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Philip Taylor, Christoph Federrath, Chiaki Kobayashi, Star formation in simulated galaxies: understanding the transition to quiescence at 3 × 1010 M⊙, Monthly Notices of the Royal Astronomical Society, Volume 469, Issue 4, August 2017, Pages 4249–4257, https://doi.org/10.1093/mnras/stx1128
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Abstract
Star formation in galaxies relies on the availability of cold, dense gas, which, in turn, relies on factors internal and external to the galaxies. In order to provide a simple model for how star formation is regulated by various physical processes in galaxies, we analyse data at redshift z = 0 from a hydrodynamical cosmological simulation that includes prescriptions for star formation and stellar evolution, active galactic nuclei, and their associated feedback processes. This model can determine the star formation rate (SFR) as a function of galaxy stellar mass, gas mass, black hole mass, and environment. We find that gas mass is the most important quantity controlling star formation in low-mass galaxies, and star-forming galaxies in dense environments have higher SFR than their counterparts in the field. In high-mass galaxies, we find that black holes more massive than ∼ 107.5 M⊙ can be triggered to quench star formation in their host; this mass scale is emergent in our simulations. Furthermore, this black hole mass corresponds to a galaxy bulge mass ∼ 2 × 1010 M⊙, consistent with the mass at which galaxies start to become dominated by early types ( ∼ 3 × 1010 M⊙, as previously shown in observations by Kauffmann et al.). Finally, we demonstrate that our model can reproduce well the SFR measured from observations of galaxies in the Galaxy And Mass Assembly and Arecibo Legacy Fast ALFA surveys.
1 INTRODUCTION
Understanding how galaxies evolve is a central problem in astronomy. It has been recognized for some time that energy feedback from stars and supermassive black holes (BHs) in active galactic nuclei (AGN) is responsible for suppressing star formation in low- and high-mass galaxies, respectively. The transition between these two regimes occurs at a galaxy stellar mass, M* ∼ 3 × 1010 M⊙ (e.g. Kauffmann et al. 2003; Baldry et al. 2006). The aim of this study is to improve our understanding of the physical processes that influence star formation in all galaxies and give rise to this transition scale.
Stars are formed according to an initial mass function (IMF), which describes the distribution of stellar masses formed (e.g. Salpeter 1955; Chabrier 2003; Kroupa 2008; Kroupa et al. 2013). Very massive, but short-lived, stars explode as core-collapse supernovae, heating the surrounding interstellar medium (ISM), and enriching it with metals. Lower mass stars can also produce supernovae on longer (Gyr) time-scales via white dwarf or neutron star progenitors. The injection of energy into the ISM by both stellar feedback (e.g. Heckman et al. 2000; Pettini et al. 2000; Ohyama et al. 2002) and AGN feedback (Lynds 1967; Kraft et al. 2009; Feruglio et al. 2010; Cicone et al. 2012; Tombesi et al. 2013; Teng et al. 2014) can suppress further star formation on galactic scales (see also, e.g. Bicknell et al. 2000; Silk 2013; Zubovas et al. 2013; Shabala, Crockett & Kaviraj 2015; Bieri et al. 2016, who discuss the possibility of AGN-induced star formation on small scales). Stellar and AGN feedback can drive galactic outflows, which have been reproduced in cosmological simulations (e.g. Taylor & Kobayashi 2015b). By quenching star formation, AGN feedback in massive galaxies causes them to have lower specific star formation rates (SFRs), enhanced [α/Fe], and redder colours than lower mass galaxies, which may lead to the observed downsizing phenomenon (Cowie et al. 1996; Juneau et al. 2005; Bundy et al. 2006; Stringer et al. 2009).
Stars form in the densest regions of giant molecular clouds in the ISM (Elmegreen & Scalo 2004; Mac Low & Klessen 2004; McKee & Ostriker 2007; Federrath & Klessen 2012; Padoan et al. 2014). AGN activity, too, relies on the accretion of gas on to the central supermassive BH. The availability of gas for both star formation and AGN therefore plays an important role in the subsequent evolution of a galaxy.
Most previous theoretical work has sought to explain observed SFRs as a fraction of the molecular gas mass per (density-dependent) free-fall time (Krumholz, Dekel & McKee 2012; Federrath 2013; Salim, Federrath & Kewley 2015; Semenov, Kravtsov & Gnedin 2016). Such studies can explain local SFRs on the scale of giant molecular clouds, but a full understanding of the processes that affect the global galactic SFR is lacking. Observational data showed that there are two important, empirical relations: the Kennicutt–Schmidt (KS) law (Schmidt 1959; Kennicutt 1989; Kennicutt & Evans 2012) and the star formation main sequence (SFMS; e.g. Elbaz et al. 2011; Zahid et al. 2012; Renzini & Peng 2015). Most previous theoretical work has sought to explain the former, but a full understanding of the latter is lacking. Feldmann et al. (2016) use high-resolution simulations from the Feedback in Realistic Environments project (Hopkins et al. 2014) to study the processes governing the SFR of galaxies at z ∼ 2, but do not explore the relative importance of these processes.
