On the radio luminosity distribution of active galactic nuclei and the black hole fundamental plane

Abstract

1 INTRODUCTION

In recent years, in many galaxy formation models active galactic nuclei (AGN) are considered to be related to mechanisms capable of switching off star formation in the most massive galaxies, thus reproducing both the observed shape of the galaxy luminosity function and the red, early-type, passive evolving nature of local massive galaxies. It is expected that AGN and galaxy evolution are closely connected to each other (AGN/galaxy co-evolution) through feedback processes coupling both star formation and black hole (BH) accretion-rate histories. Some of these models assume that the AGN feedback into the host galaxy is due to the kinetic energy released by the radio jets and is, therefore, dependent on the AGN radio luminosity (Cattaneo et al. 2006; Croton et al. 2006; Marulli et al. 2008). Indeed, it has already been demonstrated that conversion of the AGN radio luminosity function into a kinetic luminosity function provides an adequate amount of energy (Best et al. 2006; Körding, Jester & Fender 2008; Merloni & Heinz 2008; Shankar et al. 2008; Cattaneo & Best 2009; Smolčić et al. 2009; La Franca et al. 2010).

In this context, in order to build up more realistic AGN/galaxy co-evolutionary models, it is very useful to measure the dependence of the AGN core radio luminosity on other physical quantities related to AGN/galaxy evolutionary status, such as the BH and galaxy star masses and their time derivatives: accretion and star formation rates. These quantities can be measured either directly or indirectly.

A good estimate of galaxy star masses can be obtained by spectral energy distribution (SED) analyses in the optical and near-infrared (NIR) domains (see e.g. Merloni et al. 2010; Pozzi et al. 2012) or, still satisfactorily, from NIR (e.g. K-band) luminosity measures, as the mass-to-light ratio in the K band has a 1σ scatter of 0.1 dex (Madau, Pozzetti & Dickinson 1998; Bell et al. 2003).

BH masses can be estimated using reverberation mapping techniques or measuring the width of broad emission lines observed in optical and NIR spectra (the single-epoch method: see e.g. Vestergaard 2002). Less direct estimates are obtained using the scaling relations observed between the BH mass and the bulge or spheroid mass or between the BH mass and the bulge luminosity of the host galaxies (e.g. Dressler 1989; Kormendy & McClure 1993; Kormendy & Richstone 1995; Magorrian et al. 1998). In this framework even the total host galaxy K-band luminosity, if converted into the bulge luminosity, can be used as a good proxy of the BH mass (see e.g. Fiore et al. 2012).

The AGN radio luminosity distribution and its relationship to either the optical or X-ray luminosity have been studied by many authors. Some of these studies discussed the AGN radio luminosity in terms of a bimodal distribution where two separate populations of radio-loud and radio-quiet objects exist (e.g. Kellermann et al. 1989; Miller, Peacock & Mead 1990). Many other studies have alternatively measured a relationship between the radio and X-ray luminosities (e.g. Brinkmann et al. 2000; Terashima & Wilson 2003; Panessa et al. 2007; Bianchi et al. 2009; Singal et al. 2011). However, almost all these studies are based on incomplete samples due to the lack of deep radio observations. AGN samples selected in other bands (typically optical or X-ray) are therefore not fully detected in the radio band.

However, in order to study the AGN radio luminosity properties and their relation to X-ray and optical luminosities properly, it is necessary to use fully (or almost) radio-detected and complete AGN samples and, when needed, to take the radio upper limits properly into account. More recently, using deep radio observations, it has been shown that both AGN radio/optical and radio/X-ray luminosity ratios span more than five decades continuously (Best et al. 2005; La Franca et al. 2010; Singal et al. 2011; Baloković et al. 2012), without evidence of bimodal distributions, and it is therefore inaccurate to deal with the AGN radio properties in terms of two separate populations of radio-loud and radio-quiet objects. La Franca et al. (2010) used a sample of about 1600 hard X-ray (mostly 2–10 keV) selected AGN to measure (also taking into account the presence of censored radio data) the probability distribution function (PDF) of the ratio between the nuclear radio (1.4 GHz) and X-ray luminosity R_X = log[ν L_ν(1.4 GHz)/L_X(2–10 keV)] (see Terashima & Wilson 2003, for a discussion on the difference between the radio-to-optical and radio-to-X-ray ratio distributions in AGN). The probability distribution function of R_X was functionally fitted as dependent on the X-ray luminosity and redshift, P(R_X|L_X, z). Measurement of the probability distribution function of R_X eventually allowed us to compute the AGN kinetic luminosity function and the kinetic energy density.

