The Infrared Medium-deep Survey. VIII. Quasar Luminosity Function at z ∼ 5

The initial photometric data set is a combination of the optical imaging data from the CFHTLS Wide Survey and the NIR imaging data from IMS and DXS covering four extragalactic fields: the XMM-Large Scale Structure survey region, the CFHTLS Wide Survey second region, the Extended Groth Strip, and the Small Selected Area 22h. The average 5σ limiting magnitudes for point sources are $u^{\prime} =26.1$, $g^{\prime} =26.4$, $r^{\prime} =25.9$, $i^{\prime} =25.6$, $z^{\prime} =24.6$, and J = 22.9 mag. Note that the optical photometric system here is based on the SDSS filter system transformed from those of the CFHTLS.¹⁴ The total survey area used for the z ∼ 5 quasar search covers ∼85 deg² of the sky, calculated from the full overlapping region of the surveys. For the i'-band detected sources, we selected z ∼ 5 quasar candidates using the broadband color selection criteria: (1) $i^{\prime} \lt 23$, (2) signal-to-noise ratio (S/N) ($u^{\prime} )\lt 2.5$, (3) $g^{\prime} -r^{\prime} \gt 1.8$ or S/N ($g^{\prime} )\lt 3.0$, (4) $r^{\prime} -i^{\prime} \gt 1.2$, (5) $i^{\prime} -z^{\prime} \lt \,0.625\times ((r^{\prime} -i^{\prime} )-1.0)$, (6) $i^{\prime} -z^{\prime} \lt 0.55$, and (7) $i^{\prime} \,-J\lt ((r^{\prime} -i^{\prime} )-1.0)+0.56$. In the visual inspection stage, seven sources that are unusually elongated or extended were rejected, giving the final 69 z ∼ 5 quasar candidates.¹⁵

To narrow down the number of plausible z ∼ 5 quasar candidates, follow-up observations of these broadband-selected candidates were carried out in medium bands with the SED Camera for Quasars in Early Universe (SQUEAN; Choi et al. 2015; Kim et al. 2016) on the Otto Struve 2.1 m telescope at McDonald Observatory. At z ∼ 5, the Lyα λ1216 line is located at ∼7300 Å, so we have obtained m675, m725, and m775 images, where the filter names are chosen by combining "m" for medium band and the number of the central wavelength of the filter in nanometers. These 50 nm width filters finely sample the redshifted Lyα line to improve quasar identification and determine their redshifts to nearly 1% accuracy (Jeon et al. 2016; K19). The total exposure times in medium bands are proportional to the i'-band magnitude of each target, as are the resultant imaging depths. These inhomogeneous imaging depths between targets are important when we calculate the medium-band selection functions in Section 3.2. Most of the broadband-selected candidates were observed in the m725 and m775 bands (63/69), while 50 were also observed in the m675 band. We introduced the medium-band color selection criteria of Jeon et al. 2016: (1) m675 − m725 > 1.0 and m675 − m725 > m725 − m775 + 1.5 (4.7 < z < 5.1) and (2) m725 − m775 > 1 (5.1 < z < 5.5). Among the 50 medium-band observed candidates, 33 satisfy the criteria. In Figure 2, we plot an m675 − m725 versus m725 − m775 diagram with the medium-band color criteria shown as dotted lines. For simplicity, we only present the 37 spectroscopically identified 4.7 < z < 5.4 quasars that are also observed in the three medium bands. There are five quasars excluded by our selection (open circles), possibly due to their m675 imaging depths being shallower than expected for a given i'-band magnitude, as described in Section 3.2, and/or their marginal redshift of z ≃ 4.7.

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We obtained the optical spectra of the plausible candidates, which satisfied either the above broadband or medium-band color selection criteria, with the Gemini Multi-Object Spectrograph (GMOS; Hook et al. 2004) on the Gemini-South 8 m telescope on 2018 September 17 (PID: GS-2018B-Q-217), 2019 January 2–11, and 2019 February 14–24 (PID: GS-2019A-Q-218) under the seeing condition of  ≲10. The observing configurations were set to increase the S/N of the spectrum; we used the nod-and-shuffle mode for sky subtraction, an R150_G5326 grating with a resolution of R ∼ 315, and spectral/spatial binning of 4 × 4. In total, we obtained the spectra of 10 candidates observed in at least two medium bands. For spectral data reduction, we followed the procedure in K19, which is briefly summarized here. We used the Gemini IRAF package for the basic reductions: the bias subtraction, flat-fielding, sky-line subtraction, wavelength calibration with CuAr arc lines, and flux calibration with standard stars. After the 1D extraction with an aperture with a diameter of 10, the overall flux of each spectrum was scaled with the i'-band magnitude of each target. To maximize S/N, we binned the spectra along the wavelength taking account of the instrumental resolution of ∼300 using the inverse-variance weighting method (e.g., Kim et al. 2018).

Through these spectroscopic observations, eight z ∼ 5 quasars were newly discovered, the optical spectra of which are shown in Figure 3. All of them show strong Lyα emission lines with sharp breaks, and some also show other possible emission lines like C iv, while there are broad absorption lines (BALs) in the spectrum of IMS J090508−021523.

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We list the broadband and medium-band photometry of the new IMS z ∼ 5 quasars in Table 1, and their spectroscopic observations are summarized in Table 2. The other two candidates identified as nonquasars are listed in Appendix A.