Galaxies form from massive gas clouds at high redshift, but can both gain and lose gas over the course of their life. Simulations based on the cold dark matter (CDM) cosmology predict large-scale flows of cold gas along filaments in the cosmic web (e.g. Dekel, Sari & Ceverino 2009; Cen 2014) that can efficiently supply a galaxy with gas without being shock heated to the virial temperature (e.g. Birnboim, Padnos & Zinger 2016). Groups and clusters are found at more massive nodes in the web, and can be fed by a greater number of filaments.
Gas is also transported into galaxies via mergers. So-called wet (two gas-rich galaxies) and damp (one gas-rich and one gas-poor galaxy) mergers can provide a galaxy with gas for both star formation and AGN activity, as well as potentially altering the morphology. During a merger, gas can be efficiently transported to the centre of the new galaxy, providing fuel for star formation and AGN activity (Toomre & Toomre 1972; Combes et al. 1990; Mihos & Hernquist 1994, 1996; Barnes & Hernquist 1996; Hopkins et al. 2008). Major mergers, in which the galaxies have similar masses, are typically thought to be the main drivers of galaxy evolution (in terms of morphology, SFR, and AGN activity) in the local Universe (e.g. Darg et al. 2010), though minor mergers contribute significantly to the cosmic star formation budget at low redshift (e.g. Kaviraj 2014). In cosmological simulations, both major and minor mergers occur according to the hierarchical clustering of galaxy haloes.
Note that, as well as the feedback processes described above, galaxies can lose gas when falling into a cluster. Tidal interactions with cluster members and the dark matter halo itself can have an effect (e.g. Moore et al. 1996), and ram pressure stripping of the ISM by the hot intracluster medium can remove much of the halo gas (e.g. Gunn & Gott 1972; Dressler & Gunn 1983; Gavazzi, Randone & Branchini 1995; Boselli & Gavazzi 2006). These effects are not well reproduced in smoothed particle hydrodynamics (SPH) simulations.
It is clear that there are a number of complex and competing processes that alter the amount of gas in a galaxy that can form stars or fuel BHs. What is less apparent is which of these processes is more important, and in what circumstances, in determining the SFR? In this paper, we aim to quantitatively answer this using cosmological hydrodynamical simulations that self-consistently solve the relevant physics. Section 2 gives a brief overview of the simulations analysed. In Section 3, we describe the analysis methodology including model formulae, and determine the model parameters in Section 4. Section 5 presents a discussion of the results, and in Section 6 we test our model against observational data. Finally, we give our conclusions in Section 7.
2 SIMULATIONS
The simulations used in this paper were introduced in Taylor & Kobayashi (2015a); they are a pair of cosmological, chemodynamical simulations, one of which includes a model for AGN feedback (Taylor & Kobayashi 2014), but otherwise have identical initial conditions and physics. Our simulation code is based on the SPH code gadget-3 (Springel 2005), updated to include: star formation (Kobayashi, Springel & White 2007), energy feedback and chemical enrichment from supernovae (SNe II, Ibc, and Ia; Kobayashi 2004; Kobayashi & Nomoto 2009) and hypernovae (Kobayashi et al. 2006; Kobayashi & Nakasato 2011), and asymptotic giant branch stars (Kobayashi, Karakas & Umeda 2011); heating from a uniform, evolving UV background (Haardt & Madau 1996); metallicity-dependent radiative gas cooling (Sutherland & Dopita 1993); and a model for BH formation, growth, and feedback (Taylor & Kobayashi 2014), described in more detail below. We use the IMF of stars from Kroupa (2008) in the range 0.01-120 M⊙, with an upper mass limit for core-collapse supernovae of 50 M⊙.
The initial conditions for both simulations consist of 2403 particles of each of gas and dark matter in a periodic, cubic box 25 h−1 Mpc on a side, giving spatial and mass resolutions of 1.125 h−1 kpc and MDM = 7.3 × 107 h− 1 M⊙, Mg = 1.4 × 107 h− 1 M⊙, respectively. We employ a WMAP-9 ΛCDM cosmology (Hinshaw et al. 2013) with h = 0.7, Ωm = 0.28, |$\Omega _\Lambda =0.72$|, Ωb = 0.046, and σ8 = 0.82.
