jsons_folder/2013MNRAS.428.2529H.json

{
  "ABSTRACT ": [
    "(<http://arxiv.org/abs/1209.2413v2>)arXiv:1209.2413v2  [astro-ph.CO]  15 Oct 2012Accepted 2012 October 15. Received 2012 October 9; in original form 2012 September 10 ",
    "High-redshift submillimetre galaxies (SMGs) are some of the most rapidly star-forming galaxies in the Universe. Historically, galaxy formation models have had dif•culty explaining the observed number counts of SMGs. We combine a semi-empirical model with 3-D hydrodynamical simulations and 3-D dust radiative transfer to predict the number counts of unlensed SMGs. Because the stellar mass functions, gas and dust masses, and sizes of our galaxies are constrained to match observations, we can isolate uncertainties related to the dynamical evolution of galaxy mergers and the dust radiative transfer. The number counts and redshift distributions predicted by our model agree well with observations. Isolated disc galaxies dominate the faint (S1.1 . 1 mJy, or S850 . 2 mJy) population. The brighter sources are a mix of merger-induced starbursts and galaxy-pair SMGs; the latter subpopulation accounts for ˘ 30 −50 per cent of all SMGs at all S1.1 & 0.5 mJy (S850 & 1 mJy). The mean redshifts are ˘ 3.0−3.5, depending on the •ux cut, and the brightest sources tend to be at higher redshifts. Because the galaxy-pair SMGs will be resolved into multiple fainter sources by ALMA, the bright ALMA counts should be as much as 2 times less than those observed using single-dish telescopes. The agreement between our model, which uses a Kroupa IMF, and observations suggests that the IMF in high-redshifts starbursts need not be top-heavy; if the IMF were top-heavy, our model would over-predict the number counts. We conclude that the dif•culty some models have reproducing the observed SMG counts is likely indicative of more general problems • such as an under-prediction of the abundance of massive galaxies or a star formation rate•stellar mass relation normalisation lower than that observed • rather than a problem speci•c to the SMG population. ",
    "Key words: galaxies: high-redshift • galaxies: starburst • infrared: galaxies • radiative transfer • stars: luminosity function, mass function • submillimetre: galaxies. "
  ],
  "1 INTRODUCTION ": [
    "Submillimetre galaxies (SMGs; (<>)Smail et al. (<>)1997; (<>)Barger et al. (<>)1998; (<>)Hughes et al. (<>)1998; (<>)Eales et al. (<>)1999; see (<>)Blain et al. (<>)2002 for a review) are amongst the most luminous, rapidly star-forming galaxies known, with luminosities in excess of 1012 L⊙and star formation rates (SFR) of order ˘ 102 − 103 M⊙yr−1 (e.g., (<>)Kov´acs et al. (<>)2006; (<>)Coppin et al. (<>)2008; (<>)Micha›owski et al. ",
    "(<>)2010, (<>)2012; (<>)Magnelli et al. (<>)2010, (<>)2012; (<>)Chapman et al. (<>)2010). They have stellar masses of ˘ 1011 M⊙, although recent estimates ((<>)Hainline et al. (<>)2011; (<>)Micha›owski et al. (<>)2010, (<>)2012) differ by a factor of ˘ 6, and typical gas fractions of ˘ 40 per cent ((<>)Greve et al. (<>)2005; (<>)Tacconi et al. (<>)2006, (<>)2008; but cf. (<>)Narayanan, Bothwell, & Dav´e (<>)2012a). ",
    "The most luminous local galaxies, ultra-luminous infrared galaxies (ULIRGs, de•ned by LIR > 1012 L⊙), are almost exclusively late-stage major mergers (e.g., (<>)Lonsdale et al. (<>)2006) because the strong tidal torques exerted by the galaxies upon one another when they are near coalescence cause signi“cant gas in”ows and, consequently, bursts of star formation (e.g., (<>)Hernquist ",
    "(<>)1989; (<>)Barnes & Hernquist (<>)1991, (<>)1996; (<>)Mihos & Hernquist (<>)1996). Thus, it is natural to suppose that SMGs, which are the most luminous, highly star-forming galaxies at high redshift, are also late-stage major mergers undergoing starbursts. There is signi•cant observational support for this picture (e.g., (<>)Ivison et al. (<>)2002, (<>)2007, (<>)2010; (<>)Chapman et al. (<>)2003; (<>)Neri et al. (<>)2003; (<>)Smail et al. (<>)2004; (<>)Swinbank et al. (<>)2004; (<>)Greve et al. (<>)2005; (<>)Tacconi et al. (<>)2006, (<>)2008; (<>)Bouch´e et al. (<>)2007; (<>)Biggs & Ivison (<>)2008; (<>)Capak et al. (<>)2008; (<>)Younger et al. (<>)2008, (<>)2010; (<>)Iono et al. (<>)2009; (<>)Engel et al. (<>)2010; (<>)Bothwell et al. (<>)2010, (<>)2012; (<>)Riechers et al. (<>)2011a,(<>)b; (<>)Magnelli et al. (<>)2012). However, there may not be enough major mergers of galaxies of the required masses to account for the observed SMG abundances ((<>)Dav´e et al. (<>)2010). Consequently, explaining the abundance of SMGs has proven to be a challenge for galaxy formation models. ",
    "Much observational effort has been invested to determine the number counts and redshift distribution of SMGs (e.g., (<>)Chapman et al. (<>)2005; (<>)Coppin et al. (<>)2006; (<>)Knudsen et al. (<>)2008; (<>)Chapin et al. (<>)2009; (<>)Weiß et al. (<>)2009; (<>)Austermann et al. (<>)2009, (<>)2010; (<>)Scott et al. (<>)2010; (<>)Zemcov et al. (<>)2010; (<>)Aretxaga et al. (<>)2011; (<>)Banerji et al. (<>)2011; (<>)Hatsukade et al. (<>)2011; (<>)Wardlow et al. (<>)2011; (<>)Roseboom et al. (<>)2012; (<>)Yun et al. (<>)2012) because this information is key to relate the SMG population to their descendants and to understand SMGs in the context of hierarchical galaxy formation models. Various authors have attempted to explain the observed abundance of SMGs using phenomenological models (e.g., (<>)Pearson & Rowan-Robinson (<>)1996; (<>)Blain et al. (<>)1999b; (<>)Devriendt & Guiderdoni (<>)2000; (<>)Lagache et al. (<>)2003; (<>)Negrello et al. (<>)2007; (<>)B´ethermin et al. (<>)2012), semi-analytic models (SAMs; e.g., (<>)Guiderdoni et al. (<>)1998; (<>)Blain et al. (<>)1999a; (<>)Granato et al. (<>)2000; (<>)Kaviani et al. (<>)2003; (<>)Granato et al. (<>)2004; (<>)Baugh et al. (<>)2005; (<>)Fontanot et al. (<>)2007; (<>)Fontanot & Monaco (<>)2010; (<>)Lacey et al. (<>)2008, (<>)2010; (<>)Swinbank et al. (<>)2008; (<>)Lo Faro et al. (<>)2009; (<>)Gonz´alez et al. (<>)2011), and cosmological hydrodynamical simulations ((<>)Fardal et al. (<>)2001; (<>)Dekel et al. (<>)2009; (<>)Dav´e et al. (<>)2010; (<>)Shimizu et al. (<>)2012). ",
    "(<>)Granato et al. ((<>)2000) presented one of the •rst SAMs to self-consistently calculate dust absorption and emission by coupling the GALFORM SAM ((<>)Cole et al. (<>)2000) with the GRASIL spectrophotometric code ((<>)Silva et al. (<>)1998). This was a signi•cant advance over previous work, which effectively treated the dust temperature as a free parameter. Self-consistently computing dust temperatures made matching the submm counts signi•cantly more dif•cult: the submm counts predicted by the (<>)Granato et al. model were a factor of ˘ 20 − 30 less than those observed ((<>)Baugh et al. (<>)2005; (<>)Swinbank et al. (<>)2008). ",
    "The work of (<>)Baugh et al. ((<>)2005, hereafter B05) has attracted signi•cant attention to the •eld because of its claim that a •at IMF is necessary to reproduce the properties of the SMG population, which we will discuss in detail here. B05 set out to modify the (<>)Granato et al. ((<>)2000) model so that it would reproduce the properties of both z ˘ 2 SMGs and Lyman-break galaxies (LBGs) while also matching the observed z = 0 optical and IR luminosity functions. Adopting a •at IMF1 (<>)in starbursts rather than the (<>)Kennicutt ((<>)1983) IMF used in (<>)Granato et al. ((<>)2000) was the key change that enabled B05 to match the observed SMG counts and redshift distribution while still reproducing the local K-band luminosity function. A more top-heavy IMF results in both more ",
    "luminosity emitted and more dust produced per unit SFR; consequently, the submm •ux per unit SFR is increased signi•cantly (see B05 and (<>)Hayward et al. (<>)2011a, hereafter H11, for details). The B05 modi•cations increased the S850 per unit SFR for starbursts by a factor of ˘ 5 (G.-L. Granato, private communication), which caused starbursts to account for a factor of ˘ 10 3 times more sources at S850 = 3 mJy than in (<>)Granato et al. ((<>)2000). As a result, in the B05 model, ongoing starbursts dominate the counts for 0.1 . S850 . 30 mJy. Interestingly, these starbursts are triggered predominantly by minor mergers (B05; (<>)Gonz´alez et al. (<>)2011). (<>)Swinbank et al. ((<>)2008) present a detailed comparison of the properties of SMGs in the B05 model with those of observed SMGs. The far-IR SEDs, velocity dispersions, and halo masses (see also (<>)Almeida et al. (<>)2011) are in good agreement; however, recent observations suggest that the typical redshift of SMGs may be higher than predicted by the B05 model and, contrary to the B05 prediction, brighter SMGs tend to be at higher redshifts ((<>)Yun et al. (<>)2012; (<>)Smolˇci´c et al. (<>)2012). Furthermore, the rest-frame K-band •uxes of the B05 SMGs are a factor of ˘ 10 lower than observed; the most plausible explanation is that the masses of the SMGs in the B05 SAM are too low ((<>)Swinbank et al. (<>)2008), but the top-heavy IMF in starbursts used by B05 makes a direct comparison of masses dif•cult. These disagreements are reasons it is worthwhile to explore alternative SMG models. ",
    "(<>)Granato et al. ((<>)2004) presented an alternate model, based on spheroid formation via monolithic collapse, that predicts submm counts in good agreement with those observed and reproduces the evolution of the K-band luminosity function. However, the typical redshift they predict for SMGs is lower than recent observational constraints ((<>)Yun et al. (<>)2012; (<>)Smolˇci´c et al. (<>)2012), and this model does not include halo or galaxy mergers. ",
    "The (<>)Fontanot et al. ((<>)2007) model predicts SMG number counts in reasonable agreement with those observed using a standard IMF; they argue that the crucial difference between their model and that of B05 is the cooling model used (see also (<>)Viola et al. (<>)2008 and (<>)De Lucia et al. (<>)2010). However, their SMG redshift distribution peaks at a lower redshift than the redshift distribution derived from recent observations ((<>)Yun et al. (<>)2012; (<>)Smolˇci´c et al. (<>)2012). Furthermore, the (<>)Fontanot et al. ((<>)2007) model produces an overabundance of bright galaxies at z < 1. However, this problem has been signi•cantly reduced in the latest version of the model ((<>)Lo Faro et al. (<>)2009), which provides a signi•cantly better •t to the galaxy stellar mass function at low redshift ((<>)Fontanot et al. (<>)2009). In the revised model, the submm counts are reduced by ˘ 0.5 dex, primarily because of the change in the IMF from Salpeter to Chabrier, but the redshift distribution is unaffected. Thus, the submm counts for the new model are consistent with the data for S850 . 3 mJy, but they are slightly less than the observed counts at higher •uxes ((<>)Fontanot & Monaco (<>)2010). No •ne-tuning of the dust parameters has been performed for the new model. ",