Table 1.  Broadband and Medium-band Photometry of New IMS z ∼ 5 Quasars

ID	R.A.	Decl.	$r^{\prime}$	i'	$z^{\prime}$	J	m675	m725	m775
	(J2000)	(J2000)	(mag)	(mag)	(mag)	(mag)	(mag)	(mag)	(mag)
IMS J021507−051849	02:15:07.31	−05:18:49.2	24.16 ± 0.08	22.87 ± 0.03	22.65 ± 0.06	>22.65	23.99 ± 0.25	23.38 ± 0.21	>22.83
IMS J084906−030304	08:49:06.33	−03:03:03.9	22.31 ± 0.03	20.87 ± 0.01	20.67 ± 0.02	21.21 ± 0.07	>22.65	21.25 ± 0.12	21.53 ± 0.13
IMS J085711−021957	08:57:10.66	−02:19:57.3	23.85 ± 0.07	22.28 ± 0.02	22.40 ± 0.08	22.30 ± 0.16	>23.01	22.88 ± 0.20	23.12 ± 0.34
IMS J090037−052549	09:00:36.89	−05:25:49.4	22.66 ± 0.03	21.02 ± 0.01	21.15 ± 0.02	21.20 ± 0.10	>23.15	20.97 ± 0.08	21.79 ± 0.18
IMS J090226−022923	09:02:25.81	−02:29:22.8	24.52 ± 0.06	22.67 ± 0.02	22.26 ± 0.05	22.61 ± 0.32	⋯	22.77 ± 0.14	22.71 ± 0.15
IMS J090508−021523	09:05:08.10	−02:15:23.4	22.93 ± 0.04	21.44 ± 0.01	21.07 ± 0.02	20.92 ± 0.07	22.79 ± 0.30	21.83 ± 0.18	>22.40
IMS J090706−031931	09:07:06.12	−03:19:30.6	22.30 ± 0.02	20.62 ± 0.01	20.20 ± 0.01	20.34 ± 0.04	22.61 ± 0.37	21.75 ± 0.11	20.27 ± 0.05
IMS J221055+013430	22:10:54.87	+01:34:30.5	22.98 ± 0.03	21.32 ± 0.01	21.27 ± 0.03	21.88 ± 0.08	>23.65	21.19 ± 0.19	21.71 ± 0.19

Note. All magnitudes are given in the AB system. The detection limits (∼3σ) are given as the upper limits for objects that were undetected or fainter than the limit.

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We measured the spectral properties of the eight new quasars by fitting their spectra with a refined version of our high-redshift quasar model (K19). The model is based on the composite quasar spectrum of SDSS quasars (Vanden Berk et al. 2001). Since no IGM correction has been applied in this spectrum, we replaced the spectrum at λ < 1216 Å with that of Lusso et al. (2015). Note that the two spectra were normalized at 1450 Å before merging. For this composite spectrum, the IGM attenuation by neutral hydrogen was applied in the form of a function of redshift (Madau et al. 1996). The model has four parameters: redshift (z), absolute magnitude (M₁₄₅₀), continuum slope (α_λ), and equivalent width (EW) of Lyα λ1216 and N v λ1250 (EW_Lyα+NV). The SDSS composite spectrum is first decomposed into the emission line and continuum components by fitting a power-law continuum to it. The slope of the continuum component is allowed to vary to a given α_λ by multiplying a factor of ${({\lambda }_{\mathrm{rest}}/1000\mathring{\rm A} )}^{{\alpha }_{\lambda }+1.54}$. The EW_Lyα+NV, estimated from the composite continuum-subtracted flux at 1160 ≤ λ_rest (Å) ≤ 1290, is also scaled with ${({\lambda }_{\mathrm{rest}}/1290\mathring{\rm A} )}^{p}$, where p is the appropriate power to adjust the EW_Lyα+NV value of the model to an arbitrary one. In the original K19 model, only the EW of Lyα+N v was allowed to scale. On the other hand, in the refined K19 model, we also adjusted the other emission lines, such as C iv λ1549 and C iii] λ1909, by scaling the fluxes over the continuum with EW_Lyα+NV. This is because the EWs of these emission lines are known to scale with EW_Lyα+NV (Dietrich et al. 2002) and have EW distributions with scatters similar to that of Lyα+N v (0.2–0.3 dex; Diamond-Stanic et al. 2009; Shen et al. 2011).

We found the best-fit model for each quasar spectrum by finding the minimum χ² values. Considering the narrow wavelength coverage of our spectra, we fixed α_λ to −1.54, same as K19. In Figure 3, the best-fit models for the eight new quasars are shown as the red solid lines, and we list the best-fit values in Table 3. For IMS J090508−021523, a BAL quasar, the fluxes at BAL wavelengths are excluded from the fitting. Note that this model is also used to determine the selection functions in Section 3.2.