3 A NEW SIMPLE MODEL FOR THE SFR
However, as discussed in Section 1, other factors, such as the mass of gas in a galaxy, and the environment the galaxy exists in, may also affect the SFR. Feedback from star formation, supernovae, and AGN activity can remove gas from galaxies (Taylor & Kobayashi 2015b), depending on the galaxy mass and strength of feedback. Although the feedback energy in our AGN model is calculated from the instantaneous accretion rate, the current mass of gas in the galaxy likely depends more on the BH mass, since this reflects the entire accretion history of the BH. Similarly, the total stellar mass better reflects the integrated influence of stellar feedback on the gas than the current SFR.
The function f (MBH) in equations (4) and (5) reflects the fact that BHs must grow sufficiently massive before their feedback energy cannot be efficiently radiated away. This is illustrated in Fig. 2, which shows the residuals of SFR in the simulation with AGN compared with the SFR estimated from equation (3), as a function of BH mass. There are two clear regimes: at MBH ≲ 107 M⊙, the residuals are distributed around 0, while at MBH ≳ 107 M⊙ the residuals decrease with increasing MBH, and the SFR can be as much as two orders of magnitude below the SFMS.
Values of C, α, β, γ, k1, k2, log Mb, and Δlog M are found using the amoeba routine (Press et al. 1992), minimizing the quantity ∑i(log SFRi − log SFRi,fit)2. In order to estimate uncertainties on the fitted parameters, we employ a bootstrap resampling technique whereby the parameters are re-derived for a random selection of the data, with repeats, the same size as the original data set. We do this for 5 × 106 resamplings, in order to fully sample the parameter space of the models with the most free parameters.
4 MODEL FITS
Fitting the SFR of our simulated galaxies using the method described in Section 3 to the models described in equations (5)–(8) results in the parameters given in Table 1; the full parameter distributions for f1 are shown in Fig. 4. The uncertainties are derived from the bootstrap resampling procedure described in Section 3; the first column of Table 1 gives the best-fitting parameters to the simulated data, the second column gives the mean and standard deviation from the bootstrap distributions, and third column shows the modal value with asymmetric errors denoting the 16th and 84th percentiles. For all three models, the values of C, α, β, and γ are consistent with one another, and are well constrained by the data.
x . | Best fit . | |$\bar{x} \pm \sigma _x$| . | Mode |$^{+}_{-}$| distribution width . |
---|---|---|---|
Model 1 (equation 6) | |||
C | −2.50 | −2.59 ± 0.18 | |$-2.58^{+0.18}_{-0.18}$| |
α | 0.21 | 0.21 ± 0.03 | |$0.21^{+0.03}_{-0.03}$| |
β | −0.17 | −0.15 ± 0.04 | |$-0.16^{+0.04}_{-0.03}$| |
γ | 1.03 | 1.04 ± 0.05 | |$1.04^{+0.05}_{-0.05}$| |
k1 | 0.00 | −0.02 ± 0.03 | |$-0.01^{+0.03}_{-0.04}$| |
k2 | −1.11 | −1.99 ± 3.13 | |$-1.12^{+0.52}_{-0.82}$| |
log Mb | 7.31 | 7.79 ± 0.64 | |$7.29^{+0.46}_{-0.15}$| |
Model 2 (equation 7) | |||
C | −2.72 | −2.72 ± 0.15 | |$-2.72^{+0.15}_{-0.14}$| |
