AGN EVOLUTION FROM A GALAXY EVOLUTION VIEWPOINT

As noted above in Equation (3), the QLF will, with our Ansätze, be a convolution of the black hole mass function ϕ_BH(m_bh) (itself derived from the stellar mass function of SF galaxies ${\phi }_{\mathrm{SF}}({m}_{\ast })$) and the distribution of Eddington ratios $\xi (\lambda ),$ which gives the distribution of luminosities for black holes of a given mass. In this section we look at the general features of the ξ(λ) distribution that are needed to produce a double power-law QLF from a Schechter-like mass function and derive the connections that will exist between the parameters of these different functions.

First, we immediately note that in a model with a mass-independent ξ(λ) and an input Schechter mass function, the only way to produce a power law at the bright end of the QLF (with slope γ₂) is to have a ξ(λ) that is also a power law of the same slope at high values of the Eddington ratio. This power-law will then ensure that there is a high-end power law in the QLF even as the mass function drops off exponentially. This is shown in Appendix A. Given the limited dynamic range of the data, other representations of the QLF instead of a double power law are possible, e.g., a log-normal bright end or a modified Schechter function (e.g., Hopkins et al. 2007; Aird et al. 2012; Veale et al. 2014). We choose the conventional double power law for analytic simplicity and because it certainly provides a reasonable representation of the available data. Denoting the slope of the high end of Eddington ratio distribution by δ₂ we can thus equate

$$\begin{eqnarray}&&{\gamma }_{2}={\delta }_{2},\end{eqnarray} \tag{ 4 }$$

with γ_2, the bright end slope of the QLF.

At the faint end, we may also expect a power-law QLF but now the QLF faint end slope γ₁ will be given by the steeper of the low end slope of ϕ_BH(m_bh) and the low end slope of ξ(λ), which we denote by δ₁ (see also the discussion in Veale et al. 2014). Figure 1 illustrates what is happening with the faint end of the QLF (with slope) for different Eddington ratio assumptions.

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We can express this as

$$\begin{eqnarray}&&{\gamma }_{1}={\rm{max}}({-\alpha }_{\mathrm{BH}},{\delta }_{1}).\end{eqnarray} \tag{ 5 }$$

The natural conclusion of a mass-independent Eddington rate distribution is that the faint end of the QLF is set up by either the low end slope of the underlying black hole mass function or by the low end slope of the Eddington ratio function. If the logarithmic slope of the m_bh versus m_* relation is β as above, then the faint end slope of the black hole mass function will be related to that of the SF galaxies by ${\alpha }_{\mathrm{BH}}={\alpha }_{\mathrm{SF}}/\beta .$

We note at this point that there is good observational evidence that the faint end QLF slope, γ₁, is similar to the observed low mass slope of the mass function of SF galaxies, α_SF, allowing for the reversal of sign in our definition. The QLF faint end slope, ${\gamma }_{1}$ is usually observed to be between 0.3 and 0.9 (Hasinger et al. 2005; Hopkins et al. 2007; Aird et al. 2010; Masters et al. 2012; Ueda et al. 2014). On the other hand, the observed values of α_SF range from −0.6 to −0.2 (Baldry et al. 2008; Pérez-González et al. 2008; Peng et al. 2010; González et al. 2011; Kajisawa et al. 2011; Baldry et al. 2012; Lee et al. 2012) with most newer studies converging around α_SF ∼ −0.4.

The data are therefore consistent with the idea that ${\gamma }_{1}\sim {-\alpha }_{\mathrm{SF}}.$ This suggests that γ₁ is indeed being set by the low end slope of the black hole mass function (and not the low end of the Eddington ration distribution) and further that β ∼ 1. We will henceforth assume that this is the case, i.e., that ${\alpha }_{\mathrm{BH}}={\alpha }_{\mathrm{SF}}\approx {-\gamma }_{1}.$ This then means that the low end slope δ₁ of ξ(λ) can take any value that is shallower than this, i.e., ${\delta }_{1}\lt {-\alpha }_{\mathrm{SF}}\sim 0.4.$ The high- and low-end power laws of the Eddington ratio distribution will meet in a knee at a characteristic Eddington ratio, which we denote by ${\lambda }^{\ast }$, at which point the value of ξ(λ) has a characteristic value ${\xi }_{\lambda }^{\ast }.$

To further demonstrate our approach we use the simplest function possible for the Eddington ratio distribution that reproduces, analytically, the required broken power-law shape of QLF. We use a "triangular" distribution in which some fraction, f_d, of objects is active above a certain threshold value ${\lambda }^{\ast }$ with an Eddington ratio distribution that is a single power law with slope δ₂, while (1 – f_d) of the objects are completely inactive. Effectively this sets the low end slope ${\delta }_{1}\sim -\infty .$

The exact functional form of $\xi (\lambda )$ can then be written as

$$\begin{eqnarray}&&\xi (\lambda )=\displaystyle \frac{{dN}}{{Nd}\mathrm{log}\lambda }={\xi }_{\lambda }^{\ast }{\left(\displaystyle \frac{\lambda }{{\lambda }^{\ast }}\right)}^{-{\delta }_{2}},\quad \lambda \gt {\lambda }^{\ast },\end{eqnarray} \tag{ 6 }$$

where ${\xi }_{\lambda }^{\ast }$ in constrained by the requirement that all of the objects have to be either active or inactive, so ξ in this case has to integrate to a duty cycle f_d.

In Appendix A we show that the knee of the QLF, ${L}^{\ast },$ will be related to the parameters of the original distribution functions through a formula describing the population, which is analogous to Equation (2) of individual black holes, i.e.,

$$\begin{eqnarray}&&{L}^{\ast }={10}^{38.1}\cdot {\lambda }^{\ast }\cdot {M}_{\mathrm{BH}}^{\ast }\cdot {{\rm{\Delta }}}_{L}({\gamma }_{2}),\end{eqnarray} \tag{ 7 }$$

where the Δ_L factor denotes a small multiplicative factor that is weakly dependent on γ₂ and varying by less than 0.15 dex.

In the same appendix, we also show that the ${\phi }_{\mathrm{QLF}}^{\ast }$ normalization of the QLF, i.e., the value of ϕ_L at L = L^∗ will be linked to the normalization of the galaxy mass function ${\phi }_{\mathrm{SF}}^{\ast }$ and the normalization of the Eddington ratio distribution, ${\xi }_{\lambda }^{\ast },$

$$\begin{eqnarray}&&{\phi }_{\mathrm{QLF}}^{\ast }={\phi }_{\mathrm{SF}}^{\ast }\cdot {\xi }_{\lambda }^{\ast }\cdot {{\rm{\Delta }}}_{\phi }({\gamma }_{2})\end{eqnarray} \tag{ 8 }$$

where the Δ_ϕ denotes once again a small multiplicative factor, weakly dependent on γ₂ and varying by less than 0.15 dex. This is not surprising and is stating the fact that the normalization of objects at a given luminosity L^∗ is connected with the normalization of the contributing functions at ${M}_{\mathrm{BH}}^{\ast }$ and ${\lambda }^{\ast }.$ Simply put, if there are more objects at mass M^∗ and more of them are active at ${\lambda }^{\ast },$ then we expect also more objects with a corresponding luminosity ${L}^{\ast }.$

Even though the triangular distribution of ξ(λ) is a simple and analytically tractable one, it is unlikely to describe the real AGN population. In the remainder of this paper we will adopt a more realistic broken power-law distribution of the Eddington ratio given by

$$\begin{eqnarray}&&\xi (\lambda )=\displaystyle \frac{{dN}}{{Nd}\mathrm{log}\lambda }=\displaystyle \frac{{\xi }_{\lambda }^{\ast }}{{(\lambda /{\lambda }^{\ast })}^{{\delta }_{1}}+{(\lambda /{\lambda }^{\ast })}^{{\delta }_{2}}},\end{eqnarray} \tag{ 9 }$$

where we set δ₁ = 0 for further analysis. This function fits both observations and hydrodynamical simulations better, which show no sudden cutoff in the Eddington ratio distribution at some single value (e.g., Novak et al. 2011; Kelly & Shen 2013). This distribution diverges logarithmically at the low λ end. Since the integral of ξ(λ) will therefore reach unity at some low value of $\lambda \lt \lt {\lambda }^{\ast },$ all black holes are "active" at some very low level and we need no longer consider "inactive" ones.

