2.1. Overview and history

Stars and star clusters form when dense gas collapses through gravitational or turbulent processes. The physical state of the gas (including temperature, pressure, metallicity, and turbulence) determines which pockets of gas fragment and collapse, and so ultimately the masses of the stars formed. Since the evolution of a single star is almost entirely determined by its initial mass (although binary effects also play a role), and the distribution of mass within a bound system defines its kinematics, the evolution of a cluster of coeval stars is determined almost entirely by its stellar IMF. The evolution of a galaxy composed of such clusters depends intimately on this (potentially varying) IMF in combination with its SFH.

The IMF establishes the fraction of mass sequestered in sub-solar-mass stars (down to masses as little as 0.1 M_⊙) with lifetimes much greater than the age of the Universe, and the high-mass fraction (stars up to 120 M_⊙ or perhaps more) that rapidly become supernovae, returning chemically enriched gas to the interstellar medium to support subsequent generations of star formation. The less numerous higher mass stars dominate the light from a star cluster or a galaxy, but the more numerous lower mass stars dominate the mass. This results in a need for different tracers to probe the high- and low-mass regimes of the IMF. It also means that the mass-to-light ratio is sensitive to the IMF shape.

The IMF is consequently the fundamental concept linking each of: (1) the process of star formation itself through the conversion of molecular clouds (enriched to some degree by heavy elements) into a population of stars; (2) feedback and chemical enrichment processes arising from the radiative and mechanical energy returned to the interstellar medium through stellar winds and supernovae from existing stellar populations that influence subsequent generations of stars and their metallicity; (3) the measurements used to convert observables (such as broadband luminosities or spectral line measurements), to underlying physical quantities (such as the current rate of star formation and total stellar mass), in order to enable studies of star formation and galaxy evolution.

The IMF was first measured by Salpeter (Reference Salpeter1955) while working at the Australian National University, by measuring the luminosity distribution of stars in the solar neighbourhood. It was shown to be consistent with a power law over the mass range 0.4 ≲ m/M_⊙ ≲ 10. Numerous measurements of the IMF over the subsequent sixty years (e.g. Miller & Scalo Reference Miller and Scalo1979; Scalo Reference Scalo, De Loore, Willis and Laskarides1986; Basu & Rana Reference Basu and Rana1992; Kroupa, Tout, & Gilmore Reference Kroupa, Tout and Gilmore1993) show that this power law does not extend to the lowest masses but has a flatter slope below about half a solar mass. The original power law slope for high-mass stars found by Salpeter extends up to about 120 solar masses (e.g. Scalo Reference Scalo, De Loore, Willis and Laskarides1986, Reference Scalo, Gilmore and Howell1998; Kroupa Reference Kroupa2001; Chabrier Reference Chabrier2003a), although with some variation (but also large observational uncertainties) in the reported high-mass slope and upper mass limit, and some debate about the value for the characteristic or ‘turn over’ mass.

While much of the observational work on the IMF in the late 20^th century focused on this same method of using resolved star counts as the most robust and direct approach available, Kennicutt (Reference Kennicutt1983) pioneered an approach using integrated galaxy light. Many alternatives were also explored, as summarised by Scalo (Reference Scalo, De Loore, Willis and Laskarides1986) and Kennicutt (Reference Kennicutt, Gilmore and Howell1998). These include a range of approaches such as ultraviolet (UV) luminosities of galaxies (Donas & Deharveng Reference Donas and Deharveng1984), indirect approaches related to chemical evolution and abundance ratios (Audouze & Tinsley Reference Audouze and Tinsley1976), and others like galaxy mass-to-light ratios that are now more routinely used to estimate IMF properties (e.g. Treu et al. Reference Treu, Auger, Koopmans, Gavazzi, Marshall and Bolton2010).

The IMF was also used as a probe of cosmology and dark matter. For example, constraints on the IMF and cosmology were inferred from the evolution of galaxy colours (Tinsley Reference Tinsley1972), number counts of galaxies (Guiderdoni & Rocca-Volmerange Reference Guiderdoni and Rocca-Volmerange1990), and the form of the IMF was invoked to explore the extent to which stellar remnants (e.g. Dantona & Mazzitelli Reference Dantona and Mazzitelli1986) or substellar objects (e.g. Staller & de Jong Reference Staller and de Jong1981) could explain the ‘missing matter’ in the solar vicinity (Bahcall Reference Bahcall1984). The cosmological constraints associated with the IMF are no longer compelling in the age of precision cosmology (e.g. Schmidt et al. Reference Schmidt1998; Planck Collaboration et al. 2016). Likewise, as the numbers of substellar objects have been progressively constrained by observations (e.g. Tinney Reference Tinney1993; Kroupa et al. Reference Kroupa, Tout and Gilmore1993) and other approaches matured in ruling out stellar-related contributions to possible baryonic dark matter (Graff & Freese Reference Graff and Freese1996a, Reference Graff and Freese1996b; Alcock et al. Reference Alcock2000, Reference Alcock2001), this aspect of the IMF has also become less important. With the establishment of the now standard ΛCDM model, the focus on the IMF now is primarily connected to the physics of star formation and galaxy evolution.

Part of the challenge in understanding the IMF as currently conceived is that it is a fundamentally statistical concept and not directly observable. Elmegreen (Reference Karlsson, Bromm and Bland-Hawthorn2009) notes that when estimating the IMF for star clusters, ‘no cluster IMF has ever been observed throughout the whole stellar mass range’. He explains that to probe the upper mass range of the IMF needs a very massive cluster, which are rare systems, with the nearest being too far away (a few kpc) to see the low-mass stars. Conversely the nearest clusters, required for measuring the low-mass end of the IMF, are all low-mass clusters having few high-mass stars. He concludes: ‘Until we can observe the lowest mass stars in the highest mass clusters an IMF makes sense only for an ensemble of clusters or stars’. It is notable that the science cases for the next generation of major telescope facilities, James Webb Space Telescope (JWST), Giant Magellan Telescope (GMT), Thirty Metre Telescope (TMT), European Large Telescope (ELT), all include the goal of studying resolved star formation in such high-mass Galactic star clusters. Kroupa et al. (Reference Kroupa, Weidner, Pflamm-Altenburg, Thies, Dabringhausen, Marks and Maschberger2013) take this concept a step further and detail why the IMF is not ever a measurable quantity, by noting that star formation occurs on Myr timescales. This means that for stellar systems younger than about 1Myr, star formation has not ceased and so the IMF is not yet assembled, while for systems older than about 0.5 Myr, higher mass stars are lost through stellar evolutionary effects, while dynamical processes can also cause the loss of lower mass stars. This means that there is no single time at which the full ensemble of masses is present and measurable within a discrete spatial volume. There is hence a need to address the issue of the short but finite time of formation, together with the fact that star clusters do not form in isolation (typically) but within a complex, multiphase interstellar medium that is also influenced by, and influencing, adjacent sites of star formation.

A possible solution to this issue arises through considering how many independent samples are required, and over what spatial scale they must be probed, in order to infer the IMF robustly. By sampling a sufficiently large number of star-forming regions it might be expected that each evolutionary stage is captured and the ensemble can be used to infer the underlying IMF. Kruijssen & Longmore (Reference Kruijssen and Longmore2014) describe a general formalism, which they apply to star formation scaling relations in galaxies, that links the timescale of different phases of a process with the number of independent samples required to capture all temporal phases and the spatial scale on which the processes are measured. They note that ‘[star formation] relations measured in the solar neighbourhood are fundamentally different from their galactic counterparts’ and conclude that ‘… when a macroscopic correlation is caused by a time evolution, then it must break down on small scales because the subsequent phases are resolved’. Considering the temporal dependencies of star formation and the range of spatial scales over which we are interested in characterising it, it may be that the formalism and concept of the IMF itself may need to be restructured (Scalo Reference Scalo, Gilmore and Howell1998).

Despite these difficulties, a range of the early approaches towards inferring the IMF have been refined over the past decade and are now used routinely. These include an update of the Kennicutt (Reference Kennicutt1983) approach used by Hoversten & Glazebrook (Reference Hoversten and Glazebrook2008) and Gunawardhana et al. (Reference Gunawardhana2011), use of the Wing–Ford band to infer dwarf-to-giant ratio (e.g. Cenarro et al. Reference Cenarro, Gorgas, Vazdekis, Cardiel and Peletier2003; van Dokkum & Conroy Reference van Dokkum and Conroy2010; Smith, Lucey, & Carter Reference Smith, Lucey and Carter2012) following the early work of Whitford (Reference Whitford1977), use of kinematics (e.g. Cappellari et al. Reference Cappellari2012), gravitational lensing observations (e.g. Treu et al. Reference Treu, Auger, Koopmans, Gavazzi, Marshall and Bolton2010; Smith & Lucey Reference Smith and Lucey2013), chemical abundance constraints (e.g. Portinari, Sommer-Larsen, & Tantalo Reference Portinari, Sommer-Larsen and Tantalo2004a; Komiya et al. Reference Komiya, Suda, Minaguchi, Shigeyama, Aoki and Fujimoto2007; Sliwa et al. Reference Sliwa, Wilson, Aalto and Privon2017), and more. The 2.3 μm CO index has also been proposed for probing the dwarf-to-giant ratio (Kroupa & Gilmore Reference Kroupa and Gilmore1994; Mieske & Kroupa Reference Mieske and Kroupa2008). In the same period other novel approaches have been developed, such as those using cosmic census measurements to place constraints on the IMF (e.g. Baldry & Glazebrook Reference Baldry and Glazebrook2003; Hopkins & Beacom Reference Hopkins and Beacom2006; Wilkins et al. Reference Wilkins, Trentham and Hopkins2008a, Reference Wilkins, Hopkins, Trentham and Tojeiro2008b).

With this explosion in the range of approaches now being used to measure or infer the IMF, there has been a related growth in the tension between apparently conflicting results. One example is a need for so-called ‘top-heavy’ IMFs (a relative excess of high-mass stars compared to the nominal Salpeter IMF) in regions of elevated star formation rate (SFR) (e.g. Gunawardhana et al. Reference Gunawardhana2011) that contrasts with the so-called ‘bottom-heavy’ IMFs (a relative excess of low-mass stars) inferred in the cores of massive elliptical galaxies (e.g. van Dokkum & Conroy Reference van Dokkum and Conroy2012). It is less clear whether such results are actually in conflict or not. The different approaches measure different things, and the spatial scales probed are different as is the epoch for the star formation activity. The current review is aimed at assessing the available wealth of different metrics and their results in a self-consistent fashion, to begin to unify our approach to understanding the IMF.

With this context in mind it is first necessary to review the terminology used in discussing the IMF and to explore conventions of nomenclature.

2.2. IMF definitions and terminology

At its most concrete, the IMF can be defined as the mass distribution of stars arising from a star formation event. It has been described and inferred in this sense from observational measurements by innumerable authors over more than sixty years (e.g. Salpeter Reference Salpeter1955; Miller & Scalo Reference Miller and Scalo1979; Kennicutt Reference Kennicutt1983; Scalo Reference Scalo, De Loore, Willis and Laskarides1986; Kroupa Reference Kroupa2001; Chabrier Reference Chabrier2003a), who have found that the IMF in many cases follows a similar form and established the broad properties of this distribution. In general, the IMF has a declining power law shape for masses above about 1 M_⊙, with a flatter slope at lower masses down to some minimum mass. Below the stellar/sub-stellar boundary brown dwarfs are often now included in IMF estimates (e.g. Kroupa et al. Reference Kroupa, Weidner, Pflamm-Altenburg, Thies, Dabringhausen, Marks and Maschberger2013), with a more positive slope below the hydrogen burning mass limit (although the shape at the lowest masses may be more complex, e.g. Drass et al. Reference Drass, Haas, Chini, Bayo, Hackstein, Hoffmeister, Godoy and Vogt2016). The observed mass function across the stellar/sub-stellar boundary may be a superposition of two physically distinct IMFs, inferred from the deficit in models compared to observations of brown dwarfs that form through direct gravitational collapse in molecular clouds (Thies et al. Reference Thies, Pflamm-Altenburg, Kroupa and Marks2015). The general shape and key parameters of the IMF are illustrated in Figure 1.

Figure 1. An illustration of the key aspects of the IMF as it has been parameterised, either as a piecewise series of power law segments (e.g. Kroupa Reference Kroupa2001) or a log-normal at low masses with a power law tail at high masses (e.g. Chabrier Reference Chabrier2003a).

Many authors have summarised the range of functional forms used to parameterise the IMF, with the common choices being piecewise power laws (e.g. Kroupa et al. Reference Kroupa, Weidner, Pflamm-Altenburg, Thies, Dabringhausen, Marks and Maschberger2013, their equations 4 and 5) or a log-normal form (e.g. Chabrier Reference Chabrier2003b, Reference Chabrier, Corbelli, Palla and Zinnecker2005). Alternative functional forms have been proposed with varying motivations (e.g. De Marchi, Paresce, & Portegies Zwart Reference De Marchi, Paresce, Portegies Zwart, Corbelli, Palla and Zinnecker2005; Parravano, McKee, & Hollenbach Reference Parravano, McKee and Hollenbach2011; Maschberger Reference Maschberger2013) that largely provide the same practical functionality as the more commonly used forms.

Key parameters are: (1) the lower mass limit, m_l, typically chosen as m_l = 0.08 M_⊙ or m_l = 0.01 M_⊙ (unless substellar objects are included, in which case m_l = 0.01 M_⊙ is common); (2) the upper mass limit, m_u, with typical values of m_u = 100 M_⊙, m_u = 120 M_⊙ or m_u = 150 M_⊙; (3) m_c, the characteristic mass, which is the peak in the lognormal form, or the ‘turn over’ mass where the slope of the power law representation changes (although as seen in Figure 1 this isn’t necessarily an actual turn over in the relation), with m_c ranging from about 0.2–1 M_⊙ depending on the formalism chosen, and m_c = 0.5 M_⊙ common in the power law representation; (4) the slope parameters for each segment of a piecewise power law relation, or the equivalent in the lognormal relation defining the width of the relation at low masses, and the power law slope at high masses. Here and throughout I use α_s for the substellar power law slope, α_l for the low-mass slope, and α_h for the high-mass slope (noting m_c for clarity when relevant). This choice avoids a numerical sequence in which α ₁ (say) is ambiguous depending on the value of m_l, i.e., whether the IMF in question includes substellar masses or not. Where a single power law spanning more than one of these segments is assumed, I use α and define the mass range explicitly.

The same functional form for the mass distribution of stars formed in a single star-forming region and called ‘the IMF’ is also used to describe the average mass distribution of stars formed across a galaxy, a concept sometimes referred to as the ‘integrated galaxy IMF’ or IGIMF (Kroupa & Weidner Reference Kroupa and Weidner2003), as well as to the effective average stellar mass distribution for a population of galaxies, referred to as the ‘cosmic IMF’ (Wilkins et al. Reference Wilkins, Hopkins, Trentham and Tojeiro2008b). If the IMF is universal, these quantities may be identicalFootnote ^b but not otherwise. Such broad application of the term ‘IMF’, validated through an underlying presumption of an IMF that is ‘universal’ until proven otherwise, may actually be hampering attempts to further understand the properties of the IMF including whether or not it varies. I return to this point below in Sections 2.4 and 8.

Other issues that hamper progress arise from inconsistent conventions describing the IMF. This field suffers from a wide range of such inconsistencies, and these seem to be growing in number rather than converging as the breadth of investigations increases. In an effort to stem this flow, I explicitly address these next.

2.3. Conventions and language usage

Ambiguities in the way the IMF is discussed unnecessarily complicate what is already a complex problem. Different authors adopt different conventions or approaches to the description of the IMF. Different language is used to describe the same quantity, mass ranges are omitted or assumed implicitly, and ambiguous terms are introduced. While not a fundamental problem, this definitely leads to confusion and the potential for misinterpretation, which can be easily avoided if clear conventions and unambiguous language are used. That this point has been made repeatedly by different authors (e.g. Kennicutt Reference Kennicutt, Gilmore and Howell1998; Bastian et al. Reference Bastian, Covey and Meyer2010) and still bears repeating is evidence that it deserves attention. Similar issues of convention and usage have been recognised by the cosmology community (Croton Reference Croton2013), emphasising the importance of striving for clarity.

Here I recount a number of sources of potential confusion and make recommendations for avoiding ambiguity, while acknowledging that the majority of authors do tend to be diligent. The bottom line, though, is that because of the many potential sources of confusion in this field there is an especial need for authors, referees, and editors to make extra effort to ensure clarity and consistency.

2.3.1. Sign conventions

Different authors have adopted a variety of nomenclature to represent the shape and power law slope(s) of the IMF, and in particular whether or not a negative sign is given in the power law definition or appears in the parameter. Opposing sign conventions may even appear within a single publication (e.g. Elmegreen Reference Elmegreen2009; Turner Reference Turner2009). This can lead to unnecessary confusion, especially when discussing the exponent of a power law slope in a distribution that has opposite signs at the low-mass and high-mass ends. It is worth noting that opposing conventions for the use of the negative sign have existed almost as long as work in this field. Salpeter (Reference Salpeter1955) did not use a pronumeral descriptor for the power law slope at all, but provided the power law value explicitly in his Equation (5), a choice followed by Kennicutt (Reference Kennicutt1983). Audouze & Tinsley (Reference Audouze and Tinsley1976) and Tinsley (Reference Tinsley1977) give the negative sign in the equation, a choice subsequently recommended by Kennicutt (Reference Kennicutt, Gilmore and Howell1998), but Miller & Scalo (Reference Miller and Scalo1979), Scalo (Reference Scalo, De Loore, Willis and Laskarides1986), and Kennicutt, Tamblyn, & Congdon (Reference Kennicutt, Tamblyn and Congdon1994) use the convention that any sign is incorporated into the slope parameter.

I strongly recommend that, to minimise ambiguity, all authors adopt the latter convention (following, e.g. Scalo Reference Scalo, De Loore, Willis and Laskarides1986; Kennicutt et al. Reference Kennicutt, Tamblyn and Congdon1994):

(1) \begin{equation} {{{\rm{d}}N} \over {{\rm{d}}m}} \propto {\left( {{m \over {{{\rm{M}}_ \odot }}}} \right)^\alpha }{\rm{a}}nd \end{equation}

(2) \begin{equation} {{{\rm{d}}N} \over {{\rm{d}}\log m}} \propto {\left( {{m \over {{{\rm{M}}_ \odot }}}} \right)^\Gamma }, \end{equation}

where Γ = α + 1 and the original Salpeter slope is α = −2.35 or Γ = −1.35. Contrary to some recent usage (e.g. Bastian et al. Reference Bastian, Covey and Meyer2010; Kroupa et al. Reference Kroupa, Weidner, Pflamm-Altenburg, Thies, Dabringhausen, Marks and Maschberger2013), the negative sign is not included in the relations adopted here and appears in the quantities α and Γ explicitly. This convention has the advantage that the sign of the parameter and of the power law itself are the same, not opposing. It ensures that a sign change from the lowest masses to the highest masses (e.g. in a piecewise power law description) is in the sense that intuition would suggest. It avoids inconsistencies or clumsy presentation when discussing the value of the power law slope as contrasted with the value of the parameter, or when inequalities are used to describe slopes flatter or steeper than some nominal parameter value. It eliminates confusion over the need to swap the sense of asymmetric errors in estimates of the parameter as opposed to the actual slope. For internal self-consistency and ease of comparison between published results, I present all IMF slopes discussed throughout using α as defined above.

2.4. The confusion wrought through ‘universality’

There are few areas of astrophysics as emotionally charged as the argument over whether the IMF is ‘universal,’ that is, the same unchanging distribution regardless of environment and over the entirety of cosmic history. With conflicting lines of evidence and apparently inconsistent conclusions, emotional attachments to a particular viewpoint, as opposed to evidence-based conclusions, easily develop and can strongly influence discussion in person and also in published work. Such an environment by itself makes work in this area challenging and can limit the depth or scope of investigations and interpretation, independent of any actual observational limitations.