α | 0.21 | 0.21 ± 0.03 | |$0.21^{+0.03}_{-0.03}$| |
β | −0.12 | −0.12 ± 0.04 | |$-0.12^{+0.04}_{-0.03}$| |
γ | 1.05 | 1.05 ± 0.05 | |$1.04^{+0.06}_{-0.05}$| |
k1 | −0.05 | −0.05 ± 0.02 | |$-0.04^{+0.02}_{-0.03}$| |
Model 3 (equation 8) | |||
C | −3.12 | −3.08 ± 0.19 | |$-3.08^{+0.19}_{-0.18}$| |
α | 0.21 | 0.22 ± 0.03 | |$0.21^{+0.03}_{-0.03}$| |
β | −0.16 | −0.16 ± 0.03 | |$-0.16^{+0.03}_{-0.03}$| |
γ | 1.05 | 1.05 ± 0.05 | |$1.06^{+0.05}_{-0.05}$| |
k1 | 0.04 | 0.06 ± 0.05 | |$0.08^{+0.03}_{-0.08}$| |
k2 | −1.19 | −1.25 ± 0.85 | |$-0.82^{+0.31}_{-0.53}$| |
log Mb | 7.68 | 7.66 ± 0.15 | |$7.58^{+0.14}_{-0.07}$| |
Δlog M | 1.22 | 1.55 ± 0.76 | 2.06† |
x . | Best fit . | |$\bar{x} \pm \sigma _x$| . | Mode |$^{+}_{-}$| distribution width . |
---|---|---|---|
Model 1 (equation 6) | |||
C | −2.50 | −2.59 ± 0.18 | |$-2.58^{+0.18}_{-0.18}$| |
α | 0.21 | 0.21 ± 0.03 | |$0.21^{+0.03}_{-0.03}$| |
β | −0.17 | −0.15 ± 0.04 | |$-0.16^{+0.04}_{-0.03}$| |
γ | 1.03 | 1.04 ± 0.05 | |$1.04^{+0.05}_{-0.05}$| |
k1 | 0.00 | −0.02 ± 0.03 | |$-0.01^{+0.03}_{-0.04}$| |
k2 | −1.11 | −1.99 ± 3.13 | |$-1.12^{+0.52}_{-0.82}$| |
log Mb | 7.31 | 7.79 ± 0.64 | |$7.29^{+0.46}_{-0.15}$| |
Model 2 (equation 7) | |||
C | −2.72 | −2.72 ± 0.15 | |$-2.72^{+0.15}_{-0.14}$| |
α | 0.21 | 0.21 ± 0.03 | |$0.21^{+0.03}_{-0.03}$| |
β | −0.12 | −0.12 ± 0.04 | |$-0.12^{+0.04}_{-0.03}$| |
γ | 1.05 | 1.05 ± 0.05 | |$1.04^{+0.06}_{-0.05}$| |
k1 | −0.05 | −0.05 ± 0.02 | |$-0.04^{+0.02}_{-0.03}$| |
Model 3 (equation 8) | |||
C | −3.12 | −3.08 ± 0.19 | |$-3.08^{+0.19}_{-0.18}$| |
α | 0.21 | 0.22 ± 0.03 | |$0.21^{+0.03}_{-0.03}$| |
β | −0.16 | −0.16 ± 0.03 | |$-0.16^{+0.03}_{-0.03}$| |
γ | 1.05 | 1.05 ± 0.05 | |$1.06^{+0.05}_{-0.05}$| |
k1 | 0.04 | 0.06 ± 0.05 | |$0.08^{+0.03}_{-0.08}$| |
k2 | −1.19 | −1.25 ± 0.85 | |$-0.82^{+0.31}_{-0.53}$| |
log Mb | 7.68 | 7.66 ± 0.15 | |$7.58^{+0.14}_{-0.07}$| |
Δlog M | 1.22 | 1.55 ± 0.76 | 2.06† |
x . | Best fit . | |$\bar{x} \pm \sigma _x$| . | Mode |$^{+}_{-}$| distribution width . |
---|---|---|---|
Model 1 (equation 6) | |||
C | −2.50 | −2.59 ± 0.18 | |$-2.58^{+0.18}_{-0.18}$| |
α | 0.21 | 0.21 ± 0.03 | |$0.21^{+0.03}_{-0.03}$| |
β | −0.17 | −0.15 ± 0.04 | |$-0.16^{+0.04}_{-0.03}$| |
γ | 1.03 | 1.04 ± 0.05 | |$1.04^{+0.05}_{-0.05}$| |
k1 | 0.00 | −0.02 ± 0.03 | |$-0.01^{+0.03}_{-0.04}$| |
k2 | −1.11 | −1.99 ± 3.13 | |$-1.12^{+0.52}_{-0.82}$| |
log Mb | 7.31 | 7.79 ± 0.64 | |$7.29^{+0.46}_{-0.15}$| |
Model 2 (equation 7) | |||
C | −2.72 | −2.72 ± 0.15 | |$-2.72^{+0.15}_{-0.14}$| |
α | 0.21 | 0.21 ± 0.03 | |$0.21^{+0.03}_{-0.03}$| |
β | −0.12 | −0.12 ± 0.04 | |$-0.12^{+0.04}_{-0.03}$| |
γ | 1.05 | 1.05 ± 0.05 | |$1.04^{+0.06}_{-0.05}$| |
k1 | −0.05 | −0.05 ± 0.02 | |$-0.04^{+0.02}_{-0.03}$| |
Model 3 (equation 8) | |||
C | −3.12 | −3.08 ± 0.19 | |$-3.08^{+0.19}_{-0.18}$| |
α | 0.21 | 0.22 ± 0.03 | |$0.21^{+0.03}_{-0.03}$| |
β | −0.16 | −0.16 ± 0.03 | |$-0.16^{+0.03}_{-0.03}$| |
γ | 1.05 | 1.05 ± 0.05 | |$1.06^{+0.05}_{-0.05}$| |
k1 | 0.04 | 0.06 ± 0.05 | |$0.08^{+0.03}_{-0.08}$| |
k2 | −1.19 | −1.25 ± 0.85 | |$-0.82^{+0.31}_{-0.53}$| |
log Mb | 7.68 | 7.66 ± 0.15 | |$7.58^{+0.14}_{-0.07}$| |