As discussed in Section 3.1, this distribution will also reproduce the same shape of the QLF as the "triangular" distribution just discussed, provided that ${\delta }_{1}\lt {-\alpha }_{\mathrm{BH}}.$

We note that such a distribution, (with a sharp break at ${\lambda }^{\ast },$ instead of the "smooth" version above), would naturally arise if individual AGNs are boosted to some initial Eddington ratio above ${\lambda }^{\ast }$ (the distribution of which is given by γ₂) and thereafter decay exponentially with a constant decay time τ. In this case, the concept of the "duty cycle" f_d in the previous section is replaced by the value of ${\xi }_{\lambda }^{\ast }.$ If the "boost plus decay" scenario is relevant, then it is easy to see that, if AGNs are activated at some fractional rate η per SF galaxy per unit time (e.g., Hopkins et al. 2005), then

$$\begin{eqnarray}&&{\xi }_{\lambda }^{\ast }=\eta \cdot \tau ,\end{eqnarray} \tag{ 10 }$$

so that ${\xi }_{\lambda }^{\ast }$ is the fraction of black holes that are activated in the time interval corresponding to their subsequent decay time. This is quite a useful definition of duty cycle.

Of course, other scenarios could also produce this ξ(λ) distribution and a more general "duty cycle" could be defined as the fraction of black holes accreting above ${\lambda }^{\ast }$ (as in the triangular distribution above). In this particular case, as shown in Appendix A, this definition of duty cycle would be given by

$$\begin{eqnarray}&&{f}_{d}=\displaystyle \frac{{\xi }_{\lambda }^{\ast }}{{\delta }_{2}\mathrm{ln}(10)}.\end{eqnarray} \tag{ 11 }$$

The point is that the value of ${\xi }_{\lambda }^{\ast }$ is a good indicator of a generalized duty cycle.

If ${\delta }_{1}\lt {-\alpha }_{\mathrm{BH}},$ then the exact choice $\xi (\lambda )$ below ${\lambda }^{\ast }$ is of no great importance to the convolution model and the connections between the parameters, which we can derive analytically in the simplified model as in Equations (4), (5), (7), and (8) will still hold. Specifically, we can write

$$\begin{eqnarray}{\gamma }_{2} & = & {\delta }_{2},\\ {\gamma }_{1} & = & {\rm{max}}({-\alpha }_{\mathrm{BH}},{\delta }_{1}),\\ {L}^{\ast } & = & {10}^{38.1}\cdot {\lambda }^{\ast }\cdot {M}_{\mathrm{BH}}^{\ast }\cdot {{\rm{\Delta }}}_{L}({\delta }_{1},{\gamma }_{2}),\\ {\phi }_{\mathrm{QLF}}^{\ast } & = & {\phi }_{\mathrm{SF}}^{\ast }\cdot {\xi }_{\lambda }^{\ast }\cdot {{\rm{\Delta }}}_{\phi }({\delta }_{1},{\gamma }_{2}),\end{eqnarray} \tag{ 12 }$$

where we have now also explicitly shown a δ₁ dependence in the Δ factors because this corrective factor will also depend on our exact choice of low slope of the Eddington ratio distribution. Now using the m_bh/m_* relation, we can also connect the observed L^∗ of the QLF to the M^∗ of the SF galaxy mass function with

$$\begin{eqnarray}&&{L}^{\ast }={10}^{38.1}\cdot {\lambda }^{\ast }\cdot {M}^{\ast }\cdot \left(\displaystyle \frac{{m}_{\mathrm{bh}}}{{m}_{\ast }}\right)\cdot {{\rm{\Delta }}}_{L}({\delta }_{1},{\gamma }_{2}).\end{eqnarray} \tag{ 13 }$$

If the QLF is indeed produced by the convolution described in the previous two sections, then we can immediately see in general terms how the mass distribution of AGNs, or host galaxies of AGNs, selected at a given luminosity will vary with that selection luminosity. This is shown in Figure 2 where we show the mass functions of galaxies and black holes. The masses and number densities are normalized respectively by the M^∗ and ${\phi }_{\mathrm{SF}}^{\ast }$ of the input Schechter mass function of the SF galaxies. These distributions are plotted for different AGN luminosities relative to the knee luminosity L^∗ in the QLFs (see also Equation (1)). To generate this plot we used the model for ξ(λ) considered in the previous section, i.e., with a flat ξ below ${\lambda }^{\ast }$ and with 0.5 dex scatter in the black hole to stellar mass relation. With this normalization the figures apply at all redshifts. Several points should be noted on this figure.

First, all of the mass functions (of both black holes and host galaxies) always show a peak at some mass. This is quite unlike the input mass function of SF galaxies, and thus also of black holes, which increase monotonically to lower masses. The peak arises because the low-mass end of the input mass function is modulated by the high λ part of ξ(λ), simply because it is the highest Eddington ratios that pull the lowest mass black holes (and hosts) into a given luminosity bin. The low mass end of these mass distributions should therefore have a slope of ${\gamma }_{2}+{\alpha }_{\mathrm{SF}},$ i.e., with ${\alpha }_{\mathrm{SF}}\sim -0.4$ and ${\gamma }_{2}\sim 2$ (see below) we predict a low mass slope of +1.6. This is a quite generic and robust prediction of the convolution model and is independent of the choice of δ₁, the low λ behavior of ξ(λ) discussed above. This is because for most luminosities, the high-mass end of the mass functions in Figure 2 is dominated by the exponential fall-off of the input mass function.

Second, at high luminosities, above ${L}^{\ast },$ the effect of the AGN luminosity on the host galaxy mass function is to change the number of hosts, but not to change the peak mass or, to first order, the shape of the mass function. This is because of the steep exponential fall off in the input mass function of galaxies above ${M}^{\ast }.$ Above ${L}^{\ast },$ the peak host galaxy mass is always close to M^∗ because there are so few more massive galaxies. The number of M^∗ galaxies at the peak is determined by how high of Eddington ratios are required to bring these objects into the luminosity range in question. For these high AGN luminosities, the shape of the mass function of (SF) hosts is very similar to that of passive galaxies except that the faint end slope is significantly steeper (${\alpha }_{\mathrm{SF}}+{\gamma }_{2},$ with ${\gamma }_{2}\sim 2$) than that of the passive galaxies, which is given by ${\alpha }_{\mathrm{SF}}+1$ (see Peng et al. 2010). In mathematical form, the mass function of galaxies at any AGN luminosity above L^∗ will have a Schechter shape

$$\begin{eqnarray}&&\phi ({m}_{\ast };L\gt {L}^{\ast })\propto {\left(\displaystyle \frac{{m}_{\ast }}{{M}^{\ast }}\right)}^{{\alpha }_{\mathrm{SF}}+{\gamma }_{2}}\mathrm{exp}\left(-\displaystyle \frac{{m}_{\ast }}{{M}^{\ast }}\right).\end{eqnarray} \tag{ 14 }$$