In the absence of compelling evidence to the contrary the IMF is typically assumed to be universal. Partly this is an issue of convenience, as it makes interpreting the observations of galaxies easier and allows the direct calculation of quantities such as stellar mass and SFR that can then easily be compared among galaxy populations and over cosmic history. Also, there is generally an understandable reluctance to invoke the more complex scenario of an IMF that varies if it is not warranted, and a strong aversion to what is sometimes characterised as giving the theorists and simulators yet more free parameters to play with. Given this underlying tendency to default to the ‘universal’ assumption, there is a preferential inclination for authors to present results as being consistent with a ‘universal’ IMF, rather than using measured uncertainties instead to place limits on the scale of any possible variations for the given mass range, epoch, spatial scale, and physical conditions being probed. This approach hampers efforts to unify IMF studies because of the need to independently extract the relevant spatial scale and other physical properties, which may not be a trivial process and serves to provide further opportunity for error. It supports a tendency to acknowledge but then dismiss a host of observational challenges in inferring the IMF (such as accounting for mass segregation, metallicity effects in the mass–luminosity relation, dynamical effects, SPS limitations, SFHs, and more), by drawing a conclusion that is consistent with ‘universality’. This attitude may also lead to a tendency to downplay or dismiss evidence inconsistent with a ‘universal’ IMF as arising from observational systematics or model limitations. Such results may also be relegated to the status of a special case, as with the ‘non-standard IMFs in specific local or extragalactic environments’ noted in the abstract of the review by Bastian et al. (Reference Bastian, Covey and Meyer2010). A related issue is that the various published IMFs for Milky Way stars can easily be conflated when arguing that observations are consistent with a ‘universal’ IMF. As noted by Bastian et al. (Reference Bastian, Covey and Meyer2010), the Miller & Scalo (Reference Miller and Scalo1979) and Scalo (Reference Scalo, De Loore, Willis and Laskarides1986) IMFs are steeper at high masses than the more recently determined Kroupa (Reference Kroupa2001) or Chabrier (Reference Chabrier2003a) IMFs (e.g.), and observations consistent with the former are not necessarily also consistent with the latter.

The assumption of ‘universality’ has been questioned for about as long as the IMF has been observationally measured (see discussions in Scalo Reference Scalo, De Loore, Willis and Laskarides1986; Kennicutt Reference Kennicutt, Gilmore and Howell1998; Larson Reference Larson1998), and arguments for a varying IMF have been put forward since the early 1960s (e.g. Schmidt Reference Schmidt1963). Much of the discussion in the 1980s and 1990s touches on the need for an evolving or variable IMF to explain a variety of puzzles, including some that still remain unresolved. These include the so-called G-dwarf problem (the deficiency of metal-poor stars in the Solar neighbourhood, e.g. Worthey, Dorman, & Jones Reference Worthey, Dorman and Jones1996), the correlation between stellar M/L (ϒ_*) and Mg/H abundance in ellipticals that both increase with galaxy stellar mass (e.g. Worthey, Faber, & Gonzalez Reference Worthey, Faber and Gonzalez1992; Larson Reference Larson1998), the iron abundance in intracluster gas (e.g. Elbaz, Arnaud, & Vangioni-Flam Reference Elbaz, Arnaud and Vangioni-Flam1995; Wyse Reference Wyse1997), and others well summarised in the reviews of Scalo (Reference Scalo, De Loore, Willis and Laskarides1986), Kennicutt (Reference Kennicutt, Gilmore and Howell1998), and Larson (Reference Larson1998). Associated with these observational lines of evidence, an extensive number of different models for varying IMFs have been proposed, both to characterise their impact on different aspects of galaxy evolution and to explore different physical mechanisms motivating the IMF variation. While still maintaining the preference for a ‘universal’ IMF, there developed some degree of consensus by the early 2000s that an IMF that was over-represented in high-mass stars (through having a low-mass cut-off at several solar masses) at early times, or in high SFR events, could explain many of these different astrophysical results (e.g. Larson Reference Larson1998; Chabrier Reference Chabrier2003a). Subsequently, explaining the observed 850 μm galaxy source counts with semi-analytic models (Baugh et al. Reference Baugh, Lacey, Frenk, Granato, Silva, Bressan, Benson and Cole2005) required invoking such a ‘top-heavy’ IMF in starbursts, with a flatter high-mass slope (α = −1 over 0.15 < m/M_⊙ < 125) compared to quiescent star formation (α_l = −1.4, m < 1 M_⊙, and α_h = −2.5, m > 1 M_⊙). This IMF modification reduces the total SFR necessary to produce the observed 850 μm flux due to the increased number of high-mass stars for a given SFR (see Section 7). Such a requirement continues to be developed and refined (Lacey et al. Reference Lacey2016), although recent observations may reduce this need somewhat (see Section 6).

There are many systematics involved in estimating an IMF, though, making it challenging to unambiguously conclude that the IMF is different in different regions. This is highlighted, for example, by Massey (Reference Massey, Treyer, Wyder, Neill, Seibert and Lee2011), who demonstrates that within realistic uncertainties estimates of the slope of the high-mass end of the IMF in the Small Magellanic Cloud (SMC), Large Magellanic Cloud (LMC), and the Milky Way are all consistent with the Salpeter value, α = −2.35. But appealing to the ‘universality’ of the IMF based only on the similarity of observed IMFs within nearby regions of the Milky Way or even within nearby neighbouring galaxies is not justified. The range of physical conditions being probed in these systems is limited and does not encompass the extremes seen, for example, in starburst galaxies or in the early Universe (z > 2, say). The large observational and systematic uncertainties, too, place very broad constraints that are equally consistent with the scale of some published claims for IMF variations. The range of uncertainties for the compilation of measurements shown by Massey (Reference Massey, Treyer, Wyder, Neill, Seibert and Lee2011), for example, means that those results are also consistent with the variations proposed by Gunawardhana et al. (Reference Gunawardhana2011), with a high-mass IMF slope −2.5 < α_h < −1.8, seen over a range of almost 2 dex in SFR surface density, from analysing a sample of more than 40 000 galaxies. It would be enlightening to compare local IMF results as a function of some underlying physical property directly with the extragalactic results, to see whether or not the same trends hold. This is one example of the limitations on our investigations that arise from an underlying assumption of, or a tendency to prefer a conclusion for, the ‘universality’ of the IMF.

More than simply limiting the scope of investigation or interpretion, though, the tendency to default to an assumed ‘universal’ IMF has other insidious effects. There is an ever-present danger for investigations to be internally inconsistent if some elements (such as an SPS model or a numerical or semi-analytic simulation) make a ‘universal’ IMF assumption, but the analysis is testing IMF variations. Any ‘variation’ identified must be self-consistently present in the underlying models used to infer it.

In addition, because any potential variation of the IMF means that there can be a different effective IMF as a function of spatial scale, it now becomes important to discriminate between analyses that probe the scale of star clusters, larger H ii complexes or dust and molecular gas clouds, galaxies or even entire galaxy populations. It is only relevant to compare these directly if the IMF is indeed ‘universal’, but if not then such comparisons may easily be misleading. Any comparison must adequately account for any putative variation with the relevant physical quantity. It also means that measured PDMFs for m ≲ 0.8 M_⊙ may not necessarily correspond, as typically assumed, to the IMF (Jeffries Reference Jeffries, Reylé, Charbonnel and Schultheis2012).

There are other confusions that arise through the use of the term ‘universal’. It is easy, for example, to conflate the concepts of a ‘universal’ IMF and a ‘universal’ physical process that gives rise to an IMF that itself may or may not be ‘universal’. There are now numerous published models demonstrating how a common underlying physical process may lead to different IMFs (e.g. Narayanan & Davé Reference Narayanan and Davé2012; Hopkins Reference Hopkins2013b) and result in IMF variations between galaxies and as a function of time. So a ‘universal’ physical process does not necessarily imply a ‘universal IMF’, and care must be taken to distinguish the two.

Occam’s razor is commonly invoked by scientists because there is an elegance to the simplest possible solution, leading us to prefer not to invoke additional parameters unless clearly warranted by the data. In the case of the IMF, this leads to the well-established assumption that the IMF should be ‘universal’ in the absence of compelling evidence otherwise, but I now argue that this approach has been carried too far. It is clear that the simplest explanations are not always the most accurate or correct, although they may have the benefit of ease of use (e.g. compare Newtonian and Einsteinian formulations of gravity) and at some level the definition of ‘simplest’ is itself a subjective one. There are some physical motivators for supposing that the IMF is universal, such as the turbulent power spectrum in molecular clouds apparently having a universal form, which in turn leads to a prediction for a constant high-mass IMF slope (e.g. Hopkins Reference Hopkins2013b). Even this argument, though, leaves open whether the low-mass end of the IMF may vary. Accordingly, while there may be some physical expectation for some elements of the IMF to be universal, there is also a large selection of data that question this picture. As a consequence, I suggest that it is time to turn the basic assumption around. A better assumption would be the most general scenario, rather than the simplest, that the IMF is not universal. This approach echoes the sentiment expressed by Scalo (Reference Scalo, Gilmore and Howell1998), almost twenty years ago! Many of the conclusions by Scalo (Reference Scalo, Gilmore and Howell1998) are still quite pertinent today, in particular his statement that ‘… we are in the rather uncomfortable position of concluding that either the systematic uncertainties are so large that the IMF cannot yet be estimated, or that there are real and significant variations of the IMF index at all masses above about 1 M_⊙’.

In adopting the default assumption that the IMF may be variable, we should be aiming to pose research questions that can assess how and the extent to which it varies, what physical processes are responsible, and couching discussions in language that places constraints on variations rather than merely asserting that our evidence is consistent with ‘universality’. Broadly adopting this attitude would lead to authors presenting the relevant physical scale, mass range, metallicity, SFR, epoch, and other relevant quantities over which their results hold, making it easier to assess the degree of consistency or not between different analyses, and improving the community’s ability to make progress in this field.

To help with this endeavour it is valuable to develop an ensemble of reference observations that provide a well-defined set of boundary conditions that future measurements can be tested against. It is also critical to summarise the current state of the constraints on the IMF as a function not only of mass range, but also spatial scale, epoch and as many relevant additional physical parameters as possible such as metallicity, SFR, or SFR surface density, in order to extend the visual summary introduced by Scalo (Reference Scalo, Gilmore and Howell1998) and referred to by Kroupa (Reference Kroupa2002) and Bastian et al. (Reference Bastian, Covey and Meyer2010) as the ‘alpha plot’.Footnote ^d Producing such a suite of IMF diagnostics will be invaluable in order to begin the task of quantitatively establishing whether and the degree to which the IMF may vary.

3.1. Resolved star counts and luminosity functions

Measuring the IMF directly, even within the Milky Way and nearby galaxies where individual stars can be resolved, is challenging for several reasons, including: (1) stellar luminosities need to be converted to stellar masses, requiring information about their ages and metallicities, with more uncertainty at the low-mass end (e.g. Kroupa et al. Reference Kroupa, Tout and Gilmore1993); (2) account needs to be taken of the ‘missing’ stars, those high-mass stars that have already evolved off the main sequence, using a relation between the stellar mass and main-sequence lifetime (e.g. Reid, Gizis, & Hawley Reference Reid, Gizis and Hawley2002; Elmegreen & Scalo Reference Elmegreen and Scalo2006); (3) assumptions need to be made for the fraction of stars that are unresolved binary systems (e.g. Bochanski et al. Reference Bochanski, Hawley, Covey, West, Reid, Golimowski and Ivezić2010; Luhman Reference Luhman2012; De Marco & Izzard Reference DeMarco and Izzard2017), with the intrinsic IMF slope being steeper (proportionally fewer higher mass stars) than nominally inferred if this fraction is underestimated (Scalo Reference Scalo, De Loore, Willis and Laskarides1986; Sagar & Richtler Reference Sagar and Richtler1991), although Weidner, Kroupa, & Maschberger (Reference Weidner, Kroupa and Maschberger2009) argue that this effect is minor for high-mass stars, but significant at the low-mass end; (4) the degree to which mass segregation (the effect of high-mass stars in a gravitationally bound system moving towards the centre of a cluster over time) affects the results in stellar clusters or associations (Zinnecker & Yorke Reference Zinnecker and Yorke2007; Tan et al. Reference Tan, Beltrán, Caselli, Fontani, Fuente, Krumholz, McKee and Stolte2014; DeMarchi, Paresce, & Portegies Zwart Reference De Marchi, Paresce and Portegies Zwart2010). The reviews by Bastian et al. (Reference Bastian, Covey and Meyer2010), Jeffries (Reference Jeffries, Reylé, Charbonnel and Schultheis2012), Luhman (Reference Luhman2012), and Offner et al. (Reference Offner, Clark, Hennebelle, Bastian, Bate, Hopkins, Moraux and Whitworth2014) provide a more detailed discussion of these and related limitations.

Only a relatively small number of stellar systems are accessible to measure directly in this fashion, either within the Milky Way or in nearby galaxies, with many fewer being the very young systems where high-mass stars are able to be probed directly. In consequence, much of the work on the IMF in the Milky Way to date has focused on the low-mass end (e.g. Jeffries Reference Jeffries, Reylé, Charbonnel and Schultheis2012; Luhman Reference Luhman2012). The small number of systems available also gives rise to issues of stochasticity and sampling, which can limit the accuracy when attempting to infer the IMF for individual star clusters, associations, or dispersed field populations (Elmegreen Reference Elmegreen1999; Kruijssen & Longmore Reference Kruijssen and Longmore2014). Apparent variations between inferred IMFs for different systems may at some level just be a consequence of these observational limitations, although De Marchi et al. (Reference De Marchi, Paresce and Portegies Zwart2010) argue that all star clusters in the Milky Way, young and old, are consistent with having a common underlying mass function when dynamical effects are accounted for. The IGIMF approach (Kroupa & Weidner Reference Kroupa and Weidner2003; Kroupa et al. Reference Kroupa, Weidner, Pflamm-Altenburg, Thies, Dabringhausen, Marks and Maschberger2013) presents an alternative explanation, where the variations for star clusters are real and depend on, for example, a relationship between the cluster mass and the highest mass star in the cluster.

Broadly, the IMF shape for field stars in the Milky Way demonstrates a slope somewhat steeper than Salpeter (α_h ≈ −2.7) at high mass (m ≳ 0.7 M_⊙), with a flatter slope (α_l ≈ −0.5 to α_l ≈ −1) at lower masses, as summarised by Bastian et al. (Reference Bastian, Covey and Meyer2010) and Offner et al. (Reference Offner, Clark, Hennebelle, Bastian, Bate, Hopkins, Moraux and Whitworth2014). There are many studies of the local low-mass (m ≲ 1 M_⊙) IMF, as reviewed by Chabrier (Reference Chabrier2003a) and Jeffries (Reference Jeffries, Reylé, Charbonnel and Schultheis2012) for example, but there are few Galactic studies of the field star IMF in the mass range 1 < m/M_⊙ < 10. These use assumptions about the Milky Way SFH to infer an IMF with α_h = −2.65 ± 0.2, as described, for example, by Bastian et al. (Reference Bastian, Covey and Meyer2010).

A limitation arises from the need to assume a recent SFH in estimating an IMF. Elmegreen & Scalo (Reference Elmegreen and Scalo2006) demonstrate how an assumption of a constant or slowly varying SFH can distort the inferred IMFs from observed PDMFs if the true underlying SFH is more stochastic. In particular, an SFH decreasing with time can be misinterpreted as a steeper IMF if a constant SFH has been assumed. Elmegreen & Scalo (Reference Elmegreen and Scalo2006) show that this explanation can account for apparently steep IMF slopes (α_h ≈ −5 ± 0.5 for 25 < m/M_⊙ < 120) found for OB associations in the LMC and SMC (Massey et al. Reference Massey, Lang, Degioia-Eastwood and Garmany1995; Massey Reference Massey2002). This demonstrates the need for realistic SFHs to be adopted, and for SFH uncertainties to be incorporated into uncertainties on the inferred IMF.

There are challenges in constraining the higher mass IMF (m > 1 M_⊙) for the field star population due to the short lifetimes of the highest mass stars. These are best studied in OB associations and massive young clusters (e.g. Bastian et al. Reference Bastian, Covey and Meyer2010). At m ≳ 3 – 10 M_⊙ Offner et al. (Reference Offner, Clark, Hennebelle, Bastian, Bate, Hopkins, Moraux and Whitworth2014) summarise recent results that suggest such star clusters and associations in the Milky Way have slopes that scatter around the Salpeter value, α_h = −2.35. Mass segregation, the most massive stars tending to be found in a cluster’s central regions, is often invoked as the origin of much of the scatter. Haghi et al. (Reference Haghi, Zonoozi, Kroupa, Banerjee and Baumgardt2015) use simulations to argue that the lack of low-mass stars observed in some globular clusters may arise through mass segregation at birth combined with the process of gas expulsion (see also, e.g. Zonoozi et al. Reference Zonoozi, Haghi, Kroupa, Küpper and Baumgardt2017). If mass segregation is primordial, i.e., that the stars form in these locations, then the IMF must trivially be a variable property, although such segregation is perhaps most easily attributable to dynamical effects (Zinnecker & Yorke Reference Zinnecker and Yorke2007; Tan et al. Reference Tan, Beltrán, Caselli, Fontani, Fuente, Krumholz, McKee and Stolte2014). The existence of mass segregation leads to a necessity for observations to sample sufficiently large cluster radii in estimating the IMF, in order not to be biased by the prevalence of high-mass centrally located stars.

It can be seen already from this brief and incomplete summary that the broad range of observational challenges in estimating the IMF for stars in various regions within the Milky Way reinforces a tendency to invoke a ‘universal’ IMF. There is an understandable preference to conclude that the observations are not inconsistent with a ‘universal’ IMF, given the subtleties involved in accounting for the broad range of systematics and observational limitations.

Figure 2. The stellar mass-to-light ratio at 10 Gyr, ϒ_*,10, as a function of metallicity and age, showing two distinct populations. Clusters (blue data points) with younger ages, or higher metallicities, tend to show higher mass-to-light ratios, indicative of an IMF similar to Salpeter (α = −2.35) over the full mass range. Older, more metal-poor, clusters have mass-to-light ratios consistent with an IMF having proportionally fewer low-mass stars, such as that of Kroupa et al. (Reference Kroupa, Tout and Gilmore1993). The red data points represent the mass-to-light ratios for early-type galaxies, while the blue box indicates the range of ϒ_*,10 for disk galaxies. See Zaritsky et al. (Reference Zaritsky, Colucci, Pessev, Bernstein and Chandar2014b) for details. (Figure 9 of ‘Evidence for two distinct stellar initial mass functions: Probing for clues to the dichotomy’, Zaritsky et al. (Reference Zaritsky, Colucci, Pessev, Bernstein and Chandar2014b), © AAS. Reproduced with permission.)

3.2. Stellar clusters

In their review of the formation of young massive clusters, Portegies Zwart et al. (Reference Portegies Zwart, McMillan and Gieles2010) conclude by noting that ‘globular clusters are simply old massive clusters, the logical descendants of young massive clusters in the early Universe’, a view supported by Kruijssen (Reference Kruijssen2015). Zaritsky et al. (Reference Zaritsky, Colucci, Pessev, Bernstein and Chandar2012, Reference Zaritsky, Colucci, Pessev, Bernstein and Chandar2013, Reference Zaritsky, Colucci, Pessev, Bernstein and Chandar2014b) use the stellar mass-to-light ratios in Milky Way and Local Group stellar and globular clusters to infer, in contrast to the results above, two distinct stellar IMFs in such systems, which they describe as a ‘bimodal’ IMF (Figure 2).

This result was initially established by measuring the stellar mass-to-light ratio in the V-band, ϒ_*, based on observed velocity dispersions of four key clusters (Zaritsky et al. Reference Zaritsky, Colucci, Pessev, Bernstein and Chandar2012), ultimately extended to a sample of 29 clusters among 4 different host galaxies (Zaritsky et al. Reference Zaritsky, Colucci, Pessev, Bernstein and Chandar2014b). The quantity ϒ_*,10 was introduced, being the stellar mass-to-light ratio that a cluster would have at the age of 10 Gyr, based on simple evolutionary models, in order to more accurately compare between clusters of different ages. After exploring the impact of stellar binarity on the measured velocity dispersions (Zaritsky et al. Reference Zaritsky, Colucci, Pessev, Bernstein and Chandar2012), and the effects of internal dynamical evolution and relaxation driven mass loss (Zaritsky et al. Reference Zaritsky, Colucci, Pessev, Bernstein and Chandar2013), they conclude that such effects are not enough to account for the observed differences in mass-to-light ratio. The resulting conclusion is that the bimodality seen in ϒ_*,10 is evidence for two distinct IMFs, with young stellar clusters (log(t/yr) ≲ 9.5) favouring IMFs similar to Salpeter (α = −2.35) over the full mass range (0.1 < m/M_⊙ < 120, Zaritsky et al. Reference Zaritsky, Colucci, Pessev, Bernstein and Chandar2012), with older clusters favouring IMFs similar to that of Kroupa et al. (Reference Kroupa, Tout and Gilmore1993), with proportionally fewer low-mass stars, and a steeper high-mass slope (α_h = −2.7). They are careful to note, though, that neither of these IMFs is a unique solution, given that the constraint is on total mass arising from the measured mass-to-light ratios.

