The 2QDES Pilot: the luminosity and redshift dependence of quasar clustering Free

Abstract

1 INTRODUCTION

Quasars are a very luminous subset of the active galactic nuclei (AGNs) population. Due to their high intrinsic luminosities they can be exploited in a wide variety of cosmological and astrophysical studies. Quasars possess an ultraviolet excess (UVX) of emission with respect to stars. The UVX property has previously been exploited by large area surveys to preform quasar selection, such as 2QZ (Smith et al. 2005), 2SLAQ (Richards, Croom & Anderson 2005) and SDSS (Richards, Fan & Newberg 2002). In addition to their UVX property, quasars possess an excess of emission with respect to stars in the infrared. This method of selecting quasars is sometimes known as the KX (K-band excess) technique and has also been used to photometrically select quasars (see Maddox et al. 2012). The aim of the 2QDESp survey is to maximize the measured sky density of quasars between 0.8 < z < 2.5 and hence minimize the correlation function errors for both cosmological and quasar physics studies.

The 2QZ (Croom et al. 2005, hereafter C05), 2SLAQ (da Ângela et al. 2008, hereafter dA08) and SDSS (Ross et al. 2009, hereafter R09) surveys all measured the quasar correlation function in approximately the same redshift interval (0.8 < z < 2.5). As each of the surveys had different magnitude limits but similar redshift distributions we can compare these directly to one another to measure the luminosity dependence of quasar clustering. Shanks et al. (2011, hereafter S11) discuss the implications of these three surveys measuring a consistent value of r₀, the clustering scale. Of particular interest to galaxy formation models is the apparent independence (see S11; see dA08, and Shen et al. 2009) of quasar clustering with luminosity. S11 examines the results from optical clustering measurements (C05, dA08, R09) and attempts to reconcile these results in context of existing models and finds no evidence for strong luminosity dependence of quasar clustering. This is surprising given the measured relation between optical luminosity and black hole mass, M_BH, (Peterson, Ferrarese & Gilbert 2004) and black hole mass and dark matter halo mass, M_Halo, (Ferrarese 2002; Fine et al. 2006).

However, Fanidakis et al. (2013) predict that less luminous quasars inhabit higher mass haloes and reported that X-ray selected quasars inhabit higher mass haloes (∼10¹³ M_⊙) than found in optical studies (∼10¹² M_⊙). If optically and X-ray selected quasars sample distinct populations there may be no contradiction to the conclusions of S11. We aim to measure ∼80 quasars deg⁻², comparable to the sky density quasars in deep X-ray surveys (Allevato et al. 2011). At these space densities the significant overlap between X-ray and optical quasar samples should result in larger correlation lengths if X-ray selected quasars inhabit higher mass haloes.

Here we describe the first quasar survey (2QDESp) using VLT survey telescope (VST)-ATLAS (Shanks et al. 2015) optical photometry with follow-up spectroscopic observations made using 17 nights of AAT time using 2dF and the AAOmega spectrograph. Combining 2QDESp with several previous quasar surveys we measure the luminosity dependence of quasar clustering for the combined sample. In Section 2 we describe the imaging data and quasar selection techniques. In Section 5 we describe our spectroscopic follow-up of targets and the resulting quasar catalogue. In Section 6 we discuss the methods used to measure quasar clustering and the results from the 2QDESp sample before incorporating the other surveys into our analysis. From this combined sample we make measurements of the luminosity and redshift dependence of quasar clustering in Sections 6.3 and 6.4. We discuss our results and their implications in Section 7. We assume H₀ = 100 h km s⁻¹ Mpc⁻¹ and a flat cosmology from Planck Collaboration XVI (2014) with Ω_M = 0.307. ugriz magnitudes are quoted in the AB magnitude system unless stated otherwise, Wide-field Infrared Survey Explorer (WISE) magnitudes are left in their native Vega system.

2 IMAGING

2.1 Imaging

2.1.1 VST-ATLAS

The VLT Survey Telescope (VST) is a 2.6 m wide-field survey telescope with a 1° × 1° field of view and hosts the OmegaCAM instrument. OmegaCAM (Kuijken, Bender & Cappellaro 2004) is an arrangement of 32 CCDs with 2k × 4k pixels, resulting in 16k × 16k image with a pixel scale of 0.21 arcsec. The VST-ATLAS is an ongoing photometric survey that will image ≈4700 deg² of the southern extragalactic sky with coverage in ugriz bands. The survey takes two sub-exposures (exposure time varies across filters) per 1 degree field with a 25 × 85 arcsec dither in X and Y to ensure coverage across interchip gaps. The sub-exposures are then processed and stacked by the Cambridge Astronomy Survey Unit (CASU). The CASU pipeline outputs catalogues that are cut at approximately 5σ and provides fixed aperture fluxes and morphological classifications of detected objects. The u-band catalogue comprises ‘forced photometry’ at the position of g-band detections; no other band is forced. The processing pipeline and resulting data products are described in detail by Shanks et al. (2015). Bandmerged catalogues were produced using topcat and stilts software (Taylor 2005, 2006). Unless otherwise stated, for stellar photometry we use a 1 arcsec radius aperture (aper3 in the CASU nomenclature). ATLAS photometry is calibrated using nightly observations of standard stars. The calibration between nights can vary by ±0.05 mag (see Shanks et al. 2015 for details). We performed a further calibration on the fields we observed prior to target selection to ensure agreement between VST-ATLAS fields and the SDSS stellar locus, as described in Section 4.0.6. With the VST-ATLAS survey under halfway complete during our spectroscopy, the selection of 2dF pointing positions was governed by the progress of ATLAS. The fields are not generally distributed over a spatially contiguous region, although their seeing and magnitude limits are representative of the survey as a whole. The morphological star-galaxy classification we use is that supplied as default in the CASU catalogues. This classification is discussed in detail (by González-Solares, Walton & Greimel 2008). We test the morphological completeness for different colour–colour selections in Section 4.0.3.

2.1.2 WISE

The NASA satellite WISE (Wright, Eisenhardt & Mainzer 2010), mapped the entire night sky in four passbands between 3.4 and 22 μm. The survey depth varies over the sky but approximate 5σ limits for point sources are W1 = 16.83 and W2 = 15.60 mag. in the Vega system. The W1 and W2 bands have point spread functions (PSFs) of 6.1 arcsec and 6.4 arcsec respectively compared with ≈1 arcsec in the VST-ATLAS bands. A comparison¹ between WISE catalogue positions and the USNO CCD Astrograph Catalog (UCAC3) catalogue shows that even at the faintest limits of W1 there is <0.5 arcsec rms offset between the two catalogues. We matched ATLAS optical photometry to the publicly available WISE All-Sky Source Catalogue using a 1 arcsec matching radius. Given the size of the WISE PSF we examine the possibility of WISE-ATLAS mismatching. For sky density of WISE sources at |b| > 30° we calculate that 1 in 25 quasars identified in WISE will have a blended WISE source within 3 arcsec. Compared to this value, the contribution from quasar–quasar pairs will be smaller. Given the other advantages of using WISE selection, we view this effect as essentially negligible.

3 OTHER QUASAR REDSHIFT SURVEYS

Here we introduce three additional quasar surveys that were used to measure the clustering of optically selected quasars. To aid comparison between these surveys and our own we summarize the method of quasar selection for each survey, the measured space density, area and size. In Section 6 we remeasure the correlation function for these surveys, verifying our measurement against previously published values (see Table 3). We then combine these survey with the 2QDESp sample to better constrain the autocorrelation function.

3.1 2QZ

The 2QZ survey (Croom et al. 2004) covers approximately 750 deg² of the sky in two contiguous areas of equal size. The quasar sample consists of over 22 500 spectroscopically confirmed sources at redshifts less than 3.5 and apparent magnitudes 18.25 < b_J < 20.85. Quasars are selected based on their broad-band optical colours from automated plate measurement (APM) scans from United Kingdom Schmidt Telescope (UKST) photographic plates. Colour selection is performed using u − b_J versus b_J − r. The measured quasar density is ≈30 quasars deg⁻².

3.2 SDSS DR5

The SDSS DR5 uniform sample (Schneider, Hall & Richards 2007) contains 30 000 spectroscopically confirmed quasars between redshifts 0.3 ≤ z ≤ 2.2 and an apparent magnitude limit of i_SDSS ≤ 19.1 over ≈4000 deg². This sample was selected using single epoch photometry from the SDSS using the algorithm given in Richards et al. (2002). The sample is described in detail by R09 and has a measured quasar density of ≈8 quasars deg⁻².

