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The VLA Survey of the Chandra Deep Field South. V. Evolution and Luminosity Functions of sub-mJy radio sources and the issue of radio emission in radio-quiet AGN

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The VLA Survey of the Chandra Deep Field South. V. Evolution and Luminosity Functions of sub-mJy radio sources and the issue of radio emission in radio-quiet AGN
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    a  r   X   i  v  :   1   1   0   7 .   2   7   5   9  v   1   [  a  s   t  r  o  -  p   h .   C   O   ]   1   4   J  u   l   2   0   1   1 Accepted for publication in The Astrophysical Journal (July 12, 2011) Preprint typeset using L A TEX style emulateapj v. 11/10/09 THE VLA SURVEY OF THE CHANDRA DEEP FIELD SOUTH. V. EVOLUTION AND LUMINOSITYFUNCTIONS OF SUB-MJY RADIO SOURCES AND THE ISSUE OF RADIO EMISSION IN RADIO-QUIETAGN P. Padovani European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748 Garching bei M¨unchen, Germany N. Miller Department of Astronomy, University of Maryland, College Park, MD 20742-2421, USA K. I. Kellermann National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903-2475, USA V. Mainieri, P. Rosati European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748 Garching bei M¨unchen, Germany P. Tozzi INAF, Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, I-34131, Trieste, Italy Draft version July 15, 2011 ABSTRACTWe present the evolutionary properties and luminosity functions of the radio sources belonging to theChandra Deep Field South VLA survey, which reaches a flux density limit at 1.4 GHz of 43 µ Jy at thefield center and redshift ∼ 5, and which includes the first radio-selected complete sample of radio-quietactive galactic nuclei (AGN). We use a new, comprehensive classification scheme based on radio, far-and near-IR, optical, and X-ray data to disentangle star-forming galaxies from AGN and radio-quietfrom radio-loud AGN. We confirm our previous result that star-forming galaxies become dominantonly below 0.1 mJy. The sub-millijansky radio sky turns out to be a complex mix of star-forminggalaxies and radio-quiet AGN evolving at a similar, strong rate; non-evolving low-luminosity radiogalaxies; and declining radio powerful ( P   3 × 10 24 W Hz − 1 ) AGN. Our results suggest that radioemission from radio-quiet AGN is closely related to star formation. The detection of compact, highbrightness temperature cores in several nearby radio-quiet AGN can be explained by the co-existenceof two components, one non-evolving and AGN-related and one evolving and star-formation-related.Radio-quiet AGN are an important class of sub-millijansky sources, accounting for ∼ 30% of thesample and ∼ 60% of all AGN, and outnumbering radio-loud AGN at  0 . 1 mJy. This implies thatfuture, large area sub-millijansky surveys, given the appropriate ancillary multi-wavelength data, havethe potential of being able to assemble vast samples of radio-quiet AGN by-passing the problems of obscuration, which plague the optical and soft X-ray bands. Subject headings: galaxies: active — galaxies: starburst — radio continuum: galaxies — infraredradiation: galaxies — X-rays: galaxies 1. INTRODUCTION The relationship between star formation and AGN inthe Universe is one of the hottest topics of current extra-galactic research, at two different levels. On cosmologicalscales, the growth of supermassive black holes in AGNappears to be correlated with the growth of stellar massin galaxies (e.g.,Merloni, Rudnick & Di Matteo 2008).On nuclear scales, the accreting gas feeding the blackhole at the center of the AGN might trigger a starburst.The black hole, through winds and jets, can in turn feedenergy back to its surroundings, which can compress thegas and therefore accelerate star formation but can alsoblow it away, thereby stopping accretion and star forma-tion altogether. The general consensus is that nuclear ppadovan@eso.org activity plays a major role in the co-evolution of super-massive black holes and galaxies through the so-called“AGN Feedback” and indeed radio emission from AGNhas been recently suggested to play an important role ingalaxy evolution (Croton et al. 2006). Moreover, radioobservations afford a view of the Universe unaffected bythe absorption, which plagues observations made at mostother wavelengths, and therefore providea vital contribu-tion to our understanding of this co-evolution. These twopoints imply that studies of the evolution of star-forminggalaxies (SFG) and AGN in the radio band should pro-vide a better understanding of the link between the twophenomena. These are obviously done best by reachingrelatively faint (  1 mJy) flux densities, and hence theimportance of characterizing the radio faint source pop-ulation.  