Highly accreting quasars: sample definition and possible cosmological implications

Abstract

We propose a method to identify quasars radiating closest to the Eddington limit, defining primary and secondary selection criteria in the optical, UV and X-ray spectral range based on the 4D eigenvector 1 formalism. We then show that it is possible to derive a redshift-independent estimate of luminosity for extreme Eddington ratio sources. Using preliminary samples of these sources in three redshift intervals (as well as two mock samples), we test a range of cosmological models. Results are consistent with concordance cosmology but the data are insufficient for deriving strong constraints. Mock samples indicate that application of the method proposed in this paper using dedicated observations would allow us to set stringent limits on Ω_M and significant constraints on Ω_Λ.

1 INTRODUCTION

The concordance cosmology model (Spergel et al. 2003; Riess et al. 2009; Komatsu et al. 2011) favours a flat Universe (Ω_M+ Ω_Λ = 1) and significant energy density associated with a cosmological constant (Ω_Λ = 0.72). Inference of a significantly non-zero Ω_Λ rests on the use of supernovae as standard candles (Perlmutter et al. 1997, 1999) and studies of the cosmic microwave background radiation (e.g. Tegmark et al. 2004). The most recent Wilkinson Microwave Anisotropy Probe (WMAP) results indicate that the baryonic acoustic oscillation scale, the Hubble constant and the densities are determined to a precision of ≈1.5 per cent (Hinshaw et al. 2013). Even if the large-scale distribution of galaxies (Schlegel et al. 2011; Nuza et al. 2013) and galaxy clusters (e.g. Allen, Evrard & Mantz 2011) provide additional constraints, it is urgent that independent lines of investigation are devised to test these results. It remains important because Ω_M is not very tightly constrained by supernova surveys (Conley et al. 2011): Ω_M ≲ 0.5 at 2σ confidence level from supernovae mainly at z ≲ 1.5 (Campbell et al. 2013). Little information on Ω_M has been extracted from the redshift range 1 ≲ z ≲ 3, where Ω_M is most strongly affecting the metric. In addition, recently announced cosmological parameter values from the Planck-only best-fitting six-parameter Λ-cold dark matter model differ from the previous WMAP estimate, and yield Ω_Λ = 0.69 ± 0.01 (Planck Collaboration et al. 2013), significantly different from Ω_Λ = 0.72, Ω_M = 0.28 of the concordance cosmology adopted in the past few years (Hinshaw et al. 2009).

It is perhaps appropriate that we reconsider the cosmological utility of quasars at the 15th anniversary of their discovery (e.g. D'Onofrio, Marziani & Sulentic 2012). It is well known that use of quasar properties for independent measurement of cosmological parameters is fraught with difficulties. Quasars show properties that make them potential cosmological probes (see e.g. Bartelmann et al. 2009): they are plentiful, very luminous and detected at early cosmic epochs (currently out to z ≈ 7; Mortlock et al. 2011). The downside is that they show a more than 6 dex spread in luminosity and are also anisotropic radiators. Quasars are thought to be the observational manifestation of accretion on to supermassive black holes. Accretion phenomena in the Universe show a scale invariance with respect to mass and, indeed, we observe similar quasar spectra over the entire luminosity/mass range. It is not surprising that pre-1990 quasar research tacitly assumed that high- and low-luminosity active galactic nuclei (AGNs) were spectroscopically similar as well. The key to possible cosmological utility lies in realizing that quasars show a wide dispersion in observational properties which is a reflection of the source Eddington ratio.

1.1 Quasar systematics

The goal of systematizing observational properties of quasars has made considerable progress in the past 20 years and this makes possible discrimination of sources by L/L_Edd. An important first step involved a principal component analysis of line profile measures from high S/N spectra of 87 PG quasars (Boroson & Green 1992). That study revealed that eigenvector 1 (E1) correlates including a trend of increasing optical Fe ii emission strength with decreasing FWHM Hβ and peak intensity of [O iii] λ5007. Source luminosity was found to be part of the second eigenvector and is therefore not directly correlated with the eigenvector 1 parameters. A second step in quasar systematization attempted to identify more key line and continuum diagnostic measures. 4D eigenvector 1 (4DE1; Sulentic, Marziani & Dultzin-Hacyan 2000a) included two E1 broad-line measures: (1) full width half-maximum of broad Hβ (FWHM Hβ) and (2) optical Fe ii strength defined by the equivalent width W or by the intensity ratio |$R_{\rm Fe\,{\small II}}$| = W(Fe iiλ4570)/W(Hβ) ≈ I(Fe iiλ4570)/I(Hβ), where Fe iiλ4570 indicates the blend of Fe ii emission between 4434 and 4684 Å. Fig. 1 shows the optical plane of 4DE1 as defined by the 470 brightest quasars from SDSS DR5. Domain space in the figure is binned in such a way that all sources within a bin are the same within measurement uncertainties, and can be assigned a well-defined spectral type (Sulentic et al. 2002; Zamfir et al. 2010). The majority of quasars occupy bins A2 and B1 with tails extending towards bins with sources showing stronger Fe ii emission or broader Hβ profiles. Currently eight bins in intervals of FWHM = 4000 km s⁻¹ and |$R_{\rm Fe\,{\small II}}$| = 0.5 are needed to fully map source occupation. 4DE1 added two additional principal parameters: (3) a measure discussed in Wang, Brinkmann & Bergeron (1996) of the soft X-ray photon index (Γ_soft) and (4) a measure of the C iv λ1549 broad-line profile shift (at half-maximum; Sulentic et al. 2007). Points of departure from BG92 involve: (a) removal of [O iii] λ5007 measures as 4DE1 correlates, (b) consideration of differences in parameter space occupation between radio-quiet (RQ) and radio-loud (RL) sources (Sulentic et al. 2003; Zamfir, Sulentic & Marziani 2008) as well as (c) division of sources into two populations (A and B) designed to emphasize source spectroscopic differences.

Population (Pop.) A sources show FWHM Hβ < 4000 km s⁻¹, stronger |$R_{\rm Fe\,{\small II}}$|⁠, a soft X-ray excess and C iv λ1549 blueshift/asymmetry. Pop. A includes sources often called Narrow line Seyfert 1s (NLSy1s). The Hβ broad-line profile shows an unshifted Lorentzian shape (Véron-Cetty, Véron & Gonçalves 2001). Pop. B sources show FWHM Hβ > 4000 km s⁻¹ with weaker |$R_{\rm Fe\,{\small II}}$|⁠, no soft X-ray excess and usually no C iv λ1549 blueshift. Most RL sources are found in the Pop. B domain of Fig. 1. Pop. B sources usually require a double Gaussian model (broad unshifted + very broad redshifted components) to describe Hβ. Labels on the bins shown in Fig. 1 identify regions of occupation for the two source populations following the spectral type classification of Sulentic et al. (2002). Other important 4DE1 correlates exist. The present 4DE1 parameters represent those for which large numbers of reasonably accurate measures are available and for which clear correlations can be seen.

4DE1 parameters measure: FWHM Hβ – the dispersion in low-ionization broad-line region (BLR) gas velocity (it is the virial estimator of choice at low-z), |$R_{\rm Fe\,{\small II}}$| – the relative strengths of Fe ii and Hβ emission – likely driven by density n_H, ionization and metallicity, Γ_soft – the strength of a soft X-ray excess viewed as a thermal signature of accretion (Shields 1978; Malkan & Sargent 1982) and C iv λ1549 shift – the amplitude of systemic radial motions in high-ionization BLR gas, possibly due to an accretion disc wind (Konigl & Kartje 1994; Proga 2003; Königl 2006). If we ask what might drive the source distribution in Fig. 1 (or any of the other planes of 4DE1), the answer is most likely Eddington ratio. The idea that Eddington ratio drives 4DE1 goes back to the first E1 study (Boroson & Green 1992) and has received considerable support in the past 20 years (e.g. Boroson 2002; Marziani et al. 2003c; Baskin & Laor 2004; Grupe 2004; Yip et al. 2004; Ai et al. 2010; Wang et al. 2011; Matsuoka 2012; Xu et al. 2012). Black hole mass, and orientation are sources of scatter in the 4DE1 sequence. The value L/L_Edd ≈ 0.2 ± 0.1 corresponds to the boundary between Pop. A and B sources (for a black hole mass log  M_BH ∼ 8.0; Marziani et al. 2001, 2003c), and may be ultimately related to a transition between a geometrically thin, optically thick disc and an advection dominated, ‘slim’ disc (e.g. Różańska & Czerny 2000; Collin et al. 2002; Chen & Wang 2004). Identification of the most extreme accretors (potentially ‘Eddington standard candles’) offers the best hope towards finding a cosmologically useful sample of quasars.

1.2 Finding ‘Eddington standard candles’

Any attempt to use quasars as redshift-independent distance estimators must be tied to identification of a special class with some well-defined observational properties. These properties should be chosen to provide an easy link to a physical parameter related to source luminosity. If L/L_Edd is known, then standard assumptions can lead to a z-independent estimate of source luminosity since source luminosity estimation is connected to estimation of M_BH (Section 3).

