Star formation and stellar mass assembly in dark matter haloes: from giants to dwarfs

Abstract

1 INTRODUCTION

Given that the hierarchical formation of dark matter haloes is the backbone of galaxy formation, and that dark matter is the dominant mass component in the Universe, understanding the origin and diversity of the galaxy population requires a solid understanding of the connection between galaxies and dark matter haloes. The last decade has seen tremendous progress in this area. By combining the galaxy–dark matter connection as a function of redshift with halo merger trees, which describe the hierarchical assembly of dark matter haloes, one obtains a statistical description of how galaxies assemble their stellar mass over time. Such an ‘empirical’ approach is intuitive and transparent in that it describes galaxy evolution within the natural framework of hierarchical structure formation. This method was developed by Conroy, Wechsler & Kravtsov (2007), Conroy & Wechsler (2009) and Yang, Mo & van den Bosch (2009), and has since been applied in numerous studies (e.g. Moster et al. 2010; Behroozi, Conroy & Wechsler 2010; Avila-Reese & Firmani 2011; Yang et al. 2012, 2013; Behroozi, Wechsler & Conroy 2013a,b; Béthermin et al. 2013; Moster, Naab & White 2013; Wang et al. 2013). An important new insight that has resulted from this ‘empirical modelling’ approach is that there is an amazing amount of ‘regularity’ and ‘simplicity’ in galaxy demographics. In particular, the relatively mild evolution in the  M_⋆– M_h relation with redshift is found to translate into an instantaneous star formation efficiency (i.e. the star formation rate divided by the baryon accretion rate) that peaks at a characteristic halo mass of ∼10¹² h⁻¹ M_⊙, over the entire epoch from z = 8 to the present day, and almost all star formation has occurred inside haloes that lie within the narrow range 10¹¹ h^{− 1} M_⊙ ≲ M_h ≲ 10¹² h^{− 1} M_⊙ (Leauthaud et al. 2012a,b; Yang et al. 2012, 2013; Behroozi et al. 2013a,b; Moster et al. 2013). In addition, these studies have shown that virtually all stellar mass is assembled in situ; merging is only a significant channel of mass assembly for the most massive galaxies. In what follows, we will refer to this picture of galaxy formation as the ‘Slow-Evolution’ model, to emphasize that it suggests little evolution in the halo mass dependence of the star formation efficiency.

A recent study by the authors (Lu et al. 2014a) has argued for an important revision to this ‘Slow-Evolution’ picture. Using data on the faint end of the luminosity function of cluster galaxies, they show that low-mass haloes ( M_h ≲ 10¹¹ h^{− 1} M_⊙) have to form stars efficiently, but only at high redshift (z ≳ 2). In particular, they identified z ≃ 2 as a new characteristic epoch in galaxy formation, where there is a fairly sudden transition in the star formation efficiency of low-mass haloes. This transition leads to some interesting predictions, e.g. a significant old stellar population in present-day dwarf galaxies with M_⋆ ≤ 10⁸ h⁻² M_⊙ and steep low end slopes at high redshift of the galaxy stellar mass and star formation rate (SFR) functions. Interestingly, recent work based on deeper high-redshift surveys provide some evidence for such a steepening.

The goal of the paper is to update the model of Lu et al. (2014a) with more recent observational data and to characterize in detail the star formation and merger histories of galaxies across cosmic time with the updated model. We address questions related to in situ star formation versus accretion, downsizing both in star formation and assembly and mergers of galaxies and their roles in shaping galaxy morphology, and the prevalence of halo stars. Wherever possible we will contrast the differences between the ‘Slow-Evolution’ model and the modified picture advocated by Lu et al. (2014a).

This paper is organized as follows. Our method to model the star formation–halo mass connection is described in Section 2. The overall trends in the star formation and assembly histories are described in Section 3. In Section 4, we describe the total star formation and assembly histories as a function of halo (stellar) mass, paying particular attention to in situ star formation versus accretion, the breakdown of downsizing, and the global SFR density in connection to the ionization state of the Universe. In Section 5, we focus on galaxy merger rates and their implications for the morphological transformation of galaxies. In Section 6, we address the stellar populations of galaxies as a function of their stellar mass, paying particular attention to the properties of halo stars. Finally, we summarize our results in Section 7.

Throughout the paper, we use a Λ cold dark matter cosmology with Ω_{m, 0} = 0.273, Ω_{Λ, 0} = 0.727, Ω_{b, 0} = 0.0455, h = 0.704, n = 0.967 and σ₈ = 0.811. This set of parameters is from the seven year Wilkinson Microwave Anisotropy Probe observations (Komatsu et al. 2011). In addition, unless stated otherwise, we adopt the stellar population synthesis model of Bruzual & Charlot (2003) and a Chabrier (2003) initial mass function (IMF).

2 AN EMPIRICAL MODEL OF STAR FORMATION IN DARK MATTER HALOES

2.1 The model

In this section, we provide a brief description of the model of Lu et al. (2014a), which we adopt here to make model predictions, referring the reader to the original paper for details.

The hierarchical assembly of individual dark matter haloes is modelled using halo merger trees generated with the Monte Carlo model of Parkinson, Cole & Helly (2008), which is based on a modified treatment of the extended Press–Schechter formalism calibrated using N-body simulations (see Cole et al. 2008). As shown in Jiang & van den Bosch (2014), the merger trees obtained with this method are as accurate as those obtained from high-resolution N-body simulations.

Each satellite galaxy is assumed to merge with the central galaxy of its host halo in a time given by the dynamical friction time-scale. During the merger, only a fraction f_TS of the stellar mass of the satellite is added to the central galaxy; the remainder is considered ‘stripped’ and becomes part of the ‘halo stars’ (Monaco et al. 2006; Conroy et al. 2007). The fraction f_TS is treated as a free parameter.