There are conflicting results regarding the mass function of globular clusters in the low-mass range (m < 1 M_⊙). Using mass-to-light ratios of 200 globular clusters in M31, Strader, Caldwell, & Seth (Reference Strader, Caldwell and Seth2011) find a deficit of low-mass stars compared to a Salpeter slope, with −1.3 < α_l < −0.8 for m < 1 M_⊙ (although mass underestimates may change this conclusion, see Shanahan & Gieles Reference Shanahan and Gieles2015). Zonoozi, Haghi, & Kroupa (Reference Zonoozi, Haghi and Kroupa2016) argue that an excess of high-mass stars as a function of metallicity can account for the lower than expected mass-to-light ratios of metal-rich globular clusters in M31. van Dokkum & Conroy (Reference van Dokkum and Conroy2011) use stellar absorption features (see Section 5.3) measured for four globular clusters in M31 to infer an IMF consistent with that of Kroupa (Reference Kroupa2001), with no low-mass excess. In contrast, Goudfrooij & Kruijssen (Reference Goudfrooij and Kruijssen2014) show that some globular cluster systems (at least the metal-rich population) in elliptical galaxies have a low-mass excess, requiring −3.0 < α_l < −2.3 over 0.3 < m/M_⊙ < 0.8. Zaritsky et al. (Reference Zaritsky, Colucci, Pessev, Bernstein and Chandar2014b) go on to show that the high and low mass-to-light ratios for their stellar clusters are well matched to those of early- and late-type galaxies, respectively (Figure 2), and potentially also consistent with the variations in IMF proposed for such systems (e.g. Gunawardhana et al. Reference Gunawardhana2011; van Dokkum & Conroy Reference van Dokkum and Conroy2012). This suggests an observational approach that can be used for directly linking and comparing studies of stellar and galactic systems.

Accounting for dynamical evolution is important in understanding how the observed mass function of a star cluster changes with time and may be related to its IMF. Some simulations demonstrate that the impact of tidal fields on star clusters, or that gas expulsion from initially mass segregated globular clusters, ejects predominantly low-mass stars (e.g. Baumgardt & Makino Reference Baumgardt and Makino2003; Haghi et al. Reference Haghi, Zonoozi, Kroupa, Banerjee and Baumgardt2015). Others demonstrate the mechanisms by which star clusters can eject high-mass stars (e.g. Oh & Kroupa Reference Oh and Kroupa2016; Banerjee & Kroupa Reference Banerjee and Kroupa2012). It is clear that there are many subtleties and details that need to be considered carefully in inferring an IMF from complex dynamically and astrophysically evolving systems.

The challenges in measuring star counts and accounting for mass segregation in star clusters when investigating the high-mass end of the IMF can be sidestepped through measuring the hydrogen ionising photon rate (Calzetti et al. Reference Calzetti, Chandar, Lee, Elmegreen, Kennicutt and Whitmore2010; Andrews et al. Reference Andrews2013, Reference Andrews2014). Andrews et al. (Reference Andrews2014) demonstrate the presence of high-mass stars in young (t < 8 Myr) low-mass clusters (down to cluster masses of M _cl ≈ 500 M_⊙) in M83. They conclude that star clusters are populated stochastically, or randomly, in stellar mass, allowing the existence of high-mass stars (up to 120 M_⊙) even in the lowest mass clusters (M _cl ≈ 500 M_⊙), in direct contrast with the m_u − M _cl relation of Weidner, Kroupa, & Bonnell (Reference Weidner, Kroupa and Bonnell2010) (but see also Weidner, Kroupa, & Pflamm-Altenburg Reference Weidner, Kroupa and Pflamm-Altenburg2014). They conclude that the population of M83 star clusters they observe have a total Hα luminosity to cluster mass ratio consistent with that expected from a Kroupa (Reference Kroupa2001) IMF with 0.08 < m/M_⊙ < 120.

Also exploring star clusters in a nearby galaxy, Weisz et al. (Reference Weisz2015) have undertaken a large systematic study of the colour-magnitude diagrams for 85 young, intermediate mass stellar clusters in M31. Using a framework to infer the distribution of mass function slopes given a set of noisy measurements (Hogg, Myers, & Bovy Reference Hogg, Myers and Bovy2010; Foreman-Mackey, Hogg, & Morton Reference Foreman-Mackey, Hogg and Morton2014), they find that the high-mass slope of the IMF is best described by $ {\alpha _h} &#x003D; - 2.45_{ - 0.03}^{ + 0.06} $, somewhat steeper than the high-mass slope of Kroupa (Reference Kroupa2001) (α_h = −2.3) inferred by Andrews et al. (Reference Andrews2014). This is similar to the result of Veltchev, Nedialkov, & Borisov (Reference Veltchev, Nedialkov and Borisov2004) who found α_h = −2.59 ± 0.09 (for m ≳ 7 M_⊙) using colour-magnitude diagrams for 50 OB associations in the south-western region of M31.

The conclusions of Andrews et al. (Reference Andrews2014) and Weisz et al. (Reference Weisz2015) reveal part of the challenge in discussing and understanding the IMF. Both Andrews et al. (Reference Andrews2014) and Weisz et al. (Reference Weisz2015) present their results as being consistent with a ‘universal’ IMF. The claim is that the IMF for the population of clusters as a whole produces an IMF consistent with that measured for the Milky Way, although the individual clusters demonstrate observed ‘variations’. Any of the individual low-mass star clusters of Andrews et al. (Reference Andrews2014), for example, that contain a very high-mass star will necessarily demonstrate an IMF skewed to the high-mass end, and be different to the IMF of other clusters in the M83 ensemble, even though their ensemble IMF is consistent with that of Kroupa (Reference Kroupa2001). Equally, the M31 cluster IMF slopes found by Weisz et al. (Reference Weisz2015) show significant scatter individually (see their Figure 4), while the ensemble is well described by IMFs having high-mass slopes drawn from a normal distribution with a very narrow intrinsic dispersion.

What the results of Andrews et al. (Reference Andrews2014) demonstrate, but which is not made explicit, is that stochastic or random sampling of the IMF for a given cluster leads directly to variations in the IMF between clusters. The importance of stochasticity in the IMF is also highlighted by Barker, de Grijs, & Cerviño (Reference Barker, de Grijs and Cerviño2008). The ambiguity here is deeply buried in the difference between a ‘universal’ process of star formation compared to a ‘universal’ mass distribution produced from any given star formation event, a complication that has led to an entire industry exploring how the IMF is populated (see, e.g. discussion in Kroupa et al. Reference Kroupa, Weidner, Pflamm-Altenburg, Thies, Dabringhausen, Marks and Maschberger2013). The conclusions of Andrews et al. (Reference Andrews2014) and Weisz et al. (Reference Weisz2015) are in support of the former (a ‘universal’ process), but not the latter. Given these consistent conclusions, the small but measurable difference in α_h between M31 (Weisz et al. Reference Weisz2015) and M83 (Andrews et al. Reference Andrews2014) is worth noting. Both cases here also highlight the difference between an IMF inferred for any individual star cluster and that for a galaxy taken as a whole, a distinction that will be a recurring theme in this review.

While Weisz et al. (Reference Weisz2015) also use their technique to show α_h = −2.15 ± 0.1 for the Milky Way and α_h = −2.3 ± 0.1 for the LMC, they argue that to be robust these values would need to be calculated using the same homogeneous and principled approach as they applied to M31. They go on to recommend that their steeper $ {\alpha _h} &#x003D; - 2.45_{ - 0.03}^{ + 0.06} $ slope for m > 1 M_⊙ be used in the ‘universal’ IMF shape. When calculating SFRs this IMF slope leads to values 30–50% higher than assuming the Kroupa (Reference Kroupa2001) IMF. I return to this point in Section 6.

Other observations of starburst clusters and super star clusters provide conflicting evidence for the shape of the IMF. Such systems, containing numerous and unresolved stellar components, have been investigated using a range of techniques, including spectroscopic observations, in some cases analysed using SPS models, mass-to-light ratios, and dynamics. A deficit of low-mass stars has been identified in the super star cluster M82F within the galaxy M82 (Smith & Gallagher Reference Smith and Gallagher2001; McCrady, Graham, & Vacca Reference McCrady, Graham and Vacca2005; Bastian et al. Reference Bastian, Konstantopoulos, Smith, Trancho, Westmoquette and Gallagher2007). The impact of a deficit of low-mass stars on the dynamical evolution of such clusters appears relatively mild (Kouwenhoven et al. Reference Kouwenhoven, Goodwin, de Grijs, Rose and Kim2014). The Milky Way’s nuclear star cluster was analysed by Lu et al. (Reference Lu, Do, Ghez, Morris, Yelda and Matthews2013), who find an IMF slope of α_h = −1.7 (for 1 < m/M_⊙ < 150) and an age of around 3.3 Myr. There is evidence for a Salpeter IMF (α = −2.35 for the full mass range 0.1 < m/M_⊙ < 100) in a massive young cluster in the Antennae (Greissl et al. Reference Greissl, Meyer, Christopher and Scoville2010), which would imply an excess of low-mass stars relative to a Kroupa (Reference Kroupa2001) or Chabrier (Reference Chabrier2003a) IMF, but other dynamical mass estimates suggest ‘standard Kroupa IMFs’ (although without detailing whichFootnote ^e) in star clusters in the Antennae and NGC 1487 (Mengel et al. Reference Mengel, Lehnert, Thatte, Vacca, Whitmore and Chandar2008). Banerjee & Kroupa (Reference Banerjee and Kroupa2012) use simulations to argue that the true IMF of R136 must have had an excess of high-mass stars, given that their dynamical ejection is efficient, and that the observed mass function is consistent with that of Kroupa (Reference Kroupa2001). The early suggestions of a high value for m_l in starburst nuclei (e.g. Rieke et al. Reference Rieke, Lebofsky, Thompson, Low and Tokunaga1980) have not been supported by more recent work, with Rigby & Rieke (Reference Rigby and Rieke2004) finding evidence using mid-infrared Ne line ratios for either m_c ≈ 40 M_⊙ (or equivalently a strong steepening in the high-mass IMF slope above this value), or that high-mass stars in starbursts are embedded within ultra-compact Hii regions, preventing the nebular lines from forming and escaping, the solution they favour (see also summaries by Elmegreen Reference Elmegreen, de Grijs and González Delgado2005, Reference Elmegreen, Kang, Lee, Leung and Cheng2007). Recent results analysing the 30 Doradus star forming region in the LMC (Schneider et al. Reference Schneider2018) show strong evidence for an IMF well populated up to m_u ≈ 200 M_⊙, and with $ {\alpha _h} &#x003D; - 1.90_{ - 0.37}^{ + 0.26} $ for 15 < m/M_⊙ < 200. Taken together, such results appear to provide evidence for a relative excess of high-mass stars in some, but not all, starburst clusters.

3.3. Chemical abundance measurements

Another common stellar technique used in inferring an IMF relies on the chemical abundances of stars. Since different elements are produced by stars of different mass, the present-day elemental abundances can be used to infer an indirect measure of the IMF. The summary by Wyse (Reference Wyse, Gilmore and Howell1998) provides an excellent overview of this approach and its issues and limitations. Broadly, oxygen and the α-elements are produced predominantly in core-collapse (Type II) supernovae, while iron is produced in both core-collapse and thermonuclear (Type Ia) supernovae. The ratio of [O/Fe] can then be used to probe the high-mass star IMF, with a ‘plateau’ in this ratio arising from stars pre-enriched only by type-II SNe, the quantitative value of which would change with a change in the high-mass IMF slope. A limitation is that the value of this type-II plateau depends on the theoretical yields assumed for different elements as a function of supernova progenitor mass. Wyse & Gilmore (Reference Wyse and Gilmore1992) note that varying the IMF slope from α_h = −2.1 to α_h = −3.3 (for m > 1 M_⊙) results in [O/Fe] changing by ≈ 0.4 dex, although the difference for the smaller range of −2.5 < α_h < −2.1 is only Δ[O/Fe] ≈ 0.1 – 0.15 dex depending on the elemental yields assumed (their Tables 1 and 2). Wyse (Reference Wyse, Gilmore and Howell1998) argues that for the Milky Way stellar halo, thick disk, and bulge populations, the measured abundances are consistent with a Salpeter high-mass IMF slope. A compilation of abundances was used by Nicholls et al. (Reference Nicholls, Sutherland, Dopita, Kewley and Groves2017) in introducing a new approach to scaling abundances with total metallicity, reinforcing the detection of the type-II plateau for the Milky Way, LMC, and the Sculptor Dwarf (shown using [Mg/Fe], Nicholls et al. (Reference Nicholls, Sutherland, Dopita, Kewley and Groves2017, Figure 9)).

Combining abundance constraints with mass-to-light ratio constraints, Tsujimoto et al. (Reference Tsujimoto, Yoshii, Nomoto, Matteucci, Thielemann and Hashimoto1997) find −2.6 < α_h < −2.3 for m > 1 M_⊙ for stars presently in the solar neighbourhood. They also argue for m_u = 50 ± 10 M_⊙, although this m_u is not necessarily the highest mass of stars formed, but instead is the highest mass of stars that return chemically enriched material to the interstellar medium. In their analysis stars may exist above this mass, but those that do must end as black holes without ejecting processed material into the interstellar medium. This result has been called into question by Gibson (Reference Gibson1998), though, who demonstrates that using different chemical yield models relaxes this outcome to a much less stringent constraint of m_u ≈ 30 – 200 M_⊙.

The chemical abundances of stars in some dwarf spheroidal (dSph) satellites of the Milky Way show measurable differences from Milky Way stars. Early work suggested that any abundance differences were still consistent with an IMF having a Salpeter high-mass slope (e.g. Tolstoy et al. Reference Tolstoy, Venn, Shetrone, Primas, Hill, Kaufer and Szeifert2003; Venn et al. Reference Venn, Irwin, Shetrone, Tout, Hill and Tolstoy2004). A high-mass truncation of the IMF was discussed as a possible scenario to explain the differences, but a solution arising from the contributions of metal-poor AGB stars was favoured. More recently, Tsujimoto (Reference Tsujimoto2011) argue that the deficiency of α-elements combined with an enhancement in s-process elements (Ba) found in dSph galaxies provides evidence of a lack of high-mass stars (m ≳ 25 M_⊙) in these systems, a result in keeping with the idea of a lower value for m_u in lower SFR environments (Weidner & Kroupa Reference Weidner and Kroupa2005). This result is supported by a different type of constraint, explored by Portinari et al. (Reference Portinari, Sommer-Larsen and Tantalo2004a), who find that IMFs typical of the Milky Way (Kennicutt Reference Kennicutt1983; Larson Reference Larson1998; Chabrier Reference Chabrier2001) can explain the mass-to-light ratios seen in Sbc/Sc galaxies but then overestimate the metallicities. They argue that unless the observed metallicity is underestimated (due to expulsion into the intergalactic medium or through being locked up in dust grains), the IMF needs to be truncated at high masses. This result was extended by Portinari et al. (Reference Portinari, Moretti, Chiosi and Sommer-Larsen2004b), showing that a ‘standard solar neighbourhood IMF’ (Kroupa Reference Kroupa, Rebolo, Martin and Zapatero Osorio1998) cannot provide sufficient heavy elements to account for the observed metallicity of galaxies in clusters. None of these analyses account for galactic winds, which, if included, may help explain the observed chemical signatures without the need for the lower value of m_u, although Portinari et al. (Reference Portinari, Moretti, Chiosi and Sommer-Larsen2004b) note that this would require ‘substantial loss of metals from the solar neighbourhood and from disk galaxies in general’.

At the other extreme, a recent analysis of starburst galaxies by Romano et al. (Reference Romano, Matteucci, Zhang, Papadopoulos and Ivison2017), using updated chemical models to track CNO isotopes and accounting for stellar rotation, finds a need for an excess of high-mass stars (α_h = −1.95 for m > 0.5 M_⊙) to reproduce observed isotope abundances. This result is consistent with the conclusions of Sliwa et al. (Reference Sliwa, Wilson, Aalto and Privon2017) who find a need for an excess of high-mass stars to explain the CO isotopic abundances in the starburst galaxy IRAS 13120-5453.

A related approach links the cosmic microwave background (CMB) to a directly observable stellar chemical signature, the carbon-enhanced metal-poor (CEMP) stars (Tumlinson Reference Tumlinson2007), probing stars in the mass range 1 < m/M_⊙ < 8. The CMB defines a temperature minimum that may translate to a characteristic fragmentation scale for star-forming gas (Larson Reference Larson2005). The time dependence of the CMB can hence have an impact on the fraction of CEMP stars as a function of metallicity, and Tumlinson (Reference Tumlinson2007) found that m_c should increase towards higher redshift. This result is supported by analyses of observed CEMP stars (Komiya et al. Reference Komiya, Suda, Minaguchi, Shigeyama, Aoki and Fujimoto2007) explained as arising from binary systems, and as modelled by binary population synthesis (Suda et al. Reference Suda2013). Tumlinson (Reference Tumlinson2007) pointed out that such an evolution of the IMF would lead to two clear systematic errors if it is not accounted for. Early-time SFHs for local galaxies derived from colour-magnitude diagrams assuming a non-evolving IMF would be systematically underestimated, and SFRs from high-z luminosity tracers, such as UV, would be systematically overestimated. I return to these points in Section 6.

While constraints from stellar chemistry in this fashion may turn out to be quite powerful probes of the IMF, in particular its properties at high redshift, it would be valuable to explore how limitations or systematics in our understanding of stellar yields and binary star evolution may influence or limit such measurements. One of the important potential advantages of stellar chemistry, and in particular the most metal-poor stars, is their potential for probing the first generation of stars, called ‘Population III’ stars (Frebel Reference Frebel2010), through the preserved signatures of their supernova chemical yields in subsequent generations of stars. I briefly discuss this in the next section.

3.4. Population III stars

Simulations and physical arguments have demonstrated for many years that Population III stars are likely to be dominated by high-mass objects (e.g. Abel, Bryan, & Norman Reference Abel, Bryan and Norman2002; Bromm, Coppi, & Larson Reference Bromm, Coppi and Larson2002), with typical masses m > 100 M_⊙ and few or no low-mass stars (see review by Bromm & Larson Reference Bromm and Larson2004, and references therein). Further work has explored the impact of such Population III star properties for the earliest generations of galaxies (Bromm & Yoshida Reference Bromm and Yoshida2011). The physical processes involved are also summarised well, in the broader context of structure formation and evolution, by Loeb (Reference Loeb2006). More recently, some simulations now appear to extend the lower mass limit for Population III stars to lower values (e.g. Hirano et al. Reference Hirano, Hosokawa, Yoshida, Umeda, Omukai, Chiaki and Yorke2014; Susa, Hasegawa, & Tominaga Reference Susa, Hasegawa and Tominaga2014). Clearly the IMF for such a population would be radically different to that in the local Universe and in that sense there is trivially an evolution in the IMF. This does not address the question that is usually meant regarding the ‘universality’ of the IMF, though, which instead is focused on whether the IMF may be different between coeval galaxies, or between different star-forming regions within a galaxy. With that in mind, it is instructive to briefly touch on some of the results associated with the current measurements probing Population III stars.

Observational probes of the first stars are summarised in the review by Bromm & Larson (Reference Bromm and Larson2004), who highlight their reionisation signature, chemical enrichment of subsequent stellar generations (stellar archaeology), and gamma-ray bursts as opportunities then developing. This builds on a significant body of earlier work to explore and explain the lack of low metallicity stars in the Milky Way and its halo (e.g. Bond Reference Bond1981; Jones Reference Jones1985; Cayrel Reference Cayrel1986) and other novel probes of Population III stars (e.g. Tarbet & Rowan-Robinson Reference Tarbet and Rowan-Robinson1982). Subsequently, gamma-ray bursts have been used to constrain the high-z SFH (Yüksel et al. Reference Bertelli Motta, Clark, Glover, Klessen and Pasquali2008; Kistler et al. Reference Kistler, Yüksel, Beacom, Hopkins and Wyithe2009; Kistler, Yuksel, & Hopkins Reference Kistler, Yüksel, Beacom, Hopkins and Wyithe2013), finding a higher value of the SFR density for z > 5 than commonly inferred from deep imaging data (see summary in Madau & Dickinson Reference Madau and Dickinson2014), with implications for reionisation (Wyithe et al. Reference Wyithe, Hopkins, Kistler, Yüksel and Beacom2010), although with no direct constraints yet on the Population III IMF. Ma et al. (Reference Ma, Maio, Ciardi and Salvaterra2015) show that none of the gamma-ray bursts detected at 5 ≲ z ≲ 6 show abundance ratios consistent with an environment dominated by Population III stars. Opportunities for probing the first stars through stellar archaeology are summarised by Frebel (Reference Frebel2010) and in an extensive review by Karlsson, Bromm, & Bland-Hawthorn (Reference Karlsson, Bromm and Bland-Hawthorn2013). Strong abundance ratio signatures are expected, with Heger & Woosley (Reference Heger and Woosley2010), for example, demonstrating that increasing m_l from 10 to 40 M_⊙ strongly suppresses the production of elements heavier than aluminium.

