GLOBAL STAR FORMATION REVISITED

The Tully–Fisher (TF) relation is also controlled by disk surface density. We use the empirical I-band TF relation: L_* = C_TFv^α_r, with α ≈ 4 in the K band (Masters et al. 2008) and v_r the maximum rotation velocity, and where the virial theorem requires that C_TF = (3/4π)G⁻²Σ⁻¹_tot(L_*/M_tot). We find using Equation (5) that

$$\begin{equation} C_{\rm SK} = {3 \over 4}C_{\rm TF}^{-1/2} f_c f_{\rm cl}\left({m_{\rm SN}v_c}\over {E_{\rm SN}}\right)\left[{p_g / p_{\rm cl}}\right]^{1 \over 2} \left({L_{\ast } \over M_{\rm tot}}\right)^{1/2}. \end{equation} \tag{ 7 }$$

We infer that the SK law residuals should anticorrelate with the TF law residuals. The TF normalization is correct, by assumption: what is new is the predicted inverse correlation between SK and TF law residuals.

The global star formation law can be applied to regions that are gas-dominated. When gas dominates the self-gravity, the cloud collision model suggests that $\dot{\Sigma }_\ast \propto { \Sigma _g}^{3/2}{\Sigma _{\rm tot} }^{1/2}\;^{\propto}\!\!\!\! _{\sim}\;{\Sigma _{\rm gas} }^2,$ and the KS law steepens. There are indications of such a steepening in several environments.

• 1.  
  The extended H i spiral structure in NGC 6946 (Boomsma 2007) shows that global gravitational instability is not a sufficient condition for forming stars. In the case of M83, the H i disk extends to more than twice the optical scale. Deep UV imaging reveals very low level star formation in the outer H i disk, well below the SK threshold. The cloud collision model provides a possible explanation of these phenomena, although the observed radial dependence of star formation rate is too steep to be explained by the simplest models (Leroy et al. 2008).
• 2.  
  Individual young star complexes in M51 fall on the SK law, although with increased dispersion (Kennicutt et al. 2007) and a slightly steeper slope.
• 3.  
  Damped Lyman alpha systems at z ∼  2 (Wolfe & Chen 2006) underproduce stars by up to a factor of 10 in star formation rate as predicted from the SK law.
• 4.  
  Steepening also occurs in the inner regions of disks at extreme star formation rates. This is found in intensely star-forming galaxies at high redshift (Gao et al. 2007). Krumholz & Thompson (2007) account for the linear relation found for the local HCN data in terms of the critical density for excitation of the HCN transition in dense gas, that effectively samples only the densest molecular clouds and thereby bypasses the sensitivity to dynamical timescale.

Steepening in a cloud collision model is not a unique explanation for any of these phenomena. For example, the outer parts of disks are more thermally stable (Schaye 2004), and the star formation rate in DLAs could be suppressed because of the low H₂ content due to a combination of a low dust content plus a high radiation field.

If the porosity is high, as may happen transiently in starbursts, we assume that disk outflows occur. These may be winds from dwarf galaxies or fountains in the case of more massive disks. Numerical simulations of star formation in the multiphase interstellar medium of a disk galaxy are able to model the disk outflows and global star formation history (Tasker & Bryan 2006). It is useful however to provide an analytic formulation. Let f_L be the hot gas loading factor. It is measured to be around 10 for the outflow from NGC 1569 (Martin 2005).

Now the outflow from the disk is

$$\begin{eqnarray} &&\dot{M}_{\rm out}=(1-e^{-Q})f_L \dot{M}_\ast \approx Q f_L \dot{M}_\ast. \end{eqnarray} \tag{ 23 }$$

This tells us that $\dot{M}_{\rm out} / \dot{M}_\ast =f_L Q\;^{\bm \propto}\!\!\!\! _{\bm \sim}\;\sigma _g^{-1.7}.$ Low porosity suppresses outflows so that outflows are suppressed in massive potential wells.

We can also express the outflow rate as

$$\begin{eqnarray} &&\dot{M}_{\rm out}\approx Q^2f_L({\sigma _g}/{\sigma _{\rm fid}})^{2.7}f_g^{1/2}M_g/t_d. \end{eqnarray} \tag{ 24 }$$

This shows that outflows could indeed occur from massive potential wells if porosity can somehow be maintained. We argue below that AGN-triggered star formation fulfills this role.

Outflows are important for dwarf galaxies, but are seen to be quenched in deep potential wells as well as at extreme porosity. If σ_g is high, the porosity is low, feedback is suppressed and supernova-driven winds are quenched. Even if σ_g is low, outflows may be suppressed if the gas is dense, leading to low porosity. Dwarf galaxy outflows play an important role in IGM enrichment at high redshift. The mass ejected during dwarf formation is comparable to the mass retained in stars formed. Nearby starbursts display this trend, suggesting that the effect may be generic to powerful starbursts in dwarf galaxies.

Meurer et al. (1997) and Hathi et al. (2008) report a bolometric upper limit on the disk surface brightness in starbursts of 2 × 10¹¹ L_☉ kpc⁻² over 0.1 ≲ R_e ≲ 10 kpc. We interpret this upper limit on disk luminosity for our model galaxy in terms of radiation pressure and mechanical pressure from limiting the gas surface density (compare Thompson et al. 2005).

To avoid lift-off via application of the Eddington condition requires:

• 1.  
  for radiation pressure:

  $$\begin{equation} \Sigma _L <(\pi /2) G c \Sigma _{\rm tot}\Sigma _g \end{equation} \tag{ 25 }$$

  or $\dot{\Sigma }_\ast <c\pi Gf_l^{-1}\Sigma _{\rm tot}\Sigma _{g}/2,$ where f_l = _lc² ≈ 10⁻³–10⁻⁴c² for massive stars. Let us evaluate this expression after using the earlier expression for ${\dot{\Sigma }_{\ast }}$ to eliminate Σ_g. We find using Equation (25) that star formation is reduced at low gas surface density and lift-off can be avoided if

  $$\begin{eqnarray} &&\left[ \left({\epsilon _l m_{\rm SN} c^2 \over E_{\rm SN}}\right)\left({v_c \over c}\right)\right]^2 \left[{p_g \over p_{\rm cl}}\right]\le {\Sigma _{\rm tot} / \Sigma _g}. \end{eqnarray} \tag{ 26 }$$
• 2.  
  For mechanical energy input: we have instead no lift-off provided that

  $$\begin{eqnarray} &&\left[\left({\epsilon _{m} m_{\rm SN} c^2 \over E_{\rm SN}}\right) \left({v_c \over v_w}\right)\right]^2 \left[{p_g \over p_{\rm cl}}\right] \le {\Sigma _{\rm tot} \over \Sigma _g}. \end{eqnarray} \tag{ 27 }$$