Δlog M | 1.22 | 1.55 ± 0.76 | 2.06† |
x . | Best fit . | |$\bar{x} \pm \sigma _x$| . | Mode |$^{+}_{-}$| distribution width . |
---|---|---|---|
Model 1 (equation 6) | |||
C | −2.50 | −2.59 ± 0.18 | |$-2.58^{+0.18}_{-0.18}$| |
α | 0.21 | 0.21 ± 0.03 | |$0.21^{+0.03}_{-0.03}$| |
β | −0.17 | −0.15 ± 0.04 | |$-0.16^{+0.04}_{-0.03}$| |
γ | 1.03 | 1.04 ± 0.05 | |$1.04^{+0.05}_{-0.05}$| |
k1 | 0.00 | −0.02 ± 0.03 | |$-0.01^{+0.03}_{-0.04}$| |
k2 | −1.11 | −1.99 ± 3.13 | |$-1.12^{+0.52}_{-0.82}$| |
log Mb | 7.31 | 7.79 ± 0.64 | |$7.29^{+0.46}_{-0.15}$| |
Model 2 (equation 7) | |||
C | −2.72 | −2.72 ± 0.15 | |$-2.72^{+0.15}_{-0.14}$| |
α | 0.21 | 0.21 ± 0.03 | |$0.21^{+0.03}_{-0.03}$| |
β | −0.12 | −0.12 ± 0.04 | |$-0.12^{+0.04}_{-0.03}$| |
γ | 1.05 | 1.05 ± 0.05 | |$1.04^{+0.06}_{-0.05}$| |
k1 | −0.05 | −0.05 ± 0.02 | |$-0.04^{+0.02}_{-0.03}$| |
Model 3 (equation 8) | |||
C | −3.12 | −3.08 ± 0.19 | |$-3.08^{+0.19}_{-0.18}$| |
α | 0.21 | 0.22 ± 0.03 | |$0.21^{+0.03}_{-0.03}$| |
β | −0.16 | −0.16 ± 0.03 | |$-0.16^{+0.03}_{-0.03}$| |
γ | 1.05 | 1.05 ± 0.05 | |$1.06^{+0.05}_{-0.05}$| |
k1 | 0.04 | 0.06 ± 0.05 | |$0.08^{+0.03}_{-0.08}$| |
k2 | −1.19 | −1.25 ± 0.85 | |$-0.82^{+0.31}_{-0.53}$| |
log Mb | 7.68 | 7.66 ± 0.15 | |$7.58^{+0.14}_{-0.07}$| |
Δlog M | 1.22 | 1.55 ± 0.76 | 2.06† |
We find that SFR depends most strongly on the amount of gas in a galaxy, scaling linearly with Mgas (γ ≈ 1). Additionally, galaxies with large stellar mass and high environmental density (low s5) show greater SFR, with the dependence on galaxy mass being the more important (i.e. α > |β|). However, with f1 and f3, this is tempered by the strong negative correlation between SFR and MBH in galaxies with BHs more massive than Mb (i.e. k2 is large and negative). Such galaxies tend to be found in the densest environments, at the centre of clusters, implying that satellite galaxies within the cluster have the highest SFRs, while field galaxies tend to have lower SFRs at a given stellar mass. This is consistent with the observational result of Koyama et al. (2013), who found that the SFR of star-forming galaxies increases with environmental density (see also Peng et al. 2010; Wijesinghe et al. 2012, who find weak or no dependence of SFR on environment for star-forming galaxies). In all models, star formation in galaxies with low-mass BHs is not strongly affected, as indicated by the value k1 ∼ 0, that is, SFR follows the SFMS for MBH ≲ Mb (see Fig. 2).
The relatively large uncertainties on the values of both k2 log Mb in f1 are due to their distributions, shown in Fig. 4, being highly extended, and, in the case of log Mb, nearly bimodal. In addition to the main peak around log Mb ≈ 7.3, there is an extended plateau of values near log Mb ∼ 8. log Mb ≳ 8 is found when very few of the 38 galaxies with MBH > 107 M⊙ are included in a bootstrap realization of the data set. In such cases, none of the data constrains Mb other than being larger than the largest MBH, and so Mb can take any arbitrarily large number without affecting how well the model fits the data. This is also responsible for the very extended distribution of k2 seen in Fig. 4.