Other observations are now starting to constrain the possible IMF shapes for this earliest generation of stars. Sobral et al. (Reference Sobral, Matthee, Darvish, Schaerer, Mobasher, Röttgering, Santos and Hemmati2015) argue for a ‘flat or top-heavy IMF’ (lacking stars below 10 M_⊙) for Population III stars in a high-redshift (z = 6.6) Lyα system, although Bowler et al. (Reference Bowler, McLure, Dunlop, McLeod, Stanway, Eldridge and Jarvis2017) dispute this conclusion based on deeper infrared observations. They argue for a low-mass narrow-line active galactic nucleus (AGN) or low metallicity starburst to explain the observed infrared colours. Fraser et al. (Reference Fraser, Casey, Gilmore, Heger and Chan2017) use abundances in extremely metal-poor stars to infer a Salpeter IMF slope, $ {\alpha _h} &#x003D; - 2.35_{ - 0.29}^{ + 0.24} $, with a maximum supernova progenitor mass of $ m &#x003D; 87_{ - 33}^{ + 13}{\kern 1pt} {{\rm{M}}_ \odot } $ and a value of m_l below the minimum mass for Population III supernovae (m_l ≲ 9 M_⊙). Hartwig et al. (Reference Hartwig, Bromm, Klessen and Glover2015) demonstrate a technique for constraining the lower mass limit of Population III stars, finding that they can exclude stars with m < 0.65 M_⊙ at 95% confidence.

Future observations hold great promise for constraining the high-z IMF. Using simulations, de Souza et al. (Reference de Souza, Ishida, Whalen, Johnson and Ferrara2014) show that a few hundred supernovae detections with the JWST could be sufficient to discriminate between a ‘Salpeter and flat mass distribution for high-redshift stars’. Jeřábková et al. (Reference Jeřábková, Kroupa, Dabringhausen, Hilker and Bekki2017) demonstrate the redshift-dependent photometric properties of globular clusters and ultra-compact dwarf galaxies to provide observable IMF diagnostics for anticipated JWST observations.

With this review focused primarily on the consistency of approaches to observational constraints of the IMF I do not discuss Population III stars further. It is clear that this area will see rapid growth of a variety of observational constraints in the near future, and I hope that the framework presented here will be applied to these approaches to aid in understanding this earliest generation of stars.

3.5. Review

With the broad range of approaches, techniques, and results described above, it is worth briefly summarising. Star clusters sample physical scales on the order of a few pc and a broad range of ages and SFRs, although most of those observed are high SFR surface density objects. The field star population in principle probes the full galaxy-wide scale (tens of kpc) and is sensitive to the galaxy-wide SFH.

Although the large uncertainties and systematics involved in these measurements understandably lead to a conclusion that the IMFs are all similar and broadly consistent with (e.g.) the IMFs of Kroupa (Reference Kroupa2001) or Chabrier (Reference Chabrier2003a), it is tantalising to note that the field star and stellar association population have high-mass (m > 1 M_⊙) slopes on average somewhat steeper (α_h ≈ −2.5) than the cluster stars (α_h ≈ −2.2, see Figure 2 in Bastian et al. Reference Bastian, Covey and Meyer2010). The populations of globular clusters and extragalactic star clusters, in contrast, show evidence for some differences between IMFs. Probing spatial scales on the order of pc, a range of IMFs are inferred, with some globular clusters consistent with the IMF of Kroupa et al. (Reference Kroupa, Tout and Gilmore1993) and its steeper high-mass slope (α_h = −2.7), and others consistent with a Salpeter slope over the full mass range extending to the lowest masses. It is worth noting the link between the mass-to-light ratio approaches for globular clusters and galaxy systems as a technique that may lend itself to being applied self-consistently across a broad range of different physical scales and conditions.

Extragalactic star clusters demonstrate IMFs consistent with Kroupa (Reference Kroupa2001) in M83 (α_h = −2.3), or with high-mass slopes somewhat steeper (α_h = −2.45) in M31. While this difference is small, the level of precision of these results begins to suggest that it is not negligible. At least some starburst and super star clusters tend to show evidence for an excess of high-mass stars, either through an increase in m_c or a flatter α_h compared to the IMF of Kroupa (Reference Kroupa2001). Approaches using abundance measurements reinforce the slightly steeper high-mass slope seen in field stars and further suggest the possibility that the high-mass limit m_u may need to be somewhat lower than the typical Milky Way value in nearby dSph galaxies.

Taken as a whole, while the range of uncertainties and systematics makes it easy to assert that there is no strong evidence measured for IMF variations, adopting the converse approach of placing constraints on the scale of possible variations opens a different line of argument. Referring below to the Kroupa (Reference Kroupa2001) or Chabrier (Reference Chabrier2003a) IMF as the ‘typical’ IMF for the Milky Way, it can be said that:

• For stars and star clusters within Local Group galaxies, the high-mass IMF slope does not vary more than ±0.3 around α_h = −2.5.

• For extragalactic globular clusters, there are conflicting observations of mass-to-light ratios, which are parameterised as a constraint on the low-mass (m < 1 M_⊙) slope ranging across −3.0 < α_l < −0.8, from a deficit to an excess compared to the typical Milky Way values.

• In high SFR regions (starburst or super star clusters), there is an apparent variation with some having m_c ≳ 1 M_⊙ or α_h > −2.3, higher than the typical values in the Milky Way.

• In low SFR regions (dSph galaxies), there may be an apparent variation with m_u needing to be somewhat lower than the typical value in the Milky Way.

It is worth recalling here that there may be degeneracies between m_u and α_h, and that the measured constraints may be able to be reproduced by different parameter combinations. It remains the case, though, that there does appear to be some difference between the form of the IMF in high and low SFR regions. These lines of evidence suggest that the IMF is not ‘universal’, but that there are differences in different regions, although the details are only qualitatively constrained.

5.1. Background

Techniques for inferring an IMF for a galaxy, as opposed to a star cluster or stellar population, date back to early population synthesis work in the 1960s (Spinrad Reference Spinrad1962; Wood Reference Wood1966; Spinrad Reference Spinrad1966; Spinrad & Taylor Reference Spinrad and Taylor1971) that claimed an excess of M dwarfs in the nuclei of nearby galaxies. Wood (Reference Wood1966) also claim a proportionally larger number of giants for the spiral arms of NGC 224, NGC 3031 (M81), and NGC 5194, ‘essentially identical with that for the Solar neighbourhood’. This approach synthesised galaxy spectra by combining stellar spectra to reproduced the observed galaxy colours and certain spectral features including Mg b and Na D, as well as molecular bands including TiO and CN. These early analyses were focused largely on understanding the stellar populations and explaining the mass-to-light ratio for these systems, rather than constraining the IMF explicitly. Subsequently, Whitford (Reference Whitford1977) showed that the Wing–Ford band was also a sensitive probe of the dwarf-to-giant ratio and demonstrated that it could be used to constrain the IMF for old stellar populations within galaxies. This approach has been revived (e.g. van Dokkum & Conroy Reference van Dokkum and Conroy2012) and is now used routinely to infer IMF properties for early-type galaxies.

A different kind of approach was used by Kennicutt (Reference Kennicutt1983), who introduced a diagnostic comparing the equivalent width of Hα (sensitive to the presence of high-mass stars) to an optical colour index (sensitive to low-mass stars), as a probe of the IMF. Using this approach, Kennicutt et al. (Reference Kennicutt, Tamblyn and Congdon1994) showed that the high-mass slope of the IMF in nearby spiral galaxies is flatter (–2.5 ≲ α_h ≲ –2.35) than the Solar neighbourhood IMF of Miller & Scalo (Reference Miller and Scalo1979) or Scalo (Reference Scalo, De Loore, Willis and Laskarides1986) (–3.3 ≲ α_h ≲ –2.7). They attribute this at least in part to a deficiency of high-mass stars in the small volume of the Galaxy used to construct the local IMF, noting that Parker & Garmany (Reference Parker and Garmany1993) find flatter slopes for the high-mass end of the IMF in 30 Dor in the LMC.

Yet another approach is the Hα-to-UV flux ratio, originally proposed by Buat, Donas, & Deharveng (Reference Buat, Donas and Deharveng1987) as an IMF probe. They used a sample of 31 late type galaxies to find a high-mass (m > 1.8 M_⊙) IMF slope ranging over –3.1 < α_h < –2.3. They note that the dispersion in slope is smaller, ±0.25, for the sub-sample of 17 galaxies most similar to the Milky Way. Other approaches, including UV luminosities and mass-to-light ratio techniques, have been summarised by Scalo (Reference Scalo, De Loore, Willis and Laskarides1986) and Kennicutt (Reference Kennicutt, Gilmore and Howell1998). Subsequently many of these approaches have been refined and used extensively. More recent approaches include the IGIMF approach of Kroupa & Weidner (Reference Kroupa and Weidner2003), which has been explored extensively over the past decade (e.g. Yan, Jerabkova, & Kroupa Reference Yan, Jerabkova and Kroupa2017).

Different kinds of integrated galaxy techniques have been used for different kinds of galaxies. The Kennicutt (Reference Kennicutt1983) and Buat et al. (Reference Buat, Donas and Deharveng1987) diagnostics, and the use of UV luminosities directly as an IMF probe have focused on currently star-forming galaxies, while the mass-to-light ratio and dwarf-to-giant ratio approaches have focused on passive galaxies. This is natural given the requirements or limitations of each technique, but it causes difficulty in directly comparing the results between the approaches. For now I summarise the various methods based on the galaxy type they are used to probe and explore the comparisons between them further in Sections 5.4 and 9.

5.2. Star-forming galaxies

The IMF for galaxies with active star formation, even at a low level, is able to be probed with diagnostics relying on emission associated with young high-mass stars. The early work by Kennicutt (Reference Kennicutt1983) and Kennicutt et al. (Reference Kennicutt, Tamblyn and Congdon1994) was extended by Hoversten & Glazebrook (Reference Hoversten and Glazebrook2008) using data from the Sloan Digital Sky Survey DR4 (Adelman-McCarthy et al. Reference Adelman-McCarthy2006). They concluded that the sample as a whole was best fit by an IMF with high-mass (m > 0.5 M_⊙) slope α_h = –2.45 with negligible random error and systematic error of ±0.1. While noting that a constraint on α_h is degenerate with one on m_u, they also showed that brighter galaxies are better fit with a slightly flatter slope of α_h ≈ –2.4, and that fainter galaxies are not well described by a ‘universal’ IMF, inferring fewer massive stars leading either to steeper α_h or lower m_u. They point out that stochastic SFHs may mimic or contribute to this kind of signature, although by testing models for such stochastic SFHs they conclude that to reproduce the measurements for the observed population of bright galaxies would require an ‘implausible coordination of burst times’.

Figure 3. Following Kennicutt (Reference Kennicutt1983) and Hoversten & Glazebrook (Reference Hoversten and Glazebrook2008), this diagnostic shows how IMFs with different α_h can be discriminated using the equivalent width of Hα and an optical colour, in this case (g – r). The solid tracks are the evolutionary paths followed through a star formation event, showing (top to bottom) the location expected for α_h = –2, α_h = –2.35, α_h = –3. The data correspond to galaxies in the highest redshift bin of the three volume-limited samples, with ⟨z⟩ = 0.29, split into eight bins of SFR. This illustrates the tendency for the higher SFR systems to favour IMFs with flatter high-mass slopes (more positive α_h). See Gunawardhana et al. (Reference Gunawardhana2011) for details. Reproduced from Figure 6a of ‘Galaxy and Mass Assembly (GAMA): the star formation rate (SFR) dependence of the stellar initial mass function’, Gunawardhana et al. (Reference Gunawardhana2011).

The same method was used by Gunawardhana et al. (Reference Gunawardhana2011) with data from the Galaxy and Mass Assembly survey (Driver et al. Reference Driver2011; Hopkins et al. Reference Hopkins2013; Liske et al. Reference Medling2015). Using three independent volume-limited galaxy samples spanning 0.1 ≲ z ≲ 0.35, they find a trend between the high-mass (m > 0.5 M_⊙) IMF slope and SFR (Figure 3) or SFR surface density (Σ_SFR). With the large sample available they are able to explore in detail the dependency of α_h on these properties, which they quantify through the relations α_h ≈ 0.36 log (SFR / M_⊙ yr ^–1) – 2.6 or α_h ≈ 0.3 log (Σ_SFR / M_⊙ yr ^–1 kpc ^–2) –1.7. For the range of SFR or Σ_SFR they probed, this leads to a variation in the high-mass IMF slope from α_h ≈ –2.5 for the least active star-forming systems (SFR ≈ 0.1 M_⊙ yr ^–1) to α_h ≈ –1.7 for the most active (SFR ≈ 50 M_⊙ yr^–1). They point out that for the current SFR of the Milky Way, these results would imply a value of α_h ≈ –2.35, which is an encouraging consistency check. They also comment that the SFR dependence would imply a flatter high-mass IMF slope (α_h > –2.35) in the Milky Way’s early history, given that it had an elevated SFR in the past.

Gunawardhana et al. (Reference Gunawardhana2011) note that while the SFR is an IMF-dependent quantity, applying an IMF-dependent SFR calibration would not qualitatively alter their conclusions, since the variation seen is monotonic, and would only have the effect of reducing the range of SFR sampled. Gunawardhana et al. (Reference Gunawardhana2011) also explore degeneracies in the variation of the IMF that would lead to the same observed combination of Hα equivalent width and g – r colour. They show that while the high-mass slope can be fixed at the Salpeter value (α_h = –2.35) and the results explained by allowing m_c to increase, it requires a very high value of m_c ≈ 10 M_⊙ to account for the highest star-forming systems, reminiscent of the early results for starburst nuclei (Rieke et al. Reference Rieke, Lebofsky, Thompson, Low and Tokunaga1980) that are no longer favoured (Elmegreen Reference Elmegreen, Kang, Lee, Leung and Cheng2007).

Again to rule out the possibility that stochastic SFHs could be the origin of the observed signature, Gunawardhana et al. (Reference Gunawardhana2011) extended the analysis of Hoversten & Glazebrook (Reference Hoversten and Glazebrook2008) by demonstrating that both stellar mass and mass-doubling time for the observed galaxies vary smoothly along the SPS model tracks. They found no signature corresponding to the significant bursts of star formation that would be seen if stochastic SFHs were the dominant effect. Subsequently, Nanayakkara et al. (Reference Nanayakkara2017), too, demonstrated quantitatively that stochastic star formation histories could not explain their measurements, in a sample at much higher redshift. Gunawardhana et al. (Reference Gunawardhana2011) further demonstrate that the result is robust to the implementation of dust correction and choice of population synthesis model. While there is a degeneracy between a varying high-mass α_h and a varying m_c, it is possible to exclude a variation in m_l (Figure 4). It is clear that a varying low-mass cut-off has a very different signature in this diagnostic than a varying high-mass slope or m_c. This confirms that the results of Gunawardhana et al. (Reference Gunawardhana2011) are not attributable to variations in m_l while maintaining a Salpeter value for the high-mass slope.

Figure 4. The IMF diagnostic used by Gunawardhana et al. (Reference Gunawardhana2011) to identify variations in the slope of the high-mass (m > 0.5 M_⊙) IMF. The black solid lines show the evolutionary tracks expected for galaxies with, from top to bottom, α_h = –2, –2.35, –3. The additional tracks (dashed coloured lines) illustrate the effect of a fixed high-mass slope (α = –2.35) but varying m_l. The tracks become shorter as m_l increases, due to the shorter lifetimes of the higher mass stars. Figure courtesy of M. Gunawardhana.

These results were extended to z ≈ 2 by Nanayakkara et al. (Reference Nanayakkara2017), finding that systems with the highest Hα equivalent widths could only be explained either by rotating extremely metal-poor stars with binary interactions, or IMFs with α_h > –2 (0.5 < m / M_⊙ <120). They note that in the latter case, no single IMF slope could reproduce the data, implying a need for a stochastically varying high-mass IMF slope. They explore the trend with SFR derived from Hα and from the combined UV and far-infrared (FIR) luminosities, finding a weak trend with the Hα SFR in the same sense as that of Gunawardhana et al. (Reference Gunawardhana2011), but none with the SFRs from the UV+FIR (Figure 21 of Nanayakkara et al. Reference Nanayakkara2017). This arises from the lowest Hα SFR systems having higher SFRs as measured by the UV+FIR. In order to avoid internal inconsistencies, it would be valuable for all analyses of this kind to present results in terms of IMF-independent quantities (such as luminosities) or to calculate SFRs or other IMF-dependent properties self-consistently assuming whatever IMF-dependency on α_h or other parameter is being tested.

At similar redshifts, Zhang et al. (Reference Zhang, Romano, Ivison, Papadopoulos and Matteucci2018) measure the ¹³C/¹⁸O abundance ratio, probed through the rotational transitions of the ¹³CO and C¹⁸O isotopologues, for four starburst galaxies. These galaxies are gravitationally lensed submillimetre galaxies spanning 2.3 < z < 3.1. They find low ratios compared to chemical evolutionary models for the Milky Way, implying considerably more high-mass stars in such starbursts than in typical spiral galaxies, and with a high-mass slope flatter than that of Kroupa (Reference Kroupa2001).

At the opposite end of the scale, the Buat et al. (Reference Buat, Donas and Deharveng1987) approach relying on Hα-to-UV flux ratios has more recently been used to focus on potential variations to the IMF for low-mass or low SFR galaxies. The results of Meurer et al. (Reference Meurer2009), Lee et al. (Reference Lee2009), and Boselli et al. (Reference Boselli, Boissier, Cortese, Buat, Hughes and Gavazzi2009) all suggest that for low SFR (or low luminosity, or low mass) galaxies, there is evidence of a deficiency of high-mass stars, characterised as an IMF slope steeper than Salpeter (α < –2.35 for 0.1 < m/M_⊙ < 100). As the Hα luminosity (or surface density) decreases within various samples of nearby galaxies, these authors show that the Hα luminosity (or flux) declines faster than that of the UV. This is in the opposite sense than could be explained through dust obscuration, which affects the UV proportionally more than Hα, although different levels of attenuation for the line and continuum emission may play some role (e.g. Charlot & Fall Reference Charlot and Fall2000). Pflamm-Altenburg, Weidner, & Kroupa (Reference Pflamm-Altenburg, Weidner and Kroupa2009) demonstrate how this result can arise within the IGIMF formalism.

After exploring and ruling out a range of possible scenarios, including dust obscuration, SFH, stochastic population of the IMF, metallicity, and escape fraction, Meurer et al. (Reference Meurer2009) conclude that the most likely scenario is a variation in the high-mass end of the IMF, with either m_u ranging over 30 ≲ m_u/M_⊙ ≲ 120 or having a varying IMF slope spanning –3.3 ≲ α ≲ –1.3 (for 0.1 < m/M_⊙ <100), in the sense that the lower SFR systems have either lower m_u or steeper α. It is worth noting the use of the single power law IMF slope extending to the lowest masses here, and questioning whether a slope change below some m_c would affect the results. Since the diagnostic uses flux ratios of indicators sensitive only to high-mass stars, though, it seems unlikely that changing this assumption would alter the IMF conclusions significantly.

The results of Meurer et al. (Reference Meurer2009) are qualitatively similar to those of Gunawardhana et al. (Reference Gunawardhana2011), with a high-mass IMF slope that gets progressively steeper as Σ_SFR becomes lower. Quantitatively, though, the results are somewhat different, with Meurer et al. (Reference Meurer2009) inferring steeper IMF slopes at a given value of Σ_SFR (or equivalently, Σ_Hα) than Gunawardhana et al. (Reference Gunawardhana2011), when retaining the same m_u. Specifically, as an example, at log(Σ_SFR/M_⊙ yr ^–1 kpc ^–2) = –2.7 Gunawardhana et al. (Reference Gunawardhana2011) find α_h ≈ –2.4 (Gunawardhana et al. Reference Gunawardhana2011, Figure 13c). This value of Σ_SFR corresponds to log (Σ_Hα / Wkpc ^–2) = 31.4, for which Meurer et al. (Reference Meurer2009) find a typical value of log ((F_Hα/ f_FUV) / Å) ≈ 0.65 (Meurer et al. Reference Meurer2009, Figure 3), and which leads to α_h ≈ –3.0 (Meurer et al. Reference Meurer2009, Figure 10b). I return to exploring the details of these quantitative differences below.

Again in a qualitatively similar fashion, Boselli et al. (Reference Boselli, Boissier, Cortese, Buat, Hughes and Gavazzi2009) find that the IMF slope (for 0.1 < m/M_⊙ <100) spans –2.6 ≲ α ≲ –2.3, with steeper slopes for the lower mass galaxies. Comparing quantitatively to Gunawardhana et al. (Reference Gunawardhana2011), these slopes are closer than inferred by Meurer et al. (Reference Meurer2009) but still rather steeper for a given stellar mass or sSFR. Based on Figure 10 of Boselli et al. (Reference Boselli, Boissier, Cortese, Buat, Hughes and Gavazzi2009), α = –2.5 is favoured for galaxies with 9.2 <log (M/M_⊙), for example. This corresponds broadly to a range of –10 ≲ log(sSFR/yr ^–1) ≲ –9 (see Figure 4 of Bauer et al. Reference Bauer2013) for the GAMA sample in the redshift range analysed by Gunawardhana et al. (Reference Gunawardhana2011). For this range of sSFR, Gunawardhana et al. (Reference Gunawardhana2011) find high-mass IMF slopes spanning –2.4 ≲ α_h ≲ –2.1 (Gunawardhana et al. Reference Gunawardhana2011, Figure 13b), somewhat flatter than the α = –2.5 of Boselli et al. (Reference Boselli, Boissier, Cortese, Buat, Hughes and Gavazzi2009).

