolution at high redshifts according to Fan et al. (2001b). Compared to previous results, our binned QLF has more luminous quasars and extends the measurement of the z ∼ 5 QLF to M₁₄₅₀ = −29, and thus gives a smaller error bar in each bin at M₁₄₅₀ < −27.05. Our data show a similar result, but suggest a higher value at the bright end. The binned QLF in the brightest bin has a large error bar due to the fact that there is only one quasar in this bin.

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The binned QLF result, while non-parametric, is dependent on the choice of binning. Here we derive a parametric QLF by performing a maximum likelihood fit for each quasar in the sample. We model the QLF using the most common double power-law form (Boyle et al. 2000):

$$\begin{eqnarray}&&{\rm{\Phi }}(M,z)=\displaystyle \frac{{{\rm{\Phi }}}^{* }(z)}{{10}^{0.4(\alpha +1)(M-{M}^{* })}+{10}^{0.4(\beta +1)(M-{M}^{* })}}\,,\end{eqnarray} \tag{ 6 }$$

where α and β are the faint end and the bright end slopes, M* is the break magnitude, and ${{\rm{\Phi }}}^{* }(z)$ is the normalization. These four parameters have been suggested to evolve with redshift. Following previous work, we adopt the rapid decline in quasar number density at high redshift from Fan et al. (2001b) as the QLF evolution within our narrow redshift interval, ${{\rm{\Phi }}}^{* }(z)={{\rm{\Phi }}}^{* }(z=6)\times {10}^{k(z-6)}$, where k = −0.47 (Fan et al. 2001b).¹¹ Here we also normalize Φ* to z = 6 for easier comparison to the higher redshift results.

Due to the fact that our quasar sample covers the magnitude range of M₁₄₅₀ ≤ −26.8, and the break magnitude given by M13 is around −26 to −27, our sample cannot be used to constrain the faint end slope. For measurement of the break magnitude M*, we combine our luminous quasar sample with the S82 and DR7 quasar samples from M13 and then carry out parametric fits for the QLF for all observed quasars in the combined sample. The DR7 sample has a large number of overlaps with our luminous quasar sample. Therefore, we select DR7 quasars only in the magnitude range −26.8 < M₁₄₅₀ < −25.8 to construct the combined sample. For the S82 and DR7 samples, we use the same incompleteness corrections as in M13. We use maximum likelihood estimation to derive the fit. The maximum likelihood fit (Marshall et al. 1983) for a luminosity function aims to minimize the log likelihood function S, which is equal to $-2\mathrm{ln}L$, where L is the likelihood function:

$$\begin{eqnarray}&&S=-2\displaystyle \sum _{i}^{N}\mathrm{ln}[{\rm{\Phi }}({M}_{i},{z}_{i})]+2\int \int {\rm{\Phi }}(M,z)p(M,z)\displaystyle \frac{{dV}}{{dz}}{dMdz},\end{eqnarray} \tag{ 7 }$$

where the first term is the sum over all observed quasars in the sample, and the second term is integrated over the full range of absolute magnitude and redshift of the sample (Marshall et al. 1983; Fan et al. 2001a); $p(M,z)$ is the probability for a quasar to be observed by the survey at given absolute magnitude M₁₄₅₀ and redshift z. It includes all incompleteness corrections discussed above. The second term represents the total number of expected quasars in the survey with a given luminosity function, and provides the normalization for the likelihood function. The confidence intervals are determined from the likelihood function by assuming a χ² distribution of ΔS (=S −S_min) (Lampton et al. 1976).