2 Padovani et al.After years of intense debate, the contribution to thesub-millijansky population from synchrotronemission re-sulting from relativistic plasma ejected from supernovaeassociated with massive star formation in galaxies ap-pears not to be overwhelming, at least down to ∼ 50 µ Jy,contrary to the (until recently) most accepted paradigm.Our deep ( S  1 . 4GHz ≥ 43 µ Jy) radio observations with theNRAO Very Large Array (VLA) of the Chandra DeepField South (CDFS), complemented by a variety of dataat other frequencies, imply in fact a roughly 50/50 splitbetween SFG and AGN (Padovani et al. 2009), in broadagreement with other recent papers (e.g.,Seymour et al.2008;Smolˇci´c et al. 2008). The purpose of this paper is to study the evolutionand luminosity functions (LFs) of sub-millijansky radiosources through the VLA-CDFS sample. Apart from thetopics mentioned above, this is important also for otherissues, including:1. predictions for the source population at radio fluxdensities < 1 µ Jy, which are relevant, for example,for the Square Kilometre Array (SKA). All existingestimates, in fact, had to rely, for obvious reasons,on extrapolations and are based on high flux den-sity samples. This affects particularly the highestredshifts, which can better be probed at fainter fluxdensities;2. the radio evolution of radio-quiet AGN. No radio-selected sample of radio-quiet AGN is currentlyavailable and this is badly needed to shed light onthe mechanism behind their radio emission and al-low a proper comparison with radio-loud quasars;3. the fact that number counts by themselves do notnecessarily reflect the relative intrinsic abundanceof astrophysical sources, which requires the deter-mination of the evolution and LF(Padovani et al.2007).We note that the evolution and LFs of sub-millijanskyradio sources have been studied, so far, only in two fields:the COSMOS field (Smolˇci´c et al. 2009a,b) and the Deep Spitzer  Wide-area InfraRed Extragalactic (SWIRE) field(Strazzullo et al. 2010), in both cases up to a maxi-mum redshift of 1 . 3 and without differentiating betweenradio-quiet and radio-loud AGN. Source classificationwas based on a rest-frame optical color scheme and onspectral energy distribution (SED) fitting to photometricdata covering the UV to near-IR range respectively.We define as AGN sources in which most of the en-ergy is produced through physical processes other thanthe nuclear fusion that powers stars. In practice, thismeans that electromagnetic emission most likely related,directly or indirectly, to a supermassive black hole is pre-dominant in at least one band. A small fraction of AGNhave, for the same optical power, radio powers three tofour orders of magnitude higher than the rest. Theseare called “radio-loud” quasars and most of the energythey emit is non-thermal and is associated with powerfulrelativistic jets, although thermal components associatedwith an accretion disk may also be observed, especiallyin the optical/UV band. Radio galaxies are also char-acterized by strong radio jets (manifested also throughradio lobes), typically laying in or near the plane of thesky, and a fraction of them (the most powerful ones)are thought to be radio-loud quasars, which have insteadtheir jets oriented with a small angle to the line of sight(e.g.,Urry & Padovani 1995). We define as “radio-quiet”AGN in which jets are either not present or make a tinycontribution to the total energy budget over the wholeelectromagnetic spectrum, which is dominated by ther-mal emission. All other AGN we call radio-loud. Notethat radio-quiet AGN are not radio-silent. Indeed, theradio power of many low-luminosity radio galaxies, theso-called Fanaroff-Riley (FR) Is (“low-power radio-loudAGN” according to our nomenclature) overlap with thatof radio-quiet AGN, which can generate some confusionand requires great care during the classification process,which needs to involve also the X-ray and far-IR bands(Section2.4). However, the two classes are physicallydistinct (see Sections5.3and5.8), although the srcin of  radio emission in radio-quiet AGN is still not clear (butsee Section5.7).Translating these high-level definitions into a classi-fication scheme requires a wealth of multi-wavelengthdata, which were described in our previous papers.Kellermann et al.