BLACK HOLE MASS AND GROWTH RATE AT z ≃ 4.8: A SHORT EPISODE OF FAST GROWTH FOLLOWED BY SHORT DUTY CYCLE ACTIVITY^*

Sources were selected to allow the measurement of the Mg ii λ2798 emission line and the continuum flux at 3000 Å (F₃₀₀₀), thus providing reliable M_BH and L/L_Edd estimates. The sources were selected from the sixth data release (DR6; Adelman-McCarthy et al. 2008) of the Sloan Digital Sky Survey (SDSS; York et al. 2000). We first limited our search to quasi-stellar object-class ("QSO"-class) objects with z ∼ 4.65–4.95, so that the Mg ii line and F₃₀₀₀ could be observed within the H band. The completeness rate of SDSS spectroscopy for QSOs at this redshift is above 98%, down to a limiting magnitude of i ≃ 20 (Richards et al. 2006a). This initial search resulted in 177 objects. Next, we verified that the SDSS-observed C iv lines are of acceptable S/N and that no broad absorption features are present, since these are also expected to appear in the Mg ii profile. These criteria reduced the list of candidates to 129 sources.

To estimate the H-band flux of the targets, we extrapolated the rest-frame flux density at 1450 Å to the observed-frame flux density at 1.65μm, using the f_ν ∝ ν^−0.44 spectral energy distribution (SED) presented by Vanden Berk et al. (2001). We applied a flux limit of f_λ(1450 Å) ≳ 6 × 10⁻¹⁸ erg cm⁻² s⁻¹ Å⁻¹, which translates to f_λ(1.65 μm) ≳ 2.1 × 10⁻¹⁸ erg cm⁻² s⁻¹ Å⁻¹, to include only those sources which would provide NIR spectra with $\rm S/N\gtrsim 5\hbox{--}10$ within reasonable exposure times. This formal flux limit translates to L_bol ≃ 5 × 10⁴⁶ erg s⁻¹ (see below). Our flux limit results in the omission of only six candidates, leaving 123 sources. Different observational constraints (a combination of declination, observational seasons etc.; see Section 2.2) forced us to observe only 40 of these 123 candidates. We verified that the distribution of f_λ(1450 Å) for the observed sample is very similar to that of the flux-limited sample of 123 sources, as well as the entire initial sample of 177 SDSS objects. In particular, we find four sources (∼11% of the observed sample) with f_λ(1450 Å) ≲ 7.5 × 10⁻¹⁸ erg cm⁻² s⁻¹ Å⁻¹ and 15 sources (∼43%) with f_λ(1450 Å) ≲ 1.5 × 10⁻¹⁷erg cm⁻² s⁻¹ Å⁻¹. Given this and the high level of completeness in the SDSS, we consider this sample to be the representative of the complete sample of luminous AGNs at z ≃ 4.8.

Several studies suggest that the mass and accretion rate evolution of SMBHs may be connected with their radio properties (see, e.g., McLure & Jarvis 2004; Shankar et al. 2010b, but also Woo & Urry 2002). To determine the radio properties of the z ≃ 4.8 sources, we utilized the cross-matched catalog Faint Images of the Radio Sky at Twenty centimeters radio survey (FIRST; Becker et al. 1995; White et al. 1997). Due to the high redshift of our sample, and the relatively low sensitivity of FIRST, most candidates have only upper limits on their radio fluxes, and thus upper limits on the radio loudness ($R_{\rm L}\equiv \frac{ f_{\nu }\left(5\, {\rm GHz}\right)}{f_{\nu }(4400\,{\rm\AA})}$; Kellermann et al. 1989). For those sources, we estimated R_L based on a 3σ FIRST upper flux limit of f_ν(5 GHz) = 0.6 mJy and assuming a radio SED of f_ν ∝ ν^−0.8. f_ν(4400 Å) was estimated from f_λ(1450 Å) and the Vanden Berk et al. (2001) composite. All but five observed targets (see Table 1) have upper limits on R_L which are below ∼40. Two of the remaining sources (J0210-0018 & J1235-0003) have firm FIRST detections (f_ν[5 GHz] = 9.75 and 18.35 mJy, respectively) and are thus considered as radio-loud AGNs (R_L ≃ 104 and 1080, respectively). Three additional targets were not observed by FIRST and thus have no viable radio data. In what follows, we verified that the inclusion or exclusion of these five sources does not significantly affect our results. In summary, the 40 z ≃ 4.8 AGNs presented here comprise a flux-limited sample which represents a large fraction of all such SDSS sources. In terms of bolometric luminosity, it is complete down to L_bol ≃ 5 × 10⁴⁶ erg s⁻¹ and most of the sources are not radio loud. The basic properties of the 40 sources are given in Table 1.

The H-band spectra of our z ≃ 4.8 sample were obtained using the Gemini-North and the Very Large Telescope (VLT) observatories. The campaign was split over several semesters, with brighter targets predominantly observed with the Near Infra-Red Imager instrument (NIRI; Hodapp et al. 2003) on Gemini-North (as part of programs GN-2007B-Q-56 and GN-2008B-Q-75) and fainter ones with the more sensitive SINFONI instrument (Eisenhauer et al. 2003) on the VLT-UT4 (as part of programs 081.B-0549, 082.B-0520, and 085.B-0863). The log of observations is given in Table 1. All the observations were performed in queue/service modes, requested not to exceed an airmass of ∼1.7 and seeing of ∼1'', during clear nights and "gray-time" lunar phase. The 11 Gemini-North/NIRI targets were observed through a 075 × 110'' slit and the G5203 grism at the f/6 setup, providing R ∼ 520. The sub-integrations were of ∼300 s, with dithers of 6'' along the slit after each sub-integration. The 29 VLT/SINFONI targets were observed through the 8'' × 8'' field of view (FOV; aka "250 mas spaxel⁻¹") We used SINFONI's "H+K" mode, since the broad Mg ii λ2798 line can be resolved well with the resulting R ∼ 1500. The K-band spectra assisted in determining the continuum flux of the SINFONI targets. Sub-integrations were of 150 s or 300 s and (diagonally) dithered by ∼6'' across the FOV. Despite this type of dithering, the spectroscopically resolved OH sky emission contributes a major source of noise to our reduced data (see below). In both observatories, telluric standards of spectroscopic types B, A, and G were observed immediately before or after the science targets, at a similar airmass.

