It is natural when investigating the IMF to turn from already existing stars to the progenitor clouds in which they form, to probe the earliest stages of the formation process. This is explored most efficiently through measurements of cold gas and dust in molecular clouds. Due to observational practicalities, much of the work here has focused on clouds within the Milky Way and nearby galaxies, and is well-summarised in reviews by Zinnecker & Yorke (2007), Tan et al. (2014) and Offner et al. (2014). There is ample evidence demonstrating that the mass function for the molecular “cores” (those gas regions of sufficient density to go on to form stars) has a similar shape to the stellar IMF in the Milky Way (e.g., André et al., 2010, Könyves et al., 2010, Montillaud et al., 2015, Marsh et al., 2016a). Many core mass funtions (CMFs) show an offset in mass compared to the IMFs of Kroupa (2001) or Chabrier (2003a), with the break in the power law at masses larger by a factor of ≈ 3−4, and with a similar range of variations in the high mass slope between different molecular clouds as seen in the various stellar analyses (see summary by Offner et al., 2014). Despite the range of variations, such results have for decades been similarly interpreted as consistent with a “universal” form, and have led naturally to the idea that the CMF and IMF are linked physically through some star formation efficiency factor.
The connection between pre-stellar CMFs and the IMF is still not clear, although many models have been proposed to explain it (e.g., Hopkins, 2013b, Guszejnov & Hopkins, 2015, Zhou et al., 2015). One issue is how a core is defined observationally, and that observational limitations and different threshold levels for defining a core lead to different results (see discussions by, e.g., Elmegreen, 2009, Offner et al., 2014). As noted by Offner et al. (2014), “Different algorithms …even when applied to the same observations, do not always identify the same cores, and when they do, they sometimes assign widely different masses.” Even if cores can be adequately identified, there is evidence questioning the fragmentation models leading from cores to stars, and hence linking the IMFs of each (Holman et al., 2013, Bertelli Motta et al., 2016).
A more robust approach than discrete core identifications is to use the full probability distribution function (PDF) of observed column densities within a star forming cloud, in order to identify which regions may have sufficient density to be star forming (e.g., Rathborne et al., 2014). There is an open question over whether there exists some threshold in column density of molecular hydrogen above which star formation proceeds efficiently. A universal threshold of N(H2) ≳ 1.4 × 1022 cm2 was proposed by Lada et al. (2012), although Krumholz et al. (2012) argue against the existence of such a threshold. At least one counterexample, the Galactic centre molecular cloud G0.253+0.016, questions the idea of a universal threshold (Rathborne et al., 2015). Dust temperature measurements of this cloud suggest that star formation may have recently begun, with detection of a cool filament whose hot central region is undergoing gravitational collapse and fragmentation to form a “line of protostars” (Marsh et al., 2016b). Despite this, the central molecular zone of the Milky Way appears to support substantially less star formation than might be expected from a column density threshold (Longmore et al., 2013). These results bring the idea of a universal threshold into question, at least for environments with the extreme high pressures found in the Milky Way central molecular zone, which may mimic the conditions of star formation at high redshift.
Broadly, the studies of pre-stellar clouds suggest that turbulence and hierarchical fragmentation are dominant processes in driving the star formation. Turbulence as a dominant contributor to the star formation process has also been shown to be effective in high pressure environments (Rathborne et al., 2014), and may therefore be significant in starburst nuclei and high redshift galaxies. High mass stars and clusters can form in filamentary molecular clouds (Contreras et al., 2016), although Contreras et al. (2017) note that high mass protoclusters are very rare in the Galaxy. Young high mass clusters in the Milky Way have been shown to form hierarchically rather than through monolithic collapse (Walker et al., 2015), a result seen also in the arms of the grand-design spiral NGC 1566, where Gouliermis et al. (2017) demonstrate hierarchical star formation driven by turbulence. Grasha et al. (2017) show that star cluster formation in eight local galaxies is hierarchical both in space and time, and that the ages of adjacent clusters are consistent with turbulence driving the star formation. In contrast, there is evidence for monolithic collapse in the formation of some young Galactic star clusters (e.g., Banerjee & Kroupa, 2014, Banerjee & Kroupa, 2015, Banerjee & Kroupa, 2018).
