© ESO, 2011

1. Introduction

Modelling the evolution of dust in galaxies is a key ingredient in understanding the origin of the high observed dust masses in high-redshift galaxies and quasars (QSOs). Dust masses  ≥ 10⁸ M_⊙ have been derived from observations in QSOs at redshift z ≳ 6 (e.g., Bertoldi et al. 2003a; Robson et al. 2004; Beelen et al. 2006; Hines et al. 2006; Michałowski et al. 2010b) along with high star formation rates (SFR) up to a few times 10²⁻³ M_⊙ yr^-1 (e.g., Walter et al. 2004; Wang et al. 2010). Additionally, QSOs at z > 6 harbour supermassive black holes (SMBHs) with masses  > 10⁹ M_⊙ (e.g., Willott et al. 2003; Vestergaard 2004; Jiang et al. 2006). Kawakatu & Wada (2008, 2009) showed that to form a SMBH  >  10⁹ M_⊙, a high mass supply of  > 10¹⁰⁻¹¹ M_⊙ is needed. These requirements, together with derived molecular gas masses from CO line emission measurements in excess of 10¹⁰ M_⊙ (e.g., Cox et al. 2002; Carilli et al. 2002; Bertoldi et al. 2003b; Walter et al. 2003; Riechers et al. 2009; Wang et al. 2010), set significant constraints on the physical properties of the host galaxies. These in turn have implications for the origin and evolution of dust. Furthermore, a tendency toward increased dust attenuation with higher galaxy masses for systems at z ~ 6–8 has been found by Schaerer & de Barros (2010).

Analytical and numerical models for dust evolution have been developed. Dwek et al. (2007) propose 1 M_⊙ of dust per SN will be necessary to account for dust masses in high-z QSOs, while contemplating SNe as the only source. Such high dust masses for SNe contradict derived dust masses from nearby SNe and SN remnants, which on average reveal a few times 10^-4–10^-2 M_⊙ of dust (e.g., Wooden et al. 1993; Elmhamdi et al. 2003; Temim et al. 2006; Meikle et al. 2007; Rho et al. 2008; Kotak et al. 2009; Sibthorpe et al. 2010; Barlow et al. 2010). A review of observationally and theoretically derived dust from stellar sources is provided by Gall et al. (in prep., herafter GAH11).

The issue of whether SNe produce large amounts of dust is unclear. Other sources of dust such as AGB stars have been taken into account (Morgan & Edmunds 2003; Valiante et al. 2009) in chemical evolution models of high-redshift galaxies. Valiante et al. (2009) claim that with the contribution of AGB stars, 10⁸ M_⊙ of dust can be reached, with AGB stars dominating the dust production. AGB stars (0.85–8 M_⊙) are the main source of dust in the present universe, but only stars with masses  ≳ 3 M_⊙ are likely to contribute at z > 6 (e.g., Marchenko 2006). Evidence that metal-deficient AGB stars also undergo strong mass loss and are able to efficiently produce dust is supported observationally (e.g., Zijlstra et al. 2006; Groenewegen et al. 2007; Lagadec et al. 2007; Matsuura et al. 2007; Sloan et al. 2009) and theoretically (e.g., Wachter et al. 2008; Mattsson et al. 2008). However, the theoretical models (Dwek et al. 2007; Morgan & Edmunds 2003; Valiante et al. 2009) greatly differ with respect to the assumptions made for the mass of the galaxy and dust contribution from stellar sources, as well as the treatment of the star formation. Thus the origin of dust and its evolution remain unclear.

In GAH11 we discuss plausible dust production efficiency limits for stellar sources between 3 M_⊙ and 40 M_⊙, and determined the dust productivity of these sources for a single stellar population. In this paper, we investigate the evolution of dust in high-z galaxies. We develop a numerical chemical evolution model, which allows exploration of the physical parameter space for galaxies at z > 5–6. Different types of core collapse supernovae are specified and the contribution from AGB stars and the impact of SMBHs are taken into account. We furthermore follow the evolution of some physical properties of these galaxies. The main parameters varied in the model are the IMF, the mass of the galaxy, yields for SNe, and the strength of dust destruction in the ISM, as well as the dust production efficiency limits. Models with or without the SMBH formation are considered.

The paper is arranged as follows. In Sect. 2 the equations used to construct the model are developed. We discuss the model parameters and their possible values in Sect. 3. A detailed analysis of the results is presented in Sect. 4, which is followed by a discussion in Sect. 5 and our conclusions of this work in Sect. 6.

2. Modelling the evolution of dust in starburst galaxies

In this section we formulate the equations needed to follow a galaxy’s time-dependent evolution in a self-consistent numerical model. The main basic logic is adopted from Tinsley (1980, and references therein), which has also been used in other chemical evolution models to study dust in galaxies (e.g., Morgan & Edmunds 2003; Dwek et al. 2007). We focus on an elaborate treatment of dust from different types of CCSNe and AGB stars. In order to calculate the amount of dust from these sources we use the dust production efficiencies described in Sect. 3.4. In particular, the lifetime-dependent delayed dust and gas injection from AGB stars and SNe is taken into account. The metallicity-dependent lifetimes of all stars are taken from Schaller et al. (1992), Schaerer et al. (1993), and Charbonnel et al. (1993). In GAH11 we show that the variation of the lifetime with metallicity is minimal. We therefore calculate and use a metallicity-averaged lifetime for all stars. The recycled gaseous material is defined as the remaining ejected stellar yields from all massive stars in the mass range 3–100 M_⊙, which has not been incorporated into dust grains. This also includes the stellar feedback from very massive stars (≳30–40 M_⊙) in the form of stellar winds. Dust and gas are assumed to be released instantaneously after the death of the stars.

We strictly treat the elements in the gas and solid phases separately, while paying attention to their interplay. Thus, we define M_g(t) as the total mass of elements in the ISM, which are in the gas phase and M_d(t) as the total amount of elements in the solid dust phase. The mass of the ISM is defined as M_ISM(t) ≡ M_g(t) + M_d(t), which in the literature is often referred to as the “gas mass”. The total amount of dust in our models is solely calculated from the dust contributions from SNe and AGB stars. Thus, no further growth in the ISM is contemplated. However, dust destruction in the ISM through SN shocks is taken into account. The formation of the SMBH is considered as a simple sink for dust, gas, and metals.

We assume a so-called “closed box” model; i.e., the effect of infalling and outflowing gas in the galactic system is neglected. Although galaxies may not evolve in such a simple manner, this assumption is plausible since massive starburst galaxies are assumed to have SFRs ψ(t) ≳ 10³ M_⊙ yr^-1. Infall of neutral gas would only affect the system when the infall rate is comparable to the SFR. In this case a large gas reservoir needs to be present in the vicinity of the galaxy already. Besides, infall rates for high-z galaxies are not known. We further assume the ISM to be homogeneously mixed, and we evolve our model only up to the first Gyr. Examples of “closed box” models being sufficiently accurate include the works of Frayer et al. (1999) and Tecza et al. (2004) for the luminous and massive submillimetre galaxy SMMJ14011+0252 at z = 2.565.

2.1. Basic considerations

The initial mass functions (IMF) φ(m) is normalized to unity in the mass interval [m₁, m₂] as (1)where m₁ and m₂ are the lower and upper limits of the IMF (see Sect. 3.2).

The relation between the total mass M_ISM(t) and the SFR ψ(t) is given by the Kennicutt law (Kennicutt 1998), where ψ(t) ∝ M_ISM(t)^k. Analogously to Dwek et al. (2007), we apply the following notation to calculate the SFR (2)where ψ_ini is the initial SFR, M_ini the initial gas mass of the galaxy, and M_ISM(t) the mass of the ISM. The value of k is between 1 and 2, so we set k = 1.5. This value has often been assumed in other models (e.g., Dwek 1998; Dwek et al. 2007; Calura et al. 2008).

2.2. Equations for AGB stars and supernovae

The amount of dust from all stellar sources released into the ISM per unit time is simply calculated as the amount of dust produced by the stellar sources times the source rate. The considered dust producing stellar sources are AGB stars and CCSNe. We account for a potentially diverse dust contribution by different SNe subtypes and distinguish between type IIP SNe and the remaining type II subtypes. Types Ib and Ic SNe, collectively referred to as Ib/c, are not considered as dust producing SNe. However they inject their stellar yields into the ISM. A specification of these sources and their lower and upper stellar mass limits, m_L(i) and m_U(i), are discussed in Sect. 3.3. In the following equations these sources are indicated by the subscript i = AGB, IIP, II, Ib/c.

The AGB and SN rate R_i(t) calculates as (3)where τ = τ(m) is the lifetime of a star with a zero-age main sequence (ZAMS) mass m; i.e., the star was born at time (t − τ) when it dies at time t. It is evident that stars only contribute when the condition t − τ ≥ 0 is fulfilled.

For SNe, the time of releasing the total produced elements is assumed to take place right after explosion. The main sequence lifetime for AGB stars is defined as the time until the end of the early AGB phase, which can take up to several 100 Myr. However, the most efficient mass loss phase itself is less than 1 Myr at the very end of the AGB phase and is relatively short compared to the total lifetime of AGB stars. Thus, we make the same approximation as for SNe: all produced elements and dust are released instantaneously after the main sequence lifetime.

For each kind of source the total produced dust per unit time is calculated as (4)where Y_Z = Y_Z(m,Z) with Z = Z(t − τ) is the mass (m) and metallicity (Z) dependent amount of ejected heavy elements per star. The metallicity Z is defined as the metallicity with which the star was born at a time (t − τ). The parameter ϵ_i(m,Z) is defined as the dust production efficiency. The assumed efficiencies are described further in Sect. 3.4. For types Ib/c SNe, .

The total amount of mass lost in stellar winds prior to explosion, Y_WIND, caused by mass loss during stellar evolution of SNe, is calculated as (5)where Y_WIND = Y_WIND(m,Z) and M_fin = M_fin(m,Z). The latter is the final mass of a SN before explosion. Type II SN suffer strong mass loss leading to the formation of dense circumstellar disks. Dust has been found in these disks for some SNe (GAH11 and references therein). However, the actual amount of metals in these disks is not known and data are not available, thus we cannot account for them. In the case of AGB stars and for SN models where no mass loss prescription is available Y_WIND = 0.

Besides the elements bound in dust grains, elements in the gas phase are also released into the ISM. The produced amount of elements in gaseous form is calculated as (6)where Y_E = Y_E(m,Z) is the amount of all ejected elements per star.

The total mass of heavy elements released into the ISM per unit time is calculated as

(7)We include the possibility of dust destruction in the SN remnant (SNR) due to reverse shock interaction. Dust destruction time scales up to 10⁴ yr have been predicted by Bianchi & Schneider (2007) and Nozawa et al. (2007, 2010). However, this timescale is relatively short in comparison to the lifetime of a star, so we make the approximation that dust is destroyed immediately after formation.

We define the parameter ξ_SN as the SN dust destruction factor; i.e., the destroyed mass of dust per unit time of all SNe is (8)This term applies only to type IIP and type II SNe, thus i = IIP, II.

The final SN dust injection rate per unit time is calculated as

(9)while the final AGB dust injection rate is

(10)

2.2.1. Recycled gaseous material

The recycled material from SNe, and AGB stars consists of all the mass of the elements not being incorporated into dust grains, thus the material is in the gas phase. Very massive stars ending as BHs may contribute with their stellar winds to the recycled material. We refer to stars that directly form a BH as the “remaining stars”. Pertaining to the short lifetime of very massive stars and the resulting short duration of the wind phase, we assume that all the elements lost in the wind phase are released after the death of the star. The total mass of the released elements per unit time of the remaining stars is (11)where X_g,R = X_g,R(m,Z) is the mass of elements released into the ISM per star, while the subscript “R” stands for “remaining stars”.