The reduction of the raw data was carried out using the standard pipelines of the respective facilities. We used the gemini v.1.8 package in IRAF to co-align and combine the sub-integrated frames while subtracting the sky emission. The one-dimensional spectra were extracted through a typical aperture of 20 pixels which correspond to ∼23. The SINFONI v.2.0.5 pipeline (under the Esorex environment) was used to extract three-dimensional "data cubes" from each sub-integrated frame, which include an individual wavelength calibration. The data cubes were then co-aligned and combined, while the sky emission was estimated from the appropriate dithered pointings. The combined data cube of each source was examined to estimate the actual point-spread function, and an appropriate circular aperture was used to extract the one-dimensional spectra. Due to the varying flexure of the instrument during the long (1 hr) SINFONI observing blocks, there is a noticeable "wavelength flexure" effect in the extracted one-dimensional spectra. This effect results in P-Cygni-like spectral features and can be treated separately (see Davies 2007). We note that the observable properties critical to this work (i.e., L₃₀₀₀ and FWHM(Mg ii)) are not very sensitive to such narrow (≲400 km s⁻¹) OH-originated features.

The spectra of the standard stars were reduced and extracted in the same manner as above, using the same apertures as those used for the science targets. The published spectral types of these stars were used to correct for the instrumental response of each science observations and their published Two Micron All Sky Survey (2MASS; Cutri et al. 2003) magnitudes were used to flux-calibrate the science spectra. The typical photometric error associated with the 2MASS magnitudes is ≲10%.

Wide-field H- and K-band imaging for 11 of the z ≃ 4.8 sources was obtained at the CTIO Blanco telescope, with the ISPI instrument (van der Bliek et al. 2004). We aimed at having at least three bright 2MASS stars observed in the same field for each of the science targets. Each field was imaged to achieve S/N ≳ 20 for the science targets, through a series of dithered short exposures, to avoid saturation. The CTIO observations were carried out with mostly clear conditions, with seeing of ∼1'', during a single run on the night of 2010 March 4. The raw imaging data were reduced following standard procedures, including bad pixel removal, dark and flat calibration, and co-alignment of the sub-exposures. Aperture photometry was performed over the science targets and the several visible 2MASS stars in each field. Whenever possible, we used only stars with best-quality 2MASS photometry for the relative scaling of the science target fluxes. The magnitudes thus obtained are listed in Table 1, for the relevant sources.

Since the spectroscopy and imaging observations were not simultaneous, there exists a real uncertainty regarding possible flux variability of our z ≃ 4.8 sources. Our targets are relatively luminous, and thus are not expected to have large variability amplitudes (e.g., Bauer et al. 2009 and references therein). We compared the photometric and the spectroscopic fluxes, achieved by synthetic photometry of the SINFONI and NIRI spectra. The median discrepancy is ≲0.09 mag, in the sense that the (later observed) photometric fluxes are on average fainter than the spectroscopy-deduced fluxes. The standard deviation is close to 0.2 mag. Photometric fluxes for 10 additional sources were obtained through the fourth data release of the UKIRT Infrared Digital Sky Survey (UKIDSS/DR4; Lawrence et al. 2007). The photometry of these sources is also consistent with the fluxes deduced from our spectrophotometric calibration. Excluding one significant outlier (J0210–0018), the standard deviation between the two methods is, again, close to 0.2 mag.

With this evidence of little flux variation, and reliable spectrophotometry for a large fraction (∼50%) of arbitrary selected sources, we conclude that the accuracy of our spectrophotometric flux calibration is high, with uncertainties of about 10%–15%. In order to correctly probe the instantaneous emission from the sources, we thus adopt the spectrophotometric fluxes to measure luminosities, even for those sources which were photometrically observed. The full set of calibrated spectra is presented in Figure 1.

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To determine M_BH and L/L_Edd, we fit the observed spectra with a model that combines a continuum component, Fe ii and Fe iii lines, and the two doublet Mg ii lines. The procedure used here is similar to the one presented in several other studies (e.g., Shen et al. 2008 and references therein). It is based on a new code that was developed and extensively tested on a large sample of 0.5 ≲ z ≲ 2 SDSS type-I AGNs, and will be described in detail in a future publication (B. Trakhtenbrot et  al. 2011, in preparation). In particular, this large sample was used to verify that important quantities such as narrow emission lines and bolometric corrections are consistent with those derived from the well-studied Hβ-[O iii] λ5007 emission region (rest-frame wavelengths of 4600–5100 Å). This was done using a subsample of ∼5000 sources at 0.5 ≲ z ≲ 0.75, where the SDSS spectra show both the Hβ and Mg ii lines.

The fitting of the z ≃ 4.8 AGNs was preformed individually, verifying a satisfactory match between the data and the model. As a first step, a linear pseudocontinuum is fit to the flux around 2655 and 3020 Å. Next, we fit the Fe ii and Fe iii emission complexes redward and blueward of Mg ii. For this we use a template made of the composite prepared by Vestergaard & Wilkes (2001) that was supplemented by several predicted Fe ii lines (kindly provided by G. Ferland). These are mainly Fe ii emission lines which coincide in wavelength with the Mg ii line, where the observationally based template of Vestergaard & Wilkes (2001) is incomplete. This part of the template is very similar to the models presented in Sigut & Pradhan (2003, their Figure 13) and in Baldwin et al. (2004, their Figure 5). In both cases, the Fe ii flux under the Mg ii line is dominated by the red extension of the ∼2750 Å emission complex. The additional Fe ii flux tends to flatten in sources with very broad lines, but the overall effect on the Mg ii line fitting is very small. We consider this template to be more reliable than the addition of constant flux under the Mg ii line, adopted by Kurk et al. (2007) and Fine et al. (2008). Several other studies, such as Salviander et al. (2007) and Shen et al. (2008), use templates very similar to ours, following the models in Sigut & Pradhan (2003).