At lower luminosities well below ${L}^{\ast },$ the behavior changes and a region of intermediate slope appears. For black hole masses between ${({10}^{38.1})}^{-1}{({m}_{\mathrm{bh}}/{m}_{\ast })}^{-1}{({\lambda }^{\ast })}^{-1}L$ and ${({10}^{38.1})}^{-1}{({m}_{\mathrm{bh}}/{m}_{\ast })}^{-1}{({\lambda }^{\ast })}^{-1}{L}^{\ast },$ the mass function will have the slope given by $\alpha +{\delta }_{1}.$ At very low masses we will again always recover the slope of ${\alpha }_{\mathrm{SF}}+{\gamma }_{2}.$

The location of the peak in the black hole mass function therefore depends on the slopes of both underlying distributions (Eddington ratio and galaxy mass function), i.e., on the sign of α + δ₁. Not surprisingly, this intermediate zone also appears in the host galaxy mass function with the same slope. This produces a peak host galaxy mass of

$$\begin{eqnarray}&&{m}_{\mathrm{peak}}\sim \displaystyle \frac{{M}^{\ast }}{{10}^{38.1}{m}_{\mathrm{bh}}{\lambda }^{\ast }}L\qquad | {\alpha }_{\mathrm{SF}}| \gt | {\delta }_{1}| \\ &&{m}_{\mathrm{peak}}\sim \displaystyle \frac{{M}^{\ast }}{{10}^{38.1}{m}_{\mathrm{bh}}{\lambda }^{\ast }}{L}^{\ast }\qquad | {\alpha }_{\mathrm{SF}}| \lt | {\delta }_{1}| .\end{eqnarray} \tag{ 15 }$$

This difference in behavior can be understood as follows: In the $| {\alpha }_{\mathrm{SF}}| \gt | {\delta }_{1}|$ case the dominant contribution to the luminosity function will come from low-mass object accretion at high Eddington ratios, which is more numerous than high-mass object accretion at low Eddington ratios. In the second case, the roles are simply reversed and the main contributor to the QLF will be high-mass AGNs with low Eddington ratios.

If black hole masses were very tightly correlated with host galaxy masses, then clearly the same behavior would be seen in the black hole mass function(s) since these would always be a simple (mass-)scaling of the galaxy mass function(s). However, the effect of scatter in the black hole to host galaxy mass relation is quite marked. This is visible in Figure 2. The reason is clear: the mass function of black holes will now be much smoother than that of the host galaxies since the log normal scatter smooths out the sharp exponential drop of the galaxies at high masses (recall that Figure 2 used a 0.5 dex scatter). Of course, it might be argued that the black hole mass is somehow more fundamental, and that the galaxy mass function should be obtained by smoothing it. However, we know the galaxy mass function quite well, and it has an exponential cutoff (see Peng et al. 2010; Baldry et al. 2012). We do not, at this stage, know the underlying mass function of black holes in SF galaxies. With this scatter, there is a much smoother variation of the peak mass with AGN luminosity.

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The dashed lines show the variation of the peak mass with luminosity. The fact that the difference between these dashed curves on the two panels decreases with luminosity illustrates the well-known "Lauer effect" (Lauer et al. 2007) whereby the typical objects at high AGN luminosities will generally have been scattered above the mean black hole host mass relation. It should be noted, however, that there is no change in the low-mass slope of the mass function(s) due to scatter and this remains a very robust prediction of our model.

It is worth noting in Figure 2 that the peak of the black hole mass distribution varies quite weakly with luminosity, i.e., a change in luminosity of 1 dex is associated with a smaller increase in the peak m_bh. This means that we will generally see higher Eddington ratios in higher luminosity quasars even though the Eddington ratio distribution is taken to be strictly independent of black hole mass. We return to this point in Section 6.1 below.

We stress that the (solid) curves in Figure 2 that show the mass functions of AGN host galaxies (with masses normalized to the Schechter M^∗) at different AGN luminosities (computed relative to the knee of the luminosity function) are an easily testable prediction of the convolution approach, modulo the effects linked to the choice of α and δ₁ discussed above, which can shift the peak of $\phi (m).$

As explained previously, we need to know the mass function of SF galaxies to deduce the SMBH mass function in SF galaxies, ϕ_BH(m_bh), which is an integral part of predicting the QLF in Equation (3). We also want to verify the predictions of Peng et al. (2010, 2012) for the time evolution of the parameters of the galaxy mass function (e.g., Equation (B3) from Peng et al. 2012).

In order to do this, we fit the data for SF galaxies from Ilbert et al. (2013) at all redshifts with a single Schechter function in which the parameters (${\phi }_{\mathrm{SF}}^{\ast }\equiv {\phi }_{\mathrm{SF}}^{\ast }(z),$ ${M}^{\ast }\equiv {M}^{\ast }(z)$) are smoothly varying functions of redshift. Fitting all the data in this fashion, instead of fitting at single redshifts, enables us to follow better the evolution of the parameters. It also reduces the sensitivity to small variations that may compromise fits at a single redshift and add to the degeneracies between parameters. This procedure of simultaneously fitting all the data of the galaxy mass function at different redshifts is analogous to the standard procedure often used in determining the evolving QLF (e.g., Hopkins et al. 2007; Ueda et al. 2014). The functional form that we use for the redshift evolution of the galaxy mass function is

$$\begin{eqnarray}{\phi }_{\mathrm{SF}}({m}_{\ast },z) & = & \displaystyle \frac{{dN}}{d\mathrm{log}{m}_{\ast }}\\ & = & {\phi }_{\mathrm{SF}}^{\ast }(z){\left(\displaystyle \frac{{m}_{\ast }}{{M}^{\ast }(z)}\right)}^{{\alpha }_{\mathrm{SF}}}\mathrm{exp}\left(-\displaystyle \frac{{m}_{\ast }}{{M}^{\ast }(z)}\right),\end{eqnarray} \tag{ 16 }$$

where we model the redshift dependence as

$$\begin{eqnarray}\mathrm{log}{\phi }_{\mathrm{SF}}^{\ast }(z) & = & {a}_{0}+{a}_{1}\kappa +{a}_{2}{\kappa }^{2}+{a}_{3}{\kappa }^{3},\\ \mathrm{log}{M}^{\ast }(z) & = & {a}_{0}+{b}_{1}\kappa +{b}_{2}{\kappa }^{2}+{b}_{3}{\kappa }^{3},\end{eqnarray} \tag{ 17 }$$

with $\kappa \equiv \mathrm{log}(1+z).$ The slope of the low-mass end, α_SF, was kept constant at the local value of α_SF = −0.4 as there is considerable evidence that there is little or no change in the low-mass slope for the redshift range considered (Peng et al. 2014).

The evolution of parameters of mass function in Figure 3 confirms that M^∗ is more or less constant up until at least redshift 2.5, i.e., it verifies the Ansätz of Peng et al. (2010), which establishes a single quenching mass scale M^∗ that does not change with redshift. It is important to stress that our results will not depend critically on the exact functional evolution of M^∗ above $z\gtrsim 2.5,$ but will use the by now well-established observational fact that M^∗ does not change at ${SS}\lesssim 2.5$ and that the evolution in the galaxy population since that time is associated with a "vertical" evolution in ${\phi }_{\mathrm{SF}}^{\ast }.$ We have also performed this analysis with the data set from Muzzin et al. (2013) and the compilation of galaxy mass functions from Behroozi et al. (2013b) and our conclusions presented in the rest of this paper do not change.

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Hopkins et al. (2007) combined measurements of QLFs in different bands, fields, and redshifts in order to characterize the bolometric QLF at epochs $0\lt z\lt 6.$ In their work, the best-fit luminosity-dependent bolometric correction and luminosity and redshift-dependent column density distributions are used in order to construct an estimate of the bolometric QLF, which should be consistent with all of the various individual surveys. Although now several years old, we believe that this synthesized QLF compilation remains the most comprehensive and is used as the basis of the current paper.