For these remaining stars the following two scenarios are possible: (1) a supernova without display (Eldridge & Tout 2004) occurs even if a BH is formed and elements are ejected or (2) no SN occurs because no SN shock is launched. For the first case, the term X_g,R can either be substituted with (Y_E + Y_WIND), or if Y_E and Y_WIND are not known, the common approximation of X_g,R = m − M_rem can be made. The mass M_rem = M_rem(m,Z) is the remnant mass of a star. In the second case we assume that no nucleosynthesis products will be ejected. A contribution to the gas household in the ISM comes solely from the stellar winds, thus X_g,R = Y_WIND.

The mass of released metals from these stars per unit time is given by

(12)where X_Z,R = X_Z,R(m,Z) is the mass of the ejected heavy elements per star. For the first case, X_Z,R = Y_Z + Y_WIND   Z(t − τ). For the second, X_Z,R = Y_WIND   Z(t − τ).

By extending the subscript i to i = AGB, IIP, II, Ib/c, R, the total returned mass of gaseous material to the ISM per unit time calculates as

(13)It is important to note that elements locked up in dust grains will return to the gas phase when dust destruction takes place. As a result, the amount of destroyed dust in SNRs must be added to the total amount of gas in the ISM (second term in Eq. (13)).

2.3. The evolution of dust and gas in the galaxy

The chemical evolution of a galaxy is mainly determined by the equations balancing the net amount of gas, dust, and heavy elements.

2.3.1. Effect of a super massive black hole

We take the effect of a SMBH into account as an additional sink for the gas, dust, and heavy elements. We base our assumptions for the treatment of the BH on the theoretical work of Kawakatu & Wada (2008, 2009). Super Eddington growth is required to form a SMBH. This necessitates a large mass supply of  ~10¹⁰⁻¹¹ M_⊙ on a short supply timescale of t_SMBHsup ~ 10⁸ yr from the host galaxy to a massive circumnuclear disk. Kawakatu & Wada (2009) find that the final SMBH mass is around 1–10% of the supply mass M_BHsup.

We do not treat the formation of the disk in detail, but assume that the supply mass needed is equal to the initial mass M_ini of our considered systems. We therefore only take the overall mass of the ISM needed to form the SMBH into account. A simple constant growth rate is calculated as (14)where M_SMBH is the mass of the final SMBH, and Ψ_SMBH = const. for t ≤ t_SMBH, whereas Ψ_SMBH = 0 for t > t_SMBH. The growth timescale to build up the SMBH is equal to the supply timescale t_SMBH = t_SMBHsup. The onset of the SMBH formation coincides with the onset of starburst of the whole galaxy. Where the SMBH is not taken into account, the growth rate is set to zero (Ψ_SMBH = 0).

2.3.2. The amount of dust in the ISM

The evolution of the amount of dust M_d(t) in the galaxy is

(15)The first and second terms are the dust injection rates from SNe and AGB stars contributing to the increase in the dust household in the ISM. The third term E_D(t) is defined as the total dust destruction rate. It determines the dust reduction through astration, as well as through the SMBH formation and destruction in the ISM caused by SN shocks, if considered. The total dust destruction rate is calculated as

(16)The variable η_d(t) = M_d(t)/M_ISM(t) is the fraction of dust in the ISM and will be referred to as the “dust-to-gas mass ratio”. For simplicity we make the assumption that the produced dust will be immediately mixed with the material in the ISM. The dust destruction in the ISM through SN shocks will be discussed in more detail in Sect. 3.5.

2.3.3. The amount of gas in the ISM

The temporal evolution of the gas content in the galaxy is calculated as

(17)Here, E_g(t) is the recycled gaseous material (see Eq. (13)). The second term is the amount of destroyed dust in gaseous form (see Sect. 2.3.2). The third term accounts for the depletion of the gas in the ISM through incorporation into stars and the loss to the SMBH. The fraction of gas in the ISM is expressed as (1 − η_d(t)).

2.3.4. Metallicity

Owing to the separation of the ISM mass into the material locked in dust and gas, respectively, it is necessary to formally take care of the transition of elements from the dust phase to the gas phase (due to destruction). For the evolution of heavy elements we do not distinguish between their chemical states.

The equation for the evolution of the total amount of heavy elements is formulated as (18)where

(19)is the total ejected mass of heavy elements per unit time from all considered sources. The last term in Eq. (18) determines the reduction of heavy elements due to astration and the loss to the SMBH.

The total metallicity of the system is defined as Z(t) = η_Z(t) = M_Z(t)/M_ISM(t). The fraction of metals in the ISM which are bound in dust grains is calculated as η_Zd(t) = M_d(t)/M_Z(t). The amount of metals in the gas phase is (20)The gas phase metallicity is given as η_Zg(t) = M_Z,g(t)/M_g(t).

3. Model parameters

In this section we describe the prime model parameters, along with the values used in this study. In particular, we consider the initial conditions of the galaxy, the IMF, stellar yields, and the destruction rates of dust in the ISM. These characterize the system and significantly influence the evolution of gas, dust, and metals. In addition we define some switches, which specify various possibilities for some model parameters. All parameters and their considered values, as well as the possibilities for the switches, are listed in Table 2.

3.1. Initial conditions

The model is defined by the initial values of the parameters. It neither depends on nor is influenced by additional input from other models; i.e., it does not depend on a cosmological model. Therefore it can be applied to any galaxy within the accuracy limit of a “closed-box” treatment. We are mainly interested in massive high-redshift galaxies in which high dust masses, stellar masses, and H₂ masses have been inferred from observations. The parameters of our computed models are therefore tuned to such galaxies.

One of the main parameters is the baryonic initial gas mass M_ini, which is equal to the total mass of the galaxy in baryons. A relation between M_ini and the mass of the dark matter halo M_DM hosting such systems is given through M_ini = Ω_b/Ω_m   M_DM. In this work we consider four different massive galaxies with M_ini = 1.3 × 10¹² M_⊙, M_ini = 5 × 10¹¹ M_⊙, M_ini = 1 × 10¹¹ M_⊙, and M_ini = 5 × 10¹⁰ M_⊙.

In GAH11 we argue that the very first population of stars (so-called Pop III stars) are not likely to be the main sources of high dust masses at high redshift. Thus we consider only the next generations of stars (Pops II or I). The formation of these stars takes place as soon as a critical metallicity of Z_cr ~ 10^-6–10^-4 Z_⊙ (Bromm & Loeb 2003; Schneider et al. 2006; Tumlinson 2006) is reached in the star-forming region. In this regard we assume an initial metallicity in accordance with the critical metallicity of Z_ini = Z_cr = 10^-6 Z_⊙.

Pertaining to the rather high derived star formation rates from observations of some high-z massive galaxies and QSOs (e.g. Frayer et al. 1999; Bertoldi et al. 2003a; Riechers et al. 2007), we consider an initial SFR of ψ_ini = 1 × 10³   M_⊙ yr^-1. The evolution is determined using the Kennicutt law as described in Sect. 2.1, Eq. (2). We also consider a case of constant SFR where ψ(t) = ψ_ini = 1 × 10³ M_⊙ yr^-1.

In our model the onset of starburst is not directly connected to redshift, so, the age of the galaxy is identical to the evolutionary time after starburst. For dusty galaxies seen at redshift 5–6 the earliest onset of starburst with very high SFRs can be considered to have taken place at z ⋍ 10. For a ΛCDM universe with H₀ = 70 km s^-1 Mpc^-1, Ω_Λ = 0.73 and Ω_m = 0.27 and Ω_b = 0.04 (Spergel et al. 2003), the evolutionary time of interest for building up high dust masses possibly lies then within 400–500 Myr. We have computed all models presented in this paper up to an age of the galaxy of t_max = 1 Gyr.

3.2. The initial mass function

The initial mass function is an important parameter influencing the evolution of dust, gas, and metals in a galaxy. It determines the mass distribution of a population of stars with a certain ZAMS mass. In models of dust evolution in galaxies and high-z quasars (e.g., Morgan & Edmunds 2003; Dwek et al. 2007), an IMF first proposed by Salpeter (1955) is often used. However, for starburst galaxies there is also observational evidence of a top-heavy IMF (e.g. Doane & Mathews 1993; Tumlinson 2006; Dabringhausen et al. 2009; Habergham et al. 2010).

In view of this, we consider a set of five different IMFs. The power-law IMFs (Salpeter, mass heavy and top-heavy) have the form φ(m) ∝    m ^− α. The lognormal Larson IMFs (Larson 1998) are given as φ(m) ∝    m ^{− (α + 1)}exp(−m_ch/m), where m_ch is the characteristic mass. Their parameters are presented in Table 1.

The influence of the diverse IMFs, particularly on the dust production rates from stellar sources, is demonstrated in GAH11.

3.3. Stellar yields

The model is adapted to published results from stellar evolution models. Using these models the metallicity dependent upper and lower mass limits of the mass range of diverse SNe subtypes can be determined. For comparison to the usually adopted treatment of SNe we calculate simplified models with a fixed mass range for CCSNe between 8–40 M_⊙ and use stellar yields from nucleosynthesis calculations for SNe. For SNe we adopt three different structural models.

3.3.1. SN yields from stellar evolution models

We adopt the stellar yields for the hydrogen mass M_H(m,Z) in the envelope, the He core mass M_core(m,Z), and the final mass M_fin(m,Z) of the progenitors, as well as Y_E(m,Z) and Y_Z(m,Z) from the single stellar evolution models by Eldridge et al. (2008, herafter EIT08. These yields are based on previous works (Eldridge & Tout 2004, 2005). EIT08 adopt a mass loss prescription using the rates of de Jager et al. (1988) with the rates of Vink et al. (2001) for pre-Wolf-Rayet (WR) and from Nugis & Lamers (2000) for WR evolution including overshooting. In their study, this model is closest to a set of SN progenitor observations.

Eldridge & Tout (2004, 2005) also show that all stars end up as II-P SN at metallicities roughly below Z = 10^-4, while with increasing metallicity the upper mass limit for II-Ps decreases; i.e., at solar metallicity it is at roughly 28 M_⊙. type II-P SNe are per definition all stars that have retained at least 2 M_⊙ of hydrogen in their envelopes at their pre SN stage (Heger et al. 2003). Since mass loss is more efficient at higher metallicities, the upper mass limit for II-P supernovae decreases.

With increasing metallicity further types of CCSNe such as IIL, Ib, and Ic SNe arise. The upper limit for IILs is defined by the small hydrogen fraction of  ~0.1 M_⊙ in the envelope. In case little or no hydrogen in the envelope of the progenitor is present, SNe appear as types Ib or Ic. The SNe arising from the higher mass end of all type IIs are the subtypes IIb or IIn. These types are difficult to fit into a quiescent mass loss prescription and are not specified by EIT08. Thus we simply assume that these subtypes may be part of the fraction of SNe, which is determined as IIL in the models of Eldridge & Tout (2004). These SNe will be collectively referred to as the remaining type II supernovae.

We define the conditions for determining the lower m_SL,(i) and upper m_SU,(i) boundaries of the different SN subtypes analogous to Eldridge & Tout (2004) as given in Table 3. The masses M_H(m,Z) and M_core(m,Z) are dependent on metallicity. Therefore the lower and upper mass limits for these SN types are m_SL,(i) = m_SL,(i)(Z) and m_SU,(i) = m_SU,(i)(Z). The lower mass limit for type II-P SNe is fixed at m_SL,IIP(Z) ≡ m_SL,IIP = 8   M_⊙. The absolute upper mass limit for the most massive stars is the cutoff mass defined by the IMF, thus m_SU,(i)(Z) ≡ m_SU,(i) = m₂.

The criterion for direct BH formation is based on the system used by Heger et al. (2003) and Eldridge & Tout (2004). A direct BH forms when the final He core mass M_core(m,Z) exceeds the critical He core mass,  M_⊙. Models using yields from EIT08 will be referred to as “EIT08M”.