Our basic iron template is based on a single line profile with FWHM ≃ 1175 km s⁻¹. To account for the range of observed line widths, we created a grid of broadened Fe templates, by convolving the basic template with single Gaussian profiles of varying width, ranging from FWHM = 1200 to 10, 000 km s⁻¹. We note that the templates that correspond to FWHM ≳ 4000 km s⁻¹ lack almost any distinguishable emission features. The grid of broadened Fe templates is fitted to the continuum-subtracted spectra over the rest-frame wavelength regions of 2600–2700 Å and 2900–3030 Å, and the best-fit template is chosen by standard χ² minimization. We allow the Fe template to be shifted with respect to the systemic redshift and for its width to differ from that of the Mg ii line. After the subtraction of the best-fit Fe template, the pseudocontinuum is re-fitted and we iterate the Fe-fitting process once again. This is done to ensure the convergence of the best-fit pseudocontinuum and Fe lines. The final measure of the pseudocontinuum flux, f_λ(3000 Å), provides the continuum luminosity L₃₀₀₀ which is further used to calculate M_BH and L/L_Edd. It is important to note that the L₃₀₀₀ thus obtained is not the intrinsic, underlying AGN continuum as it also includes a contribution from the Balmer continuum emission, that is not accounted for by the fitting procedure.

Once the pseudocontinuum and Fe contributions are subtracted, we fit the Mg ii line itself. The model for the line consists of three Gaussians, two broad and one narrow component for each of the Mg ii doublet lines. The intensity ratio of the doublet components is fixed to 1:1, suitable for optically thick lines. The broad components are limited to line widths in the range 1200 km s⁻¹ < FWHM < 10, 000 km s⁻¹, while the width of the narrow components is bound to 300 km s⁻¹ < FWHM < 1200 km s⁻¹. Although the relative contribution of the narrow components is very small (<10%), the procedure reproduces well the narrow-line region (NLR) width. This was verified by comparing the NLR–FWHM resulting from the (separate) Mg ii and Hβ fitting for the test sample of ∼5000 sources at 0.5 ≲ z ≲ 0.75 mentioned above. We have also included a single absorption feature (also a Gaussian) to account for the appearance of blueshifted Mg ii troughs. The best-fit model is used to measure the FWHM of the total broad single Mg ii line and the total luminosity of both Mg ii components. We stress that in earlier studies, the line width was calculated using the combined doublet profile, which has implications to the deduced M_BH (see Section 5.1). The best-fit continuum and Mg ii line parameters are given in Table 2. The uncertainties that also appear in Table 2 reflect the true uncertainties associated with the spectroscopic reduction and line-fitting procedure. These were estimated from varying several of the parameters involved in these processes⁴ and examining the range of possible outcomes. We consider these uncertainties to be much more realistic than those derived solely from the flux noise.

Table 2. Observed and Derived Properties

Object ID (SDSS J)	zSDSS	$z_{\rm Mg\,{\mathsc{ii}}}$a	log  L1450b	log  L3000	L-qual.c	log  Lbol	FWHM(Mg ii)	FWHM-qual.d	log  MBH	log  L/LEdd
			(erg s−1)	(erg s−1)		(erg s−1)	(km s−1)		(M☉)
000749.17+004119.4	4.837	4.786	46.39	46.04	3	46.57	3665	3	8.90	−0.51
003525.28+004002.8	4.757	4.759	46.35	46.38	3	46.91	1805	3	8.49	0.24
021043.15-001818.2	4.733	4.713	46.66	46.04	2	46.56	4583	3	9.09	−0.70
033119.67-074143.1	4.738	4.729	46.76	46.55	1	47.09	2360	1	8.83	0.08
075907.58+180054.7	4.861	4.804	46.46	46.54	1	47.07	2717	1	8.95	−0.05
080023.03+305100.0	4.687	4.677	46.82	46.73	1	47.26	1404	1	8.49	0.59
080715.12+132804.8	4.874	4.885	46.71	46.53	2	47.07	3837	3	9.24	−0.35
083920.53+352457.6	4.777	4.795	46.70	46.24	2	46.77	1971	2	8.49	0.11
085707.94+321032.0	4.776	4.801	46.94	46.72	2	47.25	2851	2	9.10	−0.03
092303.53+024739.5	4.660	4.659	46.33	46.14	1	46.67	2636	1	8.68	−0.18
093508.50+080114.5	4.699	4.671	46.62	46.33	2	46.87	2714	2	8.82	−0.13
093523.32+411518.7	4.836	4.802	46.66	46.58	2	47.12	3447	3	9.18	−0.24
094409.52+100656.7	4.748	4.771	46.64	46.40	2	46.93	2128	2	8.65	0.11
101759.64+032740.0	4.917	4.943	46.27	46.10	1	46.63	2822	1	8.71	−0.26
105919.22+023428.8	4.735	4.789	46.65	46.36	2	46.89	2899	2	8.89	−0.18
111358.32+025333.6	4.882	4.870	46.49	46.35	2	46.89	3793	3	9.12	−0.41
114448.54+055709.8	4.793	4.790	46.13	46.11	3	46.63	3215	2	8.83	−0.37
115158.25+030341.7	4.698	4.687	46.05	45.91	3	46.44	3741	3	8.84	−0.57
120256.44+072038.9	4.785	4.810	46.28	46.27	1	46.80	2171	1	8.59	0.04
123503.04-000331.6	4.723	4.700	46.07	46.12	2	46.65	4422	3	9.11	−0.64
130619.38+023658.9	4.852	4.860	46.57	46.82	1	47.35	5340	1	9.71	−0.54
131737.28+110533.1	4.810	4.744	46.39	46.34	2	46.87	3144	2	8.95	−0.25
132110.82+003821.7	4.716	4.726	46.47	46.17	2	46.70	3651	3	8.98	−0.45
132853.67-022441.7	4.695	4.658	46.42	46.28	2	46.81	3815	2	9.08	−0.45
133125.57+025535.6	4.737	4.762	46.15	46.02	3	46.55	3445	3	8.83	−0.46
134134.20+014157.8	4.670	4.689	46.87	46.73	2	47.26	6480	2	9.82	−0.74
134546.97-015940.3	4.714	4.728	46.62	46.08	2	46.60	3412	2	8.86	−0.43
140404.64+031404.0	4.870	4.903	46.55	46.49	2	47.02	5360	2	9.51	−0.66
143352.21+022714.1	4.721	4.722	47.14	46.84	2	47.37	2622	2	9.11	0.09
143629.94+063508.0	4.850	4.817	46.62	46.44	2	46.98	3052	2	8.99	−0.19
144352.95+060533.1	4.879	4.884	46.44	46.16	3	46.69	3609	3	8.96	−0.45
144734.10+102513.2	4.686	4.679	46.29	45.99	1	46.51	1407	1	8.03	0.30
151155.98+040803.0	4.686	4.670	46.62	46.32	2	46.86	1735	2	8.42	0.26
161622.11+050127.7	4.872	4.869	47.08	46.80	2	47.33	3910	1	9.43	−0.27
165436.86+222733.7	4.678	4.717	47.14	46.48	1	47.02	5637	1	9.55	−0.70
205724.15-003018.0	4.663	4.680	47.04	46.83	2	47.36	3050	2	9.23	−0.05
220008.66+001744.8	4.818	4.804	46.70	46.51	2	47.04	2404	2	8.82	0.04
221705.72-001307.7	4.689	4.676	46.44	46.28	3	46.81	2274	3	8.63	0.00
222509.16-001406.8	4.888	4.890	46.97	46.70	1	47.23	3504	1	9.27	−0.21
224453.06+134631.8	4.657	4.656	46.30	46.06	2	46.58	2516	2	8.58	−0.17