Empirical studies show that the L/L_Edd distribution truncates near L/L_Edd ≈ 1 (e.g. Woo & Urry 2002; Shen et al. 2011). We observe a few super-Eddington outliers but their masses have likely been significantly underestimated due to a special line-of-sight orientation (e.g. sources oriented face-on where virial motions make little or no contribution to line width Marziani et al. 2003c and Marziani & Sulentic 2012). The range of L/L_Edd is 0.01 ≲ L/L_Edd ≲ 1 for luminous Seyfert 1 nuclei and quasars (Woo & Urry 2002; Marziani et al. 2003c; Steinhardt & Elvis 2010; Shen & Kelly 2012) with a much broader range if low-luminosity sources (of less interest for the present study) are included. The flux-limited distribution of L/L_Edd is subject to a strong, mass-dependent Malmquist bias (Shen & Kelly 2012) that may explain claims of a significantly narrower range. The minimum detectable L/L_Edd for a limiting magnitude m_{B, lim} can be written as |$L/L_{\rm {Edd,min}} \propto \frac{1}{M_{\rm {BH}}} 10^{-0.4 m_\rm{B,lim}} f(z)^2 (1+z)^{(1-a)}$|⁠, where f(z) is the redshift function appearing in the comoving distance definition, and a is the visual continuum spectral index. The above expression shows that only lower L/L_Edd values are lost at increasing redshift and that the effect is M_BH dependent.

A physical motivation underlies our search for sources radiating at extreme Eddington ratio. When accretion becomes super-Eddington, emitted radiation is advected towards the black hole, so that the source luminosity tends to saturate if accretion rate |$\dot{m} \gg \dot{m}_\rm{Edd}$|⁠, where |$\dot{m}_\rm{Edd} = L_\rm{Edd}/ \eta c^2$| is the Eddington accretion rate for fixed efficiency. The parameter η can be assumed |$\approx \frac{1}{6}$|⁠, as for α accretion discs (e.g. Shakura & Sunyaev 1973), or equal 1 (Mineshige et al. 2000). Radiative efficiency is expected to decrease with increasing Eddington ratio and luminosity to increase with log accretion rate (Abramowicz et al. 1988; Szuszkiewicz, Malkan & Abramowicz 1996; Collin & Kawaguchi 2004; Kawaguchi et al. 2004). Current models indicate a saturation value of a few times the Eddington luminosity for the bolometric luminosity (Mineshige et al. 2000; Watarai et al. 2000). The observed L/L_Edd distribution of high Eddington ratio candidates considered in this paper is consistent with a limiting L/L_Edd→ 1 (Section 2.3), as found in earlier empirical (e.g. Marziani et al. 2003c).

The primary goal of this paper is to find sources radiating near the limit L/L_Edd using emission line intensity ratios. These sources should be the quasars most easily found in flux-limited surveys, i.e. the bias described above will only result in selective loss of quasars radiating at lower Eddington ratios for fixed M_BH. We suggest that the 4DE1 formalism provides a set of parameters currently best suited to identify extreme Eddington radiators (Section 2). A second goal is to propose a method to for using high L/L_Edd quasars as redshift-independent luminosity estimators where the constant parameter is Eddington ratio rather than luminosity (Section 3). We report explorative calculations and preliminary results (Section 4) and discuss possible improvements (Section 5) along with limits and uncertainties (Section 6).

2 SAMPLE SELECTION AND MEASUREMENTS

2.1 I Zw 1 as a prototype of highly accreting quasars

The much studied NLSy1 source I Zw 1 is considered to be a low-z (z ≈ 0.0605) prototype of quasars radiating close to the Eddington limit. It is located in bin A3 of Fig. 1. The 4DE1 properties of I Zw 1 are as follows:

• FWHM Hβ = 1200 ± 50 km s⁻¹ (Boroson & Green 1992);

• |$R_{\rm Fe\,{\small II}}$| = 1.3 ± 0.1 (Boroson & Green 1992);

• C iv λ1549 blueshift at FWHM (relative to rest frame) Δ v_r= −1670 ± 100 km s⁻¹ (Sulentic et al. 2007);

• soft-X photon index Γ_soft = 3.050 ± 0.014 (Wang et al. 1996).

All 4DE1 parameters for I Zw 1 are extreme making it a candidate for the sources we are seeking. Conventional M_BH and L/L_Edd estimates for this source yield log  M_BH≈ 7.3–7.5 in solar units and log L/L_Edd≈ −0.13 to +0.10 (Vestergaard & Peterson 2006; Assef et al. 2011; Negrete et al. 2012; Trakhtenbrot & Netzer 2012).

If we consider the average M_BH of Vestergaard & Peterson (2006) and Trakhtenbrot & Netzer (2012) adopting two different bolometric corrections (L = 10 λL_λ(5100) and the |$\text{B.C.}$| of Nemmen & Brotherton 2010) we obtain log  L/L_Edd≈−0.11 ± 0.17, consistent with unity.

We need a statistically useful sample of extreme sources for any attempt at cosmological application. It is premature to discuss here all the corrections that must be applied; however, we seek sources similar to, or more extreme than I Zw 1.

FWHM Hβ will be the least useful 4DE1 parameter because all Pop. A sources – 50 per cent of the low-z quasar population – show FWHM Hβ < 4000 km s⁻¹, and only a small fraction of this population is likely to involve extreme Eddington radiators based upon current L/L_Edd estimates (Marziani et al. 2003c). This is also true for higher z sources. In addition, FWHM of Hβ increases slowly but systematically with L, and there is a minimum FWHM possible if gas is moving virially and L/L_Edd ≤ 1 (Marziani et al. 2009) that is ≈ 4000 km s⁻¹ at log L ∼ 48 [erg s⁻¹]. Currently only very high L sources can be studied with any accuracy at high-z. Therefore, we will relax any FWHM limit introduced in the definition of spectral types in low-redshift samples (z ≲ 0.7). This concerns only spectral types A3 and A4 for which there is no danger of confusion with broader sources, as shown by Fig. 1. The spectral types based only on |$R_{\rm Fe\,{\small II}}$| will be indicated with A3^m (1 ≤ |$R_{\rm Fe\,{\small II}}$| < 1.5) and A4^m (1.5 ≤ |$R_{\rm Fe\,{\small II}}$| < 2.0), or together, with ‘xA’ (⁠|$R_{\rm Fe\,{\small II}}$| ≥ 1.0).

At this stage, we adopt |$R_{\rm Fe\,{\small II}}$|>1.0 as a primary selector of low-redshift extreme Eddington radiators which are by definition Pop. A sources. Clearly significant numbers of such candidates can be identified, ≈10 per cent of all low-z quasars (cf. Zamfir et al. 2010; Shen et al. 2011). In a recent analysis of Mg ii λ2800 line profiles for an SDSS-based sample of 680 quasars (z in the range 0.4–0.75), we found n = 58 candidate A3^m/A4^m sources. They are high-confidence candidates because spectral S/N is high enough to be certain about the extreme Fe ii emission. Γ_soft and C iv λ1549 measures exist for almost none of these sources at this time but most PG candidates at lower redshift (z < 0.5) show extreme C iv λ1549 and soft X-ray properties (Boroson & Green 1992; Wang et al. 1996; Sulentic et al. 2007).

The left-hand panel of Fig. 2 shows profile model fitting to the median bin A3 spectrum (sample of Marziani et al. 2013a). The |$R_{\rm Fe\,{\small II}}$| value is consistent with the one of I Zw 1 reported in Negrete et al. (2012). Details of model fitting procedures can be found in Section 2.4 and Marziani et al. (2010).

2.2 Selecting high-z samples of Eddington candles

Without IR spectra we lose the |$R_{\rm Fe\,{\small II}}$| selector employed at z < 1.0. We must take advantage of quasar-abundant rest-frame UV spectra. The first choice would be to use measures of C iv λ1549 shift, asymmetry and equivalent width because extreme sources tend to show low EW C iv λ1549 profiles that are blueshifted and blue asymmetric. A measure of C iv λ1549 shift was chosen as a 4DE1 parameter because it is the best measure of extreme C iv λ1549 properties. At low-z, we rely on satellite observations where the Hubble Space Telescope (HST) archive provides the highest S/N and resolution UV spectra for our low-z candidates. Data exist in the HST archive for about 130 low-z sources of which n = 11 are bin A3 and A4 sources (⁠|$R_{\rm Fe\,{\small II}}$| > 1.0). 10 of the 11 Fe ii extreme sources show a significant C iv λ1549 blueshift confirming its utility as an extreme Eddington selector (Sulentic et al. 2007). The majority of Pop. A sources with |$R_{\rm Fe\,{\small II}}$| < 1.0 and Pop. B sources show a smaller blueshift and in many cases zero shift or a redshift (n = 3 with shift > 10³ km s⁻¹).

However, we do not know the geometry of the outflows implied by the profile blueshift. We may miss a considerable number of candidates where orientation diminishes the blueshift. There is also the issue of whether C iv λ1549 blueshifts are quasi-ubiquitous in high-z quasars (Richards et al. 2011). If C iv λ1549 blueshifts are more common at high-z, then they may not be useful as a clearcut selector. A further problem is that a selector involving a line shift requires a more accurate estimate of the quasar rest frame which is often missing in high-redshift sources.