    "A compelling reason to model the SMG population in an alternative manner is to test whether a top-heavy IMF is required to explain the observed SMG counts. Matching the submm counts is the primary reason B05 needed to adopt a •at IMF in starbursts.2 (<>)Using the same model, (<>)Lacey et al. ((<>)2008) show that the •at IMF is necessary to reproduce the evolution of the mid-IR luminosity function. Others (e.g., (<>)Guiderdoni et al. (<>)1998; (<>)Blain et al. (<>)1999a; (<>)Dav´e et al. ",
    "(<>)2010) have also suggested that the IMF may be top-heavy in SMGs, but they do not necessarily require variation as extreme as that assumed in B05. However, the use of a •at IMF in starbursts remains controversial: though there are some theoretical reasons to believe the IMF is more top-heavy in starbursts (e.g., (<>)Larson (<>)1998, (<>)2005; (<>)Elmegreen & Shadmehri (<>)2003; (<>)Elmegreen (<>)2004; (<>)Hopkins (<>)2012; (<>)Narayanan & Dav´e (<>)2012), there is to date no clear evidence for strong, systematic IMF variation in any environment ((<>)Bastian et al. (<>)2010 and references therein). Furthermore, in local massive ellipticals, the probable descendants of SMGs, the IMF may actually be bottom-heavy (e.g., (<>)van Dokkum & Conroy (<>)2010, (<>)2011; (<>)Conroy & van Dokkum (<>)2012; (<>)Hopkins (<>)2012). Finally, the large parameter space of SAMs can yield multiple qualitatively distinct solutions that satisfy all observational constraints ((<>)Bower et al. (<>)2010; (<>)Lu et al. (<>)2011, (<>)2012), so it is possible that a top-heavy IMF in starbursts is not required to match the observed submm counts even though it enables B05 to match the submm counts.3 (<>)Thus, it is useful to explore other methods to predict the submm counts and to determine whether a match can be achieved without using a top-heavy IMF. ",
    "Another reason to model the SMG population is to investigate whether, like local ULIRGs, they are predominantly merger-induced starbursts. Some observational evidence suggests that some SMGs may be early-stage mergers in which the discs have not yet coalesced and are likely not undergoing starbursts (e.g., (<>)Tacconi et al. (<>)2006, (<>)2008; (<>)Engel et al. (<>)2010; (<>)Bothwell et al. (<>)2010; (<>)Riechers et al. (<>)2011a,(<>)b), and massive isolated disc galaxies may also contribute to the population (e.g., (<>)Bothwell et al. (<>)2010; (<>)Carilli et al. (<>)2010; (<>)Ricciardelli et al. (<>)2010; (<>)Targett et al. (<>)2011, (<>)2012) In H11 and H12, we suggested that the inef•cient scaling of (sub)mm •ux with SFR in starbursts results in an SMG population that is heterogeneous: major mergers contribute both as coalescence-induced starbursts and during the pre-coalescence in-fall stage, when the merging discs are blended into one (sub)mm source because of the large (˘ 15•, or ˘ 130 kpc at z ˘ 2 − 3) beams of the single-dish (sub)mm telescopes used to perform large SMG surveys. We refer to the latter subpopulation as •galaxy-pair SMGs•. Similarly, compact groups may be blended into one source and can thus also contribute to the population. The most massive, highly star-forming isolated discs may also contribute (H11). Finally, it has been observationally demonstrated that there is a contribution from physically unrelated galaxies blended into one source ((<>)Wang et al. (<>)2011). It is becoming increasingly clear that the SMG population is a mix of various classes of sources; if one subpopulation does not dominate the population, physically interpreting observations of SMGs is signi•cantly more complicated than previously assumed. ",
    "Narayanan et al. ",
    "2010b",
    "Micha•owski et al. ",
    "2012",
    "In previous work, we demonstrated that major mergers can reproduce the observed 850-µm •uxes and typical SED ((<>)Narayanan et al. (<>)2010b); CO spatial extents, line-widths, and excitation ladders ((<>)Narayanan et al. (<>)2009); stellar masses (; H11; ); LIR• effective dust temperature relation, IR excess, and star formation ef•ciency ((<>)Hayward et al. (<>)2012, hereafter H12) observed for SMGs. In this work, we present a novel method to predict the (sub)mm counts from mergers and quiescently star-forming disc galaxies. We utilise a combination of 3-D hydrodynamical simulations, on which we perform radiative transfer in post-processing to ",
    "calculate the UV-to-mm SEDs, and a semi-empirical model (SEM) of galaxy formation • both of which have been extensively validated in previous work • to predict the number counts and redshift distribution of SMGs in our model. We address four primary questions: 1. Can our model reproduce the observed SMG number counts and redshift distribution? 2. What are the relative contributions of merger-induced starbursts, galaxy pairs, and isolated discs to the SMG population? 3. How will the number counts and redshift distribution of ALMA-detected SMGs differ from those determined using single-dish surveys? 4. Does the SMG population provide evidence for a top-heavy IMF in high-redshift starbursts? ",
    "The remainder of this paper is organised as follows: In Section (<>)3, we present the details of the simulations we use to determine the time evolution of galaxy mergers and to translate physical properties of model galaxies into observed-frame (sub)mm •ux densities. In Section (<>)4, we discuss how we combine the simulations with a SEM to predict the (sub)mm counts for merger-induced starburst SMGs (Section (<>)4.1) and isolated disc and galaxy-pair SMGs (Section (<>)4.2). In Section (<>)5, we present the predicted counts and redshift distribution of our model SMGs and the relative contribution of each subpopulation. We discuss implications for the IMF, compare to previous work, and highlight some uncertainties in and limitations of our model in Section (<>)6, and we conclude in Section (<>)7. "
  ],
  "2 SUMMARY OF THE MODEL ": [
    "Predicting SMG counts requires three main ingredients: 1. Because SFR and dust mass are the most important properties for predicting the (sub)mm •ux of a galaxy (H11), one must model the time evolution of those properties for individual discs and mergers. 2. The physical properties of the model galaxies must be used to determine the observed-frame (sub)mm •ux density of those galaxies. 3. One must put the model galaxies in a cosmological context. Ideally, one could combine a cosmological hydrodynamical simulation with dust radiative transfer to self-consistently predict the (sub)mm counts. However, this is currently infeasible because the resolution required for reliable radiative transfer calculations cannot be achieved for a cosmological simulation large enough to contain a signi•cant number of SMGs (see, e.g., (<>)Dav´e et al. (<>)2010).4 (<>)",
    "Here, we develop a novel method to predict the number counts and redshift distribution of high-z SMGs while still resolving the dusty ISM on scales of ˘ 200 pc. We predict (sub)mm counts using a combination of a simple SEM ((<>)Hopkins et al. (<>)2008a,(<>)c) and ide-alised high-resolution simulations of galaxy mergers. The method we use for each of the three model ingredients depends on the subpopulation being modelled. The physical properties of the isolated disc galaxies and early-stage mergers are determined using the SEM. For the late-stage mergers, hydrodynamical simulations are used because of the complexity of modelling a merger•s evolution. Dust radiative transfer is performed on the hydrodynamical simulations to translate the physical properties into observed (sub)mm •ux density. For the isolated discs and early-stage mergers, •tting functions derived from the simulations are used, whereas ",
    "Table 1. Summary of methods ",
    "for the late-stage mergers, the (sub)mm light curves are taken directly from the simulations. Finally, the isolated galaxies are put in a cosmological context using an observed stellar mass function (SMF). For the mergers, merger rates from the SEM and duty cycles from the simulations are used. The methods are summarised in Table (<>)1, and each component of the model is discussed in detail below. ",
    "We emphasize that we do not attempt to model the SMG population in an ab initio manner. Instead, we construct our model so that the SMF, gas fractions, and metallicities are consistent with observations. This will enable us to test whether, given a demographically accurate galaxy population, we are able to reproduce the SMG counts and redshift distribution. If we are not able to reproduce the counts and redshift distribution, then our simulations or radiative transfer calculations must be incomplete. If we can reproduce the counts and redshift distribution, then it is possible that the failure of some SAMs and cosmological simulations to reproduce the SMG counts may be indicative of a more general problem with those models (e.g., a general under-prediction of the abundances of massive galaxies) rather than a problem speci•c to the SMG population. ",
    "In the next two sections, we describe our model in detail. Readers whom are uninterested in the details of the methodology may wish to skip to Section (<>)5. "
  ],
  "3 SIMULATION METHODOLOGY ": [],
  "3.1 Hydrodynamical simulations ": [
    "We have performed a suite of simulations of isolated and merging disc galaxies with GADGET-2 ((<>)Springel et al. (<>)2001; (<>)Springel (<>)2005), a TreeSPH ((<>)Hernquist & Katz (<>)1989) code that computes gravitational interactions via a hierarchical tree method ((<>)Barnes & Hut (<>)1986) and gas dynamics via smoothed-particle hydrodynamics (SPH; (<>)Lucy (<>)1977; (<>)Gingold & Monaghan (<>)1977; (<>)Springel (<>)2010).5 (<>)It explicitly conserves both energy and entropy when appropriate ((<>)Springel & Hernquist (<>)2002). Beyond the core ",