Notes. ^aRedshift measured from the best-fit models of the Mg ii lines. ^bMonochromatic luminosity at rest wavelength 1450 Å, obtained from the SDSS/DR6 spectra and redshifts. ^cQuality flag associated with L₃₀₀₀. Quality flags of "1," "2," and "3" correspond to calibration and/or continuum measurement uncertainties of ∼10%, ∼20%, and ∼40%, respectively. ^dQuality flag associated with FWHM(Mg ii). Quality flags of "1," "2," and "3" correspond to uncertainties of ∼10%, 20%–30% and ∼50%, respectively.

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The L_bol and M_BH estimates used in this work are based on the measured L₃₀₀₀ and FWHM(Mg ii). For the latter, we adopt the McLure & Dunlop (2004) relationship:

$$\begin{equation} M_{\rm BH} =3.2\times 10^6 \left[\frac{L_{3000} }{10^{44}\,{\rm erg\,s}^{-1} }\right]^{0.62} \left[\frac{{\rm FWHM}\!\left({\rm Mg}\,{\mathsc {ii}} \right) }{10^3 \,{\rm km\,s}^{-1} }\right]^2 \,\,M_{\odot } \,. \end{equation} \tag{ 1 }$$

To estimate L_bol, we have to assign a bolometric correction factor, f_bol(λ) = L_bol/λL_λ. For this we adopt the Marconi et al. (2004) luminosity-dependent SED, which also provides a polynomial prescription for estimating f_bol(4400 Å). The prescription can be used with other UV–optical monochromatic luminosities, by adopting the Vanden Berk et al. (2001) UV continuum (f_ν ∝ ν^−0.44). Our procedure relies on our own empirically calibrated f_bol(3000 Å) versus L₃₀₀₀ relation, derived using the 0.5 ≲ z ≲ 0.75 type-I SDSS AGN test sample. For each source, we measured both L₅₁₀₀ and L₃₀₀₀ and converted L₅₁₀₀ to L_bol using the relation of Marconi et al. (2004) and the Vanden Berk et al. (2001) template. This provides, for each of the ∼5000 sources, both L₃₀₀₀ and f_bol(3000 Å) = L_bol/L₃₀₀₀. The best-fit relation is

$$\begin{eqnarray} f_{\rm bol}(3000 \, \hbox{\AA}) & = & {-}0.58{\cal L}_{3000,44}^3 + 3.85 {\cal L}_{3000,44}^2 \nonumber \\ && -8.38 {\cal L}_{3000,44} +9.34, \end{eqnarray} \tag{ 2 }$$

where ${\cal L}_{3000,44}\equiv \log (L_{3000} /10^{44}\,{\rm erg\,s}^{-1})$. The bolometric corrections for our z ≃ 4.8 sample range between 3.35 for the least luminous source and 3.43 for the most luminous one. The typical f_bol(3000 Å) is lower than those used in other studies (e.g., Elvis et al. 1994; Richards et al. 2006b) by a factor of about 1.5. This is mostly due to the fact that the Marconi et al. (2004) estimates of L_bol do not include most of the mid- to far-IR emission, which originates from dust around the central source. The normalized accretion rate is L/L_Edd = L_bol/(1.5 × 10³⁸ M_BH M⁻¹_☉) that is appropriate for ionized gas with solar metallicity. The deduced M_BH and L/L_Edd are given in Table 2.

We have also estimated M_BH from the C iv λ1549 line, that is observed in the SDSS spectrum of each source. The line-fitting procedure was similar to the one described above for Mg ii and also to those discussed in other studies (e.g., Shen et al. 2008; Fine et al. 2010). We calculated M_BH from L₁₄₅₀ and FWHM(C iv) following the prescription of Vestergaard & Peterson (2006). We find that the C iv line is systematically broader than Mg ii, similar to the recent finding of Fine et al. (2010). The differences in widths translate to higher C iv-based M_BH estimates, with respect to those derived from Mg ii. These findings confirm the suspicion about the problematic use of the C iv-based method (see Section 1). A full analysis of these issues is deferred to a future publication. In what follows, we rely solely on the Mg ii-based measurements of M_BH and L/L_Edd.