In this paper, using the same sample and a similar method to that used by La Franca et al. (2010), we describe the measure of dependence of the AGN radio core luminosity distribution, L_R, on both the X-ray (2–10 keV) luminosity L_X and the host galaxy (AGN-subtracted) K-band luminosity L_K. This measurement, in the context of the AGN/galaxy co-evolution scenario (see the discussion above), is vey useful in order to relate the kinetic (radio) feedback to the accretion rate (L_X) and the galaxy assembled star mass (L_K). In order to take the presence of censored data in the radio band accurately into account, an ad hoc Bayesian maximum likelihood (ML) method and a three-dimensional Kolmogorov–Smirnov (KS) test have been developed.

Many authors have observed the existence of an analogous relationship between the radio luminosity, X-ray luminosity and black hole mass (M), the BH fundamental plane: namely, log L_R = ξ_RX log L_X + ξ_RM log M + constant (see Merloni, Heinz & di Matteo 2003; Falcke, Körding & Markoff 2004; Gültekin et al. 2009a). The measurement of such a relation is very useful in order to discriminate between several theoretical models of jet production in AGN, as it suggests that BH regulate their radiative and mechanical luminosity in the same way at any given accretion rate scaled to the Eddington one (Falcke & Biermann 1995; Heinz & Sunyaev 2003; Churazov et al. 2005; Laor & Behar 2008).

As relations have been observed between the BH mass and the K-band bulge luminosity (see discussion above), in Section 6 we convert our measure of the relation in log L_R–log L_X–log L_K space into a relation in log L_R–log L_X–log M space and discuss how important it is to take the presence of radio upper limits properly into account in order to measure the BH fundamental plane.

Unless otherwise stated, all quoted errors are at the 68 per cent confidence level. We assume H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_m = 0.3 and |$\Omega _\Lambda$| = 0.7.

2 THE DATA

In our analysis we used the same data set used by La Franca et al. (2010), where radio (1.4-GHz) observations (either detections or upper limits) were collected for 1641 AGN (both type 1 (AGN1) and type 2 (AGN2), i.e. showing or not showing the broad-line region in their optical spectra) belonging to complete (i.e. with almost all redshift and N_H measures available) hard X-ray (>2 keV; mostly 2–10 keV) selected AGN samples,¹ with unabsorbed 2–10 keV luminosities larger than 10⁴² erg s⁻¹.

As the goal was to use the radio luminosity in order to estimate the kinetic luminosity, La Franca et al. (2010) measured radio emission that was, as much as possible, causally linked (contemporary) to the observed X-ray activity (accretion). Radio fluxes were measured in a region as close as possible to the AGN, therefore minimizing the contribution of objects like the radio lobes in Fanaroff–Riley type II (FR II) sources (Fanaroff & Riley 1974). For this reason they built up a large data set of X-ray-selected AGN (where redshift and N_H column densities estimates were available) observed at 1.4 GHz with ∼1 arcsec typical spatial resolution (1 arcsec corresponds, at maximum, to about 8 kpc at z ∼ 2). The cross-correlation of the X-ray and radio catalogues was carried out inside a region with 5 arcsec radius (approximately less than or equal to the size of the central part of a galaxy like ours), following a maximum-likelihood algorithm as described by Sutherland & Saunders (1992) and Ciliegi et al. (2003). The offsets between the X-ray and radio positions of the whole sample resulted in a root-mean-square (rms) of 1.4 arcsec (similar to the typical values obtained in X-ray to optical cross-correlations, e.g. Cocchia et al. 2007).

In order to measure the K-band galaxy luminosity, the 1641 AGN from La Franca et al. (2010) have been cross-correlated with already existing K-band photometric catalogues, as explained below.

