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Edited by: Fabio La Franca, Roma Tre University, Italy

Reviewed by: Jian-Min Wang, Institute of High Energy Physics, Chinese Academy of Sciences, China; Dragana Ilic, University of Belgrade, Serbia

This article was submitted to Extragalactic Astronomy, a section of the journal Frontiers in Astronomy and Space Sciences

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Quasars accreting matter at very high rates (known as extreme Population A [xA] quasars, possibly associated with super-Eddington accreting massive black holes) may provide a new class of distance indicators covering cosmic epochs from present day up to less than 1 Gyr from the Big Bang. At a more fundamental level, xA quasars are of special interest in studies of the physics of AGNs and host galaxy evolution. However, their observational properties are largely unknown. xA quasars can be identified in relatively large numbers from major optical surveys over a broad range of redshifts, and efficiently separated from other type-1 quasars thanks to selection criteria defined from the systematically-changing properties along the quasars main sequence. It has been possible to build a sample of ~250 quasars at low and intermediate redshift, and larger samples can be easily selected from the latest data releases of the Sloan Digital Sky Survey. A large sample can clarify the main properties of xA quasars which are expected—unlike the general population of quasars—to radiate at an extreme, well-defined Eddington ratio with small scatter. As a result of the small scatter in Eddington ratio shown by xA quasars, we propose a method to derive the main cosmological parameters based on redshift-independent “virial luminosity” estimates from measurements of emission line widths, roughly equivalent to the luminosity estimates based from line width in early and late type galaxies. A major issue related to the cosmological application of the xA quasar luminosity estimates from line widths is the identification of proper emission lines whose broadening is predominantly virial over a wide range of redshift and luminosity. We report on preliminary developments using the AlIIIλ1860 intermediate ionization line and the Hydrogen Balmer line Hβ as virial broadening estimators, and we briefly discuss the perspective of the method based on xA quasars.

Quasars show a rich spectrum of emission lines coming from a side range of ionic species (e.g., Netzer,

Quoting Marziani et al. (_{Edd}

The Main Sequence stems from the definition of the so-called Eigenvector 1 of quasars by a Principal Component Analysis of PG quasars (Boroson and Green, _{FeII} (the intensity ratio between the FeII blend at λ4570 and Hβ) and by the full-width half maximum (FWHM) of Hβ(Boroson and Green,

Since 1992, the Eigenvector 1 MS has been found in increasingly larger samples, involving more and more extended sets of multifrequency parameters (Sulentic et al., _{FeII} and FWHM(Hβ), several multifrequency parameters related to the accretion process and the accompanying outflows are also correlated (see Fraix-Burnet et al.,

The MS allows for the definition of spectral types (Sulentic et al., _{FeII} end of the main sequence (_{FeII}> 1). The optical spectra of the prototypical Narrow Line Seyfert 1 (NLSy1) I Zw 1 and the Seyfert-1 nucleus NGC 5548 illustrate the changes that occur along the sequence. I Zw 1 (a prototypical xA source) shows a low-ionization appearance, with strong Fe

The comparison between LILs and HILs has provided insightful constraints of the BLR at low-^{47} erg s^{−1}, the Hβ profile remains (almost) symmetric and unshifted with respect to rest frame.

The broad component (BC), also known as the intermediate component, the core component or the central broad component following various authors (e.g., Brotherton et al., ^{1}

The blue shifted component (BLUE). A strong blue excess in Pop. A C

The line width of the LILs clearly increases with luminosity passing from ~ 1, 000 to 5,000 km s−1 from log

Intercomparison of line profiles of Hβ (left), C

The data point occupation in the plane _{FeII}– FWHM(Hβ) make it expedient to define spectral types; A1, A2…in order of increasing _{FeII}; A1, B1, B1+.…in order of increasing FWHM(Hβ). This has the considerable advantage that a composite over all spectra within each bin should be representative of objects in similar dynamical and physical conditions. In principle, a prototypical object can be defined for each spectral type to analyze systematic changes along the quasar MS. Before summarizing some basic trends, it is helpful to recall the main motivations behind the distinction between Population A and B.