We first fix the faint-end slope α to be −2.03, as given by M13. We find the luminosity function parameters to be log ${{\rm{\Phi }}}^{* }(z=6)$ = −8.82 ± 0.15, ${M}_{1450}^{* }=-26.98\pm 0.23$ and β = −3.58 ± 0.24. This result is plotted in Figure 9 and shows excellent agreement with our binned QLF. In order to investigate how the different values of α affect our result, we also assume the faint-end slope α to be −1.8, similar to what was measured from quasar samples at z ≥ 4 (Glikman et al. 2010; Willott et al. 2010; Masters et al. 2012; McGreer et al. 2013), and to −1.5, typical for lower redshift measurements at z ≲ 3 (Croom et al. 2009). The values of parameters assuming different faint-end slopes are listed in Table 3. When we change the faint-end slope α from −2.03 to −1.8 and −1.5, the bright end slope β is flattened but only at the ≲1σ level. The break magnitude becomes fainter more significantly following the change of β. If we allow all four parameters to be unconstrained, we derive a steeper bright end slope β = −3.80 and a very bright break magnitude of ${M}_{1450}^{* }=-27.33$ with significantly larger error bars. Allowing all parameters to be free has many degeneracies, which is why the uncertainty ranges are larger. In this case, the faint-end slope α also becomes steeper (α = −2.15). We need more data to better constrain a four-parameter fit, especially for M₁₄₅₀ < −28.3. Considering that α = −2.03 derived by M13 is a strong constraint on the faint-end slope, we adopt the result based on fixed α = −2.03 as our best fit. The fitted QLFs for different cases are plotted in Figure 9.

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Table 3.  Parameters of Fits

α	β	${M}_{1450}^{* }$	log${{\rm{\Phi }}}^{* }$(z = 6)
−2.03	−3.58 ± 0.24	−26.98 ± 0.23	−8.82 ± 0.15
−1.80	−3.26 ± 0.18	−26.28 ± 0.29	−8.35 ± 0.17
−1.50	−3.03 ± 0.12	−25.56 ± 0.29	−7.94 ± 0.15
−2.14 ± 0.16	−3.80 ± 0.47	−27.32 ± 0.53	−9.07 ± 0.40

Note. We fix the faint slope α to be −2.03, −1.8, and −1.5 respectively. Then we allow all four parameters to be free. α = −2.03 is measured from the combination of the SDSS S82 and DR7 samples in M13. We adopt the result with fixed α = −2.03 as our best fit.

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We calculate the confidence regions to investigate the degeneracy between the bright end slope and break magnitude; the results are shown in Figure 10. The regions filled with different colors illustrate 1σ (68.3%), 2σ (95.4%), and 3σ (99.7%) regions, respectively. We generate the probability contours by calculating S_min for each (M₁₄₅₀, β) point and allowing $\mathrm{log}{\rm{\Phi }}{(z=6)}^{* }$ to be free at each point with the fixed α = −2.03. Figure 10 shows that our data constrain β to the range  of −4.83 < β < −2.78 at 95% confidence; this is flatter than the result from M13, which shows β < −3.1 at 95% confidence, although the best fit from M13, β = −4, lies within our 2σ region.

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We compare the parameters of our result with previous work at different redshifts to study the evolution of the QLF. In Figure 11, we plot the evolution of the normalization Φ*, the break magnitude ${M}_{1450}^{* }$, and the bright end slope β with redshift. Ross et al. (2013) measured the QLF at 2.2 < z < 3.5 using the BOSS DR9 quasar sample and concluded that the QLF can be described well by an LEDE model at this redshift range. In this model, the evolutions of normalization and break luminosity with redshift are expressed in a log–linear relation, and slopes of the double power law are fixed. M13 add a point at z = 4.9 and combine the result from Masters et al. (2012) at z = 4 and the result from Willott et al. (2010) at z = 6 to modify this model. They found that the slope of the normalization evolution was steeper (c1 = −0.7) and the slope of the break magnitude evolution was shallower (c₂ = −0.55). Now we add our new measurement at z = 5.05 and the point at z = 6 from Kashikawa et al. (2015) in case 1. The result from Kashikawa et al. (2015) includes the discovery of new faint z ∼ 6 quasars and places stronger constraints on the faint-end slope and break magnitude of the z ∼ 6 QLF. Then we use all of these points to fit the LEDE model.