We propose an alternative high-z selector involving the 1900 emission line blend of Al iii λ1860, Si iii] λ1892 and C iii]λ1909. The blend involving these lines constrains the physical conditions in the broad-line emitting gas much the same way as measures of very strong optical Fe ii. The definition of this selector comes from three sources: (1) measures from a composite spectrum of HST archival data for 10 bin A3 sources (right-hand panel of Fig. 2; Bachev et al. 2004), (2) line ratio measures for I Zw 1 (our bin A3 prototype) from a high S/N and resolution HST spectrum (Laor et al. 1997; Negrete et al. 2012) and (3) measures of SDSS1201+0116 (a high-z bin A4 analogue of I Zw 1 from a high S/N VLT spectrum; Negrete et al. 2012).

Bachev et al. (2004) noted that the Si iii] λ1892/C iii]λ1909 intensity ratio decreased monotonically along the 4DE1 sequence by a factor ≈4 from bin A3 (≈0.8) to bin B1+ (≈0.2). This implies a selection constraint for extreme sources Si iii] λ1892/C iii]λ1909 > 0.6-0.7. Measures using very high S/N spectra and on the 1900 blend of the A3 composite spectrum (Fig. 2) yield a selection criterion based on two related ratios:

• Al iii λ1860/Si iii] λ1892/ ≥ 0.5 and

• Si iii] λ1892/C iii] λ1909 ≥1.0.

Both our low- and high-redshift selectors are based on more than empiricism because they effectively constrain the range of physical conditions in the line-emitting region of the extreme quasars (Baldwin et al. 1996, 2004; Marziani et al. 2010; Negrete et al. 2012). The extreme Fe ii emission from these sources has been discussed in terms of the densest broad-line emitting region (n_H∼10¹² cm⁻³; Negrete et al. 2012) and extreme metallicity again plausibly connected with high accretion rates.

2.3 Identification of preliminary samples

Equipped with low- and high-redshift selection criteria, we identify three samples of extreme Eddington sources over the range z = 0.4–3.0.

• Sample 1: 58 sources with Hβ spectral coverage from an SDSS DR8 sample of 680 quasars in the range z ≈ 0.4–0.75 (Marziani et al. 2013b). Their spectral bins are A3 and A4. Table 1 lists the 30 and 13 sources in bin A3 and A4 with S/N ≥ 15 at 5100 Å. Formats for Tables 1 and 2 are: ID, redshift, rest-frame flux λf_λ, S/N of spectrum, FWHM of virial estimator used, FWHM uncertainty and sample ID. FWHM uncertainties for sample 1 Hβ profiles were assumed to be 10 per cent.

• Sample 2: 7 sources from a sample of 52 Hamburg-ESO quasars (Marziani et al. 2009) in the range z = 1.0–2.5 (all but two 1 ≲ z ≲ 1.6) with high S/N spectra of the Hβ region, obtained with the IR spectrometer ISAAC at the Very Large Telescope (VLT) of the European Southern Observatory (ESO). They all satisfy the criterion |$R_{\rm Fe\,{\small II}}$| ≥ 1.0 within observational uncertainty; 3 of them are however borderline sources with |$R_{\rm Fe\,{\small II}}$| ≈ 1.0. The ISAAC sources are meant to cover a redshift range where Hβ observations are very sparse.

• Sample 3: 63 SDSS sources (additional candidates were identified but require higher S/N data) are listed in Table 2. We extracted spectra for ≈3000 sources from SDSS DR6 with coverage of the 1900 blend (2.0 ≲ z ≲ 2.6). Selected sources show emission line ratios Al iii λ1860/Si iii] λ1892 and C iii]λ1909/Si iii] λ1892 satisfying our criterion. A further restriction was imposed by excluding sources with low S/N (<15) spectra that might have biased FWHM measures as clearly shown by Shen et al. (2008). This yielded a subsample 3 of 42 sources with S/N ≥ 15 on the continuum at 1800 Å. FWHM uncertainties were estimated from simulated spectra as a function of Al iii λ1860 EW, FWHM and S/N.

Samples 1 and 2 are based on flux-limited samples analysed in the studies cited above. Sample 3 has no pretence of completeness: the lower fraction of identified xA sources is due to the low S/N of most spectra.

2.4 Emission-line measurements

Line ratios yield 4DE1 bin assignments. In the optical, |$R_{\rm Fe\,{\small II}}$| is retrieved from the intensity of the Lorentzian profile representing Hβ and from Fe ii_opt flux in the integrated over the wavelength range 4434–4684 Å (Boroson & Green 1992; Marziani et al. 2003b). In the UV, it is especially the Al iii λ1860/Si iii] λ1892 ratio that, with or without C iv λ1549 measures, provides a selector for high- z extreme sources. Fig. 3 shows model fits to the 1900 blends for two high-z extreme source candidates.

Measurements were made using a non-linear multicomponent fitting routine that seeks χ² minimization between observed and model spectra, i.e. specfit incorporated into iraf (Kriss 1994). The procedure allows for simultaneous fit of continuum, blended iron emission, and all narrow and broad lines identified in the spectral range under scrutiny. Singly and doubly ionized iron emission (the latter present only in the UV) were modelled with the template method. For the optical Fe ii emission, we considered the semi-empirical template used by Marziani et al. (2009). This template was obtained from a high-resolution spectrum of I Zw 1, with a model of Fe ii emission computed by a photoionization code in the wavelength range underlying Hβ. For the UV Fe ii and Fe iii emission, we considered templates provided by Brühweiler & Verner (2008) and Vestergaard & Wilkes (2001), respectively. The fitting routine scaled and broadened the original templates to reproduce the observed emission. Broad Hβ and the 1900 Å blend lines were fitted assuming a Lorentzian function. This assumption follows from analysis of large samples of Pop. A sources (Véron-Cetty et al. 2001; Marziani et al. 2003c; Zamfir et al. 2010; Marziani et al. 2013a): Negrete et al. (2013) verified that Hβ, Al iii λ1860 and Si iii] λ1892 can be fitted with a Lorentzian function whose width is the same for the three lines in Pop. A sources studied with reverberation mapping. The broad Hβ line profile is often asymmetric towards the blue. The line profile has been modelled adding to the Lorentzian component an additional, shifted line component associated with outflows (cf. Marziani et al. 2013a). Narrow lines ( [O iii]λλ4959,5007 and Hβ_NC) were fitted with Gaussian functions.

The uncertainty associated with FWHM Hβ was 10 per cent for sample 1, and as reported in Marziani et al. (2009) for the sample 2 (usually ≲ 10 per cent). For the UV virial estimator, i.e. in cases of lower S/N, synthetic blend profiles were computed as a function of S/N, equivalent width, and individual line width for typical values in our sample (e.g. S/N = 14,22; W(Al iii λ1860) = 5,10 Å; FWHM = 3000,4000 km s⁻¹). Statistical measurement errors on each source FWHM were assigned to the value computed for the synthetic case with closest lower S/N, closest lower equivalent width and closest FWHM.

2.5 Consistency of criteria

If optical and UV rest-frame range are both covered, it becomes possible to test that the condition |$R_{\rm Fe\,{\small II}}$| ≳ 1 is fully interchangeable with Al iii λ1860/Si iii] λ1892 ≳ 0.5 and Si iii] λ1892/C iii]λ1909 ≳ 1 (i.e. |$R_{\rm Fe\,{\small II}}$| ≳ 1 ⇔ Al iii λ1860/Si iii] λ1892 ≳ 0.5 and Si iii] λ1892/C iii]λ1909 ≳ 1). If only xA sources are considered, at low-z not many spectra covering the 1900 blend are available in the Mikulski Archive for Space Telescopes (MAST) archive. At high-z, few objects have Hβ covered in IR windows with adequate resolution spectroscopy. At present, data available to us encompass six low-z (including I Zw 1, and excluding all RL sources) and three HE sources of sample 2.

Low-z

We measured with specfit both UV ratios on ≈ 100 sources of Bachev et al. (2004) for which the 1900 blend is covered from HST/FOS observations, as a test of consistency for the two criteria. Among them, six A3 sources (1H 0707–495, HE 0132–4313, I Zw 1, PG 1259+593, PG1415+451, PG1444+407) are included. Spectral type assignments and references to the used data can be retrieved from Sulentic et al. (2007).

All Pop. B (56) sources save one¹ show C iii]λ1909/Si iii] λ1892 ≳ 2.0 and are therefore immediately excluded by the selection criteria; many sources appear dominated by C iii]λ1909 emission, with C iii]λ1909/ Si iii] λ1892 ≫1. Fig. 4 shows the distribution of 37 Pop. A sources in the plane C iii]λ1909/Si iii] λ1892 versus Al iii λ1860/Si iii] λ1892, with special attention to sources that are ‘borderline’ and for which error bars are also shown. All type A3 objects fall at the border or within the shaded region defined by the conditions Al iii λ1860/Si ii λ1814 ≥ 0.5 and C iii]λ1909/Si iii] λ1892 ≤ 1. Sources falling close to the border show |$R_{\rm Fe\,{\small II}}$| ≈ 1, while the two sources with |$R_{\rm Fe\,{\small II}}$| >1 are well within the region. The only source (Mark 478) that is not formally of spectral type A3 and falls close to the area Al iii λ1860/Si ii λ1814 ≥ 0.5 C iii] λ1909/Si iii] λ1892 ≤ 1 is of type A2 with |$R_{\rm Fe\,{\small II}}$| ≈ 0.92. Fig. 4 confirms the consistency of the optical and UV criteria: no source of |$R_{\rm Fe\,{\small II}}$| < 0.9 falls in the domain C iii] λ1909/Si iii] λ1892 ≲ 1 and Al iii λ1860/Si iii] λ1892/ ≥ 0.5, and no source with |$R_{\rm Fe\,{\small II}}$| ≳ 0.9 falls, within the error, outside of it.