Why choose a FWHM limit for Hβ at 4,000 km s−1? Many studies even in recent times distinguish between NLSy1s (FWHM Hβ ≲ 2, 000 km s−1) and rest of type 1 AGNs (e.g., Cracco et al., _{FeII} parameter being between almost 0 and 2 in the quasi-totality of sources, as in the case of the rest of Population A.

A complete tracing of the MS at high _{BH} can be predicted. If the motion in the LIL-BLR is predominantly virial, we can write the central black hole mass _{BH} as a virial mass :

which follows from the application of the virial theorem in case all gravitational potential is associated with a central mass. δ_{K} is the virial velocity module, _{BLR} the radius of the BLR, _{K} to the observed velocity dispersion, represented here by the FWHM of the line profile:

via the structure factor (a.k.a. _{S} form or virial factor) whose definition is given by:

If the BLR radius follows a scaling power-law with luminosity (^{a}, Kaspi et al.

Or, equivalently,

If _{BH}^{0.25}, meaning a factor of 10 (!) for log_{BH}, the MS becomes displaced toward higher FWHM values in the optical plane of the MS; the displacement most likely accounts for the wedge shaped appearance of the MS if large samples of SDSS quasars are considered (Shen and Ho,

If we consider a physical criterion for the distinction between Pop. A and B a limiting Eddington ratio (log _{Edd} ~ 0.1–0.2), then the separation at fixed _{Edd} becomes luminosity dependent. In addition FWHM(Hβ) is strongly sensitive to viewing angle, via the dependence on θ of the _{S} (section 4.2). An important consequence is that a strict FWHM limit has no clear physical meaning: its interpretation depends on sample properties. At low and moderate luminosity (log^{−1}]), the limit at 4,000 km s−1 selects mainly sources with log _{Edd} ≳ 0.1–0.2 i.e., above a threshold that is also expected to be relevant in terms of accretion disk structure. However, the selection is not rigorous, since a minority of low _{Edd} sources observed pole-on (and hence with FWHM narrower because of a pure orientation effect) are expected to be below the threshold FWHM at 4,000 km s−1. Such sources include core-dominated radio-loud quasars whose viewing angle θ should be relatively small (Marziani et al., _{Edd} sources.

As mentioned in section 1, Eddington ratio is a parameter that is probably behind the _{FeII} sequence in the optical plane of the MS. It is still unclear why it is so. At a first glance, the low-ionization appearance of sources whose Eddington ratio is expected to be higher is puzzling. Marziani et al. (_{FeII}. More recent results include a careful assessment of the role of metallicity (Panda et al., _{FeII} and _{Edd}, but the issue is compounded with the strong effect of orientation on the line broadening, and hence on _{BH} and _{Edd} computation. As an estimate of the viewing angle is not possible for individual radio-quiet sources (even if some recent works provide new perspectives), conventional _{BH} (Equation 1) and _{Edd} estimates following the definitions in the previous section suffer of a large amplitude scatter (Marziani et al., _{FeII} as mainly related to _{Edd} is correct, then the strongest Fe_{BH}) is anti correlated with _{FeII}, implying that _{Edd} increases with Fe_{Edd} and dimensionless accretion rate with _{FeII} and with the “Gaussianity” parameter _{FeII} corresponds to the highest _{Edd}.

If we select spectral types along the sequence satisfying the condition _{FeII} ≳1 (A3, A4, …), we select spectra with distinctively strong Fe

The simple selection criterion

_{FeII}= FeIIλ4570 blend/Hβ > 1.0

corresponds to the selection criterion:

AlIII λ1860/SiIII]λ1892>0.5 & SiIII]λ1892/ CIII]λ1909>1

The _{FeII}>1 and the UV selection criteria are believed to be equivalent. Marziani and Sulentic (_{FeII} ≳1 in the plane AlIII λ1860/SiIII]λ1892 vs. SiIII]λ1892/ CIII]λ1909, and found that these sources were confined in a box satisfying the boundary conditions AlIII λ1860/SiIII]λ1892>0.5 & SiIII]λ1892/ CIII]λ1909>1, as expected if the two selection criteria are consistent. The number of sources for which this check could be carried out is however small, and several xA sources are located in borderline positions. Work is in progress to test the consistency on a larger sample.

