RADIO SOURCE FEEDBACK IN GALAXY EVOLUTION

Gas will be accreted onto the halo along with the dark matter (e.g., White & Frenk 1991; Scannapieco et al. 2005). We take the mass of baryonic matter accreted at each point in the halo's evolution to be a fraction f_b of the total accreted mass. For high-mass halos $f_{\rm b} \approx \frac{\Omega _{\rm B}}{\Omega _{\rm M}}$. Photoionization heating from the UV background reduces this fraction in low-mass halos; this is discussed in Section 3.1. The gas is shocked to its virial temperature and density as it is accreted at the halo virial radius,

$$\begin{eqnarray} \rho _{{{\rm vir,gas}}} & = & \Delta _{{{\rm vir}}} \frac{3 H_{\rm 0}^2}{8 \pi G} \Omega _{\rm B} (1+z)^3 \nonumber \\ R_{{{\rm vir}}} & = & \left[ \frac{3 M_{{{\rm vir}}}}{4 \pi \rho _{{{\rm vir,gas}}} \big(\frac{\Omega _{\rm M}}{\Omega _{\rm B}} \big)} \right]^{1/3} \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} T_{{{\rm vir}}} & = & \frac{1}{2} \frac{G M_{{{\rm vir}}}}{R_{{{\rm vir}}}} \frac{\mu m_{\rm H}}{k_{\rm b}}, \nonumber \end{eqnarray}$$

where Δ_vir = 18π² + 82(Ω(z) − 1) − 39(Ω(z) − 1)² (Bryan & Norman 1998) is the mean overdensity at the virial radius, and μm_H is the mean particle mass, with μ = 0.62 for a fully ionized gas. The halo virial velocity is

$$\begin{equation} V_{{{\rm vir}}} = \left(\frac{G M_{{{\rm vir}}}}{R_{{{\rm vir}}}} \right)^{1/2}. \end{equation} \tag{ 8 }$$

The potential well of a spherical dark matter halo is assumed to follow the Navarro, Frenk, and White (NFW; Navarro et al. 1997) profile,

$$\begin{equation} \rho _{{{\rm DM}}}(r)=\frac{\rho _{{{{\rm DM}}},0}}{\left(\frac{r}{R_{\rm s}} \right) \left(1+\frac{r}{R_{\rm s}} \right)^2}, \end{equation} \tag{ 9 }$$

where scale factor $R_{\rm s}=\frac{R_{{{\rm vir}}}}{C}$ for concentration index $C=\frac{8}{1+z} \big(\frac{M_{{{\rm vir}}}}{1.4 \times 10^{14} \;M_{\odot }} \big)^{-0.13}$ (Bullock et al. 2001). The central density ρ_DM,0 is set by the constraint that the halo mass M_halo is contained within the virial radius. The NFW profile is amenable to integration in closed form, yielding

$$\begin{equation} \rho _{{{{\rm DM}}},0}=\left(\frac{\Omega _{\rm M}-\Omega _{\rm B}}{\Omega _{\rm M}} \right) \left(\frac{M_{{{\rm halo}}}}{4 \pi R_{{{\rm vir}}}^3 f \big(\frac{1}{C} \big)} \right), \end{equation} \tag{ 10 }$$

where $f(x)=x^3 \left[ {\rm ln} \left(1+\frac{1}{x} \right) - \frac{1}{1+x} \right]$.

We assume that the density distribution of hot X-ray gas also follows Equation (10). As with the dark matter, the constraint that all the gas is located within the virial radius determines the central density ρ_gas,0. In this fashion, the density profile can be determined at each point in the halo's evolution.

The gas subsequently cools, with the cooling time of each gas parcel being given by $t_{{{\rm cool}}} = \frac{3}{2} \frac{\rho _{{{\rm gas}}} k_{\rm B} T}{\mu m_{\rm H} n_e^2 \Lambda (T,Z)}$, where the electron density $n_e = \frac{2+\mu }{5 \mu } \frac{\rho _{{{\rm gas}}}}{m_{\rm H}}$ for a fully ionized gas. This yields

$$\begin{equation} t_{{{\rm cool}}} = \frac{75 \mu }{2 (2+\mu)^2} \frac{m_{\rm H} k_{\rm B} T}{\rho _{{{\rm gas}}} \Lambda (T,Z)}. \end{equation} \tag{ 11 }$$

The cooling function Λ(T, Z) depends on gas temperature and metallicity. The detailed cooling models of Sutherland & Dopita (1993) for Z = 0.1 Z_☉ are used here.