Function f3 produces fairly consistent values with those of f1, favouring a slightly higher value log Mb ≈ 7.6, as well as predicting a wide range of BH masses over which AGN feedback quenches star formation with Δlog M ∼ 1–2. Similar caveats apply to k2, log Mb, and Δlog M in this model as for k2 and log Mb in f1. Fortunately, however, the conclusions we draw in subsequent sections do not depend on the exact values of these parameters. Fig. 5 shows the residuals for each of the models f1 to f3 as a function of MBH. The mean and standard deviation are shown for bins with width 0.5 dex, and the red band shows the range of Mb from Table 1. At log MBH ≲ 7, the three models have residuals that are distributed fairly uniformly around 0, though f2 may show a very slight increasing trend with MBH. At higher masses, f2, which assumes that there is no break mass Mb, shows a significant systematic decrease in residuals with mass that is not seen in the residuals of f1. A similar trend is seen for the residuals of f3, though the residuals are consistent with 0 over the full range of MBH.
5 DISCUSSION
5.1 BH growth and AGN feedback
We focus now on understanding the implications of the value of Δlog M from f3. It is instructive to estimate a time-scale over which BHs grow by a factor of |$10^{\Delta _{{\rm log\,}_{\rm M}}}\sim 10{\rm -}100$|. In Fig. 6, we show the mass assembly history of all simulated BHs with final masses greater than 106 M⊙, and the horizontal dashed and dot–dashed lines show |$M_{\rm BH}=M_{\rm b}\times 10^{\pm \frac{1}{2}\Delta _{{\rm log\,}_{\rm M}}}$| assuming the best-fitting and modal values, respectively. It is clear that, regardless of the exact values of Mb and Δlog M used, the time-scale for BHs to grow by a factor of |$10^{\Delta _{{\rm log\,}_{\rm M}}}$| is at least several Gyr. This should not be interpreted, however, as the time-scale over which an individual BH quenches star formation in its host, since we showed in Taylor & Kobayashi (2015a) that star formation in individual galaxies can be quenched abruptly. Rather, this shows that once they grow to ∼Mb, BHs have the potential to exert influence over the evolution of their galaxy.
The fact that an individual BH can quench star formation in its host much more quickly than it takes to grow by a factor of |$10^{\Delta _{{\rm log\,}_{\rm M}}}$| once it reaches MBH ≈ Mb implies that attaining a mass Mb is a necessary but not sufficient requirement for AGN feedback to become effective. This may suggest that local properties within a galaxy can affect exactly when AGN feedback shuts off star formation, or that a ‘trigger’ for strong AGN feedback may be required once the BH is sufficiently massive, with gas-rich galaxy mergers being a likely candidate (e.g. Comerford et al. 2015; Gatti et al. 2016). The cause of strong AGN activity, such as the AGN-driven outflows described in Taylor & Kobayashi (2015b), will be investigated in detail in a future work.
5.2 Galaxy transition at 3 × 1010 M⊙
Fig. 5 shows that the model that does not include a BH break mass, f2, does not describe our simulated data well for MBH ≳ Mb. Therefore, we can conclude that the influence of BHs on star formation changes once the BH grows above Mb. There is an empirical relationship between MBH and the stellar bulge mass of its host galaxy, Mbulge. Converting the range of fitted values of Mb from both f1 and f3 (Mb ≈ 2-5 × 107 M⊙) into a stellar bulge mass using the relation given in Graham & Scott (2015) gives Mbulge ≈ 1-2 × 1010 M⊙. At these stellar masses, galaxies typically have bulge-to-disc (B/D) ratios of 0.3–0.4 (Shen et al. 2003), corresponding to bulge-to-total (B/T) ratios of 0.23–0.29 (note that ETGs can have much larger B/T∼0.7; see e.g. Morselli et al. 2017). For such values of B/T, Mb corresponds to a galaxy mass that is consistent with the galaxy mass of M* = 3 × 1010 M⊙ found by Kauffmann et al. (2003) to separate ETGs and LTGs.
This consistency between our simulations and observational data implies that feedback from supermassive BHs may be responsible for the transition from LTGs to ETGs at M* = 3 × 1010 M⊙. Such a result is in broad agreement with other studies; Keller, Wadsley & Couchman (2016) find that even very strong supernova feedback in the form of superbubbles is inefficient in galaxies M* ≳ 4 × 1010 M⊙, arguing that AGN feedback must dominate in more massive galaxies. Similarly, Dubois et al. (2015); Bower et al. (2017) find that BH accretion is suppressed by supernova feedback in galaxies less massive than 3 × 1010 M⊙, with AGN feedback becoming important once supernova feedback can no longer remove gas from the galaxy potential. In this paper, we have not investigated why strong AGN feedback is triggered in massive galaxies, finding only that it is triggered once a BH grows to Mb ≈ 2-5 × 107 M⊙. The reason for the triggering will be investigated in detail in a future work.