(2008) (Paper I) presented the radio data of the VLA-CDFS sample, together with optical im-ages and X-ray counterparts, whileMainieri et al.(2008) (Paper II), discussed the optical and near IR counter-parts to the observed radio sources and, based on rest-frame colors and the morphology of the host galaxies,found evidences for a change in the sub-millijansky ra-dio source population below ≈ 80 microJy.Tozzi et al.(2009) (Paper III), dealt with the X-ray properties, whilePadovani et al.(2009) (Paper IV), discussed the source population. This turned out to be made up of SFG andAGN at roughly equal levels, with the AGN including ra-dio galaxies, mostly low-power (FR Is), and a significant( ∼ 50%) radio-quiet component. Paper IV made alsoclear that the “standard” definitions of radio-loudness,based on radio-to-optical flux density ratios, R , and ra-dio powers, were insufficient to identify radio-quiet AGNwhen dealing with a sample, which included also star-forming and radio-galaxies, as both classes are or can be(respectively) characterizedby low R and radio powers aswell. R , for example, is useful for quasar samples, whereit can be assumed that the optical flux is related to theaccretion disk, but obviously loses its meaning as an in-dicator of jet strength if both the radio and the opticalband are dominated by jet emission, as might be the casein FR Is (Chiaberge et al. 1999). Source classification inPaper IV was then based on radio, optical, and X-raydata, and was meant to provide a robust upper limit tothe fraction of SFG at sub-millijansky levels. SFG can-didates were selected based on their (low values of) R ,(low) radio power, (non-elliptical or S0) optical morphol-ogy, and (low) X-ray power ( L x ). The fact that X-rayupper limits above the AGN threshold (10 42 ergs s − 1 )were also included was conservative in the sense that itmaximized the number of SFG, as some of these sourcescould still be AGN. Furthermore, the selection of radio-quiet AGN candidates was only approximate, as it wasbased solely on R and L x and suffered from uncertain-ties in the optical K-correction, possible contaminationby radio galaxies, and the exclusion of X-ray upper lim-its (see Paper IV for details). In order to deal with the  VLA Survey of the CDFS. V. Evolution and Luminosity Functions 3 18 20 22 24 26 280.040.050.060.070.080.090.050.10.20.30.40.50.60.70.80.90.512345 Fig. 1.— Redshift versus V  mag for our sources with redshift in-formation. The solid line is the best fit, while the dashed linesrepresent the scatter. See text for details. evolution and luminosity functions of the various classesof sources, we need to refine our classification. In partic-ular, the CDFS field has been observed by Spitzer  andtherefore near-(Section2.2) and mid/far-IR (Section2.3) data are available for our sample. The source classifica-tion used in this paper relies then on a combination of radio, IR, optical, and X-ray data (Section2.4).Section 2 describes the updated classification of theVLA-CDFS sample, while Section 3 studies its evolu-tion. Section 4 derives the LFs for various classes, whileSection 5 discusses our results. Finally, Section 6 sum-marizes our conclusions. Throughout this paper spectralindices are defined by S  ν ∝ ν  − α and the values H  0 = 70km s − 1 Mpc − 1 , Ω M = 0 . 3, and Ω Λ = 0 . 7 have been used. 2. THE UPDATED SAMPLE CLASSIFICATION 2.1. Redshifts Our sample includes all VLA-CDFS sources with reli-able optical counterparts and eight empty fields, for to-tal of 256 objects, 193 of which belong to a completesample 1 (see Paper IV for details). 