Though, the UV xA spectrum is very easily recognizable, as shown by the composite spectrum of Mart́ınez-Aldama et al. (_{H} defined by CLOUDY simulation, UV line intensity ratio converge toward extreme values for density (high, ^{3}), and ionization (low, ionization parameter ^{−3} − 10^{−2.5}). Extreme values of metallicity are also derived from the intensity ratios CIV/AlIII CIV/HeII AlIII/SiIII] (Negrete et al.,

In addition to the selection criteria, a virial broadening estimator equivalent to Hβ should be defined from the emission line in the rest frame UV spectrum. The C

FWHM of Al

The Eddington ratio precise values depend on the normalization applied; the relevant result is that xA quasars radiate at extreme _{Edd} along the MS with small dispersion (Marziani and Sulentic, _{Edd} saturates toward a limiting values (Abramowicz et al., _{Edd} distribution shown by Marziani and Sulentic (_{FeII} and _{Edd} might be correlated, as mentioned above, if _{FeII} ≳1, _{Edd} apparently scatters around a well-defined value with a relatively small scatter ≈ 0.13 dex. Clearly, this results should be tested by larger sample and, even more importantly, by Eddington ratio estimators not employing the FWHM of the line used for the _{BH} computation. Another important fact is the self similarity of the spectra selected by the _{FeII} criterion: as ^{−1}], covering local type 1 quasars in the local Universe as well as the most luminous quasars that are now extinct but that were shining bright at redshift 2 when the Universe had one quarter of its present age.

Accretion disk theory predicts that at high accretion rate a geometrically thick, advection dominated disk should develop (Abramowicz et al.,

From the discussion of the previous sections we gather that three conditions are satisfied for xA quasars:

Constant Eddington ratio _{Edd}; xA quasars radiate close to Eddington limit:

Note that the precise value or _{Edd} depends on the normalization applied for _{BH} and on the bolometric correction to compute the bolometric luminosity. Applying widely employed scaling laws, _{Edd} ≲ 1.

The assumption of virial motions of the low-ionization BLR, so that the black hole mass _{BH} can be expressed by the virial relation (Equation 1):

Spectral invariance: for extreme Population A, the ionization parameter

Netzer (

Taking the three constraints into account, the virial luminosity equation derived by Marziani and Sulentic (

where the energy value has been normalized to 100 eV (_{i,0.5} is the fraction of bolometric luminosity belonging to the ionizing continuum scaled to 0.5, the product (_{H}^{9.6}cm^{−3} (Padovani and Rafanelli, _{S} is scaled to the value 2 following the determination of Collin et al. (

Recent works involved similar ideas concerning the possibility of “virial luminosity" estimates, and confirm that extremely accreting quasars could provide suitable distance indicators because their emission properties appear to be stable with their luminosity scaling with black hole mass at a fixed ratio (Wang et al., _{Edd} ~ 1). However, it can be applied to

The virial luminosity equation is analogous to the Tully-Fisher and the early formulation of the Faber Jackson laws for early- type galaxies (Faber and Jackson, ^{n}, where σ is a measurement of the velocity dispersion and

The effect of orientation can be quantified by assuming that the line broadening is due to an isotropic component + a flattened component whose velocity field projection along the line of sight is ∝ sin θ:

If we considered a flattened distribution of clouds with an isotropic δ_{iso} and a velocity component associated with a rotating flat disk δ_{K}, the structure factor appearing in Equation (8) can be written as

which can reach values ≳1 if _{S} = 2 implies that we are seeing a highly flattened system (if all parameters in Equation (8) are set to their appropriate values); an isotropic velocity field would yield _{S} = 0.75.

The virial luminosity equation may be rewritten in the form:

where _{vir} is the true virial luminosity (which implies _{S} = 1) with _{S} = 2.

Following Weinberg (_{L} can be written as:

where Ω_{M} is the energy density associated to matter and Ω_{Λ} the energy density associated to Λ, and _{H} ≈ 28.12[~cm], for _{0} = 70 km s^{−1} Mpc^{−1}.

