Unseen Progenitors of Luminous High-z Quasars in the R_h = ct Universe

The ongoing detection of quasars at high (z ≳ 5) redshift provides vital information regarding the growth of black hole (BH) mass and, in turn, informs our understanding of how the universe has evolved since the beginning of the stelliferous era. To be sure, the assembly of the high-z quasar sample has been painstaking work as these objects have proven to be quite elusive. As an example, Weigel et al. (2015) estimated that a search for z ≳ 5 AGNs in the Chandra Deep Field South (0.03 deg² field of view) should have lead to a discovery of ∼20 AGNs, yet no convincing identifications were made. Such non-detections, while consistent with the seeming strong evolution at the faint-end of the AGN luminosity function with increasing redshift from z ∼ 3 to z ∼ 5 (Georgakakis et al. 2015), do put existing models of early BH evolution at odds with observational constraints on their growth rate (Treister et al. 2013). Several possible explanations have been proposed for the very limited number of detections, including dust obscuration (Fiore et al. 2009), low BH occupation fraction, super-Eddington accretion episodes with low duty cycles (Madau et al. 2014; Volonteri et al. 2015), and BH merging scenarios.

At the same time, the handful of z > 6.5 quasars that have been observed (e.g., Mortlock et al. 2011) provide supermassive black hole (SMBH) mass estimates that are hard to reconcile with the timeline of a ΛCDM universe (Melia 2013; Melia & McClintock 2015). Specifically, BHs in the local universe are produced via supernova explosions with masses ≈5–20 M_⊙. But Eddington-limited accretion would require ∼10⁵ M_⊙ seeds in order to produce the billion-solar-mass quasars seen at redshift z ≳ 6.5. Accretion scenarios operating within the ΛCDM paradigm thus require either anomalously high accretion rates (Volonteri & Rees 2005) or the creation of massive seeds (Yoo & Miralda-Escudeé 2004), neither of which has actually ever been observed.

In recent work, Melia (2013) and Melia & McClintock (2015) present a simple and elegant solution to the SMBH anomaly by viewing the evolution of SMBHs through the age–redshift relation predicted by the R_h = ct universe, a Friedmann–Robertson–Walker cosmology with zero active mass. In their scenario, cosmic reionization lasted from t ≈ 883 Myr (z ∼ 15) to t ≈ 2 Gyr (z ∼ 6) (see also Melia & Fatuzzo 2016). As such, 5–20 M_⊙ BH seeds that formed shortly after the beginning of reionization would have evolved into ∼10¹⁰ M_⊙ quasars by z ∼ 6–7 via the standard Eddington-limited accretion rate. It should be noted that these SMBH results are but one of many comparative tests completed between the R_h = ct and ΛCDM paradigms, the results of which show that the data tend to favor the former over the latter with a likelihood of ∼90% versus ∼10%, according to the Akaike (AIC) and Bayesian (BIC) Information Criteria (see, e.g., Melia & Maier 2013; Wei et al. 2013; Melia 2014; Wei et al. 2014a, 2014b, 2015a, 2015b; Melia et al. 2015). A summary of 18 such tests may be found in Table 1 of Melia (2017).

Clearly, our understanding of SMBH evolution remains an open question, with even the basic questions on how these objects were seeded and the mechanism through which they evolved remaining unanswered. But the next decade promises to be transformative. The eROSITA mission, scheduled for launch in 2018, will perform the first imaging all-sky survey in the medium energy X-ray range with unprecedented spectral and angular resolution. Likewise, the ATHENA X-ray observatory mission scheduled for launch in 2028 is expected to perform a complete census of BH growth in the universe tracing to the earliest cosmic epochs.

Motivated by the feasibility of testing the R_h = ct and ΛCDM paradigms with this upcoming wealth of observational data, we here extend the analysis of Melia (2013) and Melia & McClintock (2015) by using accretion-based evolutionary models of SMBHs to determine the expected soft X-ray flux values observable at earth as a function of redshift. The analysis is carried out for both Planck ΛCDM, with optimized parameters Ω_b = 0.308, w = −1, and H₀ = 67.8 km s⁻¹ Mpc⁻¹, and the R_h = ct universe with the same Hubble constant for ease of comparison. We shall demonstrate that the quasar mass function for SMBHs at z ≳ 7 is considerably different between the two scenarios, indicating that the next generation of quasar observations at high-redshifts will allow us to discriminate between these two cosmologies.