$$\begin{eqnarray}&&\mathrm{log}[{{\rm{\Phi }}}^{* }(z)]=\mathrm{log}[{{\rm{\Phi }}}^{* }(z=2.2)]+{c}_{1}(z-2.2),\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{M}_{i,2}^{* }(z)={M}_{i,2}^{* }(z=2.2)+{c}_{2}(z-2.2),\end{eqnarray} \tag{ 9 }$$

where ${M}_{i,2}\equiv {M}_{i}(z=2)={M}_{1450}-1.486$ is the absolute i-band magnitude at z = 2 (Richards et al. 2006), corresponding to rest-frame ∼2600 Å in the assumption of a spectral index of α_ν = −0.5. We obtain values of log ${{\rm{\Phi }}}^{* }(z=2.2)\,=-5.87\pm 0.07$ and c1 = −0.81 ± 0.03; ${M}_{i,2}^{* }(z=2.2)\,=-26.68\pm 0.15$ and c2 = −0.50 ± 0.08. Note that the errors of parameters are standard deviation errors of fit. We only use these points without uncertainties to do the fit because the uncertainties of the best fit in Willott et al. (2010) are not reported. The real errors should be larger than the fitting errors explored here. Our result is consistent with the LEDE model but prefers a steeper slope of log ${{\rm{\Phi }}}^{* }(z)$ evolution and a flatter slope of the break magnitude evolution.¹²

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Previous measurements of the QLF at z = 5 and 6 have shown evidence that quasars cannot produce the entire required ionizing photon background (M13; Meiksin 2005; Bolton & Haehnelt 2007; Willott et al. 2010; Kashikawa et al. 2015) at those redshifts. It is suggested that quasars can contribute ∼30%–70% of the ionizing photons required to maintain full ionization at z = 5 (M13), and produce about several percent to 15% at z = 6 (Willott et al. 2010; Kashikawa et al. 2015), depending on the assumed IGM clumping factor C. Here we update the quasar contribution to the high-redshift ionizing background using our new QLF at z ∼ 5.

We calculate the comoving emissivity of quasars at the Lyman limit by $\epsilon (z)=\int \phi ({L}_{\nu },z){L}_{\nu }{{dL}}_{\nu }$ erg s⁻¹ Hz⁻¹ Mpc⁻³, assuming the escape fraction of ionizing radiation from quasars f = 1. We integrate our parametric QLF ${\rm{\Phi }}({M}_{1450},z)$ and then convert it into emissivity at λ = 912 Å. For the conversion, we adopt the UV slopes from Stevans et al. (2014), who suggest a gradual break wavelength at 1000 Å, with the index α_ν = −1.41 in the extreme ultraviolet and a spectral index α_ν = −0.83 at wavelengths above the break. By integrating our best-fit QLF to M₁₄₅₀ = −20, we derive an ionizing photon density ${\dot{N}}_{Q}$ = 6.06 × 10⁴⁹ Mpc⁻³ s⁻¹. When we use the QLF result with a fixed faint-end slope of α = −1.8, the photon number density changes to ${\dot{N}}_{Q}$ = 4.73 × 10⁴⁹ Mpc⁻³ s⁻¹. Using the QLF generated from the four-parameter fit, we get ${\dot{N}}_{Q}$ = 7.37 × 10⁴⁹ Mpc⁻³ s⁻¹. The change of ionizing photon density is dominated by the change of the break magnitude ${M}_{1450}^{* }$ and the faint-end slope α. Luminous quasars make little contribution to the ionizing background, so a survey of faint quasars is required to give a more accurate measurement.

