Here we describe how we calculate the amount of the cold gas, which gets accreted to a central galaxy. We note that we define a central galaxy as a central galaxy of the most massive progenitor halo.

We firstly calculate cooling radius r_cool(t). Same as Makiya et al. (2016), we assume Navarro-Frenk-White (NFW) density profile (Navarro et al., 1997) for dark matter (DM) haloes and isothermal density profile with a finite core radius, r_c for hot gas haloes;

where r_s is the specific radius of the DM halo, which is described by using concentration parameter, c, and virial radius, R_vir, as R_vir/r_s ≡ c. We assume r_c = 0.22r_s (Makino et al., 1998). We use the analytical formulae of c obtained from fitting of cosmological N-body simulations (Prada et al., 2012). After the collapse of a DM halo, the hot gas gradually cools via radiative cooling. Then, the cooling time is described with r_cool as

where μ, m_p, k_B, and n_e are the mean molecular weight, proton mass, Boltzmann constant, and electron number density, respectively. We employ a cooling function, Λ, provided by Sutherland and Dopita (1993), which is a function of hot gas metallicity, Z_hot, and virial temperature, T_vir. Virial temperature is calculated from circular velocity of the host DM halo, V_circ, as

r_cool(t) is defined at which t_cool of Equation (3) is equal to the time elapsed since the halo formation epoch.

We next calculate free fall radius, r_ff(t) with ρ_NFW(r);

where G is the gravitational constant. Now t_ff = t_cool, in order to compare the size of r_cool with r_ff at the same time.

We then evaluate the accretion radius, r_acc(t), in which gas can actually cool and get accreted to the central galaxy. We set r_acc as the minimum value among r_cool, r_ff, and R_vir, where R_vir is the virial radius of the halo. The case with r_acc = r_cool means the gas cooling is not so efficient and gas can free-fall rapidly. This case occurs only for the massive (>10¹³ M_⊙) haloes. Since Makiya et al. (2016) assume r_acc = MIN{r_cool, R_vir}, they would overestimate the amount of cold gas especially at z < 1.0 if they employ the same parameter set as that of in this paper. We note that we assume that the existence of a “cooling hole” same as Makiya et al. (2016); the radial profile of hot gas remains unchanged until the DM halo mass doubles.

We introduce the so-called radio-mode AGN feedback process in order to prevent gas in massive haloes from cooling and forming stars. Following Bower et al. (2006), gas cooling in a halo is quenched when the following two conditions are satisfied:

where L_Edd is the Eddington luminosity, L_cool is the cooling luminosity of the gas, α_cool and ϵ_SMBH are free parameters which are determined in order to reproduce the luminous end of the luminosity function of galaxies at z ~ 0. In this paper, we fiducially set (α_cool, ϵ_SMBH) = (1.00, 0.012).

From Equation (6), we can see that the radio-mode AGN feedback is more efficient for more massive galaxies at lower redshifts since t_ff and t_cool are roughly proportional to (1 + z)^−3/2 and (1 + z)⁻³, respectively. Galaxies with radio-mode AGN feedback might correspond to FR-I radio galaxies (Fanaroff and Riley, 1974). It is however uncertain how the FR-I radio galaxies have triggered their activity. We thus cannot define galaxies with radio-mode AGN feedback in our model as mock FR-I radio galaxies. We have not included the model of FR-II radio galaxies, which would have intense cold gas accretion. We leave it for future work.

The model which employed for this study includes some revisions from Makiya et al. (2016). We skipped the details of the revisions because of the limitation of the numbers of characters for this paper. For more details appear in Shirakata et al. (2016) and Shirakata et al. in preparation. We find that the fiducial model can explain observational galaxy properties well. In Figure 1, we compare K-band luminosity functions obtained from the model with observations (see the figure caption). The bright end slope of K-band luminosity function of galaxies at z ~ 0 is sensitive to the strength of the radio-mode AGN feedback. If the AGN feedback is weak, we overproduce bright galaxies with M_K < −22.0 and the luminosity function has a single-power law. We also find that the fiducial model can explain SMBH mass function, Faber-Jackson relation, Tully-Fisher relation at z ~ 0.

Figure 2 shows the M_BH − σ relation at z ~ 0 with the fiducial model. We select the galaxies with M_V < −20, where M_V is the absolute AB magnitude in V-band. We derive the best fit function with the least square method (black line):

For comparison, we also depict the best fit function obtained from van den Bosch (2016) (gray line):

We classify galaxies between over-massive BH galaxies and under-massive BH galaxies following our best fit function of M_BH − σ relation with 1σ errors. We then depict M_BH−age relation in Figure 3. We find that galaxies with larger σ have older stellar ages and over-massive BH galaxies have older stellar ages. These results are roughly consistent with the result obtained from Martín-Navarro et al. (2016). On the other hand, when we employ the model with weak AGN feedback, in which α_cool set 0.1 times smaller than the fiducial value, the model cannot explain the result of Martín-Navarro et al. (2016) (Figure 4); all galaxies have younger stellar age than those obtained from Martín-Navarro et al. (2016) and galaxies with under-massive BHs become older. This result would be explained as follows. From Equation (6), more massive (i.e., more luminous) galaxies tend to be quenched their star formation by the AGN feedback. We thus get the steeper slope for the fiducial model compared to the model with the weak AGN feedback (α_cool is smaller). In addition, from Equation (7), galaxies with more massive SMBHs should have so much heating luminosity with the AGN activity that compensates the cooling luminosity. Therefore, we can get the same trend as Martín-Navarro et al. (2016), over-massive black hole galaxies have older stellar ages. From these results, we have concluded that radio mode AGN feedback model might play a role for quenching the star formation of massive galaxies.

We find that the difference between galaxies with over-massive BHs and under-massive BHs is smaller than those obtained from the observation (Martín-Navarro et al., 2016). For observations, as noted in Martín-Navarro et al. (2016), their estimation of the galactic ages have large observational errors especially for the galaxies with smaller velocity dispersion. It would thus be important to use a larger amount of observational data.

For the theoretical model, the smaller difference between galaxies with over- and under- massive BHs would be partly because of the radio-mode AGN feedback modeling. Since more massive haloes tend to have more massive SMBHs, under-massive galaxies with larger velocity dispersion (and with larger bulge mass) could be quenched their star formation because of the AGN feedback. The ages of over- and under- massive galaxies with large velocity dispersion in the model thus overlap. We then test the effect of the galaxy selection and find the relation between M_BH, σ, and stellar age is sensitive to the sample selection; when we select only brighter galaxies with M_V < −22.0, the difference between galaxies with over- and under- massive BHs becomes clear. It might suggest that the efficiency of the radio-mode AGN feedback should more strongly depend on the BH mass.

Another possible way to explain the results of Martín-Navarro et al. (2016) would be to introduce quasar-mode AGN feedback. Since the number density of bright quasars peaks at higher redshift (e.g., Ueda et al., 2014), quasar-mode AGN feedback could be effective at higher redshift while the radio-mode becomes efficient at lower redshift. We will leave it for future work.

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