The paper is organized as follows. We present our SMBH mass evolutionary model in Section 2, and relate it to redshift evolution in both ΛCDM and R_h = ct. We then present our emission model in Section 3, which links the mass of a BH to its X-ray emissivity. In Section 4, we combine the results of Sections 2 and 3 to calculate the flux expected as a function of redshift during the evolutionary history of an SMBH for both the ΛCDM and R_h = ct cosmologies. These results are then combined with the known quasar mass function at z = 6 in order to calculate the expected number of observable quasars as a function of redshift, again for both cosmologies. Our summary and conclusions are presented in Section 5.

We adopt a streamlined model wherein the early universe (6 ≲ z ≲ 10) is comprised of non-rotating BHs of mass M_bh that grow continuously through mass accretion via a thin or slim disk (see below), maintaining a constant radiative efficiency _r over that time (see, e.g., Chan et al. 2009). The ensuing bolometric luminosity is parametrized in terms of the Eddington ratio λ_Edd ≡ L_bol/L_Edd, which is also assumed to remain constant throughout this time. The disk accretion rate is therefore given by $\dot{M}={L}_{\mathrm{bol}}/({\epsilon }_{r}{c}^{2})$, and with the further assumption that a BH steadily accretes a fraction (1 − _r) of the infalling material (see, e.g., Ruffert & Melia 1994), the mass-growth rate is

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{bh}}=\displaystyle \frac{(1-{\epsilon }_{r})}{{\epsilon }_{r}}\displaystyle \frac{{\lambda }_{\mathrm{Edd}}{L}_{\mathrm{Edd}}}{{c}^{2}}=\displaystyle \frac{(1-{\epsilon }_{r})}{{\epsilon }_{r}}\displaystyle \frac{{\lambda }_{\mathrm{Edd}}}{{t}_{\mathrm{Edd}}}\,{M}_{\mathrm{bh}},\end{eqnarray} \tag{ 1 }$$

where t_Edd = 0.45 Gyr. Integrating Equation (1), one obtains an expression for the BH mass as a function of the age of the universe,

$$\begin{eqnarray}&&{M}_{\mathrm{bh}}(t)={M}_{0}\,{e}^{\displaystyle \frac{(1-{\epsilon }_{r})}{{\epsilon }_{r}}\displaystyle \frac{{\lambda }_{\mathrm{Edd}}}{{t}_{\mathrm{Edd}}}(t-{t}_{0})},\end{eqnarray} \tag{ 2 }$$

where M₀ is the mass observed at redshift z₀, corresponding to an age t₀.

Connecting Equation (2) to observational cosmology requires a relation between redshift and the age of the universe. In ΛCDM, this relation is given via the well known integral expression

$$\begin{eqnarray}&&{t}^{{\rm{\Lambda }}}(z)=\displaystyle \frac{1}{{H}_{0}}{\int }_{z}^{\infty }\displaystyle \frac{{du}}{\sqrt{{{\rm{\Omega }}}_{m}{(1+u)}^{5}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}{(1+u)}^{5+3w}}},\end{eqnarray} \tag{ 3 }$$

where the radiation contribution has been omitted given the redshift of interest, thus leaving Ω_Λ = 1 − Ω_m. The corresponding expression for R_h = ct takes on the simpler form

$$\begin{eqnarray}&&{t}^{{R}_{{\rm{h}}}}(z)=\displaystyle \frac{1}{{H}_{0}(1+z)}.\end{eqnarray} \tag{ 4 }$$

We adopt the Planck parameters (Ade et al. 2016), H₀ = 67.8 km s⁻¹ Mpc⁻¹, Ω_m = 0.308, and w = −1, throughout this work.