    "Schmidt ",
    "1959",
    "Kennicutt ",
    "1998",
    "gravitational and gas physics, the version of GADGET-2 we use includes radiative heating and cooling ((<>)Katz et al. (<>)1996). Star formation is implemented using a volume-density-dependent Kennicutt-Schmidt (KS) law (; ), ˆSFR / ˆNgas, with a low-density cutoff. We use N = 1.5, which reproduces the global KS law and is consistent with observations of high-redshift disc galaxies ((<>)Krumholz & Thompson (<>)2007; (<>)Narayanan et al. (<>)2008, (<>)2011; but see (<>)Narayanan et al. (<>)2012b). ",
    "Furthermore, our simulations include a two-phase sub-resolution model for the interstellar medium (ISM; (<>)Springel & Hernquist (<>)2003) in which cold dense clouds are in pressure equilibrium with a diffuse hot medium. The division of mass, energy, and entropy between the two phases is affected by star formation, radiative heating and cooling, and supernova feedback, which heats the diffuse phase and evaporates the cold clouds ((<>)Cox et al. (<>)2006b). Metal enrichment is calculated assuming each particle behaves as a closed box the yield appropriate for a (<>)Kroupa ((<>)2001) IMF. The simulations also include the (<>)Springel et al. ((<>)2005) model for feedback from active galactic nuclei (AGN), in which black hole (BH) sink particles, initialised with mass 10 5 h−1 M⊙, undergo Eddington-limited Bondi-Hoyle accretion ((<>)Hoyle & Lyttleton (<>)1939; (<>)Bondi & Hoyle (<>)1944; (<>)Bondi (<>)1952). They deposit 5 per cent of their luminosity (L = 0.1 ˙ 2mc , where m˙ is the mass accretion rate and c is the speed of light) to the surrounding ISM. This choice is made so that the normalisation of the MBH − ˙ relation is recovered ((<>)Di Matteo et al. (<>)2005). Note that our results do not depend crucially on the implementation of BH accretion and feedback for two reasons: 1. the AGN typically do not dominate (but can still contribute signi“cantly to) the luminosity of our model SMGs because the SEDs during the phase of strong AGN activity tend to be hotter than during the starburst phase (e.g., (<>)Younger et al. (<>)2009, Snyder et al., in preparation), so the mergers are typically not SMGs during the AGN-dominated phase. 2. Even in the absence of AGN feedback, the SFR decreases sharply after the starburst simply because the majority of the cold gas is consumed in the starburst. ",
    "Each disc galaxy is composed of a dark matter halo with a (<>)Hernquist ((<>)1990) pro•le and an exponential gas and stellar disc in which gas initially accounts for 80 per cent of the total bary-onic mass. At merger coalescence, the baryonic gas fractions are typically 20-30 per cent, which is consistent with the estimates of (<>)Narayanan et al. ((<>)2012a). The mass of the baryonic component is 4 per cent of the total. The galaxies are scaled to z = 3 following the method described in (<>)Robertson et al. ((<>)2006). Dark matter particles have gravitational softening lengths of 200h−1 pc, whereas gas and star particles have 100h−1 pc. We use 6×104 dark matter, 4×104 stellar, 4 × 104 gas, and 1 BH particle per disc galaxy. The detailed properties of the progenitor galaxies are given in Table (<>)2. Note that we have chosen galaxy masses such that most of the mergers, based upon our simulations, will contribute to the bright SMG population (i.e., at some time during the simulation they have observed 850-µm •ux density S850 > 3 mJy). More massive galaxies ",
    "Table 2. rogenitor disc gl properties ",
    "Table 3. erger prmeters ",
    "ill lso contriute ut re incresingl more rre so our simultions should e representtie o ll ut the rightest rrest s (<>)ichosi et l. (<>). ote lso tht e he included some slightl loer mss mergers or completeness. ",
    "",
    "hochr  urert ",
    "",
    "e simulte ech disc gl listed in le (<>) in isoltion or 1.5h−1 r nd use these isolted disc simultions s prt o our simultion suite. ur suite lso includes  numer o simultions o mor nd minor gl mergers. or the merger simultions to o the progenitor disc glies re plced on prolic orits hich re motited  cosmologicl simultions (<>)enson   ith initil seprtion Rinit = 5R200/8 nd pericentricpssge distnce eul to tice the disc scle length Rperi = 2Rd(<>)oertson et l. (<>). he eolution o the sstem is olloed or 1.5h−1 r hich is sucient time or the glies to colescence nd or signicnt str ormtion nd  ctiit to cese. he detils o the merger simultions re gien in le (<>). or ech comintion o progenitor discs in le (<>) e simulte  suset o the ip orits o (<>)o et l. (<>). pecicll e use the ip orits or the mor mergers  .. nd  nd the i nd  orits or the uneulmss mergers .  nd  ecuse the ltter he shorter dut ccles nd the rition in dut ccles mongst orits is not  primr source o uncertint. onseuentl e use  totl o   simultions. "
  ],
  "3.2 Dust radiative transfer ": [
    "n postprocessing e use the  onte rlo rditie trnser code  6 (<>)to clculte the tomm s o the simulted glies. e he preiousl simulted glies ith colours/s consistent ith locl  (<>)ennicutt et l. (<>) (<>)le et l. (<>) glies (<>)onsson roes  o (<>) locl s (<>)ounger et l. (<>) mssie uiescent compct ",
    "z ˘ 2 glies (<>)uts et l. (<>) (<>)  µmselected glies (<>)rnn et l. (<>) /poststrurst glies (<>)nder et l. (<>) nd etended  discs (<>)ush et l. (<>) mong other popultions so e re condent tht  cn e used to model the high  popultion. s discussed oe preious or hs demonstrted tht mn properties o our simulted s gree ith osertions (<>)rnn et l. (<>) (<>) (<>)rd et l. (<>) (<>) ut e he et to put our simulted s in  cosmologicl contet. e rie reie the detils o  here ut e reer the reder to (<>)onsson et l. (<>) (<>)onsson et l. (<>) nd (<>)onsson  rimc (<>) or ull detils o the  code. ",
    " uses the output o the  simultions to speci the detils o the rditie trnser prolem to e soled specicll the input rdition eld nd dust geometr. he str nd  prticles rom the  simultions re used s sources o emission. tr prticles re ssigned  (<>)eitherer et l. (<>) s ccording to their ges nd metllicities. e consertiel use the (<>)roup (<>)  hen clculting the simple stellr popultion  templtes. tr prticles present t the strt o the  simultion re ssigned ges  ssuming tht their stellr mss s ormed t  constnt rte eul to the str ormtion rte o the initil snpshot ut the results re insensitie to this choice ecuse e discrd the erl snpshots nd the strs present t the strt o the simultion ccount or  smll rction o the luminosit t lter times. he initil gs nd stellr metllicities re Z = 0.015 ˘ Z⊙ (<>)splund et l. (<>). e he chosen this lue so tht the strursts lie roughl on the osered mssmetllicit reltion hoeer the results re irl roust to this choice ecuse  ctor o  chnge in dust mss chnges the summ u  onl ˘ 50 per cent ecuse summ u scles pproimtel s Md0.6 utions (<>) nd (<>).  prticles re ssigned luminositdependent templtes deried rom osertions o unreddened usrs (<>)opins et l. (<>) here the luminosit is determined using the ccretion rte rom the  simultions s descried oe. ",
    "he dust distriution is determined  proecting the totl gsphse metl densit in the  simultions on to   dptie mesh renement grid ssuming  dusttometl rtio o . (<>)e (<>) (<>)mes et l. (<>). e he used  mimum renement leel o  hich results in  minimum cell sie o h−1 pc. his renement leel is sucient to ensure the s re conerged to ithin  e per cent ecuse the structure present in the  simultions is sucientl resoled i the resolution o the  simultions ere ner the rditie trnser ould reuire correspondingl smller cell sies. ote tht e ssume the  is smooth on scles elo the  resolution nd do not me use o the (<>)roes et l. (<>) suresolution photodissocition region model. he detils o motition or nd implictions o this choice re discussed in sections .. nd . ",
    "o . e ssume the dust hs properties gien  the il  R = 3.1 dust model o (<>)eingrtner  rine (<>) s updted  (<>)rine  i (<>). he summ ues re similr i the  or  dust models re used. ",
    "nce the str nd  prticles re ssigned s nd the dust densit eld is specied  perorms the rditie trnser using  onte rlo pproch  emitting photon pcets tht re scttered nd sored  dust s the propgte through the . he energ sored  dust is rerdited in the . ust tempertures hich depend on oth grin sie nd the locl rdition eld re clculted  ssuming the dust is in therml euilirium. he  o our simulted glies cn oten e opticll thic t  elengths so  clcultes the eects o dust selsorption using n itertie method. his is crucil or ensuring ccurte dust tempertures. ",
    "he  clcultion ields sptill resoled s nlogous to integrl eld unit spectrogrph dt o the simulted glies ieed rom dierent ieing ngles. ere e use  cmers distriuted isotropicll in solid ngle. e use the  µm  .mm nd  nds  nd  lter response cures to clculte the summ u densities. epending on the mss nd the  u cut the simultions re selected s s or ˘ 0 − 80 snpshots nd ech is ieed rom  ieing ngles. onseuentl e he  smple o ˘ 3.7 × 104 distinct snthetic  s tht e use to derie tting unctions or the isolted disc nd glpir s nd dut ccles or the strurst s. "
  ],
  "4 PREDICTING (SUB)MM NUMBER COUNTS ": [
    "o clculte the totl  numer counts predicted  our model e must ccount or ll supopultions including the inllstge glpir s discussed in  nd  ltestge mergerinduced strursts nd isolted discs. o clculte the counts or the to supopultions ssocited ith mergers e must comine the dut ccles o the mergers the time the merger hs summ u greter thn some u cut ith merger rtes ecuse the numer densit is clculted  multipling the dut ccles  the merger rtes. or the isolted discs e reuire the numer densit o  disc gl s  unction o its properties nd the summ u ssocited ith tht gl. e descrie our methods or predicting the counts o ech supopultion no. "
  ],
  "4.1 Late-stage merger-induced starbursts ": [
    "o predict the numer counts or the popultion o ltestge mergerinduced strurst s e comine merger rtes  hich depend on mss mss rtio gs rction nd redshit  rom the  ith summ light cures rom our simultions. e use the summ light cures rom the simultions directl ecuse it is dicult to nlticll model the dnmicl eolution o the mergers hich cn depend on the gl msses merger mss rtio progenitor redshit gs rction nd oritl properties. or the  supopultion ttriutle to mergers the numer densit o sources ith u densit greter thn Sλt redshit z is ",