3.3.2. SN yields from rotating stellar models

We take the yields for M_H(m,Z), M_fin(m,Z), Y_E(m,Z), and Y_Z(m,Z) from Georgy et al. (2009). Rotationally enhanced mass loss becomes very efficient with increasing metallicity. The mass loss description used in their models are from Meynet & Maeder (2003, 2005) with a rotational velocity of 300 km s^-1. Provided that a SN occurs even if a BH is formed, the resulting range for type II SN (without further subdivision) is between 8–54 M_⊙ at a metallicity Z = 0.004   Z_⊙. However, only stars between 8–25 M_⊙ form neutron stars. At Z_⊙ the type II mass range becomes narrower (8–25 M_⊙) and stars above the upper limit will become WR stars and explode as Ib/c SNe. The range for neutron stars at this metallicity extends up to 35 M_⊙.

To determine the lower m_SL,(i)(Z) and upper m_SU,(i)(Z) mass limits for the considered SNe types and remaining stars (i = IIP, II, Ib/c, R), we appled the same definitions as described for the EIT08M models. However, the direct BH cut is constrained by the mass of the remnant star M_rem(m,Z) instead of M_core(m,Z). We follow the notation for the BH formation by Georgy et al. (2009), where a BH forms when M_rem(m,Z) > 2.7 M_⊙. Models where these yields and the above SN type division and BH formation are applied, are referred to as “G09M”.

3.3.3. Models with fixed SN mass range

A treatment in which the boundaries of the mass range for all considered dust-forming supernovae and remaining stars are fixed throughout the evolution is quite common and has been used in previous models (e.g. Dwek 1998; Morgan & Edmunds 2003; Dwek et al. 2007; Valiante et al. 2009).

We used the stellar yields for Y_E(m,Z), Y_Z(m,Z) and certain elements Y_Q(m,Z) with Q = C, O from nucleosynthesis models of either Woosley & Weaver (1995, hereafter WW95 or Nomoto et al. (2006) (herafter N06). All SNe are of type II and considered to be within the mass interval m_L,(II) = 8   M_⊙ and m_U,(II) = 40. Yields for the final mass prior to explosion are not available, thus in all equations Y_WIND = 0.

The remaining very massive stars between m_L,(R) = m_U,(II) = 40 and the upper limit m_U,(R) = m₂ are assumed to turn into BHs. We apply the yields from EIT08 to these remaining stars in order to account for the gas return into the ISM from their mass loss phase according to Eqs. (11) and (12). Thus, the option of whether a SN explosion occurs or not is retained.

The models with stellar yields from WW95 will be referred to as “WW95M” and the models with yields from N06 will be referred to as “N06M”.

3.3.4. AGB stars

For high-mass AGB stars we use the stellar yields for the total ejected mass Y_E(m,Z), the metals Y_Z(m,Z) and certain elements Y_Q(m,Z) with Q = C, O from van den Hoek & Groenewegen (1997). The mass interval is set to a fixed mass range of m_L,(i) = 3 M_⊙ and m_U,(i) = 8 M_⊙, i = AGB. AGB stars with masses below 3 M_⊙ are not considered.

3.4. AGB star and SN dust production efficiencies

The values for the AGB star and SN dust production efficiencies are derived from observationally and theoretically determined dust yields. The dust production efficiency per stellar mass and metallicity, ϵ_i(m,Z), is calculated as (21)where Y_d(m,Z) is assumed to be the final amount of dust ejected into the ISM (i.e., produced and possibly processed through shock interactions).

For AGB stars, we adopted for Y_d(m,Z) the theoretically derived total dust yields from Ferrarotti & Gail (2006) and for Y_Z(m,Z) the yields from van den Hoek & Groenewegen (1997). We calculated the AGB star efficiency for different metallicities. For illustration, we show the metallicity-averaged efficiency, ϵ_AGB(m), which is representative of the trend of the metallicity-dependent ϵ_AGB(m,Z) in Fig. 1. One notices that, on average, AGB stars between 3–4 M_⊙ are the most efficient dust producers.

For the dust production efficiency for SNe we used two plausible limits: (i) an upper limit that was motivated by theoretical SN dust formation models; and (ii) a lower limit based on dust masses inferred from observations of older SN remnants, such as Cas A, B0540 − 69.3, Crab nebula, and 1E0102.2 − 7219 (e.g., Douvion et al. 2001; Bouchet et al. 2004; Green et al. 2004; Krause et al. 2004; Temim et al. 2006; Rho et al. 2008; Sandstrom et al. 2008; Williams et al. 2008; Dunne et al. 2009; Barlow et al. 2010). For the upper limit we applied two different combinations of the dust and metal yields, which are (i) Y_d(m,Z) from Todini & Ferrara (2001) with Y_Z(m,Z) from WW95 and (ii) Y_d(m,Z) from Nozawa et al. (2003) with Y_Z(m,Z) from N06. Averaging the obtained efficiencies over Z leads to the metallicity independent “maximum” SN dust production efficiency, ϵ_max(m). For the lower limit an average dust mass of 3 × 10^-3 M_⊙ inferred from the SN remnants is applied to all SNe in the mass range 8–40 M_⊙. To calculate the SN dust production efficiency, stellar yields from WW95, N06, and EIT08 are used. The efficiencies obtained with these yields are averaged, constituting the “low” SN dust production efficiency, ϵ_low(m).

Applying a dust destruction factor (see Sect. 2.2) to ϵ_max(m) results in a reduced dust production efficiency. We refer to this efficiency as “high” SN dust-production efficiency, ϵ_high(m) = ξ_SNϵ_max(m). The amount of dust (~2–6 × 10^-2 M_⊙) produced per SN with this efficiency is comparable to the higher dust masses derived in older SN remnants.

We used a dust destruction coefficient, ξ_SN = 0.93, following Bianchi & Schneider (2007). These SN efficiencies may possibly also arise from different scenarios of dust formation, such as grain growth and shock interactions in SN remnants. The averaged SN efficiencies ϵ_max(m) and ϵ_low(m), as well as the “high” SN efficiency, ϵ_high(m), are seen in Fig. 1.

Fig. 1 Dust-production efficiencies of massive stars. The dashed line represents the “maximum” SN efficiency ϵmax(m) and the dashed dotted line the “high” SN efficiency ϵhigh(m). The solid line is the “low” SN efficiency ϵlow(m). The left dotted line represents the metallicity-averaged AGB efficiency ϵAGB(m). The vertical line marks the least massive (3 M⊙) AGB star considered.

3.5. Dust destruction in the ISM

Once a dust grain is injected into the interstellar medium, it is subject to either growth or to disruptive or destructive processes. We here focus on the destructive and disruptive ones due to supernova shocks.

Disruptive processes are those that lead to fragmentation of large dust grains (radius  >  1000   Å) into smaller dust grains (radius  <  500   Å). Jones et al. (1996) found that shattering due to grain-grain collisions dominates vaporization and therefore also determines the grain size redistribution, which is shifted towards smaller grains.

Destructive processes return dust grains back into the gas phase. The main destruction of dust grains in the ISM comes from the sputtering caused by interstellar shock waves with shock velocities  ≥ 100 km s^-1 (Seab 1987; Jones et al. 1996). The destruction takes place because of high-velocity gas-grain impacts of smaller projectiles (radius  < 100   Å), such as energetic He⁺ ions, onto dust grains. This results in the removal of dust species at or near the surface of the grains.

A dust grain is exposed to thermal and nonthermal sputtering, as well as to vaporization during the passage of the shock. Jones et al. (1996) also found that graphite grains are mainly destroyed by thermal sputtering. Silicates are equally affected by thermal and non-thermal sputtering in high-velocity shocks with velocities v_s > 150 km s^-1, while vaporization is negligible. These processes determine the lifetime of dust grains in the ISM. The timescale τ_dl(t) of the dust grains against destruction, or simply the lifetime of the dust grains, and the dust destruction rate in the ISM E_D,ISM(t) are given (McKee 1989; Dwek 1998; Dwek et al. 2007) through (22)where M_ISM(t) is the mass of the ISM and R_SN(t) the supernova rate of all supernovae causing the destruction. The mass M_cl is the mass of the ISM, which is completely cleared of dust through one single supernova remnant. These two equations combined lead to an expression for E_D,ISM(t) in the form (23)which shows that besides R_SN(t) and M_cl, the dust destruction rate in the ISM also depends on the dust-to-gas ratio η_d(t).

An expression for M_cl which is dependent on the shock velocity is given by Dwek et al. (2007). For a homogenous ISM and under the assumption that silicon and carbon grains are equally mixed, Dwek et al. (2007) obtains M_cl = 1100–1300 M_⊙. However, the ISM is inhomogeneous, characterized by cold, warm, and hot phases with different densities (e.g., McKee 1989, and references therein). The density contrast between the cold and warm phases and the hot phase can be relatively large. Shocks traveling through these phases are found to be very inefficient in destroying dust (Jones 2004). The destruction process is solely effective in the warm (T > 100 K) phase of the interstellar medium, while SN shocks propagating through either a hot ISM with low density or cold clouds (atomic and molecular) do not destroy dust effectively (e.g., McKee 1989).

Another important parameter is the injection timescale of stellar yields and dust into the ISM. Following McKee (1989), we estimate the injection timescale of the dust from stellar sources as

(24)In GAH11 we made a rough estimate of the minimum averaged dust injection rate from SNe and AGBs of 0.5 M_⊙ yr^-1 based on data for high-z QSOs. Using this dust injection rate we obtained an average injection time τ_in(t) = 4 × 10⁸ yr.

For comparison we estimate the lifetime τ_dl(t) of the dust grains. The mass of the ISM is assumed to be M_ISM = 2 × 10¹⁰ M_⊙. The SN rates are calculated for a constant SFR of 500 M_⊙ yr^-1 and for a Larson 2 IMF. This results in a timescale of τ_dl(t)  ~  1.3  ×  10⁷ yr for M_cl = 100 M_⊙. A shorter timescale τ_dl(t) ~ 1.6 × 10⁶ yr is obtained for M_cl = 800   M_⊙. When using a Salpeter IMF for the SN rates, the timescales are usually longer (τ_dl(t) ~ 6.6 × 10⁷ yr, for M_cl = 100 M_⊙).

We note that this is a rather rough estimate. Typically τ_in(t) and τ_dl(t) are strongly dependent on the IMF, M_ISM(t) and the SNe dust production efficiency ϵ_i(m,Z). Hence, the values will deviate from the above estimated average during evolution.

However, this example demonstrates that the dust injection timescale can be longer than the lifetimes of the dust grains. The difference between these timescales is influenced by the value of M_cl. Pertaining to the formation of high dust masses in galaxies, this has significant consequences. An injection timescale τ_in(t) longer than the destruction timescale τ_dl(t) does not allow the build up of high dust masses in the galaxy. This implies that the dust injection rate must be higher than the dust destruction rate. A lowering of the dust destruction rate could be achieved if dust grains are shielded from destruction. Alternatively, a rapid dust grain growth in the ISM might be an option, if SNe and AGBs cannot generate the necessary high dust injection rate. Grain growth, however, is not incorporated into our model and remains to be investigated.

Furthermore, we assume that the starburst occurs in an initially dust free galaxy. Consequently dust produced by the first generations of SNe might be ejected into the ISM unhindered. The epoch at which the first SN shocks are able to sweep up ISM gas mixed with dust remains elusive. In view of these considerations, we stress that dust destruction in the ISM is uncertain, particularly when considering galaxies with conditions as described in Sects. 3.1 and 3.2.

Despite our simple assumption of dust being immediately homogeneously mixed with the gas in the ISM, we account for the uncertainty of the lifetime of dust grains against destruction by using M_cl as a parameter. This has also been done by Dwek et al. (2007). The considered cases are for M_cl = 800   M_⊙ as the highest destruction, M_cl = 100   M_⊙ for modest destruction, and M_cl = 0 for no dust destruction.