The cool gas is deposited onto an accretion disk after a dynamical time $t_{{{\rm dyn}}} (r) = \frac{r}{V_{{{\rm vir}}}}$, and subsequently star formation can take place. Kennicutt (1989) showed that SFRs in a sample of nearby spiral galaxies have a sharp cutoff below a certain surface density, in line with gravitational instability considerations. This critical surface density is representative of the disk gas density (except at the very edges), and thus one can define a critical disk mass below which no star formation takes place. Kauffmann (1996) gives the threshold density as $\Sigma _{{{\rm crit}}} = 0.59 \big(\frac{V_{{{\rm vir}}}}{\mbox{km s}^{-1}} \big) \big(\frac{R_{{{\rm disk}}}}{\mbox{kpc}} \big)\;M_{\odot }$ pc⁻². Integrating over the surface of the disk, the critical mass is $M_{{{\rm crit}}} = 7.5 \times 10^8 \big(\frac{V_{{{\rm vir}}}}{\mbox{km s}^{-1}} \big) \big(\frac{R_{{{\rm disk}}}}{\mbox{kpc}} \big)\;M_{\odot }$. Adopting disk radius R_disk = 0.1R_vir (Kauffmann 1996; Croton et al. 2006), this yields

$$\begin{equation} M_{{{\rm crit}}}=7.5 \times 10^7 \bigg(\frac{V_{{{\rm vir}}}}{\mbox{km s}^{-1}} \bigg) \bigg(\frac{R_{{{\rm vir}}}}{\mbox{kpc}} \bigg) M_{\odot }. \end{equation} \tag{ 12 }$$

Following Croton et al. (2006), the instantaneous SFR is taken to be

$$\begin{equation} \dot{M}_{\star }= \alpha _{\rm SF} \frac{M_{{{\rm cold}}}-M_{{{\rm crit}}}}{t_{{{\rm dyn,disk}}}}, \end{equation} \tag{ 13 }$$

where the disk dynamical time is $t_{{{\rm dyn,disk}}}=\frac{R_{{{\rm disk}}}}{V_{{{\rm vir}}}}$ and the star formation efficiency α_SF ≈ 0.1 (Croton et al. 2006). Here, stellar mass refers to the total stellar mass within the dark matter halo.

Observations (e.g., Leahy et al. 1989; Subrahmanyan et al. 1996) suggest radio sources are self-similar, meaning the bow-shock and cocoon radii are related by R_cocoon = λR_shock. We adopt the models of Kaiser & Alexander (1997) and Alexander (2000, 2006) to describe the evolution of the cocoon radius R_cocoon with time. The gas density profile in each halo is approximated as a flat core–double power-law atmosphere. In other words, the source initially evolves in a flat atmosphere, followed by two power-law profiles of the form $\rho (r) = \rho _{{{\rm core}}} \big(\frac{r}{R_{{{\rm core}}}} \big)^{-\beta }$. This would correspond to an expansion through a high-density core, into the galaxy, and then the cluster gas.

Both analytical work (e.g., Kaiser & Alexander 1997) and numerical simulations (Reynolds et al. 2001) suggest the swept-up gas lying between the shock and cocoon radii is isobaric. Such an arrangement is facilitated by backflow of swept-up gas from the hotspot to the sides of the cocoon. We therefore follow Heinz et al. (1998) and Alexander (2002) and assume the cocoon to be spherical.

We further assume the swept-up gas evolves isothermally, consistent with cluster observations. Following the treatment of Alexander (2002), the supersonic expansion of the radio source shocks this gas to a temperature

$$\begin{equation} T_{{{\rm shocked}}} = \frac{15}{16} \frac{3-\beta }{11-\beta } \left(\frac{\mu m_{\rm H}}{k_{\rm B}} \right) \frac{1}{\lambda ^2} \dot{R}_{{{\rm cocoon}}}^2, \end{equation} \tag{ 22 }$$

where λ and β are related by $1-\lambda ^3 = \frac{15}{4(11-\beta)}$.