2.1 The local sample: SWIFT and Grossan

At the lowest redshift we have used a sample of 33 AGN belonging to the 22-month SWIFT catalogue (Tueller et al. 2008). These AGN have been detected at high galactic latitude (|b| > 15°) in the 14–195 keV band with fluxes brighter than 10⁻¹¹ erg s⁻¹ cm⁻²; all objects have N_H column density and optical spectroscopic classification available. The radio luminosity at 1.4 GHz has been derived using the Faint Image of the Radio Sky at Twenty Centimetres (FIRST) Very Large Array (VLA) survey (Becker, White & Helfand 1995). In the case of no radio detection, a 5σ upper limit of 0.75 mJy was adopted. The correlation between catalogues was made through the likelihood-ratio technique (Sutherland & Saunders 1992; Ciliegi et al. 2003). K_S magnitudes have been associated with the sources using the Two-Micron All-Sky Survey (2MASS) catalogue, which has a ∼14.3 mag completeness limiting magnitude (Skrutskie et al. 2006). Cross-correlation with the K-band data has been carried out by comparing the positions of the the 2MASS sources on the K-band image of the AGN counterpart.

To enlarge the local sample, La Franca et al. (2010) used the hard X-ray selected AGN catalogue detected by the High-Energy Astronomy Observatory 1 (HEAO-1) mission (2–10 keV fluxes brighter than 2 × 10⁻¹¹ erg s⁻¹ cm⁻²) described by Grossan (1992) and revised by Brusadin (2003). As for the SWIFT sample, these sources have been cross-correlated with the FIRST and 2MASS radio and K-band catalogues, respectively.

In summary, the local sample contains 43 X-ray sources, all with K_S band detection.

2.2 Hard Bright Sensitivity Survey

We have used the 32 AGN selected by the Hard Bright Sensitivity Survey (HBSS) of the XMM–Newton satellite, carried out at 4.5–7.5 keV fluxes brighter than 7 × 10⁻¹⁴ erg s⁻¹ cm⁻² (Della Ceca et al. 2008). K_S detections and upper limits were obtained for 19 and 13 sources, respectively, from the 2MASS catalogue. The cross-correlation was carried out using the same technique used for the local sample. Those sources missing a K_S detection were eventually excluded from the analysis.

2.3 The ASCA surveys: AMSS and ALSS

Two samples come from observations of the Advanced Satellite for Cosmology and Astrophysics (ASCA) satellite: the ASCA Medium Sensitivity Survey (AMSS: Akiyama et al. 2003), which is composed of 43 AGN, and the ASCA Large-Sky Survey (LASS: Ueda et al. 1999), which is composed of 30 AGN. For both of these catalogues we used the K_S photometric measures by Watanabe et al. (2004). Only one source (belonging to the AMSS) is missing a K_S detection.

2.4 COSMOS

The largest sample used in this work comes from X-ray observations of the field of the Cosmic Evolution Survey (COSMOS). The catalogue by Cappelluti et al. (2009) of XMM–Newton sources with 2–10 keV fluxes brighter than ∼3 × 10⁻¹⁵ erg s⁻¹ cm⁻² was used. K-band magnitudes for 648 out of 677 sources were taken from Brusa et al. (2010, and private communication).

2.5 Chandra Lockman Area North Survey

K-band photometry for 125 out of 139 AGN belonging to the Chandra Lockman Area North Survey (CLANS) were obtained from Trouille et al. (2008). The radio data come from Owen & Morrison (2008).

2.6 European Large Area ISO Survey

In the field S1 of the European Large Area ISO Survey (ELAIS-S1) we used the catalogue of X-ray sources detected using XMM–Newton by Puccetti et al. (2006), which reaches a 2–10 keV flux limit of 2 × 10⁻¹⁵ erg s⁻¹ cm⁻². Radio data were taken from Middelberg et al. (2008), while spectroscopic and photometric identifications were taken from La Franca et al. (2004), Berta et al. (2006), Feruglio et al. (2008) and Sacchi et al. (2009). In Feruglio et al. (2008), K_S detections for 363 objects out of the 421 sources used by La Franca et al. (2010) are available.

2.7 Deep samples: Chandra Deep Field South and North

The deepest X-ray catalogues used in this work are those available in the Chandra Deep Field South (CDFS) and North (CDFN). These are subsamples of the Great Observatories Origins Deep Survey (GOODS) GOOD-S and GOOD-N multi-wavelength surveys with 2–10 keV flux limits of 2.6 × 10⁻¹⁶ erg s⁻¹ cm⁻² and 1.4 × 10⁻¹⁶ erg s⁻¹ cm⁻², respectively.