Table 3.  Spectral Fitting Results for New IMS z ∼ 5 Quasars

ID	z	M1450	log EWLyα+NV
IMS J021507−051849	${4.716}_{-0.003}^{+0.003}$	$-{23.28}_{-0.17}^{+0.19}$	${1.78}_{-0.17}^{+0.16}$
IMS J084906−030304	${4.790}_{-0.225}^{+0.095}$	$-{25.50}_{-0.45}^{+0.75}$	${1.50}_{-1.50}^{+0.49}$
IMS J085711−021957	${4.904}_{-0.003}^{+0.003}$	$-{24.10}_{-0.06}^{+0.06}$	${1.43}_{-0.12}^{+0.11}$
IMS J090037−052549	${4.840}_{-0.023}^{+0.004}$	$-{25.04}_{-0.23}^{+0.36}$	${1.90}_{-0.21}^{+0.21}$
IMS J090226−022923	${4.939}_{-0.020}^{+0.003}$	$-{23.72}_{-0.09}^{+0.14}$	${1.38}_{-0.27}^{+0.17}$
IMS J090508−021523	${4.707}_{-0.022}^{+0.031}$	$-{25.05}_{-0.22}^{+0.24}$	${1.29}_{-0.59}^{+0.32}$
IMS J090706−031931	${5.154}_{-0.004}^{+0.017}$	$-{26.20}_{-0.03}^{+0.07}$	${1.39}_{-0.14}^{+0.14}$
IMS J221055+013430	${4.968}_{-0.015}^{+0.004}$	$-{24.60}_{-0.35}^{+0.57}$	${2.12}_{-0.27}^{+0.27}$

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Including our eight newly discovered quasars, 49 z ∼ 5 quasars with $i^{\prime} \lt 23$ mag have been spectroscopically identified in our survey area (McGreer et al. 2013; Wang et al. 2016; M18; K19). With these quasars, we constructed the QLF at z ∼ 5. However, for the QLF construction, we excluded four quasars that were not selected by broadband color (see K19) and two quasars outside the redshift range of 4.7 < z < 5.4. Figure 4 shows the sky distributions of the remaining 43 quasars, color-coded by their magnitudes. We used the z and M₁₄₅₀ values of the eight new quasars measured in Section 2.4, while those of the other quasars were sourced from the literature (McGreer et al. 2013; Wang et al. 2016; M18; K19).

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To examine the effect of adding the medium-band color selection, we considered either the broadband selection only (case 1) or a combination of broadband and medium-band selection (case 2). The numbers for each case are 43 and 32 in cases 1 and 2, respectively, and their z and M₁₄₅₀ values are given in Table 4. The five quasars observed in the three medium bands but outside the selection boxes (open circles in Figure 2) could have been selected if we obtained deeper images. However, we excluded them from the case 2 sample and consider this issue in Section 3.2.

Using our high-redshift quasar model described in Section 2.4, we calculated the selection efficiency of our color selection criteria. We generated mock spectra of 100,000 quasars randomly distributed in the range of 4.4 < z < 6.0 and −28 < M₁₄₅₀ < −22, while α_λ and EW_Lyα+NV were randomly generated by Gaussian distributions with mean and standard deviation values of α_λ = −1.6 ± 1.0 (Mazzucchelli et al. 2017) and logEW_Lyα+NV = 1.803 ± 0.205 (Diamond-Stanic et al. 2009), respectively, including the Baldwin effect of Lyα (Dietrich et al. 2002). We integrated the mock quasar spectra over the broadband and medium-band transmission curves to obtain the magnitudes used for color selection. In Figure 5, we show the color distributions of these mock quasars. They are in line with not only the IMS quasars but also the known z ∼ 5.5 quasars in the same photometric system (Yang et al. 2017, 2019a). Note that the relatively larger scatter in $r^{\prime} -i^{\prime}$ color appears to be due to the various proximity zone sizes of high-redshift quasars according to their ages (e.g., Eilers et al. 2017).

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The imaging survey depths are not always homogeneous, and the varying imaging depths at different locations can affect the selection function. For case 1, we generated broadband depth maps resampled at a pixel scale of 10. For each pixel of the broadband depth maps and z–M₁₄₅₀ bin with a size of Δz = 0.05 and ΔM₁₄₅₀ = 0.1 mag, we calculated the selection completeness, a ratio of the number of quasars satisfying our broadband color selection criteria to the total number of mock quasars in the bin (the average number of mock quasars in each bin is ∼50). In this process, we applied additional Gaussian noise to the model magnitudes, considering the image depths and model magnitudes. In Figure 6, the calculated selection functions are shown as gray contours in the left panel. Although our J-band data were inhomogeneous (K19), the differences between the selection functions of the four extragalactic fields of our survey were negligible. We used the combined selection function weighted by the area of each survey field.

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For case 2, we calculated a selection function including our medium-band color selection criteria. The imaging depths of the follow-up medium-band observations are not uniform but depend on the i'-band magnitudes of the targets, because we gave exposures proportional to the i'-band magnitude as much as we could, although the detection limits vary depending on the observing conditions at the time of observation and whether the object was easily detected or not. If the object was detected with a sufficient S/N (≳10), we terminated observation of the target in the corresponding band to save observing time. Figure 7 shows the detection limits (2.7σ) for a point source in the m675, m725, and m775 images of 32 quasars of the case 2 sample as a function of their i'-band magnitude. We fitted a linear regression model to these imaging depths, shown as the solid lines with confidence intervals calculated with the seaborn Python package.¹⁶ Out of 32 quasars, 22 are not detected in m675 (open circles), implying small fluxes below the redshifted Lyα. However, we used all of them for linear regression fitting, because we consider such undetected cases in determining the selection function of case 2.