The luminosity can be calculated in the same way as the stellar mass by replacing R(t_i − t) with the light-to-mass ratio of a simple stellar population. We use the metallicity–stellar mass relation from Gallazzi et al. (2005). Note that the z- and r-band luminosities are quite insensitive to the assumed metallicity.

2.2 Updating model parameters with recent observational data

In Lu et al. (2014a), two sets of observational data were used to constrain the models described above. One is the galaxy SMFs at different redshifts and the other is the z-band cluster galaxy luminosity function from Popesso et al. (2006). The SMFs were those from Baldry et al. (2012) for z ∼ 0, from Perez-Gonzalez et al. (2008) for 1.0 < z < 1.3, from Marchesini et al. (2009) for 2.0 < z < 3.0, and from Stark et al. (2009) for 3.19 < z < 4.73. The results presented in Lu et al. (2014a) are based on Model II constrained by the SMFs alone and Model III constrained by both the SMFs and the cluster galaxy luminosity function. As mentioned above, Model II is unable to match the steep, faint-end upturn observed in the cluster galaxy luminosity function.

As shown in Lu et al. (2014a), one way to distinguish Model II and Model III is the difference in their predicted faint-end slope of the SMF at high redshift. The SMF predicted by Model II is quite flat at the low-mass end at all redshifts, while Model III predicts a steepening of the SMF at the low-mass end at high z. More recently, analyses based on deeper surveys provide some evidence for such a steepening. It is, therefore, interesting to use the new data to update our model parameters. In this paper, we use two sets of SMFs recently published. The first is that of Santini et al. (2012) based on the Early Release Science data of the Wide Field Camera 3 in The Great Observatories Origins Deep Survey (GOODS)-S Field, and the SMFs for galaxies in the redshift range between 0.6 and 4.5 are estimated to a stellar mass limit of a few times 10⁹ M_⊙. The second set is that of Tomczak et al. (2014), who estimated the galaxy SMFs in the redshift range 0.2–3.0 using data from the FourStar Galaxy Evolution Survey (ZFOURGE) and the Cosmic Assembly Near-IR Deep Extragalactic Legacy Survey (CANDELS). Unfortunately, the two sets are not completely consistent with each other. While at z < 2.0, the results obtained by both are similar, at z > 2 the SMFs at the low-mass ends obtained by Tomczak et al. (2014) are lower than those obtained by Santini et al. (2012) by a factor of 2. In this paper, we use the two sets of data separately to update our model parameters, and to check the reliability of our results against uncertainties in the observational data.

In practice, we replace the SMFs at z > 1 in Lu et al. (2014a) with the new SMFs to constrain the models. As in Lu et al. (2014a), we use the multinest method developed by Feroz, Hobson & Bridges (2009), which implements the nested sampling algorithm of Skilling (2006), to explore the model parameter space and to make posterior predictions. The new inferences of the model parameters are shown in Table 1 for Model II and in Table 2 for Model III. The comparison between the posterior predictions and the constraining data is shown in Figs 1 and 2. We find that without the cluster data the new data sets alone cannot distinguish decisively between Model II and Model III, because at z > 3, the new SMFs are only complete for M_⋆ > 3 × 10⁹ M_⊙ where the two models are quite similar. The model inferences based on the Santini et al. (2012) data are similar to those obtained in Lu et al. (2014a). The inferences from Tomczak et al. (2014) data are qualitatively the same, except that the inferred SFRs at z ≥ 3 in low-mass haloes are lower by a factor of ∼2. Note that both models overpredict slightly the SMF in the intermediate redshift range, and even Model III underpredicts the cluster galaxy luminosity function at the faint end somewhat, indicating that either the faint-end slope in the cluster galaxy luminosity function is overestimated, or the SMF is underestimated in the intermediate redshift range, or our Model III is still not flexible enough to match the details of the data.

Figure 1.

Comparison between the posterior predictions and the constraining data (points with error bars). The high-z SMFs are from Santini et al. (2012). The red curves (best fit) and the bands (95 per cent credible intervals) are predictions of Model III, while the green curves are the predicted means of Model II.

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Figure 2.

The same as Fig. 1 but the high-z constraining data are from Tomczak et al. (2014).

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Fig. 3 demonstrates the major differences between Model II and Model III. The upper panels show the SMFs and  M_⋆/M_h at different redshifts predicted by Model II constrained by the z ≈ 0 SMF of Baldry et al. (2012) together with the SMFs of Santini et al. (2012) at z > 1. The thick lines in the lower panels show the predictions of Model III constrained by the z ≈ 0 SMF of Baldry et al. (2012), the SMFs of Santini et al. (2012) at z > 1, and the cluster galaxy luminosity function of Popesso et al. (2006). As found in Lu et al. (2014a), the main difference between Model II and Model III lies in the predicted faint end, especially at high redshift. For comparison, the thin lines in the lower panels show the predictions of Model III but constrained with the SMFs of Tomczak et al. (2014). The model constrained by the Santini et al. (2012) data predicts systematically higher and steeper SMFs at z ≥ 3 than that constrained by Tomczak et al. (2014). Clearly, accurate observations of the SMFs at z > 3 down to a stellar mass limit M_⋆ < 10⁹ M_⊙ are crucial to discriminate the ‘Slow Evolution’ Model II and Model III advocated by Lu et al. (2014a). In what follows, we will use the constrained Model II and Model III to make model predictions for a number of statistical properties of the galaxy population at different redshifts. For clarity, our presentation is based on model parameters constrained by the Santini et al. (2012) SMFs at z > 1, i.e. the values listed in the second columns of Tables 1 and 2. We emphasize, however, that none of our results will change qualitatively if the Tomczak et al. (2014) data are used instead to constrain the models.

Figure 3.