3.3 2SLAQ

The 2SLAQ survey (Croom, Richards & Shanks 2009) overlaps two subregions within the 2QZ survey area, with an average quasar density ≈45 quasars deg⁻² and redshifts of z ≲ 3. The 2SLAQ survey is based on SDSS photometry and measures redshifts for quasars of apparent magnitudes 20.5 < g_SDSS < 21.85. This sample was designed to be used in conjunction with the observations from the 2QZ survey, (see dA08).

4 2QDESP QUASAR SELECTIONS

4.0.1 Quasar density g ≤ 22.5

Previously, 2QZ measured a completeness corrected sky density of 30 quasars deg⁻² at b_J < 20.85. 2SLAQ reached a nominal density of 45 deg⁻² at 20.5 < g_SDSS < 21.85. dA08 combined the 2QZ and 2SLAQ samples to produce a higher density sample of ≈80 deg⁻². However, the high incompleteness of 2SLAQ meant this high density was only achieved after the application of completeness corrections. In this survey we aim to measure 80–100 quasars deg⁻² in the redshift range 0.8 < z < 2.5 in ∼1 h 2dF exposures; we demonstrate the feasibility of our aims below.

4.0.2 Quasar luminosity function

The first concern of the 2QDES pilot is whether or not the luminosity function of quasars predicted 80 + quasars deg⁻² within our targeted redshift (0.8 < z < 2.5) and magnitude (16 < g < 22.5) range. A small number of quasar redshift surveys have explored this redshift range to fainter limits than 2SLAQ although always in relatively small areas. Fine et al. (2012) made a survey based on Pan-STARRS Medium Deep Survey imaging. As well as using colour selection, Fine et al. (2012) also used variability from many epochs of imaging to select their quasar candidates. To a magnitude limit of g = 22.5 their measured quasar density was 88 ± 6 deg⁻² (0.8 < z < 2.5).

In SDSS Stripe 82 Palanque-Delabrouille, Magneville & Yèche (2013) covered ≈15 deg² and measured a completeness corrected quasar density of 99 ± 4 quasars deg⁻². This was to the same depth as 2QDESp (g ≤ 22.5) and in a narrower redshift range (1 ≤ z ≤ 2.2). However, both these authors relied on multi-epoch photometry reaching 50 per cent completeness for point sources at g = 24.6 (c.f. g ∼ 23 for VST-ATLAS).

Finally, spectroscopic follow-up of X-ray sources in the XMM-COSMOS field (Brusa et al. 2010) has measured a quasar density of 110 quasars deg⁻² within our redshift interval to a depth of g < 22.5 (i  ≲  22.2).

Thus from these comparisons to other surveys we can be confident that there exist a sufficiently high space density of quasars within the g ≤ 22.5 limit of the survey. However, these complete samples are often selected from deeper imaging with the added benefit of selecting quasars from their variability.

4.0.3 Photometric incompleteness from VST-ATLAS

The second question we address is whether the VST-ATLAS catalogues are of sufficient depth to select 80–100 quasars deg⁻². As an approximate test of our photometric completeness we rely on the u and g bands where quasars have the colour u − g < 0.5. In the g band the limit g < 22.5 which is ∼0.7 mag brighter than the median 5σ depth of VST-ATLAS as such we assume we are always complete in this band. The 5σ limits are based on sky noise but as the u band is forced this limit may not provide a good estimate of the image depth. We match to the deeper KIDS survey (de Jong, Kuijken & Applegate 2013) in an area of VST-ATLAS with u_5σ = 21.7 (90 per cent of tiles have fainter limits, see Shanks et al. 2015). We find that the use of forced photometry in the u results in 50 per cent completeness (c.f. KIDS) at u = 22 which is 0.3 mag deeper than the 5σ limit. Applying the limits g < 22.5 and u < 22 (with 0.8 < z < 2.5) to the Fine et al. (2012) data we recover 80 ± 6 quasars deg⁻². Assuming median depth limits (u = 21.99) gives 87 ± 6 quasars deg⁻².

The number counts of the Fine et al. (2012) suggest the sample is complete to g ≈ 21.9. This incompleteness at fainter magnitudes will lower the estimated return of quasars in the VST-ATLAS data. We note that the more complete data of Palanque-Delabrouille et al. (2013) returns ∼10 per cent more quasars than Fine et al. (2012).

We have taken a conservative approximation of our photometry and a conservative estimate of the true quasar density and found the VST-ATLAS photometry is sufficiently deep to return our minimum target density (80 quasars deg⁻²). By assuming more representative photometry and comparing to a more compete quasar sample we expect these estimated densities to increase. We finally note that in the 2QDESp survey (see Sections 5.2 and Table A1) that 90 per cent of 2dF pointings have a u_5σ > 21.85 i.e. the u imaging is slightly better than found in VST-ATLAS as a whole.

4.0.4 ugri selection

The UVX property of quasars was successfully used by both 2QZ and SDSS to select quasars in our target redshift range (0.8 < z < 2.5). As our photometric bands are the same as those used by SDSS, we can base our colour selection on work from the SDSS collaboration. We used known 2QZ quasars within the ATLAS footprint to determine the colour cuts suitable for use with VST-ATLAS aperture photometry.

For reference, we show the location of our cuts in ugr colour space in the left-hand panel of Fig. 1. Our selection criteria were as follows.

• −1 ≤ (u − g) ≤ +0.8

• −1.25 ≤ (g − r) ≤ 1.25

• (r − i) ≥ 0.38 − (g − r)

Figure 1.

In the left-hand panel we show the ugr colour space of the field centred at 23^h16^m − 26^d01^m. We show all objects identified in the g band as point-sources (between 16 ≤ g ≤ 20.5) as grey dots. We show the SDSS Stripe 82 stellar locus (dotted blue line) and our ugr colour cuts (purple dashed lines) from Section 4.0.4. Spectroscopically confirmed quasars within our target redshift range (0.8 < z < 2.5) are shown as green triangles and confirmed stars are shown as black five-point stars. Sources without a positive identification are outlined with a red circle. In the middle panel we show the same objects in the giW1 colour space and in the right-hand panel we show the gW1W2 colour space.

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We applied this colour selection in a 2dF field with typical VST-ATLAS depth and seeing and found ≈600 candidates deg⁻², where we considered only point sources (in the g band) and targets between 16 < g < 22.5. These cuts selected a large area in colour space (minimizing the effect of colour incompleteness) and therefore resulted in a high sky density of targets but with significant stellar contamination. We relied on the combination of these cuts and the |${\small \tt XDQSO}$| algorithm (see Section 4.0.6) to minimize this stellar contamination, particularly for fainter targets 21.5 < g < 22.5.

Due to the proximity of the quasar locus to main-sequence (MS) stars, photometric errors are a concern for optical quasar selection. Galaxies may be incorrectly identified as point-sources from their morphology but galaxy colours are sufficiently non-quasar like that galaxy contamination is not considered to be problematic. Morphological incompleteness may be introduced, however, by identifying point-sources as extended sources and therefore not selecting them as quasar candidates. To mitigate this effect we rely on the deeper VST-ATLAS bands to perform our morphological cuts (the g and r bands). We relied on two bands to account for the possibility of poor image quality affecting the morphological classification in a single band.

4.0.5 Optical and mid-IR selection

By combining the mid-infrared photometry from WISE with the optical bands from VST-ATLAS we achieve a larger separation between the stellar locus and our target quasars than is possible using optical colours alone (see Fig. 1). Unlike the ugri colours this selection relies on the infrared excess of emission to differentiate between stars and quasars.

Quasars in our target redshift range have a mean g −W1 = 4 with a large dispersion ≈± 1. The 5σ limits are g ≈ 23.25 and W1 ≈ 16.83. As such the depth of our mid-IR selection is limited by the depth of the WISE photometry.

In the centre panel of Fig. 1 we show the g − i colour plotted against the i −W1 colour. The stellar locus is clearly identifiable. The right-hand panel shows the mid-IR colour W1 −W2 as a function of g-band magnitude. The latter colour selection helped to remove any remaining stellar contamination that was left by the g − i: i −W1 colour cut.

The colour cuts we applied are given here.

• (i − W1) ≥ (g − i) + 1.5

• −1 ≤ (g − i) ≤ 1.3

• (i − W1) ≤ 8

• ( W1 − W2) > 0.4  and  g < 19.5

• ( W1 − W2) > −0.4g + 8.2  and  g > 19.5

Within a typical 2dF pointing, this target selection returns ≈100 candidates deg⁻². This algorithm therefore supplies optimal target density to observe all candidates on the 2dF in a single exposure. However, to meet our target density we required this algorithm to be both highly complete and free from contamination. We used the giW1 colours to test our morphological classification of sources from their g-band imaging, by separating the galaxy and stellar loci in colour space. We determined that of the stars identified by their colour, 91.5 ± 0.5 per cent were identified as point sources by the g-band morphological classification. We tested the morphological classification in the r band with the rzW1 colours and found a similar value.