Important parameters governing the radio source expansion and hence the heating of the gas are the gas density profile, which is given by Equation (9); the cocoon axial ratio R_T, which is related to the jet opening angle and is fixed at R_T = 2, a value appropriate for Cygnus A (Kaiser & Alexander 1997; Begelman & Cioffi 1989); and jet power Q_jet. Following the dynamical model of Kaiser & Alexander (1997), the cocoon radius is

$$\begin{equation} R_{{{\rm cocoon}}} (t) = a_D R_{{{\rm core}}} \left(\frac{t}{\tau } \right)^{3/(5-\beta)}, \end{equation} \tag{ 23 }$$

where $\tau = \big(\frac{2 R_{\rm core}^5 \rho _{\rm core}}{Q_{\rm jet}} \big)^{1/3}$ is a convenient timescale, and the dimensionless constant a_D is given by (Kaiser & Alexander 1997)

$$\begin{equation} a_D = \left[ \frac{(\Gamma _{\rm x}+1) (\Gamma _{\rm c}-1) (5-\beta)^3}{18 \pi \big(9 \big[ \Gamma _{\rm c} + (\Gamma _{\rm c}-1) R_{\rm T}^2 \big] -4-\beta \big)} \right]^{1/(5-\beta)} \nonumber. \end{equation} \tag{ 24 }$$

The adiabatic indices for the cocoon and external gas, Γ_c and Γ_x respectively, are equal to 5/3, corresponding to non-relativistic material.

Theoretical work (Narayan et al. 1998; Meier 2001) as well as observations of black hole X-ray binaries (Körding et al. 2006; Fender et al. 2004) suggest that in both these objects and AGNs at least two different accretion states exist. In both cases, angular momentum is removed from the accretion disk by viscosity, allowing the gas to spiral in toward the central black hole. At high inflow rates (compared to the Eddington rate), the accretion flow is described by a standard thin disk solution (Shakura & Sunyaev 1973). Here, the accretion disk is geometrically thin and optically thick, and produces a quasi-blackbody spectrum. This flow is radiatively efficient, as all the energy released through viscous dissipation can be radiated away (Narayan 2002). By contrast, at low accretion rates the flow is geometrically thick and optically thin. As a result, the cooling times are long, and instead of being radiated away the thermal energy of the inflowing gas is advected inward. The resultant advection dominated accretion flow (ADAF; Narayan & Yi 1995) is thus radiatively inefficient, and can produce powerful jets.

Meier (2001) gives the jet power generated by a geometrically thick, optically thin ADAF disk as

$$\begin{eqnarray} Q_{{{\rm jet,ADAF}}} & = & 1.3 \times 10^{38} \left(\frac{\alpha _{\rm ADAF}}{0.3} \right)^{-1} g_{\rm ADAF}^2 (0.55 f_{\rm ADAF}^2 \nonumber \\ && + 1.5 f_{\rm ADAF} j_{{{\rm BH}}} + j_{{{\rm BH}}}^2) \left(\frac{M_{{{\rm BH}}}}{10^9 \mbox{${M}_{\odot }$}} \right) \nonumber\\ &&\times\left(\frac{\dot{M}_{{{\rm BH}}}}{\epsilon _{{{\rm rad}}} \dot{M}_{{{\rm edd}}}} \right) {\rm W} \end{eqnarray} \tag{ 25 }$$

for $\dot{m}_{{{\rm BH}}} \equiv \frac{\dot{M}_{{{\rm BH}}}}{\dot{M}_{{{\rm edd}}}} < \dot{m}_{{{\rm crit}}}$, where $\dot{m}_{{{\rm crit}}}$ is a parameter discussed in the following section.