High-z

Three sources of sample 2 have newly obtained observations covering the C iv λ1549–C iii]λ1909 emission lines (Martínez-Carballo et al., in preparation): HE0122–3759 (log L ≈ 47.7 [erg s⁻¹]), HE0358–3959 (log L ≈ 47.3), HE1347– 2457 (log L ≈ 47.9). In all cases, the condition on the UV lines is satisfied.

The prototype source I Zw1 has log L ≈ 45.6 [erg s⁻¹]; the lowest luminosity and nearest source is 1H 0707–495 with log L ≈ 45.04 at z ≈ 0.04; the luminosity of the three PG sources is in the range |$\log L \approx 45.5\text{-}47.0$|⁠. The condition |$R_{\rm Fe\,{\small II}}$| ≥ 1 implied Al iii λ1860/Si iii] λ1892 ≥0.5 in all cases, and all individual spectra consistently showed the features typical of A3 sources (low C iv λ1549 equivalent width, large C iv λ1549 blueshift, etc.). Within the limit of the available data, the optical and UV criteria are consistent over a very large range in luminosity and redshift, 45 ≲ log L ≲ 48, 0.04 ≲ z ≲ 3.

2.6 The a posteriori distribution of Eddington ratios

An estimate of the spread associated with the L/L_Edd distribution comes from the a posteriori analysis of our sample. We computed M_BH following Assef et al. (2011) and applied the bolometric correction indicated by Richards et al. (2006, optical) and derived using the Mathews & Ferland (1987) continuum (UV, see discussion on B.C. later in this section). The resulting L/L_Edd distribution is shown in Fig. 5. The dispersion in the full sample is σ ≈ 0.13. This is consistent with current search techniques that allow us to identify sources radiating within ± 0.15 dex in log L/L_Edd (Steinhardt & Elvis 2010).² Inter-subsample systematic differences are smaller than the full sample dispersion. A4 Hβ sources of sample 1 (sample 1a in Table 1; cross-hatched histogram in Fig. 5) do not differ significantly from A3 sources. There is a systematic difference between the Hβ and Al iii λ1860 sample, by Δlog λ_Edd ≈ 0.07 < 1σ that is dependent on the assumed ratio of the |$\text{B.C.}$| at 1800 Å and 5100 Å (≈ 0.63). This can give to an important systematic effect discussed in Section 6.2.

Armed with selection criteria to isolate sources clustering around L/L_Edd ≈ 1 and having defined preliminary samples, we now discuss how we can derive luminosity information without prior redshift knowledge.

3 MEASURING LUMINOSITY FROM EMISSION LINE PROPERTIES

Equation (6) (hereafter the ‘virial’ luminosity equation) is formally valid for any L/L_Edd; the key issue in the practical use of equation (6) is to have a sample of sources tightly clustering around an average L/L_Edd (whose value does not need to be 1, or to be accurately known). At present, we can identify sources with λ_Edd → 1, but it is still possible that an eventual analysis may employ different spectral types representative of much different L/L_Edd average values. In practice, an approach followed in this paper has been to consider equation (6) in the form |$L \approx \mathcal {L}_{0} \delta v^{4}$|⁠, where |$\mathcal {L}_{0}$| has been set by the best guess of the quasar parameters with λ_Edd → 1. This will imply a value of H₀, and to ignore source-by-source diversity. A second possibility is to compute |$L \approx \mathcal {L}_{0}$| to yield the concordance value of H₀, especially for a sample of sources at very low-z (≲0.05) where Ω_M and Ω_Λ are insignificant in luminosity computations. Statistical and systematic errors will be discussed in Section 6.

4 PRELIMINARY RESULTS

4.1 Comparison of virial luminosity and luminosity derived from redshift

Using our three combined samples, we initially consider four cases: (1) ‘concordance cosmology’ (Ω_M + Ω_Λ = 1.0, Ω_M = 0.28, Ω_k = 0.0), (2) an ‘empirical’ open model (Ω_M = 0.1, Ω_Λ = 0.0, Ω_k = 0.0), (3) a matter-dominated Universe (Ω_M = 1.00, Ω_Λ = 0.0, Ω_k = 0.0) and (4) a Λ-dominated model Ω_M = 0.0, Ω_Λ = 1.0, Ω_k = 0.0). Expected bolometric luminosity differences as a function of redshift with respect to an empty Universe are shown in Fig. 6.

We compare two sets of luminosity values as a function of redshift: one derived from the redshift L (equation 8) and the other from the virial luminosity (equation 6). We assume |${\cal L}_0 \approx 1.16 \times 10^{45}$| erg s⁻¹ from the best guess of parameters entering equation (6), for the standard continuum of Mathews & Ferland (1987): κ ≈ 0.6, n_HU ≈ 10^9.6 cm⁻³, |$h\bar{\nu _\rm{i}} \approx 41$| eV and f_S ≈ 1.5 as recommended by Collin et al. (2006). Fig. 7 shows data points computed from the virial equation (blue circles) and values computed from the redshift in the case of concordance cosmology (H₀ = 70 km s⁻¹/Mpc⁻¹, Ω_M = 0.28, Ω_Λ = 0.72). Error bars are 1σ uncertainties from FWHM uncertainty estimates given in Section 2 and are reported in Tables 1 and 2 (Fig. 7). Residuals defined as Δ = log L(v) − log L are shown in the bottom panel.

We consider the conventional χ² computed from the squared Δs as a goodness of fit estimator. In principle, the residuals Δ(z, H₀, Ω_M, Ω_Λ) could be fitted with an analytical approximation of ζ(z, Ω_M, Ω_Λ) as complex as needed. Our preliminary data allow only a simple linear least-squares fit to the residuals: Δ(z) =a + b · z. The sought-for model requires a = b = 0. In the following analysis, we found expedient to consider the normalized average |$\bar{\Delta }/\sigma _{\bar{\Delta }}$| and the normalized slope b/σ_b. |$\bar{\Delta }/\sigma _{\bar{\Delta }}$| and b/σ_b are both t-distributed estimators that for our sample size can be considered normally distributed. They both provide statistical confidence limits |${\propto} 1/({\rm{rms}}/\sqrt{N-1})$|⁠, where N = 92. The normalized average and the slope estimator have been computed under the assumption that the rms scatter of our data is an estimate of uncertainty for individual measures (Press et al. 1992, chapter 15). The slope estimator is not a very tightly constraining parameter; however it has the considerable advantage to be fully independent on H₀, and to provide a straightforward representation of the systematic trends associated with Ω_M and Ω_Λ as function of z.

The rms value (≈0.365) is intrinsic to our data set and will probably not change much with larger samples unless FWHM measures with substantially higher accuracy, and a reduction of other sources of statistical errors, are obtained (Section 6). Given the redshift range of our data, a change in Ω_M gives rise to a change in slope of the residuals Δ(z) that strongly affects |$\bar{\Delta }$| (as can be glimpsed from Fig. 6).

4.2 Selected alternative cosmologies

Table 3 reports normalized average |$\bar{\Delta }/\sigma _{\bar{\Delta }}$|⁠, b, and χ² values for the selected cosmological models of Fig. 6, assuming fixed H₀= 70 km s⁻¹/Mpc⁻¹. Normalized |$\chi ^{2}/\chi ^{2}_\rm{min}$| values of 1.1, 1.41 and 1.79 limit acceptable fits at confidence levels of 1σ, 2σ and 3σ, respectively. Note that χ² and |$\bar{\Delta }/\sigma _{\bar{\Delta }}$| values are affected by H₀, while b is not dependent on H₀.

Extreme implausible cases such as a matter (Ω_M = 1) or Λ-dominated (Ω_Λ = 1) Universe are ruled out. The case with Ω_M = 1 (Ω_Λ = 1) will yield a large positive (negative) slope meaning that redshift based estimates are underluminous (overluminous) with respect to virial luminosities. The concordance case is favoured by our data set, with χ² ≈ 1.05 (Fig. 7). The ‘empirical’ model with Λ = 0 and Ω_M = 0.1 is not disfavoured by our data, with χ² ≈ 1.08 if H₀= 70 km s⁻¹/Mpc⁻¹. The two cases, concordance and ‘empirical’ Ω_M = 0.1 (along with the unrealistic case of a fully empty Universe) cannot be statistically distinguished since their slopes b are close to 0 and |$|b/\sigma _\rm{b}| \lesssim 1$|⁠.

Obtaining a solution close to the one of concordance cosmology is not a consequence of how the parameters entering equation (6) were obtained. Parameters related to continuum shape and ionizing photon flux were not derived assuming any particular luminosity. The assumed value of |${\cal L}_0$| depends from H₀ since the assumed L/L_Edd depends on H₀. However, effects of Ω_M and Ω_Λ are negligible since L/L_Edd was calibrated on a very low-z source, I Zw 1 (Fig. 6).