Boselli et al. (Reference Boselli, Boissier, Cortese, Buat, Hughes and Gavazzi2009) invoke bursty SFHs to explain their findings in preference to IMF variations, contradicting the claim by Meurer et al. (Reference Meurer2009) that the ‘gasps’ between such bursts would have to be unrealistically synchronised between galaxies to produce the range of measured flux ratios. Boselli et al. (Reference Boselli, Boissier, Cortese, Buat, Hughes and Gavazzi2009) note that the star formation in dwarf galaxies can be dominated by a single H ii region, which can lead to larger scatter in the Hα to UV flux ratio than for high-mass galaxies, arising from the different lifetimes of the OB stars (∼10⁶ yr) responsible for the Hα emission and the A stars (∼10⁷ yr) responsible for the UV emission.

Lee et al. (Reference Lee2009) use a nearby sample of galaxies to probe to extremely low levels of SFR, down to SFR ≈ 10^–4 M_⊙ yr ^–1 and see very similar effects to the results of Meurer et al. (Reference Meurer2009) and Boselli et al. (Reference Boselli, Boissier, Cortese, Buat, Hughes and Gavazzi2009). They do not attempt to constrain the IMF directly but instead test (and rule out) a variety of explanations, while noting that the IGIMF predictions of Pflamm-Altenburg et al. (Reference Pflamm-Altenburg, Weidner and Kroupa2009) match their observations surprisingly well, before also implying a preference for SFH variations (bursty or ‘flickering’) as a possible explanation (see also Lee et al. Reference Lee, Gil de Paz, Tremonti, Kennicutt, Treyer, Wyder, Neill, Seibert and Lee2011). In contrast, Fumagalli, da Silva, & Krumholz (Reference Fumagalli, da Silva and Krumholz2011) argue, based on a stochastic code for synthetic photometry, that these observed Hα-to-UV flux ratios in low SFR galaxies can be explained by random sampling from a ‘universal’ IMF, combined with stellar evolution. A similar result was found by Weisz et al. (Reference Weisz2012) who note that such stochastic SFHs can explain the observed Hα to UV flux ratios in low-mass galaxies, without invoking IMF variations.

Bearing in mind the likely contributions of stochastic SFHs, there does seem to be some evidence for the idea that systems with lower luminosities, masses, or SFRs favour IMFs lacking in high-mass stars relative to a Kroupa (Reference Kroupa2001) or Chabrier (Reference Chabrier2003a) IMF. This is reinforced by studies of the LMC, SMC, and other dwarf, low metallicity, or low surface brightness galaxies. Using UV imaging of the LMC and models jointly constraining IMF slope, obscuration, and age, Parker et al. (Reference Parker1998) find α_h = –2.80 ±0.09 (for m ≳ 7 M_⊙). Lamb et al. (Reference Lamb, Oey, Graus, Adams and Segura-Cox2013) find α_h = –3.3±0.4 (for m > 20 M_⊙) in the SMC based on spectra for a spatially complete census of field OB stars. The dwarf starburst galaxy NGC 4214 was found by Úbeda, Maíz-Apellániz, & MacKenty (Reference Úbeda, Maíz-Apellániz and MacKenty2007) to have α_h < –2.83±0.07 (for 20 < m/M_⊙ <100), the upper limit arising due to the presence of unresolved binaries, the neglect of which acts to flatten the inferred IMF. Lee et al. (Reference Lee, Gibson, Flynn, Kawata and Beasley2004) use the high M/L ratios for seven low surface brightness disk galaxies to estimate α = –3.85 (for 0.1 < m/M_⊙ < 60). Garcia et al. (Reference Garcia, Herrero, Najarro, Camacho, Lennon, Urbaneja, Castro, Eldridge, Bray, McClelland and Xiao2017) summarise studies of high-mass stars in Local Group dwarf galaxies with very low metallicity, oxygen abundances less than 1/7 of the solar value. Although relying on a sample of only four galaxies, with incomplete observations of the high-mass stellar population, they note that the highest mass stars so far identified have initial masses of m ≈ 60 M_⊙. Bruzzese et al. (Reference Bruzzese, Meurer, Lagos, Elson, Werk, Blakeslee and Ford2015) explore the IMF of the dark-matter dominated, extremely low SFR, blue compact dwarf galaxy NGC 2915, and reinforce the possibility of high-mass IMF slopes rather steeper than Salpeter for such systems. They combine colour magnitude diagrams for the stellar populations with an assumed recent SFH to find a high-mass IMF slope α_h = –2.85 (for m ≳ 4 M_⊙) and a poorly constrained upper mass limit of m_u = 60 M_⊙. Noting the impact of assuming a constant recent SFH (Elmegreen & Scalo Reference Elmegreen and Scalo2006), the IMF may not be quite this extreme, although it is not inconsistent with the other results highlighted here, in similar environments. The same approach, assuming a constant SFR over the dynamical timescale, was used in inferring the IMF of the dwarf irregular galaxy DDO 154 by Watts et al. (Reference Watts, Meurer, Lagos, Bruzzese, Kroupa and Jerabkova2018), to derive α_h = –2.45 with a similarly poorly constrained m_u = 16 M_⊙.

For the population of ultra-compact dwarf galaxies (UCDs), known to have high V-band mass-to-light ratios (ϒ_V), Dabringhausen, Kroupa, & Baumgardt (Reference Dabringhausen, Kroupa and Baumgardt2009) use population synthesis modelling to infer α_h, after arguing that the non-baryonic dark matter contribution is too low to influence the dynamics of these systems (Murray Reference Murray2009). Contrary to the results above for other dwarf galaxy systems, they show that an IMF with a relative excess of high-mass stars, α_h ≈ –1.6 to α_h ≈ –1.0 (depending on the assumed age), is required to account for the observed ϒ_V. This perhaps highlights SFR density as a significant factor in shaping the IMF. In contrast to the local dwarf systems, which are typically low surface density galaxies, the UCDs are expected to have formed rapidly, with SFRs as high as perhaps 10–100 M_⊙ yr ^–1, and a correspondingly high Σ_SFR given their small physical sizes. For a physical scale of 10 pc this would give 5 ≲ log (Σ_SFR/M_⊙ yr ^–1kpc ^–2)≲6, substantially higher than in the population of higher mass galaxies discussed above. The elevated values of α_h inferred are qualitatively consistent with the results for higher mass star-forming galaxies, although any potential dependence of α_h on Σ_SFR is quantitatively inconsistent with the nominal linear relations of Meurer et al. (Reference Meurer2009), Gunawardhana et al. (Reference Gunawardhana2011), or Nanayakkara et al. (Reference Nanayakkara2017).

Using UV spectral lines as a constraint, Leitherer (Reference Leitherer, Treyer, Wyder, Neill, Seibert and Lee2011) combines measurements for a sample of 28 nearby galaxies (distances less than 250 Mpc) to construct the average UV spectrum and concludes that the high-mass (m > 0.5 M_⊙) IMF slope is constrained to lie between –2.6 ≲ α_h ≲ –2.0. This range encompasses most of the variation in α_h inferred from the studies described above, and consequently does not substantially rule out the proposed variations.

With the exception of the dwarf galaxies and low surface brightness galaxies, which show rather steeper high-mass (m > 0.5 M_⊙) IMF slopes (although noting the opposite result for UCDs, which are perhaps more analogous to starburst systems), the range spanned by α_h from these investigations is not large, with most galaxies having −2.5 ≲ α_h ≲ − 1.8 (e.g. Gunawardhana et al. Reference Gunawardhana2011), similar to the range seen in nearby stellar populations and star clusters, and variously attributed to observational or astrophysical limitations. It is notable that Gunawardhana et al. (Reference Gunawardhana2011) place the Milky Way on their relationship between α and SFR or Σ_SFR, finding a Salpeter IMF slope (α_h = −2.35) consistent with the bulk of analyses of Galactic stellar populations.

In summary, the observations for star-forming galaxies suggest a broadly consistent picture in favour of IMF variations, with α_h larger (flatter) for higher SFR or Σ_SFR, and smaller (steeper) for lower SFR or Σ_SFR. This is in line, qualitatively at least, with the results seen above for starburst or super star clusters.

5.3. Passive galaxies

The descriptor ‘passive’ is intended here to refer to the degree of star formation, or rather its absence at any significant level, and is independent of the existence or not of an AGN. The methods of Kennicutt (Reference Kennicutt1983) and Buat et al. (Reference Buat, Donas and Deharveng1987) are not typically able to be applied to passive galaxies, as they have little or no Hα and UV emission arising from high-mass young stars. Where such emission is present it is likely to arise from, and be dominated by, an AGN rather than star formation. Also, since the stars being probed in these passive galaxies are only those low-mass objects remaining from star formation episodes much earlier, any IMF measured in such systems is in a sense a ‘relic’ IMF, analogous to the PDMF in Milky Way stellar systems, and in many cases this has led naturally to a focus on the low-mass end of the IMF.

A novel approach was explored by van Dokkum (Reference van Dokkum2008), noting the opposing effect of the IMF on luminosity and colour evolution identified by Tinsley (Reference Tinsley1980), such that an IMF with proportionally more high-mass stars would lead to stronger luminosity evolution and weaker color evolution. He compared the rate of luminosity and colour evolution for high-mass elliptical galaxies in clusters spanning 0 ≤ z ≤ 0.83 and found a need for an IMF with $ \alpha {\kern 1pt} &#x003D; {\kern 1pt} - 0.7_{ - 0.4}^{ + 0.7} $ at masses around m ≈ 1 M_⊙, much flatter than a Kroupa (Reference Kroupa2001) or Chabrier (Reference Chabrier2003a) IMF at these masses, to explain the observations. Casting this as an estimate of $ {m_c} &#x003D; 1.9_{ - 1.2}^{ + 9.3}{\kern 1pt} {{\rm{M}}_ \odot } $ at $ z &#x003D; 3.7_{ - 0.8}^{ + 2.3} $ (the estimated formation redshift of stars at this stellar mass) for a Chabrier-like IMF, and comparing with the lower inferred values of m_c at lower redshift, he goes on to explore the impact of an evolving m_c. He notes the effect of reducing the apparent discrepancy between the cosmic SFH and SMD, confirming similar results (Hopkins & Beacom Reference Hopkins and Beacom2006; Fardal et al. Reference Fardal, Katz, Weinberg and Davé2007; Wilkins et al. Reference Wilkins, Trentham and Hopkins2008a), and which I discuss in detail in Section 6. The result of van Dokkum (Reference van Dokkum2008) was questioned by van Dokkum & Conroy (Reference van Dokkum and Conroy2012) who show instead that when comparing galaxies at a given velocity dispersion, rather than stellar mass, the observed luminosity and colour evolution is consistent with the standard Salpeter slope.

More recent approaches have focused on using spectral signatures sensitive to the dwarf-to-giant ratio, stellar mass-to-light ratios, and kinematics to explore the inferred IMF.

The dwarf-to-giant ratio approach evolved from early work using Na i D lines as a tracer of dwarf star populations (Spinrad Reference Spinrad1962) which concluded that the most luminous and high-mass elliptical galaxies showed evidence for proportionally more dwarf stars than found in lower mass galaxies. Additional spectral features (Mg i, Ca i, Ca ii, TiO, CN, CH, CaH, MgH) sensitive to different stellar populations (Spinrad Reference Spinrad1966; Wood Reference Wood1966; Spinrad & Taylor Reference Spinrad and Taylor1971) enabled improvement of the ability to characterise the stellar populations present in integrated galaxy spectra. Whitford (Reference Whitford1977) incorporated measurements of the Wing–Ford molecular band at 9910 Å to argue against the earlier results that favoured an excess of dwarf stars in such galaxies, ruling out IMFs with α ≤ −3, and finding results supporting an IMF with α ≈ −2.

Figure 5. Three spectral regions showing features sensitive to the presence or absence of low-mass stars (upper panels), and the trend in the absorption strength of those features seen with velocity dispersion of the galaxies (lower panels). This demonstrates that galaxies with higher velocity dispersion, and hence higher stellar mass, have a tendency to favour an excess of dwarf, or low-mass, stars. See van Dokkum & Conroy (Reference van Dokkum and Conroy2012) for details. (Figure 10 of ‘The stellar initial mass function in early-type galaxies from absorption line spectroscopy. I. Data and empirical trends,’ van Dokkum & Conroy (Reference van Dokkum and Conroy2012), © AAS. Reproduced with permission.)

This general technique has been developed significantly through recent work (e.g. van Dokkum & Conroy Reference van Dokkum and Conroy2010, Reference van Dokkum and Conroy2012) and is illustrated by Figure 5, which has been reproduced from van Dokkum & Conroy (Reference van Dokkum and Conroy2012). A key result was that of van Dokkum & Conroy (Reference van Dokkum and Conroy2010), who apply the high resolution SPS models of Conroy, Gunn, & White (Reference Conroy, Gunn and White2009) to estimate the proportion of low-mass dwarf stars (m < 0.3 M_⊙) necessary to explain the observed absorption in the Na iλλ 8183,8185 Å doublet and the Wing–Ford FeH band around λ 9916Å for eight nearby elliptical galaxies in the Virgo and Coma clusters. They infer α_l ≈ −3 for the low-mass range (0.1 < m/M_⊙ < 0.3), significantly steeper than the Salpeter slope. As these stars would have formed at high redshift (z = 2−5), they argue against high redshift IMFs with a deficit of low-mass stars such as the truncated IMF of Baugh et al. (Reference Baugh, Lacey, Frenk, Granato, Silva, Bressan, Benson and Cole2005). Recall that for the Milky Way −1.4 ≲ α_l ≲ −0.6 from the compilation of Bastian et al. (Reference Bastian, Covey and Meyer2010). van Dokkum & Conroy (Reference van Dokkum and Conroy2012) and Conroy & van Dokkum (Reference Conroy and van Dokkum2012) extended this result to the bulge of M31 and 34 galaxies from the SAURON integral field spectroscopy sample (Bacon et al. Reference Bacon2001), adding the Ca ii λλ 8498,8542,8662 Å triplet to the diagnostics, and concluding that the low-mass m < 0.3 M_⊙ IMF slope varies systematically with galaxy velocity dispersion and α-enhancement, with steeper slopes in more massive and high-abundance galaxies (Figure 5).

Using similar approaches, both Spiniello et al. (Reference Spiniello, Trager, Koopmans and Chen2012) and Ferreras et al. (Reference Ferreras, La Barbera, de la Rosa, Vazdekis, de Carvalho, Falcón-Barroso and Ricciardelli2013) infer IMF slopes for the same mass range steeper than Salpeter (α_l ≈ −3) for high velocity dispersion systems (σ ≳ 300 km s^–1), and La Barbera et al. (Reference La Barbera, Ferreras, Vazdekis, de la Rosa, de Carvalho, Trevisan, Falcón-Barroso and Ricciardelli2013) also find IMF slopes steeper than Salpeter for systems with σ ≳ 220 km s^–1. Zaritsky, Gil de Paz, & Bouquin (Reference Zaritsky, Gil de Paz and Bouquin2014a) find a strong correlation between UV colour and ϒ_* for the same sample of galaxies analysed by van Dokkum & Conroy (Reference van Dokkum and Conroy2012) and conclude that this correlation is attributable to varying populations of extreme horizontal branch stars arising from the IMF variations.

For lower stellar mass systems, Smith et al. (Reference Smith, Lucey and Carter2012) analysed 92 red-sequence galaxies in the Coma cluster, including more galaxies with lower velocity dispersions than van Dokkum & Conroy and (Reference van Dokkum and Conroy2012) and Conroy & van Dokkum (Reference Conroy and van Dokkum2012). They found no clear dependence of IMF slope on velocity dispersion with a Salpeter slope adequate for 100 ≲ σ/kms ^–1 ≲ 250 (although they do not rule out steeper IMF slopes at higher velocity dispersions), but they do see a dependence on α-enhancement (Mg/Fe ratio), concluding that the IMF variation arises as a result of star formation mode, with rapid bursts leading to proportionally more low-mass stars (m < 0.3 M_⊙).

In parallel with these analyses, gravitational lensing and dynamical constraints were being explored as a tool for inferring mass-to-light ratios (ϒ) and placing associated constraints on IMF slopes. This approach was refined by Treu et al. (Reference Treu, Auger, Koopmans, Gavazzi, Marshall and Bolton2010) who infer ϒ for 56 galaxies from the SLACS survey (Bolton et al. Reference Bolton, Burles, Koopmans, Treu and Moustakas2006). They introduce an IMF mismatch parameter, also denoted α, which is defined to be the ratio of ϒ inferred independently from the lensing and dynamical analysis compared to that from SPS modelling. For clarity, I refer to the IMF mismatch parameter as α_mm = ϒ_LD/ϒ_SPS throughout.

Treu et al. (Reference Treu, Auger, Koopmans, Gavazzi, Marshall and Bolton2010) find that, with the assumption of an NFW dark matter profile (Navarro, Frenk, & White Reference Navarro, Frenk and White1996), such galaxies tend to favour IMFs such as Salpeter that provide a higher ϒ_* than those of Chabrier (Reference Chabrier2003a). While stellar mass range is not discussed, the Bruzual & Charlot (Reference Bruzual and Charlot2003) models used span a mass range of 0.1 ≤ m/M_⊙ ≤ 100, and from the scale of the IMF mismatch parameter it is reasonable to infer that the Salpeter IMF slope (α = −2.35) is applied over that full range for their Salpeter SPS mass estimates (confirmed by T. Treu, personal communication). Treu et al. (Reference Treu, Auger, Koopmans, Gavazzi, Marshall and Bolton2010) also note that their sample is limited to relatively high velocity dispersion systems, and show that, while for galaxies with σ ≈ 200 km s^–1 a Chabrier (Reference Chabrier2003a) IMF provides consistent ϒ_* with the lensing and dynamical estimate, ‘heavier’ IMFs (i.e. with a greater total mass normalisation) are required for higher velocity dispersion systems. They conclude that either the IMF varies towards one with a Salpeter-like mass normalisation for the most massive galaxies, or that dark matter profiles are not universal and the inner slope is systematically steeper than NFW for the most massive galaxies, or possibly a combination of both. This result is still not as extreme as the α_l ≈ −3 found spectroscopically by van Dokkum & Conroy (Reference van Dokkum and Conroy2010).

Similarly, Läsker et al. (Reference Läsker, van den Bosch, van de Ven, Ferreras, La Barbera, Vazdekis and Falcón-Barroso2013) use a combination of dynamical models and the SPS and IMF approach of Vazdekis et al. (Reference Vazdekis, Casuso, Peletier and Beckman1996, Reference Vazdekis, Ricciardelli, Cenarro, Rivero-González, Díaz-García and Falcón-Barroso2012) to infer an IMF with a steep high-mass slope, (α_h = −4.2±0.1 for m > 0.6 M_⊙ constraining α_l = −1.3 for 0.1 < m/M_⊙ < 0.6) in a low redshift (z = 0.116) high-mass (σ=360 km s^–1) early-type galaxy with a putative extremely high-mass nuclear black hole. In apparent contrast to these results, though, Smith & Lucey (Reference Smith and Lucey2013) used gravitational lensing mass estimates to demonstrate that ϒ_* for a high-mass (σ ≈ 330 km s^–1) low-redshift (z = 0.035) giant elliptical is consistent with a Kroupa (Reference Kroupa2001) IMF. This result was subsequently reinforced by Smith, Lucey, & Conroy (Reference Smith, Lucey and Conroy2015a) who analysed three high-mass (σ > 300 km s^–1) low redshift (z ≲ 0.05) galaxies, showing that the inferred IMF mass normalisation is consistent with that of Kroupa (Reference Kroupa2001), and excluding a Salpeter IMF (α = −2.35 over 0.1 < m/M_⊙ < 100) at the 3.5 σ level.

These results, though, are not inconsistent with the scatter seen by Treu et al. (Reference Treu, Auger, Koopmans, Gavazzi, Marshall and Bolton2010) in their IMF mismatch parameter. It is worth reiterating that Treu et al. (Reference Treu, Auger, Koopmans, Gavazzi, Marshall and Bolton2010) use the observed small range of scatter on α_mm to argue that ‘the absolute normalization of the IMF is uniform to better than 25%’, a result that echoes the relatively small range of potential variations found for the star-forming galaxy population, although with a different sense in the variation itself for the passive galaxies (i.e. higher mass passive galaxies favouring proportionally more low-mass stars, but higher mass star-forming galaxies favouring proportionally more high-mass stars).