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As shown in Figure 2, for objects without detection, we used their detection limits as their magnitudes for the color selection. Similarly, in the case of mock quasars, medium-band magnitudes fainter than the detection limit expected from their i'-band magnitudes (i.e., linear regression models) are replaced with the expected values. The color selection with these modified colors agreed with our medium-band color selection process, which could miss some quasars in marginal conditions (e.g., open circles in Figure 2). The right panel of Figure 6 shows the selection function for case 2. Note that there is a valley at z ∼ 5.1, which corresponds to a specific redshift for which the medium-band selection fails (see Figure 2).

It has been suggested that the EW_Lyα+NV values of z > 5.7 quasars are lower (e.g., logEW_Lyα+NV = 1.542 ± 0.391;  Bañados et al. 2016) than their low-redshift counterparts, which might be due to the decrease in Lyα transmission at λ > 1216 Å by neutral hydrogen (e.g., Davies et al. 2018). At z ∼ 5, the neutral hydrogen fraction is much lower than 0.1 (Fan et al. 2006; McGreer et al. 2015); thus, it has a negligible effect on the redward Lyα fluxes. If we introduce the EW distribution of Bañados et al. (2016), the selection efficiencies of cases 1 and 2 are reduced by about 30% and 50% overall, respectively.

We first consider the completeness for case 1. In the process of selecting the 69 broadband-selected candidates, we used the magnitude cut of $i^{\prime} \lt 23$ mag, at which the CFHTLS i'-band (or $i{{\prime} }_{\mathrm{CFHTLS}}$) images we used for the source detection were almost complete (Hudelot et al. 2012; M18). Hudelot et al. (2012) provided the 50% and 80% completeness limits for a point source of each $i{{\prime} }_{\mathrm{CFHTLS}}$-band image. We fitted these limits with an analytic completeness function (Fleming et al. 1995), resulting in ≳99% completeness at $i{{\prime} }_{\mathrm{CFHTLS}}\lt 23$ mag for all CFHTLS images used. Even if we consider the transformation from $i{{\prime} }_{\mathrm{CFHTLS}}$ to i' (i.e., from CFHTLS to SDSS), the point-source detection at $i^{\prime} \lt 23$ mag is almost complete. Therefore, for the broadband photometric completeness, we assume that our detection is complete. The point-source separation process was not included in our selection,¹⁷ but note that we performed the visual inspection as mentioned in Section 2.1. The left panel of Figure 8 shows the number and completeness as a function of the i'-band magnitude for case 1. Of the 69 candidates, 51 were spectroscopically observed, and there are 43 quasars at 4.7 < z < 5.4, as mentioned above. We used the ratio of the candidates with spectroscopy to the broadband-selected candidates as the spectroscopic completeness.

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For case 2, the completeness for both broadband and medium-band photometry should be considered. As with case 1, we also assume complete broadband photometry. In addition, we adopted medium-band photometric completeness. According to our criteria, quasars at 5.1 < z < 5.4 could be selected with the m725 and m775 bands alone, while 63 of the 69 broadband-selected candidates (91%) were observed in these two bands. However, the selection of quasars at 4.7 < z < 5.1 requires the m675, m725, and m775 bands, in which 50 of 69 candidates (72%) were observed. To avoid complex and ambiguous calculations, the candidates with observations in the three medium bands were considered as the main sample in case 2. Therefore, we used the ratio of the number of candidates with observations in the three medium bands to the total number of broadband-selected candidates as the medium-band photometry completeness in case 2 (green histogram in the bottom right panel of Figure 8). As described in Section 2, 33 of the 50 candidates observed in the three medium bands satisfy our medium-band color selection criteria (yellow histogram in Figure 8). We also obtained spectra for 32 of the candidates, all of which were identified as 4.7 < z < 5.4 quasars. The red histogram in the bottom right panel of Figure 8 shows the spectroscopic completeness of the case 2 sample.

To apply these photometric/spectroscopic completenesses to each case, we generated the photometric/spectroscopic completeness functions of (z, M₁₄₅₀) using our high-redshift quasar model described in Section 3.2. For the mock quasars in each bin of the selection functions, we took the number-weighted mean values of their photometric/spectroscopic completenesses.

We first calculated the binned QLF at z ∼ 5 using the 1/V_a method from Avni & Bahcall (1980). Given a bin with sizes of ΔM₁₄₅₀ and Δz, a specific comoving volume of our survey V_a can be calculated as

$$\begin{eqnarray}&&{V}_{a}=\displaystyle \frac{1}{{\rm{\Delta }}{M}_{1450}}{\int }_{{\rm{\Delta }}{M}_{1450}}{\int }_{{\rm{\Delta }}z}p({M}_{1450},z)\displaystyle \frac{{dV}}{{dz}}{{dzdM}}_{1450},\end{eqnarray} \tag{ 1 }$$

where p(M₁₄₅₀, z) is the total selection probability combining the quasar selection functions (Section 3.2) and the photometric/spectroscopic completeness functions (Section 3.3), and dV/dz is the comoving volume element of our survey area. The binned QLF Φ_bin(M₁₄₅₀) can then be written as

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{bin}}({M}_{1450})=\displaystyle \frac{1}{{\rm{\Delta }}{M}_{1450}}\displaystyle \sum ^{{N}_{Q}}\displaystyle \frac{1}{{V}_{a}},\end{eqnarray} \tag{ 2 }$$

where N_Q is the number of quasars in the bin. The bin size was set to ΔM₁₄₅₀ = 0.5 mag (see Figure 6). After experimenting with several different bin sizes, we settled on this size, which gave a good compromise between the number of available quasars per bin and the fine sampling of the QLF. Note that the uncertainties of Φ_bin are calculated using the root-sum-square method. We have listed the derived Φ_bin for cases 1 and 2 in Table 5, which are also denoted by red and blue points in Figure 9, respectively.