4.3 H₀, Ω_M and Ω_Λ in a flat Universe

Assuming a flat geometry (Ω_M + Ω_Λ = 1, Ω_k= 0), the normalized average is influenced by changes in H₀ and Ω_M (Ω_Λ is set by Ω_M). Fig. 8 shows χ² (top), |$\bar{\Delta }/\sigma _{\bar{\Delta }}$| (middle) as a function of H₀ and Ω_M, and |b/σ_b| as a function of Ω_M. The χ² contour lines trace four values of the ratio |$\chi ^{2}/\chi ^{2}_\rm{min}$|⁠: |$\chi ^{2} \approx \chi ^{2}_\rm{min}$|⁠, and three values meant to represent the 0.32, 0.05 and 0.003 probability interval considering that the probability of the |$\chi ^{2}/\chi ^{2}_\rm{min}$| ratio follows an F-distribution (Bevington 1969). The χ² diagram indicate that Ω_M is constrained within 0.05 and 0.8 if H₀ is not chosen a priori. The lines in the |$\bar{\Delta }/\sigma _{\bar{\Delta }}$| plot (middle panel of Fig. 8) identify a curved strip close to 0, and values 1, 2, 3. The lines of value 1,2,3 trace the corresponding 1,2,3 σ confidence levels. The bottom panel of Fig. 8 shows the behaviour of |b/σ_b| as a function of Ω_M. The minimum identifies the condition b = 0. Changing Ω_M will increase the slope that will become significantly different from 0 at 1 and 2 σ confidence levels when |b/σ_b| = 1 and 2. It is possible to define only –2, –1, and +1, +2σ uncertainties: Ω_M ≈ 0.19|$^{+0.16}_{-0.08}$|⁠, and ≈ 0.19|$^{+0.76}_{-0.15}$| at 1 and 2σ, respectively.

Once an H₀ value is assumed as indicated by |$\chi ^{2}_\rm{min}$| or b = 0, uncertainties on Ω_M are significantly reduced if the normalized average is used as a confidence limit estimator. equation (6) in the form |$L = {\cal L}_{0} \delta v^{4}_{1000}$|⁠, with |${\cal L}_{0} \approx 1.16 \times 10^{45}$| erg s⁻¹, implies b = 0 and a minimum χ² at H₀ ≈ 71 km s⁻¹/Mpc⁻¹, yielding Ω_M ≈ 0.19|$^{+0.07}_{-0.05}$| at 1σ, and 0.19|$^{+0.21}_{-0.12}$| at 3σ (middle panel of Fig. 8). Broader limits are inferred from the normalized χ² once H₀ is prefixed because χ² includes the uncertainty in H₀ that we may instead assume a priori. Uncertainties from χ², |b/σ_b| and |$\bar{\Delta }/\sigma _{\bar{\Delta }}$| reflect statistical errors only. Constraints are already significant although uncertainties are not as low as those derived from the combination of recent surveys (Hinshaw et al. 2013; Planck Collaboration et al. 2013).

4.4 Constraints on Ω_M and Ω_Λ

Rather loose constraints are also obtained where Ω_M and Ω_Λ are left free to vary once H₀ is assumed (H₀= 70 km s⁻¹/Mpc⁻¹). The top panel of shows the χ² behaviour in the plane Ω_M and Ω_Λ. Present data yield Ω_M ≈ 0.19|$^{+0.24}_{-0.19}$| and Ω_M ≈ 0.19|$^{+1.04}_{-0.19}$| at 1 and 3σ confidence level. Ω_Λ is unconstrained within the limits of Fig. 9.

5 PROSPECTS FOR IMPROVEMENT: ANALYSIS OF SYNTHETIC DATA SETS

The limits on Ω_M, and Ω_Λ in Fig. 9 are not strong compared to previous studies despite the fact that Ω_Λ and Ω_M are close to currently accepted ones. Our preliminary sample is too small and inhomogeneous. Extreme Eddington sources are estimated to include a sizeable minority of the quasar population – most likely ≈10 per cent. Significant improvements can come from an increase in sample size with high S/N observations, and by reducing the (large) statistical error in the data set analysed here (Section 6.1).

To quantify the improvement that can be expected from the reduction in statistical errors, synthetic data were created by adding Gaussian deviates to the luminosity computed for the concordance case (H₀= 70, km s⁻¹/Mpc⁻¹, Ω_M =0.28, Ω_Λ =0.72), assuming a uniform redshift distribution.³ In other words, artificial luminosity differences were made to scatter with a Gaussian distribution around the luminosities derived assuming concordance cosmology. This approach is meant to model sources of statistical error that may be present in the virial and z-based computations (i.e. FWHM uncertainty, errors in spectrophotometry and bolometric luminosity).

Fig. 10 (top-left panel) shows the residual as a function of redshift and reports average and dispersion for a mock sample of 200 sources with rms = 0.2. The top-right, lower-left and lower-right panels of Fig. 10 show the behaviours of χ², |b/σ_b| and |$\bar{\Delta }/\sigma _{\bar{\Delta }}$| respectively, computed assuming Ω_M+Ω_Λ = 1. Meaningful limits can be set in this hypothetical case. The |b/σ_b| estimator yields Ω_M |$\approx 0.30^{+0.20}_{-0.12}$| at a 2σ confidence level, with no assumption on H₀. If H₀ = 70 km s⁻¹/Mpc⁻¹, |$\bar{\Delta }/\sigma _{\bar{\Delta }}$| (top-right panel of Fig. 10) indicates Ω_M|$\approx 0.28^{+0.02}_{-0.02}$| (1σ).

The middle panel of Fig. 9, computed for unconstrained Ω_M + Ω_Λ, indicates Ω_M|$\approx 0.30^{+0.12}_{-0.09}$| at 1σ confidence level, with poor constraints on Ω_Λ. The χ² distribution for the mock samples was computed as for the real data, varying Ω_M and Ω_Λ with no constraints on their sum.

The upper-left panel of Fig. 11 (organized as Fig. 10 for the Ω_M + Ω_Λ = 1 case) shows a mock sample of 100 sources with rms ≈ 0.1 that is otherwise identical to the previous case. The |b/σ_b| estimator yields Ω_M|$\approx 0.28^{+0.13}_{-0.09}$| at a 2σ confidence level, with no assumption on H₀. If H₀ = 70 km s⁻¹/Mpc⁻¹, |$\bar{\Delta }/\sigma _{\bar{\Delta }}$| (top-right panel of Fig. 11) yields Ω_M with an uncertainty ±0.05 at a 3σ confidence level. From the χ² distribution (lower-right panel of Fig. 11), we derive Ω_M|$\approx 0.28^{+0.07}_{-0.06}$|⁠, and Ω_M|$\approx 0.28^{+0.005}_{-0.015}$| if H₀ = 70 km s⁻¹/Mpc⁻¹ and Ω_M+Ω_Λ = 1. The bottom panel of Fig. 9 (Ω_M + Ω_Λ unconstrained) indicates Ω_M|$\approx 0.28^{+0.04}_{-0.08}$| at 1σ confidence level, again with poor constraints on Ω_Λ.

The mock sample improvement over our real sample comes from the larger sample size N (a factor |$\sqrt{\frac{N}{92}}$|⁠), lower intrinsic rms (a factor 2–4), and the uniform distribution in redshift. A reduction of errors associated with the method is therefore tied to an increase in sample size since uncertainties of |$\bar{\Delta }$| and |b/σ_b| decrease with |$1/\sqrt{N}$|⁠. Achieving rms ≈ 0.1 would require careful consideration of several systematic effects summarized below, or the definition of a large sample (∼ 1000) of quasars (Section 6.1).

The value of Ω_Λ is affected by larger uncertainty than Ω_M, but significant constraints (for example, the verification that Ω_Λ > 0 at a 2σ confidence level) are within the reach. The weak constraints on Ω_Λ stem from (1) the large errors, (2) the redshift range. Mock sample sources are assumed to spread uniformly over the z range 0.1–3.0, but only z ≲ 1.2 is strongly affected by Ω_Λ (Fig. 6) so that only |$\frac{1}{3}$| of the sources are in the relevant range.

We will set more stringent limits on the requirements for an observational program after discussing sources of statistical and systematic uncertainty.

6 LIMITATIONS AND SOURCES OF ERROR/UNCERTAINTY

6.1 Statistical error budget

Estimation of the Ωs is affected by statistical errors that will be significantly reduced with larger samples and better source spectra. The residual distribution is affected by errors on the parameters entering into the virial equation (equation 6) and into the customary determination of the quasar luminosity from redshift (equation 8). In the following each source of statistical error is discussed separately, first for the virial equation and then for the z-based L determination. Column 2 of Table 4 reports the 1σ errors as estimated in the following paragraphs, propagating them quadratically to obtain an estimate of statistical error that should be compared with the rms derived from our sample.

6.1.1 Virial equation

6.1.1.1 Eddington ratio λ_Edd

– The L/L_Edd value of I Zw 1 is consistent with unity (Section 2), although the exact value depends on the normalization assumed for M_BH and on the bolometric correction. The L/L_Edd distribution of Fig. 5 shows σ ≈ 0.13. This dispersion value includes orientation effects. It is unlikely that a larger sample can be obtained with a significantly lower scatter unless scatter is reduced if part of the dispersion can be accounted for by systematic trends (Section 6.2).