  Here, the symbol _m denotes the efficiency of conversion of rest mass energy to mechanical energy in massive stars. For the lift-off limits to be reached the left-hand side of the above two equations must be greater than unity since Σ_g ⩽ Σ_tot. For the radiation case, unity is not normally reached on the left-hand side unless some of the parameters such as _l are significantly different from their normal values. This is what led Thompson et al. (2005) to argue for an AGN luminosity driving this Eddington limit for disks. For mechanical winds, the left side is boosted over the equivalent radiation driving term by a factor of (c/v_w)² making it reasonable that such mechanically driven feedback occurs. For completeness and because of our poor knowledge of star formation at high redshift we will keep both the radiation and mechanical estimates in our calculations. This upper limit on Σ_g corresponds to an upper limit on global star formation of 45 M_☉ kpc⁻² yr⁻¹. By writing the disk surface brightness as $\Sigma _L=\epsilon _l c^2\dot{\Sigma }_\ast,$ we find using Equation (26) that the disk surface brightness satisfies

  $$\begin{eqnarray} &&\Sigma _L<\left(\frac{\pi }{2}\right)^2 Gc\Sigma _{\rm tot}^2\left[\frac{E_{\rm SN}}{m_{\rm SN} c^2\epsilon _l}\left({c \over v_c}\right)\right]^2 \left(\frac{p_{\rm cl}}{p_{g}}\right). \end{eqnarray} \tag{ 28 }$$

  For mechanical input we have a similar equation using Equation (27)

  $$\begin{equation} \Sigma _L<\left(\frac{\pi }{2}\right)^2 Gc\Sigma _{\rm tot}^2\left[\frac{E_{\rm SN}}{m_{\rm SN} c^2\epsilon _m}\left(v_w \over v_c\right)\right]^2 \left(\frac{p_{\rm cl}}{p_{g}}\right). \end{equation} \tag{ 29 }$$

  We see that the absolute maximum global star formation-driven surface brightness, Σ_L,max, is

  $$\begin{equation} \Sigma _{L, {\rm max}}=(\pi /2)Gc \Sigma _{\rm tot}^2. \end{equation} \tag{ 30 }$$

  This expression can be derived directly from the previous equation by setting Σ_g = Σ_tot. Note that in this case all the photon energy is used up to support the disk and the photon bubbles can occur with the signature of light and dark patches on the disk. This suggests that saturation of the star formation rate occurs at high surface density. Inserting typical numbers, Σ_L < 10¹¹ L_☉ kpc⁻²E²₅₁v⁻²_c,400⁻²_n,−3Σ²_tot,1000. Note that Komugi et al. (2005) report an offset roughly equivalent to an effective steepening of the SK law for ultra-luminous starbursts, as do Gao et al. (2007). We now find that for an extreme starburst at the Eddington luminosity with Thomson scattering, σ_T, as the dominant opacity the relation between surface brightness and mass surface density is

  $$\begin{equation} \Sigma _L =4 \pi Gc \Sigma _{\rm tot} \left({m_p \over \sigma _T}\right), \end{equation} \tag{ 31 }$$

  where m_p is the proton mass. From Equations (28) and (29) we deduce that the total surface density, which is essentially all gas in initial situations, satisfies

  $$\begin{eqnarray} \Sigma _{\rm tot} >\Sigma _\ast \equiv && \left(\frac{16}{\pi }\right)\left[\frac{m_{\rm SN}\epsilon _l c^2}{E_{\rm SN}} \left({v_c \over c}\right)\right]^2\left(\frac{p_{g}}{p_{\rm cl}}\right) \left(\frac{m_p}{\sigma _T}\right)\nonumber\\ &&\sim 10^3\;M_\odot \;{\rm pc}^{-2}, \end{eqnarray} \tag{ 32 }$$

  and similarly for mechanical input

  $$\begin{eqnarray} &&\Sigma _{\rm tot} >\Sigma _\ast \equiv \left(\frac{16}{\pi }\right) \left[\frac{m_{\rm SN}\epsilon _m c^2}{E_{\rm SN}} \left(v_c \over v_w\right)\right]^2 \left(\frac{p_{g}}{p_{\rm cl}}\right) \left(\frac{m_p}{\sigma _T}\right).\nonumber\\ \end{eqnarray} \tag{ 33 }$$

  A direct and simple way to derive these limits is to take Equations (31) and (26) and find that

  $$\begin{equation} \Sigma _{\rm tot} \ge 8 \left(m_p \over \sigma _T\right)\sim 4 \times 10^{5}\;M_{\odot }\;{\rm pc}^{-2} \end{equation} \tag{ 34 }$$

  which is a derivation of Fish's Law. The densest galactic molecular clouds, traced by H₂O masers, have a similar surface density (Plume et al. 1997). By z ∼  2, there is a substantial population of galaxies with star formation rates of 500–1000 M_☉ yr⁻¹, ultraluminous infrared galaxies (ULIRGs) and submillimeter galaxies (SMGs). A recent study of SMGs at 0.5 arcsec resolution (Tacconi et al. 2008) finds that they are gas-rich (molecular gas fraction ∼ 30%) with the gas in compact disks at a density of ∼ 10⁴ M_☉ pc⁻², and are undergoing major mergers.

Negative feedback helps account for the black hole mass–σ correlation (Di Matteo et al. 2008) and for the luminosity function of massive galaxies (Bower et al. 2006; Croton et al. 2006). More physics must be added however to account for downsizing and efficient star formation in massive galaxies. The key may be AGN outflows that can trigger star formation by compressing dense clouds. These would precede the outflow phase which in this case is due to the combined effect of AGN outflow and triggered SNe. A prior phase of positive feedback is a possible new ingredient in feedback modeling and is motivated by evidence (admittedly sparse but compelling: see, e.g., Feain et al. 2007) for AGN triggering of star formation.