Upper panels: the evolution of the SMF and the  M_⋆/M_h–M_h relation predicted by Model II. Lower panels: predictions of Model III constrained using the SMFs from Santini et al. (2012, thick lines) and Tomczak et al. (2014, thin lines).

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3 GENERAL TRENDS AND CHARACTERISTIC SCALES

Galaxies can grow their stellar mass either via in situ star formation or mergers. In this section, we identify critical halo masses and redshifts to characterize the different stages of stellar mass acquisition as their host haloes grow.

The plots in Fig. 4 show ϵ_SFR as a function of M_h(z) and z. As one can see, Model II predicts that the in situ star formation is most efficient (with ϵ_SFR > 0.3) in a narrow band between 10¹¹ and 10¹² h⁻¹ M_⊙, with a tilt towards lower mass haloes at lower redshifts. Outside the band, the star formation efficiency drops rapidly towards both higher and lower masses without depending strongly on redshift. This trend of star formation efficiency with halo mass and redshift agrees with the results obtained earlier by Bouche et al. (2010), Behroozi et al. (2013b), Béthermin et al. (2013), Tacchella, Trenti & Carollo (2013) and Yang et al. (2013), using different constraints and methods. Model III predicts a similar trend except for haloes with masses <10¹¹ h⁻¹ M_⊙. Instead of a strong suppression of star formation with decreasing halo mass, the star formation efficiency remains ∼1/30 at z > 3 for all M_h < 10¹¹ h⁻¹ M_⊙ haloes.

Figure 4.

Star formation efficiency and merger rate in the halo mass–redshift plane. The blue shading is the star formation efficiency as indicated in the colour bar on the right. The blue solid lines correspond to star formation efficiency 0.1 (inner two lines) and 0.03 (outer lines). The solid red lines are the loci of one merger event per Hubble time, with the hedged sides corresponding to a higher merger frequency. The left-hand panels are for major mergers (with mass ratios larger than 1/3) and the right-hand panels are for minor mergers with mass ratios between 1/10 and 1/3. The solid grey lines are the mean mass assembly histories of dark matter haloes with present-day masses equal to 10¹¹, 10¹², …, 10¹⁵ h⁻¹ M_⊙, which are obtained by averaging over the main branches of the merger trees with the same present-day halo mass. The upper panels are predictions of Model II and the lower panels are the predictions of Model III.

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For central galaxies, another potentially important process that can affect their stellar mass and perhaps the size and morphology is the accretion of satellites. To characterize this process, we calculate the mean galaxy merger rate for haloes of a given mass at a given redshift. We distinguish two different types of mergers: (i) major mergers for which the stellar mass ratio between the merging satellite and the central galaxy is ≥1/3 and (ii) minor mergers for which the ratio is between 1/10 and 1/3. When calculating the mass ratios, we use the original stellar mass of the satellites before they deposit a fraction 1 − f_TS into the stellar halo component. We discuss this choice in the last paragraph of Section 5. We define the merger rate as the number of mergers per unit time multiplied by the Hubble time t_H(z) ≡ H(z)⁻¹. In Fig. 4, the red lines are the loci of one merger event per t_H(z) in the halo mass–redshift plane; haloes on the hedged sides of the loci on average have more than one merger per t_H(z). We show results for both major (left-hand panels) and minor (right-hand panels) mergers, and separately for Model II (upper panels) and Model III (lower panels).

For Model II, central galaxies at z = 0 have experienced at least one major merger per Hubble time if they are hosted by haloes with masses larger than ≈10¹³ h⁻¹ M_⊙. At z > 1, frequent major mergers only occur for centrals hosted by haloes with masses higher than ∼3 × 10¹² h⁻¹ M_⊙. The predictions of Model III are quite similar for massive haloes, but an additional branch of high major merger rate is also predicted for low-mass haloes (10¹⁰ h⁻¹ M_⊙ < M_h < 10¹¹ h⁻¹ M_⊙) at z > 3. Comparing the red hedged lines indicating mergers with the star formation efficiency, one can see that most star-forming galaxies are not associated with major mergers, except for central galaxies in low-mass haloes with M_h = 10¹⁰–10¹¹ h⁻¹ M_⊙ at z > 3, where galaxies can experience major mergers while actively forming stars. Minor mergers are more common. In particular, Model III predicts that active star-forming central galaxies in all haloes with M_h > 10¹⁰ h⁻¹ M_⊙ may have experienced at least on minor merger at z > 2.

The following set of functions and scales characterize the star formation efficiency and merger frequency.

• The ridge of the highest star formation efficiency is well described by

  \begin{equation} M_{\rm h} (z) \approx 3\times 10^{11}\, h^{-1}\,{\rm M_{\odot }}(1+z)^{0.3}\,, \end{equation}

  (11)

  with a height ϵ_mx ∼ 0.5 and a full width at half-maximum Δlog₁₀(M_h) ≈ 1.0.
• The line separating frequent from infrequent major mergers for massive haloes can be approximated by

  \begin{equation} M_{\rm h} (z) \approx 10^{13}\, h^{-1}\,{\rm M_{\odot }}\left[0.7\exp (-z/0.6)+0.3\right]\,. \end{equation}

  (12)
• The line separating frequent from infrequent minor mergers in the entire redshift range for Model II, and at z < 2 for Model III, can be approximated by

  \begin{equation} M_{\rm h} (z) \approx 10^{13}\, h^{-1}\,{\rm M_{\odot }}\left[ \exp (-z/0.8)+0.06\right]\,. \end{equation}

  (13)

• For Model-III, there is a characteristic redshift, z_c ∼ 2, above which the star formation efficiency and the major merger frequency are boosted in low-mass haloes with M_h(z) ≲ 10¹¹ h^{− 1} M_⊙, and the minor merger frequency is boosted in all haloes with M_h(z) > 10¹⁰ h⁻¹ M_⊙.