The distributions of M_BH and L/L_Edd for the z ≃ 4.8 sample are shown in Figure 2. They cover the range 8.03 < log(M_BH/M_☉) < 9.82 and 0.18 < L/L_Edd < 3.92. The median values and 68% percentiles (taken to be symmetric around the medians) correspond to <log(M_BH/M_☉)> = 8.89^+0.32_−0.34 and <L/L_Edd = 0.59^+0.63_−0.30.

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As explained in Section 3, both M_BH and L/L_Edd are derived directly from L₃₀₀₀, and both distributions are affected by the flux limit of the sample. The 1450 Å flux limit corresponds to f_λ(3000 Å) ≳ 2.1 × 10⁻¹⁸ erg cm⁻² s⁻¹ Å⁻¹ which, at z ≃ 4.8, gives a limiting luminosity of L₃₀₀₀ ≃ 10⁴⁶ erg s⁻¹. The limiting M_BH is estimated by combining the limiting L₃₀₀₀ and FWHM = 1500 km s⁻¹ using Equation (1). This line width is the minimal width of any emission line for which the SDSS pipeline labels the target as a spectroscopic target of type "QSO;" a pre-requirement of our sample selection. We thus obtain log(M_BH/M_☉)_limit ≃ 8. Indeed, one of our sources (J1447+1025) has very similar properties, in terms of L₃₀₀₀, FWHM(Mg ii) and M_BH. However, this source looks as an outlier in the M_BH distribution. The next smallest source has log(M_BH/M_☉) ≃ 8.4, which represents the low end of the M_BH distribution much better. We conclude that our sample is complete down to log(M_BH/M_☉) ≃ 8, and the fact that we observe (almost) no sources with log(M_BH/M_☉) < 8.5 is a real characteristic of the population of spectroscopically observed, optically selected type-I AGNs at z ≃ 4.8.

The limiting L/L_Edd is deduced by combining the lowest L₃₀₀₀ with ${\rm FWHM}\!\left({\rm Mg}{\mathsc {ii}} \right) =10{,}000\,{\rm km\,s}^{-1}$, the upper boundary that FWHM(Mg ii) can take in our line-fitting procedure (see Section 3.1). The resulting limiting value is (L/L_Edd)_limit ≃ 0.04. Clearly, the limiting L/L_Edd is a factor of ∼4.5 below the lowest L/L_Edd we actually observe. This reflects the fact that the broadest Mg ii line we observe has ${\rm FWHM}\!\left({\rm Mg}{\mathsc {ii}} \right) \simeq 6{,}500\,{\rm km\,s}^{-1}$.

The highest observable L/L_Edd is determined by a combination of the narrowest observable FWHM(Mg ii) and the highest luminosities. For instance, the faint and narrow-lined source J1447+1025 is one of the higher L/L_Edd sources in our sample. On the other hand, the highest L/L_Edd source (J0800+3051) has a similarly narrow Mg ii line but is among the most luminous sources in the sample (a factor of ∼5.6 more luminous than J1447+1025), thus approaching L/L_Edd ≃ 4. Since there are no brighter z ≃ 4.8 sources in the entire SDSS/DR6 database, we conclude that our sample unveils the highest L/L_Edd (≃4) sources at z ≃ 4.8. We also note that ∼1/4 of the AGNs in our sample have 1 < L/L_Edd ≲ 4. Several studies (e.g., Mineshige et al. 2000; Wang & Netzer 2003; Kurosawa & Proga 2009) have shown that such high L/L_Edd can be produced in several types of accretion disks.

Given the observed M_BH, L_bol, and L/L_Edd, we can constrain the lifetimes of the SMBHs in our sample. We first assume that accretion proceeds with a constant L/L_Edd. This results in an exponential growth of M_BH, with an e-folding time (Salpeter time) of

$$\begin{equation} \tau = 4 \times 10^8 \frac{\eta /(1- \eta) }{L/L_{\rm Edd} } \,\, {\rm yr}, \end{equation} \tag{ 3 }$$

where η is the radiative efficiency ($L_{\rm bol} =\eta \dot{M}_{\rm infall} c^2$). The time required to grow by such accretion from a seed BH of mass M_seed to M_BH is

$$\begin{equation} t_{\rm grow} = \tau \, \ln \left(\frac{M_{\rm BH} }{M_{\rm seed} } \right) \,\, {\rm yr}. \end{equation} \tag{ 4 }$$

Despite several attempts to estimate η (e.g., Volonteri et al. 2005; King et al. 2008; and references therein), its typical value and redshift dependence are poorly constrained. A broad range of η ≃ 0.05–0.3 is required to match the local BH mass function to the integrated accretion history of SMBHs (e.g., Shankar et al. 2004; Wang et al. 2009; Shankar et al. 2010a). There is an even broader range of possible values of M_seed, depending on the various mechanisms to produce such objects. Remnants of population-III stars would result in M_seed ∼ 10–100 M_☉ (e.g., Heger & Woosley 2002), while collapse models of either dense stellar clusters or gas halos predict seed masses as large as M_seed ∼ 10³–10⁶ M_☉ (e.g., Begelman et al. 2006; Devecchi & Volonteri 2009; see Volonteri 2010 for a review). For the sake of consistency with earlier z2 studies (N07), we assume here η = 0.1 and M_seed = 10⁴ M_☉. With this choice of η, the e-folding times in our sample range from ∼10 to ∼240 Myr and the faster accreting 50% of the sources show τ ≲ 75 Myr. Such SMBHs can increase their mass by a factor of ∼750 within ∼500 Myr. We compared the growth of the SMBHs in our sample up to z ≃ 4.8 with the corresponding age of the universe at that redshift (t_Universe ≃ 1.21 Gyr). The assumption of continuous, constant L/L_Edd growth for such a long period is, of course, somewhat simplistic, given the timescales of standard accretion disks (a review of this and related issues can be found in King 2008). About 65% of the sources have t_growth < t_universe. This is a larger fraction than observed in lower redshift samples, which suggests that z ≃ 4.8 might represent the sought after episode of fast growth for most SMBHs.