The required number of photons to balance hydrogen recombination and maintain full ionization was estimated by Madau et al. (1999) as a function of redshift. The number of required photons at z = 5 is ${\dot{N}}_{{\rm{ion}}}=3.38\,\times {10}^{50}(C/5)\,{{\rm{Mpc}}}^{-3}\,{{\rm{s}}}^{-1}$ in our adopted cosmology. The clumping factor C is crucial to estimate the contribution of quasars to the ionizing background. Madau et al. (1999) considered a recombination-dominated IGM and suggested a high value for the clumping factor C = 30. Recent work provides a lower clumping factor C < 10. Meiksin (2005) suggests that C ≈ 5 and a C ≈ 2–3 at z = 6 is suggested by some reionization models (McQuinn et al. 2011; Finlator et al. 2012; Shull et al. 2012). For C = 2, based on the result from our best-fit QLF, quasars are estimated to provide ∼45% of the required photons; while for C = 5, the fraction changes to 18%. This result agrees with previous work, suggesting that quasars may play some role in maintaining ionization at z ∼ 5 but have low possibilities of being the dominant source of ionizing photons (M13).

Traditionally, quasars have been divided into two populations, radio-loud and radio-quiet (Kellermann et al. 1989). The similarity and difference between the evolution of radio-loud and radio-quiet quasars are thought to be related to black-hole mass, accretion, and spin. (Rees et al. 1982; Wilson & Colbert 1995; Laor 2000). The radio-loud fraction (RLF) has been suggested to evolve with optical luminosity and redshift by some work (e.g., Padovani 1993; La Franca et al. 1994; Hooper et al. 1995; Jiang et al. 2007; Kratzer & Richards 2015). In contrast to this, no evolution of the RLF is also found (e.g., Goldschmidt et al. 1999; Stern et al. 2000; Ivezić et al. 2002; Cirasuolo et al. 2003). Our luminous quasar sample is selected only by optical and near-infrared colors and thus it can be considered to be an unbiased sample for the study of the RLF.

We cross-match all 99 quasars in our sample with catalogs from Faint Images of the Radio Sky at Twenty-cm (FIRST; Becker et al. 1995) and the NRAO VLA Sky Survey (NVSS; Condon et al. 1998) and find 8 quasars with radio detections. We use 3'' matching radius for FIRST data, and 5'' for NVSS due to the lower resolution of NVSS. We calculate the radio loudness for these 8 quasars by assuming an optical spectral index of −0.5 and radio spectral index of −0.5 (${f}_{\nu }\propto {\nu }^{\alpha }$ ) and list them in Table 4. To compare with previous work at higher redshifts (Bañados et al. 2015), we adopt the radio/optical flux density ratio R₄₄₀₀ = ${f}_{\nu ,5\mathrm{GHz}}$/${f}_{\nu ,4400{\mathring{\rm{A}}} }$ (Kellermann et al. 1989) and the criterion R > 10 for the definition of a radio loud quasar, where ${f}_{\nu ,5\mathrm{GHz}}$ is the radio flux density at rest-frame 5 GHz, and ${f}_{\nu ,4400{\mathring{\rm{A}}} }$ is the optical flux density at rest-frame 4400 Å. There are 7 quasars considered to be radio loud quasars among the 99 quasars. Therefore, we find a radio loud fraction RLF ∼ 7.1%. Considering FIRST with its 1mJy flux limit and NVSS with its 2.5 mJy flux limit are not deep enough to detect all radio loud quasars, especially for quasars at z_SDSS > 19, 7.1% is a lower limit. This result shows agreement with the result from Bañados et al. (2015), which constrains the RLF at z ∼ 6 to be ${8.1}_{-3.2}^{+5.0}$%. We also do the calculation using a radio spectral index of −0.75. The radio loudness based on α = −0.5 to −0.75 increases 12%–15%. The radio loud fraction has no change.