The following model is necessarily schematic pending fully three-dimensional simulations of jet propagation into a clumpy protogalactic interstellar medium. We speculate that the triggering works as follows. Jet propagation into a clumpy medium develops into an expanding, overpressurized cocoon (Saxton et al. 2005; Antonuccio-Delogu & Silk 2008). This adds a potentially large multiplier to the efficacy of the BH-driven outflow for the following reason. The jet-driven plasma-filled radio lobe drives a cocoon that expands into the hot virialized component of the protogalaxy at a speed v_co that is much larger than the velocity field associated with the gravitational potential well. Protogalactic clouds that are above or near the Jeans mass will be induced to collapse. The cloud overpressuring and resulting triggered star formation occurs at a rate much larger, by 1–2 orders of magnitude, than is associated with normal gravitationally driven fueling, as would be appropriate to a star-forming disk galaxy. We show below that the star formation rate enhancement amount to a factor ∼ v_co/σ. Hence central massive black holes that have not yet grown by accretion to the limiting mass controlled by radiation outflow should still have a considerable impact on the evolution of the protogalaxy core. Incidentally, this early phase of black hole feedback makes observational confirmation difficult, as discussed below.

Numerical studies of radiative shock-induced cloud collapse reveal the complex interplay of hot and cold gas (Mellema et al. 2002; Fragile et al. 2004). Here, we focus on the implications for star formation via analytic considerations. For a simple ansatz, suppose that the triggered star formation occurs over the cocoon propagation time, t_co that is much less than the dynamical time, t_d, t_co ≪ t_d . We show below that jet triggering of the cocoon expansion gives a star formation rate enhancement factor t_d/t_co ∼  v_co/σ .

Although we shall generally assume v_co is a parameter of the wind, consider the case in which the injected momentum flux is L/c for a wind that is mechanically driven but is originated via radiation pressure. We expect the mechanical jet luminosity to be a fraction v_w/c ∼  0.1 of the luminosity. More precisely, the ratio (L_mech/v_w)/(L/c) is equal to the optical depth τ, due to a combination of Thomson scattering, line opacity and/or dust. To estimate τ, an effective Rosseland mean opacity can be defined from these opacity contributions. This is valid if the jet is radiation pressure-driven. Now

$$\begin{equation} v_{\rm co}= \sigma \tau ^{1/2} f_g^{-1/2}\left({L^{\rm AGN} \over L_{\rm SR}}\right)^{1/2}, \end{equation} \tag{ 39 }$$

where we have set ρ_g = σ²/2πGr². The previously derived star formation rate can be generalized for round systems with velocity dispersion σ to

$$\begin{eqnarray} &&{\dot{M}_{\ast }} =(\epsilon _{\rm SN}/\sigma)f_c f_{\rm cl} M_g(Gp_g)^{1/2}. \end{eqnarray} \tag{ 40 }$$

Thus, if the gas pressure is replaced by the AGN-driven pressure, we can see how a central AGN can boost the star formation rate by writing the pressure as

$$\begin{eqnarray} &&p_{\rm AGN} = {L_{\rm AGN} {4\pi v_{\rm co} r^2}}^{-1}. \end{eqnarray} \tag{ 41 }$$

The star formation rate is now given generally using Equations (40) and (41) by

$$\begin{equation} {\dot{M}_{\ast }}^{\rm AGN} \approx \left[ f_c f_{\rm cl} \epsilon _{\rm SN}\left[M_g \over t_d \right]\right]{f_g}^{-1/2} \left({ c \over V_{\rm AGN}}\right)^{1/2} \left(L_{\rm AGN} \over L_{\rm SR}\right)^{1/2}. \end{equation} \tag{ 42 }$$

Here, we have left V_AGN as an arbitrary variable to be applied to the appropriate situation such as jet, wind, outflow, etc. In the radiation-driven case discussed above V_AGN = v_co, the cocoon velocity, and using Equations (39) and (42) we find

$$\begin{equation} {\dot{M}_{\ast }}^{\rm AGN} \approx \left[ f_c f_{\rm cl} \epsilon _{\rm SN}\left[M_g \over t_d \right]\right]{f_g}^{1/4} {\tau }^{-1/4}\left({ c \over \sigma }\right)^{1/4} \left(L_{\rm AGN} \over L_{\rm SR}\right)^{1/4}. \end{equation} \tag{ 43 }$$

The AGN-driven enhancement factor is (p_AGN/p_g)^1/2 ≈ (v_co/σ)τ^1/2. Note that _SN ∝ σ, so that the star formation efficiency coefficient (fraction of stars formed per dynamical time) is boosted considerably for spheroids relative to disks. The AGN luminosity explicitly drives star formation. It is the AGN-triggered star formation multiplier rather than the AGN itself that drives the feedback. The boost effect is generally important in the innermost spheroid, and globally important for super-Eddington AGN luminosities.

Note that we must be careful that the AGN pressure does not blow away the ISM completely. For a SNe-regulated ISM, the pressure relates to Q and includes details of the SNe bubble evolution. But when the AGN jet provides the pressure it will overpressure the clouds and turn them into stars with some efficiency. We need to relate the star formation from the triggered clouds to the AGN accretion rate and this would then affect the jet luminosity and pressure. Essentially, we raise the external pressure (here using the AGN) on the star-forming clouds. This allows us to regulate Q for the SN bubbles. If the porosity is maintained to be constant by the triggering of massive star formation, we have achieved the desired self-regulation between AGN activity and star formation. If Q is given by other means (and is of order unity), then star formation-regulated feedback follows naturally. To show this point explicitly in our formulation of AGN-driven star formation we return to the pressure-regulated model of the structure of the ISM, and instead of calculating the final size of SNeII-driven remnant bubbles by setting the remnant's velocity, v_a equal to σ_g we set the external pressure ρ_gσ_g² equal to the mechanical pressure driven by the AGN, L/4πr²v_co. Assuming, as above, that ρ = σ_g²/Gr² with an associated pressure p_g = σ_g⁴/Gr² and replacing this with the AGN pressure, p_AGN given by p_AGN = L_AGN/4πV_AGNr² we find that the required transformation to replace gas pressure with AGN pressure is

$$\begin{eqnarray} &&\left({ \sigma _g \over \sigma _{\rm fid}} \right) =\left({c \over V_{\rm AGN}}\right)^{1/4} \left({L_{\rm AGN} \over L_{{\rm SR}, {\rm fid}}}\right)^{1/4}. \end{eqnarray} \tag{ 44 }$$

It then follows directly that

$$\begin{equation} Q e^{Q} = f_{\rm cl}\epsilon _{{\rm SN}, {\rm fid}} \left({V_{\rm AGN}\over c}\right)^{1/4}\left({L_{{\rm SR},{\rm fid}} \over L_{\rm AGN}}\right)^{3/7}, \end{equation} \tag{ 45 }$$

and for Q ≪ 1 we find

$$\begin{equation} Q \sim f_{\rm cl}\epsilon _{{\rm SN}, {\rm fid}} \left({V_{\rm AGN}\over c}\right)^{1/4}\left({L_{{\rm SR},{\rm fid}} \over L_{\rm AGN}}\right)^{3/7}. \end{equation} \tag{ 46 }$$