We also calculated the regress of M_BH with increasing redshift assuming the aforementioned exponential growth scenario, up to z = 20. These evolutionary tracks are shown in Figure 3. The point at which such tracks cross the y-axis in Figure 3 can be regarded as the value of M_seed required to match the observed M_BH at z ≃ 4.8, given the assumption of a constant L/L_Edd growth. The diagram demonstrates how the z ≃ 4.8 sample can constrain the different BH formation and growth scenarios. For example, about 40% of the z ≃ 4.8 sources could have grown from seed BHs with M_seed < 100 M_☉, i.e., stellar-remnant BHs. A further ∼20% could have grown from BH seeds with 1000 M_☉ < M_seed < 10⁵ M_☉, and another ∼20% from BH seeds with 10⁵ M_☉ < M_seed < 10⁷ M_☉. The remaining lowest L/L_Edd sources (∼10%) have 10⁷ M_☉ < M_seed < 10⁸ M_☉.⁵ Since even the most extreme scenarios cannot produce such high M_seed (Volonteri 2010), these sources had to accrete at higher rates in the past in order to attain their measured mass. Another explanation is a combination of massive seed BHs with low spins (i.e., small η) at very early epochs. For example, if we assume η = 0.05, then at z = 20 all the sources have an implied M_seed < 10⁶ M_☉. As mentioned above, the radiative efficiency could be higher, e.g., ∼0.2–0.3. Assuming such high values of η in our calculations naturally leads to the requirement of more massive seed BHs. For example, in the extreme case of η = 0.3 in all sources, we find that 95% of the seed BHs would have M_seed500 M_☉.

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Our main goal is to identify the epoch at which most SMBHs experienced the first episode of fast growth. We focus on a comparison of our z ≃ 4.8 sample to several z2 samples with reliably measured M_BH and L/L_Edd. In S04 and N07, we studied a sample of 44 z ≃ 2.4 and ≃3.3 type-I AGNs using NIR spectroscopy to measure L₅₁₀₀ and FWHM(Hβ). The study showed that a large fraction of the high M_BH sources accrete at a rate that is well below the Eddington limit. Assuming these sources accrete at constant L/L_Edd, such accretion rates cannot explain their measured masses and are in contradiction with several theoretical predictions. Having obtained new data on the z ≃ 4.8 sample, we can now compare several groups of AGNs, at several epochs, in an attempt to follow their growth all the way from z ≃ 6.2 to z ≃ 2.4.

The samples we consider here are the new one at z ≃ 4.8 and our earlier (S04 and N07) samples at z ≃ 2.4 and ≃3.3. We also consider a small number of sources at z ≃ 6.2 from the samples studied by K07 and W10, which have five and nine reliable M_BH and L/L_Edd measurements, respectively. These small samples do not have a common flux limit and simply represent the up-to-date collection of the sources discovered at those redshifts that were also observed in follow-up NIR spectroscopy. As such, they are not representative of the AGN population at z ≃ 6.2. This, along with the small size of the samples, limits their statistical usefulness. To use the data from the z ≃ 6.2 samples, we re-calculated M_BH and L_bol using the methods described above. First, we corrected the published FWHM(Mg ii) according to the assumption of only one of the doublet components. This reduces FWHM(Mg ii) by a mean factor of ∼1.2 and the reported M_BH by a mean factor of ∼1.44. Second, the smaller f_bol(3000 Å) adopted here reduces the reported L_bol by a factor of ∼1.49. Regarding L/L_Edd, the two corrections almost completely cancel out.

In Figure 4, we present L_bol, M_BH, and L/L_Edd for the new z ≃ 4.8 sample, the z ≃ 2.4 and ≃3.3 samples of S04 and N07, and the combined sample of z ≃ 6.2 AGNs. The four samples suggest an increasing M_BH and decreasing L/L_Edd with cosmic time. This is better illustrated in Figure 5, where we present the cumulative distribution functions (CDFs) of M_BH and L/L_Edd. The increase in the median M_BH and the decrease in the median L/L_Edd as a function of decreasing redshift are evident. In particular, there is a clear shift of ∼0.5 dex between the median M_BH values of the z ≃ 4.8 and z ≃ 3.3 samples, in the sense of a lower M_BH at z ≃ 4.8. There is also an opposite shift between the median L/L_Edd values. The differences between the z ≃ 3.3 and the z ≃ 2.4 samples are much smaller, although the z ≃ 2.4 sample includes several sources with extremely massive BHs (log(M_BH/M_☉)>10) which are not observed at earlier epochs. Regarding M_BH, only ∼14% of the z ≃ 3.3 sample (two sources) lie below the median value of the z ≃ 4.8 sample (log(M_BH/M_☉) = 8.92). Similarly, only ∼13% of the z ≃ 4.8 sample (five sources) lie above the median value of the z ≃ 3.3 sample (log(M_BH/M_☉) = 9.37) and even less (∼6%; two sources) lie above the median value of the z ≃ 2.4 sample (log(M_BH/M_☉) = 9.59).

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We have further tested the significance of these differences by performing a series of two sample Kolmogorov–Smirnov tests. The null hypothesis that the observed distributions of L/L_Edd at z ≃ 4.8 and at z ≃ 3.3 (or at z ≃ 2.4) are drawn from the same parent distribution is rejected with significance levels >99%. A similar test was applied to the z ≃ 4.8 and z ≃ 6.2 samples, and could not reject the null hypothesis, suggesting that the distributions of L/L_Edd at z ≃ 4.8 and z ≃ 6.2 are statistically similar. We obtain similar results when comparing the distribution of M_BH at z ≃ 4.8 to that of the z ≃ 2.4 and ≃3.3 samples. Due to the large fraction of z ≃ 6.2 sources with log(M_BH/M_☉) ≲ 8.5, we can reject the hypothesis that the M_BH values at z ≃ 4.8 and z ≃ 6.2 are drawn from the same distribution.