Table 1.  Evolution of Star-forming Galaxy Mass Function, Data From Ilbert et al. (2013)

Parameter	Fixed ${\alpha }_{\mathrm{SF}}$ = −0.4
a0	−2.55 ± 0.04
a1	−0.26 ± 0.06
a2	−1.6 ± 0.12
a3	−0.88 ± 0.16
b0	10.9 ± 0.04
b1	−0.53 ± 0.11
b2	3.36 ± 0.11
b3	−3.75 ± 0.13

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We proceed with fitting their tabulated data with a double power-law QLF, as given by Equation (1). We fit both at each redshift individually, and carry out a "full fit" where the parameters (i.e., ${\phi }_{\mathrm{QLF}}^{\ast },$ ${L}^{\ast },$ ${\gamma }_{1}$, and γ₂) are all constrained to be smoothly varying functions of redshift, i.e., adopting a similar approach as we used earlier in our fitting of the galaxy mass function in Section 4.1. This "full" fitting of a redshift-dependent double power law is essentially the same as the approach used in Hopkins et al. (2007), with one important difference. In order to avoid degeneracies in the fits it is necessary, both here and in Hopkins et al. (2007), to fix one or more of the parameters. Hopkins et al. (2007) chose to require a redshift-independent ${\phi }_{\mathrm{QLF}}^{\ast }$ at a constant value. We now know that ${\phi }_{\mathrm{SF}}^{\ast }$ of the galaxy population changes significantly over the redshift range of interest and, as developed in the previous section, the relationship of ${\phi }_{\mathrm{QLF}}^{\ast }$ relative to ${\phi }_{\mathrm{SF}}^{\ast }$ is of great interest in the context of the duty cycle. In contrast, γ₁ is set, in our convolution model, by the low mass slope α_SF of the mass function of SF galaxies. As mentioned before there is not much evidence (see Peng et al. 2014) that this changes significantly, if at all, over the redshift range of interest, nor compelling evidence for a change in γ₁. Therefore, we choose to have a redshift-independent faint end slope of the luminosity function, γ1 and to allow ${\phi }_{\mathrm{QLF}}^{\ast }$ to vary with redshift. It should be noted that our fitting procedure is the same as a "luminosity and density evolution" model (e.g., Aird et al. 2010) except that we are allowing the bright end slope γ₂ to vary. We do this because, as we have seen, γ₂ is set by the high λ behavior of the Eddington ratio distribution ξ(λ).

The parameters in the luminosity function are allowed to vary as

$$\begin{eqnarray}{\rm{log}}\;{\rm{}}{\phi }_{\mathrm{QLF}}^{\ast } & = & {c}_{0}+{c}_{1}\kappa +{c}_{2}{\kappa }^{2}+{c}_{3}{\kappa }^{3},\\ {\rm{log}}\;{\rm{}}{L}^{\ast } & = & {d}_{0}+{d}_{1}\kappa +{d}_{2}{\kappa }^{2}+{d}_{3}{\kappa }^{3}+{d}_{4}{\kappa }^{4},\\ {\gamma }_{1} & = & {e}_{0},\\ {\gamma }_{2} & = & {f}_{0}+{f}_{1}z+{f}_{2}{z}^{2},\end{eqnarray} \tag{ 18 }$$

with again κ = log(1 + z). We follow Hopkins et al. (2007) in fitting γ₁ and γ₂ as a polynomial in z (rather than κ =log(1 + z)), but this choice is not of great importance.

Additional degrees of freedom were added to the fit until we can find no appreciable quantitative difference in the quality of the fit. The values of the double power-law parameters in the smoothly varying "full-fit" are given in Table 2 and the fits are shown with orange curves in Figure 4.

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Table 2.  Best Fits to QLF Data from Hopkins et al. (2007)

Parameter	"Full Fit"	With ${\phi }_{\mathrm{QLF}}^{\ast }\propto {\phi }_{\mathrm{SF}}^{\ast }$
c0	−4.35 ± 0.06	−1.79 ± 0.04
c1	0.059 ± 0.07	⋯
c2	−3.2 ± 0.3	⋯
c3	1.73 ± 0.2	⋯
d0	44.67 ± 0.06	44.67 ± 0.04
d1	4.02 ± 0.07	4.09 ± 0.09
d2	3.78 ± 0.08	3.5 ± 0.07
d3	−4.68 ± 0.1	−3.85 ± 0.07
d4	−5.7 ± 0.2	−5.98 ± 0.15
e0	0.5 ± 0.03	0.52 ± 0.04
f0	1.64 ± 0.05	1.63 ± 0.06
f1	0.54 ± 0.05	0.56 ± 0.06
f2	−0.12 ± 0.01	−0.106 ± 0.03

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The redshift dependences of ${\phi }_{\mathrm{QLF}}^{\ast }$ and L^∗ are shown in the panels of Figure 5 and we discuss these results in the next section.

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The first result is that we notice a strong similarity between the observed redshift dependence of ${\phi }_{\mathrm{QLF}}^{\ast }$ in Figure 5 and the observed ${\phi }_{\mathrm{SF}}^{\ast }$ in Figure 3. Both drop by ∼0.5 dex by redshift 2. Their relative evolution is explicitly compared in the bottom right panel of Figure 7, which shows that a constant ratio between ${\phi }_{\mathrm{QLF}}^{\ast }$ and ${\phi }_{\mathrm{SF}}^{\ast }$ is perfectly consistent with the data. We note that the fact that the density normalization of the QLF decreases slowly with redshift has been seen in several previous analyses, for instance in the "luminosity and density evolution" model of Aird et al. (2010), which has a 0.4 dex decrease in normalization between redshifts 0 and 2, and Croom et al. (2009) shows a very similar decrease of normalization in their "luminosity and density evolution" model, while Hasinger et al. (2005) show a decrease in normalization of 0.5 dex between redshift 0.6 and 2.4. Here we highlight the striking similarity of this behavior to the observed decline in ${\phi }_{\mathrm{SF}}^{\ast }.$

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Referring back to Section 3.3, a constant ratio between ${\phi }_{\mathrm{QLF}}^{\ast }$ and ${\phi }_{\mathrm{SF}}^{\ast }$ implies a constant duty cycle of the black holes in the context of our convolution model. To explore this further, we now repeat the fitting procedure but set the evolution of ${\phi }_{\mathrm{QLF}}^{\ast }$ to be exactly that of ${\phi }_{\mathrm{SF}}^{\ast },$ i.e., we impose that the evolution of the QLF normalization is the same as the evolution of the normalization of SF galaxies in the redshift range where we have ${\phi }_{\mathrm{SF}}^{\ast }$ available (z < 3.5), while the constant multiplicative offset between these parameters remains a free parameter to be determined by the fitting procedure, i.e.,

$$\begin{eqnarray}&&{\phi }_{\mathrm{QLF}}^{\ast }\propto {\phi }_{\mathrm{SF}}^{\ast }.\end{eqnarray} \tag{ 19 }$$

The results of this fitting are given in Tableapj519757t2 2 and the resulting QLF is shown with the red curve in Figure 4. It can be seen that the fit is extremely good, being only marginally worse then the full fit of Hopkins et al. (2007) (χ²_{this work}/dof = 2.1; χ²_Hopkins/dof=2.0), both of which have the comparable number of free parameters. The main driver for the slightly worse fits in our work is the deviation of the fit from data for low luminosities at low redshifts. The data in this regime may be contaminated by contributions from the stellar populations of the hosts, as discussed in Hopkins et al. (2007).