In the CDFS we used 94 sources from the catalogue of Alexander et al. (2003) and identified by Brusa et al. (2010). These sources were correlated with radio data taken from Miller et al. (2008). K_S-band photometry was obtained using the deepest multi-wavelength catalogue (FIREWORKS) provided by Wuyts et al. (2008), which reaches K_S ≃ 22.5 mag. The K_S-band catalogue was cross-correlated with the optical catalogue using the likelihood-ratio technique. K_S-band counterparts for 86 out of 94 sources were found.

In the CDFN we used the X-ray catalogue from Alexander et al. (2003) with the identifications from Trouille et al. (2008). The sample consists of 162 extragalactic sources for which radio information was obtained from Biggs & Ivison (2006). Trouille et al. (2008) provided K_S-band detections for all but one of the sources.

In Table 1 the breakdown of all samples used is reported. Column 1 lists the original number N of sources contained in the samples used by La Franca et al. (2010), while column 2 lists the number of sources N_K having a K-band detection.

3 K-BAND AGN AND GALAXY LUMINOSITIES

In order to estimate the host galaxy K-band luminosities (i.e. the stellar component), we subtracted the AGN contribution from the measured total luminosities. For this purpose we used the nuclear (AGN-only) infrared SEDs, normalized to the hard X-ray (2–10 keV) intrinsic luminosity and averaged within bins of absorbing N_H as published by Silva, Maiolino & Granato (2004). These AGN SEDs were obtained through interpolation, using updated models from Granato & Danese (1994), of the nuclear infrared data of AGN taken from the Maiolino & Rieke (1995) sample. The accuracy of our method has been tested by comparing our estimates with those obtained by Merloni et al. (2010) and Bongiorno et al. (2012) for the COSMOS catalogue using SED decomposition fitting techniques. As shown in Fig. 1, our estimates, although less accurate, are in good agreement with those obtained by Merloni et al. (2010) and Bongiorno et al. (2012). The average difference that results is log L_K (COSMOS)−log L_K (us) = −0.07(0.05) dex, with a 1σ spread of 0.30 (0.18) dex for AGN1 (AGN2).

Figure 1.

Histogram of the logarithmic differences between our estimates of the K-band luminosities of host galaxies (after AGN component subtraction; see text) in the COSMOS sample and the SED fitting measures from Bongiorno et al. (2012). AGN1 and AGN2 are shown by blue and red lines in the online article, respectively.

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In some cases we obtained that the expected AGN K-band luminosity was very close to (or even larger than) the total measured (AGN + host galaxy) luminosity. With the SED decomposition fitting techniques used by Merloni et al. (2010) and Bongiorno et al. (2012), no object could have an AGN luminosity larger than the total one (see also Pozzi et al. 2007, 2012). Bongiorno et al. (2012) conservatively decided that if the galaxy component were smaller than 10 per cent of the total one then only an upper limit, corresponding to 10 per cent of the total luminosity, could be assigned. Following this approach, we decided to adopt a more conservative assumption and excluded from our analysis those 249 sources where the galaxy component that resulted was smaller than 20 per cent of the total one.²

In Table 1 we report in column 3 the number N_{K − GL} of sources where it was possible to estimate the galaxy K-band luminosity.

4 THE WHOLE SAMPLE

In summary, our data set is composed of nine X-ray-selected AGN samples that contain a total of 1268 sources for which we were able to estimate the K-band stellar component luminosities of the host galaxy, L_K. For all these AGN, column densities and de-absorbed 2–10 keV luminosities, L_X, are available. The radio data allowed us to measure the ‘nuclear’ 1.4-GHz luminosity, L_R, for 293 sources while for the remaining sources 5σ radio upper limits (Table 1, column 4) are available. In total, this is the largest catalogue of AGN (both of type 1 and 2) belonging to statistically well-defined samples where radio, X and K-band information exists. The distribution of the whole AGN sample in the L_X–z plane is shown in Fig. 2, while in Figs 3 and 4 we show the 3D L_K–L_X–L_R distribution, projected on to the three 2D L_K–L_X, L_K–L_R and L_X–L_R planes, where L_K is shown before (L^tot_K) and after (L_K) the subtraction of the AGN component in the K band, respectively.

Figure 2.

2–10 keV de-absorbed luminosity, L_X, of the total sample as a function of redshift.

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Figure 3.