There is a subtlety around gas recycling in SPS that affects the M/L ratio comparison technique. Whether the gas recycled through stellar evolution is retained or not in the SPS inferred masses has a direct impact on the comparison to masses inferred from lensing and dynamics. Treu et al. (Reference Treu, Auger, Koopmans, Gavazzi, Marshall and Bolton2010) explore the two extreme cases, where all gas lost through stellar evolution is removed, and where it is retained (the ‘zero age’ SPS mass). They note that for a Chabrier (Reference Chabrier2003a) IMF the ‘zero age’ masses tend to be overestimates compared to the lensing masses, implying that at least some fraction of the gas associated with recycling is expelled and does not contribute to the baryonic mass in the region probed by the lensing and dynamical constraints. In this and subsequent work, it is typically just the mass in stars and stellar remnants derived from the SPS that is compared with the lensing and dynamical mass estimates, which for early type galaxies with very low gas fractions is likely to be a reasonable assumption. It must be noted, though, that such mass estimates may be lower limits if some gas component still contributes non-negligibly to the baryonic mass and needs to be accounted for in the uncertainties on inferred IMF constraints.

Figure 6. The mass-to-light ratio for the stellar component of ATLAS^3D galaxies estimated using dynamical models, (M/L)_stars, compared to that estimated from spectral fitting using SPS models assuming a fixed Salpeter IMF, (M/L)_Salp. This demonstrates the trend for the high mass-to-light, or high velocity dispersion, galaxies in this sample to favour IMFs with an excess of mass compared to the IMFs of Chabrier (Reference Chabrier2003a) or Kroupa (Reference Kroupa2001), approaching and exceeding that from a Salpeter slope over the full mass range (an excess of low-mass stars). See Cappellari et al. (Reference Cappellari2013) for details. Reproduced from Figure 11 of ‘The ATLAS^3D project – XX. Mass-size and mass-σ distributions of early-type galaxies: bulge fraction drives kinematics, mass-to-light ratio, molecular gas fraction, and stellar initial mass function,’ Cappellari et al. (Reference Cappellari2013).

Another significant development around the same time was the use of stellar kinematics and dynamical models to constrain the IMF (Cappellari et al. Reference Cappellari2012, Reference Cappellari2013) illustrated in Figure 6. They used the ATLAS3D sample of 260 early-type galaxies (Cappellari et al. Reference Cappellari2011) combining the measured stellar kinematics with detailed axisymmetric dynamical models, to derive accurate stellar masses and mass-to-light ratios for the population. By comparing the mass-to-light ratio measured dynamically in this way to that inferred from the photometry using SPS models that assume a Salpeter IMF over the full mass range (0.1 < m/M_⊙ < 100), they are able to show a systematic variation in the inferred IMF normalisation. This variation ranges from a mass normalisation consistent with Chabrier (Reference Chabrier2003a) or Kroupa (Reference Kroupa2001) at ϒ_* ≈ 6 M_⊙/L_⊙ to one consistent with a Salpeter slope spanning 0.1 ≤ m/M_⊙ ≤ 100 at ϒ_* ≈ 6 M_⊙/L_⊙ (Figure 6). This may extend to even higher mass normalisations at the most extreme measured values of ϒ_* ≈ 10 M_⊙/L_⊙, with an IMF characterised equally by α = −2.8 (dominated by low-mass stars) or α = −1.5 (dominated by high-mass stars). This result has been questioned by Clauwens, Schaye, & Franx (Reference Clauwens, Schaye and Franx2015), though, who note that these trends could also be produced if the kinematic mass estimates had Gaussian errors of the order of 30%.

There are significant degeneracies possible in IMF shape when only the mass normalisation is constrained. While this is highlighted for the very high-mass normalisations by Cappellari et al. (Reference Cappellari2013), that of a Salpeter IMF slope (α = −2.35) over the full mass range can also be reproduced by an IMF with a Milky Way style low-mass slope, and an excess of high-mass stars. By way of illustration, this is achieved (including only the mass in stars and stellar remnants) by an IMF with α_l = −1.5 (0.1 ≤ m/M_⊙ ≤ 0.5) and α_h=−1.70 (0.5 ≤ m/M_⊙ ≤ 100), or an IMF with the Kroupa (Reference Kroupa2001) low-mass slope α_l = −1.3 (0.1 ≤ m/M_⊙ ≤ 0.5) and α_h = −1.64 (0.5 ≤ m/M_⊙ ≤ 100). Different combinations can equally be used to match the zero-age mass normalisation. The point is that a value of ϒ_* consistent with a Salpeter IMF over the full mass range does not necessarily imply an excess of low-mass stars. To break this degeneracy, Conroy et al. (Reference Conroy, Dutton, Graves, Mendel and van Dokkum2013) quantitatively compared the scale of the IMF mass normalisation derived using the dwarf-to-giant approach with that from dynamical mass constraints and found that they are consistent, inferring that the explanation lies in an excess of low-mass stars (m ≲ 1 M_⊙).

There have also been a range of enhancements and refinements that combine lensing, dynamical, and SPS constraints. These results largely confirm the need for a low-mass IMF slope similar to the Salpeter value for high-mass early-type galaxies. Spiniello et al. (Reference Spiniello, Barnabè, Koopmans and Trager2015a), for example, find a low-mass slope of α_l = −2.37 ± 0.12 and $ {m_l} &#x003D; 0.131_{ - 0.026}^{ + 0.023}{\kern 1pt} {{\rm{M}}_ \odot } $ at a reference point corresponding to σ = 250 km s^–1 for nine early-type galaxies, confirming earlier results by Barnabè et al. (Reference Barnabè, Spiniello, Koopmans, Trager, Czoske and Treu2013). In apparent contrast Lyubenova et al. (Reference Lyubenova2016), using 27 early-type galaxies from CALIFA (Walcher et al. Reference Walcher2014) and the IMF parameterisation approach of Vazdekis et al. (Reference Vazdekis, Casuso, Peletier and Beckman1996), rule out a single power law IMF and conclude that a double power law with a varying high-mass slope is required to explain the dynamical and stellar M/L ratios. This result may not be inconsistent with most of the results above, to the degree that it is only the mass normalisation that is being constrained through this approach, but it would be inconsistent with the conclusions of Conroy et al. (Reference Conroy, Dutton, Graves, Mendel and van Dokkum2013). Testing the consistency of IMF measurements between the SPS and the lensing and dynamical approaches is clearly important, and some work in this area has already begun (Newman et al. Reference Newman, Smith, Conroy, Villaume and van Dokkum2017). Using three strongly lensed passive galaxies, Newman et al. (Reference Newman, Smith, Conroy, Villaume and van Dokkum2017) find consistent IMF estimates between the two techniques for one galaxy, but require a variable low-mass cut-off or a non-parametric form of the IMF to reconcile the approaches for the remaining two systems. Clearly, extending such analyses to larger samples is desirable.

More recently these approaches have been extended to identify radial gradients in the IMF shape (van Dokkum et al. Reference van Dokkum, Conroy, Villaume, Brodie and Romanowsky2017). They characterise their results using a variant of the α_mm IMF mismatch parameter which, instead of comparing dynamical to SPS M/L ratios, compares the ϒ_* inferred from a given SPS fit to a canonical Milky Way IMF such as Kroupa (Reference Kroupa2001) or Chabrier (Reference Chabrier2003a) as a convenient shorthand for encapsulating the relative mass normalisation. Using α_mm = ϒ_SPS/ϒ_MW in this way, they find IMFs with mass normalisations heavier than Kroupa (Reference Kroupa2001) or Chabrier (Reference Chabrier2003a) (α_mm ≈ 2.5) in the central regions for six early-type galaxies, but approaching the Milky Way value as radius increases, with α_mm ≈ 1.1 at R > 0.4 R_e. The SPS IMF constraint approach has also been refined by Conroy, van Dokkum, & Villaume (Reference Conroy, van Dokkum and Villaume2017), who developed a non-parametric approach to constraining the m < 1 M_⊙ shape. Applying this to the centre of NGC 1407, they find an IMF slope consistent with α_l=−2.7. Such radial trends may provide an explanation for the differences found in the IMF properties between the spectroscopic and the dynamical approaches. It is also a tantalising link to the postulated ‘two-phase’ formation scenario for early type galaxies (e.g. González Delgado et al. Reference González Delgado2017), which proposes that the cores of elliptical galaxies formed quickly and quenched rapidly (the high redshift ‘red nuggets’), in contrast to their outer regions. Martín-Navarro et al. (Reference Martín-Navarro, La Barbera, Vazdekis, Ferré-Mateu, Trujillo and Beasley2015b) find a ‘bottom-heavy’ IMF (1.5 ≲ α_mm ≲ 2) out to 1.5 R_e in NGC 1277 and argue that this is an example of the kind of ‘core’ that would evolve through dry merging to the characteristic masses and sizes of z ≈ 0 elliptical systems.

Martín-Navarro et al. (Reference Martín-Navarro2015d) argue that metallicity, rather than velocity dispersion, is the driver of IMF variations in early type galaxies, building on their earlier work ((Martín-Navarro et al. Reference Martín-Navarro, La Barbera, Vazdekis, Falcón-Barroso and Ferreras2015a, Reference Martín-Navarro, La Barbera, Vazdekis, Ferré-Mateu, Trujillo and Beasley2015b) using the SPS and IMF parameterisation approach of Vazdekis et al. (Reference Vazdekis, Casuso, Peletier and Beckman1996). Using five key spectral features sensitive to metallicity, age, and the IMF, they add a sample of 24 galaxies from CALIFA to the earlier work, jointly constraining the metallicity and high-mass (m > 0.6 M_⊙) IMF slope, with the low-mass (m < 0.6 M_⊙) slope fixed at the Kroupa (Reference Kroupa2001) value (α_l = –1.3). They find a metallicity dependence on the high-mass slope expressed as α_h = −3.2(±0.1) −3.1(±0.5)[M/H], consistent with a Kroupa (Reference Kroupa2001) high-mass slope (α_h = −2.3) for metallicities [M/H] ≈ −0.3, and steeper for higher metallicity. They also fit for a single power law, finding that the data could also be explained by an IMF with slope α = −2.5(±0.05)−2.1(±0.2)[M/H] over the full mass range (0.1 ≤ m/M_⊙ ≤ 100). They go on to demonstrate that the relationship observed by other authors between IMF slope and stellar velocity dispersion naturally arises, qualitatively at least, through a combination of the mass–metallicity relation (e.g. Tremonti et al. Reference Tremonti2004; Lara-López et al. Reference Lara-López2013) and their derived IMF slope relation with metallicity. Martín-Navarro et al. (Reference Martín-Navarro2015d) further assert that the evolution in metallicity during galaxy formation implies an evolving IMF, drawing on arguments for IMFs dominated by high-mass stars in low metallicity environments (Marks et al. Reference Marks, Kroupa, Dabringhausen and Pawlowski2012) transitioning to IMFs with a relative excess of low-mass stars as the metallicity rapidly increases. They also point to similar arguments by Arrigoni et al. (Reference Arrigoni, Trager, Somerville and Gibson2010) in favour of an evolving IMF for early-type galaxies, based on chemical evolution in semi-analytic models. This conclusion contrasts with the analysis of 212 ATLAS3D early-type galaxies by McDermid et al. (Reference McDermid2014) who conclude that there are no strong trends between any of the stellar population-derived parameters (age, metallicity, [α/Fe]) and the IMF mass normalisation.

More significantly, perhaps, Martín-Navarro (Reference Martín-Navarro2016) show that, in order for passive galaxies to have both enhanced [Mg/Fe] ratios and an IMF overabundant in low-mass stars relative to the Salpeter slope (α_l < −2.35), they must have had extremely short star formation episodes that imply exceptionally high SFRs at high redshift, SFR ≈ 105 M_⊙ yr^–1, which have not been observed. They present two possible scenarios to resolve this issue. The first invokes an IMF overabundant in both low and high-mass stars. The second argues for a time varying IMF, initially overabundant in high-mass stars to account for the chemical signature, evolving to one overabundant in low-mass stars at later times.

The results described above have generally used relatively low redshift galaxy samples. Exploring the higher redshift Universe (0.9 < z < 1.5) Martín-Navarro et al. (Reference Martín-Navarro2015c) use the TiO₂ IMF-sensitive spectral feature and the Vazdekis et al. (Reference Vazdekis, Casuso, Peletier and Beckman1996) SPS and IMF parameterisation, finding α_h = −4.2±0.2 (for m > 0.6 M_⊙) for the most massive galaxies (with stellar masses M _* > 10¹¹ M_⊙), and slightly less steep ($ {\alpha _h} &#x003D; - 3.7_{ - 0.3}^{ + 0.4} $) at lower stellar mass (2×10¹⁰ < M _*/M_⊙ < 10¹¹). With estimated ages of 1.7±0.3 Gyr, this population would have formed at redshifts of 1.5 ≲ z ≲ 3. These IMF slopes are similar to those found in the low redshift population for the highest metallicity galaxies by Martín-Navarro et al. (Reference Martín-Navarro2015d), which have similar formation epochs (z ≈ 2) based on the ages inferred in that analysis.

Using a different approach again, Dabringhausen et al. (Reference Dabringhausen, Kroupa, Pflamm-Altenburg and Mieske2012) measure the low-mass X-ray binary (LMXB) population in globular clusters and UCDs within 11 elliptical galaxies of the Virgo Cluster. They conclude that these are 10 times more frequent in UCDs than expected from a Kroupa (Reference Kroupa2001) IMF, implying an excess of high-mass stars. When the LMXB number is compared against an optical or infrared galaxy luminosity, this provides a direct constraint of the high-mass end of the IMF (the progenitors of the neutron stars and black holes detected as LMXBs) compared to the low-mass stellar population. In contrast, Peacock et al. (Reference Peacock, Zepf, Maccarone, Kundu, Gonzalez, Lehmer and Maraston2014) find no evidence for IMF variations in a sample of eight early-type galaxies, based on their LMXB population. Their results are based on a small sample, and are consistent with the range of scatter seen by Treu et al. (Reference Treu, Auger, Koopmans, Gavazzi, Marshall and Bolton2010). In either case, this demonstrates an important complementary approach to constraining the IMF. The presence of a low-mass companion in an LMXB also implies a degree of binarity that is often neglected in common SPS models. There is clearly scope to expand this kind of analysis to larger samples and through incorporating such SPS models (e.g. Eldridge Reference Eldridge2012; Eldridge et al. Reference Eldridge, Stanway, Xiao, McClelland, Taylor, Ng, Greis and Bray2017).

Other complementary approaches that have been recently explored for estimating integrated galaxy IMFs include the demonstration by Recchi et al. (Reference Recchi, Calura, Gibson and Kroupa2014) of how the plateau in the [α/Fe] ratio for a galaxy is sensitive to the galaxy IMF and can be used as a test of whether IMFs vary between galaxies. Brewer et al. (Reference Brewer, Marshall, Auger, Treu, Dutton and Barnabè2014) present a hierarchical modelling approach that can be used to derive upper limits on departures from universality in the IMF. Podorvanyuk et al. (Reference Podorvanyuk, Chilingarian and Katkov2013) introduce a pixel-based SPS fitting approach, similar to the ‘pixel-z’ technique (Conti et al. Reference Conti2003; Welikala et al. Reference Welikala, Connolly, Hopkins, Scranton and Conti2008, Reference Welikala, Connolly, Hopkins and Scranton2009) that fits pixel colours to infer stellar population ages, obscurations and SFHs. The approach of Podorvanyuk et al. (Reference Podorvanyuk, Chilingarian and Katkov2013) allows the IMF slope at the low-mass end to be a free parameter in the library of models generated and fit to the observed pixel colours, in principle allowing the IMF to be inferred within spatially resolved galaxy images. Geha et al. (Reference Geha2013) use resolved star counts of two nearby ultra-faint dwarf galaxies to determine $ {\alpha _l} &#x003D; - 1.2_{ - 0.4}^{ + 0.5} $ (Hercules) and α_l = −1.3±0.8 (Leo IV) in the mass range 0.52 ≤ m/M_⊙ ≤ 0.77, and argue that, in combination with resolved star counts from the Milky Way, SMC, and Ursa Minor, this suggests a trend to flatter low-mass IMF slopes (or increasing m_c) for systems with lower velocity dispersion and metallicity, qualitatively consistent with the broad results seen for early type galaxies summarised above (e.g. van Dokkum & Conroy Reference van Dokkum and Conroy2012).

As with the star-forming galaxies, the range of analyses of passive galaxies reveal a broadly consistent picture, finding an increased abundance of low-mass stars for higher velocity dispersion, or higher metallicity, galaxies, and potentially with this excess located preferentially in the galaxy cores. These IMF variations for passive galaxies do seem to contradict the sense of the variation found for star-forming galaxies, a point that I return to below. It would also be interesting to explore the extent to which the recent generation of integral field spectroscopic surveys may be able to bridge the two, applying techniques so far used only for star-forming galaxies but instead looking at passive systems with some residual star formation. With spatially resolved spectra, any AGN contributions may be identified and excluded to isolate star formation signatures in otherwise passive galaxies (e.g. Hampton et al. Reference Hampton2017; Medling et al. Reference Medling2018), allowing the Kennicutt (Reference Kennicutt1983), allowing the Kennicutt (Reference Kennicutt1983) and related approaches to be applied.

5.4. Summary and limitations

It is clear that analyses of the star-forming and passive galaxy populations have a rather different focus in terms of the IMF, of necessity, due to the available observational metrics. Work on the star-forming galaxy population has focused on the high-mass end of the IMF, and its potential dependence on luminosity, SFR, or SFR (surface) density, loosely characterised as star formation intensity. Analysis of the passive galaxy population, instead, has focused on the low-mass end of the IMF. The stars being probed in these systems are only the low-mass stars remaining from star formation episodes many Gyr earlier, analogous to the PDMF in Milky Way stellar systems. In qualitative terms the results can be summarised as:

• There is evidence for α_h (m ≲ 1 M_⊙) variation in star-forming galaxies, with flatter (more positive) values in stronger star-forming galaxies (increasing SFR, sSFR, or luminosity), and vice-versa.

• There is evidence for α_l (m ≲ 1 M_⊙) variation in passive galaxies, (or equivalently, steeper α_h when α_l = −1.3 for m < 0.6 M_⊙ is constrained) with steeper (more negative) values in (at least the centres of) more high-mass passive galaxies (higher σ or metallicity).

There is a range of uncertainty, quantitative difference, and scatter around these general conclusions, but the broad picture seems to be well established given the observational approaches used.

This broad picture leads to an apparent tension, though, because at first glance the two qualitative results seem inconsistent. The stellar population in the centres of passive galaxies, that formed at the peak of cosmic star formation (z ≈ 2, e.g. Hopkins Reference Hopkins2004; Hopkins & Beacom Reference Hopkins and Beacom2006), have a relative excess of low-mass stars (that may be characterised broadly as α < –2.35), while the star-forming galaxies seem to imply that high star formation is associated with a relative excess of high-mass stars (similarly characterised broadly by α > −2.35). Are these results actually inconsistent or not? If the IMF is not universal, they are not necessarily inconsistent, as it may be the case that different physical conditions prevailed in the progenitors of high-mass elliptical galaxy nuclei, than seen now in high SFR galaxies at low redshift. The question needs to be answered by exploring the physical properties dominating the origin of a particular stellar mass distribution, and the physical conditions prevailing in the different systems at the time of star formation. This is reviewed in Section 7.

Before delving into those issues, it is worth questioning the robustness of the various observational approaches, and assessing the degree to which they may be systematically biased.