For Q ≫ 1

$$\begin{equation} Q \sim \ln \left[f_{\rm cl}\epsilon _{{\rm SN}, {\rm fid}} \left({V_{\rm AGN}\over c}\right)^{1/4}\left({L_{{\rm SR},{\rm fid}} \over L_{\rm AGN}}\right)^{3/7}\right], \end{equation} \tag{ 47 }$$

with a good fit to the range Q ≪ 1 up to Q ≫ 1 given by

$$\begin{equation} Q \sim \ln {\left[1 + f_{\rm cl}\epsilon _{{\rm SN}, {\rm fid}} \left({V_{\rm AGN}\over c}\right)^{1/4}\left({L_{{\rm SR},{\rm fid}} \over L_{\rm AGN}}\right)^{3/7}\right]}. \end{equation} \tag{ 48 }$$

Following the simple cocoon model of Begelman & Cioffi (1989), we examine the effect of the power flow in the canonical two bidirectional jets diverted into a small nuclear cocoon and then via the cocoon pressure acting back on the central nucleus where gravitational instability and enhanced collapse and accretion may occur if the Bonnor–Ebert (BE) critical pressure is reached. For an isothermal gas distribution with velocity dispersion σ as used throughout this paper, a jet opening angle Θ_J, an approximately ellipsoidal cocoon with axes a and b with a > b for simplicity powered by two thin jets with luminosity, L_J, we find that the cocoon pressure is given by

$$\begin{equation} P_{\rm co} = \left({ L_J \sigma ^2 v_J \Theta _J \over G }\right)^{1/2} \left(a\over b \right)^2 \end{equation} \tag{ 49 }$$

and the ratio of the cocoon pressure to the gas pressure in the central region (putting a ∼  b) for simplicity is

$$\begin{equation} {P_{\rm CO} \over P_g} = \left(L_J \over L_{\rm SR}\right)^{1/2} \left({ v_J c \over \sigma ^2} \right)^{1/2} \Theta _J^{1/2}. \end{equation} \tag{ 50 }$$

Now for the central isothermal core of gas pressured by the cocoon pressure, the ratio of the BE mass to the core mass is given by

$$\begin{equation} {M_{{\rm BE}} \over M} = 1.18 \left({L_J \over L_{\rm SR}}\right)^{-{1/4}} \left({ v_J c \over \sigma ^2} \right)^{-{1/4}} {\Theta _J}^{-{1/4}}. \end{equation} \tag{ 51 }$$

Thus, if v_J ∼  0.1c, the core can be over-pressured by the cocoon if Θ_J > 10(σ/c) ∼  10⁻² when L_J ∼  L_SR. This may be another way to look at the feedback process involving the growth of black holes along the M_BH–σ line since at the Eddington luminosity L_J ∼  L_SR implies M ∼  M_SR.

The global mass loss for AGN-driven wind is given by

$$\begin{equation} \dot{M}_{\rm out}^{\rm gal}= {L_{\rm AGN}\over V_{\rm AGN}^2} \end{equation} \tag{ 52 }$$

and for the radiation-driven AGN-generated wind case

$$\begin{eqnarray} &&\dot{M}_{\rm out}^{\rm gal}= {L_{\rm AGN}\over c v_{\rm co}}\approx \left({{\sigma _g}^3\over G }\right) {\tau }^{-1/2} \left({L_{\rm AGN}\over L_{\rm SR}} \right)^{1/2}. \end{eqnarray} \tag{ 53 }$$

The outflow rate is proportional to the spheroid velocity dispersion and to the square root of the AGN luminosity. The scaling of the outflow rate with regard to AGN luminosity implies that outflows saturate. It may be compared with observations of broadened features that demonstrate the presence of massive winds in ultraluminous star-forming infrared and radio galaxies.

Making use of the AGN-enhanced star formation rate, we can express the outflow as a ratio using Equations (42) and (53) generally as

$$\begin{equation} {\dot{M}_{\rm out}^{\rm gal}\over \dot{M}_\ast ^{\rm AGN}}= \left({ {f_g}^{1/2} \over f_c f_{\rm cl} \epsilon _{\rm SN} (c/\sigma)}\right) \left(L_{\rm AGN} \over L_{\rm SR}\right)^{1/2} \left({ c \over V_{\rm AGN}}\right)^{3/2}. \end{equation} \tag{ 54 }$$

It follows that the outflow rate from AGN is always of the order of the AGN-boosted star formation rate for the protospheroid. We may also compare the star formation-boosted global outflow rate with the AGN outflow. The AGN mass outflow rate is $\eta c\dot{M}_{\rm acc}/(v_{\rm out}),$ with v_AGN ∼  0.1c. Hence it is of the order of the SMBH accretion rate. As expected, the mass flux associated with the galaxy outflow dominates that from the AGN. The mass flux ratio is

$$\begin{equation} {\dot{M}_{\rm out}^{\rm gal} \over \dot{M}_{\rm acc}} =\eta f_w^{-1} \left({c\over \sigma }\right)\left(L_{\rm SR}\over L_{\rm AGN}\right)^{1/2}. \end{equation} \tag{ 55 }$$

Hence ${\dot{M}_{\rm out}^{\rm gal} / \dot{M}_{\rm acc}} \;{\sim}\; 100$ for AGN at the Eddington luminosity and κ ∼  10³σ_T/m_p. In order to allow for dust, if a factor τ⁻¹ is incorporated into the definition of L_SR, this ratio is seen to be inversely proportional to the square root of the adopted (dust) opacity. The momentum flux ratio is

$$\begin{equation} {\dot{M}_{\rm out}^{\rm gal} \sigma \over {\dot{M}_{\rm acc} v_{\rm AGN}}} =\eta f_w^{-1} {c\over v_{\rm AGN}} \left(L_{\rm SR}\over L_{\rm AGN}\right)^{1/2}. \end{equation} \tag{ 56 }$$

We see that the momentum ejected from the AGN dominates over that in the global outflow by a factor of a few for AGN near the Eddington luminosity L_AGN ∼  L_SR.

