A FIRST-APM-SDSS survey for high-redshift radio QSOs

We selected from VLA-FIRST a sample of 94 objects starlike in SDSSS, and with APM colour O-E>2, i.e. consistent with their being high-z QSOs. 78 candidates were classified spectroscopically from published data (mainly SDSS) or observations reported here. The fractions of QSOs (51/78) and z>3 QSOs (23/78) are comparable to those found in other photometric searches for high-z QSOs. We confirm that O-E>2 ensures inclusion of all QSOs with 3.7<z<4.4. The fraction of broad absorption line (BAL) QSOs for 2<z<4.4 is 27+-10 per cent (7/26), and the estimated BAL fraction for radio loud (RL) QSOs is at least as high as for optically selected QSOs (about 13 per cent). The high BAL fraction and the high fraction of LoBALs in our sample are likely due to the red colour selection. The space density of RL QSOs for 3.7<z<4.4, MAB (1450 A)<-26.6 and P(1.4 GHz)>10^25.7 W Hz^(-1) is 1.7+-0.6 Gpc^(-3). Adopting a RL fraction 13.4+-3 per cent, this corresponds to rho = 12.5+-5.6 Gpc^(-3), in good agreement with the SDSS QSO luminosity function in Fan et al. (2001). We note the unusual QSO FIRST 1413+4505 (z=3.11), which shows strong associated Lyalpha absorption and an extreme observed luminosity, L about 2 x 10^(15) solar luminosities.


INTRODUCTION
This is the third of a series of papers presenting new samples of high-redshift radio QSOs selected by matching the FIRST catalogue of radio sources (Faint Images of the Radio Sky at Twenty-cm; Becker, White & Helfand 1995 with red starlike objects from the APM (Automated Plate Measuring Facility) catalogue of the POSS-I survey (McMahon & Irwin 1992).
Papers I and II (Benn et al. 2002, Holt et al. 2004) reported a sample of 18 z > 3.6 QSOs including the largest sample of z > 4 radio-selected QSOs then known. The search was carried out within a ∼ 7030 deg 2 region using the con-⋆ E-mail:carballor@unican.es straints: i) E ≤ 18.8 and starlike, ii) S1.4 GHz ≥ 1 mJy, and iii) colour selection O −E ≥ 3. This colour range includes an estimated 95 ± 1.5 per cent of E < 18.8 QSOs with redshift 3.8 < z < 4.5 (Vigotti et al. 2003). On the basis of the 13 QSOs with z ≃ 3.8 − 4.5 and comparing to an equivalent sample at z ≃ 2 drawn from the FIRST Bright Quasar Survey of the north Galactic cap (FBQS-2, White et al. 2000), we showed (Vigotti et al. 2003) that the decline in space density of MAB (1450Å) ≤ −26.9 QSOs (H• = 50 km s −1 Mpc −1 , ΩM = 1 and ΩΛ = 0 here and throughout the paper) was approximately a factor 2 between z ∼ 2 and z ∼ 4, significantly smaller than the value ∼ 10 found for samples including lower luminosity objects (Fan et al. 2001a moderately deep CCD imaging in five bands ugriz covering ∼ 5282 deg 2 . We present and discuss a new sample of high-redshift radio-selected QSO candidates in a 1378.5 deg 2 area of overlap between FIRST and SDSS DR3 in the north Galactic cap. The selection criteria are: i) E ≤ 19.1 and starlike in APM, ii) S1.4 GHz ≥ 1 mJy, iii) radio-optical separation less than 1.5 arcsec, iv) colour O − E ≥ 2 (including O non-detections) and v) starlike in SDSS. This new sample, with a wider colour range, has several advantages. Firstly, the SDSS photometric catalogue provides reliable morphological classification of the sources, allowing us to readily eliminate the galaxies classed as starlike in APM POSS-I. Secondly, SDSS provides ∼ 3Å-resolution spectra and spectroscopic classifications of many objects, particularly those selected as QSO candidates on the basis of their ugriz colours, or as counterparts of FIRST sources. We continue to use the APM catalogue for colour selection since our previous work showed a high efficiency and completeness in the selection of z > 3.85 QSOs using O − E ≥ 3 and with the new limit O − E ≥ 2 we can check that no z > 3.7 QSOs have O − E < 3. The paper is structured as follows. In Section 2 we present the sample and the status of the spectroscopic classification. Section 3 reports optical spectroscopy of part of the sample. The spectroscopic classification of the sample as QSOs, narrow emission line galaxies or stars is presented in Section 4.1. The distribution of optical magnitudes and O − E colours is discussed in Section 4.2. In Section 4.3 the sample is compared with previous radio-selected QSO samples from the literature, in terms of the selection criteria and the resulting QSO redshift distribution. In Section 4.4 we comment briefly on the spectra of seven QSOs exhibiting strong blueshifted broad absorption lines (BALs) and we analyse the fraction of BAL QSOs in the sample. Section 4.5 is devoted to the peculiar QSO FIRST 1413+4505. In Section 4.6 we compute the absolute magnitudes, k-corrections and radio luminosities of the 10 QSOs with z ≥ 3.7, and we discuss the completeness of a sub-sample of seven of them. In Section 5 we use this sample to calculate the space density of QSOs. Section 6 summarizes our conclusions.