6.1.1.2 Factor |$\kappa /\bar{\nu _{\rm i}}$| –

the product Un_H value entered in equation (6) comes from dedicated sets of photoionization simulations that assume Mathews & Ferland (1987) continuum which is thought to be appropriate for Pop. A sources. However, the values of κ, |$\bar{\nu _\rm{i}}$|⁠, and hence of factor |$\kappa /\bar{\nu _{\rm i}}$| depend on the shape of the assumed photoionizing continuum. Extreme sources are expected to converge towards large values of Γ_soft (e.g. Boller, Brandt & Fink 1996; Wang et al. 1996; Grupe et al. 1999; Sulentic et al. 2000a; Grupe et al. 2010; Ai et al. 2011). In order to investigate the effect of different continua, we considered a typical NLSy1 continuum as proposed by Grupe et al. (2010, Fig. 12). The average Γ_soft derived by Panessa et al. (2011) was assumed for energies larger than 20 keV. The dashed line shows a second ‘minimum’ continuum representative of sources with Γ_hard ≈ 2.2, computed on the basis of the observed dispersion in photon indexes up to 100 keV (Nikolajuk et al. 2004; Wang et al. 2013). The ‘standard’ Mathews & Ferland (1987) and (Korista et al. 1997) were also considered as suitable for Pop. A sources. The scatter in |$\log (\kappa /\bar{\nu _{\rm i}})$| derived from the values of the four continua is 0.033. We report this value in Table 4. A more appropriate range is probably between the NLSy1 continuum and the ‘minimum’ spectral energy distribution (SED) of Fig. 12: if so, the scatter will be reduced to 0.02. This prediction should be tested by an analysis of low-z sources for which X-ray data are available.

6.1.1.3 Factor n_HU –

the term n_HU is only slightly affected by the frequency redistribution of ionizing continua (within the continua assumed). However, log n_HU depends on the diagnostic ratios values. cloudy 13 simulations (Ferland et al. 2013) indicate a difference of Δlog n_HU ≈ 0.2 from ratio Al iii λ1860/Si iii] λ1892≈ 0.5 (A3^m) and Al iii λ1860/Si iii] λ1892≈ 1.0 (A4^m). Since we do not actually separate A3^m and A4^m sources, we consider the difference as a source of statistical error ±0.1 around an average value. A proper treatment of the difference as a systematic effect will require a larger sample (Section 6.2). The measures of Negrete et al. (2013) on high S/N spectra indicate that measurement errors can be reduced to Δlog n_HU ≈ 0.05.

6.1.1.4 The structure factor f_S –

the value of f_S is probably governed by λ_Edd (Netzer & Marziani 2010). An upper limit variance was estimated by considering sources observed in reverberation mapping for which M_BH has been also derived from the M_BH–bulge velocity dispersion relation (Onken et al. 2004). The resulting uncertainty δf_S/f_S ≈ 0.2 was derived from an heterogeneous sample of sources whose emission-line profiles and accretion rate are very different (Negrete et al. 2013). The Hβ profiles of xA sources can be almost always modelled with an unshifted (virial) Lorentzian component plus an additional component affecting the line base. This suggests the same, reproducible structure. We therefore expect a significantly smaller δf_S/f_S, → 0, if xA sources are considered as a distinct population in the analysis of the M_BH–bulge velocity dispersion relation, and assume δf_S/f_S = 0.1 in column 3 of Table 4.

6.1.1.5 FWHM –

equation (6) suggests that a most relevant source of statistical error may be δv that appears at the fourth power and is assumed here to be represented by the FWHM of Hβ and intermediate-ionization lines Al iii λ1860 and Si iii] λ1892 (Section 2). Indeed, a large fraction of the rms in the present sample can be accounted for by FWHM measurement errors, especially in sample 3. In Table 4, we conservatively estimate a typical error ≈15 per cent in FWHM measures. A lower, ≈ 5 per cent uncertainty in FWHM is within reach of dedicated observations.

6.1.2 z-based luminosity

6.1.2.1 Aperture effects –

aperture losses and errors associated with the spectrophotometric calibration should be carefully assessed. Here we conservatively set a 1σ confidence error of 10 per cent following the SDSS website.⁴

6.1.2.2 Redshift z –

detailed fitting can provide accurate redshift making the uncertainty on z negligible. However, z-values provided by the survey suffer from significant uncertainties (Hewett & Wild 2010). We assume a δz ≈ 0.002. This uncertainty should be easily reduced to 0 by precise z measures unless quasars show significant peculiar velocities with respect to the Hubble flow.

6.1.2.3 Bolometric correction –

the continuum shape has been represented here by a typical parametrization thought to be suitable for Pop. A sources (Mathews & Ferland 1987) that are moderate/high Eddington radiators, with a single bolometric correction applied to all sources. The dispersion around the assumed value has been estimated to be ≈20 per cent by Richards et al. (2006) for quasars of all spectral types. The dispersion around the typical SED of xA sources is probably lower. The main argument for a small scatter is again that we are considering objects thought to be producing similar emission line ratios. These ratios are dependent on the shape of the continuum (Negrete et al. 2012), making it reasonable to assume that the continua of xA sources are similar and with a dispersion significantly less than the one found in general surveys that do not distinguish spectral types. We derive a difference of ≈ 10 per cent between the NLSy1 SED and the minimum SED in Fig. 12.

6.1.2.4 Continuum anisotropy –

if the optical/UV continuum is emitted by an accretion disc, the observed luminosity should be dependent on the angle θ between the disc axis and the line of sight, which will be different for each source. The angular dependence can be written as |$\lambda L_{\lambda } \approx \lambda L_{\lambda , \theta = 0^\circ } \cos \theta (1+ a_{1} \cos \theta )$|⁠, where the second term is the limb-darkening effect (e.g. Netzer, Laor & Gondhalekar 1992; Netzer & Trakhtenbrot 2014, in the case of a thin disc). The limb darkening is wavelength dependent, and we are not aware of any exhaustive calculation in the context of slim discs (Watarai, Mizuno & Mineshige 2001). Nonetheless slim discs are optically thick thermal radiators so that a good starting point is to consider them Lambertian radiators, i.e. to first neglect the unknown but second-order limb-darkening effect, and then apply a reasonable limb darkening term a₁ ≈ 2 (Netzer & Trakhtenbrot 2014). We estimate that anisotropy introduces a scatter of δlog L ≈ 0.085, if the probability of observing a source at θ is ∝sin θ with 0° ≤ θ ≤ 45° in the case of a thin disc with no limb-darkening effect. The scatter increases to δlog L ≲ 0.15 in the case of a slim disc whose height is assumed to scale with radius for dimensionless accretion rate |$\dot{m} = 1$| (Abramowicz et al. 1988; Abramowicz, Lanza & Percival 1997).

6.1.3 Conclusion on statistical errors

The error sources described above and reported in column 2 of Table 4 provide a total error that accounts for and even exceeds the observed rms. The observed dispersion ≈0.365 implies that dispersion values reported in Table 4 might be close to the lowest values. Observational improvements can be devised to reduce, whenever possible, the main source of statistical errors that remains the FWHM of virial broadening estimator. Lowering FWHM measurements to ≈5 per cent would results in rms ≈ 0.3. This rms values is obtainable without requiring advancements in our physical understanding and in the connection of Eddington ratio to observed properties. If the scatter in Eddington ratio could be reduced by accounting for systematic and random effects, then rms ≈0.2 would become possible.

6.2 Systematic errors

6.2.1 General considerations

Equation (6) is probably not influenced by systematic errors except for an offset associated with the assumed λ_Edd, κ (n_HU), and |$\bar{\nu _\rm{i}}$|⁠. The offset will affect only H₀. There are also systematic effects that could reduce z-derived source luminosities. Light losses will also lead to systematic underestimates for luminosity which will affect H₀. Equation (6) currently provides a valid redshift-independent luminosity estimator suited for the measure of Ω_M and Ω_Λ only.

6.2.2 Orientation bias

Under the assumption that observed line broadening δv_obs is due to an isotropic component δv_iso plus a planar Keplerian component v_K, the broadening can be expressed as |$\delta v^{2}_\rm{obs} = \delta v^{2}_\rm{iso} + v^{2}_\rm{K}\sin ^{2}\theta$| (e.g. Collin et al. 2006). The probability of observing a randomly oriented source at i is sin θ, and the systematic offset in Eddington ratio can be computed by integrating the Eddington ratio values computed at δv_obs over their probability of occurrence for a fixed v_K. Assuming |$\delta v_\rm{iso}/\delta v_\rm{K} = \frac{1}{2}$|⁠, 5 ≲ θ ≲ 45 the offset in L/L_Edd is approximately a factor of 2. Therefore, the Eddington ratio we are considering (and whose distribution is shown in Fig. 5) could be subject to a substantial bias. However, since orientation effects are not included in λ_Edd computations and λ_Edd ∝ 1/v², a correction to the virial broadening estimator in equation (6) would be compensated by the change in λ_Edd, leading to a net Δlog L ≈ 0.0.

In this paper, the λ_Edd dispersion reported in Table 4 already takes into account orientation effects since δλ_Edd has been estimated from the a posteriori computed distribution of λ_Edd that is broadened by the orientation effect. A more proper approach would be to derive an orientation angle for individual sources. xA sources likely possess strong radiatively driven winds whose physics and effect on the line profile of a high-ionization line like C iv λ1549 can be modelled (Murray & Chiang 1997; Proga, Stone & Kallman 2000; Risaliti & Elvis 2010; Flohic, Eracleous & Bogdanović 2012).