4.0.6 |${\small \tt XDQSO}$| algorithm

Automated quasar selection algorithms typically compare broad-band colours to model quasar colours (or some library of previously observed quasars). As the VST-ATLAS survey has the same filter set as the SDSS survey, there exists a legacy of quasar selection code (such as Richards, Nichol & Gray 2004; Bovy et al. 2011; Kirkpatrick et al. 2011). Bovy et al. (2011) demonstrated the success of the |${\small \tt XDQSO}$|² algorithm and we applied this algorithm throughout our observing programme. The |${\small \tt XDQSO}$| algorithm takes as input the colours of a source and compares this to empirical observations of quasars and stars. The code outputs a relative likelihood (⁠|${\tt PQSO} \: \epsilon [0,1]$|⁠) that an object is either a star or a quasar. The |${\small \tt XDQSO}$| algorithm uses SDSS as its training data and so we must consider both colour terms between SDSS and VST-ATLAS filters and differences in photometric zeropoints. If these differences between SDSS and VST-ATLAS are small then we shall be able to implement the |${\small \tt XDQSO}$| algorithm without modification. At the time of our spectroscopic programme, VST-ATLAS photometry was supplied in the Vega system. To convert to the SDSS system we adjusted the zeropoints of the individual VST-ATLAS tiles to get good agreement with the SDSS Stripe 82 co-add photometry for stars. In the left-hand panel of Fig. 1 we show the outline of the stellar locus from Stripe 82; the VST-ATLAS photometry is seen to be in good agreement with this deeper photometry. We refer the reader to Shanks et al. (2015) where the SDSS-VST colour terms are shown to be small.

The output of this selection algorithm is continuous and assigns candidates with a relative quasar likelihood. As such, we are not limited by a lack of candidates but by the availability of instrument fibres. Whilst the precise sky density of |${\small \tt XDQSO}$| candidates varies with image quality (and hence sky position), selecting candidates ranked according their PQSO value limits us to observing candidates with PQSO ≳0.7.

5 SPECTROSCOPIC OBSERVATIONS

5.1 2dF and AAOmega

Spectroscopic observations were made with the 2dF-AAOmega instrument on the AAT. The 2dF is a fibre positioning system for the AAOmega multi-object spectrograph which is capable of simultaneously observing 392 objects over ≈3.14 deg² field of view. Fibres are positioned by a robotic arm and are fed to the spectrograph. The 2dF also implements a tumbling system that allows for a second plate to be configured whilst the first is being observed. AAOmega is a dual beam spectrograph that utilizes a red/blue dichroic beam splitter, splitting the light at 5700 Å. The observations were made using the 580V and 385R gratings for the blue and red arms of the spectrograph respectively. The gratings have resolving power of R = 1300 and central wavelengths of 4800 Å and 7250 Å for the blue and red arms. The useful wavelength range in our configuration is 3700–8800 Å.

We made no nightly observations of standard stars so our spectra do not have an accurate absolute calibration. The |${\small \tt 2DFDR}$|³ data reduction pipeline combines the spectra from the red and blue arms of the spectrograph. To achieve this, the spectra are calibrated to a common scale with an arbitrary normalization due to unknown aperture losses, via a transfer function derived from previous observations of the standard star EG 21.

Of the 392 2dF fibres (not including 8 for guide stars) 20 fibres for sky subtraction. The remaining ≈372 fibres were used for science targets. The fibre allocation software configure⁴ (v7.17) allows input targets to have priorities associated with them. The observing priorities allow the software to make a decision about the importance of placing fibres. This allowed us to prioritize our spectroscopic targets according to their likelihood of being a quasar. This prioritisation was one of the requirements of the target selection process.

Exposure times varied between 0.7 and 2 h to account for observing conditions. All our data was reduced using the |${\small \tt 2DFDR}$| pipeline (v5.35) using default parameters. We measured quasar redshifts of the spectra with the runz program (Saunders, Cannon & Sutherland 2004).

5.2 Resulting QSO catalogue

We developed a combination of the three techniques described in Section 4 to optimize our quasar selection over the duration of the pilot survey. We divided the selection into two major implementations. The first (chronological) selection relied solely on the optical photometry from VST-ATLAS in the form of ugri and |${\small \tt XDQSO}$|⁠. Later implementation of the selection algorithm applied these techniques in conjunction with optical-IR colour cuts. We label these algorithms in Table A1 as ‘ugriXDQSO’ and ‘ugriXDQSOW1W2’ respectively.

The ‘ugriXDQSO’ target selection was based on ugri colours with the |${\small \tt XDQSO}$| algorithm used to rank those targets. The ‘ugriXDQSOW1W2’ selection algorithm gave candidates meeting the optical+IR conditions from Section 4.0.5 the highest priority, with remaining candidates prioritized based primarily on their PQSO.

In Table A1 we present the results of our spectroscopic observations. We list the field locations and the dates of our observations, the number of quasars identified in a given pointing (also shown in Appendix C, Fig. C1), exposure times, mean spectral signal-to-noise and a guide to the target selection used.

As variation in spectral signal to noise affects our measured quasar density, we look to parametrize our different fields in a meaningful manner so that we can compare the effectiveness of our selection algorithms. To account for varying spectroscopic incompleteness between different observations we compare the number of bright g ≤ 20.5 quasars between our fields (see column ‘N_QSO ≤ 20.5’ in Table A1). At these brighter magnitudes we are approximately spectroscopically complete for quasars. We show in Table A1 the faint limit in the g band that contains 90 per cent of our spectroscopically confirmed quasar sample and see that we are suitably bright to be photometrically complete in the ugriz bands.

Having accounted for spectroscopic incompleteness we expect variation in our measured quasar density to be primarily determined by our selection algorithm and the number of background stars (see Section 5.5). Whilst the VST-ATLAS is limited to high Galactic latitudes we note that the number of stars in a 2dF pointing can vary significantly (see Table A1). We include in Table A1 the number of point sources of magnitude g ≤ 21.5 under the heading N_stars and see this density vary by up to a factor of 3. This variation is primarily determined by Galactic latitude (c.f. ≈5 per cent from zeropoint errors).

Our spectroscopic programme was awarded 17 nights of observing time to develop a QSO selection algorithm as preparation for a larger programme. We obtained redshifts for ≈10 000 quasars with apparent magnitudes g ≤ 22.5 and <z > =1.55 (80 per cent of the sample lies within 0.8 < z < 2.5). We present the redshift distribution of our quasars in Fig. 2 and include the redshift distributions for 2QZ, 2SLAQ and SDSS for comparison. We see that the SDSS n(z) has a second peak at z ∼ 3.1 which is due to a secondary colour selection designed to identify quasars at this redshift. When comparing between surveys we limit to 0.3 < z < 2.9 and so ensure good agreement between the redshift distributions.

Figure 2.

We show the redshift distribution of the 2QDESp spectroscopic quasar sample as the shaded region. For comparison we include the redshift distributions for the SDSS DR5, 2SLAQ and 2QZ samples.

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5.2.1 Redshift errors

Here we consider factors which affect the measurement of quasar redshifts. Poor-quality spectra will cause errors in redshift estimation as well as incompleteness due to failure to identify the target as a quasar. Reliance on a small number of quasar emission lines will also cause systematic errors due to mis-identification of emission features, as noted by Croom et al. (2004). We have a number of repeat observations that we can use to quantify the redshift error. The runz code allows for three quality flags qop = 5, 4 or 3 for reliable redshifts.

Restricting our analysis to the highest quality spectra (qop = 5) we find a redshift error of σ(z)/z = 0.002, comparing repeat observations as in Croom et al. (2004). This corresponds to ∼600 km s⁻¹ or ∼2 h⁻¹ Mpc (comoving) at our mean redshift. We next compare the highest quality to the intermediate quality (qop = 4) spectra. We take any difference in redshift greater than Δz = 0.01 as a redshift failure. Intermediate quality spectra have a redshift failure rate of 6 ± 2 per cent and an error of σ(z)/z = 0.002, excluding Δz > 0.01. Similarly we find σ(z)/z = 0.002 for our lowest quality spectra (qop = 3) but this time with a failure rate of 16 ± 12 per cent.

We quantify the rate of redshift failure due to line mis-identification. Having examined quasars with repeat observations we find that redshift failures occur for 9 ± 1 per cent of quasars, over all redshifts, magnitudes and spectral quality.

5.3 Effectiveness of quasar selection methods

We introduced the optical–IR colour cuts in Section 4.0.5; we review their effectiveness here and compare to the |${\small \tt XDQSO}$| technique which relies on UVX techniques to identify quasars. To compare these selection techniques we examine one of the most complete fields (23^h16^m − 26^d01^m) where we find over 80 quasars deg⁻² between 0.8 < z < 2.5 and 16 < g < 22.5.