The parameter evolution in this "${\phi }_{\mathrm{SF}}^{\ast }$-matched" fitting is shown as linked filled circles in Figure 6. As one can see, the points are typically situated on the edges of the individual 1-σ contours, which is to be expected given that χ²/dof = 2.1. We conclude that the change of normalization of the QLF is perfectly consistent with the change of normalization of the SF galaxy mass function over the entire redshift range for which we have the measurements of normalization of the SF galaxy mass function.

The fact that as far as we can tell the ${\phi }_{\mathrm{SF}}^{\ast }(z)$ and ${\phi }_{\mathrm{QLF}}^{\ast }(z)$ track each other throughout cosmic time (at least since z ∼ 3, and quite possibly since earlier epochs also, is an interesting result. It means that the factor connecting these two quantities, which in the convolution model is ${\xi }_{\lambda }^{\ast }$ (slightly modified by Δ_ϕ), remains constant.

As we have discussed above, this suggests that the general "duty cycle," f_d, stays constant over cosmic time (e.g., Equations (8) and (11); see also Conroy & White 2013). Clearly, this would not be the case for other definitions of "duty cycle" that are based, for instance, on the fraction of black holes radiating above some luminosity or accreting above some Eddington ratio, if ${\lambda }^{\ast }$ or L^∗ evolves with time, as it does (see below), but we believe that our definition of duty cycle is the most natural one (as discussed above).

The other striking feature of the QLF is the strong redshift evolution of ${L}^{\ast },$ which increases by almost two orders of magnitude back to z ∼ 2. At higher redshifts, the ${L}^{\ast }(z)$ certainly flattens out and probably declines, although this cannot be established at high significance. The strong initial rise with redshift is seen independent of whether we use the full fit, or the fit constrained to have ${\phi }_{\mathrm{SF}}^{\ast }(z)$ and ${\phi }_{\mathrm{QLF}}^{\ast }(z)$ tracking each other.

In our convolution model, the steep rise in ${L}^{\ast }(z)$ (see Equation (13)) could have been caused in principle by one or more of (a) an evolution of ${\lambda }^{\ast },$ (b) an evolution of M^∗ of the galaxy mass function, or (c) an evolution of the mass ratio ${m}_{\mathrm{bh}}/{m}_{\ast }.$

As discussed in Section 4.1, we know that the characteristic M^∗ of the galaxy population does not change, especially in the redshift range where increase of L^∗ is most prominent (z ≤ 2), so case (b) will not apply. There is however complete degeneracy between cases (a) and (c), i.e., the distribution of specific accretion rates of AGNs and black holes to stellar mass ratio. This is clear in Equation (13) and as has been pointed out by, e.g., Veale et al. (2014). This degeneracy can only be broken if we have information on the black hole masses of the AGN. We will explore this in the next section of the paper.

In this part of the paper we will compare predictions of how the black hole mass–luminosity plane of broad-line AGNs should be populated in our model, and compare these with SDSS data. After creating a mock sample of SF galaxies in an SDSS volume, we will populate them with black holes assuming different redshift-dependent ${m}_{\mathrm{bh}}/{m}_{\ast }$ scaling relations and an assumed scatter. We will then assign Eddington ratios from the evolving $\xi (\lambda ,z)$ distribution and apply an obscuration prescription from Hopkins et al. (2007). These two functions are chosen so that the QLF is reproduced as in the previous section: in other words the redshift evolution in ${m}_{\mathrm{bh}}/{m}_{\ast }$ and the redshift evolution in ${\lambda }^{\ast }$ must combine to produce the observed evolution in the QLF ${L}^{\ast }.$

For the observational distribution, we use data from two observational studies (Shen et al. 2011; Trakhtenbrot & Netzer 2012) to show that our results do not depend critically on the data choice and to give the reader a graphical impression of the uncertainties involved in this kind of measurement. By comparing our mock data with the observed distribution we will be able to see which combination of redshift-dependent ${m}_{\mathrm{bh}}/{m}_{\ast }$ and $\xi (\lambda ,z)$ best reproduces the observational data. We take into account the obscuration factor and the differences in bolometric correction between different works and apply the same bolometric correction to the data and model (namely one used in Shen et al. 2011) to make them directly comparable.

By using this mock sample approach we can fully account for biases introduced by the luminosity selection of the quasar samples. We recreate data only up to z = 2, as black hole mass estimates for higher redshifts are based on the broad C iv emission line, which was shown to be far less reliable for these purposes (Shen et al. 2008; Trakhtenbrot & Netzer2, and references therein).

We first show in the topmost panels of the Figure 8 the modeled distribution of quasars in the black hole mass–luminosity (m_bh, L) plane in three representative redshift bins if we assume a redshift-independent ${m}_{\mathrm{bh}}/{m}_{\ast }$ with the standard value of ${m}_{\mathrm{bh}}/{m}_{\ast }\approx {10}^{-2.8}.$ We introduce a log-normal scatter in this relationship of 0.5 dex to account for scatter in the ${m}_{\mathrm{bh}}/{m}_{\ast }$ relationship. The solid diagonal line indicates the Eddington limit ($\lambda =1$).

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The data from (Shen et al. 2011; Trakhtenbrot & Netzer 2012) are plotted in the two bottom rows of panels in Figure 8. In these panels, the diagonal dashed and dotted lines indicate the locus of black hole masses for two constant FWHMs (of 1000 km s⁻¹ and 1500 km s⁻¹, respectively) of the emission lines (Hβ and Mg ii) that were used to infer the black hole masses. Systems of lower FWHM will not appear in the samples, e.g., the limit used in Shen et al. (2011) is set at 1200 km s⁻¹.

We see that, although there is good agreement at low redshifts, a non-evolving ${m}_{\mathrm{bh}}/{m}_{\ast }$ relation produces far too many objects with masses that are too small or, equivalently, which have very high Eddington ratios, with around 50% being super-Eddington in the final redshift bin, while only around 2% of objects are super-Eddington in the data. This is a simple consequence of the fact that L^∗ is much higher than locally and is a reflection of the high ${\lambda }^{\ast }$ implied for an unevolving ${m}_{\mathrm{bh}}/{m}_{\ast }$ relation shown in Figure 17.

It should be noted that the problem gets worse if we reduce the scatter in the ${m}_{\mathrm{bh}}/{m}_{\ast }$ relation, as we then have even fewer high-mass black holes. We note that the comparison could not be expected to be perfect because our data is constrained to reproduce the QLF from Hopkins et al. (2007); although SDSS data and the optical QLF is the main contributor to the Hopkins QLF in this luminosity range, there are small contributions from other surveys as well as slightly different bolometric and obscuration corrections that will induce small differences. Nevertheless, the disagreement with a non-evolving ${m}_{\mathrm{bh}}/{m}_{\ast }$ ratio is too large to be due to this.

A much better agreement is obtained (second row of Figure 8) if we adopt an evolving ${m}_{\mathrm{bh}}/{m}_{\ast }$ relation. We adopt as a heuristic example the form ${m}_{\mathrm{bh}}/{m}_{\ast }\propto {(1+z)}^{n}$ with n = 2. The agreement with the observed distribution is considerably better and there are now far fewer objects crossing the Eddington limit (∼3%) at high redshifts. This means that the observed $\sim {(1+z)}^{4}$ increase in $L\ast$ back to $z\sim 2$ would be due to an equal split between a ${(1+z)}^{2}$ change in ${m}_{\mathrm{bh}}/{m}_{\ast }$ and a ${(1+z)}^{2}$ change in characteristic Eddington ratio ${\lambda }^{\ast },$ remembering that these two changes are degenerate in our convolution model without the ${m}_{\mathrm{bh}}/{m}_{\ast }$ data of Figure 8.