Distribution of the sample in the 2D L_K–L_X, L^tot_K–L_R and L_X–L_R planes, where L^tot_K is the total (galaxy + AGN) K-band luminosity. Squares represent the radio-detected sources while sources with a radio upper limit are shown by either grey arrows or grey squares (upper left panel).

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Figure 4.

As in Fig. 3, with L_K representing the AGN-subtracted K-band galaxy luminosity.

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5 THE PLANE FITTING

5.1 Maximum-likelihood fitting method

The first sum is computed for all i radio-detected sources and, as in equation (3), is proportional to the combined probability of observing the radio luminosities |$L_{{\rm R}_i}$| of all i detected sources, while the second term is the sum of the probability of radio-detecting each j observed (either radio-detected or not) AGN, with a radio luminosity larger than its radio-detection limit |$L_{\lim _j}$|⁠. In this case, analogously to the classical maximum-likelihood method (equat-ion 3), this second term corresponds to the expected total number of radio-detected sources.

5.2 The 3D Kolmogorov–Smirnov test

Although the maximum-likelihood technique is very powerful in finding the parameters of the best-fitting solution and their uncertainties, it does not allow us to quantify how good the solution is. We have therefore devised a three-dimensional Kolmogorv–Smirnov (3D–KS) test able to measure the probability of observing the 3D distribution (in L_R–L_X–L_K space) of our sample of radio-detected sources under the null hypothesis that the data are drawn from the best-fitting model distribution.

The KS test is a standard statistical test for deciding whether a set of data is consistent with a given probability distribution. In one dimension the KS statistic is the maximum difference D between the cumulative distribution functions of the data and the model (or another sample). What makes the KS statistic useful is that its distribution (in the case of the null hypothesis that the data are drawn from the same distribution) can be calculated, giving the significance of any non-zero value of D.

In order to make a 3D–KS test, we have followed the examples of generalization of the KS test to two-dimensional distributions by Peacock (1983) and Fasano & Franceschini (1987). The maximum difference D statistic has been computed by measuring the difference between the fraction of observed and expected (by the model) sources in each of the eight octants defined at the 3D positions of each radio-detected source. In order to compute the number of expected sources, 2000 Monte Carlo simulations have been used. Although the 3D–KS test is carried out on the radio-detected sample only, it properly takes into account the effects of the radio upper limits. Indeed, the simulations have been carried by extracting a radio luminosity (or not) for each observed source by taking into account its radio-detection limits and the model conditional probability distribution function P(L_R|L_X, L_K). The same 2000 simulations have been used to compute the probability (significance) of observing the measured maximum D statistic in the case of the null hypothesis that the data are drawn from the same distribution.

5.3 The fits

This spread was first modelled by a Gaussian function, G(X|0, σ), where |$X = \log (L_{\rm R}/\bar{L_{\rm R}})$|⁠, centred on X = 0 with standard deviation σ. The best-fitting parameters are shown in Table 2 (model 1). However, the 3D–KS statistic tells that there is only a 2 per cent probability that the observed distribution is drawn from the model. The probability distribution function of X is shown in Fig. 5. The data have been plotted comparing in each bin of X the number of observed (N_obs) and expected (N_exp; by the model) sources. This method (N_obs versus N_exp method) reproduces the observations and consequently takes properly into account both radio detections and upper limits (see e.g. La Franca et al. 1994; La Franca & Cristiani 1997; Matute et al. 2006, for similar applications). It is worth noting that this fit, although not satisfactory, gives an indication that the PDF of L_R is quite large: σ ≃ 1 dex, i.e. 68 per cent of cases are included in a 2-dex wide distribution. Similar results have been found by Merloni et al. (2003) and Gültekin et al. (2009a).

Figure 5.

Probability distribution functions of |$X = \log (L_{\rm R}/\bar{L_{\rm R}})$|⁠, where |$\bar{L_r}$| is the luminosity of the peak (mode) with the position shown by a vertical continuous line (orange in the online article). The four distributions correspond to the four best fits reported in Table 2. Other vertical lines (blue in the online article) show the values of the means of the distributions, the offsets from peak luminosity of which are reported in Table 2, and which are used in the next figures to represent the best-fitting solutions.