The difference in age between z ∼ 10 and z ∼ 6 for the two scenarios, as illustrated in Figure 1, has important implications for quasar evolution and detectability, and can therefore be used to test each model against present and future observations. Specifically, while the age of the universe at z = 6 is approximately 0.93 Gyr in standard (ΛCDM) cosmology, its value is approximately 2.1 Gyr for R_h = ct. As a result, an SMBH in the R_h = ct universe has ∼18 t_Edd of additional time to grow before redshift z = 6, and is therefore advantaged by a factor ∼e¹⁸  ≈ 6 × 10⁷ over its ΛCDM counterpart (assuming _r = 0.1 and λ_Edd = 1). More relevant to our discussion, the time interval between z = 6 and z = 10 in the R_h = ct universe is approximately 0.74 Gyr versus 0.46 Gyr in ΛCDM. As shown in Figure 2, mass growth during this epoch in R_h = ct is therefore advantaged by a factor of ∼e⁶ ≈ 400 over its ΛCDM counterpart for this scenario. These dramatically different growths, as measured by redshift, have important consequences on our ability to detect quasars at z ≳ 7, and is the primary focus of our work.

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3.1. Disk Emission

Our emission model follows the basic development in Pezzulli et al. (2017). In the classical model, a geometrically thin disk has an inner radius given by the last stable orbit at radius

$$\begin{eqnarray}&&{r}_{0}=3{R}_{{\rm{S}}}=\displaystyle \frac{6\,{{GM}}_{\mathrm{bh}}}{{c}^{2}},\end{eqnarray} \tag{ 5 }$$

and has the temperature profile

$$\begin{eqnarray}&&T(r)={\left(\displaystyle \frac{3{{GM}}_{\mathrm{bh}}\dot{M}}{8\pi \sigma {r}^{3}}\right)}^{1/4}{\left(1-\sqrt{\displaystyle \frac{{r}_{0}}{r}}\right)}^{1/4},\end{eqnarray} \tag{ 6 }$$

for which the maximum temperature is achieved at $r=\tfrac{49}{36}{r}_{0}$ (Shakura & Sunyaev 1973). If the disk emits a perfect blackbody, the bolometric luminosity is then given by the well know expression

$$\begin{eqnarray}&&{L}_{\mathrm{bol}}=\displaystyle \frac{1}{12}\dot{M}{c}^{2},\end{eqnarray} \tag{ 7 }$$

which sets the radiative efficiency at _r = 1/12. One thus need only specify the BH mass and the bolometric luminosity (or alternatively, the mass accretion rate $\dot{M}$) in order to calculate the emission from the disk.

To allow for a more general treatment, where both L_bol and _r can be used as model parameters, we calculate the disk emission through the expression

$$\begin{eqnarray}&&{L}_{\nu }={L}_{0}{\int }_{{r}_{i}}^{\infty }{B}_{\nu }(T[r])\,r\,{dr},\end{eqnarray} \tag{ 8 }$$

where B_ν is the Planck function and L₀ is a normalizing factor used to set the bolometric luminosity

$$\begin{eqnarray}&&{L}_{\mathrm{bol}}={\int }_{0}^{\infty }{L}_{\nu }\,d\nu .\end{eqnarray} \tag{ 9 }$$

The inner disk radius r_i is determined by considering whether a thin disk or a slim disk serves as a better representation of the system under consideration (Abramowicz et al. 1988). Specifically, for a disk to remain geometrically thin, L_bol ≲ 0.3 L_Edd. If this condition is not met, radiation pressure inflates the disk, which is then better described by a slim disk model. In this case, photons are trapped for radii $r\lesssim {r}_{\mathrm{pt}}=1.5H({R}_{{\rm{S}}}/r)(\dot{M}/{\dot{M}}_{\mathrm{Edd}})$, where H is the half-disk thickness. Assuming H/r = 2/3, and since $\dot{M}/{\dot{M}}_{\mathrm{Edd}}={L}_{\mathrm{bol}}/{L}_{\mathrm{Edd}}$, we therefore set r_i = Max[3 R_S, λ_Edd R_S].