Some of the above results may be biased by the fact that the four samples cover a different range of L_bol, which originate from the different target selection criteria. We thus repeated the statistical tests aforementioned, focusing on subsamples which share a common range of 46.4 < log(L_bol/erg s⁻¹) < 47.4 (i.e., matching the L_bol range of the z ≃ 4.8 sample). All but one comparison result in the same conclusions, with similar confidence levels. The one exception are the distributions of M_BH at z ≃ 4.8 and z ≃ 6.2, for which the null hypothesis now cannot be rejected, suggesting that these distributions represent the same parent population. This is not surprising given the lower luminosities of most of the z ≃ 6.2 sources, the dependence of M_BH on source luminosity, and the incompleteness of the z ≃ 6.2 sample.

We conclude that there is strong evidence for a rise in M_BH and a drop in L/L_Edd of about a factor of 2.8 between z ≃ 4.8 and z ≃ 3.3. This strong trend is not observed with respect to neither lower nor higher redshift samples, which span similar periods of time.⁶ It thus seems that the most massive BHs, associated with the most luminous AGNs, started the episode of fast BH growth at redshifts above z ∼ 5. By z ∼ 2–3, these SMBHs reach their peak (final) mass ( 10¹⁰ M_☉) and their mass accretion is less efficient.

Given the trends and differences in M_BH and L/L_Edd, as well as the short e-folding times of the z ≃ 4.8 AGNs (Section 4.2), it is possible to think of the z ≃ 4.8, z ≃ 3.3, and z ≃ 2.4 samples as representing different evolutionary stages of the same parent population of SMBHs. In what follows, we focus on the z ≃ 4.8 and z ≃ 3.3 samples (separated by ∼680 Myr) to test this evolutionary interpretation, assuming various scenarios.

We consider two growth scenarios: constant mass accretion rate (i.e., constant L_bol) and constant L/L_Edd.⁷ As explained, the assumption of constant L/L_Edd results in an exponential growth of M_BH and given our chosen value of η, the mass growth e-folding time is τ ≃ 45(L/L_Edd)⁻¹ Myr, which translates to τ ≲ 240 Myr (L/L_Edd0.18) for the z ≃ 4.8 sources. Thus, even the lowest L/L_Edd SMBHs at z ≃ 4.8 could have increased their M_BH by a factor of ∼20 by z ≃ 3.3, which is much larger than the typical difference in M_BH between the two samples. The assumption of a constant L_bol results in much slower growth. The luminosities of the z ≃ 4.8 sources translate to $4\, M_{\odot } \,{\rm yr}^{-1}\lesssim \dot{M}_{\rm BH} \lesssim 37 \, M_{\odot } \,{\rm yr}^{-1}$. Even this much slower growth scenario results in very high M_BH at z ≃ 3.3. This scenario also produces very low accretion rates (L/L_Edd < 0.1) at z ≃ 3.3.

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In Figure 6, we present evolutionary tracks for our z ≃ 4.8 AGNs to z ≃ 3.3 and eventually to z = 2. Clearly, continuous constant L_bol growth results in too large M_BH and cannot reproduce the lower M_BH sources at z ≃ 2.4 and ≃3.3. Specifically, the calculated distribution of M_BH for the z ≃ 4.8 sources, when evolved to z ≃ 3.3, has a median value of log(M_BH/M_☉) ≃ 10, larger by ∼0.6 dex than the observed median of the z ≃ 3.3 sample. This scenario also fails to reproduce the observed range of L/L_Edd, since the lowest L/L_Edd sources at z ≃ 3.3 have L/L_Edd ≳ 0.1. As the dotted evolutionary tracks in Figure 6 demonstrate, the constant L/L_Edd scenario produces even larger masses, and is only feasible if all the z ≃ 4.8 sources cease their accretion shortly after their observed active phase. This scenario assumes a constant L/L_Edd, hence the significant difference between the L/L_Edd distributions at z ≃ 4.8 and at z ≃ 3.3 (∼0.5 dex; see Figure 5) is not resolved. We also note that in both scenarios the z ≃ 4.8 sources would have been easily observed at z ≃ 3.3, since they would have L_bol which is either similar (linear growth) or much higher than (exponential growth) the sources observed by S04 and N07 at z ≃ 2.4 and ≃3.3. From these two simplistic growth scenarios, we conclude that the fast growth of the z ≃ 4.8 sources must either experience a shut down before z ≃ 3.3, or accrete in several short episodes, with duty cycles which are much smaller than unity.

To further constrain the evolution of the observed SMBHs, we ran a series of calculations with different duty cycles for each of the two evolutionary scenarios. In each calculation, we assembled the distribution of calculated M_BH at exactly z = 3.3 and z = 2.4. Several of these distributions are shown in Figure 7. For the fast, constant L/L_Edd scenario, the only calculated distributions which resemble the observed distribution of M_BH at z ≃ 3.3 are those with duty cycles in the range 7.5%–12.5%. The observed distribution at z ≃ 2.4, on the other hand, can only be achieved by duty cycles of 5%–7.5%. For the constant L_bol scenario, the z ≃ 3.3 distribution can be matched by assuming a duty cycle in the range of 15%–25%, while the z ≃ 2.4 distributions can be partially explained by assuming duty cycles of ∼5%–25%. The ranges of duty cycles which account for the observed distribution of L/L_Edd are somewhat different: 10%–15% for the L/L_Edd distribution at z ≃ 3.3 and 5%–10% for the one at z ≃ 2.4. All the above calculations assumed η = 0.1. Naturally, lower (higher) radiative efficiencies will require shorter (longer) duty cycles to reproduce the same calculated CDFs at z ≃ 2.4 and ≃3.3. This means that for any assumed η in the range 0.05 ⩽ η ⩽ 0.3 the aforementioned "acceptable" duty cycles can change by up to a factor of ∼2, where the exact factor scales as η/(1 − η). We note that Figure 7 suggests that, in both evolution scenarios, the more massive BHs at z ≃ 2.4 and ≃3.3 seem to have grown at higher duty cycles than the less massive ones. Alternatively, this can be interpreted as an increase in radiative efficiency with increasing resultant M_BH at z ≃ 2.4 and ≃3.3. In particular, the observed distributions of M_BH and L/L_Edd at z ≃ 2.4 are much more complex than the calculated ones and no single, fixed duty cycle can account for the shape of the observed distributions. The reason for this apparent discrepancy might also be the way the z ≃ 2.4 sources were selected in the S04 and N07 studies.