92% of the sourcesin the complete sample now have redshift information(74% spectroscopic) as compared to only 77% in Pa-per IV. We have included in this paper spectroscopicredshifts from a dedicated follow-up program performedwith the VIMOS spectrograph at the VLT (Bonzini etal., in preparation). We also used recently published red-shifts for the counterparts of Chandra sources in this field(Treister et al. 2009;Silverman et al. 2010) and the pho- 1 Four more sources belonging to the complete sample have veryuncertain counterparts (see Paper II) and for one other source,very close to a bright star, we could not get reliable photometry.The inclusion of these sources in any of the classes described belowwould change our results by much less than 1 σ . tometric redshifts published by the Multiwavelength Sur-vey by Yale-Chile (MUSYC) (Cardamone et al. 2010),which are based on 32 photometric bands.As shown in Fig.1,redshift is strongly correlatedwith magnitude, albeit with some scatter. The bestand simplest approach to estimate the redshift for the16 objects in the complete sample without observed red-shifts is then to derive it from their magnitude by usingthe relationship shown in the figure (solid line), that islog z = 0 . 166 V  mag − 3 . 85. This was derived applyingto the whole sample the ordinary least-square bisectormethod (Isobe et al. 1990), which treats the variablessymmetrically. Including only spectroscopic redshifts,or only the complete sample, or excluding sources withlikely AGN contamination in the optical band (based onSzokoly et al. 2004), all give relations within 1 σ from theadopted one. The effect of this assumption on our resultsis discussed in Section5.1.Note that while objects well to the left of the corre-lation can be explained as having an AGN componentin the optical band, the single source in the lower rightpart of the diagram is ≈ 7 magnitudes fainter than theaverage and therefore well into the dwarf galaxy regime.However, its photometry is affected by its closeness toa bright star, which might explain at least in part itsfaintness. 2.2. Near-IR data  The usage of  Spitzer  Infrared Array Camera (IRAC;Fazio et al. 2004) colors to identify AGN has beendiscussed at length in the literature (e.g.,Lacy et al.2004;Hatziminaoglou et al. 2005;Stern et al. 2005; Sajina et al. 2005;Cardamone et al. 2008). Although it is by now evident that only some classes of extragalac-tic sources occupy restricted regions of parameter spacein such plots, it is nevertheless also clear that there arebroad trends which can be used to, for example, identifypossible misclassifications.Fig.2plots the IRAC flux density ratios S  8 . 0 /S  4 . 5 vs. S  5 . 8 /S  3 . 6 for our sources classified as in Paper IV,where the flux densities refer to all four IRAC chan-nels at 3.6, 4.5, 5.8, and 8.0 µ m. The IRAC data comefrom the Spitzer  IRAC/MUSYC Public Legacy survey inthe Extended-CDFS (SIMPLE;Damen et al. 2011). Wecross-correlated the SIMPLE catalogue with the VLA-CDFS sources, accepting matches with separations lessthan 2 ′′ . The SIMPLE catalogue has convolved the im-ages associated with each IRAC channel to match thatof channel 4 (8.0 µ m), the one with the lowest resolu-tion, so that reasonably accurate colors may be obtainedfrom the four IRAC bands. We have used total fluxesand applied the prescribed normalization to produce fluxdensities in µ Jy. Fig.2shows the following: (a) mostAGN candidates fall around the locus of sources whosemid-IR spectrum can be characterized by a single powerlaw (dotted line); a significant number of AGN is alsowithin the so-called “Lacy’s wedge” (dashed lines), whichis where most unobscured, broad-lined (type 1) AGN arethought to lie (Lacy et al. 2004). Note that highly ob-scured sources might also occupy that region of param-eter space (e.g.,Dasyra et al. 2009;Prandoni 2010); (b) most SFG candidates are distributed in a vertical bandcentered around S  5 . 8 /S  3 . 6 ∼ 0 . 6 − 0 . 8, which is wherepolycyclic aromatic hydrocarbon (PAH)- and starlight-  4 Padovani et al. 0.2 0.4 0.6 0.8 1 2 4 60.20.40.60.81246810 Fig. 2.— IRAC color-color plot for the AGN and SFG candidatesselected in Paper IV and four FR Is from the SWIRE field. Thedotted line indicates the locus of sources whose spectrum can bedescribed as a power-law over the four IRAC bands. The dashedlines indicate the so-called “Lacy’s wedge”, which is where mostAGN are thought to lie. The solid lines denote a more restrictiveregion, which takes into account the fact that for z > 0 . 