SELECTION OF THE SAMPLE
The sample was selected from the 1378.5-deg 2 area defined in Table 1. This area includes most of the region covered by SDSS DR3 in the north Galactic cap, which is also covered by the FIRST survey and by the APM catalogue of POSS-I. The FIRST survey includes 122463 sources in this area with S1.4 GHz (peak) ≥ 1 mJy, of which 113 have APM E < 19.1, O − E > 2.0 and lie within 1.5 arcsec of starlike objects in SDSS DR3. Eighteen of these were undetected in APM POSS-I O but were detected in APS POSS-I O and had APS(O − E) < 2, and were therefore removed from the sample. For the source FIRST 1340+5619 we found a large difference between the SDSS and APM magnitudes (r = 23.51 versus E = 18.99) and the source was eliminated from the sample after confirming with SDSS that the APM counterpart is a blend. The final sample thus includes 94 candidate high-redshift radio QSOs.
Of these 94 candidates, spectra were first obtained for seven in papers I and II, for six in the literature (found using the NASA Extragalactic Database -NED) and for 41 by SDSS DR3 (which also reobserved nine of the thirteen previously discovered). In Section 3 we present TNG optical spectroscopy of 13 (randomly selected) of the 40 remaining candidates, and we classify an SDSS DR3 source given spectral class 'unknown' in the Sloan survey. Subsequent to these observations, SDSS DR4 (2005 June 30) reported spectroscopy of 10 of the remaining 27 candidates (and also of four of those observed here). In total, 78 of the 94 candidates (83 per cent) are now spectroscopically classified (see Section 4.1).

TNG OPTICAL SPECTROSCOPY
Spectra of 11 candidates (indicated in column 9 of Table 2) were obtained with the Telescopio Nazionale Galileo (TNG) on 2005 March 10 and 11 using the DOLORES (Device Optimized for LOw RESolution) spectrograph in long-slit mode. The LR-B grism was used, yielding a wavelength range 3000 -8800Å and dispersion 2.9Å pixel −1 . The detector was a thinned, back-illuminated Loral CCD with 15µm pixels. Exposure times were typically 900 s. Three spectrophotometric standard stars were observed in order to calibrate the instrumental spectral response. The seeing was ≈ 1.8 arcsec, and the width of the slit was set to 2 arcsec, yielding a spectral resolution of 23Å, as measured from sky lines.
Standard data reduction was carried out using the IRAF † package. Arc-lamp exposures were used for the wavelength calibration, and yielded solutions with rms residuals < 1.5Å . Small spectrum-to-spectrum wavelength shifts were corrected using sky lines in the science spectra. In most of the spectra, the two or three brightest sky emission lines were not completely removed in the sky subtraction. This did not affect spectral classification or redshift measurement, but for presentation in Figs. 1, 2 we manually cleaned some of the strongest residuals.
Spectra of two more candidates (FIRST 1343+4305 and FIRST 1405+5155) were obtained with similar instrumental set-up on the nights of 2005 July 12 and 13. Two spectra per object were taken, shifting the object along the slit to minimize the effect of detector artefacts and remove the OH sky lines. For each object, one 2D spectrum was subtracted from the other, to remove the background, then wavelength calibrated, aligned in the spatial direction and coadded.  posure times were 2 × 1800 s for 1343+4305 and 2 × 1200 s for 1405+5155.