    "here dN/dV dtd log Mbardµdfg(Mbar, µ, fg, z) is the numer o mergers per comoing olume element per unit time per de ronic mss per unit mss rtio per unit gs rction hich is  unction o progenitor ronic mss Mbar merger mss rtio µ gs rction t merger fg nd redshit z nd ˝(Sλ, Mbar, µ, fg, z) is the mount o time dut ccle or hich  merger ith mostmssieprogenitor ronic mss Mbar mss rtio µ nd gs rction fgt redshit z hs u densit > Sλ. "
  ],
  "4.1.1 Duty cycles ": [
    "e clculte the dut ccles ˝(S850) nd ˝(S1.1) or rious S850 nd S1.1 lues or the ltestge mergerinduced strurst phse o our merger simultions. e neglect the dependence o dut ccle on gs rction ecuse smpling the rnge o initil gs rctions in ddition to msses mss rtios nd orits is computtionll prohiitie. nsted s descried oe e initilise the mergers ith gs rction fg= 0.8 so tht sucient gs remins t merger colescence.7 (<>)",
    "imilrl ecuse o computtionl limittions e scle ll initil disc glies to z ˘ 3. e ill see elo tht ll else eing eul the dependence o summ u densit on z is smll . 0.13 de or the redshit rnge o interest z ˘ 1 − 6 so e ssume the dut ccles re independent o redshit nd plce the mergers t z = 3 hich is pproimtel the medin redshit or our model s hen clculting the dut ccles. ote hoeer tht the summ dut ccles or the strursts m dier or mergers ith progenitor disc properties scled to dierent redshits ut our model does not cpture this eect. oeer ecuse most s in our model he z ˘ 2 − 4 see elo this uncertint should e sudominnt. ",
    "or ech Sλ e erge the dut ccles or ech set o models ith identicl (Mbar, µ) nd then t the resulting ˝(Mbar, µ) surce ith  seconddegree polnomil in Mbar nd µ to estimte the dut ccle or (Mbar, µ) lues not eplicitl smpled  our simultions. "
  ],
  "4.1.2 Merger rates ": [
    "he other ingredient needed to predict the counts or mergerinduced strursts is the merger rtes. e use rtes rom the  descried in detil in (<>)opins et l. (<>)c (<>)(<>)(<>)c hich e ill rie summrise here. he model strts ith  hlo mss unction tht hs een clirted using highresolution Nod simultions. lies re ssigned to hloes using n osered  or strorming glies nd the hlo occuption ormlism (<>)onro  echsler (<>). e use  ducil  tht is  comintion o multiple osered s in hich ech coers  suset o the totl redshit rnge. or z < 2 e use the  o strorming glies rom (<>)lert et l. (<>). or 2.0 6 z 6 3.75 e use the  o (<>)rchesini et l. (<>) ecuse their sure is mongst the idest nd deepest ille nd ecuse the he perormed  thorough nlsis o the rndom nd sstemtic uncertinties ecting the  determintion. or z > 3.75 e use the (<>)ontn et l. (<>)  prmeterition though the onl ",
    "Figure 1. umer densit o disc glies dN/dV d log(M⋆/M⊙) pc−3 de−1 ersus M⋆(M⊙) or integer redshits in the rnge z = 0 −6 or our composite . or z < 2 e use the  or strorming glies rom (<>)lert et l. (<>). or 2 6 z 6 3.75 e use the (<>)rchesini et l. (<>)  nd or z > 3.75 e use the (<>)ontn et l. (<>) prmeteristion o the . ",
    "constrined the  out to z ˘ 4 the etrpoltion grees resonl ell ith the 4 < z < 7 constrints rom (<>)onle et l. (<>) so this etrpoltion is not unresonle. ecuse the  t z & 4 is uncertin it m e possile to constrin the  t those redshits  using the osered  redshit distriution nd reltie contriutions o the supopultions e discuss these possiilities elo. he interested reder should see (<>)rd (<>) or  detiled eplortion o ho the choice o  ects the predictions o our model. e do not correct or the pssie gl rction eond z > 2 ut this rction is reltiel smll t z ˘ 2 nd decreses rpidl t higher redshits e.g. (<>)uts et l. (<>) (<>)rmmer et l. (<>). ur composite  t integer redshits in the rnge z = 0 − 6 is plotted in ig. (<>). inll e use hlohlo merger rtes rom highresolution Nod simultions nd trnslte to glgl merger rtes  ssuming the glies merge on  dnmicl riction timescle. ",
    "he merger rtes  nd osered gs rctions re ll uncertin. he merger rtes re uncertin t the ctor o ˘ 2 leel the rious sources o uncertint nd eects o modiing the model ssumptions re discussed in detil in (<>)opins et l. (<>)(<>). t the redshits o interest the rndom nd sstemtic uncertinties in the  re comprle to the totl uncertint in the merger rtes. "
  ],
  "4.1.3 Predicted counts ": [
    "sing the oe ssumptions ution (<>) ecomes ",
    " ",
    "o clculte the oserle cumultie counts deg−2 e must multipl  dV/d dz the comoing olume element in solid ngle d nd redshit interl dz nd integrte oer redshit: ",
    " ",
    "here ",
    " "
  ],
  "4.2 Isolated discs and early-stage mergers ": [
    "e tret the isolted discs nd erlstge mergers hich re dominted  uiescent str ormtion in  semiempiricl mnner in hich e ssign gl properties sed o osertions. o clculte the osered summ u densities using scling reltions similr to those o  e must determine the  nd dust mss o  gl s  unction o stellr mss nd redshit. e then use  nd merger rtes to clculte the summ counts or these popultions. "
  ],
  "4.2.1 Assigning galaxy properties ": [
    "olloing (<>)opins et l. (<>)(<>)c e ssign gs rctions nd sies s  unction o stellr mss using osertionll deried reltions. e present the relent reltions elo ut e reer the reder to (<>)opins et l. (<>)(<>)(<>)c or ull detils including the list o osertions used to derie the reltions nd ustictions or the orms used. (<>)opins et l. (<>)c he shon tht this model reproduces glol constrints such s the  luminosit unction t rious redshits nd the str ormtion histor o the nierse mong others these results support the ppliction o the model in this or. ",
    "he ronic gs rction fgas = Mgas/(Mgas + M⋆) o  gl o stellr mss M⋆nd redshit z s determined rom the osertions listed in (<>)opins et l. (<>)c is gien  ution  o (<>)opins et l. (<>)c ",
    " ",
    "here ˝(z) is the rctionl looc time to redshit z. t  gien mss gl gs rctions increse ith redshit. t ed redshit the decrese ith stellr mss. sing fgas(M⋆, z) e cn clculte the gs mss s  unction o M⋆nd z ",
    " ",
    "imilrl e prmeterie the rdius o the gs disc s  unction o mss nd redshit using the osertions listed in (<>)opins et l. (<>)c. ote tht the stellr disc rdii re signiicntl smller. he reltion ution  o (<>)opins et l. (<>)c see lso (<>)omerille et l. (<>) is ",
    "e ssume the uiescentl strorming discs oe the  reltion (<>)chmidt (<>) (<>)ennicutt (<>) ",
    " ",
    "here ˙ ⋆nd gas re the  nd gs surce densities respectiel nd nK= 1.4 (<>)ennicutt (<>) t ll redshits ",
    "Figure 2.   M⊙yr−1 ersus M⋆( M⊙) or model disc glies t integer redshits in the rnge z = 0 −6 solid lines nd rom the osertionll deried tting unction o (<>)hiter et l. (<>) t z = 0, 1, nd  dshed lines. he normlistion o the reltion increses ith redshit oth ecuse gs rctions re higher nd glies re more compct. he model grees resonl ell ith the osertions ecept t z ∼0 ut e shll see tht the z ∼0 contriution to our model  popultion is smll.  the s rom (<>)hiter et l. (<>) ere used insted o those clculted rom ution (<>) the z ∼0 contriution ould e een less so the discrepnc is unimportnt. ",
    "roup ",
    "",
    "s is supported  osertions e.g. (<>)rnn et l. (<>) (<>)ddi et l. (<>) (<>)enel et l. (<>) (<>)rnn et l. (<>) ut c.. (<>)rnn et l. (<>). e normlise the reltion ssuming   . ssuming gas ˇ Mgas/(ˇRe2) nd ˙ ⋆ˇ M˙ ⋆/(ˇRe2) here M˙ ⋆is the  e nd ",
    "",
    "",
    " ",
    "hich cn e recst in terms o M⋆rther thn Mgas using utions  nd . ig. shos the M⋆reltion gien  ution (<>) or integer redshits in the rnge z = 0 − 6 nd the osered reltions rom (<>)hiter et l. (<>) or z ˘ 0  nd . he greement eteen our reltion nd those osered or z ˘ 1 nd  is resonle or the msses M⋆& 1011 M⊙ relent to the  popultion. he greement is less good or z ˘ 0 ut s e shll see elo this is unimportnt ecuse the rction o s t z . 1 is smll.  the (<>)hiter et l. (<>) s ere used insted o those clculted rom ution (<>) the z ˘ 0 contriution ould e een smller. ",
    "n ddition to the  e reuire the dust mss to clculte the summ u densities. o determine the dust mss e must no the gsphse metllicit. sertions he demonstrted tht metllicit increses ith stellr mss this reltionship hs een constrined or redshits z ˘ 0 − 3.5 (<>)remonti et l. (<>) (<>)glio et l. (<>) (<>)r et l. (<>) (<>)ele  llison (<>) (<>)iolino et l. (<>). (<>)iolino et l. (<>) prmeteried the eolution o the mssmetllicit reltion  ith redshit using the orm ",
    " ",
    "he determine the lues o log M0 nd K0 t redshits z = ",
    ". . . nd . using the osertions o (<>)ele  llison (<>) (<>)glio et l. (<>) (<>)r et l. (<>) nd their on or respectiel. o crudel cpture the eolution o the  ith redshit e t the lues o log M0 nd K0 gien in le  o (<>)iolino et l. (<>) s poer ls in (1 + z) the result is log M0(z) ˇ 11.07(1 + z)0.094 nd K0(z) ˇ 9.09(1 + z)−0.017 . ",
    "he ",
    "e ssume the dust mss is proportionl to the gsphse metl mss Md= MgasZfdtm. hus ",
    "here e use  dusttometl rtio fdtm = 0.4 (<>)e (<>) (<>)mes et l. (<>). ",
    "otited  eution  o  e t the summ u densities o our simulted glies s poer ls in  nd Md. e nd tht ",
    "S1.1 = 0.35 mJy ",
    " ",
    " ",
    "is ccurte to ithin 0.13 de or z ˘ 1 − 6. he u or glies t z . 0.5 is underestimted signicntl  these eutions ut such glies contriute little to the oerll counts ecuse o the smller cosmologicl olume proed nd the signicntl loer gs rctions nd s so this underestimte is unimportnt or our results. he summ u is insensitie to redshit in this redshit rnge ecuse s redshit increses the decrese in u cused  the incresed luminosit distnce is lmost ectl cncelled  the increse in u cused  the restrme elength moing closer to the pe o the dust emission this eect is reerred to s the negtie  correction see e.g. (<>)lin et l. (<>).  comining utions (<>) nd (<>) ith utions (<>) nd (<>) e cn clculte S850(M⋆,z) nd S1.1(M⋆, z) ",
    "Figure 3. seredrme µm S850 top nd .mm S1.1 ottom u densit in m ersus M⋆( M⊙) or isolted discs t integer redshits in the rnge z = 1 −6 see utions (<>) nd (<>). he summ u o  disc o ed M⋆increses ith redshit or to resons: . s shon in ig. (<>) the normlistion o the M⋆reltion increses ith redshit. . or ed M⋆ gs rction increses ith redshit. or ed Z  higher gs rction corresponds to  higher gsphse metl mss. oeer ecuse the normlistion o the  decreses s z increses the increse o the gsphse metl mss ith gs rction is prtill mitigted. oth the incresed  nd incresed dust mss cuse the summ u to increse. ",