4. Results

In this section we present the results of several models calculated within the first Gyr after starburst. We focus on the total dust mass in a galaxy. At redshift z > 6, the time to build up high dust masses of  > 10⁸ M_⊙ is limited to 400–500 Myr. Models able to exceed an amount of 10⁸ M_⊙ of dust within this time are therefore of particular interest. General evolutionary tendencies of certain quantities, such as dust injection rates, SFR, metallicity, or the amount of gas, are also discussed. In this paper we present results that assume the same value for the initial SFR in all models as given in Table 2. In Gall et al. (2011), we investigated models for different initial SFRs and discuss their applicability to particular quasars at z ≳ 6.

4.1. Evolution of dust, dust-to-gas, and dust-to-metal ratios

We study models with all considered initial gas masses M_ini. The models discussed are calculated for the case that no SN occurs when a BH is formed (see Sect. 2.2.1). Models including SMBH formation are deferred to Sect. 4.5.

4.1.1. Models with SN type differentiation

Fig. 2 Evolution of the total dust mass and dust destruction rates for EIT08M. The initial gas mass of the galaxy Mini = 5 × 1011 M⊙. Calculations are performed for a “maximum” (top row), a “high” (middle row), and a “low” (bottom row) SN dust production efficiency ϵi(m). Dust destruction is taken into account for Mcl = 800   M⊙ (left column), 100 M⊙ (middle column), and 0 M⊙ (right column). The bottom group of curves in the light blue area represents the total dust destruction rate ED(t), which is based on Eq. (16). The upper group of curves in the green or yellow zones displays the evolution of the total amount of dust Md(t) in the galaxy. The yellow area marks dust masses exceeding 108 M⊙ of dust. The grey horizontal dashed line marks the limit of 108 M⊙ of dust. The grey vertical dashed lines indicate epochs at 100, 170, 400, and 500 Myr after the onset of starburst. The black solid, green dotted, cyan dashed, magenta dashed-dotted, and blue dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1 and Larson 2 IMF, respectively.

Fig. 3 Evolution of the dust-to-gas and dust-to-metal mass ratios for EIT08M. The initial gas mass of the galaxy Mini = 5 × 1011 M⊙. Calculations are performed for a “maximum” (top row), a “high” (middle row), and a “low” (bottom row) SN dust production efficiency ϵi(m). Dust destruction is taken into account for Mcl = 800   M⊙ (left column), 100 M⊙ (middle column), and 0 M⊙ (right column). The upper group of curves signifies the dust-to-metal mass ratio ηZd(t). The lower group of curves represents the dust-to-gas mass ratio ηd(t). The gas-to-dust ratio displayed in the figures is calculated for a Larson 2 IMF at an epoch of 400 Myr. The grey vertical dashed lines indicate epochs at 100, 400, and 500 Myr after the onset of starburst. The black solid, green dotted, cyan dashed, magenta dashed-dotted, and blue dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1 and Larson 2 IMF, respectively.

Fig. 4 Evolution of the total dust mass for EIT08M. Calculations for galaxies with initial gas masses of Mini = 5 × 1010 M⊙ are presented in the first and second columns and for Mini = 1 × 1011 M⊙ in the third and fourth columns. Total dust masses Md(t) are shown for a “maximum” (top row), a “high” (middle row), and a “low” (bottom row) SN dust production efficiency ϵI(m). Dust destruction is taken into account for Mcl = 100   M⊙ (first and third columns) and 0 M⊙ (second and fourth columns). Maximum dust masses exceeding 108 M⊙ are indicated as yellow shaded zones. The grey horizontal dashed line marks the limit of 108 M⊙ of dust. The grey vertical dashed lines indicate epochs at 400, and 500 Myr after the onset of starburst. The black solid, green dotted, and blue dashed-dot-dotted lines represent the Salpeter, mass-heavy and Larson 2 IMF, respectively.

Fig. 5 Evolution of the dust-to-gas and dust-to-metal mass ratios for EIT08M. Calculations for galaxies with initial gas masses of Mini = 5 × 1010 M⊙ are presented in the first and second columns and for Mini = 1 × 1011 M⊙ in the third and fourth columns. Results are presented for a “maximum” (top row), a “high” (middle row), and a “low” (bottom row) SN dust production efficiency ϵi(m). Dust destruction is taken into account for Mcl = 100   M⊙ (first and third columns) and 0 M⊙ (second and fourth columns). The upper group of thick curves signifies the dust-to-metal mass ratio ηZd(t). The lower group of thin curves represents the dust-to-gas mass ratio ηd(t). The gas-to-dust ratio displayed in the figures is calculated for a Larson 2 IMF at an epoch of 400 Myr. The grey vertical dashed lines indicate epochs at 400, and 500 Myr after the onset of starburst. The black solid, green dotted, and blue dashed-dot-dotted lines represent the Salpeter, mass-heavy and Larson 2 IMF, respectively.

EIT08M with M_ini = 5 × 10¹¹ M_⊙:

in Fig. 2 the results are presented for the mass of dust M_d(t) in a galaxy with M_ini = 5 × 10¹¹ M_⊙. Calculations were performed for a “maximum” (top row), a “high” (middle row), and a “low” (bottom row) SN dust production efficiency ϵ_i(m). Dust destruction in the ISM is included for three values of M_cl (see Sect. 3.5). This model is considered as a reference model. We therefore describe it in detail.

Dust masses M_d(t) obtained with the “maximum” SN dust production efficiency ϵ_max(m) exceed 10⁸ M_⊙ of dust in all cases of M_cl and for all IMFs. In fact, 10⁸ M_⊙ of M_d(t) is reached within the first few Myr. This is insensitive to both the IMF and the amount of M_cl. The latter however determines the suppression of M_d(t) as the system evolves. For M_cl = 800 M_⊙ (top left), a maximum amount of  ~4 × 10⁸ M_⊙ is reached for a Larson 2 IMF and sustained until the end of the computation. Only the IMFs favouring lower masses exhibit a shallow decline in M_d(t). Dust destruction with M_cl = 100   M_⊙ (top middle) leads to nearly constant dust masses retainable over 1 Gyr of evolution. The amount of dust for most IMFs is a few times 10⁹ M_⊙. When assuming no dust destruction (top right), a broader spread of the dust masses for the different IMFs develops. The amount of dust increases with time regardless of the IMF and yields up to 10^9 − 10 M_⊙ are reached.

In the case of “high” SN dust production efficiency, ϵ_high(m), similar trends are featured. However the amount of dust is lower. Without dust destruction (M_cl = 0), dust masses up to a few times 10⁸ M_⊙ are attained for all IMFs (Fig. 2 middle right). The timescale to exceed 10⁸ M_⊙ of dust ranges from 80 Myr (Larson 2 IMF) up to 400 Myr (Salpeter IMF). Taking dust destruction with only a modest amount of M_cl = 100 M_⊙ into account decreases M_d(t) substantially (Fig. 2 middle middle). The IMFs favouring high masses are most affected, while the reduction in M_d(t) for a Salpeter IMF is small. Except for the Salpeter IMF, all remaining IMFs lead to more than 10⁸ M_⊙ of dust. The highest amount of dust is reached with either a Larson 2 or a mass-heavy IMF, and is  ~2 × 10⁸ M_⊙ at an epoch of  ~400 Myr. Considering a very high destruction with M_cl = 800   M_⊙ results in a strong reduction of M_d(t), so only  ~2–3 × 10⁷ M_⊙ of dust remains throughout the evolution (Fig. 2 middle left).

The results for M_d(t) with a “low” SN dust production efficiency ϵ_low(m) are dominated by AGB dust production (see also GAH11). Without dust destruction, dust masses up to 1–2 × 10⁸ M_⊙ is reached after  ~300–500 Myr for most of the IMFs. Applying a modest dust destruction of M_cl = 100   M_⊙ reduces M_d(t) analogously to the higher SN efficiencies ϵ_max(m) or ϵ_high(m). Dust masses for either the top-heavy or the Larson 2 IMF are efficiently decreased and comparable to M_d(t) for a Salpeter IMF. Increasing M_cl to 800 M_⊙ leads to a stronger reduction in M_d(t) for these IMFs, resulting in lower dust masses than for a Salpeter IMF.

A considerable difference in the progression of M_d(t) for the “low” SN efficiency ϵ_low(m) compared to the higher SN efficiencies is encountered during the first  ~200 Myr. While a fast rise of M_d(t) is identifiable for either ϵ_max(m) or ϵ_high(m), for ϵ_low(m) the dust mass remains between 10^6 − 7 M_⊙ during this epoch. This is caused primarily by high-mass AGB stars with short lifetimes ( > 4–5 M_⊙). A further increase in the dust mass due to the delayed AGB dust injection from the less massive but more efficient AGB stars results in a second dust bump at an epoch of  ~200–300 Myr. High dust masses at early epochs (100–200 Myr) are not possible with ϵ_low(m).

The curves at the bottom of the light-blue area in Fig. 2 represent the dust destruction rate E_D(t). For M_cl = 0 and independent of ϵ_i(m) the dust destruction rate reflects the amount of dust incorporated into stars per unit time and is calculated as E_D(t) = η_d(t)   ψ(t). For M_cl > 0 E_D(t) is given by Eq. (16) and additionally consists of the destroyed dust in the ISM, E_D,ISM(t).

We find that E_D(t) increases faster and earlier with increasing M_cl. Consequently, dust destruction rates comparable to SN injection rates are reached at earlier epochs. Given Eq. (23) for the amount of dust destroyed through SN shocks, the high SN rates R_SN(t) at the beginning of evolution in conjunction with M_cl lead to a high base value of E_D,ISM(t). This suppresses the rise in the amount of dust at early epochs. Additionally, SN rates are higher for the IMFs biased towards more massive stars affecting these IMFs most. Later in the evolution, the dust destruction rates remain almost constant in most cases, with a marginal increase for M_cl = 0 or decline for M_cl > 0. Dust destruction and injection appear to be balanced, resulting in the flat development of M_d(t).

In Fig. 3 we present the results for the evolution of the dust-to-gas mass ratio η_d(t) and the dust-to-metal mass ratio η_Zd(t). Commonly the curves for η_d(t) for all IMFs are slowly increasing with time except for M_cl = 800   M_⊙ for which they remain approximately constant. Interestingly, in this case and for ϵ_max(m), the dust-to-gas ratio for all IMFs sustains a constant value of  ~10^-3 over the whole evolution. Without dust destruction (top right), η_d(t) increases up to values above  ~10^-2 for most of the IMFs. For either a “high” or “low” SN efficiency, the values are below  ~10^-3. Only for ϵ_high(m) and M_cl = 0 (middle right) is η_d(t) further increased. These overall trends show that η_d(t) can be very low for massive galaxies even if the galaxy appears dusty.

The dust-to-metal mass ratio η_Zd(t) is significantly lower for the Larson 2 and top-heavy IMF than for the other IMFs. This feature is exhibited in all calculated models. The degree of this separation between different IMFs depends primarily on the SN dust production efficiency and secondly on the destruction rate in the ISM, and is larger for ϵ_low(m). This reflects the decrease in the SN dust production efficiency towards the higher mass end of the SN mass interval. Additionally, type Ib/c SNe produce no dust, but they do inject metals into the ISM at higher rates for the top-heavy IMFs. For M_cl = 0 and ϵ_high(m), the amount of metals bound in dust grains is between  ~1.5–3%. For M_cl = 100 M_⊙ the fraction of metals bound in dust grains for the two IMFs favouring high masses is below 1% and decreases further with increasing M_cl. Generally, for a “low” SN efficiency, η_Zd(t) remains below 10^-2 for all IMFs. For the “maximum” SN efficiency ϵ_max(m) the dust-to-metal ratio, η_Zd(t), is about a factor of 10 higher than for ϵ_high(m).