5.3 Connection to the Kennicutt–Schmidt law
The KS law is an empirical relation between SFR surface density ΣSFR, and gas surface density, Σgas (Schmidt 1959; Kennicutt 1989; Kennicutt & Evans 2012). With γ ≈ 1 (the exponent of Mgas in our model), dividing equation (4) by galaxy area gives ΣSFR ∼ Σgas. Observations typically find a superlinear relation when all gas is considered (e.g. Kennicutt 1989, 1998), as we have done, while a linear or sub-linear relation tends to be found when only dense gas is considered (e.g. Bigiel et al. 2008; Shetty, Kelly & Bigiel 2013; Shetty et al. 2014, but see also Liu et al. 2011 and Momose et al. 2013). Shetty et al. (2014) analysed disc galaxies from the Survey Toward Infrared-Bright Nearby Galaxies (STING) and concluded that galactic properties other than Σgas affect ΣSFR; such properties were included in our model, which may explain the difference between our linear relationship and observations (though there is significant scatter in the KS relation; see e.g. Krumholz et al. 2012; Federrath 2013; Salim et al. 2015). We note for completeness that if we only consider SFR and Mgas, we obtain a slope 1.54 ± 0.03, in much better agreement with the observed superlinear relation of Kennicutt (1989, 1998).
5.4 The star formation main sequence
Regardless of the form of f (MBH) used, Table 1 shows that α = 0.21, that is, log SFR ∼ 0.21log M*. This is in contrast to observational estimates, which find a present-day slope around 0.75–1 (e.g. Elbaz et al. 2007, 2011; Zahid et al. 2012; Renzini & Peng 2015). This is due to the correlations between stellar mass and other physical properties that are not accounted for in the standard SFMS, but are in this model, as was the case for the KS relation above. Our results suggest that stellar mass is not as important in determining galactic SFR as implied by the standard SFMS, and that there is an environmental dependence as well as strong gas mass dependence on the SFR of star-forming galaxies.
6 COMPARISON OF OUR NEW SFR MODEL TO OBSERVATIONAL DATA
We test our star formation models by comparing to observations. Ideally, we would wish to repeat the analysis of Section 4 on observational data to critically appraise our star formation model and our simulations. However, there are no overlapping surveys that can provide all of SFR, M*, s5, Mgas, and MBH for a large number of galaxies. By searching the literature and various current observational data bases, we were able to find a sample of galaxies for which all the required measurements of SFR, M*, s5, and Mgas were available (except MBH). We use this set of galaxies to compare their measured SFR with that estimated from our model.
The Galaxy And Mass Assembly (GAMA) survey (Driver et al. 2011) is an optical survey using the Anglo-Australian Telescope and AAOmega spectrograph (Sharp et al. 2006). It is >98 per cent complete to a depth of r < 19.4 or 19.8. The survey is spread over three 12° by 4° equatorial regions. Environmental density data are only available for 16 062 galaxies in the region centred at 15h right ascension. In this paper, we use aperture-corrected M* (Taylor et al. 2011), SFR (Gunawardhana et al. 2011), and s5 (Brough et al. 2013) data for these galaxies.
For the gas masses, we use data from the Arecibo Legacy Fast ALFA (ALFALFA) survey (Giovanelli et al. 2005). ALFALFA is a wide-field survey using the Arecibo radio telescope to map the 21-cm line of atomic hydrogen. Corresponding H i masses, |$M_{{\rm H}\,\small {i}}$| are derived in Haynes et al. (2011); to convert this to total gas mass, we make the simplistic assumptions that (1) |$M_{{\rm H}\,\small {i}}$| is equal to the total mass of hydrogen in the ISM, and (2) the ISM has primordial composition, so that |$M_{\rm gas}=M_{{\rm H}\,\small {i}}/0.75$|. We associate H i detections from ALFALFA with an optical counterpart in GAMA if they are separated by less than 0|$_{.}^{\circ}$|01 in RA and Dec. Matching the catalogues in this way gives 14 galaxies with measured M*, SFR, s5, and Mgas.
In Fig. 7, we show the measured SFR (SFRobs) and the SFR predicted by our model (SFRmodel) for these galaxies (filled circles with error bars). The agreement between the observed SFR and those predicted by our model is extremely encouraging, considering the assumptions used to calculate Mgas and the fact that s5 from the GAMA catalogue is projected, whereas we used the 3D s5 in deriving our model parameters. Also shown in Fig. 7 are the predicted SFRs if Mgas or Mgas and s5 are excluded (diamond and star symbols, respectively). In both cases, the model SFR is significantly less than is observed, and shows little evidence of correlation with the observed values. This reinforces the fact that Mgas is the most important quantity in determining SFR. The inclusion of s5 alters the predicted SFR very little, but may be more important in more extreme environments such as galaxy groups and clusters (e.g. Schaefer et al. 2017).