The piston model enables downsizing of AGN and spheroids by coupling their growth. For some fiducial AGN energy conversion efficiency η(∼  0.1), we note that $L^{\rm AGN}=\eta c^2\dot{M}_{\rm acc}$ is a measure of the BH accretion rate. Since L^AGN controls the star formation rate and is itself controlled by the black hole accretion rate, we infer that black hole growth and star formation triggering downsize together, provided Q is approximately constant due to AGN triggering of SN. The AGN driving of star formation overcomes the pressure suppression of porosity in the absence of the AGN. A large porosity also results in a wind. The required turbulent velocity field controls the accretion rate and might be specified by other physics, such as a merger, or even be due to the AGN itself. Let us try to make these assertions more quantitative.

The AGN is the ultimate driver of the porosity. We need to connect AGN-induced star formation and outflows to the black hole growth rate via the AGN luminosity. The global outflow rate is

$$\begin{equation} \dot{M}_{\rm out}^{\rm gal}= Q f_L \epsilon _{\rm SN}M_g/t_d. \end{equation} \tag{ 57 }$$

By momentum conservation, this must equal the global AGN-boosted outflow rate L^AGN/(cv_c).

4.4.1. Downsizing for Porosity-Regulated Star Formation

Incorporating the effects of porosity-driven star formation means that the outflows must satisfy

$$\begin{equation} \dot{M}_{\rm out}^{\rm gal}= Q^2f_L({\sigma _g}/{\sigma _{\rm fid}})^{2.7}f_g^{-1/2}\sigma _g^3\big/G. \end{equation} \tag{ 58 }$$

The AGN luminosity is controlled by the accretion rate onto the central black hole, $\dot{M}_{\rm acc}.$ Our next step is to evaluate the black hole growth rate, M_acc. This is the key to explaining downsizing.

To reproduce the downsizing phenomenon, observed for AGN (Hasinger et al. 2005) and their massive host galaxies (Kriek et al. 2007) to occur almost coevally, we need to understand why massive SMBH and spheroids form before their less massive counterparts. The required scaling for L^AGN or M_accr is reminiscent of the scaling found for protostellar jets. The magnetically regulated disk phenomenon plausibly obeys a universal scaling law that could equally apply to jets and outflows from disks around SMBH. The protostellar scaling is (Mohanty et al. 2005) $\dot{M}_{\rm acc}\propto M^2.$ Allen et al. (2006) find that for the black holes that power the AGNs in massive ellipticals, the Bondi accretion rate is approximately proportional to the jet power. The connection with outflows and jets that are magnetically guided by the wound-up field in the accretion disk proposed by Banerjee & Pudritz (2006) is a generic scaling in their study of protostellar jets, $\dot{M}_{\rm wind}^{\rm AGN} =f_w \dot{M}_{\rm acc},$ with f_w ∼  0.1, for outflows associated with central objects that range from brown dwarfs to supermassive black holes.

The Bondi accretion formula therefore regulates SMBH growth and, implicitly, outflow. It yields ${\dot{M}_{\rm BH}} = \pi G^2\left({p_g / \sigma ^5}\right)\break{M_{\rm BH}}^2.$ A simple interpretation of this scaling is that for Bondi accretion,

$$\begin{eqnarray} &&\dot{M}_{\rm out} /f_w= \dot{M}_{\rm acc} =4\pi (GM/\sigma ^2)^2\rho v\propto (\rho /\sigma ^3)M^2, \ \ \end{eqnarray} \tag{ 59 }$$

in combination with adiabatic compression, so that ρ ∝ σ³. For the AGN case, we write $\dot{M}_{\rm acc}=\alpha M_{\rm BH}^2$ with α ∝ G²ρ/σ³. Rewriting this we see that

$$\begin{equation} { d M_{\rm BH} \over d M_*} = Q f_L {f_w}^{-1} \end{equation} \tag{ 60 }$$

Therefore if Q reaches a self-regulating constant and f_L and f_w are also constant then the black hole and galaxy growth move together on a fixed trajectory in the Magorrian plane as discussed below. It is this fixed trajectory that forces downsizing.

4.4.2. Downsizing for SN Energy Injection

Substituting further the relevant quantities for the case of SN energy input we find for the case without AGN feedback

$$\begin{equation} {d\ln { M_{\rm BH}} \over d{\, \ln { M_{*}}}} = {f_g}^{1/2}\left({{\bar{t}_S} \over t_d}\right) = \left(G f_g \rho {\bar{t}_S}^2\right), \end{equation} \tag{ 61 }$$

where

$$\begin{equation} {\bar{t}_S} = \left({ t_S \sigma \over \eta \epsilon _{\rm SN} c f_c f_{\rm cl} } \right)= \beta t_S. \end{equation} \tag{ 62 }$$

Thus the critical parameter determining the logarithmic slope in the Magorrian plane is ${\bar{t}_S}$. There is a critical density $\rho _{\rm crit} = (G f_g {\bar{t}_S})^{-2}$ above which black hole growth dominates and below which star formation dominates. This can be rewritten in terms of a critical velocity dispersion if one takes ρ_crit to be the density at the edge of the Bondi accretion sphere (the sphere of influence of the black hole, R_BH), namely, ρ ∼ (M/r³) ∼ (M_BH/r_BH³) ∼ (G⁻³σ⁶M_BH⁻²) giving then an equivalent σ_crit

$$\begin{equation} \sigma _{\rm crit} = \left({M_{\rm SR} \over M_{\rm BH} }\right) {f_g}^{1/2} {f_c}^{-1} {f_{\rm cl}}^{-1} \left({p_{\rm cl} \over p_g}\right)^{1/2}\left({ E_{\rm SN} \over m_{\rm SN} v_c}\right), \ \ \end{equation} \tag{ 63 }$$

which is a satisfying combination of black hole galaxy and ISM properties. Continuing to the case with AGN feedback, we find

$$\begin{equation} {d\ln { M_{\rm BH}} \over d{\, \ln { M_{*}}}} = {f_g}^{1/2}\left({{\bar{t}_S} \over t_d}\right)\left({v_{\rm AGN} \over c} \right)^{1/2} \left({L_{\rm SR} \over L_{\rm AGN}}\right)^{1/2} \end{equation} \tag{ 64 }$$

and in the radiation-driven case

$$\begin{equation} {d\ln { M_{\rm BH}} \over d{\, \ln { M_{*}}}} = {f_g}^{1/4}\left({{\bar{t}_S} \over t_d}\right)\left({\sigma \over c} \right)^{1/2}\tau ^{1/4} \left({L_{\rm SR} \over L_{\rm AGN}}\right)^{1/4}. \end{equation} \tag{ 65 }$$

