Many observations (e.g., Kormendy and Richstone, 1995; Magorrian et al., 1998; Häring and Rix, 2004; McConnell and Ma, 2013) have suggested that the mass of supermassive black holes (M_BH) correlates with the properties of their host galaxies such as stellar mass of bulges (M_bulge) at z ~ 0. This M_BH – M_bulge relation might suggest that supermassive black holes (SMBHs) would have co-evolved with their host galaxies.

SMBHs grow to the current mass (≳106M⊙) from their initial mass. The initial mass and its distribution have been debating. Although, there are many theoretical suggestions of formation mechanism and mass of seed BHs (e.g., Begelman et al., 2006), we cannot obtain what is the dominant mechanism by comparing theoretical models with observations since seed BHs are not observable directly.

Here, we focus on the M_BH – M_bulge relation for galaxies with bulge mass is less than 1010M⊙ to get the constraints on mass of seed BHs. This paper is a summary of Shirakata et al. (2016) in which we investigate the effect of the seed BHs' mass on model predictions of M_BH – M_bulge relation at Mbulge≲1010M⊙ by using an semi-analytic model of galaxy formation (hereafter SA model). In section 2 we briefly review the SA model we used. Section 3 includes the main results. Finally, in section 4, we summarize this review and briefly mention future prospects.

We use a revised version of an SA model, “New Numerical Galaxy Catalogue” (ν²GC; Makiya et al., 2016, hereafter M16), where the models related to the SMBH and AGNs are described in Enoki et al. (2003), Enoki et al. (2014), and Shirakata et al. (2015). We consider star formation in galactic disk and bulge, mergers of galaxies, atomic gas cooling, gas heating by UV feedback and feedbacks via supernovae and AGNs, and the growth of SMBHs by coalescence and gas accretion from their host galaxies.

Merging histories of dark matter halos are calculated from state-of-the-art cosmological N-body simulations (Ishiyama et al., 2015). The cosmological simulations have a high mass resolution and large volume compared to previous simulations (e.g., mass resolution is roughly four times better than those of Millennium simulations, Springel et al., 2005). Here we employ a simulation with L = 70.0 [h⁻¹ Mpc] of box size and 512³ particles, which corresponds to Mmin=2.20×108[h-1M⊙] of minimum halo mass.

We assume a ΛCDM universe which have the following parameters: Ω₀ = 0.31, λ₀ = 0.69, Ω_b = 0.048, σ₈ = 0.83, n_s = 0.96, and a Hubble constant of H0=100h km s-1 Mpc⁻¹, where h = 0.68 (Planck Collaboration et al., 2014).

Figure 1 shows the main result which depicts the M_BH – M_bulge relation at z ~ 0 obtained from the model and observations. Each panels correspond to the results of massive seed model (top), random seed model (middle), and light seed model (bottom), respectively. Red solid lines represents the model result, blue and green points represents the observational data.

We find all of the models reproduce the relation at Mbulge≳1010M⊙, while the massive seed model has an inconsistency in the observational results for less massive galaxies (Mbulge≲1010M⊙). Random and light seed models, on the other hand, provide the consistent results in the range of MBH≳105.5M⊙, with observational estimates. We thus conclude that to explain recent observational data of the M_BH – M_bulge relation at z ~ 0, seed BH mass should dominate with ~103M⊙.

We note that since the number of samples of galaxies with MBH≲105.5M⊙ (corresponds to Mbulge≲1010M⊙) are not sufficient. Observational data with the mass range are thus necessary to investigate the detailed mass distribution of the seed BHs. It is however difficult to estimate BH and bulge mass of less massive galaxies. We thus investigate whether the M_BH – M_bulge relation at higher redshifts could be useful for getting further constraints on the mass of seed BHs. Figure 2 displays the ratio of the average BH masses in the light seed model (≡ 〈M_BH〉₃) and those in the massive seed model (≡ 〈M_BH〉₅), as a function of bulge masses. The difference in the seed mass significantly appears in galaxies with bulge mass below 3×109M⊙ at z ~ 0, 1, and 2. We also find that the difference becomes smaller at higher redshift for a given M_bulge. Observations of less massive bulges at z ~ 0 would thus be more important than at higher redshifts for investigating the mass distribution of seed BHs.

We investigate how the mass of the seed BHs affects model predictions of the local M_BH – M_bulge relation by using an SA model. The results suggest that seed BHs with as massive as 105M⊙ should not be dominant for reproducing the observed M_BH – M_bulge relation at z ~ 0 over a wide range of bulge masses down to Mbulge≲1010M⊙. Obtaining stronger constraints of the detailed mass distribution of seed BHs observations of MBH≲105.5M⊙ would be required.

We have shown results of the local M_BH – M_bulge relations varying the mass of seed BHs. According to Shankar et al. (2016), M_bulge obtained from observations could be biased in favor of larger stellar masses. If so, we might have to use M_BH – velocity dispersion relation instead of the local M_BH – M_bulge. We leave it for future studies.

The spheroids formed through disk instability might be classified as so-called “pseudo bulges”. There are some debates whether pseudo bulges and classical bulges follow the same M_BH – M_bulge relation (e.g., Kormendy and Ho, 2013). We might need the model of the properties of pseudo bulges in the near future.

HS, RM, MN, ME, and MK have developed ν²GC. In addition, HS analyze the output data obtained from ν²GC. TK, TakO, and TaiO gave comments for the analysis. TI provides merger trees obtained from cosmological N-body simulations for ν²GC. YM gave fruitful comments from a standpoint of AGN observations.