3.2. Soft X-Ray Emission

X-ray surveys have proven to be suitable for identifying quasars at high redshift, and are advantaged over lower wavelengths due to a smaller amount of obscuration and less contamination or dilution from the host galaxy. The X-ray emission originates from a hot corona that surrounds the disk (see also Liu & Melia 2001), and can be parametrized as a power-law with exponential cutoff at E_c = 300 keV:

$$\begin{eqnarray}&&{L}_{X,\nu }\propto {\nu }^{-{\rm{\Gamma }}+1}{e}^{-h\nu /{E}_{c}},\end{eqnarray} \tag{ 10 }$$

where the photon index is set through the empirical relation

$$\begin{eqnarray}&&{\rm{\Gamma }}=0.32{\mathrm{log}}_{10}\,\left(\displaystyle \frac{{L}_{\mathrm{bol}}}{{L}_{\mathrm{Edd}}}\right)+2.27,\end{eqnarray} \tag{ 11 }$$

(Brightman et al. 2013). We use the results of Lusso & Risaliti (2106; Figure 6) to then normalize the X-ray luminosity via the expression

$$\begin{eqnarray}&&{\mathrm{log}}_{10}{L}_{2\mathrm{keV}}=0.638{\mathrm{log}}_{10}{L}_{2500}+7.074,\end{eqnarray} \tag{ 12 }$$

where both luminosity densities are in units of erg s⁻¹ Hz⁻¹. The disk (solid) and X-ray (dashed) emission for systems with M_bh = 10⁶ M_⊙ and 10⁹ M_⊙, both with λ_Edd = 1 and with _r = 0.1, are shown in Figure 3.

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Since the greatest sensitivity to the emission produced in our model occurs in soft X-rays (see, e.g., Pezzulli et al. 2017), we consider only the soft X-ray band between _l = 0.5 keV and _u = 2 keV. For a BH at redshift z, the luminosity emitted in this band is given by the expression

$$\begin{eqnarray}&&{L}_{X}={\int }_{{\epsilon }_{l}(1+z)}^{{\epsilon }_{u}(1+z)}{L}_{X,\nu }\,d\nu .\end{eqnarray} \tag{ 13 }$$

The flux received at Earth is then given by

$$\begin{eqnarray}&&{F}_{X}=\displaystyle \frac{{L}_{X}}{4\pi {D}_{L}^{2}},\end{eqnarray} \tag{ 14 }$$

where D_L is the luminosity distance, which in ΛCDM and R_h = ct are given, respectively, by the expressions

$$\begin{eqnarray}&&{D}^{{\rm{\Lambda }}\mathrm{CDM}}=\displaystyle \frac{c}{{H}_{0}}(1+z)\\ &&\,\times {\displaystyle \int }_{0}^{z}\displaystyle \frac{{du}}{\sqrt{{{\rm{\Omega }}}_{m}{(1+u)}^{3}+{{\rm{\Omega }}}_{r}{(1+u)}^{4}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}{(1+u)}^{3+3w}}},\end{eqnarray} \tag{ 15 }$$

and

$$\begin{eqnarray}&&{D}^{{R}_{{\rm{h}}}={ct}}=\displaystyle \frac{c}{{H}_{0}}(1+z)\mathrm{ln}(1+z).\end{eqnarray} \tag{ 16 }$$

We note that these distances are within 10% of each other throughout the z = 10 to z = 6 epoch (see Figure 4), indicating that the observational differences of quasars during this epoch are due almost entirely to the difference in the temporal evolution (and subsequently, the mass growth) between the two cosmologies. In addition, since the luminosity distances are very similar at z ∼ 6, we do not compensate for observationally determined values of luminosity derived assuming ΛCDM when using those results in R_h = ct.

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We ignore the Compton reflection of X-rays by the disk since the effect is minimal in the soft X-ray band (Magdziarz & Zdziarski 1995; Markoff et al. 1997; Trap et al. 2011). To keep the analysis as simple as possible, we also ignore absorption due to interactions with the surrounding gas and dust. Our flux calculations and subsequent estimates on the number of observable quasars should therefore be taken as upper limits. However, the results of Pezzulli et al. (2017) indicate that even if absorption is important, the overall effect out to z ∼ 10 is not expected to be severe enough to impact our results.