    "here e cn sustitute the pproprite epressions or Mgas Re nd Z to epress S850 nd S1.1 in terms o M⋆nd z onl. ",
    " ",
    "ig. shos the S850 − M⋆nd S1.1 − M⋆reltions gien  utions (<>) nd (<>) respectiel or isolted discs t integer redshits in the rnge z = 1 − 6. s redshit increses glies ecome more gsrich nd compct oth eects cuse the  or  gien M⋆to increse see ig. (<>). or ed Z  higher gs rction corresponds to  higher gsphse metl mss. oeer ecuse the normlistion o the  decreses s z increses the increse o the gsphse metl mss ith gs rction is prtill mitigted.8 (<>)he incresed  nd Md oth result in  higher summ u or  gien M⋆. o produce n isolted disc  S850 & 3 − 5 m or S1.1 & 1 − 2 m t z ˘ 2 − 3 e re",
    "uire M⋆& 1011 M⊙. his lue is consistent ith the results o (<>)ichosi et l. (<>) (<>). ",
    "ote lso tht e cn use these reltions to clculte the epected S850/S1.1 rtio S850/S1.1 ˇ 2.3 this is similr to osertionl estimtes e.g. (<>)ustermnn et l. (<>). or simplicit e ill use this rtio to derie pproimte S850 lues to lso sho n S850 is on the relent plots. "
  ],
  "4.2.2 Infall-stage galaxy-pair SMGs ": [
    "uring the inll stge o  merger the discs re dominted  uiescent str ormtion tht ould occur een i the ere not merging. nl or nucler seprtion . 10 pc9 (<>)do the discs he s tht re signicntl eleted  the mutul tidl interctions  this result is consistent ith osered  eletions in mergers e.g. (<>)cudder et l. (<>). hus during the inll stge e ssume the discs re in  sted stte i.e. the he constnt  nd dust mss een ithout  source o dditionl gs this is  resonle pproimtion or the inll stge to ithin  ctor o . 2 see g.  o . or  merger o to progenitors ith stellr msses M⋆,1 nd M⋆,2 the totl u densit is Sλ= Sλ(M⋆,1) + Sλ(M⋆,2). he tpicl em sies o singledish summ telescopes re  or ˘ 130 pc t z ˘ 2 − 3 schemticll hen the proected seprtion is less thn this distnce the sources re lended into  single source. 10 (<>)o predict singledish counts e ssume the glies should e treted s  single source i the phsicl seprtion is < 100 pc. rom our simultions hich use cosmologicll motited orits e nd tht this timescle is o order ˘ 500 r. hough the timescle depends slightl on the mostmssieprogenitor mss e neglect this dependence ecuse it is sudominnt to rious other uncertinties. oeer this timescle is deried rom the z ˘ 2 − 3 simultions nd thus m e too long or mergers t higher z. ien the oe ssumptions the dut ccle or  gien Sλ′nd merger descried  moremssie progenitor mss M⋆,1 nd stellr mss rtio µ = M⋆,2/M⋆,1 is . r i Sλ(M⋆,1) + Sλ(M⋆,1µ) > Sλ′nd  otherise. ith the dut ccle in hnd e cn use utions (<>) nd (<>) to clculte the predicted numer densit nd counts. ",
    "o predict counts or  e simpl ssume tht the to discs re resoled into indiidul sources nd thus tret them s to isolted disc glies s descried elo. "
  ],
  "4.2.3 Isolated disc counts ": [
    " ",
    "",
    "or  gien Sλnd z e inert the Sλ(M⋆, z) unctions utions nd  to clculte the minimum M⋆reuired or  gl t redshit z to he summ u densit > Sλ M⋆(Sλ|z). o clculte the numer densit n(> Sλ,z) e then simpl use the strorming gl  to clculte ",
    " ",
    "Figure 4. redicted cumultie numer counts or the unlensed  popultion s osered ith singledish summ telescopes N(> S1.1) in deg−2  ersus S1.1 m. he counts re decomposed into the three  supopultions e model: the green longdshed line corresponds to isolted disc glies the lue dshed to glpir s i.e. inllstge prestrurst mergers nd the red dshdotted to mergerinduced strursts. he lc solid line is the totl or ll  supopultions e model. he model predictions o (<>)ugh et l. (<>) n dotted (<>)rnto et l. (<>) mgent dotted nd (<>)ontnot et l. (<>) mroon dotted re shon or comprison. he points re osered .mm nd nd µm counts see the tet or detils. he nd µm counts he een conerted to .mm counts  ssuming S850/S1.1 ≈S870/S1.1 = 2.3. his rtio hs lso een used to sho the pproimte S850 on the top is. he htched re shos the regime here e lensing is epected to signicntl oost the counts or oerdense elds (<>)retg et l. (<>). he counts predicted  our model gree er ell ith the counts tht re not thought to e oosted signicntl  lensing. .. he steepness o the cuto in the strurst counts t S1.1 & 4 m is rticil see the tet or detils. ",
    "nd e use ution (<>) to clculte the predicted counts. o predict counts or singledish summ telescopes here the glpir s re lended into  single source e sutrct the rction o glies ith M⋆> M⋆(Sλ|z) tht re in mergers rom the isolted disc counts to oid doule counting. "
  ],
  "5 RESULTS ": [
    "ere e present the e results o this or the  cumultie numer counts the reltie contriutions o the supopultions nd the redshit distriution predicted  our model. e ocus on the  (<>)ilson et l. (<>) .mm counts here ecuse to our noledge the estconstrined lneld counts i.e. those rom the deepest nd idest sures he een determined using tht instrument (<>)ustermnn et l. (<>) (<>)retg et l. (<>). oeer ecuse or our simulted s S850/S1.1 = 2.3 to ithin ˘ 30 per cent e cn esil conert the .mm counts to µm counts. hus e include oth S1.1 nd S850 lues on the relent plots nd conert osered µm counts to .mm counts  ssuming the sme rtio holds or rel s. "
  ],
  "5.1 SMG number counts for single-dish observations ": [
    "ig. (<>) shos the totl cumultie .mm numer counts lc solid line hich re clculted rom the cumultie numer densit using ution (<>). e decompose the counts into isolted ",
    "discs green longdshed gl pirs lue dshed nd strursts induced t merger colescence red dshdotted the reltie contriution o ech supopultion is discussed in ection (<>).. he dt points in ig. (<>) re osered counts rom rious sures: .mm counts rom (<>)retg et l. (<>) circles (<>)ustermnn et l. (<>) sures (<>)tsude et l. (<>) dimonds nd (<>)cott et l. (<>) tringles µm counts rom (<>)nudsen et l. (<>) steriss nd (<>)emco et l. (<>) plus signs nd µm counts rom (<>)ei et l. (<>) s. he nd µm counts he een conerted to .mm counts  ssuming S850/S1.1 ˇ S870/S1.1 = 2.3. he model predictions o (<>)ugh et l. (<>) (<>)rnto et l. (<>) nd (<>)ontnot et l. (<>) re shon or comprison. ",
    "he predicted nd osered counts re in good greement t the loest ues ut the predicted counts re less thn some o those osered t the right end. he (<>)ustermnn et l. (<>) nd (<>)retg et l. (<>) sures re the to lrgest ˘ 0.7 deg−2 so their counts should e lest ected  cosmic rince nd thus most roust. hus it is encourging tht the greement eteen our predicted counts nd those o (<>)ustermnn et l. (<>) is er good t ll ues. he disgreement eteen our predicted counts nd those osered  (<>)retg et l. (<>) is signicnt een or the loer u ins  ctor o ˘ 2 or the S1.1 > 2 m in. oeer (<>)retg et l. (<>) conclude tht the ecess o sources t S1.1 & 5 m compred ith the  eld osered  (<>)ustermnn et l. (<>) is cused  sources modertel mplied  glgl nd glgroup lensing. t higher ues the eect o lensing is more signicnt (<>)egrello et l. (<>) (<>)cig et l. (<>) (<>)im et l. (<>) nd it ould e incredil dicult to eplin the sources ith mm u densit >> 10 m osered  (<>)ieir et l. (<>) nd (<>)egrello et l. (<>) i the re not strongl lensed. e do not include the eects o grittionl lensing in our model so it is unsurprising tht e signiicntl underpredict the counts o (<>)retg et l. (<>) despite the ecellent greement eteen our counts nd those osered  (<>)ustermnn et l. (<>). ",
    "dditionll it is importnt to note tht the steepness o the cuto in the strurst counts t S1.1 & 4 m is rticil: ecuse e determine the ues o the isolted discs nd gl pirs in n nltic  e cn etrpolte to ritrril high msses or those popultions. or the strursts hoeer e re limited  the prmeter spce spnned  our merger simultions. one o our strurst s rech S1.1 > 6.5 m or S850 & 15 m so the dut ccle or ll strursts or S1.1 > 6.5 m is ero.  e ere to simulte  gl more mssie thn our most mssie model  the simultion ould rech  correspondingl higher u so the predicted counts or S1.1 > 6.5 m ould no longer e ero. oeer the rrit o such oects does not usti the dditionl computtionl epense. hus or S1.1 & 4 m or S850 & 9 m the strurst counts should e considered  loer limit.  simple etrpoltion rom the loeru strurst counts suggests tht our model m een oerpredict the counts o the rightest sources. oeer the osered numer densit o sources ith S1.1 & 5 m is highl uncertin ecuse o the eects o smll numer sttistics cosmic rince nd lensing nd the uncertint in the model prediction is signicnt ecuse o uncertinties in the undnces nd merger rtes o such etreme sstems. hus the counts or the rightest sources should e interpreted ith cution. ",
    "urthermore e do not ttempt to model some other potentil contriutions to the  popultion. n prticulr e do not include contriutions rom mergers o more thn to discs clus",
    "Table 4. ingledishdetected  cumultie numer counts ",
    "Figure 5. rctionl contriution o ech supopultion to the totl cumultie counts ersus S1.1. he lines re the sme s in ig. (<>). t the loest ues the isolted discs dominte heres t the highest ues the strursts dominte. he gl pirs re ∼ per cent o the popultion t ll ues plotted here. ",
    "ters or phsicll unrelted sources lended into  single summ source see (<>)ng et l. (<>) or eidence o the lst tpe. ",
    "ien these cets nd the modelling uncertinties our predicted counts re clerl consistent ith those osered nd including lensing nd the preiousl mentioned dditionl possile contriutions to the  popultion ould tend to increse the numer counts. lso e stress tht our model is consertie in the sense tht it uses  roup  rther thn tophe or t   nd is tied to osertions heneer possile. he consistenc o the predicted nd osered counts suggests tht the osered  counts m not proide eidence or  rition this ill e discussed in detil in ection (<>).. "
  ],
  "5.2 Relative contributions of the subpopulations ": [
    "n preious or  (<>)rd et l. (<>)  e rgued tht the  popultion is not eclusiel ltestge mergerinduced strursts ut rther  heterogeneous collection o strursts inll",
    "stge mergers glpir s nd isolted discs. oeer so r e he onl presented the phsicl resons one should epect such heterogeneit. t is crucil to unti the reltie importnce o ech supopultion so e do this no. ",