4 Star FORMATION AND STELLAR MASS ASSEMBLY HISTORIES

4.1 Star formation histories

This total star formation history (SFH, equation 8), which takes into account the history of the accreted stars and the stars formed in situ, is needed when modelling the stellar population of the galaxy. In Fig. 5, the black solid lines show the total SFHs of central galaxies in different dark halo mass ranges by averaging over a large number of halo merger histories using the best-fitting parameters. For comparison, the shaded band in each panel represents the variance among different merger trees of the final halo mass in question. We show results for both Model II (upper panels) and Model-III (lower panels) for five final halo masses, as indicated in each panel.

Figure 5.

The SFHs in haloes with different present-day masses (as indicated in the upper panels) predicted by the best-fitting parameters of Model II (upper panels) and Model III (lower panels). The thick solid lines are the averages; the shaded areas are the 95 per cent ranges of variance owing to different halo merger histories. The dashed black lines are the average differential assembly histories.

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It is clear that haloes of different present-day masses have different star formation histories (SFHs). For centrals in massive clusters with M_h(z = 0) ∼ 5 × 10¹⁴ h⁻¹ M_⊙, the SFR peaks at z ≈ 3 and the majority of the stars form in a narrow time range, which is about 10 Gyr ago (z > 2). In contrast, for Milky Way mass haloes, the SFR reaches a maximum between z = 2 and 1 and decreases only mildly to the present day. For dwarf galaxies in haloes with M_h(z = 0) < 10¹¹ h⁻¹ M_⊙, the predictions of Model II and Model III are significantly different. For model II, most stars in these dwarf galaxies formed quite late (at z ≤ 1) and the SFR is roughly a constant over this time interval. In contrast, Model III predicts SFHs that are bimodal, with an initial starburst at z > 2, followed by a constant SFR. As discussed in Lu et al. (2014a), although the boost in the SFR at high z in low-mass haloes is inferred from the observed upturn in the cluster galaxy luminosity function at the faint end, it also has support from the observed SFR function at z ≳ 4 (Smit et al. 2012), and from the existence of a significant old stellar population in present-day dwarf galaxies (Weisz et al. 2011).

4.2 Stellar mass assembly histories

The average differential assembly histories are shown in Fig. 5 as the dashed lines. The assembly history, defined in equation (7), takes into account in situ star formation, accretion of stars already formed, and mass-loss due to stellar evolution. The average differential evolution is obtained by first averaging over the assembly histories of galaxies with the same halo mass, and then taking the time derivative of the mean assembly histories. For both dwarf- and Milky Way-sized galaxies, the stellar mass assembly histories are almost parallel to the SFHs except at the beginning of star formation. The difference in amplitude, which is about a factor of 2, owes to the mass-loss of evolved stars. This suggests that the assembly of such galaxies is dominated by in situ star formation, rather than by the accretion of stars formed in progenitors. We will have a more detailed discussion about this in the following subsection. For massive cluster galaxies, the assembly histories start with a strong episode of in situ star formation at z > 2, which is followed by a long period of mass accretion at roughly a constant rate.

4.3 In situ star formation versus accretion

Fig. 6 shows the fraction of stars formed in situ in present-day central galaxies as a function of their host halo mass. The predictions of Model II are plotted as the green line (average) and the green shaded area (with the variance arising from different halo merger trees), while the predictions of Model III are plotted in red. The predictions of the two models are quite similar. For central galaxies in haloes with masses below 10¹² h⁻¹ M_⊙, almost all the stars are formed in situ. This fraction decreases rapidly with increasing halo mass at M_h > 10¹² h⁻¹ M_⊙. About 70 per cent of all stars in the central galaxy of a halo with M_h ∼ 10¹³ h⁻¹ M_⊙ are formed in situ; for cluster haloes with M_h ∼ 10¹⁵ h⁻¹ M_⊙ this fraction is about 15 per cent, so about 85 per cent of the stellar mass is acquired through accretion.

Figure 6.

The fraction of stars formed in situ in present-day central galaxies of haloes with different masses. Shown are the predictions of Model II (green) and Model III (red). The curves are the averages obtained from an ensemble of halo merger trees, while the shaded areas represent the variances among the different halo merger trees.

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4.4 Downsizing versus upsizing

Galaxies of different masses have different star formation (assembly) histories. To characterize these histories in a more quantitative way, we examine the characteristic redshift, z_f, by which a fraction f of the final stellar mass in a galaxy has formed (or assembled). Fig. 7 shows the distribution of galaxies in the  z_0.5– z_0.1 plane for star formation (green contours) and stellar mass assembly (red contours), with the contours delineating the isodensity lines that contain 90 per cent of all the galaxies. The results are again shown for haloes with five different present-day masses, as indicated in the panels, predicted by Model II (the upper five panels) and Model III (the lower five panels), respectively. For the most massive haloes [M_h(z = 0) > 10¹⁴ h⁻¹ M_⊙], the star formation time and the assembly time differ considerably, especially in  z_0.5. On average about 50 per cent of the stellar mass in the central galaxies of such massive haloes form before z = 4, but assemble much later at z ≈ 1. For haloes with masses lower than 10¹³ h⁻¹ M_⊙, the star formation time and assembly time are almost identical, indicating that central galaxies in such haloes acquire their stars mostly through in situ star formation, as we have already seen in the last subsection. A Milky Way mass galaxy [M_h(z = 0) ∼ 10¹² h⁻¹ M_⊙] on average formed about 10 per cent of its stars by z ≈ 2 and about 50 per cent after z = 1. For dwarf galaxies residing in haloes with M_h(z = 0) < 10¹¹ h⁻¹ M_⊙, Model III predicts diverse formation redshifts. For example, the majority of galaxies residing in 10¹¹ h⁻¹ M_⊙ haloes are predicted to form 10 per cent of their stars by about z = 4, while a fraction is predicted to form their first 10 per cent at much later times: z ≈ 1. The diversity becomes larger for smaller galaxies; for galaxies in 3 × 10¹⁰ h⁻¹ M_⊙ haloes, some show star formation as early as that in the centrals of galaxy clusters, while others formed most of their stars after z��≈ 1. This diversity owes to the transition in star formation efficiency at z ≈ 2 and the variance in the halo accretion histories. Haloes that formed early generally have experienced an early burst phase of star formation, while younger haloes that assembled most of their mass later than the transition redshift did not have such an early starburst.