6.2.3 Major expected systematic effects as a function of z, L and sample selection criteria

In the 4DE1 approach, luminosity dependences are in the parameters correlated with Eigenvector 2 (Boroson & Green 1992). The main correlate is the equivalent width of C iv λ1549 (i.e. the well-known ‘Baldwin effect’; Bian et al. e.g. 2012, and references therein). In that case, indeed, even if the most likely explanation of the decrease of W(C iv λ1549) with luminosity found in large sample is a dependence of W(C iv λ1549) on Eddington ratio plus selection effects in flux limited sample (Bachev et al. 2004; Baskin & Laor 2004; Marziani, Dultzin & Sulentic 2008), the C iv λ1549 equivalent width should be avoided because of a possible residual dependence with luminosity. The diagnostic ratios we employ are meant to minimize any possible dependence from a physical parameter that is in turn related to luminosity, like for example the covering fraction of the emitting gas that may affect W(C iv λ1549).

Neither |$R_{\rm Fe\,{\small II}}$| nor Al iii λ1860/Si iii] λ1892 depend significantly on z or luminosity in xA sources. Selecting xA sources from the Shen et al. (2011) sample, correlation coefficients are not significantly different from 0. For 378 sources with |$R_{\rm Fe\,{\small II}}$| ≥ 1, the Pearson correlation coefficients are 0.075 and 0.09 with z and L, respectively. These values are both not significant even if the sample of sources is relatively large. The Al iii λ1860/Si iii] λ1892 ratio as a function of redshift and luminosity yields least-squares fittings lines that not significantly different from 0. We have verified that the optical and UV selection criteria are mutually consistent in Section 2.5, and shown that the conditions on |$R_{\rm Fe\,{\small II}}$| and Al iii λ1860–C iii]λ1909 are simultaneously satisfied over a wide range L and z, even if for the only nine xA sources with available data: from 1H 0707−495 (the lowest luminosity A3 source, log L ≈ 44.6 [erg s⁻¹] at z ≈ 0.04) to HE1347−2457 (log L ≈ 47.9, at z ≈ 2.599).

Residual systematic effects can arise if the quantities entering equation (6) are different in high- and low-z sources, i.e. in |$R_{\rm Fe\,{\small II}}$|-selected and 1900-selected samples.

1. L/L_Edd – our present N = 92 sample suffers from a slight L/L_Edd bias. The difference in L/L_Edd between samples 1 and 3 (optical and UV based) indicates that δlog  L/L_Edd ≲ 0.08.

   Several works suggest a relation between diagnostic ratios |$R_{\rm Fe\,{\small II}}$|⁠, Al iii λ1860/ Si iii] λ1892 and Si iii] λ1892/C iii]λ1909, and Eddington ratio (Aoki & Yoshida 1999; Wills et al. 1999; Marziani et al. 2003c; Shen et al. 2011). There is no correlation at all between |$R_{\rm Fe\,{\small II}}$| and the UV ratios employed in this paper as far as the general population of AGNs is concerned (Fig. 4). However, this may not be longer true when |$R_{\rm Fe\,{\small II}}$| → 1: if |$R_{\rm Fe\,{\small II}}$| ≈ 1, Al iii λ1860/Si iii] λ1892 ≈ 0.5 and C iii]λ1909/Si iii] λ1892≈1, while for larger |$R_{\rm Fe\,{\small II}}$| Al iii λ1860 becomes stronger and C iii]λ1909 weaker. Indeed, 4DE1 suggests a relation between L/L_Edd and metallicity, with the most metal-rich system being associated only with the sources accreting at the highest rate. An important result of the present investigation is right that xA sources – that are extremely metal rich (Negrete et al. 2012) – are revealed over a broad range of L and z, but only for a very narrow range of L/L_Edd. As stressed earlier, luminosity and z cannot be major correlates.

   As long as the distribution of the measured intensity ratios Al iii λ1860/Si iii] λ1892, Si iii] λ1892/C iii]λ1909 and |$R_{\rm Fe\,{\small II}}$| are independent of z and L (absence of even a weak correlation could be enforced by resampling to avoid small effects), a relation (if any) between |$R_{\rm Fe\,{\small II}}$| and Al iii λ1860/Si iii] λ1892, Si iii] λ1892/C iii]λ1909 is established, inter-sample systematic effects should be minimized. The nine sources considered in Section 2.5 are clearly insufficient to test these conditions. A proper observational strategy could involve: (1) a large sample to define a relation between |$R_{\rm Fe\,{\small II}}$| and Al iii λ1860/Si ii λ1814, Si iii] λ1892/C iii]λ1909 (and L/L_Edd). This is especially needed since any λ_Edd deviation as a function of redshift can introduce a significant systematic effect: for example, δlog L/L_Edd≈−0.05 between samples 3 and 1 implies δΩ_M ≈ 0.05; (2) a vetted subsample with Hβ observations in the near-IR and simultaneous 1900 observations in the optical.

2. |$\kappa / h\bar{\nu }$| – a change of ionizing continuum shape as a function of redshift and/or luminosity, namely of the ratio |$\kappa / h\bar{\nu }_i$| for a fixed Al iii λ1860/ Si iii] λ1892, Si iii] λ1892/ C iii]λ1909 and |$R_{\rm Fe\,{\small II}}$|⁠. This could occur, practically, if we were selecting more and more extreme objects with the steepest continua. The properties of the ionizing continuum in quasars are not very well known; however, it is reasonable to assume that Pop. A sources show continua intermediate between the continua shown in Fig. 12. If the continua labelled as Korista et al. (1997) and the extreme, minimum NLSy1 continuum (dashed line) are considered, |$\kappa /h\bar{\nu }_i$| will change from 0.174 to 0.207, with a δlog L ≈ 0.075. xA sources are expected to show a steep X-ray continuum (Grupe et al. 2010; Panessa et al. 2011; Wang et al. 2013), so that the range of change may be more realistically bracketed by the typical NLSy1 and minimum continuum. In this case, the hypothesis of a systematic evolution implies a change in |$\kappa /h\bar{\nu }_i$| that leads to δlog L ≈ 0.035. However, the change in n_HU associated with different continua (for the same values of the observed Al iii λ1860/Si ii λ1814 ratios) tends to compensate for the change in |$\kappa /h\bar{\nu }_i$|⁠, reducing the effect to δlog L ≈ 0.026. There is no evidence for such evolution/selection effects in xA sources, but the relevance of systematic continuum changes and the relation between κ (n_HU), and |$\bar{\nu _\rm{i}}$| and diagnostic ratios should be tested with dedicated X-ray observations of high-z xA sources.

3. Ionizing photon flux n_HU – the factor n_HU is set by the Al iii λ1860/ Si iii] λ1892, Si iii] λ1892/ C iii]λ1909 and |$R_{\rm Fe\,{\small II}}$| ratios and, for a given continuum shape, we do not expect systematic changes. When selecting large samples some degree of heterogeneity is unavoidable. Differences in the equation (6) parameter values for bins A3^m and A4^m will contribute to the overall sample rms but should not introduce any systematic effects as long as the fraction A3^m/A4^m or, more properly, the Al iii λ1860/ Si iii] λ1892, Si iii] λ1892/ C iii]λ1909 and |$R_{\rm Fe\,{\small II}}$| distributions are consistent and independent from z.

4. Structure factor f_S – the self-similarity of the profiles over z and luminosity should be carefully tested on high S/N spectra since the FWHM of the lines is clearly dependent on the assumed profile shape. Available data support the assumption that the shape is not changing as a function of FWHM and z (as also shown by the data of Figs 2 and 3): a Lorentzian function yielded good fits for Hβ, Al iii λ1860, Si ii λ1814 in all cases considered in this paper. This is also true when fits are made to carefully selected composite spectra with S/N ≳ 100 (see e.g. Zamfir et al. 2010).

An important systematic effect is related to the use of equation determination of the quasar luminosity from redshift (equation 8).

• |$\text{B.C.}(1800)\text{-}\text{B.C.}(5100)$| – the ratio between the bolometric correction at 5100 Å and 1800 Å has been assumed in this paper to be λf_λ(5100)/λf_λ(1800) = 0.63 on the basis of the Mathews & Ferland (1987) continuum. A more proper value could be well 0.7, as measured on the optical and UV spectral of I Zw1. This change will shift the estimate of Ω_M by ≈0.05, to Ω_M ≈ 0.25, in the case of constrained Ω_M + Ω_Λ = 1.0. In addition, if λf_λ(5100)/λf_λ(1800) ≈ 0.7 the systematic offset in the L/L_Edd distribution between sample 1 and 3 will be reduced by ≈ 0.05. The average ratio λf_λ(5100)/λf_λ(1800) and the associated dispersion (and hence |$\text{B.C.}(1800) \text{-} \text{B.C.}(5100)$|⁠) should be carefully established by dedicated observations. Such observations are also needed because continuum anisotropy leads, in addition to a random error, to a systematic underestimate of source luminosity, as discussed below. The effect is relevant here as long as it is wavelength dependent if optical and UV data are considered together. Recent observational work on RL sources suggests that the degree of anisotropy changes very little with wavelength (Runnoe, Shang & Brotherton 2013). However, this may not be the case for slim discs of highly accreting sources.