    "he counts shon in ig. (<>) re diided into supopultions ut the reltie contriutions cn e red more esil rom ig. (<>) hich shos the rctionl contriution o ech supopultion to the totl cumultie counts. t the loest ues the isolted disc contriution is the most signicnt. t S1.1 ˘ 0.8 m S850 ˘ 2 m the three supopultions contriute lmost eull. s epected rom conentionl isdom the strursts dominte t the highest ues. oeer contrr to conentionl isdom the right s re not eclusiel mergerinduced strursts: rom ig. (<>) e see tht t ll ues plotted the gl pirs ccount or ˘  per cent o the totl predicted counts so the re  signiicnt supopultion o s in our model. s eplined in  the glpir s re not phsicll nlogous to the mergerinduced strurst s thus their potentill signicnt contriution to the  popultion cn complicte phsicl interprettion o the osered properties o s. ",
    "t is interesting to compre the reltie contriutions o the isolted disc nd glpir supopultions ecuse the reltie contriutions cn e understood  t lest schemticll  in  simple mnner. or  mor merger o to glies ith M⋆= Miso the u o the resulting glpir  is pproimtel tice tht o the indiidul isolted discs 2S1.1(Miso). ecuse S1.1 depends sulinerl on M⋆see ig. (<>) or n isolted disc to he S1.1 eul to tht o the gl pir it must he M⋆& 3Miso. hus the reltie contriution o the to supopultions depends on hether the numer densit o M⋆= 3Miso discs diided  tht o M⋆= Miso discs n(3Miso)/n(Miso) is greter thn the rction o M⋆= Miso discs undergoing  mor merger hich is the merger rte times the dut ccle o the inll phse ˘ r.  the ormer is lrger the M⋆= 3Miso discs ill dominte the pirs o M⋆= Miso discs heres i the merger rction is higher thn the reltie numer densit the gl pirs ill dominte. ",
    "he ltter scenrio is liel or right s hich re on the eponentil til o the . or emple t z ˘ 2 − 3  gl ith M⋆= 1011 M⊙undergoes ˘ 0.3 mergers per ",
    "r. hus i e ssume  dut ccle o  r or the glpir phse pproimtel  per cent o such glies ill e in gl pirs. or the (<>)rchesini et l. (<>)  the numer densit o M⋆= 3 × 1011 M⊙glies is ˘ 8 per cent tht o M⋆= 1011 M⊙glies. hereore  the oe logic the pirs o M⋆= 1011 M⊙glies ill contriute more to the summ counts thn the isolted M⋆= 3 × 1011 M⊙discs. his simple rgument demonstrtes h the gl pirs ecome dominnt oer the isolted discs or S1.1 & 0.7 m S850 & 1.6 m. oeer the threshold or dominnce depends on oth the S1.1 −M⋆scling nd the shpe o the  t the highmss end. hus osertionll constrining the rction o the  popultion tht is gl pirs cn proide useul constrints on oth the summ uM⋆reltion nd the shpe o the mssie end o the . ",
    "nortuntel the reltie contriution o the strurst supopultion cnnot e eplined in s simple  mnner. he dut ccles or the mergerinduced strursts depend sensitiel on progenitor mss nd merger mss rtio so the mpping rom merger rte to numer densit is not s simple s it is or the isolted discs nd gl pirs. ortuntel the  uncertint hich is er signicnt or the oerll counts is reltiel unimportnt or the reltie contriution o strursts nd gl pirs. hus the reltie contriutions o strursts nd gl pirs depend primril on their reltie dut ccles. o chiee  gien u densit one reuires  less mssie strurst thn gl pir ecuse the strurst increses the summ u densit modertel. hus the reltie numer densit lso mtters. oeer the inecienc o strursts t incresing the summ u densit o the sstem preents signicntl less mssie ut more common strursts rom dominting oer more mssie nd rrer gl pirs. he dut ccles re uncertin ut gien tht in our ducil model the gl pirs contriute ˘ 30 − 50 per cent o the totl counts nd the uncertint in the dut ccles is denitel less thn  ctor o 2 − 3 the prediction tht oth the strurst nd gl pir supopultions re signicnt i.e. more thn  e per cent o the popultion is roust. ",
    "hough there he een mn osertionl hints suggesting the importnce o the glpir contriution see  nd  or discussion the phsicl importnce o this supopultion hs to dte not een ull pprecited nd the rctionl contriution o glpir s to the totl counts remins reltiel poorl constrined. oeer cler osertionl eidence supporting the signicnce o this supopultion is ccumulting: o the  s presented in (<>)ngel et l. (<>)  he  emission tht is resoled into to components ith inemtics consistent ith to merging discs. n to o the cses the proected seprtion o the to components is > 20 pc such oects re prime emples o the glpir supopultion. ee lso (<>)cconi et l. (<>) (<>) (<>)othell et l. (<>) (<>)iechers et l. (<>)(<>). (<>)molcic et l. (<>) presented  lrger smple o s ith summintererometric detections. he ound tht hen osered ith intererometers ith . 2 resolution ˘ 15 − 40 per cent o singledish s ere resoled into multiple sources hich is consistent ith our prediction or the reltie contriution o the glpir supopultion.  osertions ill signiicntl increse the numer o s osered ith ˘ 0.5 resolution nd thus etter constrin the glpir contriution to the  popultion. ",
    "urther eidence or  glpir contriution consistent ith ht e predict is the rction o the s ith multiple counterprts t other elengths. ne o the erliest osertionl indictions o this popultion cme rom the  m sure: o ",
    "Figure 6. op: the predicted redshit distriution o .mm sources ith S1.1 > 1.5 m S850 & 3.5 m lc solid line compred ith the osered distriution rom (<>)un et l. (<>) lue dshed line. he men redshits re . nd . or the model nd osered s respectiel. ottom: the predicted redshit distriution or sources ith S1.1 > 4 m S850 & 9 m lc solid line compred ith the osered distriution rom (<>)molcic et l. (<>) lue dshed line. he men redshits re . nd . or the model nd osered s respectiel. n oth pnels the u limits ere chosen to pproimtel mtch those o the osertions. he righter s tend to e t higher redshits. ",
    "this smple o µm sources (<>)ison et l. (<>) ound tht ˘ 25 per cent he multiple rdio counterprts. pproimtel ten per cent o the  µm (<>)ope et l. (<>)  .mm (<>)hpin et l. (<>)  −µm (<>)ison et l. (<>) (<>)lements et l. (<>) nd  .mm (<>)un et l. (<>) sources he multiple counterprts. hese rctions re someht smller thn the ˘  per cent contriution shon in ig. (<>) ut oth the predicted nd osered rctions re uncertin. s eplined oe the predicted rction depends sensitiel on the shpe o the upperend o the  nd the reltion eteen summ u nd M⋆. sertions on the other hnd m miss the more idel seprted counterprts nd cses hen one o the counterprts is signicntl more oscured or is rdiouiet. "
  ],
  "5.3 Redshift distribution ": [
    "n ddition to the numer counts  successul model or the  popultion must reproduce the redshit distriution. ig. (<>) shos the redshit distriution o .mm sources predicted  our model or dierent .mm u cuts S1.1 > 1.5 m or S850 & 3.5 ",
    "Table 5. ingledishdetected  redshit distriution ",
    "m in the top pnel nd S1.1 > 4 m or S850 & 9 m in the ottom long ith some osered distriutions tht he similr u limits. he redshit distriutions re reltiel rod nd the pe in the rnge z ˘ 2 − 4 nd decline t loer nd higher redshits. he S1.1 > 1.5 m sources he men redshit . heres the S1.1 > 4 m sources he men redshit . so there is  tendenc or the righter sources to e t higher redshits this trend grees ith osertions (<>)ison et l. (<>) (<>)un et l. (<>) (<>)molcic et l. (<>). ",
    "or the S1.1 > 1.5 m s top pnel o ig. (<>) compred ith the osertions our model predicts  higher men redshit nd  greter rction o s t z ˘ 3 − 4. his discrepnc m suggest tht the etrpoltion o the (<>)ontn et l. (<>)  e use or z > 3.75 oerpredicts the numer o mssie glies t the highest redshits. urthermore merger timescles m e shorter t high redshit nd the dust content m e loer thn in our models oth o these eects ould decrese the highredshit contriution. dditionll constrining the redshit distriution is complicted  selection eects counterprt identiction nd in some cses  lc o spectroscopic redshits. picll the selection eects nd counterprt identiction me identiing higherredshit s more dicult. inll the signicnt dierences mongst osered distriutions see (<>)molcic et l. (<>) or discussion demonstrte the dicult o determining the redshit distriution. hus the dierences in the distriutions m not e signicnt spectroscopic olloup o  sources should clri this issue ut note tht the redshit distriution o  sources hich is shon in the net section should dier slightl. he tpicl redshit or the righter s ottom pnel o ig. (<>) is lso someht higher thn tht osered . compred ith . nd the pe t z ˘ 1 − 2 is not reproduced. oeer the osered distriution is sed on  smple o  sources so smllnumer sttistics might eplin the dierences. "
  ],
  "5.4 Predicted ALMA-detected SMG number counts and redshift distribution ": [
    "n the oe sections e he presented our model predictions or sures perormed ith singledish summ telescopes such s the ",
    "Figure 7. redicted cumultie numer counts or the  supopultion osered ith telescopes such s  tht he resolution sucient to resole the glpir supopultion into multiple sources. he green longdshed line corresponds to isolted disc glies the dshdotted red to strursts nd the lc solid to the totl counts. he singledish counts rom ig. (<>) re shon or comprison lue dsh dot dot dot. he top is shos pproimte S850 lues. ecuse some o the right sources ould e resoled into multiple sources ith  the counts o right  sources re s much s  ctor o  less thn the singledish counts. ",
    "Figure 8. redicted redshit distriutions or detected s ith S1.1 > 1.5 m S850 & 3.5 m lc solid line z¯ = 3.0 nd S1.1 > 4 m S850 & 9 m lue dshed line z¯ = 3.5. s or the singledish sources the righter s tend to lie t higher redshit. ",
    " or hich the ˘ 15 rcsec em cuses the glpir s to e lended into  single source or much o their eolution.  lrger singledish summ telescopes such s the  nd  re used then the glpir s ill e lended or  smller rction o the inll stge o the merger. hus the dut ccle or the glpir phse ill e shorter nd the numer counts o the right sources conseuentl less. or intererometers ith ngulr resolution . 0.5 rcsec or ˘ 4 pc t z ˘ 2 − 3 such s  essentill ll glpir s ill e resoled into multiple sources ecuse or such seprtions the mergers re tpicll dominted  the strurst mode . hus the gl pirs ould contriute to the numer counts s to uiescentl strorming glies rther thn one lended source. ",
    "o predict the counts nd redshit distriution o  sources e modi our model  remoing the glpir con",
    "Table 6. detected  cumultie numer counts ",
    "Table 7. detected  redshit distriution ",