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As shown in Fig. 5 (and already observed by La Franca et al. 2010), the data show an excess of high-radio-luminosity sources if compared with a symmetrical distribution. In order to take this excess better into account we have assumed an asymmetrical double Gaussian distribution with two σ values: σ_l and σ_u for radio luminosities values below (lower) and above (upper) the value of the radio luminosity, |$\bar{L_r}$|⁠, of the peak of the probability distribution³ (i.e. for X values above or below zero) respectively. Indeed, the fit gives a larger spread, σ_u = 1.0, at X > 0 than measured at X < 0, where σ_l = 0.5. However, even with this model the 3D–KS test does not gives good enough probability (3 per cent). As the PDF is asymmetrical, we show in Fig. 5 the loci of both the peak and the mean of the PDF. A better representation of the model (e.g. in figures comparing the fit with the data) should, indeed, be carried out using the position of the mean of the distributions. The offsets between the mean and the mode (offset = mean - mode) of the PDF are also listed in column 4 of Table 2.

Confidence regions for each parameter were obtained by minimizing the S function at a number of values around the best-fitting solution, while leaving the other parameters free to float (see Lampton et al. 1976). The 68 per cent confidence regions quoted correspond to ΔS (=Δχ²) = 1. The best-fitting solution is also shown in Fig. 6, where the 3D distribution of the data is shown, and Fig. 7, where the 2D edge-on view of the plane is shown. This last figure helps us to understand how important it is to take into account the effects of using censored data. The fitting solution seems, indeed, not to be a good representation of the distribution of radio detections. This is because most of the X-ray samples have no radio observations able to detect all sources and then the radio detections are biased in favour of the most luminous radio sources. Indeed, the mean radio luminosity of each of our samples decreases as a function of the radio identification completeness fraction: the average radio luminosity of the samples with about 10 per cent radio identifications is about log L_R = 40.0 erg s⁻¹, while for the 90 per cent complete samples the average luminosity is about 2 dex lower (log L_R = 38.2 erg s⁻¹; see Fig. 8). This bias is taken properly into account with our ML fitting method, which takes into account upper limits. The fitting solution, indeed, reproduces fairly well the distribution of the most radio-complete samples, such as SWIFT and CLANS (see Figs 7 and 9).

Figure 6.

3D distribution in L_R–L_X–L_K space. Filled circles represent radio detections and faint open triangles represent radio upper limits. The plane of the distribution of mean L_R as a function of L_X and L_K, according to the best-fitting solution (4) (Table 2), is shown.

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Figure 7.

Edge-on view of the 3D plane. The more complete samples, SWIFT and CLANS, are represented by squares (red and green respectively in the online article). The continuous line shows the locus of the mean radio luminosity probability distribution function of our best-fitting solution (4) (Table 2). Radio upper limits are represented by arrows.

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Figure 8.

Mean radio luminosity of the radio-detected sources of each sample as a function of the radio-detection completeness.

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Figure 9.

Same as Fig. 6 but using only the more radio-complete SWIFT (red circles in the online article) and CLANS (green circles in the online article) samples. Radio upper limits are represented by open triangles. The plane of distribution of the mean L_R as a function of L_X and L_K, according to the best-fitting solution (4), is shown.

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Figure 10.

3D distribution in L_R–L_X–M_BH space. The projection from L_R–L_X–L_K space of the plane of the best-fitting solution (4) is shown in cyan in the online article. The fundamental plane from Merloni et al. (2003) is shown in gold in the online article. Radio upper limits are represented by open triangles.

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Figure 11.

Same as Fig. 10 but using only the more radio-complete SWIFT (red circles in the online article) and CLANS (green circles in the online article) samples. Radio upper limits are represented by open triangles.

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Figure 12.

Edge on view of the BH fundamental plane as measured by Merloni et al. (2003) (dashed line, gold in the online article). The continuous line (cyan in the online article) shows our solution for a sample of AGN having average log M = 8.5. The more complete samples, SWIFT and CLANS, are represented by squares (red and green respectively in the online article). Radio upper limits are represented by arrows.

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6 THE AGN FUNDAMENTAL PLANE

7 DISCUSSION AND CONCLUSIONS

As expected (see Fig. 10), our estimate of the BH fundamental plane predicts lower radio luminosities if compared with the previous measure by Merloni et al. (2003). The typical difference, computed at 10⁸ M_⊙ BH mass and 10⁴⁴ erg s⁻¹ X-ray luminosity, is ∼0.8 dex. As already observed in L_R–L_X–L_K space, this difference is due to the inclusion in our analysis of the contribution of the radio upper limits, and indeed our best-fitting solution reproduces well the distribution of the most radio-complete samples, such as SWIFT and CLANS (Figs 11 and 12).