Table 4.  Radio Detection and Radio Loudness

Name	z	m1450	${f}_{1.4\mathrm{GHz},\mathrm{FIRST}}$ a	${f}_{\mathrm{err},F}$	${f}_{1.4\mathrm{GHz},\mathrm{NVSS}}$	${f}_{\mathrm{err},N}$	${f}_{\nu ,2500}$	${f}_{\nu ,4400}$	${f}_{\nu ,5\mathrm{GHz}}$	R2500	R4400	M2500b
J0011+1446	4.96	18.03	23.96	0.146	35.8	1.5	0.0491	0.0651	5.1933	105.8	79.7	−28.65
J0131−0321	5.18	18.09	32.83	0.123	31.4	1.0	0.0448	0.0594	6.9880	156.0	117.6	−28.66
J0741+2520	5.21	18.22	2.07	0.141	⋯	⋯	0.0396	0.0525	0.4395	11.1	8.4	−28.54
J0813+3508	4.92	19.17	20.04	0.156	35.6	1.1	0.0173	0.0229	4.3583	252.0	189.9	−27.50
J1146+4037	4.98	19.48	12.45	0.146	12.5	0.5	0.0129	0.0171	2.6940	209.3	157.8	−27.21
J1318+3418	4.82	19.09	3.73	0.148	3.5	0.4	0.0189	0.0251	0.8181	43.2	32.6	−27.54
J2329+3003c	5.24	18.83	⋯	⋯	4.9	0.4	0.0224	0.0298	1.0380	46.2	34.9	−27.94
J2344+1653	5.00	18.54	⋯	⋯	15.3	0.6	0.0305	0.0404	3.3052	108.4	81.7	−28.15

Notes.

^aFlux density and flux density error are in units of mJy. ^bData in this table are calculated based on cosmology H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_m = 0.3, and Ω_Λ = 0.7. ^cThe two objects only detected by NVSS without FIRST detection are not covered by the FIRST footprint.

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Jiang et al. (2007) suggest that the RLF is a function of absolute magnitude M₂₅₀₀ and redshift at z < 4, and give the best fit for the function. To compare with Jiang et al. (2007), we also calculate the radio/optical flux density ratio R₂₅₀₀ = ${f}_{\nu ,5\mathrm{GHz}}$/${f}_{\nu ,2500{\mathring{\rm{A}}} }$, where ${f}_{\nu ,2500{\mathring{\rm{A}}} }$ is the optical flux density at rest-frame 2500 Å. We convert M₁₄₅₀ to M₂₅₀₀ by M₂₅₀₀ = M₁₄₅₀ – 0.3 on the assumption of an optical spectral index of approximately −0.5. For comparison, here we use the same cosmology as Jiang et al. (2007), which is H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_m = 0.3, and Ω_Λ = 0.7. Our sample covers magnitudes of −27.03 < M₂₅₀₀ < −29.22. Due to the fact that there are only 8 quasars with radio detection covering a narrow redshift range, we use only one redshift bin and roughly divide our sample into two magnitude bins, one for M₂₅₀₀ < −28 and the other one for M₂₅₀₀ > −28. Here we use R > 30 as the definition of the radio loud quasar so that FIRST with limiting flux of ∼1 mJy will be deep enough for our quasar sample. NVSS, with an ∼2.5 mJy flux limit, is still not deep enough and is able to detect radio-loud quasars (R > 30) at z ∼ 5 only down to M₂₅₀₀ ∼ −27.7. Therefore, we calculate the RLF for quasars within the area covered by FIRST and NVSS, respectively. In the bright magnitude bin, there are 13 quasars in the FIRST coverage and 19 quasars in the NVSS coverage. In the faint bin, there are 68 quasars in the FIRST coverage and 80 quasars in the NVSS coverage. The details and radio loudness of radio detected quasars are given in Table 4. We compare our result with the RLF evolution function log (RLF/(1-RLF)) = −0.218 – 2.096log(1 + z) −0.203(M₂₅₀₀ + 26) from Jiang et al. (2007). As shown in Figure 12, our points show an evolution of the RLF with magnitude, but prefer a higher RLF at z ∼ 5 than the prediction. Our result suggests that the RLF may evolve with optical luminosity, but it may not decline as rapidly with increasing redshift as measured in Jiang et al. (2007) at high redshift, though this could be affected by the small number of radio-loud quasars in our sample.

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