    "triution nd redistriuting those glies into the isolted disc supopultion. he predicted cumultie numer counts re shon in ig. (<>) in hich the totl singledish counts re lso plotted or comprison. he lues o the counts nd the rctionl contriutions o the isolted discs nd strursts or rious u ins re gien in le (<>). o cilitte comprison ith osertions the tle includes pproimte u densities or  nds  nd  clculted using the men rtios or our simulted s SALMA−6/S1.1 = 0.8 ± 0.06 nd SALMA−7/S1.1 = 1.6 ± 0.2. ",
    "s or the singledish counts the isolted discs dominte t the loest ues S1.1 . 1 m or S850 . 2 m nd the rightest sources re predominntl strursts. ecuse the glpir s re resoled into multiple inter sources the cumultie numer counts or detected s re loer t ll ues  ˘ 30 − 50 per cent nd the dierentil counts should e steeper. t ues inter thn those shon or hich the isolted discs completel dominte nd the right sources contriute negligil ",
    "to the cumultie counts the singledish nd  cumultie counts ill e lmost identicl. his eect hs lso een discussed  (<>)ocs et l. (<>) nd (<>)molcic et l. (<>). ",
    "he redshit distriutions or to u cuts re shon in ig. (<>) nd the lues re gien in le (<>). he men redshits or the S1.1 > . nd  m S850 & 3.5 nd  m ins re . nd . respectiel. he men lues re lmost identicl to those or the singledish counts nd there is lso the sme tendenc or the rightest s to e t higher redshits. he redshit distriutions re similr ut the distriution or the S1.1 > 4 m sources is less strongl peed thn or the singledish sources. he ltters redshit distriution is more strongl peed ecuse the redshit distriution o the right gl pirs pes t z ˘ 3.5. "
  ],
  "6 DISCUSSION ": [],
  "6.1 Are modi•cations to the IMF required to match the observed SMG counts? ": [
    "ne o the primr motitions or this or is to reemine the possiilit tht  numer counts proide eidence or  t   (<>)inn et l. (<>) (<>)e et l. (<>). o test this clim e he ssumed the null hpothesis  tht the  in s does not dier rom ht is osered locll  nd used  roup . urthermore our model is constrined to mtch the presentd mss unction  or more ccurtel to not oerproduce z ˘ 0 mssie glies   construction this is n importnt litmus test or puttie models o the  popultion. he counts predicted  our model gree ell ith the osered counts or elds elieed not to e signicntl ected  grittionl lensing e.g. (<>)ustermnn et l. (<>). hus our model does not reuire modictions to the  to mtch the osered counts. s n dditionl chec e crudel pproimte the eect o ring the  in strursts in our model  multipling the summ ues o our strursts  ctors o  to represent  mildl tophe  nd  to represent  t  ecuse this is the ctor pproprite or the  used in  .. rnto prite communiction. ",
    "ig. (<>) shos the rnge in numer counts predicted or this ",
    "Figure 9. redicted cumultie numer counts or the unlensed  popultion s osered ith singledish summ telescopes N(> S1.1) in deg−2  ersus S1.1 m here e he crudel pproimted the eects o using  mildl tophe t  in strursts  multipling the .mm u o the strurst s   ctor o  . he region lled ith cn lines indictes the rnge spnned   oost ctors eteen  nd . he counts or our stndrd model hich uses  roup  re shon or comprison. he points nd gre htched region re the sme s in ig. (<>). he strurst nd totl counts re incresed signicntl  the tophe  ut the isolted disc nd gl pir counts re not ected ecuse their ues re not ltered. he strursts dominte the  popultion or S1.1 & 1 m S850 & 2 m. or S1.1 & 2 m S850 & 5 m the mildl tophe  model oerpredicts the counts tht re thought not to e oosted  lensing nd the t  model oerpredicts ll the osered counts hich suggests tht  signicntl tophe  is ruled out. .. he lc o  oost in the counts or S1.1 . 1 m is rticil see the tet or detils. ",
    "rnge o  rition. his modiction cuses strursts to completel dominte the counts or S1.1 (S850) & 1 (2) m s is the cse or the  model nd the predicted counts re signicntl greter thn most o the osered counts or unlensed s.11 (<>)t the loer ues the similrit eteen the strursts counts or the rious  ctors is n rticil eect cused  the limited prmeter spce spnned  our simultion suite.  the suite included mergers o loermss glies dopting  tophe  ould cuse them to contriute to the counts t these ues nd thus oost the counts t ll ues. urthermore the strength o the eect t the highest ues is underestimted ecuse o the orementioned rticil cuto o our strurst counts t S1.1 & 4 m see (<>). or detils. ppling the t  ctor cuses the model to signicntl oerpredict ll the osered counts or S1.1 & 2 m S850 & 5 m nd s eplined oe the lc o n eect or loer ues is n rticil conseuence o our merger simultion suite not including loermss glies. he cler conclusion is tht in our model signicnt modiction to the  in strursts is unustied. ",
    "dmittedl the oe rguments ginst  rition depend on the detils nd ssumptions o our model. oeer there is  much simpler rgument ginst the  in strursts eing t or signicntl tophe: s discussed oe nd in detil ",
    "in  nd  there is groing osertionl eidence tht  signicnt rction o the singledishdetected  popultion is ttriutle to multiple sources lended into one summ source. n some cses the s re erlstge mergers ith components seprted  & 10 pc see (<>)ngel et l. (<>) nd (<>)iechers et l. (<>)(<>) or ecellent emples. drodnmicl simultions suggest tht t such seprtions  strong strurst is tpicll not induced  this is consistent ith osertions o locl gl pirs e.g. (<>)cudder et l. (<>). hen the glies eentull merge their s ill increse   ctor o  e t the lest tpicll signicntl more or mor mergers ut e ill e consertie nd i the  is t in strursts the summ u per unit  produced ill increse   ctor o ˘ 5. hus the summ u o the glies should increse   ctor o & 10 during the strurst. he reltie dut ccles o the glpir nd strurst phses re similr to ithin  ctor o  e so or  gien mss nd mss rtio the numer densit o strursts nd mergers in the glpir phse should e similr. hus i the summ u is & 10 times greter in the strurst phse thn in the uiescentl strorming glpir phse mergers during the glpir phse should contriute negligil to the right  popultion s is the cse in ig. (<>) een hen  more modest oost  ctor o  is used. his is in str contrdiction ith osertionl eidence. ndeed the eistence o the glpir supopultion is  nturl conseuence o strursts eing er inecient t oosting the summ u o merging glies see the discussion in ection (<>).. he onl mens to circument this rgument is to rgue tht the multiplecomponent s osered re ll strursts ut s eplined oe this is unliel. ",
    "he oe rgument does not rule out the possiilit o  mildl tophe  in strursts or sstemtic glol eolution o the  ith redshit s suggested  e.g. (<>)n oum (<>) (<>)e (<>) (<>)rnn  e (<>) it onl reuires tht the s in strurst nd glpir s t  gien redshit e similr. urthermore the rgument does not rule out the possiilit o  ottomhe  in strursts hich hs een suggested recentl or locl mssie ellipticls the descendnts o high strursts e.g. (<>)n oum  onro (<>) (<>) (<>)onro  n oum (<>) (<>)opins (<>).  the  ere ottomhe in strursts the scling eteen summ u nd  ould e eer thn ht e he shon in  nd here. he eer scling ould urther increse the contriution o glpir s to the  popultion. nortuntel the model nd osertionl uncertinties re sucientl lrge tht e cnnot use the osered glpir rction to rgue or or ginst  ottomhe  in strursts ut the oe rguments suggest tht  signicntl tophe  is unliel. "
  ],
  "6.2 Differences between our model and other work ": [
    "ecuse e nd tht contrr to some preious suggestions  (<>)inn et l. (<>) (<>)e et l. (<>)  tophe  is not reuired to mtch the osered  counts it is orthhile to emine h our results dier rom those ors. here re multiple resons our results m dier: the cosmologicl contet undnces nd merger rtes the eolution o  nd dust mss or indiidul mergers the rditie trnser clcultion the eects o lending nd dierences in the osered counts used. e eplore these in turn no. "
  ],
  "6.2.1 Radiative transfer calculation ": [
    "n  e demonstrted tht the summ u densit o our simulted glies cn e ell prmeteried s  poer l in  nd dust mss see utions (<>) nd (<>).  the sme reltion does not hold in the  model then dierences in the rditie trnser m e one cuse o the discrepnc eteen our counts nd theirs. hile e he een unle to compre directl ith the  model e he compred our results ith those o   tht uses  similr rditie trnser tretment (<>)enson (<>). e nd tht reltions similr to utions (<>) nd (<>) hold or the s in the  so it ppers tht the rditie trnser is not the primr cuse o the discrepnc een though some spects o the rditie trnser dier signicntl e.g. the geometr used in the  clcultions is ten directl rom the   simultions heres the geometr ssumed   is nlticll specied nd imuthll smmetric the  clcultions include  suresolution model or oscurtion rom the irth clouds round oung strs ut e opt not to use the corresponding suresolution model tht is implemented in  (<>)roes et l. (<>) or the resons discussed in sections .. nd . o . ",
    "(<>)e et l. (<>) did not perorm rditie trnser insted the ssumed tht the most rpidl strorming glies in their simultion ere s. oeer s noted preiousl  this simple nst is not necessril true ecuse o dierences in dust mss mongst glies nd the eects o lending. hus i dust rditie trnser ere perormed in or our tting unctions ere pplied to the (<>)e et l. (<>) simultions the conclusions might dier. "
  ],
  "6.2.2 Merger evolution ": [
    "erhps the time eolution o the  or dust mss in the   nd our model diers.  prmeterie the  in ursts s M˙ ⋆= Mgas,c/˝⋆ here Mgas,c is the cold gs mss nd ˝⋆is   timescle gien  ˝⋆= max[fdyn˝dyn, ˝⋆burst,min]. ere fdyn = 50 ˝dyn is the dnmicl time o the nel ormed spheroid nd ˝⋆burst,min = 0.2 r. he mor merger shon in g.  o  hs Mgas ˘ 1011 M⊙hen the glies re t colescence. et us suppose tht ll the gs is cold. hen the mimum  possile gie the  prescription is 1011 M⊙/0.2 r   M⊙yr−1  ˘ 9 times less thn tht o the simultion.  the dust mss is ept constnt utions (<>) nd (<>) impl tht  ctor o  decrese in  results in  ctor o ˘ 2.5 decrese in summ u hich ould signicntl ect the predicted counts. his is o course onl  crude comprison ut it demonstrtes tht the s o strursts in the  model m disgree ith those in our simultions nd dierences in the dust content m lso e importnt. ",