EIT08M with M_ini = 1  ×  10¹¹M_⊙:

in Fig. 4 (two right columns) the results are presented for M_d(t) in cases of either no or a modest destruction and all three SN dust formation efficiencies ϵ_max(m), ϵ_high(m), ϵ_low(m).

The top row of Fig. 4 depicts the evolution of the amount of dust M_d(t) for ϵ_max(m). It is evident that high dust masses beyond 10⁸ M_⊙ are obtained for both cases of M_cl. The maximum value for M_d(t) is already reached shortly after the onset of the starburst. Thereafter M_d(t) follows a negative slope, which is steepest for the IMFs biased towards low-mass stars. This decline is generally observable in all cases of M_cl and ϵ_i(m). The cause of this is that dust is incorporated into stars on higher rates than is replenished by stellar sources. This dust decrease is amplified by dust destruction in the ISM as is demonstrated for M_cl = 100 M_⊙. For ϵ_high(m) and M_cl = 0, a high total dust mass  ≥ 10⁸ M_⊙ can be achieved with a Larson 2 IMF after  ~200 Myr (Fig. 4 middle row, right column). For all remaining cases of either M_cl or for ϵ_low(m) and regardless of the IMF, M_d(t) stays below 10⁸ M_⊙. The tendencies for the various IMFs resemble those identified for M_ini = 5 × 10¹¹ M_⊙.

The curves of the dust-to-gas ratio η_d(t) and the dust-to-metal ratio η_ZD(t) are also similar to the system with M_ini = 5 × 10¹¹ M_⊙, although the amount is higher (see Fig. 5 two left columns). For M_cl = 0 and “maximum” SN efficiency, η_d(t) is roughly independent of the IMF, while exhibiting a wider range for ϵ_low(m). The fraction of metals bound in dust grains is  ≲ 4% (M_cl = 0) for IMFs favouring low-mass stars and ϵ_high(m); for ϵ_max(m) it is roughly 20–30%.

EIT08M with M_ini = 5  ×  10¹⁰M_⊙:

The evolution of M_d(t) within the first Gyr exhibits similar, but more strongly pronounced, tendencies for the various IMFs as the galaxy system with M_ini = 1 × 10¹¹ M_⊙. Results are presented in Fig. 4 (two left columns). Even with a “maximum” SN efficiency ϵ_max(m), large dust masses of a few times 10⁸ M_⊙ cannot be reached or sustained when dust destruction in the ISM is considered. In Fig. 5 (two left columns) it is evident that the dust-to-gas ratios η_d(t) and the dust-to-metal ratios η_ZD(t) result in higher values than for the higher mass galaxy systems (see Figs. 3, 5 two right columns). The tendencies are the same in general.

G09M:

Fig. 6 Evolution of the total dust mass for G09M, presented for a galaxy with Mini = 5 × 1011 M⊙. Total dust masses Md(t) are shown for a “high” (top row) and a “low” (bottom row) SN dust production efficiency ϵi(m). Dust destruction is taken into account for Mcl = 800   M⊙ (left column) and 0 M⊙ (right column). Maximum dust masses exceeding 108 M⊙ are indicated as yellow shaded zones. The grey horizontal dashed line marks the limit of 108 M⊙ of dust. The grey vertical dashed lines indicate epochs at 400 and 500 Myr after the onset of starburst. The black solid, green dotted, cyan dashed, magenta dashed-dotted, and blue dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1, and Larson 2 IMF, respectively.

in Fig. 6 the results are shown for models with a rotationally enhanced mass-loss prescription. The galaxy under consideration has M_ini = 5 × 10¹¹ M_⊙. The outcome is similar to the reference EIT08M (Fig. 2), although the obtained dust yields are increased for the IMFs favouring higher masses. For a Larson 2 IMF, M_d(t) exceeds 10⁹ M_⊙ after  ~650 Myr. Interestingly, even though the dust yields are high for no dust destruction in the ISM, for M_cl = 800   M_⊙ the dust mass also stays below 10⁸ M_⊙. In case of a “low” SN dust production efficiency, 10⁸ M_⊙ cannot be reached within the first 400 Myr. However, for the IMFs biased towards high masses an amount of dust  > 10⁸ M_⊙ is attained later in the evolution.

4.1.2. Models with fixed SNe mass range

Fig. 7 Evolution of the total dust mass for WW95M (top row) and N06M (bottom row). Both models are presented for a galaxy with Mini = 5 × 1011 M⊙. Total dust masses Md(t) are shown for “high” SN dust production efficiency ϵhigh(m). Dust destruction is taken into account for Mcl = 100   M⊙ (left column) and 0 M⊙ (right column). Maximum dust masses exceeding 108 M⊙ are indicated as yellow shaded zones. The grey horizontal dashed line marks the limit of 108 M⊙ of dust. The grey vertical dashed lines indicate epochs at 400 and 500 Myr after the onset of starburst. The black solid, green dotted, cyan dashed, magenta dashed-dotted, and blue dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1, and Larson 2 IMF, respectively.

In Fig. 7 the results of M_d(t) are shown for WW95M (top row) and N06M (bottom row) for a galaxy with M_ini = 5 × 10¹¹ M_⊙ and “high” SN efficiency ϵ_high(m). The amount of dust in N06M early in the evolution is greater than for WW95M for the IMFs favouring massive stars, but flattens later, leading to slightly lower dust masses than achieved with WW95M. The dust masses achieved with either the “maximum” or the “low” SN dust production efficiencies are nearly identical to EIT08M and are therefore not shown. Generally, the evolution of these models is similar to EIT08M.

4.1.3. Very high M_ini and the case of constant SFR

Fig. 8 Evolution of the total dust mass for WW95M with constant SFR. WW95M is presented for a galaxy with Mini = 1.3 × 1012 M⊙. Total dust masses Md(t) are shown for “high” (top row) and a “low” (bottom row) SN dust production efficiency ϵi(m). Dust destruction is taken into account for Mcl = 800   M⊙ (left column) and 0 M⊙ (right column). Maximum dust masses exceeding 108 M⊙ are indicated as yellow shaded zones. The grey horizontal dashed line marks the limit of 108 M⊙ of dust. The grey vertical dashed lines indicate epochs at 400 and 500 Myr after the onset of starburst. The black solid, green dotted, cyan dashed, magenta dashed-dotted, and blue dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1, and Larson 2 IMF, respectively.

In Fig. 8 the results are displayed for a case with constant SFR ψ(t) = 1 × 10³   M_⊙ yr^-1 and a galaxy with a high initial gas mass of M_ini = 1.3 × 10¹² M_⊙. Such a high M_ini is necessary when assuming a constant SFR, since in the lower mass galaxies the available mass for star formation gets exhausted before an age of 1 Gyr is reached.

We find that a build up of high dust masses also for such massive galaxies is greatly suppressed with a high amount of destruction. With none of the IMFs, 10⁸ M_⊙ of dust is attained, apart from the models with ϵ_max. High dust masses with either ϵ_high or ϵ_low are only possible in case of modest or no dust destruction. For the latter the results are shown in the right column of Fig. 8. Interestingly, also for “low” SN dust production efficiency, dust masses  > 10⁸ M_⊙ are possible after  ~300–400 Myr.

The results are very similar to an EIT08M with such high M_ini and an evolving SFR. A slight difference appears for M_d(t) and IMFs that favour low-mass stars. These models remain close to constant after 500–600 Myr, while the models for constant SFR slowly decline. The similarity between these models can be explained by the very high mass of the galaxy. In neither model does the ISM mass get significantly reduced.

4.2. Evolution of dust production rates, SFR and metallicity

In Fig. 9 we present the evolution of quantities such as the total dust injection rates E_d,SN(t), E_d,AGB(t), the AGB and SNe rates R_AGB(t), R_SN(t), the SFR and the metallicity Z(t) for a range of initial galaxy gas masses. The rates and the metallicities are independent of the assumed dust formation efficiency and can be discussed for each initial mass M_ini. The results for these quantities, which come from EIT08M, WW95M, and N06M with identical parameter setting, are very similar. Consequently these quantities are discussed based on EIT08M. The upper row of Fig. 9 shows the dust injection rates of AGB stars E_d,AGB(t) and SNe E_d,SN(t). For supernovae E_d,SN(t) is highest for a Larson 2 IMF and lowest for a Salpeter IMF.

The SN dust production rates are approximately one order of magnitude lower for ϵ_low(m) than for ϵ_high(m). Similar difference is the case between ϵ_high(m) and ϵ_max(m). In all cases a clear separation of the values of E_d,SN(t) amongst the various IMFs is pronounced throughout the evolution.

The dust injection rates for AGB stars, E_d,AGB(t), are considerably influenced by the long lifetimes. After  ~200 Myr the AGB dust production rates M_d,AGB(t) are comparable to the dust injection rates for SNe with ϵ_high(m). This is caused by the higher dust production efficiency for AGB stars between  ~3–4 M_⊙, leading to larger amounts of dust produced than for AGB stars in the mass range of 4–8 M_⊙ (see Sect. 3.4). The small variations in E_d,AGB(t) are mainly due to alteration of the stellar yields and dust formation efficiencies at different metallicities.

The SNe and AGB dust injection rates decline as a consequence of decreasing SNe and AGB rates, as shown in Fig. 9 (second row). Both rates exhibit a faster decline for the lower massive galaxies and for IMFs biased towards the intermediate and low mass stars.

This behaviour is determined by the slope of the SFR, which depends on the initial mass of the galaxy (see Fig. 9 third row). The decline in the SFR for systems with M_ini ≥ 3 × 10¹¹ M_⊙ is shallower. For these galaxies, a high SFR of a few hundred M_⊙ yr^-1 can be sustained over at least 1 Gyr of evolution. For the lower mass galaxies, the SFR declines faster and exhibits a strong dependence on the IMFs. At a time of 400 Myr the SFR for a galaxy with M_ini = 1 × 10¹¹   M_⊙ is between 60 (Salpeter) to 500 (Larson 2) M_⊙ yr^-1. When M_ini = 5 × 10¹⁰ M_⊙, ψ(t) = 20–400 M_⊙ yr^-1. After  ~1 Gyr the difference between the SFRs obtained with a Larson 2 IMF and a Salpeter IMF is more than an order of magnitude.

In Fig. 9 (bottom row) it is seen that the metallicity Z(t) in the two lower mass galaxies rises quickly within the first 100–200 Myr up to values of more than 5 Z_⊙ for either a top-heavy or a Larson 2 IMF. Thereafter it remains rather constant. For the system with M_ini = 5 × 10¹¹ M_⊙, the metallicity increases more slowly than in the lower mass galaxies. At an epoch of 400 Myr, a metallicity of 2–3   Z_⊙ can be observed for the top-heavy and Larson 2 IMFs, while the Salpeter IMF achieves only a bit less than half solar. In the most massive galaxies, the metallicity exceeds Z_⊙ only in case of the top-heavy IMFs before an age of 400–500 Myr, but remains below Z_⊙ for the IMFs favouring lower mass stars.

4.3. Evolution of gas, metals and stellar masses

The evolution of quantities, such as the gas mass, mass of metals, and stellar masses is discussed based on EIT08M. Figure 10 illustrates the evolution of the H + He gas mass M_G(t). As a consequence of the scaling of the SFR with the mass of the ISM, M_ISM(t) ≡ M_G(t)  +  M_Z(t), the progression of M_G(t) is identical to the SFR described in Sect. 4.2.

The most massive galaxy with M_ini = 1.3  ×  10¹² M_⊙ has a residual gas mass M_G(t) of  ~10¹² M_⊙ after 1 Gyr. This explains the very low dust-to-gas ratio for such a system even though very high dust masses can be reached. The galaxy with M_ini = 5 × 10¹¹ M_⊙ gets more exhausted, but also retains a gas mass M_G(t) of 3–4 × 10¹¹ M_⊙ after  ~400–500 Myr.