The 13 objects include eight QSOs, one z = 0.2705 emission line galaxy (ELG), and four late-type stars. The redshift of each QSO was estimated as the average of the values measured from individual emission-line centroids (excluding Lyα, which is often affected by Lyα forest absorption). Six of the QSOs have 3.1 < z < 3.9, the other two have z = 0.897 and z = 0.943.
The TNG spectra of the eight QSOs and the ELG are shown in Figs. 1 and 2. Fig. 1 also shows the SDSS DR3 spectrum of FIRST 1039+4931, classified 'unknown' in SDSS, but which we identify as a QSO at z = 1.09. Four of the eight QSOs were also observed in SDSS DR4, published af-ter our TNG observations took place. The TNG spectra of these four objects are shown in Fig. 2. Table 2 lists the basic optical and radio properties of the 94 objects in the sample, together with the spectroscopic classification and redshift, mostly from SDSS DR3 and DR4. The spectra were classified as one of: QSO (broad emission lines), emission-line galaxy (narrow emission lines), late-type star or early-type star. The sample of 94 objects includes 51 QSOs (0.27 < z < 4.31), 25 stars, two ELGs (z = 0.27, 0.31) and 16 objects with no optical spectrum, i.e. 78 out of 94 (83  per cent) are classified. Of the 78 objects with spectra, 38 per cent (30/78) are QSOs with z ≥ 2, 29 per cent (23/78) are QSOs with z ≥ 3 and 13 per cent (10/78) are QSOs with z ≥ 3.7, confirming the efficiency of the adopted selection criteria for identifying moderate-to high-z QSOs. The efficiencies of previous radio QSO surveys are compared in Section 4.3.

Distribution of optical magnitudes and O − E colours
In Table 2 and hereafter, unless otherwise indicated, the quoted POSS-I APM magnitudes are recalibrated with respect to APS (McMahon et al. 2002), and are corrected for Galactic reddening. This results in some of the sample having final O − E < 2 and E > 19.1 (the recalibrated E magnitudes are typically 0.3 ± 0.3 mags fainter than the original APM values). The histogram of the E magnitudes is shown in Fig. 3. Fig. 4 shows the distribution O − E versus E. QSOs have a broader O − E colour range than stars. The ratio QSOs/stars increases from 21/12 in the range 17.1 < E ≤ 18.6 to 29/5 for E > 18.6 (see also Fig. 3). Assuming that the unclassified sources in Fig. 4 are either QSOs or stars, the distribution in O − E versus E suggests that ∼ 60 per cent are QSOs. Fig. 5 (see also Fig. 6b) shows that the QSO colours redden significantly with redshift: the four measured values of O − E and three out of six lower limits imply O − E > 3 for z > 3.7. This is due to the drop in intensity in the O band as it becomes more dominated by the region blueward of the Lyman limit, and the increase in intensity in the E band as Lyα enters the band.
An important conclusion from Fig. 5 is that the colour cut O−E ≥ 2 includes most of the QSOs with 3.7 ≤ z < 4.4, and this result is in agreement with previous studies. In Vigotti et al. (2003) we estimated a fraction of z > 3.8 QSOs with APM(O − E) > 3 of 95 ± 1.5 per cent, using a sample  per cent of the QSO candidates were spectroscopically classified, and these included 636 QSOs, none of which had z > 3.5, although 12 had 3 < z < 3.5.
We are therefore confident that the number of QSOs we find with 3.7 ≤ z ≤ 4.4 is limited only by the radio and E band flux density limits, with no QSOs missed due to the colour selection.

Comparison with other radio samples
The survey reported here can be considered complementary to FBQS-2. Both surveys select FIRST radio sources with starlike optical counterparts, but the colour criteria are O − E ≤ 2 for FBQS-2 (blue-excess objects) and O − E ≥ 2 for our work. In FBQS-2 objects were selected with O−E ≤ 2 to eliminate galaxies, since catalogues based on photographic plates provide poor discrimination between stellar and nonstellar objects. The survey selection criteria, and the QSO Col. (1): reference, optical and radio catalogues, and optical and radio flux-density and spectral-index (Sν ∝ ν α ) limits.
(2): survey area. (3,4,5,6): number of candidates, number of candidates with spectroscopic classification and the fraction of the latter at redshifts z > 3 and z > 3.7. fractions as a function of redshift, are compared in Table 3. Figs. 6a and 6b show O − E versus E and O − E versus z for the QSOs in the two surveys. Whereas the redshifts covered by FBQS-2 range up to 3.4, our sample includes redshifts up to 4.3 (Fig. 6b, Table 3), and this is a consequence of both the colour cut and the fainter magnitudes.