In Fig. 1 we show the distribution of our quasar sample in the ugr, giW1 and gW1W2 colour spaces. We include only point sources with g < 20.5. As noted in Section 4.0.5 the left-hand and middle panels of Fig. 1 show a difference in distance between the quasars and the stellar locus. At fainter magnitudes (g > 20.5) the photometric scatter will become larger and so that the effective separation between the stars (mainly type A and F) and quasars will be reduced. The increased distance from the stellar locus seen in giW1 (compared to the distance in ugr colour space) suggests that this selection may suffer from less stellar contamination than using ugr photometry and that any stellar contamination might come from different spectral types of stars.

We examine the apparent purity of the giW1W2 colour selection by comparing its effectiveness against the |${\small \tt XDQSO}$| algorithm. We are limited in the giW1W2 selection by the depth of the WISE photometry and so must take this into account when comparing to the |${\small \tt XDQSO}$| algorithm. We limit our comparison to g < 20.5 and treat photometric and spectroscopic incompleteness for quasars as negligible. In Table 1 we take all giW1W2 quasar candidates with g < 20.5 and find that 3 per cent of these sources are stars, based on spectroscopic observations. If we assume all of the non-identified sources are stars, our stellar contamination rises to 14 per cent. We test |${\small \tt XDQSO}$| by taking the same target density as identified by giW1W2 and find contamination rates of 17–42 per cent, again depending on the nature of the non-ids.

Within our target redshift range we expect to find ≃75 quasars per 2dF pointing at this (g < 20.5) magnitude limit. In Table 1 we show the number of quasars identified by both algorithms as well as showing (in brackets) the number of quasars common to both. In the brighter regime (g < 20.5), we find that both algorithms identify at least 74 quasars within our target redshift interval and so both are consistent with being complete. However, we also note a further nine quasars from the giW1W2 selection which corresponds to a 12 per cent increase against the performance of |${\small \tt XDQSO}$|⁠.

The single quasar (g < = 20.5 & 0.8 < z < 2.5) ‘missed’ by giW1W2 is not detected in the All-Sky release and so was not missed due to incompleteness introduced as a result of our chosen colour selection. However, subsequent to our observations, an improved analysis of the WISE data (the ALLWISE data release) results in a detection for this target (W1 = 17.07, S/N = 8.5 and would be selected by our algorithm). This missing target suggests that our WISE photometry has an incompleteness for quasars within our target redshift range of ∼1 per cent.

Some quasars are only identified by giW1W2 but not by |${\small \tt XDQSO}$|⁠. Many of these would be found by our ugr colour selection, or a simpler colour–magnitude cut. The mean ‘probability’ of these targets is PQSO = 0.3 and so would not be observable without a much higher fibre density. The |${\small \tt XDQSO}$| algorithm provides a ‘PQSO_ALLZ’ parameter, that gives the ‘likelihood’ of a target being a quasar at any redshift. For these ‘missed’ quasars the mean ‘likelihood’ is PQSO_ALLZ = 0.8. The low PQSO of these quasars is apparently caused by |${\small \tt XDQSO}$| attempting to estimate the redshift of the quasar candidates. The two red (u − g > 0.7) quasars detected by WISE are given low ratings by |${\small \tt XDQSO}$| (PQSO_ALLZ = 0.81, 0.01 and PQSO = 0.26, 0.01) and are therefore too lowly ranked by |${\small \tt XDQSO}$| to be observed.

In the fainter regime, |${\small \tt XDQSO}$| identifies a single quasar redder than u − g = 0.7 of the 74 quasars within this redshift range. The giW1W2 selection recovers 9 (out of 78) with red optical colours. Combining the two selections we find that 9 per cent of our sample in the fainter regime is ‘reddened’ beyond the approximate limits of our ugri colour selection.

We widen the redshift interval from 0.8 < z < 2.5 to 0.3 < z < 3.5, in the expectation that errors in redshift estimation performed by |${\small \tt XDQSO}$| will result in a high number of quasars outside the 0.8 < z < 2.5 interval. We find that both algorithms select a significant number of quasars outside our targeted redshift range. For astrophysical studies of the quasar population this may not be an issue. However, to make the highest precision measurements of clustering, surveys require the highest density of quasars within as narrow a redshift interval as possible. Targeting quasars outside our preferred redshift interval lowers the efficiency of the survey.

We now examine spectroscopically confirmed quasars that are ranked as likely stars by the |${\small \tt XDQSO}$| algorithm. Given the continuous nature of the likelihood we make a cut in the output likelihood. We cut at |${\tt PQSO} < 0.4$| as the target density above this value is ≈250 deg⁻², well above what is observable in a single epoch of spectroscopy with 2dF. After the giW1W2 selection, we find ≃10 quasars deg⁻² (across all magnitudes), within our targeted redshift range that lie within this low PQSO region.

The mid-IR excess demonstrated by quasars places them in a region of colour space with a lower contamination rate than we see from |${\small \tt XDQSO}$|⁠. If this contamination rate were to continue to fainter limits, then a mid-IR selection alone may be sufficient to achieve the target quasar density. With the current limits from the mid-IR photometry, which introduce photometric incompleteness, we must supplement the mid-IR selection with |${\small \tt XDQSO}$| to achieve higher quasar densities. In our sample field |${\small \tt XDQSO}$| recovers a quasar density of 54 quasars deg⁻² (0.8 < z < 2.5, g < 22.5) compared to 74 quasars deg⁻² from combining WISE, VSTATLAS and |${\small \tt XDQSO}$|⁠.

5.4 The nature of mid-IR non-ids

Here we examine the contaminants within the giW1W2 colour space and attempt to discern the nature of the unidentified targets. We look both at the confirmed contaminants from a highly complete field and at the contaminants from the whole survey. Within the highly complete region (from Section 5.3) we have three spectroscopically confirmed stars with g < 20.5. These stars are identified as A and K-type stars from their spectroscopy and have been scattered up from the stellar locus. They are anomalously red in the i −W1 colour and hence included within our colour selection. Over the course of the survey we identified a number of white dwarfs (WDs) and M-type stars within our giW1 colour space. However, neither of these type of stars have broad-band colours consistent with being identified by our giW1 selection. WDs have colour i −W1 ≲ 1 and M stars have g − i > 1.5 so neither of these should contaminate the giW1 colour space.

The WIRED survey, Debes et al. (2011), categorized the infrared emission from UKIDSS Z band to WISEW4 band of SDSS DR7 WDs. WDs with an infrared excess (mostly due to a contaminating M star) were identified as a potential source of the observed WD contamination of our colour space (see Debes et al. 2011). In Fig. 3 we show that the WD+M star locus overlaps with the quasar locus in the giW1 colour space.

Figure 3.

We show the distribution of our spectroscopic quasar sample, from the entire survey, in the giW1 colour space (green contour). We include morphological point-sources (identified by the g band with 16 < g < 20.5; shown as grey contour) and our giW1 colour cuts (dashed purple lines) from Section 4.0.5 for reference. We show that the WD+M binaries from Debes et al. (2011) (red contour) directly overlap with the quasar locus in this colour space acting to reduce the efficiency of this colour selection.

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Debes et al. (2011) suggested that this may be due to the M-star contributing flux at longer wavelengths than the WD and thereby giving the system quasar colours. These authors found that 28 ± 3 per cent WDs have M dwarf companions and approximately a further ≈2–10 per cent have either associated dust or a brown dwarf that would give them excess emission in the W1 band. Given the similar depths between SDSS and ATLAS we expect a similar rate of contamination as found by Debes et al. (2011).

To better examine the overlap between the quasar locus and the WD+M-star locus, we take our entire quasar sample and map its distribution in giW1 colour space in Fig. 3. We show that the WD+M binaries directly overlap with the quasar locus in this colour space. Fig. 3 explains the appearance of WDs and M stars in the giW1 colour cut. Whilst these systems will account for the occasional appearance of the spectroscopically confirmed contaminants, however, they do not have a sufficiently high sky density to account for the majority of the unidentified sources. We attempt to find a colour space that separates the stars and quasars that appear in the giW1W2 colour selection. If we are able to separate the stars and quasars by colour selection we may be able to infer the nature of the unidentified targets. We use all our observed targets to better characterize the contamination. In the right-hand panel of Fig. 4 we show the gW1W2 plane. We show as reference the position of the stellar locus, the locations of spectral type A and M stars, as well as the WDs with excess IR emission. We also show the quasar locus (derived from our whole sample). Spectroscopic stars and non-ids with g < 20.5 that obey our giW1W2 colour selection are also plotted. We see from the right-hand panel of Fig. 4 that the majority of stars may be removed by a cut in the W1 −W2 colour. Due to the close proximity of the stellar locus, a cut in this colour may improve our efficiency but will affect our completeness as well. Where we have J-band coverage from the VHS survey (McMahon et al. 2013), we see from the left-hand panel of Fig. 4 that the J −W1 colour increases the separation between the stellar locus and the quasar locus. The majority of the non-ids lie within the quasar locus, although some do lie closer to the stellar locus. Table 2 quantifies that their mean J −W1 colours are consistent with quasar colours. Given that the number counts for non-ids peak a magnitude fainter than the peak of the identified targets, this suggests that many of these non-ids are quasars where positive identification has failed due to spectroscopic incompleteness.