In analogy with the curves of the SFR, the curves of the gas mass for the system with M_ini = 1 × 10¹¹ M_⊙ feature a pronounced separation between the IMFs. A Larson 2 IMF shows a flat evolution leading to a slower depletion of the gas than for the Salpeter IMF and after 400–500 Myr  ~  5 × 10¹⁰ M_⊙ of gas is still present. The lowest mass system (M_ini = 5 × 10¹⁰ M_⊙) exhibits the strongest dependence on the IMF. After 400 Myr, M_G(t) is  ~2 × 10¹⁰ M_⊙ for a Larson 2 IMF. The difference between a Salpeter IMF and a Larson 2 IMF after one Gyr is an order of magnitude. The residual gas mass with a Salpeter IMF is a few times 10⁸ M_⊙ at this time.

Fig. 9 Evolution of total dust injection rates for AGB stars and SNe, the AGB and SNe rates, the SFR, and the metallicity. Results are shown for Mini = 1.3 × 1012 M⊙ (first column), 5 × 1011 M⊙ (second column), 1 × 1011 M⊙ (third column), and 5 × 1010 M⊙ (fourth column). The solid, dotted, dashed, dashed-dotted, and dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1, and Larson 2 IMF, respectively. First row: total SN dust injection rates Ed,SN(t) for a “low” (black lines), “high” (blue lines), and “maximum” (green lines) SN efficiency ϵi(m), and AGB dust injection rates Ed,AGB(t) (red lines). SN dust injection rates are only shown for a Salpeter (solid) and a Larson 2 (dashed-dot-dotted) IMF. Second row: SNe rates RSN(t) (black lines) and AGB rates RAGB(t) (red lines). SN rates are only shown for a Salpeter (solid), mass-heavy (dotted), and a Larson 2 (dashed-dot-dotted) IMF. Third row: evolution of the SFR. The blue area marks the region between a SFR of 100–1000 M⊙ yr-1. Fourth row: evolution of the metallicity Z(t). The green regions mark a metallicity between 1–2 Z⊙ (light green), 2–3 Z⊙ (dark green), 3–4 Z⊙ (grass green). The grey vertical dashed lines indicate epochs at 400 and 500 Myr after the onset of starburst.

The steepness of the decline in the progression of M_G(t) is influenced by the stellar feedback. For either a Larson 2 or a top-heavy IMF, more massive stars are formed. Such stars live short lives and release most of their mass back into the ISM, through either stellar winds or in explosive events. The IMFs favouring lower masses, however, lock most of the gas used for star formation into intermediate-to-low-mass stars. These stars are formed at higher rates than massive stars. The low-mass stars live a long time and also do not inject much material back into the ISM during our considered time span of 1 Gyr. As a consequence, the material available for star formation gets more rapidly depleted for the IMFs favouring low-mass stars than for the top-heavy IMFs. Hence, the evolution of the gas mass and the SFR results in a steeper decline.

This contrasts with the stellar masses M _∗ (t) (Fig. 11 top row) obtained from the relation M _∗ (t) = M_ini − M_ISM(t). For the IMFs favouring lower mass stars, M _∗ (t) rises steeply and approaches the initial mass of this galaxy in the case of the lower massive galaxy systems. The slope of M _∗ (t) for the top-heavy IMFs is shallower, and lower stellar masses M _∗ (t) are achieved.

Figure 11 (bottom row) depicts the mass of the ejected heavy elements M_Z(t). For systems with M_ini > 1 × 10¹¹ M_⊙, the metal enrichment in the ISM is considerable. The mass of the heavy elements increases up to a few times 10¹⁰ M_⊙.

In the lower mass galaxies M_ini ≤ 1 × 10¹¹ M_⊙, the amount of metals M_Z(t) attained reaches a maximum within the first 200 Myr whereafter M_Z(t) declines. This is caused by astration. The maximum mass of metals obtained with a Salpeter IMF is only a few times 10⁸ M_⊙. For the lowest mass galaxy the amount of metals after  ~600 Myr for a Salpeter IMF is less than 10⁸ M_⊙. This implies that dust masses M_d(t) in excess of 10⁸ M_⊙ for this IMF are unfeasible, even if dust-grain growth in the ISM is invoked.

4.4. Comparison to other dust evolution models for high-z galaxies

The models with a “maximum” SN dust production efficiency for a system with M_ini of 5 × 10¹⁰ M_⊙ (Fig. 4 two left columns, first row) can be compared to models of Dwek et al. (2007). For this efficiency the AGB dust production is negligible and the SN dust yields are similar to the dust yields used by Dwek et al. (2007). Our results for a top-heavy IMF and M_cl = 0 at an epoch of 400 Myr agrees with their results, while they disagree for a Salpeter IMF and for cases with M_cl = 100 M_⊙.

The origin of this disagreement can be traced to the neglect of the lifetime-dependent mass injection of stars and a different treatment of the mass recycled into the ISM in the models of Dwek et al. (2007). The latter is approximated as 0.5ψ(t) in their models, which leads to an IMF-independent evolution of the ISM mass. In the case of a Salpeter IMF, this simple approximation has consequences for the evolution of all physical properties. It implicitly presupposes that even stars, which are formed between the lower mass limit of the IMF and the lower SN mass limit, immediately return half of their stellar mass back into the ISM. However, stars between 3–8   M_⊙ eject their elements up to a few 100 Myr delayed and stars  ≲ 3 M_⊙ do not eject a significant amount of elements within the first Gyr. For a Salpeter IMF, where more low-mass stars than high-mass stars are formed, we have shown that the mass of the ISM gets exhausted more rapidly than for the IMFs favouring higher masses (see Sect. 4.3). This results in a steeper decline in the SFR, the SN rates and the SN dust injection rates. Thus, our models lead to lower dust masses than the models by Dwek et al. (2007) for a Salpeter IMF.

The model for a system with M_ini = 1.3 × 10¹² M_⊙ and M_cl = 800   M_⊙ (Fig. 8 top left), a Larson 1 IMF and the “high” SN efficiency is directly comparable to Valiante et al. (2009). Our results are in good agreement with their results, although our model does not quite reach 10⁸ M_⊙. This discrepancy may stem from differences in the SN dust yields used. Additionally, the treatment of stars between 40–100 M_⊙ by Valiante et al. (2009) is not unambiguously traceable.

Fig. 10 Evolution of the gas mass MG(t) based on EIT08M. Results are shown for Mini = 1.3 × 1012 M⊙ (black lines), 5 × 1011 M⊙ (red lines), 1 × 1011 M⊙ (green lines), and 5 × 1010 M⊙ (magenta lines). The grey vertical dashed lines indicate epochs at 400 and 500 Myr after the onset of starburst. The solid, dotted, dashed, dashed-dotted, and dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1, and Larson 2 IMF, respectively.

4.5. Results for models including a SMBH

We have investigated whether including the formation of the SMBH leads to differences in the evolution of either the total amount of dust or the physical properties of a galaxy. Based on the observed SMBH masses  ≥ 10⁹ M_⊙ for high-redshift QSOs (e.g., Willott et al. 2003; Vestergaard 2004; Jiang et al. 2006; Wang et al. 2010), we studied two cases for the SMBH mass, M_SMBH = 3 × 10⁹ M_⊙ and M_SMBH = 5 × 10⁹ M_⊙. According to Kawakatu & Wada (2009), the final SMBH mass takes up approximately 1–10% of the supply mass M_SMBHsup. This implies that to grow a SMBH of 3(5)  ×  10⁹ M_⊙ the minimum required supply mass ranges between 3(5)  ×  10¹⁰ to 3(5)  ×  10¹¹ M_⊙. In view of our assumption that M_SMBHsup ≡ M_ini, we included the SMBH formation in galaxies with M_ini = 5 × 10¹⁰ M_⊙ and M_ini = 1 × 10¹¹ M_⊙.

In Fig. 12 we show the results of M_d(t) for a Salpeter and Larson 1 IMF for an EIT08M and M_ini = 5 × 10¹⁰ M_⊙ where the SMBH formation has been included. These results are compared to those of a model with the same M_ini, but without a SMBH.

Taking the SMBH into account leads to a steeper decline in M_d(t) and therefore to a lower amount of dust than for the model without a SMBH after  ~100 Myr. All models are similar within the first  ~100 Myr. For M_SMBH = 5 × 10⁹ M_⊙ the difference is at most  ~50% for a Salpeter IMF and  ~30% for a Larson 1 IMF at an epoch of 400 Myr. At the same epoch but for M_SMBH = 3  ×  10⁹ M_⊙, the dust mass for a Salpeter IMF is reduced by only about 30%, and 20% for the Larson 1 IMF. In cases of IMFs biased towards higher masses and in the more massive galaxy systems, the formation of the SMBH, as introduced here, does not noticeably effect the progression of M_d(t) and the physical properties of a galaxy.

5. Discussion

We have shown that, depending on the assumptions for certain model parameters, dust masses in excess of 10⁸ M_⊙ can be obtained in our model. All models leading to dust masses  ≥ 10⁸ M_⊙ within the first 400 Myr are listed in Table 4. In Fig. 13 we show the resulting relations between the dust mass and stellar mass, SFR, and metallicity at 400 Myr.

It is evident that the dust yields are related to the mass of the galaxy. For a given combination of IMF, SN dust production efficiency and M_cl higher dust yields are obtained with increasing M_ini. In the discussion below we consider ϵ_high with M_cl = 100   M_⊙ as the reference case.

The left panel in Fig. 13 depicts the dust mass versus the stellar mass. For a given IMF the stellar masses increase with M_ini. However for a given M_ini, a wide spread of the stellar mass is found with varying IMF. We find that for the reference case, galaxies with M_ini ≳ 5 × 10¹¹ M_⊙ and an IMF biased towards higher masses reproduce dust masses  >10⁸ M_⊙, while this is only possible in the most massive system with a Salpeter IMF. The stellar masses for these galaxies and with the more top-heavy IMFs are in good agreement with observations of galaxies at high-z. Santini et al. (2010) and Michałowski et al. (2010a) shows that, out of a sample of high-z submillimetre galaxies and local ULIRGs, most of these are assembled at stellar masses around 10¹¹ M_⊙ and have dust masses of 10^8 − 10 M_⊙.

Fig. 11 Evolution of the stellar mass and mass of metals based on EIT08M. Results are shown for Mini = 1.3 × 1012 M⊙ (first column), 5 × 1011 M⊙ (second column), 1 × 1011 M⊙ (third column), and 5 × 1010 M⊙ (fourth column). Top row: stellar mass M ∗ (t). Bottom row: mass of metals MZ(t). The red lines mark a metal mass of 108 M⊙. The solid, dotted, dashed, dashed-dotted, and dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1, and Larson 2 IMF, respectively.

Alternatively, a “maximum” SN dust production efficiency can account for the required dust mass. For this SN efficiency, even a higher dust destruction in the ISM of M_cl = 800   M_⊙ can be accomodated. However, assumptions of either ϵ_max or M_cl = 800 M_⊙ are both controversial. So far only in the SN remnants Cas A (Wilson & Batrla 2005; Dunne et al. 2009) and Kepler (Gomez et al. 2009) have dust masses been claimed that are consistent with “maximum” SN efficiency, ϵ_max. Most SN dust observations reveal dust masses implying efficiencies in the range of ϵ_low–ϵ_high (see GAH11). Additionally, theoretical models (Nozawa et al. 2010) predict that depending on the density and geometry of the CSM and shocks, only a part of the dust may survive the destructive reverse shock.