5 PAH-and star-light-dominated sources can be inside “Lacy’s wedge”(Dasyra et al. 2009). See text for details. dominated sources are expected to lie (e.g.,Sajina et al.2005); (c) we also plot four “bona fide” FR I from theSWIRE field (Vardoulaki et al. 2008), which fall in theregion where galaxies with an old stellar population arelocated (e.g.,Sajina et al. 2005). We have quite a fewsources in the same area, which is consistent with one of the main results of Paper IV, that is the dominance of low-luminosity radio galaxies amongst radio-loud AGN.This is reassuring and shows that the SFG/AGN di-vision derived in Paper IV is overall correct. The mostinteresting features in Fig.2, however, are the excep-tions to the above, namely: (a) the eight AGN candi-dates in the top left part of the diagram; these are allbut one at low redshift ( z ≤ 0 . 25), low radio power(log P  1 . 4 GHz ≤ 22 . 6), low X-ray (2 − 10 keV) power(log L x ≤ 41 . 6) sources, which had been classified asAGN solely because their optical morphology was S0 (5)or elliptical (2). A closer look at their images showsthat two of them show (weak) signs of spiral arms andfour more (all S0) have only low-resolution Wide FieldImager (WFI) data, which means that the presence of spiral arms cannot be excluded. Their location in thePAH-dominated region (Sajina et al. 2005) suggests are-classification as SFG for all of them apart from oneAGN with two spiral galaxies at a distance of  ∼ 3 ′′ ,which means its IRAC flux is most likely contaminated(its rest-frame radio-to-optical flux density ratio is also ∼ 2, which is typical of radio-loud AGN: see below);(b) the ten SFG candidates with S  5 . 8 /S  3 . 6 > 1 and1 < S  8 . 0 /S  4 . 5 < 3; this is more restrictive than theLacy’s wedge as it takes into account the fact that for z > 0 . 5 PAH- and star-light-dominated sources can beinside that wedge (Dasyra et al. 2009). Most of thesesources have X-ray upper limits larger than 10 42 erg/s,which makes sense since this was one of the reasons theywere classified as SFG in the first place. The locationof these sources suggests a re-classification as AGN. Insummary, seven sources were re-classified from AGN toSFG and ten sources previously classified SFG are nowclassified as AGN. 2.3. Far-IR data  It is well know that the global far-IR and radioemission are tightly and linearly correlated in star-forming systems (e.g.,Sargent et al. 2010, and refer-ences therein). This is usually expressed through theso-called q parameter, that is the logarithm of theratio of far-IR to radio flux density, as defined byHelou, Soifer, & Rowan-Robinson(1985). We take ad- vantage of the relatively narrow dispersion of  q for star-forming systems to further refine our SFG/AGN separa-tion and also to improve on our radio-quiet – radio-loudAGN division, as the latter do not follow the IR - radiocorrelation typical of SFG (e.g.,Sopp & Alexander 1991;Sargent et al. 2010). This is vital to separate radio-quietAGN from radio galaxies, as R is not very useful in thiscase (Section1) and, like radio-quiet AGN, radio galaxiescan also have relatively large X-ray powers.We have used a catalog of 70 µ m Multiband ImagingPhotometer for Spitzer  (MIPS) flux densities from theFar- Infrared Deep Extragalactic Survey (FIDEL; Dick-inson et al. in preparation) for our evaluation of  q . Wecross correlated the VLA-CDFS radio sources with theFIDEL catalogue using a radius of 8 ′′ (about half the Spitzer  70 µ m point-spread-function). For those sourcesundetected by the FIDEL survey (but still within the FI-DEL coverage), we assume an upper limit of 2.5 mJy asthis is approximately the 5 σ survey limit. To these datawe add 24 µ m flux densities from the Great Observato-ries Origins Deep Survey (GOODS) whenever available,and thus we obtain SEDs sampled at up to eight wave-lengths: 20 and 6 cm in the radio from our VLA surveys;70 µ m and 24 µ m in the IR from FIDEL and GOODS;and 8 . 0 µ m, 5 . 6 µ m, 4 . 5 µ m, and 3 . 6 µ m in the near-IRfrom SIMPLE.We then proceeded to find the template SED fromtheDale et al.(2001) SFG models that best matches the Spitzer  data. We use the source redshifts to placeeach of the 64 models into the observed frame for thatsource, and set the normalization by requiring that eachmodel SED pass through the measured 70 µ m flux den-sity for that galaxy. This, in effect, places an extraweight on 70 µ m data since it is our only measure-ment of the smooth modified blackbody portion of theSED. We then select the model which minimizes the leastsquares fit to the photometry of the four IRAC chan-nels and the MIPS 24 µ m data (when available). Oncethe best-fitting model has been selected, we derive therest-frame 60 µ m and 100 µ m flux densities to determineFIR, the total far-IR flux between 42 . 