Figure 7.

The distribution of the redshift at which 50 per cent ( z_0.5) and 10 per cent ( z_0.1) of the stars in a central galaxy have formed (green solid contours) and have assembled (red dashed contours). The contours are the isodensity lines that enclose 90 per cent of the halo merger trees. We show results for haloes with five masses at z = 0, as indicated. The upper panels are the predictions of Model II, while the lower panels are for Model III.

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In Fig. 8, we show how the averages of  z_0.1 and  z_0.5 change with halo mass for both star formation (thin lines) and stellar mass assembly (thick lines). For haloes with M_h(z = 0) > 10¹¹ h⁻¹ M_⊙, both Model II and Model III predict that the centrals of more massive haloes on average form a fixed fraction of their stars earlier, a trend usually referred to as ‘downsizing’ (Fontanot et al. 2009; Weinmann et al. 2012). A similar downsizing trend is also seen in the stellar mass assembly for haloes with 10¹¹ h⁻¹ M_⊙ < M_h(z = 0) < 10¹³ h⁻¹ M_⊙. However, this trend breaks down in two ways. First, for the most massive haloes [M_h(z = 0) > 10¹³ h⁻¹ M_⊙], more massive centrals actually assemble their stars later. At late times, these galaxies tend to build up their mass hierarchically by accreting satellite galaxies, resembling the mass assembly history of the dark matter haloes themselves. Secondly, although the downsizing trend holds for dwarf galaxies in Model II, the trend predicted by Model III is completely the opposite for these galaxies; on average smaller galaxies tend to be older. This again owes to the boost of star formation in low-mass haloes at high z in Model III.

Figure 8.

The average characteristic redshifts, z_0.5 and  z_0.1, as a function of halo mass for star formation (thin lines) and for stellar mass assembly (thick lines).

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4.5 The global SFH

The solid green and red lines in Fig. 9 show the star formation rate density (SFRD) predicted by Model II and Model III, respectively. These results take into account star formation in haloes down to a mass of 2 × 10⁹ h⁻¹ M_⊙. As shown in Lu et al. (2014a), the boost in the SFRD at z > 3 of Model III relative to Model II is dominated by star formation in low-mass galaxies hosted by haloes with masses M_h < 10^10.5 h⁻¹ M_⊙. These galaxies are missed in the current observational data used to derive the SFRD, and so the discrepancy between the prediction of Model III and the observational results at z > 3 (shown as error bars in the figure) probably owes to incompleteness in the data. Indeed, if we use the same lower limit of UV magnitude adopted in Bouwens et al. (2012), which is M_{UV, AB} = −17.7, to predict the SFRD, we get the results shown by the thin lines, which brings the prediction of Model III into much better agreement with the data. The change in the prediction of Model II is small, because in this model galaxies below the limit do not make a significant contribution. This demonstrates clearly the importance in observing and modelling very faint galaxies to understand the SFRD at high z.

Figure 9.

The predicted SFR density (solid and dashed lines) and the minimal SFRD required to keep the Universe ionized (dot–dashed lines). The hedged band and the data points are from Bouwens et al. (2012). For the model predictions, the solid lines are obtained by integrating down to the merger tree resolution, which is 2 × 10⁹ h⁻¹ M_⊙, and the dashed lines are obtained by integrating down to the detection limit in Bouwens et al. (2012), which is M_{UV, AB} = −17.7 or 0.36 M_⊙ yr⁻¹.

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5 MERGER HISTORY AND THE TRANSFORMATION OF GALAXIES

In this section, we characterize the galaxy merger histories in more details and use a simple model to study their implications for the morphological transformation of galaxies.

We make the simple assumption that stellar discs form only through in situ star formation, and that a major merger can transform a stellar disc into a spheroid. Here, major mergers are defined in the same way as in Section 3. The mass of the stellar disc is then simply the total mass of stars formed in situ after the last major merger. It should be cautioned, however, that a galactic bulge can be formed in other ways, such as the secular evolution of the disc. The bulge mass defined here, therefore, can only be taken as a lower limit and serves as a simple indicator as to how a central galaxy may be disturbed by infalling satellites. Fig. 10 shows the distribution of ‘bulge-to-total’ ratio, B/T, for present-day central galaxies in four halo mass bins.

Figure 10.

The bulge-to-total mass ratio of local central galaxies as a function of host halo mass. The green histograms are the predictions of Model II while the red hedged histograms are those of Model III.

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In cluster haloes, the central galaxies are identified as brightest cluster galaxies (BCGs) in observations. They have been bombarded quite frequently by satellites of different masses. As shown in Fig. 11, such galaxies on average have experienced about five major mergers and an even larger number of minor mergers during the last period of low in situ star formation (defined, quite arbitrarily, as the period after the in situ SFR declines to be 1/3 of the peak value). Major mergers contribute about 50 per cent of the total stellar mass. This casts doubt on the scenario in which the late growth and structure of BCGs are assumed to be determined by minor mergers (e.g. Bezanson et al. 2009; van Dokkum et al. 2010; Hilz, Naab & Ostriker 2013). The last major mergers happened quite recently, at z ∼ 0.5, as shown in Fig. 12. By inspecting images of BCGs at low z, McIntosh et al. (2008) and Tran et al. (2013) did find that a significant fraction of them indeed show signatures of recent major mergers, consistent with our predictions.