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We also cross-checked our binned QLF with another nonparametric QLF for the IMS quasar sample using the C⁻ estimator (Lynden-Bell 1971). The comparison shows that the QLFs derived from different methods agree with each other (see details in Appendix B).

We also derived the parametric QLF at z ∼ 5 (Φ_par), which has a functional form with faint- and bright-end slopes (α and β, respectively; e.g., M18),

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Phi }}}_{\mathrm{par}}({M}_{1450})\\ =\ \displaystyle \frac{{{\rm{\Phi }}}^{* }}{{10}^{0.4(\alpha +1)({M}_{1450}-{M}_{1450}^{* })}+{10}^{0.4(\beta +1)({M}_{1450}-{M}_{1450}^{* })}},\end{array}\end{eqnarray} \tag{ 3 }$$

where ${{\rm{\Phi }}}^{* }={{\rm{\Phi }}}_{z=6}^{* }\times {10}^{k(z-6)}$ with a normalization factor at z = 6 (${{\rm{\Phi }}}_{z=6}^{* }$) and k = −0.47¹⁸ (Fan et al. 2001b), and ${M}_{1450}^{* }$ is the break magnitude. Accounting for the lack of bright quasars (M₁₄₅₀ < −27 mag) in our sample, we used an ancillary SDSS quasar sample (Y16). The IMS and Y16 samples overlap at M₁₄₅₀ ∼ −27 mag. We simply assigned the two samples to cover different ranges; we used the IMS sample at M₁₄₅₀ ≥ −27 mag and the Y16 sample at M₁₄₅₀ < −27 mag. This is because of the potentially underestimated number of Y16 quasars at M₁₄₅₀ ≳ −27 mag, which is lower than that of M18 by a factor of 2–5. For ease of comparison, we rebinned the QLF of the Y16 sample (4.7 < z < 5.4 and M₁₄₅₀ < −27 mag) and marked them with orange points in Figure 9. Note that we used the completeness functions for the SDSS quasars, personally provided by Jinyi Yang.

We used the maximum-likelihood method including selection efficiencies (Marshall et al. 1983) to fit the function with four parameters (Φ*, ${M}_{1450}^{* }$, α, β) by minimizing S, defined as

$$\begin{eqnarray}\begin{array}{rcl}S & = & -2\displaystyle \sum \mathrm{ln}\left[{{\rm{\Phi }}}_{\mathrm{par}}({M}_{1450},z)p({M}_{1450},z)\right]\\ & & +\,2\iint {{\rm{\Phi }}}_{\mathrm{par}}({M}_{1450},z)p({M}_{1450},z)\displaystyle \frac{{dV}}{{dz}}{{dzdM}}_{1450}.\end{array}\end{eqnarray} \tag{ 4 }$$

The first term of this equation is the sum of all of the IMS and Y16 samples. The second term stands for the expected number of quasars with given Φ_par and p, which are integrated over the ranges of  −29.0 < M₁₄₅₀ < −23.0 and 4.7 < z < 5.4. Therefore, minimizing S is equivalent to finding the appropriate Φ_par to satisfy both the observations and expectations. In Figure 9, the best-fit results of Φ_par of cases 1 and 2 are shown as red and blue solid lines, respectively, in line with the Φ_bin of each case. Note that the uncertainties of the free parameters are calibrated from the ${\rm{\Delta }}S=S-{S}_{\min }$ distribution under the assumption of ${\rm{\Delta }}S\sim {\chi }_{\nu }^{2}$ (Lampton et al. 1976), as shown in Figure 10. The resultant Φ_par parameter of case 1 (case 2) has a break magnitude of ${M}_{1450}^{* }=-{25.78}_{-1.10}^{+1.35}$ mag ($-{25.81}_{-1.01}^{+1.22}$ mag) with a faint-end slope of $\alpha =-{1.21}_{-0.64}^{+1.36}$ ($-{1.11}_{-0.68}^{+1.31}$) and a bright-end slope of $\beta =-{3.44}_{-0.84}^{+0.66}$ ($-{3.50}_{-0.81}^{+0.66}$). Interestingly, the derived slopes and break magnitudes of our QLF are similar to those of the QLFs at adjacent redshifts of z = 4 and 6 (Akiyama et al. 2018; Matsuoka et al. 2018b), and this will be discussed in a forthcoming paper (Y. Kim et al. 2020, in preparation). We also obtained the Φ_par parameters with fixed bright-end slopes of β = −3.0 and −4.0 (the dotted and dashed lines in Figure 9, respectively). These are consistent with the best-fit results within the 1σ confidence level. All of the derived values are listed in Table 6.

