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Appendix A: FIR luminosities: detection, stacking and measurement

As a direct tracer of the FIR emission we concentrate on the mean luminosity νL_ν(60   μm), estimated at a rest-frame wavelength of 60 μm. This choice is a compromise between a wavelength long enough to avoid most of the AGN contamination (see discussion below) and at the same time short enough to be sampled by PACS 160 μm observations even at the highest redshifts considered in this work.

To preclude any assumptions about the SED shape, we computed νL_ν(60   μm) though a log-linear interpolation of PACS fluxes, after converting them to luminosities following the approach of Shao et al. (2010). For the GOODS-S dataset, which includes 70 μm observations as well, we interpolated between the two PACS bands bracketing rest-frame 60 μm. We checked that wider wavelength coverage in GOODS-S (the addition of 70 μm observations) does not introduce any bias in the analysis: when interpolating rest-frame 60 μm luminosity only from 100 and 160 μm PACS photometry, our results remain unchanged, and individual νL_ν(60   μm) do not differ by more than 50%.

We also explored fitting PACS fluxes with a typical IR template (e.g. from Chary & Elbaz 2001 or Dale & Helou 2002 libraries) to derive νL_ν(60   μm). We found differences by a few tens per cent (only in 15% of the bins is the difference larger than 50%), but the global picture is unaffected. Therefore, we adopt the simple interpolated νL_ν(60   μm) estimate throughout, making no assumptions on the detailed SED shape.

For sources fully detected (i.e. detected in both PACS bands used to interpolate 60 μm luminosity) we used the corresponding fluxes and computed individual m). For PACS sources that were completely undetected in both bands, we computed average fluxes by stacking in a given L_X and redshift interval. We stacked at the X-ray positions on PACS residual maps using the Bethermin et al. (2010) libraries. The use of residual maps, from which all detected objects were removed, avoids contamination by nearby brighter sources. PSF photometry was performed on the final stacked images. For each PACS band j, we then averaged these stacked fluxes with individual fluxes of partially detected objects (i.e. detected only in band j but not in the other one), weighting by the number of sources. These stacked and averaged fluxes in each band are used to get m), in the same fashion as for the fully detected sources.

The final 60 μm luminosity in each L_X and redshift interval is computed by averaging over the linear luminosities of detections and non-detections, weighted by the number of sources. Only bins with more than 3 sources are used in our analysis.

Errors on the infrared luminosity were obtained by bootstrapping, in similar fashion as many current studies (Shao et al. 2010; Mullaney et al. 2012; Santini et al. 2012). A set of

N sources, where N is equal to the number of sources per bin in L_X and redshift, is randomly chosen 100 times among detections and non-detections (allowing repetitions), and a νL_ν(60   μm) is computed per each iteration. The standard deviation of the obtained νL_ν(60   μm) values gives the error on the average 60 μm luminosity in each bin. The error bars thus account for both measurement errors and the error on the population distribution. They do not, however, account for cosmic variance.

Appendix B: Combination across fields

To reduce scatter and the effects of cosmic variance, we combine the estimates of the average νL_ν(60   μm) in each bin in redshift and X-ray luminosity from all three fields into one mean number for each bin.

Fig. B.1 νLν(60   μm) as a function of redshift for X-ray AGNs in the individual COSMOS, GOODS-S and GOODS-N fields. The points in all panels are colored by logarithmic bins in hard band X-ray luminosity (LX), as shown. The upper panels show the mean L60 estimated by combining fluxes from PACS detections and stacks. Note the consistency across all fields between the values and variation with redshift of the mean measurements. The lower panels show individual PACS detected AGNs and the stacked points for PACS undetected AGNs (in bins of redshift and LX). The solid black lines show the approximate 3σνLν(60   μm) limit, derived from the flux limits of the 100 and 160 μm PEP photometry catalogs.

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The PACS maps in each of the fields differ in depth and area. Therefore, we begin by comparing the estimates of νL_ν(60   μm) in all three fields against redshift, as a simple test to ensure that the mean measurements in all three fields are in fact compatible. In the three upper panels of Fig. B.1, we plot the mean νL_ν(60   μm) from AGNs in each of the three fields separately. The points are colored by bins in X-ray luminosity. At a glance, one may gather that both the measurements and their redshift evolution in all three fields are very comparable.

In the lower three panels of Fig. B.1, we plot separately the νL_ν(60   μm) of PACS detected AGNs and those derived from stacks of the undetected AGNs. In addition, we have plotted the approximate rest-frame 60 μm luminosity limit as a function of redshift, derived from a log-linear interpolation of the 3σ PACS detection limits at 100 and 160 μm for each field. These luminosity limits match reasonably well the lower envelope of the PACS detected AGNs in the figure. The COSMOS detections are more luminous than those in the GOODS fields both at the luminosity limit, due to the significantly shallower PACS imaging, as well as at the upper end, because the larger area of the COSMOS field (5 ×  larger than the GOODS fields combined) brings in more of the rare, very FIR luminous galaxies into the sample. The stacked points can be up to an order of magnitude fainter than the PACS luminosity limit.

Due to the differences in the depth and noise properties of the PACS images in the three different fields, combining the data at the level of the maps poses difficulties. On the other hand, the measurements of FIR luminosity in each of the three fields show consistent values and behavior with redshift and L_X. Therefore we combined post-facto the estimates of the mean νL_ν(60   μm) in each redshift and L_X bin. This was done by taking an average of the FIR luminosity in all three fields in each bin with a valid measurement, weighted by the inverse variance (1/err²) derived from the errors on the mean FIR luminosity.

© ESO, 2012