Figure 11.

Solid curves: the number of merger events since the in situ SFR declines to 1/3 of the peak value with a mass ratio >r, where r is the stellar mass ratio between the secondary and primary merging progenitors. Dashed curves: the fraction of the total stellar mass acquired by mergers with a mass ratio >r. The left vertical line marks r = 1/3, which separates major from minor mergers. The results are shown for Model III (the results for Model II are similar for the massive haloes in question). For comparison, we also label r × f_TS on the top of the figure, and mark r × f_TS = 1/3 by the vertical line on the right. Since f_TS is the fraction of the satellite mass that ends up in the central, r × f_TS = 1/3 represents that ratio between the satellite mass that actually ends up in the central and the central mass.

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Figure 12.

The redshift of the last major merger of present-day central galaxies as a function of host halo mass. The predictions of Model II and Model III are shown in green and red, respectively. Lines are the averages, while the shaded areas enclose the 95 per cent range of variance owing to different merger histories.

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For haloes in the mass range 10¹²–10¹³ h⁻¹ M_⊙ there is strong bimodality in their B/T distribution. Significant mergers are sparse in their entire histories; the ‘ellipticals’ in such low-mass groups on average only experienced ≈1.5 major mergers and no more than three mergers with a mass ratio >0.1 (see Fig. 11). The central galaxy is either dominated by bulge, if it has experienced a recent major merger, or remains disc-dominated, if such a merger did not occur.

For a Milky Way-sized galaxy with M_h(0) = 10^11.5–10^12.5 h⁻¹ M_⊙, in situ star formation has dominated, while major mergers have been rare since z ∼ 2. These galaxies, therefore, remain disc-dominated, free of any significant major merger-driven bulge components.

The predictions of Model II and Model III differ when it comes to dwarf galaxies. Model II, consistent with many other similar models in the literature, indicates major mergers between dwarf galaxies are extremely rare. Therefore, all the stars are expected to remain in a disc (see the upper panel of Fig. 10). In contrast, Model III predicts that most of the galaxies experienced some major mergers during their initial starburst phases (z > 2). The exact time when a major merger occurs has large variations for such galaxies, as shown in Fig. 12. Major mergers are very rare at z < 2 while in situ star formation continues, allowing the growth of new discs. The fraction of stars contained in the spheroid depends both on when the last major merger occurs and on the in situ star formation that follows. This complexity in SFH results in diverse morphologies of present-day dwarfs, as shown in Fig. 10.

Major mergers at early stages of the evolution may shed light on the origin of dwarf ellipticals (dE) and dwarf spheroidal (dSph) galaxies. One popular scenario is galaxy harassment (Moore et al. 1996), in which high-speed encounters of a dwarf disc with other galaxies in a dense environment heats up the disc and transform it into a dE or a dSph. However, the predicted kinematics, which shows significant rotation, is at odds with the observational results (e.g. Geha, Guhathakurta & van der Marel 2003; Toloba et al. 2013). Our result here suggests that some of the slow rotators could be the remnants of early major mergers between dwarf galaxies. The lower panel of Fig. 13 shows B/T of both dwarf centrals and satellites predicted by Model III, with the latter skewed towards higher B/T. In contrast, Model II indicates that all dwarf galaxies are strongly disc-dominated. However, the predicted fraction of bulge-dominated dwarfs by Model III may be too low to account for the total population of dE's and dSph's. It is likely that later evolution, such as harassment, also plays a role in transforming the disc component, making the bulge components more dominant.

Figure 13.

The ‘bulge’ mass fraction of dwarf galaxies with total stellar masses in the range 10⁸–10⁹ h⁻² M_⊙, predicted by Model II (upper panel) and Model III (lower panel) for centrals (green) and for satellites (red).

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It should be pointed out that the above discussion is based on the value of r, namely the ratio between the original secondary mass and the primary mass. As shown above, our model predicts that only a fraction, f_TS ∼ 0.4, of the original mass of the secondary is added to the central after a merger while the rest is stripped and deposited as halo stars. If the stripping occurred at large distances from the central, the secondary mass at the time of merger would be r × f_TS. For comparison, the values of this quantity are labelled on the top of Fig. 11. As is clear, in this case major mergers, still defined by a mass ratio ≥1/3, would be rare for all galaxies. However, this scenario may not be realistic. Observations show that halo stars are mostly identified around central galaxies and, as we will see below, the amount of halo stars predicted by our models is consistent with observations (Bernardi et al. 2013). This suggests that stripping of stars most likely occurs when the secondary is close to the primary, and so the mass ratio, r, defined above is more relevant when considering the mutual gravitational interaction between the merging galaxies.

6 STELLAR POPULATIONS

6.1 Stellar ages

Figure 14.

The stellar age as a function of stellar mass predicted by Model II (upper panel) and Model III (lower panel). Both mass-weighted and light (luminosity)-weighted averages are shown, with the bands representing the 95 per cent percentile of halo merger histories. The lines show the light-weighted averages of stellar ages obtained by Gallazzi et al. (2005) from the SDSS, with the solid line being the median and the two dashed lines representing the 16 and 84 per cent percentile. The crosses are the mass-weighted (dark red) and light-weighted (green) ages of the LMC obtained from the SFH given by Weisz et al. (2013).

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This discrepancy may indicate an intrinsic deficiency in the approach adopted here or in similar approaches in the literature. By matching only SMFs of galaxies, our model may be insensitive to the SFH in the recent past. For example, an enhancement in recent star formation may contribute little to the total stellar mass, and so is not well captured in our model, but can significantly increase the optical luminosity, thereby decreasing the light-weighted age. As a demonstration, the upper cross in Fig. 14 shows the stellar mass weighted age of stars in the Large Magellanic Cloud (LMC) based on the SFH derived from the colour–magnitude diagram of resolved stars (Weisz et al. 2013). Our model predictions match the observation well. For comparison, the lower cross shows the r-band luminosity-weighted age obtained from the same SFH with our adopted spectral synthesis model. The age so obtained lies below the green band, because our average model underpredicts the current SFR of the LMC.