Finally we note that, quite independently of any assumptions about ${m}_{\mathrm{bh}}/{m}_{\ast },$ our convolution model naturally recreates the apparent "sub-Eddington boundary" that has been emphasized by Steinhardt & Elvis (2010b), by which we mean the flat upper envelope. This refers to the fact that at all redshifts there seems to be a lack of objects at high masses close to the Eddington limit, which can also be seen in Figure 16 of Trakhtenbrot & Netzer (2012). This can be observed in Figure 8 where the upper contours of the red regions have slopes that are noticeably shallower than the 45° slope that corresponds to a constant Eddington ratio, thereby giving the impression of an absence of high-luminosity high λ sources.

This behavior is quite counterintuitive, and was interpreted by Steinhardt & Elvis (2010b) as being caused by some new physical effect that somehow limits accretion onto more luminous quasars. Various authors have proposed alterations to the measurement methods in the mass–luminosity plane, which could reduce or eliminate this effect (e.g., Rafiee & Hall 2011a, 2011b). We show instead that this behavior is actually expected from the convolution model presented in this paper!

We commented earlier in Section 3.4 that while the typical black hole mass increases with luminosity in our convolution model, it does so sub-linearly, so that the "typical" Eddington ratio must also increase with luminosity. This can be seen also in these plots: the ridge line that is defined by the peak in the mass distributionat a given L is indeed steeper than the 45° line of constant Eddington ratio, which is why the shallower slope of the boundary defined by the contours is so counterintuitive. However, because the number of more massive black holes falls very steeply with mass, because of the decline of the galactic mass function above ${M}^{\ast },$ the contours of constant surface density, given by the contours in the plots in Figure 8 and in the Steinhardt & Elvis (2010b) analysis, are actually shallower than the 45° line.

This is more clearly seen in Figure 9 were we show the appearance of our full sample, before the application of the SDSS luminosity cut. At lower luminosities the effect gets more and more pronounced and we expect that deeper surveys of a given area would find more super-Eddington objects at small masses.

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We emphasize that the reproduction of the Steinhardt & Elvis (2010b) effect in Figure 8 is achieved within our convolution model with an Eddington ratio distribution $\xi (\lambda )$ that is completely independent of black hole mass. The effect noted by Steinhardt & Elvis (2010b) would not be seen (in our model) if the distribution of points in the $({m}_{\mathrm{bh}},L)$ plane was normalized to the total number of black holes (in SF galaxies) at a given mass, which is, of course, unfortunately not observable. Our explanation for the sub-Eddington boundary in Steinhardt & Elvis (2010b) is therefore a kind of "plotting bias" arising from what is plotted, rather than a "selection effect" per se, coming through the construction of the sample via, e.g., emission line widths.

In this section we show how it is possible to reproduce the observed tight correlation between ${m}_{\mathrm{bh}}/{m}_{\mathrm{bulge}}$ in the local universe even if the black hole to stellar mass relation in the SF galaxies is strongly evolving (see also Croton 2006; Hopkins et al. 2006a). In what follows, we loosely equate m_bulge with the stellar mass of an SF galaxy when star formation quenches, which is also the point at which we assume that the black hole ceases growing. Readers uncomfortable with making this association may simply substitute m_bulge in what follows with m_quench.

For this analysis, we use results from the simple galaxy evolution model of Birrer et al. (2014) to construct a set of evolving SF galaxies and their quenched descendants. The details of the Birrer et al. (2014) model are not important for the present purpose since it reproduces well the overall evolution of the galaxy population, which is all that we require here.

For each quenched galaxy seen in the model at the present epoch, we know the mass and redshift at which it quenched and can therefore compute the black hole mass from the adopted redshift-dependent ${m}_{\mathrm{bh}}/{m}_{\ast }$ relation for (SF) galaxies, adding also the adopted scatter. We assume that there is no stellar mass growth after quenching and that there is no central SMBH mass growth after quenching.

After this procedure we are left with a mock sample of quenched galaxies, with their black hole masses known, after which we can account for sample selection effects. This is not trivial, as measurements of masses of black holes are from heterogeneous sources and no single luminosity or other cut is possible.

We therefore decided to create an empirical selection in galaxy mass so that the distribution of galaxy masses in the mock sample broadly matches that of the passive early-type galaxies that have had their SMBH mass measured. We show results for two situations, where we first only consider galaxies that quenched after z = 2 and then include all galaxies that have quenched since z = 5. We differentiate between these two cases since we expect merging, which is not explicitly accounted for here, to have a larger impact for galaxies that quench earlier, and because the assumed mass ratio scaling, which we think is a reasonable approximation (see above) at redshift $z\lt 2,$ may break down at higher redshifts. When we ignore quenching at $2\lt z\lt 5,$ we are losing only about 10% of today's quenched population, but the model is on a much firmer footing.

We have fitted the simulated data, derived from 40 random realizations of the model, with a relation of the form

$$\begin{eqnarray}&&100\left(\displaystyle \frac{{m}_{\mathrm{bh}}}{{m}_{\ast ,\mathrm{quench}}}\right)=a\cdot {\left(\displaystyle \frac{{m}_{\ast ,\mathrm{quench}}}{{10}^{11}{M}_{\mathrm{sun}}}\right)}^{b}\end{eqnarray} \tag{ 20 }$$

by regressing the black hole mass on the stellar mass, and compute the scatter of the simulated galaxies around this relation. We derive values of $(a,b)=(0.40,0.09)$ and a scatter of 0.53 dex for the case of quenching only from z = 2, and values of $(a,b)=(0.63,0.18)$ with a scatter of 0.6 dex for the case of quenching from z = 5. These values should be compared with the observed values of $(a,b)=({0.49}_{-0.05}^{+0.06},0.14\pm 0.08)$ and an intrinsic scatter of 0.29 dex derived in Kormendy & Ho (2013). While the predicted scatter appears to be slightly larger than observed, subsequent merging of galaxies, which has not been modeled here, will act to reduce the scatter (Hirschmann et al. 2010; Jahnke & Macciò 2011).

The origin of the ${m}_{\mathrm{bh}}/{m}_{\mathrm{bulge}}$ relation is the underlying ${m}_{\mathrm{bh}}/{m}_{\ast }$ relation in the SF galaxies assumed in the model, with added scatter due to the fact that galaxies of a given stellar mass today have quenched at various redshifts and, for a given stellar mass, the spread in black hole masses is amplified by the spread in the quenching redshifts. The mean of the ${m}_{\mathrm{bh}}/{m}_{\mathrm{bulge}}$ relation is positioned roughly at the value of ${m}_{\mathrm{bh}}/{m}_{\ast }$ at the mean redshift of quenching at that mass, which is $z\sim 1-1.5$ over a wide range of galaxy masses.

It is interesting to note that objects that have quenched more recently would be expected on average to have a lower ${m}_{\mathrm{bh}}/{m}_{\ast }$ value, because of the evolution in ${m}_{\mathrm{bh}}/{m}_{\ast }.$ It is possible that these recently quenched galaxies could be associated with pseudo-bulges instead of bulges. Kormendy & Ho (2013) has indeed argued that pseudo-bulges do indeed have a lower ${m}_{\mathrm{bh}}/{m}_{\mathrm{bulge}}$ ratio.

We also note in passing that if there is a trend for the typical quenching redshift to increase with increasing stellar mass (e.g., because of interplay of mass and environment quenching, Peng et al. 2010) then this would introduce a tilt in the local ${m}_{\mathrm{bh}}/{m}_{\mathrm{bulge}}$ relation ($b\ne 0$) even though the input ${m}_{\mathrm{bh}}/{m}_{\ast }$ in SF galaxies was perfectly linear.