Turbulence alone, though, does not seem to be a sufficient mechanism. Using high resolution observations of molecular gas in M51, Leroy et al. (2017) note that observed measures of star formation efficiency are in some tension with turbulent star-formation models, finding an anticorrelation between the star formation efficiency per free-fall time with the surface density and line width of molecular gas. Vutisalchavakul et al. (2016), based on star forming regions in the Galactic plane, argue that observed relations between SFR and molecular cloud properties are inconsistent with those seen in extragalactic relations or the model by Krumholz et al. (2012). Similarly, Heyer et al. (2016) find low values of star formation efficiency per free-fall time in a sample of Galactic young stellar objects, noting that the strongest correlations of SFR surface density are with the dense gas surface density normalized by the free-fall and clump crossing times. They state that models accounting for such local gas conditions provide a reasonable description of these observations. Lee et al. (2016) find a rather higher observed scatter in star formation efficiency for star forming giant molecular clouds in the Milky Way, which they also note is unable to be explained by constant (Krumholz & McKee, 2005) or turbulence-related (Hennebelle & Chabrier, 2011) star formation. They argue instead for a time-variable rate of star formation noting that “sporadic small-scale star formation will tend to produce more massive clusters than will steady small-scale star formation.” By analysing the dense gas in star forming clusters, Hacar et al. (2017) argue that both clustered and non-clustered star forming regions might be naturally explained through the spatial density of dense gas “sonic fibres” (Hacar et al., 2013). Hacar et al. (2018) extends this approach to propose a unified star formation scenario that leads naturally to the observed differences between low and high mass clouds, and the origin of clusters. Walker et al. (2016) show that the mass surface density profiles are shallower for gas clouds than for young massive star clusters in the Milky Way. They argue that this implies an evolution requiring mass to continue to accumulate toward cloud centres in highly star forming clouds after the onset of star formation, in a “conveyor-belt” scenario.
It is beyond the scope of this review to explore in depth the range of detailed models of star formation, and their strengths and limitations. Summaries, however, of some models describing the star formation process and linking the CMF to the IMF, or that aim to explain the IMF shape, are presented below in § 7. A detailed review of the formation of young high mass star clusters is given by Portegies Zwart et al. (2010), and Tan et al. (2014) provide a thorough review of high mass star formation. Interestingly, Zinnecker & Yorke (2007) find strong support for an IMF upper mass limit of mu ≈ 150 M⊙, and make the case that high mass star formation proceeds differently from low mass star formation, not just as a scaled up version, but “partly a mechanism of its own, primarily owing to the role of stellar mass and radiation pressure in controlling the dynamics.” This conclusion has been questioned by more recent work, summarised by Tan et al. (2014), who argue that most observations support a common mechanism for star formation from low to high masses.
A different approach linking star forming gas to the IMF was used by Hopkins et al. (2008) to explore the link between gas consumption and star formation in a cosmic global average sense. They use the Kennicutt-Schmidt law linking SFR and gas surface densities, following Hopkins et al. (2005) who convert such a surface density relation to a volume density relation using the observed redshift distributions of damped Lyman α absorbers. The IMF dependency arises through the SFR density measurement. Different assumed IMFs will alter the SFR density calculated from observed luminosity densities, and consequently the corresponding volume density of gas necessary to sustain such star formation levels. Hopkins et al. (2008) infer that the cosmic mass density of HI at high redshift (z > 1) implies SFR densities that are not consistent with an IMF typical of the Milky Way such as Kroupa (2001) or Chabrier (2003a). Instead they require an IMF with a high mass slope flatter than Salpeter (αh > −2.35), such as that proposed through the evolving IMF of Wilkins et al. (2008a). It would be valuable to revisit this alternative style of approach in light of more recent work on the relationship between SFR and gas density, as reviewed for example by Kennicutt & Evans (2012).
In summary, as with the stellar techniques, the approaches used in measuring the CMF in order to link it to the IMF are limited by the relatively small samples available within the Milky Way and nearby galaxies, and the link itself may be unclear (Holman et al., 2013, Bertelli Motta et al., 2016). As with the various stellar cluster measurements, a range of CMF high mass slopes is found for different molecular clouds, with a similar span of uncertainty, and for much the same reason. There are similar levels of variation measured for the low mass slope, and for the characteristic mass where the CMF slope changes. It is worth reiterating the argument of Kruijssen & Longmore (2014) regarding the number of independent samples required to capture all phases and the spatial scale of the processes being measured. It is also worth restating and recommending the approach of placing constraints on the scale of possible variations rather than defaulting to a “universal” conclusion.
There is a further point to be made, picking up on the PDF approach of Rathborne et al. (2014). They note that gas dense enough to form stars is dense enough to become self-gravitating and undergo runaway collapse. This shows up as a power-law tail deviation, at the high column-density end, from the otherwise log-normal form of the PDF. If the gas that will go on to form stars can be identified in this simple and direct way instead, the CMF as an entity, with all the challenges associated in measuring it, is perhaps not a physically useful quantity.
Such a conclusion reinforces the poorly-posed nature of the definition of such mass functions. What defines the star forming region of interest over which the CMF or IMF is to be measured? For gas clouds that have a continuum of densities the challenge in defining boundaries or thresholds (such as with various clump-finding software tools) is clear (e.g., Offner et al., 2014), but the PDF approach sidesteps that limitation. The problem for stars, though, may not be so readily apparent, since for a cluster it would seem straightforward to focus, for example, on the gravitationally bound stars as a single entity. But even this kind of simple scenario has been seen to suffer from issues such as mass segregation, dynamical evolution, and so on, leading to systematics affecting any IMF measurement. This raises the broader concern of whether the IMF itself is a well-posed concept. If it does not exist as a physical distribution at any given point in time (Elmegreen, 2009, Kroupa et al., 2013), and the spatial region over which it is to be measured is unclear, is there a better entity that can be more well-defined instead? I return to this point in § 8 below.
Having raised again the point regarding the spatial scale being probed, I move next to the approaches used in estimating the IMF for galaxies as a whole.