    "urther eidence tht the phsicl modelling o mergerinduced strursts m ccount or some o the discrepnc is proided  the dierent importnce o strursts in the to models. n the  model strursts dominte the summ counts   lrge mrgin nd contriute signicntl to the  densit o the unierse the dominte oer uiescentl strorming discs or z & 3. n our model isolted discs dominte the summ counts t the loest summ ues nd uiescentl strorming glpir s proide  signicnt contriution to the right counts. urthermore in our model mergerinduced strursts ccount or . 5 per cent o the  densit o the unierse t ll redshits (<>)opins et l. (<>)c this is consistent ith the strurstmode contriution inerred rom the stellr densities surce rightness ",
    "proles inemtics nd stellr popultions o osered merger remnnts (<>)opins et l. (<>) (<>)(<>) (<>)opins  ernuist (<>). ",
    "ierences in the s o mergers m lso contriute to the discrepnc eteen our model nd tht o (<>)e et l. (<>). t is not cler tht the resolution o the (<>)e et l. (<>) simultion 3.75h−1 pc comoing is sucient to resole the tidl torues tht drie mergerinduced strursts.  it is not the s during mergers o their simulted glies ould e underestimted nd prt o the discrepnc eteen their simulted s s nd those osered could e ttriuted to resolution rther thn  rition. urthermore their ind prescription m rticill suppress the  enhncement in mergerinduced strursts (<>)opins et l. (<>). ig.  o (<>)e et l. (<>) shos tht all their simulted glies lie signicntl under the osered M⋆reltion so it is perhps not surprising tht the cnnot reproduce the s o s hich re  mi o mergerinduced strursts tht re outliers rom the M⋆reltion nd er mssie uiescentl strorming glies tht lie ner the reltion  (<>)gnelli et l. (<>) (<>)ichosi et l. (<>). "
  ],
  "6.2.3 Cosmological context ": [
    "he third mor component o the models tht cn disgree is the cosmologicl contet hich includes the  nd merger rtes. n (<>)rd (<>) e demonstrted tht the predicted numer densities nd redshit distriution o isolted disc s re er sensitie to the ssumed . hus it is orthhile to compre the  in the s to the osertionll deried  e he used. hile  direct comprison o the   to those in the literture is complicted  the  models use o the t  in strursts see section . o (<>)ce et l. (<>) (<>)inn et l. (<>) he shon tht the  model underpredicts the restrme Knd ues o s hich suggests the msses o their model s re loer thn osered. his ould e  nturl result o n underprediction o the undnce o mssie glies. ",
    " the  model underpredicts the  then the must compenste  ming the strurst contriution signicntl higher the do this  enling er gsrich minor mergers to cuse strong strursts in their model minor mergers hich onl produce  strurst i the mostmssie progenitor hs ronic gs rction greter thn  per cent ccount or pproimtel threeurters o the  popultion (<>)onle et l. (<>) nd  modiing the  in strursts such tht or  gien  the produce signicntl greter summ u. n underprediction o the undnce o ll mssie glies nd suseuent need to strongl oost the strurst contriution ould eplin h the reltie contriutions o strursts nd uiescentl strorming glies dier so signicntl. "
  ],
  "6.2.4 Blending ": [
    "n dditionl reson or the discrepnc is tht neither  nor (<>)e et l. (<>) ccount or the lending o multiple glies into one summ source hich cn e signicnt or oth merging discs   nd phsicll unrelted glies (<>)ng et l. (<>). ur model suggest tht glpir s cn ccount or ˘ 30 − 50 per cent o the  popultion ttriutle to isolted discs nd mergers. he tpes o sources (<>)ng et l. (<>) osered could lso e importnt. hus not ccounting or lending could ccount or  ctor o ˘ 2 discrepnc eteen the predicted nd osered counts. "
  ],
  "6.2.5 Revised counts ": [
    "inll  compred ith nd (<>)e et l. (<>) utilised numer counts tht he since een superseded  sures coering signicntl lrger res. he ne counts re s much s  ctor o  loer thn those o e.g. (<>)hpmn et l. (<>) so the dierence in counts is nother nontriil ctor tht cn eplin prt o the discrepnc eteen our conclusion nd tht o some preious ors. "
  ],
  "6.3 Uncertainties in and limitations of our model ": [
    "ur model hs seerl dntges: the  enles us to isolte possile discrepncies eteen our model nd other or tht originte rom dierences in the dnmicl eolution o gl mergers nd the dust rditie trnser clcultion rther thn more generl issues such s n oerll underprediction o the  t z ˘ 2−4.  constrining the  nd gs rctions to mtch osertions nd including no dditionl gs in our models e mtch the z ˘ 0 mss unction  construction. ecuse e use  hdrodnmicl simultions comined ith  dust rditie trnser clcultions e cn more ccurtel clculte the dnmicl eolution o glies nd the summ u densities thn cn either s or cosmologicl simultions. urthermore e consertiel ssume  roup  rther thn inoe d hoc modictions to the . ",
    "oeer our model lso hs seerl limittions: rst computtionl constrints preent us rom running simultions scled to dierent redshits. nsted e scle ll initil disc glies to z ˘ 3. dell e ould run simultions ith structurl prmeters nd gs rctions scled to rious redshits ecuse the rition in the glies phsicl properties ith redshit m cuse the summ light cures to depend on redshit. n uture or e ill run  lrge suite o simultions tht ill more ehustiel smple the relent prmeter spce such  suite could e used to more ccurtel predict the  counts nd redshit distriution. oeer ecuse our predicted counts re dominted  glies t z ˘ 3 our conclusions ould liel not dier ulittiel. ",
    "econd to he the resolution necessr to perorm ccurte rditie trnser on  sucient numer o simulted glies e must use idelised simultions o isolted disc glies nd mergers rther thn cosmologicl simultions. n principle including gs suppl rom cosmologicl scles e.g. (<>)eres et l. (<>) could chnge the results. oeer such gs suppl is implicitl included in our model or the isolted discs nd glpir supopultions ecuse e use osertionll deried gs rctions or these supopultions. ecuse the dut ccles or mergers re clculted directl rom the merger simultions hich do not include dditionl gs suppl the gs contents nd thus s o the strurst s could e incresed i cosmologicl simultions ere used. oeer ecuse the summ u densit depends onl el on   ctor o  increse in the  increses the summ u densit  ˘ 30 per cent  this ould not chnge our results ulittiel. imilrl dierences in the dust content could ect the results ut  ctor o  decrese in the dust mss decreses the summ u densit  . 50 per cent. ",
    "hird our predictions re ected  uncertinties in the osertions used to constrin the model.  prticulr importnce re uncertinties in the normlistion nd shpe o the  hich re especill signicnt t z & 4. or z . 3 the merger rtes predicted  our model re uncertin   ctor o . 3 ut the uncertinties re higher t higher redshits.  decrese in the normlistion o the  or merger rtes ould decrese the counts ",
    "predicted  our model. oeer the most interesting conclusions ould remin: .l pirs ould still proide  signicnt contriution to the  popultion. . epending on the ctor  hich the undnces re decresed  mildl tophe  might not e ruled out. oeer  t  ould still oerpredict the counts een i the  normlistion nd merger rtes ere decresed  & 10 times hich is surel n oerestimte o the uncertint. ",
    "ourth e do not include the contriution rom phsicll unrelted i.e. nonmerging glies lended into  single summ source. rom osertions it is non tht such glies contriute to the  popultion (<>)ng et l. (<>). hether such sources contriute signicntl to the popultion is n open uestion tht should e ddressed  otining redshits or s osered ith .  the do contriute signicntl inclusion o this supopultion ould increse the numer counts predicted  our model chnge the reltie contriutions o the supopultions nd potentill lter the redshit distriutions hoeer this ould onl strengthen the eidence ginst  tophe . ",
    "inll e do not model the eects o grittionl lensing. heres the inter sources re liel dominted  unlensed s the right counts S1.1 & 5 m or S850 & 11.5 m re liel oosted signicntl  lensing (<>)egrello et l. (<>) (<>) (<>)cig et l. (<>) (<>)im et l. (<>) (<>)ieir et l. (<>). urthermore the liel cuse o the discrepnc eteen the counts osered  (<>)ustermnn et l. (<>) nd (<>)retg et l. (<>) is tht the ltter counts re oosted  glgl e lensing see the discussion in (<>)retg et l. (<>) so the discrepnc eteen our predicted counts nd the (<>)retg et l. (<>) counts might e resoled i e included the eects o lensing rom  oreground oerdensit o glies. gin our model predictions re consertie ecuse inclusion o grittionl lensing ould oost the counts thus inclusion o lensing ould strengthen the rgument ginst  tophe . "
  ],
  "7 CONCLUSIONS ": [
    "e he presented  noel method to predict the numer counts nd redshit distriution o s. e comined  simple  ith the results o  hdrodnmicl simultions nd dust rditie trnser to clculte the contriutions o isolted discs gl pirs i.e. inllstge mergers nd ltestge mergerinduced strursts to the  numer counts. ur model is constrined to osertions s much s possile conseuentl e re le to isolte the eects o uncertinties relted the dnmicl eolution o mergers nd the dust rditie trnser  hich re perhps uniuel relent to the  popultion  rom more generl issues tht ect the highredshit gl popultion s  hole such s the . urthermore e he consertiel used  roup  s opposed to t or tophe   ecuse e ish to test hether e cn mtch the osered counts ithout modiing the  rom ht is osered locll. ur principl results re: ",
    "cept t the loest ues S1.1 . 1 m or S850 . 2 m mergerinduced strursts ccount or the ul o the popultion not ccounted or  glpir s nd the rightest sources re predominntl mergerinduced strursts. hus isolted discs contriute negligil to the right  popultion. he ",
    "contriution o isolted discs to the  popultion is  roust testle prediction o our model. "
  ],
  "ACKNOWLEDGMENTS ": [
    "e thn ndre enson omeel e r inn nd oi oshid or comments on the mnuscript cott hpmn ich ichosi ierluigi onco le ope sc oseoom nd osh ounger or useul discussion rlton ugh edric ce io ontnot inuigi rnto ernes molcic el ei nd in un or proiding dt ith hich e he compred nd or useul discussion nd oler pringel or proiding the nonpulic ersion o  used or this or.  cnoledges support rom  tionl cience oundtion rnt .  s supported   through ule elloship grnt ..  cnoledges support   grnt rom the . . ec oundtion. he simultions in this pper ere perormed on the dsse cluster supported  the  eserch omputing roup t rrd niersit. "
  ]
}