We now combine the mass-growth model developed in Section 2 with the emission model from Section 3 in order to determine the observable flux of BHs evolving in ΛCDM and R_h = ct. For illustrative purposes, we first apply our model to the sample of observed quasars highlighted in Nanni et al. (2017) with the best counting statistics in X-rays: J0100+2802, J1030+0524, J1120+0641, J1148+5251, and J1306+0356. The values of z, M_BH, and λ_Edd for these sources are reproduced in Table 1. The adopted mass for J1120+0641 is based on observations of the Mg ii line, while the mass and Eddington ratio for J1030+0524 and J1306+0356 were obtained by averaging the results presented in Table 4 of de Rosa et al. (2011), based on the use of their Equation (4). The radiative efficiency is assumed to be _r = 0.1 in all cases. We calculate the observed flux in the 0.5–2 keV soft X-ray band as a function of redshift. The color coded results are shown in Figure 5 for ΛCDM and Figure 6 for R_h = ct, with data points representing the 0.5–2 keV fluxes derived by Nanni et al. (2017) based on Chandra observations.

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Our results, which are in fairly good agreement with the observations, indicate that the difference in timelines between ΛCDM and R_h = ct has clear implications for the observability of quasars between redshifts 6–10. In ΛCDM, assuming the quasars used in our analysis are representative of the broader population, a significant fraction of their counterparts would produce a soft X-ray flux above the lowest observed flux in our sample (dotted–dashed line) between redshifts z ≈ 7–7.5, and would produce a soft X-ray flux above the Chandra Deep Field South limit (dotted line) out to z ∼ 10. The situation is considerably different in R_h = ct, where relatively few quasars would be detected above our established sample threshold (dotted–dashed line) out to a redshift of z ≈ 7.5, and detection at the Chandra Deep Field South limit would be rare for z ≳ 10.

While Figures 5 and 6 provide an important insight into how SMBHs evolve in both cosmologies, we wish to put our analysis on a firmer statistical footing. The recent (and ongoing) discovery, imaging, and spectroscopic analysis of quasars at z ≈ 6 now allows us to carry out a statistical analysis of their properties over a range of luminosities, and has lead to a determination of the quasar mass function at z = 6. Specifically, the analysis of Willott et al. (2010), based on the absolute magnitude at 1450 Å for a sample of z ≈ 6 quasars, yielded a Schechter mass function of the form

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{M}[{M}_{\mathrm{bh}};z=6]={{\rm{\Phi }}}_{0}{\left(\displaystyle \frac{{M}_{\mathrm{bh}}}{{M}^{* }}\right)}^{\alpha }{e}^{-{M}_{\mathrm{bh}}/{M}^{* }},\end{eqnarray} \tag{ 17 }$$

with best-fit parameters Φ₀ = 1.23 × 10⁻⁸ Mpc⁻³ dex⁻¹, M* = 2.24 × 10⁹ M_⊙, and α = −1.03. This mass function normalized to yield the number of quasars dN(M_bh, z) per mass dex between redshift z and z + dz, evaluated at z = 6, is shown in Figure 7 for both cosmologies.

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To keep our analysis as direct as possible, we use the fairly narrow distribution in λ_Edd values observed at z ≈ 6 (see Figure 6 in Willott et al. 2010) as justification for setting λ_Edd = 1 for all quasars in the early universe. In the absence of mergers, and with accretion occurring at the Eddington rate, all BHs evolve lock-step during the early universe, and the mass function at any redshift can be easily obtained through the transformation

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{M}[{M}_{\mathrm{bh}};z]={{\rm{\Phi }}}_{M}[{M}_{\mathrm{bh}}{e}^{\displaystyle \frac{(1-{\epsilon }_{r})}{{\epsilon }_{r}}\displaystyle \frac{{\lambda }_{\mathrm{Edd}}}{{t}_{\mathrm{Edd}}}({t}_{6}-{t}_{z})};z=6],\end{eqnarray} \tag{ 18 }$$

where t_z is the age of the universe at redshift z (see Equations (2)–(4).

In order to assess the detectability of quasars at higher redshift for both cosmologies under consideration, we convert the luminosity function to a flux function at values of z = 6, 7, and 8 for each model. The results are shown in Figure 8, with the solid curves representing the ΛCDM case and the dashed curves representing the R_h = ct case. Note that the slight mismatch at z = 6 results from the slight difference in luminosity distance between the two cases, which, as discussed above, was not corrected for. As can be clearly seen, a significantly smaller fraction of the quasars detected at z = 6 can also be seen at higher redshifts for R_h = ct than for ΛCDM. This result stems almost entirely from the difference in mass growth between the two scenarios. For example, a time of 0.17 Gyr passes between z = 7 and z = 6 in ΛCDM, while the corresponding span of time is 0.26 Gyr in R_h = ct. At z = 6, an SMBH with mass M_bh = 1.5 × 10⁶ M_⊙ produces a soft X-ray flux at the Chandra limit. That result does not change much at z = 7 for either universe, since the luminosity distance changes by less than 25%. However, an SMBH evolving from z = 7 to z = 6 increases its mass by a factor of 30 in ΛCDM, and a factor of 180 in R_h = ct. The corresponding shift in the mass function, as expressed by Equation (18), is therefore much greater in the R_h = ct universe.