It should be noted that the fundamental plane of Merloni et al. (2003) was constructed by including a sample of X-ray BH binaries. While it is interesting to see that the mass scaling is still broadly consistent with that of Merloni et al. (2003) even in our study of an AGN-only sample, the radio–X-ray coefficient is very different: ∼0.4 in this study, while low-accretion-rate (and low-luminosity) X-ray BH binaries show a tight radio–X-ray correlation with slope ∼0.6. Interestingly enough, recent reanalysis of radio–X-ray correlations in X-ray binaries suggest the presence of a second, less radio-luminous branch (Gallo, Fender & Pooley 2003; Coriat et al. 2011; Gallo, Miller & Fender 2012). In this framework, our study could suggest that in AGN a second, less radio-luminous population should also be taken into account, which could correspond to those objects with low values of the |$X = \log (L_{\rm R}/\bar{L_{\rm R}})$| parameter (see Fig. 5).

As already discussed in the Introduction, the measure of the dependence in AGN of L_R on other physical quantities such as L_X and the host galaxy K-band luminosity L_K is very useful in order better to include AGN in galaxy evolution models where AGN/galaxy feedback plays a relevant role.

Our analysis has allowed us to find a good analytical solution represented by a plane in 3D log L_R–log L_X–log L_K space once a wide (1σ ∼1 dex) asymmetrical spread in the radio luminosity axis is included. This result confirms the study of La Franca et al. (2010) who, studying the dependence of PDFs of L_R on L_X and z, were able to model the 1-dex wide (1σ) spread in the AGN radio luminosity distribution. These results show that a proper study of the correlation between different band luminosities in AGN (or other sources) cannot be performed without taking into account censored data.

A clear example, in this framework, is the measure of the BH fundamental plane in 3D log L_R–log L_X–log M space. After converting our measures of the host galaxy K-band luminosity into BH masses using scaling relations, our best-fitting solution corresponds to a BH fundamental plane that on average predicts 0.8 dex lower values for the AGN radio luminosities.

It should be pointed out that, at variance with many similar statistical studies, our analysis is based on a compilation of complete, hard X-ray selected AGN samples, where both AGN1 and AGN2 are included. Therefore, our results should better represent the behaviour of the whole AGN population. However, it will be interesting to check the 3D correlation between L_R, L_X and L_K in complete, radio-selected samples. As the Merloni et al. (2003) sample was a hybrid sample without a clear selection criterion, it is plausible that some of the differences found in the present work could be due to the specific selection criterion. In very general terms, if we do not believe we know any of the three terms (X-ray luminosity, radio luminosity and BH mass) to be the primary physical driver, all should be treated as equal in a correlation study (but this is beyond the purpose of this paper).

In order to improve these analyses, it would be very useful to obtain deeper radio observations of complete samples of AGN combined with detailed optical–NIR–MIR SED observations. These data, when available, will eventually allow us to measure the dependence of the radio luminosity distribution (i.e. the feedback) on star and BH masses, their derivatives (star formation and accretion rates) and redshift. This result could be achieved by complementing multiwavelength surveys with observations carried out with new radio facilities such as the Expanded Very Large Array and the Square Kilometer Array.

We thank Andrea Merloni and Geoffrey Bicknell for discussions. We acknowledge the referee for a very careful review that allowed us to improve the quality of this work. This publication uses the NVSS and FIRST radio surveys, carried out using the National Radio Astronomy Observatory Very Large Array. NRAO is operated by Associated University Inc., under cooperative agreement with the National Science Foundation. This publication makes use of data products from the Two-Micron All-Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. We acknowledge a financial contribution from PRIN-INAF 2011.

NOTE ADDED IN PRESS

It is brought to our attention that, after the submission of this paper, a new relationship between the BH mass and K-band bulge luminosity has been published by Graham & Scott (2012). After a quick analysis, we expect that, using this new relationship, our fundamental plane estimate would be tilted such as that the distance from the plane of Merloni, Heinz & Di Matteo (2003) will remain of about 0.8 dex for logM_BH 8, while it will increase at larger BH masses and decrease at smaller BH masses.

REFERENCES