Here, $\dot{M}_{{{\rm BH}}}$ is the black hole accretion rate, and $\dot{M}_{{{\rm edd}}}=\frac{L_{{{\rm edd}}}}{\epsilon _{{{\rm rad}}} c^2}=\frac{2.3}{\epsilon _{{{\rm rad}}}} \big(\frac{M_{{{\rm BH}}}}{10^8 {{M}_{\odot }}} \big)$ M_☉ yr⁻¹ is the Eddington accretion rate for a radiative efficiency _rad. Disk viscosity is responsible for transporting angular momentum outward, and is given by the coefficient α_ADAF ∼ 0.3 (Narayan et al. 1998; Narayan 2002). The coefficients g_ADAF and f_ADAF refer to the ratios of actual angular velocity and azimuthal magnetic field to those calculated by Narayan & Yi (1995), and j_BH is the black hole spin. Following Meier (2001) and Okamoto et al. (2008b) we adopt g_ADAF = 2.3, f_ADAF = 1 and j_BH = 0.5, noting that the jet power is not very sensitive to realistic changes in these values. For this choice of parameters, Equation (25) can be rewritten as

$$\begin{eqnarray} Q_{{{\rm jet,ADAF}}} & = &4.5 \times 10^{39} \left(\frac{\dot{M}_{{{\rm BH}}}}{\mbox{${M}_{\odot }$}\;{\rm yr^{-1}}} \right) {\rm W} \nonumber \\ & = & 0.79 \dot{M}_{{{\rm BH}}} c^2. \end{eqnarray} \tag{ 26 }$$

For accretion rates $\dot{M}_{{{\rm BH}}} > \dot{m}_{{{\rm crit}}} \dot{M}_{{{\rm edd}}}$ the accretion flow is described by the standard Shakura–Sunyaev thin disk solution, which for the adopted viscosity parameter is given by

$$\begin{equation} Q_{{{\rm jet,TD}}} = 6.0 \times 10^{-4} \left(\frac{M_{{{\rm BH}}}}{10^9 \mbox{${M}_{\odot }$}} \right)^{-0.3} \left(\frac{\dot{M}_{{{\rm BH}}}}{\mbox{${M}_{\odot }$}\;{\rm yr^{-1}}} \right)^{0.2} \dot{M}_{{{\rm BH}}} c^2. \end{equation} \tag{ 27 }$$

This mode of accretion (referred to as the "quasar mode" by Croton et al. 2006) is radiatively efficient, and typically results in weaker jets than the ADAF case. Thus it is identified with a luminous AGN phase (Croton et al. 2006; Bower et al. 2006), while the ADAF phase corresponds to the "radio mode" of Croton et al. and is responsible for radio source feedback. In what follows, we assume that when the instantaneous black hole accretion rate exceeds the critical rate, the disk is in the "quasar mode;" and when the instantaneous accretion rate is below this value, a radio jet with power given by Equation (26) is generated. We further assume that in this case the actual accretion rate onto the black hole is given by $\dot{M}_{{{\rm BH}}} = \dot{m}_{{{\rm crit}}} \dot{M}_{{{\rm edd}}}$, i.e., the disk is fueled at the maximum rate (e.g., Hopkins et al. 2006). The usual radiative efficiency of _rad = 0.1 is assumed in evaluating $\dot{M}_{{{\rm edd}}}$.

Croton et al. (2006) and Bower et al. (2006) both took the "radio mode" heating provided by the jets to be continuous. Observations of black hole binaries and the apparent balance between heating and cooling rates, however, suggest otherwise. Jets will be intermittent for two reasons. Feedback limits the availability of cold gas required to fuel the central engine (Shabala et al. 2008), resulting in weaker jets that are more liable to disruption by instabilities in the dense central regions. On the other hand, when the radio cocoon expands outside the high density regions, the central gas cools rapidly. The resulting high black hole accretion rates qualitatively change the nature of the accretion flow from a radiatively inefficient ADAF to a radiatively efficient standard thin disk flow (Narayan et al. 1998). This once again results in a much weaker jet (by ∼3 orders of magnitude according to Equations (26) and (27)) which can be disrupted.