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To further illustrate this point, we next determine what fraction of quasars that produce a flux above the Chandra soft X-ray band limit at z = 6 would still do so at higher redshifts. As noted above, a BH at z = 6 would need a mass of M_bh = 1.5 × 10⁶ M_⊙ to produce a flux equal to the Chandra Deep Field South soft X-ray band flux limit in our accretion model. Taking as our parent population all BHs at z = 6 with mass greater than this limit, we then find the corresponding masses for which the population of BHs with a greater mass represent 80%, 60%, 40%, 20%, 10%, 1%, and 0.1%, respectively, of the parent population. For each of these demarking masses, we then calculate the flux evolution, as was done for Figures 5 and 6, using the same parameters _r = 0.1 and λ_Edd = 1. The results are displayed in Figure 9 for ΛCDM and Figure 10 for R_h = ct. In ΛCDM, 10% of our parent population would be observable (above the Chandra limit) beyond a redshift of z ≈ 6.55, and only around 1% would be observable beyond redshift z ≳ 7. In contrast, for R_h = ct, 10% of our parent population would be observable beyond a redshift of z ≈ 6.35, and the most massive 1% would be observable beyond a redshift of z ≈ 6.7. These results indicate that the next generation of observations, which should be able to provide a statistically significant number of z ≳ 7 detections, will be able to differentiate between ΛCDM and R_h = ct. In both cases, only the most massive 0.1% of quasars produce a flux comparable to the minimum observed flux from our Nanni et al. (2017) sample used for Figures 5 and 6 (dotted–dashed line) at a redshift of z ∼ 6.

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We conclude this analysis by estimating how many quasars should be detectable above a given flux threshold as a function of redshift by integrating the quasar mass function above that threshold out to z = 10. This number is given by the integral expression

$$\begin{eqnarray}&&N(\geqslant F;z)\equiv {\int }_{z}^{10}{\int }_{{\mathrm{log}}_{10}[{M}_{t}(z^{\prime} )]}^{\infty }{\rm{\Phi }}({M}_{\mathrm{bh}},z^{\prime} ){V}_{z^{\prime} }\,d[{\mathrm{log}}_{10}({M}_{\mathrm{bh}})]\,{dz}^{\prime} ,\end{eqnarray} \tag{ 19 }$$

where ${V}_{z}=4\pi {D}_{c}^{2}\,{{dD}}_{c}/{dz}$ is the comoving differential volume and M_t(z) represents the BH mass at redshift z required to produce a soft X-ray flux equal to F. Note that the comoving distance D_c is smaller than the luminosity distance by a factor of (1 + z). The results are presented in Figure 11. As noted already, there is a significant difference in the expected number of detectable quasars predicted by the two cosmological models.

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Based on the observed luminosity function at z = 6, which is used to normalize the expected number of detectable quasars in both models, our analysis indicates that there is a steep drop off in the number of quasars that can be observed at z ∼ 7 compared to z ∼ 6 at flux limits greater than ≳10⁻¹⁵ erg s⁻¹ cm⁻². As noted above, an SMBH evolving from z = 7 to z = 6 increases its mass by a factor of 30 in ΛCDM, and a factor of 180 in R_h = ct. Similar to what was seen in Figure 8, the result is a more pronounced downward (leftward) shift (by about an order of magnitude) in the R_h = ct curve than its ΛCDM counterpart at z = 7 compared to their common z = 6 curves. But the turnover in the mass distribution function acts to amplify the observational consequences between the two cosmologies. With a flux sensitivity ≳3 × 10⁻¹⁵ erg s⁻¹ cm⁻² (represented by the dotted–dashed vertical line in Figure 11), it seems very unlikely that eROSITA will be able to provide the observational evidence needed to discriminate between the two cosmological models under consideration. In contrast, the roughly three orders of magnitude difference in the number of z ∼ 7 quasars that can be observed in R_h = ct versus ΛCDM for the ATHENA flux sensitivity (represented by the dashed vertical line in Figure 11) makes it quite likely that the observations made by that instrument will be able to discriminate between the two cosmologies in the next decade. Specifically, while we expect that ATHENA will detect very few, if any, quasars at z ∼ 7 in R_h = ct, it should detect several hundred of them in ΛCDM—a rather compelling quantitative difference.