There are uncertainties in the observational data too. It is in general difficult to distinguish stars that formed about 8–10 Gyr ago from those that formed 4–5 Gyr ago from an analysis of the optical-NIR spectra (e.g. Bruzual & Charlot 2003; Pacifici et al. 2012). Basically the UV light provides information about recent star formation, while strong Balmer absorption lines are sensitive only to intermediate-age(1–3 Gyr old) stars, but one cannot distinguish stars older than 4 Gyr. Thus, it is possible that the average stellar ages derived from the observational data have missed the contribution of such an old population.

6.2 Halo stars

In our simple prescription for galaxy mergers, a constant fraction f_TS of the original stellar mass of the merging satellite is accreted by the central, and the rest is deposited in a diffuse component, referred to as intracluster stars in clusters or as halo stars in general. f_TS is constrained to be about 40 per cent, although the uncertainty is quite large (Tables 1 and 2). This low value of f_TS is driven predominantly by the observed SMFs at the massive end, i.e. with M_⋆ ≳ 10¹¹ M_⊙ corresponding to M_h > 10¹³ h⁻¹ M_⊙. Had we set f_TS = 1, i.e. if all the original mass in an accreted satellite were added to the central, we would get an SMF that is significantly higher than the observations of Baldry et al. (2012) at the very massive end (thick black line in Fig. 15).

Figure 15.

The SMF of galaxies predicted by Model III (grey line) compared to the prediction of the same model but assuming no stripping of satellites, i.e. f_TS = 1 (the thick black line). The data points are the observational data of Baldry et al. (2012). The green dot–dashed line is the SMF obtained by Bernardi et al. (2013) using SDSS ‘cmodel’ magnitudes, while the red dashed line is the result obtained by the same authors using the magnitudes within the entire Sérsic profiles (i.e. integrated to infinity) obtained by Simard et al. (2011).

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Recent analyses show that the most massive galaxies (with M_⋆ > 10¹¹ M_⊙) tend to have extended wings in their light profiles. This makes the luminosity (stellar mass) measurements of these galaxies quite uncertain. As shown by Bernardi et al. (2013) and He et al. (2013), using different methods and light profiles to fit galaxy images can lead to a factor of 2 difference in the estimated luminosity of a massive galaxy. The effect on the derived SMF can be seen in Fig. 15 comparing the red dashed line with the green dot–dashed line. It is interesting to note that our model prediction assuming f_TS = 1 is consistent with the SMF obtained by Bernardi et al. (2013) using the magnitudes within the entire Sérsic profiles (the red line) obtained by Simard et al. (2011). This suggests that the halo component defined in our model may simply be the extended profiles of massive galaxies that are missing in the SMFs we use as constraints.

Fig. 16 shows the ratio between the total mass of halo stars and the stellar mass of the central galaxy as a function of host halo mass. In a cluster as massive as 10¹⁵ h⁻¹ M_⊙, the mass in the diffuse component is about 2–3 times as high as that of the central galaxy and in a group-sized halo of mass ∼10¹³ h⁻¹ M_⊙, the ratio is about 1. It drops rapidly towards lower halo mass. In a Milky Way mass halo, the predicted ratio is only a few per cent. The flattening at the even lower mass end predicted by Model III owes to the boosted star formation at high z in low-mass haloes. The results for low-mass haloes should be taken with caution. As mentioned above, our model assumes a constant f_TS and it is constrained primarily by the SMFs at the massive end. It is unclear whether the same number also applies to galaxies with lower masses.

Figure 16.

The mass ratio between halo stars and the stellar mass of the central galaxy as a function of host halo mass at z = 0. The green band is the prediction of Model II and the red band that of Model III. Here, again the widths of the bands represent the variance among different halo merger trees.

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To understand the stellar population of the halo stars, we make a census of their formation time and location, in terms of redshift and the stellar mass of the host galaxy in which they formed, respectively. We show the result as the solid contours in Fig. 17. For comparison, we also plot the results for stars in the central galaxies as the dashed contours. The stellar mass can be taken as crude proxy to stellar metallicity as the two are found to be correlated for local galaxies (Gallazzi et al. 2005). The stellar metallicity increases with stellar mass for galaxies with stellar masses below 2 × 10¹⁰ M_⊙ and becomes saturated at about solar metallicity above this mass (indicated by the horizontal lines in Fig. 17).

Figure 17.

The solid contours show the formation redshift and location (i.e. the stellar mass of the galaxy) of the halo stars. Red, green and blue contours enclose 30, 60 and 90 per cent of the stars, respectively. The dashed contours are the same as the solid ones but for stars in central galaxies. We only show Model III (Model II is similar). Results are shown for Milky Way mass haloes (upper panel) and massive cluster haloes (lower panel). The horizontal lines mark  M_⋆ = 2 × 10¹⁰ M_⊙, above which the stellar metallicity begin to saturate (Gallazzi et al. 2005).

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Compared to the stars in the central galaxy, the halo stars in a Milky Way-sized halo form a distinct population; the mass weighted age of the halo stars is roughly about 9–10 Gyr, in contrast with the central galaxy, which is about 6 Gyr. These halo stars formed in progenitors with mass lower than the horizontal line, suggesting a metallicity much lower than the solar value. This is qualitatively consistent with the recent observations of M31 (Bernard et al. 2015). In massive clusters, on the other hand, the stellar populations in both the central galaxy and the halo component are quite homogeneous, with ages of ∼10 Gyr and with nearly solar metallicity. This prediction can be checked by studying the stellar age and metallicity of halo stars in clusters.