The efficiency of QSO selection in this work (51/78 = 65 per cent) is slightly larger than in FBQS-2 (636/1130 = 56 per cent). Although our selection criteria filter out most of the QSOs below z = 3.5, and these are covered by FBQS-2, our survey is better at rejecting galaxies, since we take the morphological information from SDSS whereas FBQS-2 used the less reliable APM discrimination. Although our sample has fewer sources, the efficiency for moderate to high redshift QSOs is high: 20/78 = 26 per cent for 2 ≤ z ≤ 3.7 compared to 65/1130 = 6 per cent for FBQS-2. At z ≥ 3.7 the efficiency of our selection is 10/78 = 13 per cent. The maximum redshift in FBQS-2 is z = 3.4. Table 4 compares the sizes and search efficiencies of various radio-selected high-redshift QSO samples. The successrate for high-z QSOs and the number of QSOs in this work are comparable to or larger than those of previous searches for radio-selected QSOs at much higher radio flux densities.

The fraction of BAL QSOs
Twenty-six of the QSOs with spectra have redshifts 2.0 ≤ z ≤ 4.4, so that Si iv 1397Å and C iv 1549Å are included in the observed wavelength range. Seven of these QSOs show strong broad absorption lines (BALs) with velocity widths above 2000 km s −1 and reaching velocities above 3000 km s −1 , and were classified as BALs. The balnicity index (BI) of each BAL was computed following the prescription of Weymann et al. (1991) and is presented in Table 5.
The fraction of BALs in our sample is therefore 7/26 = 27 ± 10 per cent. For comparison, from the study of the BAL QSOs in FBQS-2 (Becker et al. 2000) we derive an observed fraction of BALs for 1.5 ≤ z ≤ 3 and 15 ≤ E ≤ 17.8 of 14/134 = 10 ± 3 per cent (BALs with BI=0 excluded).  In view of the redder colours found for BALs, compared to non-BALs, and the adopted colour selection, O − E ≤ 2, Becker et al. conclude that the BAL fractions derived from their work should be considered as lower limits. Our result, 27 ± 10 per cent, confirms a larger fraction of BALs in a red-selected sample (O − E ≥ 2), and following their argument this value should be considered as an upper limit to the actual fraction. We note however, that the distribution of balnicity indices for the FBQS-2 BALs reaches lower values than the one in Table 5, which is more similar to the BI distribution of the BAL sample in Weyman et al. (1991), nowadays regarded as highly conservative. In particular we have 6/7 cases with BI ≥ 4500 km s −1 (86 per cent), Weyman et al. 24/40 (60 per cent) and Becker et al. 6/14 (43 per cent). Because of our selection of the stronger BALs, in the comparison with FBQS-2 the fraction 27 ± 10 per cent should be regarded as a lower limit. Differences in the optical magnitudes and redshift ranges, along with small number statistics could explain the differences in the BAL fractions between FBQS-2, > 10 ± 3 per cent, and our work, 27 ± 10 per cent. Reichard et al. (2003a) obtained for a QSO sample selected from SDSS Early Data Release at 1.7 ≤ z ≤ 4.2 (regardless of radio emission) an observed BAL fraction 14 ± 1 per cent. The BI distribution of these BALs reaches low values, with only 14/185 = 8 per cent of them having BI ≥ 5000 km s −1 (Reichard et al. 2003b). Reichard et al. (2003a) confirm that BAL QSOs are redder than non-BALs, and estimate a true BAL fraction of 13 ± 1 per cent, taking into  (Reichard et al. 2003b). Taken together, the results from this work, Becker et al., Reichard et al. and Trump et al. suggest that the BAL fraction in radio-selected QSO samples is at least as high as that of optically-selected QSOs. A brief description of the seven BALs is given below: • FIRST 0837+3641 shows C iv and Si iv troughs starting at the QSO redshift, with a velocity width ∼ 8500 km s −1 and sharp onset. The absorption bluewards of the expected Al iii λ1860 emission line suggests that this QSO is a low-ionization BAL (LoBAL).
• FIRST 1134+4318 shows C iv and Si iv troughs starting at the QSO redshift. The C iv absorption presents two clearly distinct components. The weak absorption bluewards of the expected Al iii emission line suggests that this QSO could be a LoBAL.