Corresponding expressions for the critical density and velocity dispersion in the AGN case can be readily obtained. At higher redshifts, galaxy systems can be denser, although much of the physics of the nuclear regions depends on local physics, and thus the dominant black hole growth phase may be more easily entered at higher redshift. The Bondi accretion formula can be rewritten as

$$\begin{equation} \left({t_{\rm BH} \over t_S}\right) = {f_g}^{-1} \left({ t_d \over t_S} \right)^2 \left({\eta c \over \sigma }\right)\left({M_{\rm SR} \over M_{\rm BH}}\right). \end{equation} \tag{ 66 }$$

Using the equation balancing inflow and outflow and the equation for the logarithmic slope in the Magorrian plane, we find

$$\begin{equation} \left({M_* \over M_{\rm BH}}\right) \left({ Q f_L \over \beta f_w}\right)= \left({t_S \over t_d} \right)\left({v_{\rm AGN} \over c} \right)^{1/2} \left({L_{\rm SR} \over L_{\rm AGN}}\right)^{1/2}. \end{equation} \tag{ 67 }$$

Eliminating (t_S/t_d) we find

$$\begin{eqnarray} &&\left({ Q f_L \over \beta f_w}\right)^2 \left({t_{\rm BH} \over t_S}\right)=\left({\eta c \over \sigma }\right)\left({v_{\rm AGN} \over c} \right) \left({L_{\rm EDD} \over L_{\rm AGN}}\right)\left({M_{\rm SR} \over M_*}\right)^2 \ \ \ \ \ \end{eqnarray} \tag{ 68 }$$

and in this case we have used the momentum balance equation with momentum injection from SN balancing dissipation. For the radiation-driven case we find

$$\begin{equation} \left({ Q f_L \over \beta f_w}\right)^2 \left({t_{\rm BH} \over t_S}\right) = \tau ^{1/2}\left({{L_{\rm EDD}}^2 \over L_{\rm AGN}L_{\rm SR}}\right)^{1/2}\left({M_{\rm SR} \over M_*}\right)^2. \end{equation} \tag{ 69 }$$

In both the above cases there is no evidence for downsizing even with constant Q. However, if there is another way to deduce Q for these turbulent AGN-driven multiphase media and the momentum injection for the medium is taken up by the AGN then, as we shall show in the following, downsizing can occur naturally as a consequence of the turbulent ISM properties.

There is an obvious but powerful constraint on the behavior of the evolutionary tracks in the observed mass ln [M_BH]–ln [M] plane. We first emphasize that this concept of tracks is implicit in our model and that there is a flow of points in the mass plane with evolutionary arrows all pointing in the direction of black hole growth. The slope p = dln [M_BH]/ln [dM] of the track cannot be negative since: (1) black holes can only grow in mass; and (2) galaxies only grow in mass (ignoring their small fractional mass loss). On dwarf galaxy scales the fractional mass loss can be easily incorporated but even there it is much less than of order unity. Therefore, for example, tracks cannot loop back from above the mean line with any slope less then zero after overshooting the mean line on a trajectory originating from below. Therefore, any nonpathological track will spend most of its life on a track with a slope close to the mean. Thus, we can assume p is approximately constant; observationally p is of order unity. We now use this slope as a parameter in our timescale and evolutionary equations and find

$$\begin{eqnarray} &&\left({p \over \beta }\right)^2 \left({t_{\rm BH} \over t_S}\right)=\left({\eta c \over \sigma }\right)\left({v_{\rm AGN} \over c} \right) \left({L_{\rm EDD} \over L_{\rm AGN}}\right)\left({M_{\rm SR} \over M_{\rm BH}}\right)^2. \ \ \ \ \ \ \end{eqnarray} \tag{ 70 }$$

For the radiation-driven case we find

$$\begin{equation} \left({p \over \beta }\right)^2 \left({t_{\rm BH} \over t_S}\right) = \tau ^{1/2}\left({{L_{\rm EDD}}^2 \over L_{\rm AGN}L_{\rm SR}}\right)^{1/2}\left({M_{\rm SR} \over M_{\rm BH}}\right)^2. \ \ \ \ \ \end{equation} \tag{ 71 }$$

We see that the parameter controlling accretion α, which is proportional to the phase space density, and also measures the specific entropy s of the initial gas distribution, is specified by the physics of accretion. Specifically, α ∝ s^−3/2, where s = kTn^−2/3. In terms of a polytropic equation of state, α is constant for γ = 5/3.

4.6.1. Hot Phase Entropy

4.6.2. Cold Phase Entropy

4.6.3. Multiple Phases

We now develop the interplay between the Bondi accretion rate parameter α and the porosity-driven star formation rate. We show that constant Q implies constant α, and vice versa. This is the key to understanding coupled downsizing for spheroids and supermassive black holes. Define the black hole growth time by $t_{\rm BH}=M_{\rm BH}/\dot{M}_{\rm acc}=1/(\alpha M_{\rm BH})$.

4.7.1. Significant Downsizing for Porosity-Regulated Star Formation

Using the above Equations (60) and (58) which relate $\dot{M}_{\rm out}$ and $\dot{M}_{\rm acc},$ we find that

$$\begin{eqnarray} &&\left({Q^2 f_L \over f_w}\right)^2\left(\eta c \over \sigma _{\rm fid}\right)\left({t_{\rm BH}\over t_S}\right) = \left({M_{\rm BH}\over M_{\rm SR}}\right)^2 \left({M_{\rm BH} \over M_g}\right)\left({\sigma _{\rm fid}\over \sigma } \right)^{31/7} \end{eqnarray} \tag{ 72 }$$

which becomes

$$\begin{eqnarray} &&\left({Q^2 f_L \over f_w}\right)^2 \left({\eta c \over \sigma _{\rm fid}}\right)\left({t_{\rm BH}\over t_S}\right)\nonumber \\ &&\quad=\left({M_{\rm BH}\over M_{\rm SR}}\right)^2 \left({M_{\rm BH} \over M_g}\right)\left({L_{{\rm SR},{fid}}\over L_{\rm SR}} \right)^{31/28}. \end{eqnarray} \tag{ 73 }$$

Alternatively using the p-parameterization discussed in Section 4.5 we find

$$\begin{eqnarray} &&\left({Q p}\right)^2\left(\eta c \over \sigma _{\rm fid}\right)\left({t_{\rm BH}\over t_S}\right) =\left({M_{g}\over M_{\rm SR}}\right) \left({M_{\rm BH} \over M_{\rm SR}}\right)\left({\sigma _{\rm fid}\over \sigma } \right)^{31/7}. \end{eqnarray} \tag{ 74 }$$