The uncertainty of dust destruction through SN shocks has been discussed in Sect. 3.5. The results obtained in this work strengthen the concerns of a high value for M_cl. Apart from the “maximum” SN efficiency, for a high value of M_cl = 800 M_⊙, only the combination of a “high” SN efficiency ϵ_high with M_ini = 1.3 × 10¹² M_⊙ leads to dust masses close to 10⁸ M_⊙, regardless of the IMF. However the remaining gas mass M_G(t) is  > 10¹² M_⊙, implying dust-to-gas mass ratios η_d(t) < 10^-4, is not supported by observations. The low dust-to-gass mass ratio is also, why dust destruction in the ISM is less efficient than in the lower mass galaxy systems.

In principle, the fairly sensitive interplay between dust-to-gas ratio η_d(t) and SN rates might be important when contemplating additional dust source in less massive galaxies. As long as SN rates are high, any increase in the dust mass, and hence in η_d(t), leads to higher destruction rates. This shows that for M_cl = 800   M_⊙ it is difficult to reach high amounts of dust, unless a rapid enrichment with high dust masses from either stellar sources or grain growth in the ISM takes place. In the latter case, grain growth rates must be comparable to SN dust injection rates for ϵ_max. Alternatively, a lowering of the dust-to-gas ratio η_d(t) due to, for example, infalling gas might be an option. However this alternative results in an increased total mass of the galaxy anyway.

Fig. 12 Comparison of the evolution of the dust mass between EIT08M including and without the SMBH formation. The initial gas mass of the galaxy Mini = 5 × 1010 M⊙. Calculations are shown for a “maximum” (top row) and a “high” (bottom row) SN dust production efficiency ϵi(m). Dust destruction is taken into account for Mcl = 100   M⊙ (left column) and 0 M⊙ (right column). Calculations including the SMBH are performed for a SMBH mass MSMBH = 5 × 109 M⊙ (solid curves) and for MSMBH = 3  ×  109 M⊙ (dotted curves). The calculations without the SMBH are illustrated as dot-dashed curves. The black lines represent the Salpeter IMF and the magenta lines the Larson 2 IMF. The grey horizontal dashed line marks the limit of 108 M⊙ of dust. The grey vertical dashed lines indicate epochs at 100, 400, and 500 Myr after the onset of starburst.

Given these uncertainties, we find that the reference case constitutes the most likely scenario. In Fig. 13 (middle panel) we have plotted the dust mass versus SFR. The highest SFRs can be sustained for the high-mass weighted IMFs in the more massive galaxies. For the less massive systems, also with top-heavy IMFs, the SFR declines rather fast. In comparison to observations of galaxies containing high dust masses at z > 6, we find that all systems more massive than 1  ×  10¹¹ M_⊙ can be brought into agreement with the SFRs derived from observations (e.g., Bertoldi et al. 2003a; Dwek et al. 2007; Wang et al. 2010). The SFRs inferred from the observed far-infrared luminosities are naturally uncertain and sensitively dependent on assumptions about the IMF, the duration of the starburst, and the contribution of the active galactic nucleus (AGN) (e.g., Omont et al. 2001). Commonly a Salpeter IMF is used, leading to high SFRs up to a few times 10³ M_⊙ yr^-1 (e.g., Walter et al. 2004; Wang et al. 2010). As pointed out by Dwek et al. (2007), the inferred SFR however decreases to about 400 M_⊙ yr^-1 if a top-heavy IMF is assumed. In Fig. 13 (middle panel) it is evident that galaxies between 1–5  ×  10¹¹ M_⊙, and top-heavy IMFs result in SFRs between about 300–700 M_⊙ yr^-1.

The right panel of Fig. 13 shows that the metallicity decreases with increasing dust mass. IMFs favouring low masses lead to lower metallicities than the more top-heavy IMFs. The IMFs favouring low masses lock most material used for star formation in low-mass stars. These stars, however, do not recycle material back into the ISM, which therefore gets rapidly exhausted. In turn this leads to a high stellar mass M_∗(t). The more top-heavy IMFs form more short lived massive stars at higher rates. These stars recycle a copious amount of their mass back into the ISM. Therefore these IMFs lead to lower stellar masses M_∗(t), since the remnant mass of SNe is low. In addition, massive stars enrich the ISM with metals, while owing to their longer lifetimes, low-mass stars do not eject a significant amount, if any at all, of heavy elements back into the ISM within 1 Gyr. This leads to lower metallicity for the lower mass star weighted IMFs, and higher metallicity for the top-heavy IMFs.

Metallicities  > Z_⊙ have been found in strong star-forming galaxies, such as ULIRGs or submillimetre galaxies, as well as in high-z QSOs (e.g., Fan et al. 2003; Freudling et al. 2003; Kawara et al. 2010). In comparison to our calculated models, such metallicities can only be reached with the IMFs biased towards higher masses. In the lowest mass systems with these IMFs, the metallicity reaches values of  ~5  ×  Z_⊙.

5.1. Caveats in our approach

The presented models are based on rather simple assumptions, such as a closed box environment, a common constant initial SFR of the starburst, and a very simple treatment of the SMBH growth. With these assumptions the models are comparable to similar works (e.g., Dwek et al. 2007; Valiante et al. 2009), and diverse evolutionary trends could be investigated in more detail, as discussed in previous sections.

However, such models usually do not capture possible impacts on the evolution of the galaxy arising from galaxy mergers or gas flows powered by, for example, SNe or the SMBH. Different evolutionary paths likely caused by these effects and the mass of the galaxy possibly result in starburst intensities different from our assumption, in turn leading to a different temporal progression of various quantities. Furthermore, the growth of the SMBH is linked to several physical processes, e.g., the energy feedback of SNe, and realistically may not take place at a constant growth rate. Despite the neglect of these effects in our model, some general remarks about their influence can be made based on our results.

For example, infall rates that could affect the systems might have to be unrealistically high because of the very high SFRs. However, merging galaxies might have an effect. The same applies to outflows in the host galaxy or the feedback from the SMBH into the ISM. In our models the loss of material to fuel the SMBH can in principle be interpreted as some constant “outflow” of the host galaxy. Although we introduced a rough treatment for the SMBH growth, we could show that “outflow” rates of the host galaxy of about 7.5–12.5 M_⊙ yr^-1 on a timescale of 400 Myr only affect the amount of dust in the least massive systems (and IMFs biased towards low-mass stars), but are negligible in the larger galaxies. In this regard, it remains to be investigated whether the energy deposition by SNe/AGNs in the ISM could be high enough to initiate sufficiently higher outflow rates to impact the total amount of dust in a galaxy.

Fig. 13 Relation between dust mass, stellar mass, SFR, and metallicity in correlation with the total mass of the galaxy and the IMF for an epoch at 400 Myr. Left panel: dust mass Md(t) versus stellar mass M∗(t). Middle panel: dust mass Md(t) versus SFR ψ(t). Right panel: dust mass Md(t) versus metallicity Z(t). The dark grey shaded area marks the dust masses Md(t) obtained with a “maximum” SN dust production efficiency ϵmax(m) and no dust destruction in the ISM (Mcl = 0 M⊙). The light grey shaded area marks dust masses Md(t) for the “low” SN dust production efficiency ϵlow(m) with dust destruction in the ISM (Mcl = 800 M⊙). Cross points of the thin white lines indicate the data points for each IMF and mass Mini of the galaxy. The coloured filled circles signify the dust masses obtained for a “high” SN efficiency ϵhigh(m) and a dust destruction in the ISM with Mcl = 100 M⊙. The size of the circles is scaled by the initial mass Mini of the galaxy. The black, green, cyan, magenta, and blue colors denote the Salpeter, mass-heavy, top-heavy, Larson 1, and Larson 2 IMF, respectively.

A change in the evolution of the dust mass and other properties may also result from quasar winds, in which dust formation has also been suggested (Elvis et al. 2002). According to Elvis et al. (2002), the estimated mass-loss rates of about  > 10 M_⊙ yr^-1 in the most luminous quasars with luminosities  > 10⁴⁷ erg s^-1 (e.g., Omont et al. 2001, 2003; Carilli et al. 2001; Bertoldi & Cox 2002) imply an amount of  ~10⁷ M_⊙ of dust produced over 10⁸ yr. Comparing these rates to our model results, we find that neither the mass-loss rates nor the dust mass seem to be high enough to be relevant for the evolution of dust and other properties.

We find that a strong impact on the evolution of dust is caused by the dust sources. Despite the included detailed treatment of the dust contribution from stellar sources in the developed model, the poorly understood dust production by SNe, but also dust destruction by SN shock interactions, very likely constitute the largest uncertainties in the evolution of dust. Although not included in the model, alternative dust sources such as dust grain growth in the ISM might be relevant (e.g., Dwek et al. 2007; Draine 2009; Michałowski et al. 2010b).

6. Conclusions

In this work we have developed a chemical evolution model for starburst galaxies at high redshift. The main purpose is to investigate the evolution of the dust content arising from SNe and AGB stars on timescales less than 1 Gyr. In addition, we elaborated on the evolution of several physical properties of galaxies. The model allows the exploration of a wide range of parameters. The main parameters varied in this work are the mass of the galaxy, the IMF, SN dust production efficiencies, and dust destruction in the ISM through SN shocks and stellar yields. The main results can be summarized as follows.

• 1.

  The total amount of dust and the physical properties of a galaxyare strongly dependent on the IMF and correlate with the mass ofthe galaxy. For many properties we find an increasing disparitybetween the IMFs with decreasing mass of the galaxy. Higherdust masses are obtained with increasing mass of the galaxy.

• 2.

  The maximum dust masses can be obtained with IMFs biased towards higher mass stars and a “maximum” SN dust production efficiency in galaxies with masses between 5  ×  10¹⁰ M_⊙ and 1.3  ×  10¹² M_⊙. This is found independently of the strength of dust destruction in the ISM. The case of “maximum” SN efficiency and no destruction constitute the maximum possible dust masses attainable with SNe and AGB dust production. The case with “low” SN efficiency and a high dust destruction (M_cl = 800 M_⊙) gives the lowest possible dust masses.

• 3.

  The contribution by AGB stars is most visible in cases where a “low” SN dust formation efficiency is considered. In this case the mass-heavy IMF dominates in all considered initial masses M_ini, and leads to the highest dust masses. However, dust production with this efficiency is found to be insufficient to fully account for dust masses in excess of 10⁸ M_⊙ within 400 Myr. In the galaxies with M_ini ≳ 5 × 10¹¹   M_⊙, this limit can be reached at epochs  ≳ 400 Myr if there is no dust destruction in the ISM.

• 4.

  The total dust mass is considerably reduced when destruction by SN shock waves is taken into account. The strength of destruction in the ISM is given in terms of the mass M_cl, which is the mass of the ISM that is swept up and cleared of the containing dust through one SN remnant. Top-heavy IMFs are sensitive to dust destruction, while IMFs favouring lower masses are more resistent to destruction due to lower SN rates. This leads to a significantly greater reduction and, in some cases, to a lower amount of dust with the top-heavy IMFs.

• 5.

  At early epochs (<200 Myr) SNe are primarily responsible for a significant enrichment by dust. For a “high” SN dust production efficiency SNe can generate 10⁸ M_⊙ within the first 100 Myr. In comparison, for “low” SN dust production efficiency, SNe are only able to increase the dust mass up to a few times 10⁶ M_⊙ in the first 100–150 Myr.

• 6.

  Taking the growth of the SMBH into account leads to a reduction of the amount of dust of at most  ~50%. This is achieved for a SMBH mass M_SMBH = 5 × 10⁹ M_⊙ in a galaxy with M_ini = 5 × 10¹⁰ M_⊙ and only for IMFs favouring low-mass stars. In more massive systems and for top-heavy IMFs no variation in the dust mass and properties of the galaxy is encountered.

• 7.