5 µ m and 122 . 5 µ m(Helou, Soifer, & Rowan-Robinson 1985) FIR = 1 . 26 × 10 − 14 [2 . 58 f  60 µm + f  100 µm ] W m − 2 (1)  VLA Survey of the CDFS. V. Evolution and Luminosity Functions 5where the flux densities, f  , are in Jy. Similarly, we con-vert the observed 1.4 GHz radio emission to the restframe using its measured spectral index between 1.4 GHzand 4.86 GHz, where available ( ∼ 80% of the sample: seePaper I), or assuming a spectral index α r = 0 . 7(the meanof the sample) otherwise. The value of  q is then calcu-lated as the logarithm of the ratio of far-IR to 1.4 GHzflux density: q = log [( FIR/ 3 . 75 × 10 12 ) /S  1 . 4 GHz ] (2)where the numeric factor is the frequency in Hz corre-sponding to a wavelength of 80 µ m.Given the large fraction ( ∼ 50%) of upper limits on q , one cannot readily look for a bimodality in its dis-tribution to define a dividing line between star-formingand non star-forming sources. The median of the detec-tions should be however quite well defined, as its valueis 2.16 and most upper limits are below 2.2. Since 96%of the detections above the median are below 2.64 andassuming a symmetric distribution, one finds a lowerend at around 2 . 16 − (2 . 64 − 2 . 16) ∼ 1 . 7. We thenassume in the following that sources characterized by q ≥ 1 . 7 are star-formers 2 (upper limits above this valueexcluded). This is the same dividing value assumed byMachalski & Condon(1999). Twenty-two of our candi- date SFG have q < 1 . 7 and therefore cannot be star-forming systems. These were then re-classified as radio-loud AGN. These sources fall in the region where passivegalaxies are found in the IRAC color-color plot, which isconsistent with this re-classification, given that most of our radio-loud AGN should be radio galaxies.Finally, eight radio-quiet AGN candidates were foundto have q < 1 . 7, while nineteen radio-loud ones had q ≥ 1 . 7, which reflects the approximation of our pre-vious classification. These objects were re-classified asradio-loud and radio-quiet respectively. 2.4. Revised classification  To summarize, based on the results presented in Pa-per IV and in the previous sub-sections, our candidatestar-forming galaxies are defined as fulfilling the follow-ing initial requirements:1. R = log( S  1 . 4GHz /S  V ) < 1 . 7 (where S  V is the V-band flux density)2. P  r < 10 24 . 5 W Hz − 1 3. optical morphology different from elliptical orlenticular4. L x (2 − 10 keV) < 10 42 ergss − 1 for X-raydetections,no limit otherwise.As discussed in Paper IV, the first two criteria include ∼ 90% of spirals and irregulars, the third one excludessources not associated with star formation at our red-shifts (  z  ∼ 1 . 1), while the fourth one excludes AGN.These are then supplemented by the following additionalrequirements, which can overrulethe previousones if nec-essary: 2 Our results are only weakly dependent on this choice. Forexample, if we defined as star-formers sources with q ≥ 1 . 8 ourSFG complete sample would only lose three objects, a 4% effect(see Tab.2). Fig. 3.— A flow chart of our classification scheme. See text fordetails. 5. IRAC constraints: the region of parameter spacedefined by S  5 . 8 /S  3 . 6 > 1 and 1 < S  8 . 0 /S  4 . 5  3 ( S  5 . 8 /S  3 . 6 ) 0 . 83 (AGN region) is excluded; sourcesnot classified as SFG by the previous criteria butwith 0 . 45 < S  5 . 8 /S  3 . 6 < 1 . 0 and S  8 . 0 /S  4 . 5 > 2 . 5(PAH-dominated region) are also included (Sec-tion2.2)6. MIPS constraints: q ≥ 1 . 7, upper limits above thisvalue excluded (Section2.3)Note that constraints number 2 and 3 have becomealmost irrelevant for our classification given these twonew requirements. Objects not fulfilling this sequence of criteria are considered to be AGN. Radio-quiet AGN aredefined initially as follows:1. R < 1 . 42. L x (2 − 10 keV) > 10 42 ergs s − 1 (detections only)As discussed in Paper IV, the first criterion is the “clas-sical” definition of radio-quiet AGN converted to the 1.4GHz and V bands. These are then supplemented by thefollowing additional requirements, which can overrule theprevious ones if necessary:3. IRAC constraints: the region of parameter spacedefined by S  8 . 0 /S  4 . 5 > 2 . 5 and 0 . 45 < S  5 . 8 /S  3 . 6 <
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