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Table 6.  Parametric QLFs

	$\mathrm{log}{{\rm{\Phi }}}^{* }$	${M}_{1450}^{* }$	α	β
Case 1
Best-fit	$-{7.36}_{-0.81}^{+0.56}$	$-{25.78}_{-1.10}^{+1.35}$	$-{1.21}_{-0.64}^{+1.36}$	$-{3.44}_{-0.84}^{+0.66}$
Fixed β	$-{7.06}_{-0.42}^{+0.28}$	$-{25.02}_{-0.58}^{+0.39}$	$-{0.79}_{-0.70}^{+0.94}$	−3.0
Fixed β	$-{7.68}_{-0.61}^{+0.41}$	$-{26.38}_{-0.63}^{+0.39}$	$-{1.48}_{-0.43}^{+0.48}$	−4.0
Case 2
Best-fit	$-{7.35}_{-0.77}^{+0.52}$	$-{25.81}_{-1.01}^{+1.22}$	$-{1.11}_{-0.68}^{+1.31}$	$-{3.50}_{-0.81}^{+0.66}$
Fixed β	$-{7.07}_{-0.42}^{+0.30}$	$-{25.04}_{-0.57}^{+0.42}$	$-{0.67}_{-0.76}^{+1.13}$	−3.0
Fixed β	$-{7.57}_{-0.62}^{+0.39}$	$-{26.28}_{-0.63}^{+0.37}$	$-{1.32}_{-0.51}^{+0.54}$	−4.0

Note. Φ* is in units of Mpc⁻³ mag⁻¹.

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When we compare the estimated QLFs of cases 1 and 2, they are consistent with each other in both the Φ_bin and Φ_par cases. The only minor differences between them are observed at M₁₄₅₀ > −25 mag, especially at the faintest bin. This possible difference at the faintest bin is due to the serendipitous identification of faint quasars, such as IMS J085028−050607 with M₁₄₅₀ = −23.5 mag at z = 5.36. As shown in Figure 6, the quasar is located at a very low completeness region, and the V_a of the faintest bin is smaller than the other bins, giving a high number density at the faintest bin after the completeness correction. In fact, in case 2, where this quasar is not included, the Φ_bin(−23.25) is more in line with our best-fit QLF. But here we stress that the faintest bin is also consistent with our best-fit QLF within the 1σ confidence level.

Considering the high success rate of our medium-band selection, especially at fainter magnitudes (K19), a future wide-field imaging survey in medium bands will allow us to estimate reliable QLFs at high redshifts even without deep spectroscopic observations. In the following discussion, the best-fit result in case 1 is used as a representative result.

For the IMS quasars only, we also performed testing with a single power-law function: Φ_par ∝ 10^−0.4(α+1). The resultant slopes are $\alpha =-{1.70}_{-0.26}^{+0.24}$ and $-{1.67}_{-0.31}^{+0.29}$ in cases 1 and 2, respectively, which is slightly steeper than our best-fit results. These values agree with the recent estimate of α = −2.1 ± 0.7, obtained from a smaller sample of z ∼ 5 faint quasars using a single power-law function (pink line in Figure 11; Shin et al. 2020).

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Figure 11 shows various QLFs at z ∼ 5 from the literature (Glikman et al. 2011; Y16; Boutsia et al. 2018; M18; G19; Shen et al. 2020; Shin et al. 2020), in comparison to our best-fit QLFs. To obtain the binned QLF of G19, we averaged their binned QLFs at z = 4.5 and 5.6 (purple crosses). Note that the upper and lower limits of each point indicate the binned QLFs at z = 4.5 and 5.6, respectively. Their parametric QLF (model 2; purple solid line) is also adjusted from z = 4.5 to 5 with k = −0.47. Considering that the binned QLFs of Glikman et al. (2011) and Boutsia et al. (2018) are used for the parametric QLF derivation of G19, for ease of comparison, they are represented by purple open circles. Similarly, the z ∼ 4.8 QLF of Shen et al. (2020; gray crosses) is also shifted to z = 5.

While the faint-end slope of our parametric QLF has a flatter value, the binned QLFs in this work are consistent with most QLFs at  −25 < M₁₄₅₀ < −23, except for the result of G19 and the results from their group in previous years. Possible reasons for the discrepancy were discussed in detail in Parsa et al. (2018), but here we discuss whether the discrepancy is caused by cosmic variance. In this magnitude range, G19 used the z ∼ 4 QLFs (purple open circles) determined from small-area surveys (3.76 deg² of Glikman et al. 2011; 1.73 deg² of Boutsia et al. 2018). We randomly selected a 1.73 deg² area in a circular shape from our survey area and counted the numbers of M₁₄₅₀ = −23.5 and −24.5 mag quasars (ΔM₁₄₅₀ = 1 mag), which is repeated 100,000 times. Then, we examined the frequency of obtaining the Φ_bin values of Boutsia et al. (2018) or higher at the two magnitude bins by chance due to cosmic variance: only 2.2% and 1.4% for M₁₄₅₀ = −24.5 and −23.5 mag, respectively. If we require the number densities to be high in these two consecutive magnitude bins, then the probability drops to zero, which can also be inferred from Figure 4. We also tested for a 5.49 deg² area, resulting in the very low probability of ≪1% even if only one magnitude bin was considered. The result shows that the likelihood of gaining a high quasar number density due to cosmic variance is small, even with an area of 1.73 deg².