If Ω_{M} + Ω_{Λ} > 1, _{M} − Ω_{Λ}; for Ω_{M} + Ω_{Λ} < 1, _{M} + Ω_{Λ} = 1,

The equation of Perlmutter et al. (_{k} + Ω_{Λ} + Ω_{M} = 1, where Ω_{k} is the energy density associated with the curvature of space-time. The luminosity distance _{L} = _{C} · (1 +

with

with log _{H} ≈ 28.12 [cm] and

The distance modulus μ is defined by

that is μ = log_{L}(_{0}, Ω_{M}, Ω_{k}, Ω_{Λ},

The distance modulus μ computed from the virial equation yielding

where the constant _{λ}λ can be the flux at 5100 Å for the Hβ sample, or the flux at 1700 Å if the δ

The δμ = μ_{vir}−μ(_{M}, Ω_{Λ}, _{0}) can be written as:

Following the notation of Marziani and Sulentic (

If we equate Equations (17) and (19) above, we obtain:

Simplifying:

we recover the expression used by Marziani and Sulentic (_{λ}λ) refers to the rest frame fluxes (the term (1 + ^{2} appears in both sides of the equations, for the virial luminosity and for the distance modulus where the luminosity distance was used).

The Hubble diagram of _{M} = 1, Ω_{Λ} = 0, red line in

Hubble diagram (distance modulus μ vs. redshift

The Hubble diagram for quasars restricted to the 92 sources of Marziani and Sulentic (_{M}. The new sample is consistent with the old one and allows to set more stringent limits to

In this paper we have described the method developed by Marziani and Sulentic (_{M} and the Hubble diagram with concordance and Einstein-de Sitter cosmologies, just to illustrate the sensitivity of the method to cosmological parameter changes. It should be also noted that a comparison between the constraints set by the supernova photometric survey described by Campbell et al. (_{M}.

An elementary error budget suggests that the main uncertainty is associated with FWHM measurement errors and with uncertainty in the structure factor (Marziani and Sulentic,

Negrete et al. (_{λ} that we receive entering in Equation (17).

The presence of a thick disk implies self-shadowing effects, and that the line emitting gas might not be exposed to the same ionizing continuum that is seen by the observer. This second effect related to the disk anisotropy is especially severe if the line emitting gas is located close to the equatorial plane of the disk: in this case, following Wang et al. (

The SED parameters entering in the relation for the virial luminosity are _{i}, and more indirectly, _{H}_{H}

The issue of anisotropy and of other factors affecting the

Summing up, we can say that the proper calibration of the data for cosmology will need large samples of quasar spectra covering the rest frame optical and UV, at high S/N, over a wide range of redshift. This is a

The quasar MS has allowed us to isolate sources which are the highest radiators among quasars. This is already an important feat, as xAs can then be selected by the application of simple criteria based on optical and UV emission line ratios. xA quasars show a relatively high prevalence (10%) and are easily recognizable in the redshift range 0–5. The present work illustrated several key aspects of the method, and provided the virial luminosity equation dependent from the viewing angle in an explicit form (a result of Negrete et al.

The raw data supporting the conclusions of this manuscript will be made available by the authors, without undue reservation, to any qualified researcher.

CN is a CONACyT Research Fellow for Instituto de Astronomía, UNAM, Mexico.

PM wrote the paper with the assistance of DD. PM and JD did the Hubble diagram analysis. JD contributed the Ω_{M} estimate. CN, AD, MM-A, MD'O, EB, NB, and GS sent comments and/or contributed to some extent during the development of the project.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

DD and CN acknowledge support from grants PAPIIT, UNAM 113719, and CONACyT221398. PM and MD'O acknowledge funding from the INAF PRIN-SKA 2017 program 1.05.01.88.04. PM also acknowledges the Programa de Estancias de Investigación (PREI) No. DGAP/DFA/2192/2018 of UNAM where this work was advanced, and the

^{1}Note that in Population B the Hβ profile can be decomposed into a BC and a broader line base, the very broad component (VBC). Attempts at fitting Fe