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To illustrate how the above duty cycles facilitate an evolutionary connection between the three samples, we present in Figure 8 the evolutionary tracks of the z ≃ 4.8 sample, similar to Figure 6, but this time assuming duty cycles of 10% and 20%, for the constant L/L_Edd and constant L_bol scenarios, respectively, and assuming again η = 0.1. We also verified that evolving the M_BH of the z ≃ 2.4 and ≃3.3 sources backwards, under the assumption of constant L/L_Edd and a duty cycle of 10% results in a distribution of M_BH which is not critically different than the one directly observed at z ≃ 4.8. However, we again find that the few highest M_BH sources at z ≃ 2.4 probably require higher duty cycles.

We conclude that all the observed measurements of M_BH, L/L_Edd, and L_bol are consistent with duty cycles of about 10% –20%. Considerably longer duty cycles can be consistent with observations only if we assume an extremely high radiative efficiency (η ≃ 0.3) for all the z ≃ 4.8 sources. Such low duty cycles are in good agreement with the models presented in Shankar et al. (2009), which are able to reproduce the growth of a similar population of SMBHs, in terms of redshift and range of M_BH (see their Figure 7). Our constrains on duty cycles are, however, in contrast with several other studies that suggest that the duty cycle at high redshift should reach unity, based on the clustering of high-redshift AGNs (e.g., White et al. 2008; Wyithe & Loeb 2009; Bonoli et al. 2010; and references therein).

All this leads us to suggest that for the most massive z>2 type-I AGNs, an epoch of fast SMBH growth took place before z ∼ 4, which we partially observe at z ≃ 4.8. This epoch is efficient enough to produce the very massive (log(M_BH/M_☉) ≳ 10) BHs observed at z ≃ 2.4. The fast growth slows down before z ≃ 3.3, and the sources observed at z ≃ 3.3 show much lower L/L_Edd. In addition, there is no significant rise in M_BH between z ≃ 3.3 and z ≃ 2.4. The growth of M_BH during this epoch can only be explained by assuming that mass accretion proceeded in short episodes, lasting an order 10%–20% of the total period. This translates to accretion episodes which cumulatively last ∼70–140 Myr (∼150–300 Myr) over the period between z ≃ 4.8 and z ≃ 3.3 (z ≃ 4.8 and z ≃ 2.4), respectively. Another possibility is that the z ≃ 4.8 SMBHs have experienced an even faster shut down, and so at z ≃ 3.3 their inactive relics have masses that do not differ significantly from those we observe at z ≃ 4.8.

Major mergers between massive, gas-rich galaxies are capable of supplying large amounts of cold gas directly to the innermost regions to be accreted by the central SMBHs. Detailed simulations of such mergers suggest that these events can fuel significant BH growth over periods of ∼1 Gyr. The mass accretion rate and source luminosity during such mergers may vary on timescales of $\rm few \times 10\, {\rm Myr}$ (Di Matteo et al. 2005; Hopkins et al. 2006; Hopkins & Hernquist 2009). According to Hopkins et al. (2006), AGNs would appear to have L_bol ≳ 2.7 × 10⁴⁶ erg s⁻¹ (i.e., matching the range we observe at z ≃ 4.8) for typically ≲100 Myr. Numerical studies suggest that massive dark matter halos may undergo more than one major merger per Gyr at z ≃ 4.8 (e.g., Genel et al. 2009 and references therein). Thus, during the ∼680 Myr between z ≃ 4.8 and z ≃ 3.3, the hosts of the z ≃ 4.8 SMBHs may have undergone a single merger, during which the period of significant accretion by the SMBHs would last for ≲100 Myr. This is in good agreement with the duty cycles of 10%–20% we find here, which correspond to 70–140 Myr. The decline in activity observed for the most massive BHs at z ≃ 3.3 and z ≃ 2.4 may then be associated with the decline in the major merger rate, which drops by a factor of ∼4 between z ≃ 4.8 and z ≃ 2.4 (Genel et al. 2009). However, it is likely that the hosts of the z ≃ 2.4 SMBHs have experienced an additional major merger during the ∼790 Myr between z ≃ 3.3 and z ≃ 2.4, especially if these SMBHs reside in the more massive dark matter halos. In such a scenario, the largest BHs at z ≃ 2.4 may have gathered their high mass during more efficient (i.e., higher duty cycle) accretion episodes. This requirement of higher duty cycles for higher M_BH sources is also reflected in our analysis (see upper right panel of Figure 7). If major mergers are indeed the main drivers of SMBH accretion history at z ∼ 2–5, the results presented here predict that the host galaxies of the z ≃ 4.8 sources would be found in the early stages of major mergers, while the hosts of z ≃ 2.4 and ≃3.3 sources would appear to be in either later stages, or in merging systems which have a lower mass ratio.

There are several other mechanisms to make large amounts of cold gas available for BH accretion. Most of these might also trigger intense star formation (SF) activity in the hosts of SMBHs. Indeed, many high-luminosity AGNs show evidence for intense SF (e.g., Netzer 2009b; Lutz et al. 2010, and references therein). New observations by Herschel suggest that SF in the hosts of the most luminous AGN peaks at z ∼ 3 and quickly decreases at later epochs (Serjeant et al. 2010). If correct, it will indicate that the amount of gas available for both BH accretion and SF has depleted by z ∼ 2–3, consistent with the decrease in SMBH accretion activity we find here. Future Herschel and ALMA observations of the hosts of the z ≃ 4.8 AGNs may be able to reveal the presence of such SF activity, and perhaps the amount and dynamical state of the cold gas. This will enable a better understanding of the galaxy-scale processes which drive the accretion history of the fast-growing z ≃ 4.8 SMBHs.