Figure 4.

In the left-hand panel we show the stellar locus in gJW1 colour space (grey contours). We also plot the locus of our quasar sample (green contours). Targets without a spectroscopic id that fulfil the giW1W2 colour cuts are shown as grey five point stars and spectroscopically confirmed stars are shown as black triangles. We also mark the location of spectral type A and M stars as well as the location of WD+IR excess stars from Debes et al. (2011). In the right-hand panel we follow the same convention for marking the quasar and stellar locus, but instead show these in the g −W1 versus W1 −W2 colour space. The majority of non-ids have colours consistent with quasars in giJW1W2 photometric bands and suggest that the failure to positively identify these targets is due to spectroscopic incompleteness.

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5.5 Background stellar density

We find that the measured spectroscopic quasar density varies across the sky, independently of the selection algorithm (see Table A1). Spectroscopic incompleteness will contribute to this variation. We minimize this by considering only sources with g < 20.5, where we are approximately complete. In Fig. 5 we show the variation of confirmed quasars with g < 20.5 as a function of the number of point sources with g < 21.5 per deg².

Figure 5.

We show the number of confirmed quasars deg⁻² (g < 20.5) against the number of stellar sources deg⁻² (with g < 21.5). We compare the two algorithms; ugriXDQSO (red circles) and ugriXDQSOW1W2 (blue triangles). By limiting the comparison to brighter quasars we assume the contribution of observational effects is negligible.

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Fig. 5 shows that the inverse correlation between stellar density and spectroscopic quasar density is the dominant cause of varying quasar density across the sky. This affect results in as much as a factor of 2 between confirmed quasars in fields with different background stellar densities. This relation indicates that the efficiency of a wide area quasar survey depends on the background stellar counts in the observed fields.

However, we also note that the ugriXDQSOW1W2 algorithm fields consistently identify a higher number density of quasars then the ugriXDQSO algorithm.

5.6 Conclusions

We combined mid-IR photometry from WISE with the ugriz photometry from the VST-ATLAS survey to improve the efficiency of our quasar selection. We found that the giW1 colour space (see Fig. 1) is approximately complete to g < 20.5. Fainter than this the colour space becomes photometrically incomplete as quasars become too faint to be detected in the W1 band of WISE. We next attempted to use broad-band colours to identify fainter (in the g band) candidates from the g − i: i −W1 colour space that we were unable to identify from spectroscopy. Further analysis with the J band failed to prove conclusively that the unidentified targets in this colour space were stars. These targets exhibit broad-band colours consistent with quasars. This could mean that the colour space is complete to fainter limits in g than found in this work. We found that the combination of optical and mid-IR photometry improved the efficiency of our quasar selection. In Fig. 5 we see that fields where WISE photometry was included in the quasar selection saw an increased yield of ≈10 quasars deg⁻². The improvement in quasar selection that we found in this survey may readily be extended to other quasar surveys in a similar redshift range such as eBOSS. Furthermore, we note that WISE photometry has proven to be a boon for quasar selection at higher redshifts (Carnall et al. 2015).

6 CLUSTERING ANALYSIS

6.1 Correlation function estimators

6.1.1 Redshift space correlation function

6.1.2 Modelling quasar clustering in redshift space

At small scales (s ≲ 5 h⁻¹ Mpc) redshift space distortions dominate the clustering signal in ξ(s). Non-linear,‘finger-of-God’ peculiar velocities of the quasars and redshift measurement errors both act to reduce ξ(s) at these scales. Justified mainly by the good fit it provides, we shall assume a Gaussian for f(w_z) (see Ratcliffe 1996) with a fixed velocity dispersion of |$\langle w_{z}^{2} \rangle ^{\frac{1}{2}} = 750 \: {\rm km s}^{-1}$|⁠.

6.2 2QDESp quasar correlation function

Figure 6.

We show the measured ξ(s) for the 2QDESp quasar sample between 0.3 < z < 3.5. We include our model with best-fitting correlation length (using Jackknife errors) of r₀ = 6.25 ± 0.30 h⁻¹ Mpc. In the top panel, we show the ratio between the Jackknife and Poisson errors.

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For comparison, we show the ratio of the jackknife error estimation to Poisson errors in the top panel of Fig. 6. We see that the Poisson error is a reasonable representation of the jackknife error out to s < 30 h⁻¹ Mpc and is still within a factor of ≈1.4 at s < 55 h⁻¹ Mpc, only reaching a factor of ≈1.8 at s ≈ 100–200 h⁻¹ Mpc. This suggests that, at least out to s < 55 h⁻¹ Mpc, pair counts are reasonably independent and this is supported by the small size of off-diagonal terms in the covariance matrix at these scales. We shall fit models in the range 5 < s < 55 h⁻¹ Mpc using both jackknife and Poisson errors.

We now fit the model from Section 6.1.2 to the 2QDESp ξ(s) data. We show our best-fitting model assuming Poisson errors in the lower panel of Fig. 6. This has r₀ = 6.25 ± 0.25 h⁻¹ Mpc which fits well with χ² = 9.4 with 10 degrees-of-freedom, (df.). Assuming jackknife errors for the fit gives a similar result, r₀ = 6.25 ± 0.30 h⁻¹ Mpc (χ², df. = 7.0, 10).

To further test the quality of our data, we divide the quasar sample into three subsets, based on the quality of their optical spectra. The three subsets consist of 1675, 4585 and 3541 quasars for the highest, intermediate and lowest quality spectra, respectively. We compare the different quality spectra in Fig. 7, where we plot the correlation function for the three quasar subsamples. We fit the model from Section 6.1.2 to each and find that the three subsamples agree at the ∼1σ level. Using this procedure we verify that our lowest quality optical spectra are suitable to use in our science measurements as the contamination by other sources is low enough not to cause significant differences in the measured clustering signal.

Figure 7.

Here we show the correlation function for 2QDESp quasars with the highest, intermediate and lowest quality spectra, qop = 5,4 and 3, respectively. We offset the high and low spectral quality correlation functions along the x-axis by 10^s ± 0.02 for clarity. The three correlation functions for each quality level agree. Hence we argue that the lowest quality sample is suitable for use in our analysis.

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6.3 Luminosity dependence of clustering

In this section we search for evidence of luminosity dependent quasar clustering. We start with the approach of S11 and compare measured r₀ values between different surveys at approximately fixed redshift. We follow this with the more precise methodology of dA08 which divides the samples by absolute magnitude and redshift. We defer measurement of redshift dependence to Section 6.4.

6.3.1 Apparent magnitude

Comparison between the 2QZ, SDSS, 2SLAQ and 2QDESp quasar surveys provides an opportunity to measure the dependence of clustering on luminosity. Whilst each survey has different selection methods and flux limits we see in Fig. 2 that the resulting redshift distributions are similar (see also Table 3). Given that each survey is flux limited, we account for photometric and spectroscopic incompletenesses by characterizing each survey by its median magnitude. For the four surveys this corresponds to; g = 18.8 (SDSS), g = 20.1 (2QZ, see Richards et al. 2005 for b_J − g conversion), g = 21.3 (2SLAQ) and g = 20.6 for the 2QDESp sample.

In Fig. 8 we show our re-analysis of the SDSS, 2QZ, 2SLAQ and 2QDESp clustering results, restricting our analysis to quasars between 0.3 < z < 2.9. We have rebinned the s-axis to a common binning across the four surveys. In each panel we show the best-fitting r₀ for each survey, assuming a fixed β. We permit use of constant β here because with the small difference in median redshifts the effect of different β(z) values will have <1 per cent effect on r₀. Our best-fitting values are shown in Table 3 and these measurements agree with the analysis of S11 with any differences in the best-fitting values due to slight difference in redshift range and fitting interval.

Figure 8.

Each panel shows our estimate of ξ(s) measured for a particular wide area survey as labelled. We annotate each panel with the median magnitude in g for comparison to our survey. Errors are Poisson. We fit the data using the model from Section 6.1.2, where we assume Gaussian velocity dispersions in real space, γ = 1.8 with a velocity dispersion, |$\langle \omega ^{2}\rangle ^{\frac{1}{2}} = 750 \: \mathrm{km} \: s^{-1}$|⁠. In each panel we show the model where r₀ = 6.1 h⁻¹ Mpc (solid line) (see Section 6.1.2). For each survey we restrict the analysis to the redshift interval 0.3 < z < 2.9 as this range is well sampled by all surveys. The best-fitting models for the individual surveys are shown in Table 3.