• FIRST 1219+4849 shows C iv and Si iv troughs, the former starting sharply at the emission redshift and extending almost to the Si iv emission line. There is absorption bluewards of the Al iii emission line, so this is probably a LoBAL, although some contamination by sky bands cannot be excluded.
• FIRST 1408+5553 shows C iv and Si iv troughs of width ∼ 5500 km s −1 , starting at the emission redshift.
• FIRST 1459+4931 shows C iv and Si iv troughs with a sharp onset at the emission redshift. The C iv absorption extends to the Si iv emission line and clearly shows two components. The strong Al iii absorption confirms this QSO as a LoBAL. We identify the emission line at 7000Å as Fe iii UV 48 (rest frame wavelengths 2062.2, 2068.9 and 2079.65 A, see Laor et al. 1997).
• FIRST 1516+4309, with deep C iv and Si iv troughs was firstly reported as a BAL in Benn et al. (2002). The absorption bluewards of the expected Al iii suggests that this QSO is a LoBAL.
The fraction of LoBALs in our sample, four to five out of seven or 55-70 per cent, is unusually high. For comparison, the observed fractions obtained from the FBQS-2 and from the SDSS BALs in Reichard et al. (2003a) are 28 ± 14 per cent (four out of fourteen) and 13±3 per cent (24/181, 1.7 ≤ z ≤ 3.9) respectively. The higher fraction in our sample is consistent with Becker et al. result that BALs in general and the LoBALs in particular are over-abundant among the reddest QSOs. Moreover, Reichard et al. (2003b) found that LoBALs have stronger CIV absorption troughs (i.e., stronger BI) than HiBALs, and the balnicity indices in our sample are in fact higher than the typical values in FBQS-2 and in Reichard et al.(2003a) SDSS sample.

The peculiar QSO FIRST 1413+4505
FIRST 1413+4505 (Fig. 2, z = 3.107) has an unusual spectrum, with a broad strong absorption starting at the expected position of the Lyα emission line and extending up to −18500 km s −1 bluewards. We identify the feature as H i on the basis of the location of the line and the detection of Lyβ absorption at this redshift. The broad absorption is not detected in metal lines. The Lyα absorption feature has two components. The first one, at z abs = 3.06, has a rest-frame EW ∼ 38Å and associated Lyβ absorption with EW ∼ 5Å. The second component, at z abs = 2.93, has EW ∼ 7Å and the corresponding Lyβ absorption is difficult to identify due to the high noise in this part of the spectrum. The EW of the first Lyα absorber is consistent with a single damped Lyα system with high H i column density, i.e. > 10 21 cm −2 (Lanzetta et al. 1991), at the QSO position. However, the Lyα/Lyβ ratio suggests an optically thin absorber, therefore the system could be a velocity-resolved BAL-like wind. A higher resolution spectrum is needed to determine the nature of this absorption system. The spectral energy distribution of the source can be obtained from NED. The FIRST integrated flux density at 1.4 GHz is 140.3 ± 0.1 mJy, and the flux density at 4.85 GHz is 125 ± 15 mJy (Becker, White & Edwards 1991). The source is unresolved in FIRST, < 1 arcsec. The SDSS magnitudes, corrected for Galactic extinction, are u = 22.32 ± 0.23, g = 20.15 ± 0.04, r = 19.27 ± 0.02, i = 19.09 ± 0.02 and z = 19.09 ± 0.05. The source is not detected in 2MASS and the magnitude limits for 99% completeness are: J = 15.8, H = 15.1 and K = 14.3 (see 2MASS Web site ‡ ). FIRST 1413+4505 is highly luminous in the radio, P1.4 GHz = 2.2 × 10 27 W Hz −1 , and has a flat radio spectral index, α = −0.09. From the E magnitude and using the TNG spectrum for the k-correction we obtained MAB (1450Å) = −25.7 (see equations in Vigotti et al. 2003).
At the position of the QSO, mid-infrared emission is detected via 1D addition of IRAS Scans (SCANPI) at 12µm and 25µm. The automatic procedure obtained via NED gives an 11σ detection at 12µm in the addition modes 'detector weighted' and 'mean', 15σ detection for 'rms weighted' and non-detection for the 'median' mode. The corresponding peak flux densities in Jy are 0.22 ± 0.02, 0.21 ± 0.02 and 0.31 ± 0.02, respectively. At 25µm coaddition gives 3.6σ detection for 'detector weighted' and 'mean' modes, 5σ for 'rms weighted' and non-detection with the 'median' mode. The corresponding peak flux densities are 0.08 ± 0.02, 0.08 ± 0.02 and 0.11 ± 0.02.