In the coming decade, vast radio surveys such as the Square Kilometre Array (SKA) and its precursors will provide H i masses for tens of thousands of galaxies, allowing for a more rigorous test of our model. However, BH masses are more difficult to measure, and are available for only a relatively small number of galaxies in the local Universe. Directly validating our full model including the effects of BHs will be a challenging, but important task for the near future.
7 CONCLUSIONS
We have analysed data from a cosmological hydrodynamical simulation that includes a detailed prescription for star formation and stellar feedback, as well as BH formation and AGN feedback. Our aim was to understand better the relative importance of factors that affect SFR of galaxies. We suggested that SFR could be influenced by the mass of the host galaxy, its environmental density, its gas content, and the mass of its BH. We proposed a relatively simple form for the relation between SFR and host galaxy properties (equation 5), with which we were able to make quantitative comparisons.
We find that once a BH has grown sufficiently massive ( ∼ 2-5 × 107 M⊙), it can be triggered to shut off star formation through AGN feedback; the details of such triggering will be investigated in future work. Such a BH mass corresponds, via the Maggorian relation (Graham & Scott 2015), to a galaxy bulge mass of ∼ 1-2 × 1010 M⊙, which is consistent with the observed galaxy mass above which ETGs make up the dominant fraction of the galaxy population ( ∼ 3 × 1010 M⊙; Kauffmann et al. 2003). It is important to note that these masses are emergent in our simulations; none of the input parameters of the baryon physics is chosen to yield such a transition mass, but are constrained by other observations.
The SFR depends strongly on the amount of gas in a galaxy, scaling approximately linearly with gas mass, while the dependence on stellar mass is weaker than predicted from the standard SFMS since we take into account the correlations between SFR and the additional physical quantities Mgas, s5, and MBH by fitting all simultaneously. Our simulation cannot resolve the cold molecular gas that would form stars, and in this analysis we have treated all gas equally, regardless of temperature or density. In light of this, it will be useful in future works to analyse galaxies simulated with sufficient resolution to distinguish between gas phases.
We find that star-forming galaxies in high-density regions have larger SFR than star-forming galaxies in the field at given mass, in agreement with observations (e.g. Koyama et al. 2013). This would most likely evolve with redshift, with massive galaxies showing the highest SFR in the past. The existence and strength of any such evolution will be investigated in a future work. Furthermore, the size of our simulation box limits the size of our most massive simulated cluster, and it would be informative to investigate if the trends seen here hold for more extreme environments.
We use observational data from the GAMA and ALFALFA surveys to compare the SFR observed with that predicted by our new model (see Fig. 7). There is excellent agreement between these two quantities, lending credence to the model we have presented. The sample is limited by the available observations, but will increase to tens of thousands in the era of the SKA. Further validation of our model will only be possible with a catalogue of BH masses for large numbers of galaxies.
Acknowledgments
We thank the referee for his/her useful comments, which improved the quality of this paper. CF gratefully acknowledges funding provided by the Australian Research Council’s Discovery Projects (grants DP150104329 and DP170100603). The simulations presented in this work used high-performance computing resources provided by the Leibniz Rechenzentrum and the Gauss Centre for Supercomputing (grants pr32lo, pr48pi and GCS Large-scale project 10391), the Partnership for Advanced Computing in Europe (PRACE grant pr89mu), the Australian National Computational Infrastructure (grant ek9), and the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia, in the framework of the National Computational Merit Allocation Scheme and the ANU Allocation Scheme. This work has made use of the University of Hertfordshire Science and Technology Research Institute high-performance computing facility. PT thanks S. Lindsay for helpful discussions. Finally, we thank V. Springel for providing gadget-3.
Throughout this paper, we adopt the standard notation log x ≡ log10x and ln x ≡ logex.
Given the normalization of MBH to 108 M⊙ in equation (5), Mb is also measured relative to 108 M⊙. In subsequent sections, the absolute value of Mb will be given, with the factor 108 taken into account.
f3 reproduces f1 to within a constant. Matching at log MBH → ±∞ gives this constant as |$\frac{1}{2}(k_2-k_1)(\Delta _{{\rm log\,}_{\rm M}}\ln 2 - \log M_{\rm b})$|. In practice, we absorb this into C to avoid unnecessary correlation between C and the other parameters.
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