In the assumed density profile, the jets are most likely to only be susceptible to disruption by Kelvin–Helmholtz instabilities in the denser central regions of the halo (Alexander 2002; Kaiser & Alexander 1997). Following the analysis of Alexander (2002), the minimum power a jet must have to traverse the flat core is taken as

$$\begin{equation} Q_{{{\rm jet,min}}} = 10^{37} \left(\frac{R_{{{\rm core}}}}{2.5\;{\rm kpc}} \right)^2 \left(\frac{\rho _{{{\rm core}}}}{1.5 \times 10^{22}\ {\rm kg\;m^{-3}}} \right) {\rm W}. \end{equation} \tag{ 28 }$$

Similarly, for a radio source that has expanded outside R_core, the jet power must exceed Q_jet,min ∝ R^2−β_cocoonρ_core in order to reach the hotspots. If this does not happen, or the instantaneous rate of accretion onto the black hole $\dot{m}_{{{\rm BH}}}$ exceeds $\dot{m}_{{{\rm crit}}}$ causing the disk to become radiatively inefficient, the jet is terminated, and a new jet inflates a new radio cocoon. For a radio source that is still within the core, i.e. when R_cocoon < R_core, the minimum required jet power is given by replacing the core radius with the cocoon radius in Equation (28).

The final important point concerning jet intermittency relates to the critical accretion rate denoting the transition between radiatively efficient and inefficient disks. Narayan et al. (1998) estimate that an ADAF will switch to a radiatively efficient (thin disk) flow when the accretion rate in Eddington units exceeds $\dot{m}_{{{\rm crit,up}}} \sim \alpha _{\rm ADAF}^2 \approx 0.1$ for the adopted viscosity value. This threshold is consistent with the observed transition from low/hard to high/soft state in X-ray binaries (Esin et al. 1998; Körding et al. 2006). Interestingly, the reverse transition is less well defined, with $\dot{m}_{{{\rm crit,down}}} \sim 0.01\hbox{--}0.1$ (Narayan 2002). The thin disk will develop a hole in its inner regions, where an ADAF is allowed. As the accretion rate drops further, the region hosting an ADAF will grow, until this becomes the dominant mode of accretion. Hence, in the present model we treat $\dot{m}_{{{\rm crit,down}}}$ as a free parameter in the range 0.01–0.1.

As Figures 1 and 2 show, low-mass halos are little affected by AGN feedback. Since these halos host galaxies with relatively low stellar masses, the low-mass end of the stellar mass function (or, correspondingly, the faint end of the K-band luminosity function) can be used to constrain the contribution of SNe to suppressing star formation.

In Figure 3 the local stellar mass function is plotted for various values of _halo. The curves are constructed by convolving the star formation histories of individual halos with the halo mass function obtained at each redshift from the Millennium Simulation (Section 2.1). Shown by open symbols are the stellar mass functions derived from the Cole et al. (2001) 2dFGRS K-band luminosity function assuming Salpeter and Bottema IMFs. These correspond to the upper and lower limits, respectively, on the mean mass-to-light ratio used to convert from luminosities to stellar masses (Bell et al. 2003). In the case of the Salpeter IMF, we use the average mass-to-light ratio of 1.32 as quoted by Cole et al. For the Bottema IMF this value is reduced to 0.82. The filled symbols correspond to the stellar mass function derived with a diet Salpeter IMF, obtained by decreasing the Salpeter mass-to-light ratio by 0.15 dex to 1.17 (Bell et al. 2003); this gives good agreement with the Cole et al. stellar mass function derived from J-band observations.

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It can be seen that _halo = 0.01 provides a good match to the local stellar mass function at the low-mass end. Although the choice of the IMF impacts the derived mass function at the bright end, in the absence of AGN feedback the number densities of the most massive galaxies are overpredicted significantly.

Low-mass halos are much more abundant than their higher mass counterparts- for example, 10¹² M_☉ halos are 5 orders of magnitude more abundant than 10¹⁵ M_☉ ones (Jenkins et al. 2001). Since massive halos host the more massive galaxies, however, the question of how the total stellar mass is distributed between halos is a non-trivial one. We plot in Figure 4 the stellar mass fraction as a function of halo mass at the present epoch. Halos with masses less than 3 × 10¹¹–3 × 10¹²M_☉ are seen to dominate the total stellar mass budget, despite the total mass distribution peaking at higher halo masses.