Mid-infrared detection at z ≥ 2 is rare and we checked if the detection with SCANPI could arise due to confusion with one of the two bright stars 52 arcsec NE and 48 arcsec SW (binary) of the QSO, but neither of them was detected at 12µm or 25µm.
Assuming that the mid-infrared emission is physically associated with the QSO, and not a chance coincidence, the luminosity of the QSO in the rest frame wavelength range 2200Å -6.1 µm (from z-band to 25 µm in the observer frame) would be 1.8 × 10 15 L⊙, i.e. FIRST 1413+4505 would be one of the most luminous objects known. To our knowledge there are five other systems with such extreme luminosities, and although they were discovered in different ways, and have different SEDs and absorption/emission-line properties, the extreme luminosity was in all cases ascribed to flux magnification due to gravitational lensing. These objects are the Seyfert 2 like object IRAS FSC 10214+4724 (Rowan-Robinson et al. 1991), the BAL QSO H1413+117 (Magain et al. 1988), the sub-mm galaxy SMM 02399-0136 (Ivison et al. 1998), the optically bright QSO APM 08279+5255 (Irwin et al. 1998) and the Lyman break galaxy MS 1512-cB58 (Yee et al. 1996).
The absolute magnitudes at rest-frame 1450-Å, MAB(1450Å), of the z ≥ 3.7 QSOs were calculated from the extinction-corrected E magnitudes using the same procedure as in Vigotti et al. (2003), except for a better estimation of the k-corrections, obtained here from the individual spectra rather than using an average. We used the optical spectra from this work (FIRST 1110+4305), Benn et al. 2002 (FIRST 0941+5119) or from SDSS (remaining QSOs). Fig. 8a shows mAB[1450(1 + z)] − E, or k-correction, for the redshift range 3.7 -4.4, using the spectra of the 10 QSOs. Fig. 8b shows the mean and standard deviation of mAB[1450(1 + z)] − E for the seven QSOs with E ≤ 19.1.

Completeness of the
There are several sources of incompleteness, summarised below.
(1) Only 78 of the 94 candidates were spectroscopically classified, giving a fraction of 83 per cent.
(2) The APM completeness for E ≤ 19.1 was estimated as the fraction of SDSS r ≤ 19.8, z ≥ 2 QSOs in the SDSS DR3 Quasar Catalog (Schneider et al. 2005) detected and starlike in APM E. The adopted r band limit was obtained from the average magnitude difference r − E = 0.69 (standard deviation 0.34) of the 10 high-z QSOs in the sample. Average r − E and standard deviation for the 94 candidates are 0.55 and 0.36 respectively. From the 2060 DR3 QSOs in the surveyed area of this work, 2002 were detected in APM and 1793 of these were starlike, giving a fraction 87 ± 2 per cent.
(4),(5) In Vigotti et al. (2003) we found for a very similar sample completeness 98 ± 1 per cent due to QSOs probably missed because of extended radio emission, and 99 per cent completeness due to QSOs that may exceed the radiooptical separation limit of 1.5 arcsec.
(6) From our work and from the literature (see Figs. 5, 6 and Section 4.2) we estimate 100 per cent completeness for the colour cut O − E ≥ 2 in the redshift range 3.7 -4.4.
(7) The completeness of the SDSS photometric survey quoted in the project web site, computed by comparing the number of objects found by SDSS to the number found by the COMBO survey (Classifying Objects by Medium-Band Observations § ), is 100 per cent for point sources with SDSS r ≤ 21, which is above the maximum r = 20.6 in our sample.
The combined completeness due to the above seven factors is 64 ± 5 per cent.