All quasars that we identified with spectroscopy (K19 and this work) have EW_Lyα+NV higher than 15.4 Å, the criterion for weak Lyα quasars (Diamond-Stanic et al. 2009). The fraction of weak Lyα quasars is known to be 13.7% at z > 5.7 (Bañados et al. 2016), higher than the fraction of 1.8% from the EW distribution adopted for our calculation of the selection function (Diamond-Stanic et al. 2009). Using the EW distribution of Bañados et al. (2016) for our calculation of the selection function instead (see Section 3.2), we find that our QLF could be underestimated by a factor of ∼1.3 at the faint end (in both cases 1 and 2). The exact value of this factor is more likely to be smaller than 1.3, assuming the gradual change in the EW distributions between z ∼ 5 and 6. Future investigation of the EW distributions is desired to improve the QLF estimate.

The G19 QLF at M₁₄₅₀ > −23 mag mainly relies on X-ray-selected quasars that do not always overlap with those selected in the UV/optical range. One possible explanation for their higher number density than UV/optical quasars is due to the high fraction of obscured X-ray quasars at high redshifts (e.g., 50%–80% at 3 < z < 6;  Vito et al. 2018), if the obscured fraction applies equally to UV/optical selected quasars. However, recent X-ray QLFs provided by Shen et al. (2020; gray crosses), corrected by their model prediction, also agree with our results, reinforcing the suggestion of the overestimation of G19 even at M₁₄₅₀ > −22 mag. We note that the Φ_bin of M18 has a value lower than those of our and other QLFs at the faintest bin they explored (M₁₄₅₀ > −23 mag). The QLFs can be easily over- or underestimated at the faintest end depending on how well the selection function is constructed. We suggest this as a possible reason for the discrepancy.

We followed the method in Kulkarni et al. (2019) to calculate the UV ionizing emissivity of z ∼ 5 quasars at 1450 (₁₄₅₀) and 912 Å (₉₁₂), which can be written as

$$\begin{eqnarray}&&{\epsilon }_{1450}={\int }_{-30}^{-18}{{\rm{\Phi }}}_{\mathrm{par}}({M}_{1450})\,{10}^{-0.4({M}_{1450}-51.60)}\,{{dM}}_{1450}\end{eqnarray} \tag{ 5 }$$

and

$$\begin{eqnarray}&&{\epsilon }_{912}={\epsilon }_{1450}\times {\left(\displaystyle \frac{912}{1450}\right)}^{0.61}.\end{eqnarray} \tag{ 6 }$$

To evaluate ₉₁₂ from ₁₄₅₀, we assume a double power-law shape of the quasar UV spectra (${f}_{\nu }\propto {\nu }^{-{\alpha }_{\nu }}$, where α_ν = 0.61 if λ ≥ 912 Å and 1.70 if λ < 912 Å;  Lusso et al. 2015). For the best-fit case 1 QLF, ₁₄₅₀ and ₉₁₂ are ${6.64}_{-1.70}^{+2.81}$ (or 4.94–9.45) and ${5.00}_{-1.28}^{+2.12}$ (or 3.72–7.12), respectively, in units of 10²³ erg s⁻¹ Hz⁻¹ Mpc⁻³. Figure 12 shows the observed ₉₁₂ values of high-redshift quasars. Our result is the lowest value at z ∼ 5, an order of magnitude lower than the Giallongo et al. (2015) estimates (brown open symbols). Considering large errors, most of the ₉₁₂ values are consistent with the synthesis model of Haardt & Madau (2012) and the best-fit model of Shen et al. (2020). Notable exceptions to the above trend are the results of Giallongo et al. (2015) and G19. However, the former were derived from X-ray candidates without spectroscopic confirmation, while the latter are also less reliable than others due to the possible overestimation of their QLF, as mentioned earlier in this section.

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The above results, however, could be sensitive to the uncertain extrapolation of the QLF down to faint magnitudes of M₁₄₅₀ = −23 to −18 mag. If we integrate Equation (5) up to M₁₄₅₀ = −23 mag, at which our QLF is well constrained, ₉₁₂ is 4.48 × 10²³ erg s⁻¹ Hz⁻¹ Mpc⁻³. This is consistent with ₉₁₂ integrated up to M₁₄₅₀ = −18 mag within the 1σ error, meaning that the additional contribution of quasars with M₁₄₅₀ > −23 mag is not significant unless their number density is unexpectedly high.

Despite the uncertainty in our emissivity estimates at the faintest end, we can explore the significance of the contribution of quasars to the UV ionizing radiation. Under the same assumptions, we calculate the number density of ionizing photons,

$$\begin{eqnarray}&&{\dot{n}}_{\mathrm{ion}}={f}_{\mathrm{esc}}{\epsilon }_{1450}{\xi }_{\mathrm{ion}},\end{eqnarray} \tag{ 7 }$$