In each case, SPS models play a crucial role in the way an IMF slope is inferred. There are some fundamental limitations in even the most sophisticated modern SPS models, each of which focuses on one particular area of strength but without necessarily incorporating other facets, or coarsely modelling them in the interests of computational efficiency. Two main limitations are the neglect of stellar rotation (e.g. Brott et al. Reference Brott2011a, Reference Brott2011b; Levesque et al. Reference Levesque, Leitherer, Ekstrom, Meynet and Schaerer2012; Leitherer et al. Reference Leitherer, Ekström, Meynet, Schaerer, Agienko and Levesque2014) and stellar multiplicity (e.g. Eldridge Reference Eldridge2012; Eldridge et al. Reference Eldridge, Stanway, Xiao, McClelland, Taylor, Ng, Greis and Bray2017). These effects are not negligible, with perhaps up to 30% of high-mass main sequence stars produced through binary interaction (Sana et al. Reference Sana2012; de Mink et al. Reference de Mink, Sana, Langer, Izzard and Schneider2014). Binarity and stellar rotation may reinforce each other in some observable properties, as the modelling of binaries can provide effects similar to the inclusion of stellar rotation for single stars (Leitherer Reference Leitherer, Treyer, Wyder, Neill, Seibert and Lee2011). In particular, extreme rotation can lead to an increase in the Hα equivalent width by up to an order of magnitude compared to the assumption of no rotation, for the same SFH, in a window between 10^6.5 < t/yr < 10⁷ (Leitherer et al. Reference Leitherer, Ekström, Meynet, Schaerer, Agienko and Levesque2014). This may lead to a need to reduce inferred SFRs (for example) in some analyses, and may well have an impact on inferred IMFs. Binaries add an extra dimension too, as binary mergers and mass transfer can lead to the presence of stars of higher mass than those in the initial population (Eldridge Reference Eldridge2012; Banerjee, Kroupa, & Oh Reference Banerjee, Kroupa and Oh2012). For example, using a binary population synthesis approach Suda et al. (Reference Suda2013) make the case for the Milky Way having an IMF dominated by high-mass stars at early times, subsequently evolving to the presently observed IMF, based on observations of CEMP stars. They note that in chemical evolution models the current Milky Way IMF would overpredict Type 1.5 SNe. While observational constraints on the degree of rotation and binarity (or multiplicity) fraction are challenging, it is clear that it is necessary to address these effects in refining any estimated IMFs.

It is also worth noting that the SPS models used to analyse star-forming galaxies are typically different from those used to analyse passive galaxies, again for the obvious reason that different codes focus on producing different diagnostics. It would be highly desirable to be able to cross-compare results between the two galaxy populations using a common SPS tool to eliminate any possibility that inconsistent conclusions regarding the IMF are related to SPS model systematics.

SPS models commonly used in the analysis of star-forming galaxies include PEGASE (Fioc & Rocca-Volmerange Reference Fioc and Rocca-Volmerange1997), STARBURST99 (Leitherer et al. Reference Leitherer1999), and GALAXEV, (Bruzual & Charlot Reference Bruzual and Charlot2003), as well as those incorporating a greater contribution from TP-AGB stars (Maraston Reference Maraston2005). The conclusions regarding IMF properties in star-forming galaxies are generally consistent between these different models, and authors often check that their results are not strongly dependent on the SPS model used (e.g. Meurer et al. Reference Meurer2009; Gunawardhana et al. Reference Gunawardhana2011). Nanayakkara et al. (Reference Nanayakkara2017) compared various diagnostics between the PEGASE, STARBURST99, and BPASS (Stanway, Eldridge, & Becker Reference Stanway, Eldridge and Becker2016) SPS tools, to confirm that their results were not sensitive to the choice of SPS model.

In the analysis of passive galaxies, the two dominant SPS tools are the FSPS code of Conroy et al. (Reference Conroy, Gunn and White2009) and its recent variants (Conroy et al. Reference Conroy, van Dokkum and Villaume2017), and that of Vazdekis (Reference Vazdekis1999) and its recent variants (Vazdekis et al. Reference Vazdekis, Sánchez-Blázquez, Falcón-Barroso, Cenarro, Beasley, Cardiel, Gorgas and Peletier2010, Reference Vazdekis, Ricciardelli, Cenarro, Rivero-González, Díaz-García and Falcón-Barroso2012) based on the MILES/MIUSCAT empirical stellar spectral libraries. The higher resolution of these libraries is important for constraining the key absorption features sensitive to the low-mass stellar populations. Analyses using the former tend to be cast in terms of single power law IMF slopes, while those of the latter explore both single and double power law forms. Broadly, both approaches tend to conclude that high-mass (or high metallicity) early-type galaxies have a need for IMFs with a relative excess of low-mass stars, although this is achieved with different IMF forms depending on the SPS tool used for the analysis. Either a single power law IMF with a slope steeper than Salpeter (α < −2.35) or a double power law with a steep high-mass (m > 0.6 M_⊙) slope, both leading to an excess of stars for m < 1 M_⊙ compared to Milky Way type IMFs of Kroupa (Reference Kroupa2001) or Chabrier (Reference Chabrier2003b). Spiniello, Trager, & Koopmans (Reference Spiniello, Trager and Koopmans2015b) compare these two SPS models explicitly, and conclude that, while quantitatively different, the result implying an excess of low-mass stars for early type galaxies is qualitatively consistent between the two. Conroy (Reference Conroy2013) reviews SPS techniques and discusses what can reliably be measured. In this comprehensive review, issues of assumed metallicity and abundances, dust obscuration, SFHs, and stellar evolution libraries are addressed. In particular, Conroy (Reference Conroy2013) describes the impact on inferred IMFs arising from limitations in SPS models in his Section 7. He focuses primarily on the results arising from the passive galaxy analyses, concluding that modest IMF variations are supported, as inferred through ϒ_* variations of a factor of 2–3. He briefly argues that the case for ‘top-heavy’ IMFs is not compelling, and does not explore the impact of SPS assumptions in those analyses.

Other potential issues affecting the use of the dwarf-to-giant ratio sensitive features are rare stellar populations and parameter degeneracies. Maccarone (Reference Maccarone2014) argue, for example, that barium stars and extrinsic S stars can explain the effects seen in the Wing–Ford band and the Na i D absorption features, without needing to invoke varying IMFs. Tang & Worthey (Reference Tang and Worthey2015) highlight strong degeneracies between the inferred IMF slope and the value of m_l, the extent of AGB populations and variations in elemental abundances. They note that ‘increasing evidence shows that single-burst, single-composition stellar populations oversimplify the underlying stellar systems’ and conclude that it is very difficult to disentangle a steepening of the IMF slope from a decreased contribution of the AGB population in young metal-rich galaxies. They note that this degeneracy can be addressed using sufficiently high precision photometry and spectroscopy for old (10 Gyr) metal-rich populations. Similarly, Smith et al. (Reference Smith, Alton, Lucey, Conroy and Carter2015b) used composite J-band spectra compiled from over 100 galaxies to show that it is not possible to jointly constrain the Na abundance and the IMF slope in the most massive galaxies. They conclude by cautioning against over-reliance on Na lines in such studies.

Another limitation was highlighted by Smith (Reference Smith2014), who analysed the dynamical and SPS IMF constraints for 34 galaxies in common between those used by Conroy & van Dokkum (Reference Conroy and van Dokkum2012) and Cappellari et al. (Reference Cappellari2013). He found that while the general results of each are consistent, there is no correlation of the IMF inferred on a galaxy-by-galaxy level between the two approaches. He argues that the results of Conroy & van Dokkum (Reference Conroy and van Dokkum2012) are explained by a trend with Mg/Fe rather than σ, (perhaps qualitatively in line 2411 with Martín-Navarro et al. Reference Martín-Navarro2015d), but that Cappellari et al. (Reference Cappellari2013) finds no such relation. He concludes that a range of confounding factors (dark matter contributions or abundance patterns) have not been disentangled from the IMF effects in one or both of the methods. Interestingly, Oldham & Auger (Reference Oldham and Auger2016) show for M87 that introducing radial stellar anisotropy has a strong impact on the derived M/L, and that this leads to an inferred IMF consistent with Chabrier (Reference Chabrier2003a), although neglecting the anisotropy would lead to a value of ϒ_* that implies a Salpeter-like IMF over 0.1 < m/M_⊙ < 100.

It is also the case that many samples analysed to date are, in most cases, limited in number to a few tens or in some cases hundreds of galaxies. Although observationally challenging, there is clearly scope for exploring large volume-limited or mass-limited samples, as done by Gunawardhana et al. (Reference Gunawardhana2011), for example, in order to account for systematics and selection effects when interpreting any putative physical dependencies for potential IMF variations.

5.5. Linking galaxies to their constituents

The link between the IMF for stars or star clusters and that inferred for galaxies has been explored extensively in the framework of the IGIMF (Kroupa & Weidner Reference Kroupa and Weidner2003; Weidner & Kroupa Reference Weidner and Kroupa2005; Kroupa et al. Reference Kroupa, Weidner, Pflamm-Altenburg, Thies, Dabringhausen, Marks and Maschberger2013). Bastian et al. (Reference Bastian, Covey and Meyer2010) nicely summarise the link between the IGIMF and the relation between m_u and cluster mass (Weidner et al. Reference Weidner, Kroupa and Bonnell2010), noting that the evidence for this relation is mixed (but see, e.g. Ramírez Alegría et al. Reference Ramírez Alegría2016; Stephens et al. Reference Stephens2017). It is worth a brief diversion here to consider a direct comparison between potential variations in the IMF for star clusters and that for galaxies, to test whether there are physical dependencies that may be consistent. From the summary of Bastian et al. (Reference Bastian, Covey and Meyer2010), Figure 2 of the reference has a suggestion that star clusters tend to have a somewhat flatter high-mass IMF slope (m > 1 M_⊙) than found in associations or the field. Can this be characterised as a function of SFR or velocity dispersion, perhaps, to compare against the results from integrated galaxy measurements?

It is not straightforward, it turns out, to compare star-forming clusters directly with galaxies. The young massive star-forming cluster Westerlund 1 in the Milky Way, for example, has α_h = −2.3 for stars in the mass range 3.4 < m/M_⊙ < 27 (Brandner et al. Reference Brandner, Beuther, Linz and Henning2008). To compare this with the star-forming galaxies of Gunawardhana et al. (Reference Gunawardhana2011), say, we need to estimate one or more of its SFR, sSFR, or SFR surface density (Σ_*). Brandner et al. (Reference Brandner, Beuther, Linz and Henning2008) quote a total initial stellar mass for Westerlund 1 of m ≈ 52 000 M_⊙, an age of t ≈ 3.6 Myr, and they measured the IMF in annuli extending out to r = 3.3 pc. We can make use of these estimates by way of illustration, although the similarity of values for the other starburst clusters in the Milky Way (Brandner Reference Brandner, Beuther, Linz and Henning2008) lead to the same conclusions. Using the values above, SFR = 0.014 M_⊙ yr^–1, and Σ_* ≈ 400 M_⊙ yr^–1 kpc^–2 for Westerlund 1. It is apparent that the sSFR (defined as the SFR divided by the total stellar mass, for a galaxy) is not really a meaningful quantity here, since the total stellar mass is the same as the stellar mass formed in the star formation event and would simply equal the inverse of the age for the cluster. This is a hint that such comparisons are not easily made, quickly supported by the fact that the SFR surface density is two orders of magnitude higher than seen in the ensemble of galaxies sampled by Gunawardhana et al. (Reference Gunawardhana2011), where −2.7 ≲ log(Σ_*/M_⊙ yr ⁻¹ kpc ⁻²)≲−0.8. Clearly the average Σ_* for a galaxy is reduced by the many regions that have no ongoing star formation. Alternatively, we could artificially define a larger area encompassing Westerlund 1 that extends out to the boundary with the closest neighbouring star-forming system in order to define the value of Σ_*, but that opens a host of related questions for how to define such a region. That leaves the direct measure of SFR itself. Here again the comparisons are not straightforward, since now the quantity for Westerlund 1 is two orders of magnitude lower than the range of SFR seen in the galaxies (0 ≲ log(SFR/M_⊙yr ^–1)≲1.7), unsurprisingly when comparing a single star cluster to a whole galaxy. If instead the stellar velocity dispersion is used as the linking factor, star clusters again have σ substantially lower than those of galaxies. The super star clusters in M82, for example, have 10 ≲ σ/km s ⁻¹ ≲ 35 (McCrady & Graham Reference McCrady and Graham2007), compared to the range of 100 ≲ σ/km s ⁻¹ ≲ 350 for the early type galaxies in the analyses described above. Said another way, approaches that parameterise the IMF as a function of SFR or σ only make sense in the case of an entire galaxy where such parameters themselves are well-defined. Any relation between the IMF and these parameters must actually reflect an underlying physical dependence on a truly local quantity such as the gas density, ionisation background, or volume density of SFR, for example.

One approach that may have potential in linking the two is that of Zaritsky et al. (Reference Zaritsky, Colucci, Pessev, Bernstein and Chandar2014b), using the stellar M/L ratio, ϒ_*, scaled to a common 10 Gyr age, and which they denote ϒ_*,10. As described in Section 3.2, Zaritsky et al. (Reference Zaritsky, Colucci, Pessev, Bernstein and Chandar2014b) directly compares ϒ_*,10 for early-type galaxies and disk galaxies, showing a general match with the values they identify for the two populations of star clusters. They argue that the different values of ϒ_*,10 for the two populations reflect variations in the underlying IMF, but that there is no characteristic physical property identified yet that maps to these IMF differences, having ruled out velocity dispersion, surface brightness, half-light radius, metallicity, age, half-mass relaxation time, central luminosity and mass densities, escape speed, binding energy, and more. Despite this lack of a clear physical origin, the use of ϒ_*,10 as a metric to compare galaxies with stellar clusters deserves further attention.

If the potential physical dependencies of the IMF for a star cluster cannot be directly compared to those for a galaxy, this reintroduces the concern about the IMF itself being a poorly-posed concept, to which I return in Sections 8 and 9.

Continuing to increase in scale, I turn next to the constraints on the IMF that have been explored through galaxy populations, rather than individual systems.

7.1. Simulating star formation

The physics of star formation is an enormous field, and I do not pretend to provide a thorough review here. The purpose of the current summary is to highlight the complexity of the field, and the challenge in directly linking fundamental astrophysical processes to the form of an IMF. For details of work in this area, interested readers are referred to reviews by Krumholz (Reference Krumholz2014), and Offner et al. (Reference Offner, Clark, Hennebelle, Bastian, Bate, Hopkins, Moraux and Whitworth2014), work by Hopkins (Reference Hopkins2013a), Bate, Tricco, & Price (Reference Bate, Tricco and Price2014), Guszejnov & Hopkins (Reference Guszejnov and Hopkins2015), Bate & Keto (Reference Bate and Keto2015), Klishin & Chilingarian (Reference Klishin and Chilingarian2016), Guszejnov, Krumholz, & Hopkins (Reference Guszejnov, Krumholz and Hopkins2016), Guszejnov, Hopkins, & Krumholz (Reference Guszejnov, Hopkins and Krumholz2017), and references therein.

For ease of readability I refer below to ‘top-heavy’ or ‘bottom-heavy’ IMFs in reference to work that uses those terms. These correspond respectively to α_h ≳ −2.35 (usually for m > 0.5 M_⊙, sometimes α ≳ −2.35 for the full mass range) and α_l ≲ −2.35 (often for m < 1 M_⊙, but about as often also α ≲ −2.35 for the full mass range). Since this summary is largely qualitative, this usage should not be too ambiguous.

The recent work by Krumholz et al. (Reference Krumholz, Myers, Klein and McKee2016) provides a concise introduction to the key elements considered by most star formation simulations. In brief, the thermal Jeans mass, turbulence, magnetic fields, radiative feedback and mechanical feedback are all considered by various authors to play more or less significant roles. That analysis extends work by Krumholz (Reference Krumholz2011), who quantifies how radiative feedback can set the stellar mass scale, in turn building on earlier work by Bate (Reference Bate2009) and Krumholz (Reference Krumholz2006). He argues that radiative processes are the dominant mechanism in determining the gas temperature and ultimately the origin of the peak in the IMF.

Early work proposed an IMF characteristic mass determined by the thermal Jeans mass (e.g. Larson Reference Larson1998, Reference Larson2005). Being temperature dependent, this would lead naturally to higher m_c in extreme environments such as super star clusters or galactic nuclei, or high redshift galaxies. Other potential drivers, such as the role of metallicity, have subsequently been explored. There are arguments that, while metallicity plays an important role in cooling for the formation of Population III stars, it is unlikely to have a direct effect on the IMF for later stellar generations (Bate Reference Bate2014, Reference Bate2005), apart from increasing the lower mass limit for lower metallicity systems. While Bate (Reference Bate2005) notes that metallicity may have an indirect impact because of its role in setting the Jeans mass during cloud fragmentation, to the degree that the Jeans mass of the cloud may affect the characteristic mass of the IMF, Bate (Reference Bate2014) find stellar mass functions that are consistent for metallicities ranging from 1/100 to 3 times solar. Similarly, using two numerical simulations corresponding to Jeans masses different by a factor of three, Bate & Bonnell (Reference Bate and Bonnell2005) argue that any potential IMF variation appears through a change in the characteristic mass of the system rather than a change in slope at the high-mass end.

It is clear that there is enormous complexity and interplay of the astrophysical processes involved in star formation. Given this complexity, it is easy to understand that a ‘universal’ IMF is an attractive end-state to aim at achieving with models and simulations, as a form of validation. Introducing IMF variations removes this touchstone, making the work of the theorists more challenging, but as the observational constraints become more complex, so too do the models in their efforts at addressing them. This has led in more recent work to the goal of testing how particular models fare in reproducing the range of popular published IMF variations.

Hopkins (Reference Hopkins2013b) uses the excursion set formalism to calculate mass functions from the density field in a supersonically turbulent interstellar medium. This analysis predicts that IMF variations are most likely to appear at the low-mass end, with remarkably uniform slope for high masses, for reasonable choices of temperature, velocity dispersion, and gas surface density. A different approach, based on a Press–Schechter formalism, by Hennebelle & Chabrier (Reference Hennebelle and Chabrier2013), extends their earlier work by including time dependence and the impact of magnetic field, and reaches similar conclusions. Chabrier, Hennebelle, & Charlot (Reference Chabrier, Hennebelle and Charlot2014) show how the turbulent Jeans mass leads to the peak of the IMF shifting towards lower masses, to reproduce ‘bottom-heavy’ IMF shapes. Subsequently Bertelli Motta et al. (Reference Bertelli Motta, Clark, Glover, Klessen and Pasquali2016) identified conditions in two suites of hydrodynamic simulations that lead to IMF variations at the high-mass end. Recently, though, Liptai et al. (Reference Liptai, Price, Wurster and Bate2017) have argued against supersonic turbulence being the primary driver in the shape of the IMF, based on two sets of simulations with different turbulent modes, finding statistically indistinguishable differences in the resulting IMFs.

It seems that despite the growing sophistication of our theoretical understanding of star formation, there is still scope for refinement in identifying the various dominant physical mechanisms in different astrophysical environments. Elmegreen (Reference Elmegreen, Kang, Lee, Leung and Cheng2007) notes that ‘most detailed theoretical models reproduce the IMF, but because they use different assumptions and conditions, there is no real convergence of explanations yet’. In the subsequent decade, although the models have become more sophisticated, subtle and complex, so have the observational constraints, and the outcome remains largely the same.

7.2. Simulating galaxy evolution

Moving from the complexity of astrophysical processes in individual star formation to the larger scale of galaxies requires a different form of modelling and simulations. As above, this is a vast field in its own right, and only briefly and incompletely summarised here with the aim of identifying some of the developments and challenges.

Galaxy populations are typically modelled through semi-analytic recipes embedded in large cosmological simulations, and individual or small numbers of galaxies through detailed hydrodynamical simulations with better physical resolution than the cosmological models. In the absence of confirmed physical drivers underlying the shape of the IMF, such simulations tend to invoke a range of empirical or phenomenological relations that have some physical motivation. The outcome is that most observational evidence for IMF variations is able to be reproduced by a suitable choice of physical dependencies for the IMF, although not all results are consistent with each other, or necessarily self-consistent.

Baugh et al. (Reference Baugh, Lacey, Frenk, Granato, Silva, Bressan, Benson and Cole2005) modelled the abundance of Lyman break galaxies and submillimetre galaxies, successfully reproducing luminosity functions and the optical and infrared properties of local galaxy populations, but found a need for a ‘top-heavy’ IMF to reproduce the observed 850 μm galaxy number counts. Without such a change to the IMF in the model, the constraint from the global SFR density led to the predicted number counts being too low. Allowing the IMF to be ‘top-heavy’ increases the 850 μm flux for a given (lower) SFR because of the relative increase in the number of high-mass stars, allowing the model to consistently reproduce both the number counts and the SFR density. Narayanan & Davé (Reference Narayanan and Davé2012) explore the impact of allowing m_c to scale with the Jeans mass in giant molecular clouds, showing that this simple assumption leads to a reduction in the SMH/SMD discrepancy, as well as reducing the tensions in several other observational constraints. Narayanan & Davé (Reference Narayanan and Davé2012) extend this work to show that such an assumption leads to galaxies experiencing both ‘top-heavy’ and ‘bottom-heavy’ IMFs at different stages of their evolution, with the bulk of stars forming in a ‘top-heavy’ phase. Marks et al. (Reference Marks, Kroupa, Dabringhausen and Pawlowski2012) use a model of rapid gas expulsion to produce more ‘top-heavy’ IMFs in systems with increasing density and decreasing metallicity. Bekki (Reference Bekki2013a) shows that such density and metallicity dependencies for the IMF can lead, among other effects, to lower SFRs than with a fixed IMF, and that [Mg/Fe] is higher for a given metallicity. Similarly, Bekki & Meurer (Reference Bekki and Meurer2013) are able to reproduce the ‘top-heavy’ IMF results of Gunawardhana et al. (Reference Gunawardhana2011) by allowing the IMF to depend on local densities and metallicities of the interstellar medium. In contrast, Bekki (Reference Bekki2013b) show that ‘bottom-heavy’ IMFs can also be reproduced with suitable choices of metallicity and gas density in the star-forming gas clouds.