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We make a comparison between the median magnitude and best-fitting values of r₀ across the four surveys in Fig. 9. We note that Shen et al. (2009) found that the brightest SDSS DR5 quasars clustered more strongly than the rest of their quasar sample. We find here that r₀ for the SDSS quasars is larger than the r₀ values from the other surveys but only at ≈1σ level. As this effect corresponds to the result reported by Shen et al. (2009) we must be cautious not to immediately dismiss the difference as purely statistical. However, we further test for the dependence on r₀ with magnitude using the Spearman rank correlation test. We find a Spearman rank order correlation of ρ = −0.19 ± 0.37 which is consistent with no correlation between apparent magnitude and clustering scale. We also find that the points in Fig. 9 are consistent with a constant |$r_{0} = 6.10^{+0.10}_{-0.10}\:h^{-1}$| Mpc with χ², df. = 3.9, 3 and p-value=0.28.

Figure 9.

We show the median depth for 2QZ, 2SLAQ, SDSS and 2QDESp surveys along with the best-fitting r₀ with the associated errors. We also show a flat r₀ = 6.1 h⁻¹ Mpc model (solid line) and an L^0.1 model (dotted line).

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In Table 3 we calculate the corresponding χ² when we compare each survey individually to a fixed r₀ = 6.1 h⁻¹ Mpc and we find that the total χ², df. = 46.8, 40 (we include the individual survey χ² values in Table 3). So from this analysis we are unable to reject the hypothesis that quasar clustering is independent of luminosity from a comparison between the individual surveys.

6.3.2 Absolute magnitude

In Section 6.3.1 we compared quasar clustering over a range of ∼3.5 mag at fixed redshift. Although further subdivision of the quasar samples will yield weaker statistical constraints, we are, however, able to probe a much larger dynamic range (−22.3 < M_i(z = 2) < −28.5) at fixed redshift by combining all four surveys. We do this by taking the error weighted mean of the four correlation functions for each subsample. Following the approach of dA08, we divide the M−z plane into non-overlapping bins. We use the sample binning of dA08 which was designed to maximize the clustering signal from the 2QZ+2SLAQ combined sample. The inclusion of the SDSS and 2QDESp data reduces the statistical errors; particularly in the highest and lowest luminosity bins. This may enable us to potentially uncover previously hidden dependencies.

We show the distribution of the 2QDESp sample in the left-hand panel of Fig. 10. We overlay the figure with the M−z divisions and include the occupancy of each division. In the right-hand panel of Fig. 10 we plot the M−z distribution for the combined (SDSS+2QZ+2SLAQ+2QDESp) sample. Again, we overlay the M−z divisions and show the total bin occupancy. In both panels of Fig. 10 the flux limited nature of the surveys is evident by the absence of lower luminosity quasars at higher redshifts. As expected, we see that the 2QDESp survey makes its largest contribution at fainter absolute luminosities.

Figure 10.

The distribution of our sample in redshift–luminosity (left) and the comparison to 2QZ, 2SLAQ, SDSS DR5 and 2QDESp surveys (right). The grids show the division in magnitude and redshift applied to the samples and the occupancy of each bin.

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Our aim in subdividing the combined sample in both magnitude and in redshift is to isolate redshift and luminosity dependent effects on the clustering amplitude. In Fig. 11 we show the signal for each of the absolute magnitude and redshift bins. To generate random samples we use R.A.–Dec. mixing (see C05), sampling from the all magnitudes and redshifts to generate the angular mask. The radial mask is generated by randomly sampling the redshift distribution of the magnitude–redshift subsample. We found that fitting the radial distribution with a polynomial provided similar results to those included here.

Figure 11.

We measure the correlation function ξ(s) for the combined sample (SDSS,2QZ, 2SLAQ and 2QDESp) in the same bins as Fig. 10. We use the error weighted mean to combine the measurements from each individual survey, where errors are Poisson (see Section 6.2). These are compared to a ξ(s) model where r₀ = 6.1 h⁻¹ Mpc (solid line). We show the fit quality for this fixed r₀ value as well as for the best-fitting value in Table 4.

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Previously we fit for r₀ at approximately fixed redshift. However, here we are fitting over Δz ∼ 1.7 and so an assumption of constant β is no longer valid. We therefore measure the correlation function for each subsample in Fig. 11 but determining β(z) from an assumed b(z) relationship (see equation 4). Whilst there is uncertainty in the precise form of the b(z), a 50 per cent increase in bias at z = 1.5 only results in a 4 per cent change in r₀.

Allowing the value of r₀ to vary between each bin we find a total χ², df. = 130.6, 130 and p-value = 0.47. We plot the best-fitting values (see Table 4) in Fig. 12. We also include in this figure our determination of r₀ from the measured ξ(s) of Eftekharzadeh et al. (2015); using our model we find their correlation function corresponds to r₀ = 7.25 ± 0.1 h⁻¹ Mpc.

Figure 12.

We show the measured correlation length (r₀) for the 13 luminosity−redshift bins from Table 4. We include our measurement of r₀ from the correlation function of Eftekharzadeh et al. (2015) who measure the clustering scale of quasars from the BOSS sample as r₀ = 7.25 ± 0.10 h⁻¹ Mpc.

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In Fig. 12 we compare across the luminosity bins at approximately fixed redshift. The fainter two magnitude bins (spanning −24.5 < M_i < −22.3) show, on average, stronger clustering at all redshifts than the brighter bins. If this trend is physical it would suggest that fainter quasars are more strongly clustered than brighter quasars, suggesting an inverse relationship between quasar luminosity and halo mass. However, we note that these magnitude–redshift bins correspond to the faintest apparent magnitudes in the 2QDESp and 2SLAQ samples and suffer from large incompleteness. So although there may be a weak underlying dependence on luminosity we are unable to claim a significant detection analysing the data in this fashion. It is possible, of course, that some effect of luminosity dependence is being masked by the redshift dependence of quasar clustering.

6.4 Redshift dependence

C05 in particular looked at the effect of both systematic and statistical uncertainties associated with integrating different radius spheres. We adopt the same radius as used by these authors (see C05, for a detailed analysis).

In Fig. 13 we show the integrated correlation function for each absolute magnitude and redshift bin from Section 6.3.2. We show the redshifts and ξ₂₀ values for these bins in Table 4. We see that the evolution of ξ₂₀(z) is flatter than one might naively expect from either Table 4 or Fig. 12. This is due to the effect of β(z), accounted for in our model, that ‘boosts’ ξ₂₀(z) more at lower redshifts than at higher redshifts and thus flattens the evolution of ξ₂₀.

Figure 13.

We show the measured |$\xi _{20}^{Q}$| for the bins from Section 6.3.2. We include model predictions for the evolution with redshift of |$\xi _{20}^{Q}$|⁠. The solid line shows the expected |$\xi _{20}^{Q}(z)$| relation assuming the empirical b(z) relationship from equation (12). For comparison we show the empirical b(z) relation from C05 as a dashed line, i.e. equation (4).

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6.4.1 Bias and halo masses

2QZ measured the quasar correlation function as a function of redshift (see C05). They reported the relationship of quasar bias with redshift described by equation (4). In this section we use the same methodology as previous works (C05; dA08 and R09) with our larger data set to more precisely determine the evolution of bias with redshift.

Figure 14.

We show our estimate of quasar bias as a function of z and absolute magnitude. We include our measurement of bias from the BOSS survey, Eftekharzadeh et al. (2015). We show the evolution for a halo of mass 2 × 10¹² h⁻¹ M_⊙ as the solid grey line. We see that our measurements of bias are consistent with quasars inhabiting the same mass haloes irrespective of magnitude or redshift. We include the 2QZ bias result (equation 4) as a black dashed line and our bias result (equation 12) as a dotted black line for comparison.

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Fig. 13 shows the difference the change in the b(z) relationship makes on ξ₂₀. The dashed line showing the prediction of ξ₂₀ from ξ_ρ(r, z = 0) and equation (4) and the solid line showing the prediction of equation (12). We also plot the independent BOSS data from Eftekharzadeh et al. (2015) which lies much closer to our b(z) result than that of 2QZ.

The clustering of the remaining three magnitude bins is best described by halo masses of 6 ± 8 × 10¹² h⁻¹ M_⊙, 1.9 ± 1.4 × 10¹² h⁻¹ M_⊙ and 2.2 ± 0.2 × 10¹² h⁻¹ M_⊙ (rms error) for the −24.5 < M_i < −23.5, −25.5 < M_i < −24.5 and −28.5 < M_i < −25.5 bins, respectively.