The most relevant systematic effects, among the ones that are listed above, are related to |$\text{B.C.}(1800)/\text{B.C.}(5100)$| and to the distributions of the optical and UV line ratios that could be linked to small – but significant – trends in L/L_Edd. We have shown that other effects, like evolution of the ionizing continuum – within reasonable limits – may yield a modest systematic effect on luminosity estimates, ≈ 0.03 dex.

The following effects are more speculative in nature and of lower relevance.

• Continuum anisotropy is also expected to give rise to a Malmquist-type bias in z-based luminosities. Close to a survey limiting magnitude the brightest (face-on) sources will be selected preferentially. Unlike relativistic beaming however, disc anisotropy is of relatively modest amplitude. In the case of a slim disc with limb darkening, the difference between a face-on source and a randomly oriented sample of objects is δlog L ≈ 0.2 dex. Since core-dominated RL sources are expected to have an additional relativistically beamed synchrotron continuum component, they should be avoided from any sample. If sources are selected from a large flux-limited sample, (1) either orientation is inferred from C iv λ1549 line profile modelling, or (2) a more elementary precaution would be to consider sources brighter than ≈0.5 mag than their discovery survey (i.e. the SDSS in the case of sample 3 limiting magnitude).

• Intervening large-scale structures are expected to produce a lensing effect on the light emitted by distant quasars. This effect is noticeable especially for sources at z ≳ 1 (Holz & Wald 1998; Holz & Linder 2005). The lensing effect is however found to be averaged out for large samples ( ≲ 100 sources) which is the case of any quasar sample that could be realistically employed for cosmology.

The present samples hint at (small) systematic differences that can be more clearly revealed and quantified only with a larger sample and/or vetted. As long as we employ the same diagnostic ratios (with consistent distributions of values as a function of z) and the same line profile model, residual systematic effects with z, L and L/L_Edd should be minimized. Any systematic change with z and L affecting the parameters entering into equation (6) will also affect the intensity ratios and will become detectable. Assessing and avoiding systematic effects would require a uniform redshift coverage, as assumed for the mock samples. More details on a possible observational strategy are given in Section 7.2.

7 DISCUSSION

The idea to use quasars as Eddington standard candles is not new (e.g. Marziani et al. 2003a; Teerikorpi 2005; Bartelmann et al. 2009; Sulentic, Marziani & D'Onofrio 2012; Wang et al. 2013; La Franca et al. 2014). Luminosity correlations were the past great hope for using quasars as standard candles. The most promising luminosity correlation involved the Baldwin effect (Baldwin 1977; Baldwin et al. 1978) which is now thought to be governed by the Eddington ratio (Bachev et al. 2004; Baskin & Laor 2004). In any case, it is too weak to provide interesting cosmological constraints. Other methods using line width measures have also been proposed (e.g. Rudge & Raine 1999) although it is unclear that the line width distribution shows real change with redshift. Recently, a somewhat similar proposal to the one presented in this paper has been advanced which advocates the hard X-ray spectra index as a selector for extreme sources (Wang et al. 2013). We considered both hard and soft X-ray measures when selecting the principal 4DE1 parameters and concluded that hard measures showed too little dispersion across the 4DE1 optical plane compared to Γ_soft (Sulentic et al. 2000b).

The concordance H₀ value has to be assigned a priori to constrain Ω_M and Ω_Λ to avoid circularity, unless the normalized slope is used as a best-fit estimator. The determination of H₀ is not fully independent of z-based distances. The Eddington ratio estimate for I Zw 1 requires a luminosity computation that assumes a value of H₀. In order to make an independent determination of λ_Edd (and hence of H₀) at least one redshift-independent luminosity determination would be needed (e.g. distance inferred from a Type Ia supernova). In this case, M_BH would follow from luminosity via the virial relation with r ∝ L^1/2 and L/L_Edd from the ratio L/M_BH. Another approach would be to derive λ_Edd from a physical model of the high-ionization outflow common to extreme Eddington sources.

The definition of ionization parameter also involves L. However, ionization parameter and density values were derived from emission line ratios. The luminosity is a theoretical luminosity that, for an assumed continuum, yields the number of ionizing photons needed to produce the emission lines. We retrieve Un_H from emission line ratios; no flux or line luminosity measurements are involved. The assumption that r ∝ L^0.5 is consistent with the assumption of Un_H = const (actually, it follows from the definition of U). Therefore, equation (6) has no implicit circularity.

7.1 Comparison with the supernova legacy survey

There are many analogies between supernova surveys (Guy et al. 2010) and our proposed method using quasars. Both methods rely on intrinsic luminosity estimates for a large number of discrete sources. The advantage of the supernova surveys is that individual supernovae show a smaller scatter in luminosity (e.g. Riess et al. 2001). However, very few supernovae have been detected at z ≳ 1 while a quasars sample can be easily extended (with significant numbers) to z ≈ 3 or possibly z ≈ 4. Differences in redshift coverage account for the different sensitivity to Λ: Ω_Λ is tightly constrained using supernovae while it remains loosely constrained using quasars. Quasars are distributed over a redshift range where Ω_M ruled the expansion of the Universe while supernovae sample epochs of accelerated expansion (Fig. 6). But quasars can sample any range covered by supernovae. Statistical errors for Ω_M derived from the mock samples (with unconstrained Ω_M and Ω_Λ) are lower than ones derived from the first three years of the supernova legacy survey that yield Ω_M|$\approx 0.19^{+0.08}_{-0.10}$| (Conley et al. 2011).

7.2 Possible observational strategies

In order to exploit a sample of high L/L_Edd radiators, both calibration observations and a larger sample of extreme Eddington sources are needed. Simultaneous rest-frame UV and optical observations covering the 1900 and Hβ range are needed (a feat within the reach of present-day multibranch spectrometers): (1) to constrain the bolometric corrections and specifically the |$\text{B.C.}(1800)/\text{B.C.}(5100)$| ratio; (2) to define systematic differences between spectral types A3^m and A4^m, including the |$\text{B.C.}$| An attempt should be also made to cover with a large Hβ sample the z range 0.1–1.5, where the effect of a non-zero Ω_Λ is most noticeable and where any rest-frame optical/UV inter calibration is not needed. A related option is to obtain |$R_{\rm Fe\,{\small II}}$| only, covering the redshifted Hβ spectral range into the near- and mid-IR (K-band observations can reach z ≈ 3.5). Alternatively, the UV 1900 blend can be easily covered by optical spectrometers over the redshift range 1.1 ≲ z ≤ 3.5. This approach would allow us to measure Ω_M without the encumbrance of an inter-calibration with Hβ data.

The best hope for accurate and precise results rests in a ‘brute force’ application of the method to a large sample. Table 4 shows that rms ≈ 0.3 can be obtained with better data. With uniform redshift coverage the precision of the method will scale with rms/|$\sqrt{N}$|⁠. This means that a precision similar to the one obtained with the mock sample (rms = 0.2) can be achieved with a sample of 400 sources while a precision similar to rms = 0.1 would require a sample near 1000 quasars. It is possible that a sample of this size (or even larger) can be selected from spectra collected by recent major optical surveys (e.g. Pâris et al. 2012).

8 CONCLUSION

We have shown that sources radiating at, or close to, L/L_Edd≈1 can be identified in significant numbers with reasonable confidence. These sources show stable emission line ratios over a very wide range of z and L. We have performed exploratory computations and shown that these sources – apart from their intrinsic importance for quasar physics – may be also prime candidates as cosmological probes. We then presented a method for using some quasars as redshift-independent distance estimators. We do not claim to present constraining results in this paper beyond showing an overall consistency with concordance cosmology and exclusion of extreme models (e.g. a flat Universe dominated by the cosmological constant) starting from an estimates of the most likely values of quasar physical parameters entering in equation (6). Our goal was to identify suitable quasars, to describe a possible approach capable of yielding meaningful constraints on Ω_M and Ω_Λ, and to identify most serious statistical and systematic sources of uncertainty. A quantitative analysis of systematic effects due to continuum shape, orientation, f_S and λ_Edd as well as an attempt at reducing statistical errors is deferred to further work. Addressing and overcoming systematic biases requires dedicated, but feasible, new observations.

We stress that the precision of our method can be greatly improved with high S/N spectroscopic observations for significant samples of quasars. This is not just the usual refrain claiming that improvement in S/N can lead to unspecified advancements: previous work shows that broad-line FWHM for Pop. A sources can be measured with a typical accuracy of 10 per cent at a 2σ confidence level. This would represent a major improvement with respect to the 20 per cent at 1σ uncertainty for many FWHM values used in this work – implying Δlog L errors decreasing from ≈0.7 to ≲0.1. Figs 10 and 11 show that cosmologically meaningful limits can be set even with currently obtainable data.

PM and JS acknowledge support from la Junta de Andalucía, through grant TIC-114, Proyectos de Excelencias P08-FQM-04205+ P08-TIC-3531 as well as the Spanish Ministry for Science and Innovation through grant AYA2010-15169.

Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the US Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS website is http://www.sdss.org/.

The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory and the University of Washington.

REFERENCES