QSO SPACE DENSITY AT HIGH-REDSHIFT
The observed space density of QSOs in the redshift range 3.7 ≤ z ≤ 4.4, over the 1378.5 deg 2 survey area, was calculated using the sub-sample of seven QSOs with E ≤ 19.1 and the 1/Va estimator (see e.g. Avni & Bahcall 1980). The space density contributed by each QSO was computed as the inverse of the available volume, using the constraints E ≤ 19.1, S1.4 GHz ≥ 1 mJy and 3.7 ≤ z ≤ 4.4. The redshift limits and the space density contribution of each QSO are listed in Table 6. The sum of these contributions yields a space density (for MAB(1450) ≤ −26.6) of 1.1 ± 0.4 Gpc −3 , assuming Poisson errors. The mean redshift of the QSOs is 4.0, the mean absolute magnitude MAB(1450) = −27.5 and the mean log radio luminosity P1.4 GHz = 10 26.3 W Hz −1 . Correcting for the 64 ± 5 per cent completeness, the space density is 1.7 ± 0.6 Gpc −3 . Assuming the radio-loud fraction 13.4 ± 3 per cent for P1.4 GHz ≥ 10 25.7 W Hz −1 from Vigotti et al. (2003), the space density of all QSOs with 3.7 ≤ z ≤ 4.4 and MAB(1450) ≤ −26.6 is 12.5 ± 5.6 Gpc −3 .

Comparison with other surveys at z ≥ 3.7
In Vigotti et al. (2003) we computed ρ[MAB(1450) ≤ −26.9] = 7.4 ± 2.6 Gpc −3 , at 3.85 ≤ z ≤ 4.45, assuming the same radio-loud fraction as above. The average redshift (4.2), optical luminosity (MAB(1450) = −27.7) and radio luminosity (P1.4 GHz = 10 26.6 W Hz −1 ) were all slightly larger than for the current sample. Although the space density derived from the new sample has a larger statistical error, this work extends the computation of the space density to lower optical luminosities and benefits from better k-corrections.
Four of the seven z ≥ 3.7 QSOs in the current work were also included in Vigotti et al. The MAB(1450) values obtained in the two analyses agree within 0.15 magnitudes, except for FIRST 1309+5733, with MAB(1450) = −26.6 compared to −27.1 in Vigotti et al. The reason for the discrepancy is the high k-correction for this source (the highest among E ≤ 19.1 QSOs in Fig. 8). Fan et al. (2001a,b) obtained the space density of QSOs at high redshift using an SDSS-selected sample of 39 QSOs with 3.6 ≤ z ≤ 5.0 and −27.75 ≤ MAB(1450) ≤ −25.5. Seven of these QSOs have redshifts and absolute magnitudes within the range used here (z = 3.7 -4.4, MAB ≤ −26.6) and in this sense our sample, with seven radio loud QSOs, has comparable quality in terms of the number of objects. Gpc −3 , compared to 12.5 ± 5.6 Gpc −3 in this work. These two values are consistent, within the errors.

CONCLUSIONS
A sample of 94 radio-emitting QSO candidates has been derived by cross-correlating the FIRST survey, APM POSS-I and the SDSS photometric survey, selecting objects with E ≤ 19.1, O − E ≥ 2, and starlike in SDSS. 78 of the 94 sources (83 per cent) are spectroscopically classified, mainly from SDSS, but also from the literature and from TNG spectroscopy presented here (13 sources). The classified sources include 51 QSOs, with redshifts 0.27 < z < 4.31. Our main results are as follows: (i) The efficiency of selection of high-redshift QSOs is 23/78 = 29 per cent for z ≥ 3, and 10/78 = 13 per cent for z ≥ 3.7, comparable to that for previous searches for high-redshift QSOs at much higher radio flux densities.
(iii) The BAL fraction for the QSOs with 2 ≤ z ≤ 4.4 in our sample is ∼ 27 ± 10 per cent (7/26), larger than found for FBQS-2 (Becker et al. 2000) and for the opticallyselected SDSS samples in Reichard et al. (2003) and Trump et al. (2006), despite these three works being more complete for lower balnicity index. Our sample also has an unusually high fraction of LoBALs, four to five out of seven, compared to the above samples. The most likely explanation for the high frequency of BALs and LoBALs in our sample is the red colour selection, O − E ≥ 2. A comparison of the three studies suggests that the BAL fraction in radio-selected QSO samples is at least as high as in optically-selected ones.
(iv) We note the unusual QSO FIRST 1413+4505 (z = 3.11), whose optical spectrum reveals a strong associated Lyα absorber (damped or BAL-like) that completely removes the Lyα emission. The source is highly luminous in the radio, has a flat radio spectral index, and is detected in IRAS 12 and 25 µm, yielding a bolometric luminosity in the range from SDSS z band to 25 µm L ∼ 1.8 × 10 15 L⊙, which places it amongst the most luminous objects known.