Constant porosity therefore guarantees downsizing, since

$$\begin{eqnarray} &&\left({t_{\rm BH} \over t_S}\right)\propto L_{\rm SR}^{-31/28} \propto M_{\rm BH}^{-{31/28}} \approx M_{\rm BH}^{-1}. \end{eqnarray} \tag{ 75 }$$

For Eddington-limited accretion, M_BH = M_SR. The ratio of black hole growth time to Salpeter time decreases with increasing black hole mass at constant Q. Note also that

$$$ 1/\alpha = t_{\rm BH}M_{\rm BH}\propto Q^{-4}M_{\rm BH}^2 M_{\rm SR}^{-59/28}\propto M_{\rm BH}^{-3/28}. $$$

Hence constant porosity also favors Bondi accretion since α≈ constant. Since σ ∼  10σ_fid for a massive spheroid, we also infer that for Eddington-limited accretion, the porosity must be of the order of 10% if the black hole growth time is of the order of the Salpeter time. Indeed, we can equate these timescales and infer that porosity depends weakly on black hole mass, Q^∝_∼ M^−1/3_BH. In the absence of AGN feedback, the derived star formation law yields a ratio of star-formation timescale to dynamical time that is proportional to σ_g and hence approximately constant, independently of galactic mass. This suggests that in starbursts when AGNs play no role, there should be no downsizing, as would be expected if internal processes such as those associated with formation of massive star clusters were to dominate.

Also, the Eddington ratio can be written as

$$$ f_{\rm Edd}\equiv {L^{\rm AGN}\over L_{\rm Edd}}=\eta c\left({\sigma _T\alpha }\over {m_pG}\right)M_{\rm BH}. $$$

Hence constant α is consistent with the observed trend measured in the Eddington ratio (Alonso-Herrero et al. 2008). The Eddington ratio is found to be lower for AGN than for quasi-stellar objects (QSOs) due to a combination of reduced host galaxy (and SMBH) mass as well as AGN feeding.

First, we give a qualitative argument for self-regulation. If the porosity Q is low, the jet is blocked, and the turbulent pressure is enhanced. Weak shocks propagate ahead of the cocoon and squeeze self-gravitating clouds over the BE stability limit. Star formation is triggered and the resulting supernovae drive up the pressure. Blow-out most likely occurs of the residual gas. This in turn increases Q ∼  1. However, dense clouds can now fall in unimpeded by intervening dense cold gas. The infall reduces Q, drives accretion and resurrects a strong jet.

Now, it is reasonable to assume that the cold phase is defined by a minimum density n_cr set by dissipative cooling and molecular formation, and that is not strongly dependent on metallicity and ionization fraction either in the molecular (Norman & Spaans 1997) or atomic (Wolfire et al. 2003) phases. The shape of the density PDF is well represented by a lognormal distribution (Wada & Norman 2007) with two parameters, dispersion of the distribution and its amplitude. When calculating the filling factor of either hot or cold gas, one integrates the lognormal PDF below or above n_cr. At fixed n_cr, the filling factor of either hot or cold gas depends only on one parameter, the dispersion. In general, for the isothermal case considered here, the dispersion is a linear function of $\ln {\cal {M}}$, for high Mach numbers (Krumholz & McKee 2005). In the adiabatic case the lognormal PDF dispersion is independent of the Mach number. The lognormal form is retained for supersonic turbulence both with and without star formation (Wada & Norman 2007). Even with significant feedback and increased Mach number, Q only varies logarithmically with Mach number, and therefore any change of filling factor in systems with developed supersonic turbulence is relatively slow as the turbulence is increased. In summary, the volume filling fractions of cold gas e^−Q and hot phase 1 − e^−Q depend only logarithmically on the Mach number. Hence the porosity is plausibly constant. Quantifying this argument we define the hot-phase filling factor in a turbulent supersonic medium to be the volume of the turbulent medium with a lower density than the mean by a factor ν_h < 1. Using the usual lognormal PDF

$$\begin{equation} f(n) = { 1 \over \sqrt{2 \pi } \sigma _t n} \exp {\left[-{{(\ln {n})}^2 \over 2 \sigma _t^2}\right]}, \end{equation} \tag{ 76 }$$

where for supersonic turbulence we use

$$\begin{equation} {\sigma _t}^2 = \ln \left[{ 1 + \lambda _t {\cal {M}}^2}\right] \end{equation} \tag{ 77 }$$

with ${\cal {M}}$ being the Mach number, and for the parameter λ_t we use λ_t = 3/4 (Krumholz & McKee 2005) for numerical estimates. We define

$$\begin{equation} \nu _h = {n_h \over n_0 \left[{ 1 + \lambda _t {\cal {M}}^2}\right]}, \end{equation} \tag{ 78 }$$

where n₀ is a reference density and we find an expression for f_h

$$\begin{equation} f_{h} = {erfc}\left[{ \ln \big[{1 \over \nu _h}\big] \over \sqrt{2} \sigma _t}\right], \end{equation} \tag{ 79 }$$

where often the first term of the asymptotic expansion for $erfc{[x]} = {{\exp {[-x^2]}}/x \sqrt{\pi }}+\cdots$ is a useful guide. For the Mach number, ${\cal {M}} = 3\hbox{--}10$ and for the under density parameter of the hot phase relative to the mean we use ν_h = 0.1 and find the hot phase filling factor is ∼ 10%–20%. As the Mach number increases, f_h increases in turn as the width of the PDF increases. Since f_h = 1 − exp − Q we find that

$$\begin{equation} Q = \ln {\left[ 1-f_{h}\right]} \end{equation} \tag{ 80 }$$

which for Q ≪ 1 becomes

$$\begin{equation} Q = \ln {\left[ 1+ f_{h}\right]} \end{equation} \tag{ 81 }$$

giving

$$\begin{equation} Q \sim \ln \left[{ 1+ {erfc}\left[\ln \left[{{1 \over {\bar{\nu }}} \over \sqrt{2} \sigma _t}\right]\right]}\right]. \end{equation} \tag{ 82 }$$

This shows how Q increases as the Mach number increases. Proceeding further, we can generalize our previous expression for Q, including both the competing effects of the star formation and supersonic turbulence, to

$$\begin{equation} Q \sim \ln {\left[ 1+ {erfc}\left[\ln \left[{{1 \over {\bar{\nu }}} \over \sqrt{2} \sigma _t}\right]\right] + f_{\rm cl}\epsilon _{{\rm SN}, {\rm fid}}\left(\frac{\sigma _{\rm fid}}{\sigma _{g}}\right)^{12/7}\right]}. \ \ \ \ \end{equation} \tag{ 83 }$$