We find that (excluding the high-z, low-M bin) the evolution of bias with redshift is well described by a mean halo mass of M = 2 ± 1 × 10¹² h⁻¹ M_⊙ (c.f. M = 3 ± 5 × 10¹² h⁻¹ M_⊙ including this bin). We show the model prediction for this halo mass in Fig. 14 as a solid line. Within the errors, our bias measurements are consistent with a single halo mass at all redshifts and luminosities.

Our measurement of the evolution of b(z) is slightly different than that of C05, the determination of halo mass has large errors. As such, our best-fitting halo mass is lower than that of C05 but remains consistent at the 1σ level.

6.5 Comparison to XMM-COSMOS quasar clustering

The semi-analytic model presented by Fanidakis et al. (2013) predict that X-ray selected quasars inhabit higher mass haloes than optically selected quasars. Fanidakis et al. (2013) present halo mass estimates from Allevato et al. (2011) and Krumpe et al. (2012) are presented as observational support to this model as these halo masses are higher (∼10¹³M_⊙) than estimates from wide area optical studies (∼10¹²M_⊙). In this section we briefly examine to whether this difference in halo mass estimates may be reconciled with the lack of dependence on optical luminosity found here. In particular, differences may occur due to differing analysis methods, and so we apply our method used for our optically selected samples to the X-ray selected sample of Allevato et al. (2011).

Allevato et al. (2011) measured the correlation function for quasars in the XMM-COSMOS field (Brusa et al. 2010) and found a clustering scale of |$r_{0} = 7.08^{+0.30}_{-0.28} \,h^{-1}$| Mpc and |$\gamma = 1.88^{+0.04}_{-0.06}$|⁠. We examine the sample of quasars used in their work and that find g_median = 21.4 (∼0.1 magnitudes fainter than the 2SLAQ sample) and their space density of quasars is ∼90 deg⁻² which is similar to that reached by 2QDESp, see Table A1. Further, the redshift distribution of their X-ray selected sources (Fig. 2; Allevato et al. 2011) is comparable to those of optically selected studies (see Fig. 2). As we find no evidence for r₀ increasing with fainter magnitude, we believe their contradictory result worthy of further scrutiny.

First, we note that an earlier clustering analysis of the XMM-COSMOS quasars (∼10 per cent fewer quasars than Allevato et al. 2011) was performed by Gilli, Zamorani & Miyaji (2009) who measured |$r_{0} = 7.03^{+0.96}_{-0.89} \,h^{-1}$| Mpc with γ = 1.8. We use the R.A.−Dec. mixing approach of Gilli et al. (2009) to generate a random catalogue. However, instead of measuring w(r_p) we measure the redshift correlation function, ξ(s), for these data, assuming γ = 1.8 as in Section 6.3.1 for the fit. Gilli et al. (2009) compared this method of random generation to modelling the angular distribution and found that it can underestimate the true correlation length. Applying the correction from Gilli et al. (2009) we find that the amplitude of clustering is described by |$r_{0}{=}6.03^{+0.80}_{-1.00} \,h^{-1}$| Mpc. This is in agreement with the measurements of quasar clustering at z ≈ 1.5 found in this work.

Both the r₀ measurements from Gilli et al. (2009) and Allevato et al. (2011) use the projected correlation function, w(r_p), as opposed to the redshift–space correlation function, ξ(s), that we use. By remeasuring the correlation function we are able to compare directly to optical results. As noted by other authors (Krumpe et al. 2012; Mountrichas & Georgakakis 2012) this approach should provide a more robust comparison than comparing between different bias or halo mass models.

We also note that our errors (and those of Gilli et al. 2009) assume Poisson statistics and still lead to a factor of 2–3 × larger errors on r₀ than the ≈± 0.3 h⁻¹ Mpc quoted by Allevato et al. (2011); it is not clear why this is the case. If the statistical errors on the XMM-COSMOS results are as large as found by Gilli et al. (2009) and ourselves then we conclude that these data contain no significant evidence for luminosity dependent clustering e.g. compared to their brighter counterparts in Fig. 9 (see also discussion in next section).

6.6 Baryonic acoustic oscillations

Here we extend our analysis of the combined quasar sample to larger scales. In Fig. 15 we show the result of combining the four correlation functions from each of the four surveys, weighting inversely according to the square of the errors at each separation (Section 6.3.1). We measure r₀ = 6.10 ± 0.15 h⁻¹ Mpc for a sample containing 70 940 quasars with <z > = 1.49. Combining these surveys gives an effective volume of ≈0.6 h⁻³ Gpc³, larger than the original SDSS LRG survey of Eisenstein et al. (2005) (≈0.55 h⁻³ Gpc³ or the 2dFGRS survey of Cole et al. (2005) (≈0.1 h⁻³ Gpc³) where BAO were detected. We use CAMB to predict the ΛCDM correlation function and scale this model to agree with the measured ξ(s) at intermediate scales, 5 < s < 55 h⁻¹ Mpc (see Fig. 15). Comparing the model to the data ξ(s) at larger scales, 60 < s < 200 h⁻¹ Mpc, we find that the model with the BAO feature is fit with χ², df. = 5.5,4 and p-value = 0.23 compared to χ², df. = 6.1, 4 and p-value = 0.19 for a similar model without BAO. Although the model with the BAO feature fits better, the reduction in χ² is not significant (Δχ² ≈ 0.6) and so it is not possible to claim that the BAO feature is detected in this combined quasar survey.

Figure 15.

At each s bin we combine the values of the correlation function ξ(s) for each of the four surveys using the error weighted mean. This combined sample consists of N_QSO = 70 940 with mean redshift |$\bar{z}=1.49$|⁠. We fit our model form Section 6.1.2 to the data and find a best-fitting value for r₀ = 6.10 ± 0.10 h⁻¹ Mpc with χ², sdf. = 15.6, 10 where we fit in the range 5 < s(h⁻¹ Mpc) < 55 (shown as the solid line). We also include the prediction of linear theory from CAMB and the ‘no wiggle’ model of Eisenstein & Hu (1998) with both normalized to our correlation function amplitude between 5 < s (h⁻¹ Mpc) < 55.

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We consider possible explanations for this lack of detection. First, the statistical errors are still relatively large, larger still once the off-diagonal covariance matrix elements are considered, motivating the need for bigger samples with larger effective volumes at the BAO scale. However, we have argued above that the effective volume should already be large enough for the detection of this feature. Secondly, it does not appear that the ±750 km s⁻¹ quasar redshift error plus intrinsic velocity dispersion affect the detectability of the BAO peak, as evidenced by convolving the ΛCDM model with Gaussians of this width. Our 9 per cent fraction of misidentified quasar redshifts will reduce the BAO signal and the small scale signal in proportion and so this effect has already been accounted for in Fig. 15 by our procedure of scaling the model to the observed small-scale ξ(s). Thus it remains unclear why the BAO peak is undetected in these data.

7 DISCUSSION

We have analysed quasar clustering using surveys covering a wide range of fluxes and luminosities. We have found that there is little evidence for an increase of clustering amplitude with luminosity within the optical surveys at fixed redshift. Even including the XMM-COSMOS survey, we still find no evidence for the dependence of the clustering scale on luminosity. Following S11 we assume a halo-black hole mass relation of the form |$M_{{\rm BH}}{\sim } M_{{\rm Halo}}^{1.82}$| (Ferrarese 2002) and the bias |$b{\sim } M_{{\rm Halo}}^{0.2}$| (Martini & Weinberg 2001) together with a fixed Eddington ratio we expect the approximate relation |$r_{0} \propto b \propto M_{{\rm BH}}^{0.1} \propto L^{0.1}$|⁠. Given the factor of ≈10 increase in luminosity between the SDSS and 2SLAQ samples, a factor of ≈1.25 increase in r₀ is predicted, corresponding to r₀ increasing from 6.20 (2SLAQ) to 7.75 h⁻¹ Mpc, significantly (≈4σ) higher than the observed value from SDSS. Thus, the observed luminosity dependence of the clustering amplitude is about a fifth of what is predicted on the basis of this simple model. This is confirmed by the formal χ² fits of the L^0.1 model in Fig. 9. Excluding XMM-COSMOS, we find χ², df. = 7.8,3 and p-value = 0.05 for the L^0.1 model compared to χ², df. = 3.8,3 and p-value = 0.23 for the L-independent case. Including XMM-COSMOS, the same preference for the L independent model is shown, although the L^0.1 model is slightly less rejected with p-value = 0.10. But if the XMM-COSMOS r₀ was closer to its corrected value of r₀ = 6.0 h⁻¹ Mpc, rather than the r₀ = 5.2 h⁻¹ Mpc we have assumed here, then the level of rejection of the L^0.1 model would again be increased. We conclude that any dependence of clustering amplitude with luminosity is smaller than expected from a simple halo model.