where f_esc is an escape fraction assumed to be unity (f_esc = 1;  e.g., Grazian et al. 2018), and ξ_ion is the ionizing photon production efficiency of a quasar normalized at 1450 Å. We obtain ξ_ion ≃ 6.05 × 10²⁵ Hz erg⁻¹ with the quasar spectral shape of Lusso et al. (2015). Using the ₁₄₅₀ value from our z ∼ 5 QLF, we obtain ${\dot{n}}_{\mathrm{ion}}$ of $({4.02}_{-1.03}^{+1.70})\times {10}^{49}$ Mpc⁻³ s⁻¹. At z = 5, the required (critical) photon density to keep balance with hydrogen recombination is ${\dot{n}}_{\mathrm{ion}}^{\mathrm{cri}}\sim 6.3\times {10}^{49}\,C$ Mpc⁻³ s⁻¹ (Madau et al. 1999; Bolton & Haehnelt 2007; thick gray lines in Figure 12), where C is the H II clumping factor recently predicted as C < 5 at z = 5 (Bolton & Haehnelt 2007; Shull et al. 2012; Jeeson-Daniel et al. 2014; D'Aloisio et al. 2020). For a plausible clumping factor of C = 3, our result suggests that z ∼ 5 quasars radiate about ${21}_{-5}^{+9} \%$ of the UV ionizing photons required to balance the ionized state of hydrogen at that time. This fraction can reach 45% if C = 2 and ${\dot{n}}_{\mathrm{ion}}$ is the 1σ upper value of 5.72 × 10⁴⁹ Mpc⁻³ s⁻¹. In contrast, if we adopt the 1σ lower limit, the quasar contribution is 9%. Therefore, z ∼ 5 quasars alone radiate only up to  ∼50% of the minimum requirement of UV photons to balance with hydrogen recombination rates (${\dot{n}}_{\mathrm{ion}}^{\mathrm{cri}}$ of Madau et al. 1999) under the extreme assumptions given above.

An alternative method to estimate ${\dot{n}}_{\mathrm{ion}}^{\mathrm{cri}}$ is by using observational inference, such as the Lyα forest. Previous studies calculated the photoionization rate of the UV ionizing background required to match the mean transmitted Lyα flux of high-redshift quasars (Bolton & Haehnelt 2007; Wyithe & Bolton 2011; D'Aloisio et al. 2018) and the corresponding critical emissivity ${\epsilon }_{912}^{\mathrm{cri}}$. The ${\dot{n}}_{\mathrm{ion}}^{\mathrm{cri}}$ can then be approximated to ${\dot{n}}_{\mathrm{ion}}^{\mathrm{cri}}\simeq ({\epsilon }_{912}^{\mathrm{cri}}/{h}_{p}{\alpha }_{\nu })$ Mpc⁻³ s⁻¹ (Meiksin 2005; Becker & Bolton 2013), where h_p is the Planck constant and α_ν = − 1.70 (Lusso et al. 2015). In Figure 12, the gray squares indicate the predicted ${\dot{n}}_{\mathrm{ion}}^{\mathrm{cri}}$ values with the recently estimated ${\epsilon }_{912}^{\mathrm{cri}}$ values of D'Aloisio et al. (2018). At z = 5, we obtain ${\dot{n}}_{\mathrm{ion}}^{\mathrm{cri}}={6.4}_{-2.3}^{+1.1}\times {10}^{50}$ erg s⁻¹, indicating that the quasar contribution with our QLF is only approximately ${6}_{-2}^{+8} \%$ (or 4%–14%) considering the 1σ errors of ${\dot{n}}_{\mathrm{ion}}$ and ${\dot{n}}_{\mathrm{ion}}^{\mathrm{cri}}$. Such a low contribution is similar to that at z ∼ 6, where the quasar contribution is considered to be less than 10% (Willott et al. 2010; Kashikawa et al. 2015; Kim et al. 2015; Onoue et al. 2017; Ricci et al. 2017; Matsuoka et al. 2018b; Dayal et al. 2020). However, note that the ${\epsilon }_{912}^{\mathrm{cri}}$ determination from the Lyα opacity of high-redshift quasars is sensitive to the mean free path of hydrogen photons, which can also be affected by the main ionizing sources (D'Aloisio et al. 2018).

Unlike at z ∼ 6, there is a discrepancy between the ${\dot{n}}_{\mathrm{ion}}^{\mathrm{cri}}$ from the two methods at z = 5. This is due to the increases in the number of ionizing sources and the volume filled with ionized hydrogen from z = 6 to 5. Considering this difference, we conclude that quasars are likely to occupy only a minor portion of the total UV ionizing background at z = 5, estimated from the Lyα opacity of high-redshift quasars. However, they can provide nearly half the quantity of photons required to balance with recombination rates. A better understanding of the quasar contribution for z = 5 IGM ionizing photons requires the construction of a QLF at M₁₄₅₀ > −23 mag, where we rely on extrapolation to estimate the QLF shape.

In addition, improving constraints on C and other physical properties of high-redshift quasars (e.g., f_esc) is also required. For instance, our calculation of ${\dot{n}}_{\mathrm{ion}}/{\dot{n}}_{\mathrm{ion}}^{\mathrm{cri}}$ depends on the assumed UV spectral slope of the quasars. The assumed value of α_λ = α_ν − 2 = −1.39 (Lusso et al. 2015) is based on quasars at z = 2.4 and may not represent the UV spectral slopes of z = 5 quasars in this work. For example, Mazzucchelli et al. (2017) found α_λ = −1.6 ± 0.1 for $\bar{z}=6.6$ quasars, which is steeper than the Lusso et al. (2015) value. We directly infer the UV slope of our sample using the spectral energy distribution fitting result of a subsample of our z = 5 quasars in K19. Using the values presented in Table 5 of K19, we find α_λ = −1.9 ± 0.1, which is the steepest UV slope among other results from various quasar samples (Laor et al. 1997; Vanden Berk et al. 2001; Telfer et al. 2002; Stevans et al. 2014; Selsing et al. 2016). If we adopt this rather extreme slope of −1.9, the exponent in Equation (6) changes to 0.1. However, the UV emissivity only increases by a factor of 1.27. So, the maximal UV ionizing photon fraction changes from 45% to 57%, but this can be considered only a modest increase.