The link with quenching redshift then prompts another interesting point. There is very good evidence that the sizes of galaxies, at a given mass, are smaller at high redshift (Daddi et al. 2005; Newman et al. 2012; Carollo et al. 2013; Shankar et al. 2013a). We would therefore expect, through virial arguments, that the velocity dispersions, at a given mass, would also be higher. This has been directly observed in a few cases (e.g., van Dokkum et al. 2014). For analytic simplicity, we assume that the scale radius of galaxies scales as $(1+z)$ so that the usual Faber–Jackson-type scaling relation would be expected to evolve as

$$\begin{eqnarray}&&{m}_{\mathrm{star}}\propto {\sigma }^{4}{(1+z)}^{2}.\end{eqnarray} \tag{ 21 }$$

If ${m}_{\mathrm{bh}}/{m}_{\ast }$ scales as ${(1+z)}^{2},$ as we have been exploring in this part of the paper, then this naturally produces an ${m}_{\mathrm{bh}}-\sigma$ relation that is independent of redshift.

This has two implications: first, we would not expect to see any significant evolution in the observed ${m}_{\mathrm{bh}}-\sigma$ relation (see, e.g., recently Shen et al. 2015). Second, we would expect the present-day ${m}_{\mathrm{bh}}-\sigma$ relation for passive galaxies to be tighter than the ${m}_{\mathrm{bh}}-{m}_{\ast }$ relation because of the aforementioned broadening of the latter from the range of z_quench.

We stress that this tighter ${m}_{\mathrm{bh}}-\sigma$ relation would be present even if the velocity dispersion is not playing a direct role in the growth of black holes. Of course, it is also possible that the evolution in the ${m}_{\mathrm{bh}}/{m}_{\ast }$ is in fact caused by the constancy of an underlying ${m}_{\mathrm{bh}}-\sigma$ causal connection. On the other hand, there could well be other causes for ${m}_{\mathrm{bh}}/{m}_{\ast }$ to evolve as ${(1+z)}^{2},$ in which case the tight ${m}_{\mathrm{bh}}-\sigma$ relation would simply be a coincidence.

Finally, we turn to estimates of the ${m}_{\mathrm{bh}}/{m}_{\ast }$ relations for AGNs in the local universe, where it is possible to estimate independently the mass of the galaxy that is hosting an optically active AGN. Matsuoka et al. (2014) made a careful decomposition into nuclear and host contributions of the images of $z\lt 0.6$ quasars in the SDSS Stripe 82. They found that the quasars are predominately hosted in massive SF galaxies, with relatively large ${m}_{\mathrm{bh}}/{m}_{\ast }$ ratios of around ${10}^{-2.5}.$ Matsuoka et al. (2014) were aware of the possibility that selection effects could bias this value upward but could not estimate their magnitude. We can now use our model to examine the expected size of this bias.

To do this we simulate sources within the same sky area as Stripe 82 and recreate objects above the AGN luminosity cut that would have been selected for quasar spectroscopy in the SDSS sample, with a scatter of $\eta =0.4$ dex. The results are shown in Figure 11, represented in the same way as in the original paper. We see that the observed quasars will have ${m}_{\mathrm{bh}}/{m}_{\ast }$ ratios that are indeed much higher than the mean value in the underlying sample, which is indicated by the shaded region in the figure.

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Finally, we notice that even though we inserted the ${m}_{\mathrm{bh}}/{m}_{\ast }\propto {(1+z)}^{2}$ redshift evolution in our model, this evolution would be quite hard to detect in the sample of Matsuoka et al. (2014), owing to the large spread of points and the selection biases connected with such a study. Nevertheless, we do still see a slight change in the mean values of the simulated sample that is not seen in the actual data. We discuss this further in Section 7.2 below.

It is important to appreciate that a quite rapid evolution in ${m}_{\mathrm{bh}}/{m}_{\ast }$ in active (SF) AGN systems does not imply an equally rapid evolution of ${m}_{\mathrm{bh}}/{m}_{\ast }$ in the quenched systems. This is because, when we are observing quenched systems, we are effectively observing the integrated history of previous activity. To illustrate this, we perform a simple heuristic exercise in which we determine ${m}_{\mathrm{bh}}/{m}_{\ast }$ in quenched systems at a given epoch, assuming as above that quenching started at redshift 2. We make the same assumptions as before, namely that there is no stellar or black hole mass growth after quenching.

Our results are shown in Figure 12. Even though SF galaxies have changed their black hole to stellar mass ratio by almost a factor of 10 from redshift $z\sim 2$ to today, the evolution in this ratio for quenched systems is much milder, more like a factor of 3 or even less (0.4 dex), simply because the quenched population includes galaxies that have quenched much earlier. This means that the redshift evolution in ${m}_{\mathrm{bh}}/{m}_{\ast }$ for quenched systems will always be much milder than the evolution in this quantity in active systems.

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The distinction between active and passive populations becomes even more important if observational studies compare actively accreting systems at high redshift with data on the quiescent population at low redshift. This is unfortunately a quite common practice (e.g., Jahnke et al. 2009; Decarli et al. 2010; Merloni et al. 2010; Bennert et al. 2011; Schramm & Silverman 2013; Schulze et al. 2015) because of the current practicalities of observations. We stress that this approach automatically produces weaker evolution since neither m_bh nor m_* will have changed once a given object becomes inactive (i.e., passive/quenched). It is therefore clear that the mean ${m}_{\mathrm{bh}}/{m}_{\ast }$ relation in the quenched systems will reflect the mass ratio in the SF galaxies at the much earlier epochs when those galaxies actually quenched (as already discussed above). For the local population, this is typically around z ∼ 1–1.5.

Clearly, if we compare the ${m}_{\mathrm{bh}}/{m}_{\ast }$ of SF systems at z ∼ 1–1.5 to the ${m}_{\mathrm{bh}}/{m}_{\ast }$ ratio of passive galaxies seen today that quenched at z ∼ 1–1.5, then we would expect to see no change in ${m}_{\mathrm{bh}}/{m}_{\ast },$ even if this ratio had changed a lot within the actively accreting population! Of course, if the high-redshift active systems are luminosity-selected, then their ${m}_{\mathrm{bh}}/{m}_{\ast }$ ratio will likely have been biased to higher values than the underlying population through the "Lauer effect" discussed in Section 3.4, which goes in the opposite direction (as shown by the parallel black lines in Figure 12). This figure shows that if nature has an underlying ${m}_{\mathrm{bh}}/{m}_{\ast }$ scaling as ${(1+z)}^{2}$ for active systems, then we would expect to see ${(1+z)}^{0.8}$ if we compare comparisons of active systems at $z\sim 2$ with quenched systems at $z\sim 0,$ once we correct for these observational biases. This is quite similar to the evolution seen in at least some of the observational studies cited above (e.g., Jahnke et al. 2009; Decarli et al. 2010; Merloni et al. 2010; Schramm & Silverman 2013; Schulze et al. 2015).