Feedback from AGN is now readily incorporated by substituting ${\cal {M}}= (\sigma /\sigma _{\rm fid})^2$ and then making the substitute, as before, for the multiphase medium under the action of the mechanical and radiative pressures originating in the central source giving

$$\begin{equation} {\cal M}_{\rm AGN}^2= \left({c \over V_{\rm AGN}}\right)^{1/2} \left({L_{\rm AGN} \over L_{{\rm SR}, {\rm fid}}}\right)^{1/2} \end{equation} \tag{ 84 }$$

and thus using Equations (77) and (84) we obtain

$$\begin{equation} {\sigma _{t,{\rm AGN}}}^2 = \ln {\left[1 + \lambda _t \left({c \over V_{\rm AGN}}\right)^{1/2} \left({L_{\rm AGN} \over L_{{\rm SR}, {\rm fid}}}\right)^{1/2} \right]}. \end{equation} \tag{ 85 }$$

In the AGN case, it is now clearly quantified how the AGN pressure is reducing f_h by confining the hot SNR bubbles but on the other hand the AGN increases the Mach number of the turbulence and therefore broadens the PDF distribution, thus increasing f_h.

Global gas consumption is dominated by star formation, and locally by SMBH growth. The two are connected via the piston model and suggest a possible self-regulation loop for both spheroid and SMBH growth. We now show that the time sequence underlying the Magorrian relation can be interpreted in terms of the ratio of spheroid to SMBH growth rates.

The star formation (or gas consumption) rate is from Equation (42)

$$\begin{eqnarray} &&{1\over t_{\ast }} ={\epsilon _{\rm SN}\over t_d}\left({ L_{\rm AGN}\tau \over L_{\rm SR}}\right)^{1/2} \end{eqnarray} \tag{ 86 }$$

$$\begin{eqnarray} &&= {\bar{\epsilon }_{\rm SN}}G{\Sigma _{\rm tot}}\left({ L_{\rm AGN}\over L_{\rm SR}}\right)^{1/2} \end{eqnarray} \tag{ 87 }$$

$$\begin{eqnarray} &&\propto p_{\rm AGN}^{1/2}, \end{eqnarray} \tag{ 88 }$$

where $\bar{\epsilon }_{\rm SN} = \epsilon _{\rm SN}/\sigma$ is a constant depending only on supernova properties and the IMF. This controls spheroid growth and demonstrates spheroid downsizing via the AGN-enhanced pressure and associated star formation rate.

Writing the AGN luminosity as $L_{\rm AGN}=\eta \dot{M}_{\rm BH}c^2,$ the Eddington luminosity can be expressed in terms of the Salpeter time as ηM_BHc²/t_S. We now rewrite the star formation rate expression and obtain

$$\begin{eqnarray} &&{M_{\rm BH}\over \sigma ^4}=\left(t_S \over t_\ast \right)^2\left({ m_p}\over {\bar{\epsilon }_{\rm SN}\sigma _T \eta c \Sigma _{\rm tot}}\right)^2. \end{eqnarray} \tag{ 89 }$$

The predicted normalization of the Magorrian relation agrees with the local value and slope for canonical parameter values (Σ_tot ∼  Σ_*, η ∼  0.001, _SN ∼  0.1). Of course, there is considerable uncertainty due to possible variations in the IMF, supernova energy, and star formation timescale.

In fact, the relevant SMBH measure in distant objects is L_AGN rather than L_Edd. Let us make use of L_SR as a fiducial luminosity, in effect a proxy for σ⁴. We now rewrite the preceding expressions to obtain

$$\begin{eqnarray} &&L_{\rm AGN} / L_{\rm SR}=\epsilon _{\rm SN}^{-2} (f_g t_d/t_\ast)^2 \;{\sim}\; (t_d/t_\ast)^2. \end{eqnarray} \tag{ 90 }$$

We also have for the AGN-boosted star formation rate

$$\begin{eqnarray} &&{\dot{M}_{\ast }} ^{\rm AGN}\approx \epsilon _{\rm SN}M_g \Omega \left(L_{\rm AGN} / L_{\rm SR}\right)^{1/2} \;{\sim}\; M_g \Omega. \end{eqnarray} \tag{ 91 }$$

This is of course the optimal rate. We also see that $\dot{M}_{\ast }^{\rm AGN} \;^{\bm \propto}\!\!\!\! _{\bm \sim}\;\sigma ^4.$ This is not inconsistent with the observed dependence of outflow velocity on star formation rate (Martin 2005).

We may consider the case of a recently detected kiloparsec scale starburst at z = 6.24, hosted by a luminous quasar which has spatially resolved [C ii] emission as well as a large reservoir of CO-detected molecular gas (Walter et al. 2009). Other similar high z objects, detected in CO, are believed to be super-Magorrian (Maiolino et al. 2007). This quasar host galaxy also has a star formation rate of ∼ 1000 M_☉ year⁻¹ kpc⁻², an order of magnitude larger than is typical of starbursts without luminous AGN. For comparison, Arp 220, a low-redshift starburst hosting an AGN of luminosity comparable to its starburst, has a similar surface brightness in star formation but only over a 100 pc scale. It is tempting to infer, admittedly with only two well mapped examples, that we may be viewing AGN boosting of star formation, with the phenomenon being greatly magnified at high redshift for the most massive objects.

We infer that if the SMBH mass is super-Magorrian, then t_* < t_S and t_* < t_d, and star formation is very efficient. This seems to be the case at high redshift. The preponderance of data indeed suggests that the local relation becomes super-Magorrian prior to z ∼  2 (McLure et al. 2006; Woo et al. 2008) and indeed persists to z ≳ 5 (Maiolino et al. 2007). Coevolution of AGN accretion and the comoving star formation rate densities occurs to z ∼  2, but the accretion rate falls off relatively toward higher redshift (Silverman et al. 2008). Comparison of the cosmic star formation history and AGN accretion rates in comoving number density as a function of luminosity suggests that the peak in massive black hole growth rate occurs several Gyr prior to the star formation peak and that downsizing at z < 1 is due to diminishing accretion rates (Babić et al. 2007). Submillimeter galaxies (SMGs) are an exception. SMGs at z ∼  2 contain SMBH that are under-massive relative to the Magorrian relation (Alexander et al. 2008). This is suggestive of triggered star formation, which reduces t_*, and may be appropriate in major mergers that generate dense central gas environments where porosity feedback is suppressed. Note also that the peak in the major merger rate also precedes the peak in cosmic star formation rate (Ryan et al. 2008), and is approximately consistent with the peak in comoving AGN accretion rate density.