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This result suggests that the star formation history of the universe is largely determined by reionization and SN, rather than AGN, feedback. This is certainly true for z > 1, when AGN feedback does not greatly affect the evolution of the stellar mass content even in the most massive halos (Figure 1). Hence a guide to the efficiency of the SN feedback can be obtained from observations of the evolution of the SFR density, plotted in Figure 5. Observations at z > 1 suggest _halo ≈ 0.01. It is also clear from Figure 5 that models without AGN feedback significantly overpredict SFRs at later epochs. The observed SFRs are derived from a combination of UV, IR, O ii, radio, and X-ray observations (see caption) assuming a diet Salpeter IMF. It is worth noting that, due to the assumed IMF, these are close to the upper limits on the actual SFRs at each epoch, and the actual SFRs can be up to 0.35 dex lower than these values.

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Figure 10 plots the SFR density as a function of cosmic epoch for the model providing the best fit to the local stellar mass function (Figure 7), _halo = 0.01, _acc = 0.02, $\dot{m}_{{{\rm crit,down}}}=0.03$. The no-AGN predictions are also plotted for comparison. As argued in Section 4.2.2, models without AGN feedback significantly overpredict the global SFR density at the present epoch. Inclusion of such feedback, on the other hand, provides results that are in good agreement with observations.

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Integrating the predicted local stellar mass function yields a total present-day stellar mass density of ρ_⋆ = 4.5 M_☉ Mpc⁻³, or Ω_⋆ = 3.3 × 10⁻³. This is consistent with observational estimates of Ω_⋆ = (2.8 ± 0.4) × 10⁻³ (Cole et al. 2001; Bell et al. 2003) and (3.5 ± 0.4) × 10⁻³ (Kochanek et al. 2001) made with the same diet Salpeter IMF that is used in the models. The integral of the predicted SFR density over the Hubble time is Ω_⋆ = 5.1 × 10⁻³, since a fraction f_rec = 0.33 of stars terminate in SNe.

A successful galaxy formation model must correctly describe the evolution of the galactic population, while simultaneously matching the local observable properties such as the mass function. This has traditionally proved very difficult, with most models either reproducing the large number of massive galaxies at high redshift but significantly overpredicting the bright end of the local luminosity function (e.g., Kauffmann et al. 1999a, 1999b), or conversely matching the local K-band luminosity function but underpredicting the bright galaxy counts at z ≳ 0.5 (e.g., Cole et al. 2000; Baugh et al. 2005).

Figure 11 plots the evolution of the stellar mass function predicted by the model. It is in remarkable agreement with the results of the K20 survey (Fontana et al. 2004), and those of the FORS Deep Field (FDF) and GOODS South (Drory et al. 2005) surveys, out to a redshift of z ≳ 1.5. The only other model in the literature to date providing a similar match is that of Bower et al. (2006).

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The last two panels of Figure 11 illustrate that the model consistently underpredicts galaxy counts at z ⩾ 2. The uncertainty in the adopted universal IMF cannot account for this as there is a good match between the model and observations at low redshift. However, observations of submillimeter and Lyman break galaxies (Baugh et al. 2005) as well as element abundances (Nagashima et al. 2005) suggest that a more top-heavy IMF is required at z ≳ 2, yielding lower mass-to-light ratios and hence rendering the derived high-redshift stellar masses as overestimates. The physical motivation for two different IMFs comes from a change in the dominant star formation mechanism with cosmic epoch. At high redshifts star formation is mainly triggered by mergers; while at the present epoch it follows the standard disk formation scenario (Kauffmann 1996; Kennicutt 1998).

Figure 12 shows the evolution of the stellar mass function under the assumption of a Bottema IMF. This gives good agreement at high redshift, but overpredicts the stellar content at low z. These results suggest a switch from the diet Salpeter IMF to a more top-heavy Bottema IMF at z ≳ 2. Baugh et al. (2005) estimate that quiescent star formation (i.e., in disks) dominates for z ≲ 2–3, consistent with our findings.

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The slight overprediction in the counts of low-mass galaxies at z > 0 is consistent with the results of other models (e.g., Bower et al. 2006). There are two possible causes of this. One possibility would be current implementations of SN feedback truncating star formation too early. Alternatively, observational constraints at the low-mass end are likely to suffer from incompleteness, as evidenced by substantial field-to-field variations in Figures 11 and 12.