  To account for dust masses  > 10⁸ M_⊙ we find that galaxies with an initial gas mass of M_ini = 1–5 × 10¹¹ M_⊙ in connection with top-heavy IMFs, “high” SN dust production efficiency ϵ_high, and dust destruction in the ISM of M_cl = 100 M_⊙ are favoured. Models with the highest amount of M_ini = 1.3 × 10¹² M_⊙ and lowest of M_ini = 5 × 10¹⁰ M_⊙ are discouraged.

More refined models including diverse flow scenarios tracing self-consistent SF histories, detailed SMBH growth scenarios, and several plausible dust sources are major future developments.

Acknowledgments

We would like to thank John Eldridge for providing tabulated values of his stellar evolution models. We also thank Justyn R. Maund, Darach Watson, and Marianne Vestergaard for informative discussions. The Dark Cosmology Centre is funded by the DNRF.

References

All Tables

All Figures

	Fig. 1 Dust-production efficiencies of massive stars. The dashed line represents the “maximum” SN efficiency ϵmax(m) and the dashed dotted line the “high” SN efficiency ϵhigh(m). The solid line is the “low” SN efficiency ϵlow(m). The left dotted line represents the metallicity-averaged AGB efficiency ϵAGB(m). The vertical line marks the least massive (3 M⊙) AGB star considered.
In the text

Fig. 2 Evolution of the total dust mass and dust destruction rates for EIT08M. The initial gas mass of the galaxy Mini = 5 × 1011 M⊙. Calculations are performed for a “maximum” (top row), a “high” (middle row), and a “low” (bottom row) SN dust production efficiency ϵi(m). Dust destruction is taken into account for Mcl = 800   M⊙ (left column), 100 M⊙ (middle column), and 0 M⊙ (right column). The bottom group of curves in the light blue area represents the total dust destruction rate ED(t), which is based on Eq. (16). The upper group of curves in the green or yellow zones displays the evolution of the total amount of dust Md(t) in the galaxy. The yellow area marks dust masses exceeding 108 M⊙ of dust. The grey horizontal dashed line marks the limit of 108 M⊙ of dust. The grey vertical dashed lines indicate epochs at 100, 170, 400, and 500 Myr after the onset of starburst. The black solid, green dotted, cyan dashed, magenta dashed-dotted, and blue dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1 and Larson 2 IMF, respectively.

In the text

Fig. 3 Evolution of the dust-to-gas and dust-to-metal mass ratios for EIT08M. The initial gas mass of the galaxy Mini = 5 × 1011 M⊙. Calculations are performed for a “maximum” (top row), a “high” (middle row), and a “low” (bottom row) SN dust production efficiency ϵi(m). Dust destruction is taken into account for Mcl = 800   M⊙ (left column), 100 M⊙ (middle column), and 0 M⊙ (right column). The upper group of curves signifies the dust-to-metal mass ratio ηZd(t). The lower group of curves represents the dust-to-gas mass ratio ηd(t). The gas-to-dust ratio displayed in the figures is calculated for a Larson 2 IMF at an epoch of 400 Myr. The grey vertical dashed lines indicate epochs at 100, 400, and 500 Myr after the onset of starburst. The black solid, green dotted, cyan dashed, magenta dashed-dotted, and blue dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1 and Larson 2 IMF, respectively.

In the text

Fig. 4 Evolution of the total dust mass for EIT08M. Calculations for galaxies with initial gas masses of Mini = 5 × 1010 M⊙ are presented in the first and second columns and for Mini = 1 × 1011 M⊙ in the third and fourth columns. Total dust masses Md(t) are shown for a “maximum” (top row), a “high” (middle row), and a “low” (bottom row) SN dust production efficiency ϵI(m). Dust destruction is taken into account for Mcl = 100   M⊙ (first and third columns) and 0 M⊙ (second and fourth columns). Maximum dust masses exceeding 108 M⊙ are indicated as yellow shaded zones. The grey horizontal dashed line marks the limit of 108 M⊙ of dust. The grey vertical dashed lines indicate epochs at 400, and 500 Myr after the onset of starburst. The black solid, green dotted, and blue dashed-dot-dotted lines represent the Salpeter, mass-heavy and Larson 2 IMF, respectively.

In the text

Fig. 5 Evolution of the dust-to-gas and dust-to-metal mass ratios for EIT08M. Calculations for galaxies with initial gas masses of Mini = 5 × 1010 M⊙ are presented in the first and second columns and for Mini = 1 × 1011 M⊙ in the third and fourth columns. Results are presented for a “maximum” (top row), a “high” (middle row), and a “low” (bottom row) SN dust production efficiency ϵi(m). Dust destruction is taken into account for Mcl = 100   M⊙ (first and third columns) and 0 M⊙ (second and fourth columns). The upper group of thick curves signifies the dust-to-metal mass ratio ηZd(t). The lower group of thin curves represents the dust-to-gas mass ratio ηd(t). The gas-to-dust ratio displayed in the figures is calculated for a Larson 2 IMF at an epoch of 400 Myr. The grey vertical dashed lines indicate epochs at 400, and 500 Myr after the onset of starburst. The black solid, green dotted, and blue dashed-dot-dotted lines represent the Salpeter, mass-heavy and Larson 2 IMF, respectively.

In the text

Fig. 6 Evolution of the total dust mass for G09M, presented for a galaxy with Mini = 5 × 1011 M⊙. Total dust masses Md(t) are shown for a “high” (top row) and a “low” (bottom row) SN dust production efficiency ϵi(m). Dust destruction is taken into account for Mcl = 800   M⊙ (left column) and 0 M⊙ (right column). Maximum dust masses exceeding 108 M⊙ are indicated as yellow shaded zones. The grey horizontal dashed line marks the limit of 108 M⊙ of dust. The grey vertical dashed lines indicate epochs at 400 and 500 Myr after the onset of starburst. The black solid, green dotted, cyan dashed, magenta dashed-dotted, and blue dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1, and Larson 2 IMF, respectively.

In the text

Fig. 7 Evolution of the total dust mass for WW95M (top row) and N06M (bottom row). Both models are presented for a galaxy with Mini = 5 × 1011 M⊙. Total dust masses Md(t) are shown for “high” SN dust production efficiency ϵhigh(m). Dust destruction is taken into account for Mcl = 100   M⊙ (left column) and 0 M⊙ (right column). Maximum dust masses exceeding 108 M⊙ are indicated as yellow shaded zones. The grey horizontal dashed line marks the limit of 108 M⊙ of dust. The grey vertical dashed lines indicate epochs at 400 and 500 Myr after the onset of starburst. The black solid, green dotted, cyan dashed, magenta dashed-dotted, and blue dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1, and Larson 2 IMF, respectively.

In the text

Fig. 8 Evolution of the total dust mass for WW95M with constant SFR. WW95M is presented for a galaxy with Mini = 1.3 × 1012 M⊙. Total dust masses Md(t) are shown for “high” (top row) and a “low” (bottom row) SN dust production efficiency ϵi(m). Dust destruction is taken into account for Mcl = 800   M⊙ (left column) and 0 M⊙ (right column). Maximum dust masses exceeding 108 M⊙ are indicated as yellow shaded zones. The grey horizontal dashed line marks the limit of 108 M⊙ of dust. The grey vertical dashed lines indicate epochs at 400 and 500 Myr after the onset of starburst. The black solid, green dotted, cyan dashed, magenta dashed-dotted, and blue dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1, and Larson 2 IMF, respectively.

In the text

Fig. 9 Evolution of total dust injection rates for AGB stars and SNe, the AGB and SNe rates, the SFR, and the metallicity. Results are shown for Mini = 1.3 × 1012 M⊙ (first column), 5 × 1011 M⊙ (second column), 1 × 1011 M⊙ (third column), and 5 × 1010 M⊙ (fourth column). The solid, dotted, dashed, dashed-dotted, and dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1, and Larson 2 IMF, respectively. First row: total SN dust injection rates Ed,SN(t) for a “low” (black lines), “high” (blue lines), and “maximum” (green lines) SN efficiency ϵi(m), and AGB dust injection rates Ed,AGB(t) (red lines). SN dust injection rates are only shown for a Salpeter (solid) and a Larson 2 (dashed-dot-dotted) IMF. Second row: SNe rates RSN(t) (black lines) and AGB rates RAGB(t) (red lines). SN rates are only shown for a Salpeter (solid), mass-heavy (dotted), and a Larson 2 (dashed-dot-dotted) IMF. Third row: evolution of the SFR. The blue area marks the region between a SFR of 100–1000 M⊙ yr-1. Fourth row: evolution of the metallicity Z(t). The green regions mark a metallicity between 1–2 Z⊙ (light green), 2–3 Z⊙ (dark green), 3–4 Z⊙ (grass green). The grey vertical dashed lines indicate epochs at 400 and 500 Myr after the onset of starburst.

In the text

Fig. 10 Evolution of the gas mass MG(t) based on EIT08M. Results are shown for Mini = 1.3 × 1012 M⊙ (black lines), 5 × 1011 M⊙ (red lines), 1 × 1011 M⊙ (green lines), and 5 × 1010 M⊙ (magenta lines). The grey vertical dashed lines indicate epochs at 400 and 500 Myr after the onset of starburst. The solid, dotted, dashed, dashed-dotted, and dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1, and Larson 2 IMF, respectively.

In the text

Fig. 11 Evolution of the stellar mass and mass of metals based on EIT08M. Results are shown for Mini = 1.3 × 1012 M⊙ (first column), 5 × 1011 M⊙ (second column), 1 × 1011 M⊙ (third column), and 5 × 1010 M⊙ (fourth column). Top row: stellar mass M ∗ (t). Bottom row: mass of metals MZ(t). The red lines mark a metal mass of 108 M⊙. The solid, dotted, dashed, dashed-dotted, and dashed-dot-dotted lines represent the Salpeter, mass-heavy, top-heavy, Larson 1, and Larson 2 IMF, respectively.

In the text

Fig. 12 Comparison of the evolution of the dust mass between EIT08M including and without the SMBH formation. The initial gas mass of the galaxy Mini = 5 × 1010 M⊙. Calculations are shown for a “maximum” (top row) and a “high” (bottom row) SN dust production efficiency ϵi(m). Dust destruction is taken into account for Mcl = 100   M⊙ (left column) and 0 M⊙ (right column). Calculations including the SMBH are performed for a SMBH mass MSMBH = 5 × 109 M⊙ (solid curves) and for MSMBH = 3  ×  109 M⊙ (dotted curves). The calculations without the SMBH are illustrated as dot-dashed curves. The black lines represent the Salpeter IMF and the magenta lines the Larson 2 IMF. The grey horizontal dashed line marks the limit of 108 M⊙ of dust. The grey vertical dashed lines indicate epochs at 100, 400, and 500 Myr after the onset of starburst.

In the text

Fig. 13 Relation between dust mass, stellar mass, SFR, and metallicity in correlation with the total mass of the galaxy and the IMF for an epoch at 400 Myr. Left panel: dust mass Md(t) versus stellar mass M∗(t). Middle panel: dust mass Md(t) versus SFR ψ(t). Right panel: dust mass Md(t) versus metallicity Z(t). The dark grey shaded area marks the dust masses Md(t) obtained with a “maximum” SN dust production efficiency ϵmax(m) and no dust destruction in the ISM (Mcl = 0 M⊙). The light grey shaded area marks dust masses Md(t) for the “low” SN dust production efficiency ϵlow(m) with dust destruction in the ISM (Mcl = 800 M⊙). Cross points of the thin white lines indicate the data points for each IMF and mass Mini of the galaxy. The coloured filled circles signify the dust masses obtained for a “high” SN efficiency ϵhigh(m) and a dust destruction in the ISM with Mcl = 100 M⊙. The size of the circles is scaled by the initial mass Mini of the galaxy. The black, green, cyan, magenta, and blue colors denote the Salpeter, mass-heavy, top-heavy, Larson 1, and Larson 2 IMF, respectively.

In the text