The detection of billion-solar-mass quasars at z ≳ 6 has created some tension with the Planck ΛCDM model, in the sense that conventional Eddington-limited accretion, as we understand it in the local universe, could not have produced such large objects in the scant 400–500 Myr afforded them by the timeline in this cosmology. Remedies to circumvent this problem have included models to create ∼10⁵ M_⊙ seeds or to permit transient super-Eddington accretion, requiring a very low duty-cycle. Both of these solutions are anomalous because no evidence for either of them has ever been seen. In fact, the quasar luminosity function toward z ∼ 6 suggests that the inferred accretion rate saturates at close to the Eddington value, with a spread no greater than about 0.3 dex. The data seem to be telling us that these SMBHs probably grew to their observed size near z = 6 by accreting more or less steadily at roughly the Eddington value.

In previous work, we demonstrated that an alternative, perhaps more elegant, solution to this problem may simply be to replace the timeline in ΛCDM with that in the R_h = ct universe. By now, these two models have been compared with each other and tested against the observations using over 20 different kinds of data. The R_h = ct model has not only passed all of these tests, but has actually been shown to account for the data analyzed thus far better than the standard model. There is therefore ample motivation to advance the study of SMBH evolution in this cosmology beyond mere demographics.

This has been the goal of this paper—to examine the progenitor statistics in both models, based on the observed luminosity function at z ∼ 6. We have sought to keep the analysis as simple and straightforward as possible, avoiding unnecessarily complicated SED contributions. For this purpose, the 0.5–2.0 keV flux, thought to be produced in the corona overlying the accretion disk, appears to be an ideal spectral component. The model for producing this emissivity is simple, and probably reliable over a large range in BH mass. In addition, one can easily compensate for a transition in the accretion rate, from low to large values.

With this basic accretion model, we have demonstrated that—for the two cosmologies examined here—the difference in the expected number of detections with the upcoming ATHENA mission is very large. According to Figure 11, and based on the observed luminosity function at z ∼ 6, the ATHENA mission is expected to detect approximately 3.9 × 10⁻⁶ quasars per square degree at z ∼ 7 in R_h = ct, i.e., approximately 0.16 over the whole sky. By comparison, this number is ∼3.9 × 10⁻³ quasars per square degree at z ∼ 7 in ΛCDM, or roughly 160 over the whole sky.

The caveat, of course, is that we have ignored the impact of mergers throughout this analysis. Superficially, one could reasonably expect that the merger rate should be about the same in both cosmologies. Nonetheless, the numbers shown in Figure 11 should be viewed as upper limits. One also needs to take into account the fact that an observed detection rate lower than that predicted by ΛCDM may be partially due to SMBH growth via these mergers, rather than it being a strong indication that the timeline in R_h = ct is preferred by the data. On the flip side, mergers cannot lead to a detection rate much higher than that shown in Figure 11, depending on how reliable our streamlined accretion model happens to be. Thus, if the number of high-z quasars detected with future surveys is closer to that predicted by ΛCDM (i.e., the solid curves in this figure), this would argue strongly against R_h = ct, particularly at z ∼ 8, where the difference is expected to be even more pronounced.

We are grateful to the anonymous referee for several important suggestions that led to an improved presentation of our results. F.M. is grateful to the Instituto de Astrofísica de Canarias in Tenerife and to Purple Mountain Observatory in Nanjing, China for their hospitality while part of this research was carried out. F.M. is also grateful for partial support to the Chinese Academy of Sciences Visiting Professorships for Senior International Scientists under grant 2012T1J0011, and to the Chinese State Administration of Foreign Experts Affairs under grant GDJ20120491013. M.F. is supported at Xavier University through the Hauck Foundation.