When we then sub-divided the combined SDSS, 2QZ, 2SLAQ and 2QDESp surveys by absolute magnitude and redshift to increase the dynamic range in luminosity, we again found no evidence for luminosity dependent clustering at fixed redshift. However, we note that we do have significant evidence for the dependence of r₀ on redshift. We introduced a new bias model for b(z) (equation 4), superseding that of C05. We find that our model for the evolution of bias with redshift is consistent with the higher r₀ = 7.25 ± 0.10 h⁻¹ Mpc measured by Eftekharzadeh et al. (2015) in the BOSS quasar survey at z ≈ 2.4.

The 2QZ results of C05 suggest that a fixed halo mass of ∼10^12.5 M_⊙ fits the z dependence of quasar clustering. Here we have confirmed the results of S11 that at approximately fixed redshift the clustering amplitude is approximately constant with luminosity. The apparent luminosity independence would suggest that the halo mass and hence black hole mass was virtually constant as a function of both luminosity and redshift.

If all quasars have the same black hole mass over a wide range of luminosity then there is a contradiction with all quasars radiating at a fixed fraction of Eddington as suggested from reverberation studies of nearby quasars (Peterson et al. 2004). To reconcile these two observations requires that the BH−halo mass relation is broken. There is some evidence for this from the work of Kormendy & Bender (2011) who found that the BH mass was more related to bulge mass than halo mass. In this view quasars would find themselves in similar sized haloes but with black hole masses more related to their luminosity.

A weak clustering dependence on luminosity is expected in ‘flickering’ models where the duty cycle for AGN activity is short and the quasar luminosity is highly variable (Lidz et al. 2006). The luminosity of a quasar may also be taken as implying a lower limit to its black hole mass via the Eddington limit. Thus low luminosity quasars must be accreting at a highly sub-Eddington rate if the halo-mass BH mass relation is to be preserved, since they have similar halo masses to their brighter counterparts. This means that quasars are preferentially seen in bright phases. S11 again noted that this model contradicts the established correlation from reverberation mapping between black hole mass and luminosity (Peterson et al. 2004) but otherwise fits the clustering data (by design).

In Section 6.4.1 we estimate the halo bias for optically selected quasars between 0.3 < z < 2.9 and −28.5 < M_i(z = 2) < −23.5. In agreement with earlier works (C05; dA08, R09, S11) we find a characteristic halo mass of M_Halo = 2 ± 1 × 10¹² h⁻¹ M_⊙. Recent measurements of quasar clustering from X-ray surveys (Gilli et al. 2009; Allevato et al. 2011; Krumpe et al. 2012) have estimated significantly higher halo masses (∼10¹³ M_⊙) than the above optically selected samples.

Semi-analytical models of AGN (Fanidakis et al. 2013) have suggested that this is a physical result caused by a difference in AGN fuelling modes between optically selected and X-ray selected samples. However, given the susceptibility of soft X-ray selection (≈0.1–2 keV) to intrinsic obscuration we would expect these two selections to sample the same population of AGN. This is supported by the similar space density and redshift distribution of unobscured X-ray AGN and optically selected quasars (see Allevato et al. 2011).

Indeed, both Krumpe et al. (2012) and Allevato et al. (2011) explicitly compare the clustering of optically selected quasars with unobscured X-ray AGN in their two samples. In both papers these authors find that the clustering of both populations (at any redshift) may be described by the same halo mass. Contrary to the claim of Allevato et al. (2011) we find consistent clustering of X-ray and optically selected samples. As such, we see little evidence for the higher halo masses in these studies (c.f. optical studies) that would support the suggestion of Fanidakis et al. (2013) that the two populations are driven by different accretion modes. The analysis by Mountrichas et al. (2013) suggests that higher X-ray AGN masses are in fact driven by X-ray AGN from groups. After excluding these AGN, Mountrichas et al. (2013) find the clustering of X-ray selected AGN is described by a halo mass |$5^{+4}_{{-}3}{\times }10^{12} \,h^{-1} \,\mathrm{M_{{\odot }}}$|⁠, consistent with the clustering results presented here.

Krumpe et al. (2012) discuss the impact of HOD versus power-law models for estimating bias from the correlation function. We agree that this may contribute to the differences in halo mass estimates. We further note that the deepest X-ray samples come from small areas on the sky and suffer from poorer statistics and greater susceptibility to sample variance than the larger area optical studies. This discrepancy could be well addressed by a large sample of deep X-ray selected AGN. Ongoing surveys such as eBOSS and the upcoming eROSITA survey have the opportunity to provide a homogeneous data set of quasars up to z ≲ 2.2. This may allow us to determine which physical processes drive accretion at different redshifts and how these processes interact to result in quasar clustering appearing largely independent of optical luminosity.

8 CONCLUSIONS

We have characterized a new quasar selection for quasars at intermediate redshifts 0.8 ≲ z ≲ 2.5 and we demonstrate that the WISE All-Sky data release is complete for quasars in the redshift range (with g < 20.5). To account for photometric incompleteness for quasars fainter than this limit, to g ≈ 22.5, requires traditional optical selection methods.

The 2QDES pilot survey has shown that a high density quasar survey is viable with the photometry from VST ATLAS and WISE. In fact the 2QDESp survey with 4 per cent of the area of SDSS has 20 per cent more effective volume to detect the BAO peak due to its ≈8 × higher quasar density. But even with 10 000 quasars from 2QDESp combined with those from SDSS, 2QZ and 2SLAQ, we still lack a large enough effective volume to measure the BAO peak in the two-point correlation function although we gain some advantage in the precision of the clustering scalelength, r₀.

Direct comparison between the quasar correlation functions of SDSS, 2QZ, 2QDESp and 2SLAQ surveys, that range over an order of magnitude in quasar luminosity, show the same hint of higher r₀ at higher (SDSS) luminosities that was seen by Shen et al. (2009) and S11. However, the errors are such that a luminosity independent r₀ cannot be rejected by these data.

We combine the clustering measurements from 2QZ, 2SLAQ, SDSS and 2QDESp in the M − z plane to search further for luminosity and redshift dependence. Contrary to the above hint, we find some tentative evidence here that fainter quasars may be more strongly clustered than brighter quasars at fixed (high) redshift (z > 1.5), albeit weakly detected. But overall the results remain consistent with a fixed quasar r₀ at fixed redshift, independent of luminosity.

We measure a significant redshift dependence of quasar clustering and see that this dependence explains the higher r₀ measurements from Eftekharzadeh et al. (2015). Comparison of the redshift dependence of quasar clustering to the halo model shows that our data (and that of Eftekharzadeh et al. (2015)) is consistent with quasars inhabiting 2 × 10¹² M_⊙ haloes irrespective of redshift or quasar luminosity. These results are usually explained by a ‘flickering’ quasar model with a short duty cycle where quasars over a wide range of luminosities have similar halo, and hence black hole, masses. However, such models are inconsistent with the strong correlation between black hole and luminosity found from reverberation mapping. S11 indicated that the quasar clustering and reverberation mapping results might only be reconciled by breaking the black hole mass-halo mass correlation, as suggested by Kormendy & Bender (2011).

We also found similar clustering scalelengths (r₀ ≈ 6 h⁻¹ Mpc for quasars in the XMM-COSMOS field, with little evidence that such quasars show a higher clustering amplitude than their more luminous, optically selected counterparts, as previously reported. This means that there is less evidence for higher halo masses at low redshift for AGN accreting in the hot halo mode, as suggested by Fanidakis et al. (2013).

The VST-ATLAS survey is based on observations made with ESO telescopes at the La Silla Paranal Observatory under programme ID 177.A-3011. We are indebted to the CASU for reducing the ATLAS images and producing catalogues. This publication makes use of data products from the WISE, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. The 2QDES pilot survey was based on observations made with the AAT and we would like to thank the staff of the Australian Astronomical Observatory for their contribution to this work. We thank Nicholas Ross for useful discussions on the quasar selection and the anonymous referee for improving the quality of this manuscript. This work makes use of the runz redshifting code developed by Will Sutherland, Will Saunders, Russell Cannon and Scott Croom. BC, TS, SF and DP are incredibly grateful for the efforts of the NSW Rural Fire Service who worked to ensure our safe evacuation of the site during the 2013 bushfire. This work was supported by the Science and Technology Facilities Council [ST/K501979/1, ST/L00075X/1].

REFERENCES

APPENDIX A: DETAILS OF OBSERVATIONS

APPENDIX B: COVARIANCE MATRIX

Figure B1.

The normalized covariance matrix (see equation B2) for ξ(s) with jackknife errors calculated from dividing our sample into the separate 2dF pointings.

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