7 SUMMARY AND DISCUSSION

In a previous paper (Lu et al. 2014a), we developed an empirical model to describe the SFRs of central galaxies in haloes of different masses at different redshifts. A series of nested models were constructed to accommodate more and more observational constraints. We found that Model II, which represents a class of ‘Slow Evolution’ models in the literature, can reproduce the SMFs since z = 4, but fails to accommodate the cluster galaxy luminosity function, the steep SFR-functions at high z, and the old stellar population seen in local dwarf galaxies. We also found that Model III is the simplest model family that can match all these observational data well. In this paper, we use the same models, but with model parameters updated using recent observational data of the galaxy SMFs at high z. The results presented here confirm and re-enforce those of Lu et al. (2014a). In particular, the constrained Model III predicts much steeper SMFs at z ≥ 4 than Model II. Since the publication of Lu et al. (2014a), two additional studies have presented evidence in support of this picture (Madau, Weisz & Conroy 2014; Weisz et al. 2014). These results suggest that the class of ‘Slow Evolution’ models in the literature (Behroozi et al. 2013a; Béthermin et al. 2013; Yang et al. 2013), in which star formation in haloes with  M_h < 10¹¹ h⁻¹ M_⊙ are suppressed at high z, are insufficient to describe the evolution of the galaxy population.

We use our constrained model parameters to characterize the star formation and stellar mass assembly histories as well as the merger histories of galaxies of different masses. The results are summarized in the following.

First, the evolution of the galaxy population is found to be characterized by a number of characteristic halo mass scales.

• A mass scale of 10¹³ h⁻¹ M_⊙ at z = 0, decreasing to ∼3 × 10¹² h⁻¹ M_⊙ at z > 2, above which in situ star formation drops rapidly, and the central galaxies experience frequent major mergers most of which will be dry.

• A mass scale of 3 × 10¹¹ h⁻¹ M_⊙ at z = 0, increasing to 10¹² h⁻¹ M_⊙ at high z, at which the efficiency of in situ star formation reaches a maximum, with an SFR as high as about half of the baryon accretion rate into the host halo. Major mergers are rare in this halo mass range.

• A mass scale of 10¹¹ h⁻¹ M_⊙ at z > 3, below which in situ star formation has a rate about 0.1 times the baryon accretion rate into the host halo. On average, one or two major mergers are expected to occur for the central galaxies.

Secondly, galaxies hosted by haloes of different masses follow distinct star formation and assembly histories. Based on the characteristic halo masses given above, central galaxies can be divided roughly into three different categories according to their formation and assembly histories.

• For haloes with M_h > 10¹³ h⁻¹ M_⊙, a strong in situ SFR declines rapidly after reaching its peak value, and is followed by significant accretion of stars from satellites. For such massive systems, more massive galaxies tend to assemble their stellar mass later, contrary to the downsizing trends observed for lower mass galaxies.

• For haloes with masses 10¹¹ h⁻¹ M_⊙ < M_h < 10¹³ h⁻¹ M_⊙, mass assembly by accretion of satellites is not important, and the star formation is delayed relative to the formation of the host halo.

• For haloes with masses below 10¹¹ h⁻¹ M_⊙, assembly by accretion is again unimportant. The SFH is characterized by a burst at z > 2 and a nearly constant SFR after z = 1. The relative importance of the early star formation increases with decreasing halo mass, and the ‘downsizing’ trend is reversed.

Thirdly, we use the merger history of the model galaxies to predict the bulge to total mass ratios of present galaxies. The average bulge mass fraction is found to depend strongly on halo mass.

• In cluster sized haloes with M_h > 3 × 10¹³ h⁻¹ M_⊙, almost all the centrals are ellipticals formed through frequent major mergers.

• In group-sized haloes with masses between 3 × 10¹² and 3 × 10¹³ h⁻¹ M_⊙, the distribution of the bulge-to-total ratio of the central galaxies is strongly bimodal. Those galaxies that experienced a recent major merger are spheroid dominated, whereas the others are free of any significant merger-driven bulge.

• For haloes with masses 3 × 10¹¹ < M_h < 3 × 10¹² h⁻¹ M_⊙, central galaxies with a significant merger-driven bulge are extremely rare.

• For dwarf galaxies, half of them have significant (with B/T > 10 per cent) spheroidal components formed during their early starburst phase (z > z_c ≈ 2). Satellite galaxies of similar masses tend to have a larger bulge fraction than centrals.

We emphasize again that bulges can form in various other ways than through major mergers, and that our prediction only applies to major merger-driven bulges.

Finally, we have made predictions for the amount of halo stars, and when and where these stars form in comparison with stars in the corresponding central galaxies. The results are as follows.

• In a Milky Way mass halo, the total mass in halo stars is 2–5 per cent of the mass of the central galaxy, and this number increases to ∼100 per cent in haloes with M_h ∼ 10¹³ h⁻¹ M_⊙, and from about 200 to 300 per cent in massive clusters.

• In a Milky Way mass halo, the stars in the central galaxy and halo stars form two distinct stellar populations, with the latter being older and poorer in metals. In contrast, these two components form a quite homogeneous population (old and with solar metallicity) in massive clusters.

All these results are obtained in an empirical way, independent of any detailed assumptions about the underlying physical processes that drive the evolution of the galaxy population. Clearly, our results should be compared with the predictions of numerical simulations and/or semi-analytical models to constrain theories of galaxy formation. We will come back to this in a forthcoming paper.

ACKNOWLEDGEMENTS

We thank Stephane Charlot for discussion, and Ryan Quadri for providing the ZFOURGE/CANDELS SMFs. We would like to acknowledge the support of NSF AST-1109354 and NSF AST-0908334.

REFERENCES