Taking a different approach, Fontanot (Reference Fontanot2014) uses a semi-analytic model to test the impact of different IMF prescriptions, broadly falling into two classes of SFR-dependent ‘top-heavy’ models and stellar mass-dependent or σ-dependent ‘bottom-heavy’ models. He finds that the ‘bottom-heavy’ models lead to variations in stellar mass and SFR functions similar to the uncertainty in the determination of those quantities, while the ‘top-heavy’ models lead to an underestimate in the high-mass end of the galaxy stellar mass function, compared to a fixed Kroupa (Reference Kroupa2001) IMF.

Blancato, Genel, & Bryan (Reference Blancato, Genel and Bryan2017) also explore the impact of observed IMF variations on models. They implement various IMF dependencies by tagging stellar particles in their simulation with individual IMFs using observationally derived dependencies on velocity dispersion, metallicity or SFR. They then find that the IMFs recovered in the simulated z = 0 galaxies no longer reproduce the imposed relations. This leads them to conclude that even more extreme physical IMF relations for some stellar populations are required to reproduce the observed level of variation. Sonnenfeld, Nipoti, & Treu (Reference Sonnenfeld, Nipoti and Treu2017) explore the evolution of α_mm (defined here as the ratio of the true stellar mass to that inferred assuming a Salpeter IMF) using cosmological N-body simulations. They find that dry mergers do not strongly impact the relation between α_mm and σ. They note, though, that the underlying dependence of the IMF on stellar mass or σ is mixed through the dry merger process, making it observationally challenging to infer which quantity was originally coupled with the IMF. Schaye et al. (Reference Schaye2010) tested the impact of a ‘top-heavy’ IMF at high gas pressures, finding that it reduced the need to invoke self-regulated feedback from accreting black holes to reproduce the observed decline in the cosmic SFR density at z < 2. Gargiulo et al. (Reference Gargiulo2015) argue that a ‘top-heavy’ IGIMF best reproduces the [α/Fe]-stellar mass relation for elliptical galaxies when there is an SFR-dependence for the IGIMF slope. Fontanot et al. (Reference Fontanot, De Lucia, Hirschmann, Bruzual, Charlot and Zibetti2017) also explore the implications of the IGIMF method in their semi-analytic model, finding that it leads to a more realistic [α/Fe]-stellar mass relation than with a fixed IMF.

It is clear that numerical simulations and semi-analytic models can provide valuable insights into the way we understand the IMF. In particular, they can be used to test how different physical prescriptions for star formation manifest, and the properties of the observational constraints on IMFs that they produce, as well as what accessible observational tracers give the most discrimination in measuring the IMF. It is important that models are used to make predictions for how different IMF prescriptions should present observationally, defining observational tests to refine or rule out particular forms of physical dependencies or underlying variation. There is perhaps more value in using the models in this way than merely through tweaking some underlying dependencies to reproduce a select subset of observational constraints. Because of the fundamental nature of the IMF it is important that models and simulations are used to test as broad a suite as possible of observational implications, rather than merely focusing on one or two in particular. This is to ensure that some observational constraints are not violated in the models while attempting to assess the impact on others.

Large cosmological hydrodynamical simulations are now available, such as Eagle (Schaye et al. Reference Schaye2015), Illustris (Vogelsberger et al. Reference Vogelsberger2014) and Magneticum (Dolag Reference Dolag2015) within which detailed galaxy simulations can be created, for example. By selecting sub-volumes sampling a broad range of galaxy environments and re-simulating those subregions at high resolution, it should be possible to identify the impact of different simulated IMFs on the physical properties of the resulting galaxies. Simulation outputs should be produced that are directly comparable to observables (e.g. luminosities as well as stellar masses or SFRs) to avoid the need to reconstruct such derived properties from observational data sets, and potentially introducing inconsistent assumptions regarding the IMF in doing so. By incorporating population synthesis approaches that link directly to the observational metrics being used in inferring IMF properties, there may be the opportunity to directly assess how underlying IMF dependencies are subsequently quantified observationally.

In summary, there is an opportunity to begin linking the numerical, semi-analytic and population synthesis model approaches to self-consistently assess whether observational approaches for inferring IMFs are actually providing the quantitative conclusions expected, or whether other underlying effects may dominate.

8.1. IMF constructs

The IMF has been used as a tool in a broad range of different contexts, as illustrated above. But if the IMF is not universal, then the quantity actually being measured in these different contexts is not necessarily the same. When inferring an IMF based on the integrated light from a galaxy, the quantity being measured is not the same as when inferring an IMF from star counts or luminosity functions. Likewise, when using a cosmic census approach such as the SFH/SMD constraint, the quantity in question is different again.

Such spatial dependence of the IMF has been recognised since the earliest work, with Salpeter (Reference Salpeter1955) describing the IMF as ‘the number of stars in [a given] mass range … created in [a given] time interval … per cubic parsec’. The spatial dependence, though, can easily be glossed over, and especially with the idea of a ‘universal’ IMF guiding the thinking, it is easy to conflate IMFs associated with different spatial volumes (star clusters, galaxies) and treat them as the same entity when they may well not be. This can lead to artificial or apparent inconsistencies that may not necessarily be in conflict.

Figure 7. A framework to aid in clarifying discussions of the IMF. If the IMF is not universal, then the sIMF, gIMF, and cIMF are not necessarily the same, and all may have a time dependence. Different measurement techniques and observational samples probe these different quantities, and what has been referred to uniformly in published work to date as ‘the IMF’ conflates these distinct properties. This may well contribute to much of the current tension between different IMF estimates in different contexts.

The notion that the IMF within a star-forming region is potentially a different quantity than the effective IMF for a galaxy, and different again from the effective IMF for a population of galaxies at a given epoch, is an important foundational concept. Here I define these three quantities as the ‘stellar IMF’ or sIMF (ξ_s), the ‘galaxy IMF’ or gIMF (ξ_g), and the ‘cosmic IMF’ or cIMF (ξ_c) as illustrated in Figure 7. Lower case prefixes and subscripts are chosen here explicitly with the aim of minimising ambiguity between other commonly used variants such as IGIMF (for a galaxy-wide IMF), or CIMF (the cluster IMF for stellar clusters, or ‘core’ IMF for dense gas cores).

We can generalise the formalism of the IMF by writing the dependence on time and spatial volume explicitly:

(3) \begin{equation} \xi (m,t,V) &#x003D; {{{\rm{d}}N(m,t,V)} \over {{\rm{d}}m{\kern 1pt} {\rm{d}}t{\kern 1pt} {\rm{d}}V}}, \end{equation}

where dN is the number of stars in mass interval dm created in the time interval dt within the spatial volume dV. In this generalisation it is important to note that the time dependence explicit to ξ allows for the form of the IMF to vary with time. It is different from the time-dependent mass function scaling that arises from a varying SFR, as defined, for example, by Schmidt (Reference Schmidt1959).

This approach describes the number of stars of a given mass that have formed up to a given time, for some spatial volume. Over a (short) finite time period and (small) spatial volume this is identically the stellar mass function (not accounting for stellar evolution):

(4) \begin{equation} \int_V \int_t \xi (m,t,V){\rm{d}}t{\kern 1pt} {\rm{d}}V &#x003D; {\rm{d}}N(m)/{\rm{d}}m \end{equation}

and corresponds to what might be considered as a ‘traditional’ IMF. This approach eliminates the ambiguity between a nominal, or Platonic ideal, IMF from which real star clusters must be populated, and the actual instantiated mass function, since what is defined here is the real physical quantity of interest, the number of stars formed as a function of mass, time and location. The ‘universal’ IMF scenario can arguably be recovered by asserting that ξ has no temporal or spatial dependence, ξ(m) = dN(m)/dm, but this reopens the issue of accounting for a finite duration for star formation, and the effects of stellar evolution, since such a ξ(m) is in principle never observable (e.g. Kroupa et al. Reference Kroupa, Weidner, Pflamm-Altenburg, Thies, Dabringhausen, Marks and Maschberger2013). In contrast, ξ(m,t,V) is directly observable in principle, although in practice doing so may be highly challenging.

Integrating ξ(m,t,V) over different volumes gives the (time dependent) sIMF, gIMF and cIMF:

(5) \begin{equation} {\xi _s}(m,t) &#x003D; \int_{{V_s}} \xi (m,t,V){\rm{d}}V &#x003D; {\left[ {{{{\rm{d}}N(m,t,V)} \over {{\rm{d}}m{\kern 1pt} {\rm{d}}t}}} \right]_{{V_s}}}, \end{equation}

where V_s is a volume characteristic of a star-forming region, for example. Relations for ξ_g and ξ_c are analogous.

Clearly there must be a lower limit to the spatial scale or volume over which an IMF is a well-defined quantity, since it makes little sense to attempt to define an IMF at the scale of individual stars, for example. A natural minimum spatial scale is that sufficient to encompass a star cluster, and much of the work on the IMF focuses on the properties of stellar clusters or uses them to probe the IMF (as reviewed by, e.g. Bastian et al. Reference Bastian, Covey and Meyer2010). Also, the volumes referred to here are not specific fixed or comoving volumes in space (although they could be defined as such, in a numerical simulation, for example), but refer instead (for convenience) to a particular spatial scale, and which I illustrate through these three representative characteristic scales.

Considering the time dimension too is a revealing mental exercise. The process of star formation is not instantaneous. As stars form within a nascent star cluster there will be different mass functions extant depending on the time step sampled (e.g. Kroupa et al. Reference Kroupa, Weidner, Pflamm-Altenburg, Thies, Dabringhausen, Marks and Maschberger2013). There is an extensive literature on the protostellar mass function, explicitly to understand this time dependence in the way that the IMF is generated. McKee & Offner (Reference McKee and Offner2010), for example, present a formalism using models for mass accretion by protostars to link the IMF to its progenitor protostellar mass function, extended to the protostellar luminosity function by Offner & McKee (Reference Offner and McKee2011). A more recent analytic model for the mass gained by protostars is presented by Myers (Reference Myers2014). Hartmann, Herczeg, & Calvet (Reference Hartmann, Herczeg and Calvet2016) review accretion onto pre-main-sequence stars, and their Figure 13 highlights the different stellar mass and luminosity functions expected based on different accretion models.

For a large ensemble of clusters, throughout a galaxy say, each at a different stage in its formation process, the stellar mass function sampled over the full ensemble at a given time step may more closely resemble the mass function expected from a well-defined physical process, such as gravo-turbulent models (e.g. Hennebelle & Chabrier Reference Hennebelle and Chabrier2013; Hopkins Reference Hopkins2013b), although the mass function for any individual region may well be rather different, due to the complex feedback effects from stellar evolution during the formation event itself, and the local environment which may be influenced by adjacent regions of star formation or other astrophysical processes.

In this way of thinking, if there is a ‘universal’ physical process that drives star formation, then it is likely to be better sampled on the scales of galaxies (the gIMF) than within individual star-forming regions. Extending this idea, and to account for the possibility of variations between galaxies (due to a range of metallicities, star formation environments, and so on), any ‘universal’ physical process might be most accurately sampled through the effective stellar mass function over an entire galaxy population (the cIMF). This leads to the need for the physical processes of star formation to be able to explain potential variations in the stellar mass function from the scale of star clusters to galaxies (which may arise through effects unrelated to the star formation process itself), ultimately converging on a model prediction when sampled over sufficiently large regions. This may be written as ξ(m,t) = ∫_V ξ(m,t,V)dV→IMF for V→V_c where V_c is some large volume encompassing one or more galaxies, and ‘IMF’ here is being used to describe the stellar mass function expected from a nominal ‘universal’ physical process.

As an alternative, rather than the sampling of a large spatial volume at a fixed time step, a small volume may be considered over a long period of time to equally ensure that all phases of the physical process of star formation are sampled. This might be summarised as ξ(m,V) = ∫_tξ(m,t,V)dt→IMF for t→t _c where t _c is large compared to the duration of a star formation event, perhaps capturing multiple such events within the volume V, and ‘IMF’ is used as above. Observationally this is not a practical approach, while the former is, but it may be of value in simulations.

This is another way of considering the arguments posed by Kruijssen & Longmore (Reference Kruijssen and Longmore2014), as this concept equally applies to the gas clouds from which the stars are forming, and the associated ‘core’ mass functions, or the mass functions of stellar clusters. Any given gas reservoir may not be representative of the full population of star-forming gas clouds throughout a galaxy, and only by sampling a sufficient number of them will the statistics of the density distribution be accurately represented.

8.2. Linking mass functions between different spatial scales

With differing stellar mass functions on different spatial scales, a natural question arises regarding how to relate individual mass functions on small scales to those measured on the larger scales, i.e., how to link the sIMF for multiple star-forming regions to the gIMF for the galaxy comprising those stars. The IGIMF method (Kroupa & Weidner Reference Kroupa and Weidner2003; Weidner & Kroupa Reference Weidner and Kroupa2005; Kroupa et al. Reference Kroupa, Weidner, Pflamm-Altenburg, Thies, Dabringhausen, Marks and Maschberger2013) is one approach, which broadly speaking considers a summation of many sIMFs to construct the gIMF. This method assumes that stars form in self-regulated embedded clusters, which follow a relationship between the total mass of a stellar cluster and the mass of its highest mass star. Their sIMFs are therefore empirically constrained by the stellar cluster mass. They can be summed to calculate the gIMF, or the IMF of a region within a galaxy containing multiple stellar clusters, and can lead to a variable gIMF (Yan et al. Reference Yan, Jerabkova and Kroupa2017). This approach allows a gIMF to be calculated given a knowledge of how the sIMF depends on the physical conditions of star formation. The early results using this technique favoured galaxy-wide IMF slopes somewhat steeper at the high-mass end than that of the individual star-forming regions. Later work incorporating constraints on the variation of the sIMF from Marks et al. (Reference Marks, Kroupa, Dabringhausen and Pawlowski2012) extended this approach to show how flatter IGIMF slopes at the high-mass end could be produced in galaxies with high SFRs (Yan et al. Reference Yan, Jerabkova and Kroupa2017).

The way that sIMFs themselves arise, or their dependencies on associated astrophysical processes, may differ from the IGIMF assumptions. Some level of variation would seem likely given that star formation happens in a complex multiphase medium, with regions of star formation potentially overlapping, and triggering or suppressing one another in highly non-linear ways, all of which may evolve with time. So the gIMF may not necessarily comprise a sum over a discrete set of identical, or even simply modelled sIMFs. If each such star formation event can be characterised by its own sIMF, though (whether or however it is influenced by, or overlapping with, neighbouring events) we can write

(6) \begin{equation} {\xi _g}(m,t) &#x003D; \int_{{V_g}} {\xi _s}(m,t,V){\rm{d}}V, \end{equation}

where V_g is the volume of the galaxy in question. It is important to distinguish this generalisation from the IGIMF method, since in the current approach each ξ_s(m, t, V) may arise from different physical dependencies or processes to those assumed in the IGIMF approach.

Likewise, the cIMF may be able to be approximated as a simple sum over gIMFs. Of course, galaxy interactions are an important channel for galaxy evolution, and they are clearly associated in many cases with significant levels of star formation. But assuming for the present argument that stars formed in this mode are a negligible fraction of all stars formed, or alternatively can be accounted for through separate characterisation with their own ξ_g(m, t, V) we can write

(7) \begin{equation} {\xi _c}(m,t) &#x003D; \int_{{V_c}} {\xi _g}(m,t,V){\rm{d}}V, \end{equation}

where V_c is the cosmic volume being probed.

In this formalism there is no analogue to the process of generating a stellar mass function by ‘populating’ or ‘drawing from’ some underlying IMF, since ξ(m,t,V) here is in effect the stellar mass function itself, incorporating its spatial and temporal variations. Instead, the link to be highlighted is over what spatial or temporal scales this mass function needs to be sampled in order to compare with predictions from various physical models of star formation. In this context, questions such as whether the sIMF is drawn from a gIMF, or how to ‘populate’ an sIMF, are poorly posed, and not helpful in developing our understanding of star formation.

8.3. Derived quantities

The SFR and total stellar mass are directly linked to the IMF (e.g. Schmidt Reference Schmidt1959), and there is some value in presenting this explicitly. The SFR, S(t,V), is the mass of stars formed in a time interval dt and volume dV:

(8) \begin{equation} S(t,V) \equiv {{{\rm{d}}m} \over {{\rm{d}}t}}(t,V) &#x003D; \int_{{m_l}}^{{m_u}} m\xi (m,t,V){\rm{d}}m, \end{equation}

and the total mass in stars ever formed in that volume over some time period t ₁ to t ₂ is then:

(9) \begin{equation} \eqalign{ {m_{total}}(V) & &#x003D; \int_{{t_1}}^{{t_2}} S(t,V){\rm{d}}t \\ & &#x003D; \int_{{t_1}}^{{t_2}} \int_{{m_l}}^{{m_u}} m\xi (m,t,V){\rm{d}}m{\kern 1pt} {\rm{d}}t. \cr} \end{equation}

The mass remaining in stars at a time τ is m_τ = (1–R)m_total, where R is the mass recycled into the interstellar medium (ISM) due to stellar evolutionary processes, and is dependent on the stellar mass distribution. Recycling fractions can be calculated for a given mass distribution if the mass returned to the ISM is known as a function of initial stellar mass (e.g. Renzini & Voli Reference Renzini and Voli1981; Woosley & Weaver Reference Woosley and Weaver1995). This is also expected to depend on metallicity.

8.4. Implementation

The value of a ‘traditional’ IMF is largely through the ability to use it as a tool to infer the presence of stellar populations not directly observed. Depending on the techniques being used, observables are often limited either to the high-mass (e.g. through Hα, UV, or infrared tracers) or the low-mass (e.g. direct star counts, or gravity sensitive spectral features) range of the stellar mass distribution, and accounting for the stellar populations not directly measured is done by invoking an IMF. With an assumed ‘universal’ IMF simply characterised through a well-defined parametric form, such extrapolations are straightforward.

In the general case posed above, a number of simplifying assumptions need to be incorporated in order to regain the utility of the simple ‘universal’ model. The value of the general approach is that these assumptions now become explicit, rather than implicit, defining the form (or absence) of any temporal or spatial variations (which may reflect other underlying physical dependencies). The same parameterisations (incorporating physical dependencies if desired) can be applied as always, using the general formalism, but assumptions about the spatial scale or epoch to which such parameterisations apply become clear. This hopefully enables distinctions to be drawn between stellar mass functions that should not necessarily be compared directly, to avoid artificial inconsistencies. It should also facilitate the exposing of internal contradictions within analyses.

The mass functions as parameterised by, say, Kroupa (Reference Kroupa2001) and Chabrier (Reference Chabrier2003a) are not inconsistent with this approach, and they can now be explicitly defined as the integrals over some spatial and temporal scale. So, to the degree that these Milky Way stellar mass functions (IMF_MW) correspond to star formation on the scale of star clusters over a period of several Myr, we may write, for example, $ {\rm{I}}M{F_{MW}} &#x003D; \int_t^{t + 5{\kern 1pt} {\rm{M}}yr} \int_{{V_s}} \xi (m,t,V){\rm{d}}V{\kern 1pt} {\rm{d}}t $, where V_s is the volume sufficient to encompass the star cluster, and 5 Myr is nominally taken as a timescale sufficient to allow the full range of masses for all stars to form. When exploring potential physical dependencies for a stellar mass function, the various observational constraints can be used in this fashion as boundary conditions.

8.5. A broader context

This more general approach can help in setting the context for the broader range of work on the IMF. In particular, by making explicit the potential spatial and temporal dependencies, which are likely a consequence of any underlying physical dependencies, the scales over which certain observational constraints apply also become explicit. It also means that analyses can be clear about the spatial and temporal scales for the stellar mass functions they are using, or making predictions for. For example, the investigation of Blancato et al. (Reference Blancato, Genel and Bryan2017) adopts an observed gIMF, which is then implemented as an sIMF in simulations. They find, perhaps not surprisingly given the discussion above, that this does not lead to the observed gIMF being reproduced in the simulated galaxies, concluding that sIMFs need to be more extreme than adopted in order to replicate the observed gIMF.

The approach presented here provides the potential for self-consistent explorations in models and simulations, enables a clearer link between what the models predict and what the observations measure, and avoids conflation between constraints that apply to different physical scales. It provides a framework in which the subtle biases associated with an implicit tendency towards a ‘universal’ IMF are eliminated, allowing for a more critical evaluation of the constraints on potential variations between stellar mass functions, and their link to the underlying physics of star formation.

With these considerations at hand, I now revisit the variety of observational constraints discussed above and position them in this self-consistent framework, in order to re-examine the extent to which the IMF may vary.