We have already commented in Section 6.2 that we would also expect to see little or no evolution in ${m}_{\mathrm{bh}}-\sigma$ for any population of galaxies, however selected. This is due to the higher σ associated with a given stellar mass at high redshift, which cancels out the ${(1+z)}^{2}$ dependence in ${m}_{\mathrm{bh}}/{m}_{\ast }.$

In this paper we have made the conventional assumption that the black hole to stellar mass relation increases as some power of (1 + z), i.e., ${m}_{\mathrm{bh}}/{m}_{\ast }\sim {(1+z)}^{n}.$ However, this is an arbitrary redshift dependence, and there are reasons why it cannot hold (with $n\sim 2$) at low redshifts (see also Hopkins et al. 2006b). Assuming that an SF galaxy is on the Main Sequence we can track the increase of its stellar mass using the Main Sequence sSFR(z),

$$\begin{eqnarray}&&\mathrm{rsSFR}(z)=\displaystyle \frac{{\dot{m}}_{\ast }}{{m}_{\ast }}=\displaystyle \frac{-1}{(1+z)\sqrt{{{\rm{\Omega }}}_{M}{(1+z)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}}\displaystyle \frac{1}{{m}_{\ast }}\displaystyle \frac{{{dm}}_{\ast }}{{dz}},\end{eqnarray} \tag{ 22 }$$

where rsSFR is the "reduced specific star formation rate" (see Lilly et al. 2013), $\mathrm{rsSFR}=(1-R)\mathrm{sSFR},$ where R is the fraction R of the stellar mass that is returned during star formation $R\sim 0.4,$ which is therefore the inverse mass doubling timescale of the stellar population. Using $\mathrm{rsSFR}(z)=0.07{(1+z)}^{3}$ Gyr⁻¹ from Lilly et al. (2013) and references therein, this can be integrated to give

$$\begin{eqnarray}{m}_{\ast }(z) & = & {m}_{\ast }(0)\left(\mathrm{exp}\left[\displaystyle \frac{2\cdot 0.07\sqrt{{{\rm{\Omega }}}_{M}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}}{3{H}_{0}{{\rm{\Omega }}}_{M}}\right.\right.\\ & & \left.\left.-\displaystyle \frac{2\cdot 0.07\sqrt{{{\rm{\Omega }}}_{M}{(1+z)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}}{3{H}_{0}{{\rm{\Omega }}}_{M}}\right]\right).\end{eqnarray} \tag{ 23 }$$

This modest increase in stellar mass for an SF galaxy sets a maximal change in ${m}_{\mathrm{bh}}/{m}_{\ast }$ ratio even in the most extreme case in which m_bh does not increase at all. We show this maximal evolution in Figure 13 and compare it with the ${(1+z)}^{2}$ evolution used elsewhere in the paper. It can be seen that the maximal evolution is actually slower than ${(1+z)}^{2}$ at redshifts $z\lesssim 0.7.$ Galaxies are growing so slowly at low redshifts that it is not possible to create a strong evolution in the ${m}_{\mathrm{bh}}/{m}_{\ast }$ ratio in a given galaxy, even if their black holes are not growing at all. This may well be why there is no evidence in Figure 11 for the "expected" increase in the observed ${m}_{\mathrm{bh}}/{m}_{\ast }.$ The maximal change in ${m}_{\mathrm{bh}}/{m}_{\ast }$ over this redshift range would have been gentle enough that it would be very hard to observe. For this reason, we emphasize that our statements elsewhere in the paper concerning the evolution in ${m}_{\mathrm{bh}}/{m}_{\ast }\propto {(1+z)}^{2}$ should best be interpreted as implying a change of a factor of about 10 to $z\sim 2,$ rather than a precise particular dependence on redshift.

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A number of authors (e.g., Barger et al. 2005; Hasinger et al. 2005; Labita et al. 2009; Li et al. 2011) have noted or discussed a "downsizing" of the quasar population. Although different authors often use this term to mean different things, it is most often used to denote the observational fact that lower-luminosity AGNs peak in comoving density at lower redshifts then higher-luminosity AGNs.

It is worth stressing that this may not have much physical significance. We have shown that it is possible to reproduce the strong observed redshift evolution in the QLF with a model based on the observed mass function of SF galaxies coupled with a mass-independent (but redshift-dependent) Eddington ratio distribution $\xi (\lambda ,z).$ This is shown more explicitly in Figure 14 where we show the comoving number density of quasars of different luminosity in the QLF, which is reproduced by our model. This emphasizes that the apparent downsizing signature in the AGN population can appear even though the distribution of Eddington ratios (and thus of specific black hole growth rates) is strictly independent of black hole mass at all redshifts in our model.

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It is clear that the apparent "downsizing" in our model arises as a natural consequence of two competing effects that are independent of mass. The first is the redshift evolution of the L^∗, which shifts the luminosity function uniformly in luminosity, but which therefore produces a differential change in number density with luminosity. This shift is produced by the degenerate combination of evolution of the ${m}_{\mathrm{bh}}/{m}_{\ast }$ mass ratio and characteristic Eddington ratio, ${\lambda }^{\ast }.$ The second is the redshift evolution of ${\phi }_{\mathrm{QLF}}^{\ast }$ that changes the number densities uniformly with luminosity. We have argued in this work that the evolution of ${\phi }_{\mathrm{QLF}}^{\ast }$ is a direct consequence of the evolution of ${\phi }_{\mathrm{SF}}^{\ast },$ coupled with a constant duty cycle. The combination of these two produces variation in the number density (at fixed luminosity) that changes with luminosity, producing a variation in the redshift of the peak in the number density as well as different rates of evolution at different luminosities.

One of the most striking results of this paper is that the observed evolution of ${\phi }_{\mathrm{QLF}}^{\ast }$ of the QLF tracks the observed evolution of ${\phi }_{\mathrm{SF}}^{\ast }$ of the SF galaxy mass function. This in turn implies that the ${\xi }^{\ast }$ knee of the Eddington ratio distribution $\xi (\lambda )$ has a more or less constant value, even though the change of the characteristic ${\lambda }^{\ast }$ "knee" is dramatic. We have argued that ${\xi }^{\ast }$ is a good measure of a generalized "duty cycle" of quasars. Indeed, if the luminosity of individual quasars decays with a timescale τ then (Equation (10)) ${\xi }_{\lambda }^{\ast }$ will be direct measure of the birthrate of quasars in SF galaxies.

By applying simple continuity equations to the observed evolution of the SF galaxy mass function, Peng et al. (2010) derived an expression for the rate of the mass-quenching process, ${\eta }_{m},$, which may be written

$$\begin{eqnarray}&&{\eta }_{m}\sim \mathrm{sSFR}(z)\cdot ({m}_{\ast }/{M}^{\ast }),\end{eqnarray} \tag{ 24 }$$

where sSFR(z) is the redshift-dependent specific star formation rate of the SF Main Sequence. It has been well established that sSFR was much higher in the past and a useful representation is (Lilly et al. 2013 and references therein)

$$\begin{eqnarray}\mathrm{sSFR}\sim \left\{\begin{array}{ll}{(1+z)}^{3}, & \mathrm{when}\ z\lt 2\\ {(1+z)}^{5/3}, & \mathrm{when}\ z\gt 2.\end{array}\right.\end{eqnarray} \tag{ 25 }$$

AGN activity in "quasar-mode" is one of the many processes that have been proposed (e.g., Granato et al. 2004; Hopkins et al. 2008a; King 2010) to drive the mass-quenching of galaxies. If a single quasar event is responsible for quenching, then this would require ${\eta }_{\mathrm{AGN}}$ (from Equation (10)) to be equal to ${\eta }_{m}.$ These can only be equated for our inferred constant ${\xi }_{\lambda }^{\ast }$ for a particular redshift and mass dependence of the decay time τ.

$$\begin{eqnarray}&&\tau (m,z)\sim \mathrm{sSFR}{(z)}^{-1}{(m/{M}^{\ast })}^{-1}{\xi }_{\lambda }^{\ast }.\end{eqnarray} \tag{ 26 }$$

We would require quasars to fade faster at high redshift and at high galaxy masses. We could well imagine ways in which this would occur, e.g., because of the shorter dynamical timescales of galaxies at high redshift.

However, the above picture is probably oversimplistic. We could well expect the physics of quenching to be more complex. It is quite plausible that the energy source for quenching is AGN activity but that the effectiveness of this energy injection depends on the stellar or halo mass of the system (the Peng et al. 2010 quenching law can be written in terms of a redshift-independent survival probability that depends only on mass). This would then break the simple link between ${\eta }_{m}$ and ${\eta }_{\mathrm{AGN}}.$